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MATHEMATICAL PAPERS
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THE COLLECTED MATHEMATICAL PAPERS
OF
JAMES JOSEPH SYLVESTER
F.R.S., D.C.L., LL.D., Sc.D.,
Honorary Fellow of Sf John's College, Cambridge j
Sometime Professor at University College, London ; at the University of Viijiiii* i
at the Royal Military Academy, Woolwich ; at the John* Hopkins Univenity, Baliimorc
and Savilian Professor in the University of Oxford
VOLUME IV
(1882— 1897)
Cambridge
At the University Press 1912
Cambttligc :
PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS
PREFATOEY NOTE
rriHE present volume contains Sylvester's Constructive Theory of Partitions, papers on Binary Matrices, and the Lectures on the Theory of Reciprocants. There is added an Index to the four volumes, and a Biographical Notice of Sylvester. The Mathematical Questions in the Educational Times are as yet unedited, but an Index to them is appended here. I have to acknowledge the kindness of Dr J. E. McTaggart, F.B.A., who secured for me the loan of the Essay on Canonical Forms, from the Library of Trinity College, Cambridge, for Vol. i, and that of Mr R. F. Scott. M.A., Master of St John's College, Cambridge, for the use of the volume called The Laws of Verse, from which the matter contained in the Appendix to Vol. ii was reprinted, who supplied also the Autograph on the Frontispiece of this Volume. To the latter gentleman, as well as to Major P. A. MacMahon, Professor E. B. Elliott and Sir Joseph Larmor, I owe my best thanks for reading through the Biographical Notice. In carrying through the task of editing the Papers, I have, in general, thought it most fitting not to oflFer any remarks of my own in regard to Sylvester's text, though many times at a loss to know how best to act. In the Appendix to Vol. I I have departed from this rule, giving there an account of Sylvester's chief theorems in regard to determinants. For two other cases the reader may find notes, Proceedings of the London Mathematical Society, Vol. iv, Ser. II (1907), pp. 131—135, and Vol. VI (1908), pp. 122—140; these refer respectively to the paper No. 36, p. 229, and to the paper No. 74, p. 452, both m Vol. II of the Reprint. Many corrections of errors in the printing of algebraical formulae have been introduced, though many, it is to be feared, still remain ; but no alterations of Sylvester's statements have been made without definite indication, by square brackets or otherwise. To the Readers and Staff of the University Press the very greatest credit and gratitude for their watchful carefulness are assuredly due, many of the corrections in the volumes being due to them.
H. F. BAKER. June 1912.
TABLE OF CONTENTS
PAGES
Portrait of J. J. Sylvester .... Frontispiece
Medallion Head of Biographical Notice
Biographical Notice xv — xxxvii
1. A constructive theory of partitions, arranged
in three acts, an interact and an exodion 1 — 83
(American Joaroal of Mathematics 1882, 1884)
2. Sur les nombres de fractions ordinaires in-
egales qu'on pent exprinier en se servant de chiffres qui n'excedent jias un nombre donne 84 — 87
(Comptes Bendus de I'Acadimie des Sciences 1883)
3. Note sur le theoreme de Legendre citS dans
une note instrde dans les Comptes
Rendus 88—90
(CompteB Rendas de I'Acad&uie des Sciences 1883)
4. Sur le prodtiit ind^fini \—x.\—af.\—af... 91
(Comptes Bendns de I'Academie des Sciences 1883)
5. Sur un theoreme de partitions ... 92
(Comptes Rendas de I'Acadimie des Sciences 1883)
6. Preuve graphique du theoreme d' Eider sur
la jHirtition des nombres pentagonaux . 93, 94
(Comptes Rendus de I'Acad^mie des Sciences 1U83)
7. Demonstration graphique d'un theoreme
d'Euler concernant les partitions des
nombres 95, 96
(Comptes Bendas de I'Acad^mie des Sciences 1883)
8. Sur un theoreme de imrtitions de nombres
complexes contenu dans un theoreme de
Jacobi 97 — 100
(Comptes Bendas de I'Acadimie des Sciences 1883) 8. IV. 5
Vlll
Contents
PAGES
9. On the number effractions cotdained in any "Farey seHes" of which the limiting number is given 101 — 109
(Philosophical Magazine 1883)
10. On the equation to the secular inequalities in
the planetary theory . . . . 110, 111
(Philosophical Magazine 1883)
11. On the involution and evolution of quater-
nions 112 — 114
(Philosophical Magazine 1883)
12. On the involution of two matrices of the
second order 115 — 117
(Soathport British Association Beport 1883)
13. Sur les quantites formant un groupe de
nonions analogues aux quaternions de
Hamilton ...... 118—121
(Comptes Bendas de I'Acad^mie des Sciences 1883)
14. On quaternions, nonions, sedenions, etc. . 122 — 132
(Johns Hopkins University Circulars 1884)
15. On involutants and other allied species of
invariants to matrix systems . . 133 — 145
(Johns Hopkins University Circulars 1884)
16. On the three laws of motion in the world of
universal algebra 146 — 151
(Johns Hopkins University Circulars 1884)
17. Equations in matrices 152, 153
(Johns Hopkins University Circulars 1884)
18. Sur les quantity formant un groupe de
nonions analogues aux quaternions de
Hamilton 154 — 159
(Comptes Bendus de 1' Academic des Sciences 1884)
19. Sur line note recente de M. D. Andre . 160, 161
(Comptes Bendus de 1' Academic des Sciences 1884)
20. Sur la solution d'une classe tres Stendue
d'iquMtions en quaternions ... 162
(Comptes Bendas de I'Acad^mie des Sciences 1884)
Contents ir
PAGES
21. Stir la coi^respondance entre deux esphces
differentes de fonctions de deux systemes de quantites, correlatifs et egalement nombreux 163 — 165
(Comptes Bendas de I'Acad^mie des Sciences 1884)
22. Sur le theoreme de M. Brioschi, relatif aux
fonctions symetriques .... 166 — 168
(Comptes Beudus de I'Acad^mie des Sciences 1884)
23. Sfiir line extension de la loi de Harriot
relative aux equations algebriques . 169 — 172
(Comptes Bendas de I'Acad^mie des Sciences 1884)
24. Sur les equations monothetiques . . . 173 — 175
(Comptes Bendus de I'Acadimie des Sciences 1884)
25. Sur Vequation en matrices px = xq . . 176 — 180
(Comptes Bendas de 1' Academic des Sciences 1884)
^26. Sur la solution du cas le plus general des Equations lin^aires en quantites binaires, c'est-d-dire en quaternions ou en matrices du second ordre 181, 182
(Comptes Bendas de I'Acad^mie des Sciences 1884)
Stir les deux methodes, cdle de Hamilton et celle de Vatiteur, pour r^soudre Vequation lindaire en quaternions .... 183 — 187
(Comptes Bendas de I'Acad^mie des Sciences 1884)
Sur la solution explicite de Vequation quad- ratique de Hamilton en quaternions ou en matrices du second ordre . . . 188 — 198
(Comptes Bendas de I'Acad^mie des Sciences 1884)
29. Sur la resolution gendrale de Vequation
lindaire en matrices d'un ordre quelcon-
qiie 199—205
(Comptes Bendus de I'Acad^mie des Sciences 1884)
30. Sur Vequation lineaire trinome en matrices
dun ordre quelconque .... 206, 207
(Comptes Bendas de I'Acad^mie des Sciences 1884)
62
Contents
31. Lectures on the principles of universal
algebra
(American Journal of Mathematics 1884)
32. On the solution of a class of equations in
quaternions
(Philosophical Magazine 1884)
33. On Hamilton's quadratic equation and the
general unilateral equation in matrices
(Philosophical Magazine 1884)
34. Note on Captain MacMahon's transforma-
tion of the theory of invariants .
(Messenger of Mathematics 1884)
35. On the D'Alembert-Camot geometrical para-
dox and its resolution ....
(Messenger of Mathematics 1885)
36. Sur line nouvelle theorie de formes algehriques
(Comptes Bendus de I'Acadfimie des Sciences 1885)
37. Note on Schwarzian derivatives .
(Messenger of Mathematics 1886)
38. On reciprocants
(Messenger of Mathematics 1886)
39. Note on certain elementary geometrical no-
tions and determinations
(Proceedings of the London Mathematical Society 1885)
40. On the trinomial unilateral quadratic equa-
tion in matrices of the second order .
(Quarterly Journal of Mathematics 1885)
41. Inaugural lecture at Oxford, on the method
of reciprocants . . . • •
(Nature 1886)
42. Lectures on the theory of reciprocants
(American Journal of Mathematics 1886 — )
43. Sur les reciprocants purs irrMuctiUes du
quatrieme ordre
(Comptes Eendus de I'Aoad^mie des Sciences 1886)
PAGES
208—224
225—230
231—235
236, 237
238—241 242—251 252—254 255—258
259—271
272—277
278—302 303—513
514
Contents xi
PAGES
44. Sur une extenmon du theorenie relatif au
nomhre d^invariants asyzygetiques dim type donne a une classe de formes ana- logues 515 — 519
(Comptes Bendas de rAcad^mie des Sciences 1886)
45. Note sur les invariants differentiels . . 520 — 523
(Comptes Bendas de I'Acad^mie des Sciences 1886)
46. Stir Tequation differentielle d'une courhe
d'ordre quelconque 524 — 526
(Comptes Bendas de I'Acad^mie des Sciences 1886)
47. Stir une extension d'un theorenie de Clebsch
relatif atix courbes du quatrieme degre 527, 528
(Comptes Bendas de 1' Academic des Sciences 1886)
48. On the differential equation to a curve of
any order 529, 530
(Natare 1886)
49. On the so-called Tschirnhausen tratuforma-
tion 531—549
(Crelle's Joamal fiir die reine nnd angewandte Mathematik 1887)
50. Stir une decouverte de M. James Hammond
relative d, une certaine s4rie de nomhres qui figurent dans la tMorie de la trans- formation Tschirnhausen . . . 550 — 552
(Comptes Bendas de I'Acad^mie des Sciences 1887)
;51. On Hamilton's numbers .... 553 — 584
(Philosophical Transactions of the Boyal Society of London 1887, 1888)
62. Sur les nombres dits de Hamilton . . 585 — 587
(Compte Benda de I'Assoc. Franijaise (Toalouse) 1887)
63. Note on a proposed addition to the voca- bulary of ordinary arithmetic . . 588 — 591
(Nature 1888)
54. On certain inequalities relating to prime numbers 592 — 603
(Nature 1888)
55. Sur les nombres parfaits .... 604 — 606
(Comptes Bendas de I'Acad^mie des Sciences 1888)
xii Contents
PAGES
56. Siir une classe sp^dale des diviseurs de la
somme d\me serie giomMrique . . 607 — 610
(Comptes BenduB de TAcad^mie des Sciences 1888)
57. Sur T impossibility de I'existence d'un nonibre
parfait impair qui ne contient pas au
moins 5 diviseurs premiers distincts . 611 — 614
(Comptes Bendus de TAcad^mie des Sciences 1888)
58. Sur les nomhres parfaits .... 615 — 619
(Comptes Eendus de 1' Academic des Sciences 1888) (Mathesis 1888)
59. Preuve Mementaire du thdoreme de Dirichlet
sur les progressions arithmStiques dans
les cas oil la raison est 8 on 12 . . 620 — 624
(Comptes Bendus de I'Acad^mie des Sciences 1888)
60. On the divisors of the sum of a geometrical
series whose first term, is unity and common ratio any positive or negative integer 625 — 629
(Nature 1888)
61. Note on certain difference equations which
possess an unique integraV . . . 630 — 637
(Messenger of Mathematics 1888 — 9)
62. Sur la reduction biorthogonale d'une forme
lineo-lineaire a sa forme canonique . 638 — 640
(Comptes Eendus de I'Acadfimie des Sciences 1889)
63. Stir la correspondance complete entre les
fractions continues qui expriment les deux racines d'une equation quadratique dont les coefficients sont des nombres rationnels 641 — 644
(Comptes Eendus de I'Acad^mie des Sciences 1889)
64. Sur la representation des fractions continues
qui expriment les deux racines d'une
Equation quadratique .... 645, 646
(Comptes Bendus de I'Acad^mie des Sciences 1889)
Contents xiii
PAGES
65. Sur la valeur d'une fraction continue finie
et purement periodique .... 647 — 649
(CJomptes Bendus de TAcad^mie des Sciences 1889)
66. A new proof that a general quadric may he
reduced to its canonical form {that is, a linear function of squares) hy means of a real orthogonal stibstittition . . 650 — 653
(Messenger of Mathematics 1890)
67. On the reduction of a bilinear quantic of the
nth order to the form of a sum of n products by a double orthogonal substi- tution 654—658
(Messenger of Mathematics 1890)
68. On an arithmetical theorem in periodic
C07itinued fractions 659 — 662
(Messenger of Mathematics 1890)
69. On a funicular solution of Buff on' s "problem
of the needle" in its most general form 663 — 679
(Acta Matbematica 1890—1)
70. Sur le rapport de la circonference an dia-
mktre 680, 681
(Ck>mptes Bendns de I'Acad^mie des Sciences 1890)
71. Preuve que ■n ne pent pas etre racine
d'une equation algebrique h coefficients
entiers . 682—686
(Comptes Bendns de I'Aoad^mie des Soienoes 1890)
72. On arithmetical series 687 — 731
(Messenger of Mathematics 1892)
73. Note on a nine schoolgirls problem . . 732, 733
(Messenger of Mathematics 1893)
74. On the Goklbach-Euler theorem, regarding
prime numbers 734 — 737
(Nature 1896—7)
xiv Contents
FAQES
75. On the number of jyroper vulgar fractions in their lowest terms that can he formed with integers not greater than a given number 738 — 742
(Messenger of Mathematics 1898)
Index to Professor Sylvester's contributions
TO "Mathematical Questions from the
Educational Times" 743 — 747
Index to the four volumes of the "Collected Mathematical Papers " of James Joseph Sylvester 748 — 756
BIOGEAPHICAL NOTICE*.
Lord of himself and blest shall prove
He who can boast "I've lived to-day, To-morrow let dispensing Jove
Cast o'er the skies what tint he may.
" Sunshine or cloud ! the work begun
And ended may his power defy, He cannot change nor make undone
What once swift Time has hurried by."
Law of Verte, p. 73 (from Horace).
James Joseph Sylvester was boru in London on 3 September 1814, 1814 of a family said to have been originally resident in Liverpool. He was among the youngest of several brothers and sisters, and the last to survive. His father, whose name was Abraham Joseph, died while he was young. His eldest brother early in life established himself in America and assumed the name of Sylvester, an example followed by all the brothers.
If we attempt to realise the scientific circumstances of the time of Sylvester's birth by recalling the dates of some of those whose work might
* The chief aathority for the oatward facts of Sylvester's life used in this record is the Obituary Notice by Major P. A. MacMahon, B.A. , F.B.8., Royal Society Proceedings, Lxni, 1898, p. ix. There is also an article in the Dictionary of National Biography, by Professor E. B. Elliott, F.B.S. and Mr P. E. Matheson, M.A., which gives a list of authorities, and an earlier article by Major MacMahon, Nature, 25 March 1897. Other sources of information are referred to in the course of the following.
rvi Biographical Notice
naturally come before him, either in connexion with his subsequent career at Cambridge, or with his own later investigations, we find it difficult to make a choice. Of Englishmen Henry Cavendish (1731 — 1810) was dead, Thomas Young (1773—1829) was forty-one, Faraday (1791—1867) was twenty-three, and had just exchanged (in 1813) a bookbinder's workshop for the laboratory of the Royal Institution, Sir John Herschel (1792 — 1871) was twenty-two, and George Green (1793 — 1841), who was afterwards to be examined with Sylvester at Cambridge, was twenty-one. Cayley, with whom he was to be so much associated, was born in 1821, and was Senior Wrangler in 1842. The year 1814 was " the year of peace," and was the year in which Poncelet (1788 — 1867) returned to Paris from the Russian prison in which he had recon- structed the theory of conic sections; Lagrange (1736 — 1813) had just died, but there were living Laplace (1749 — 1827), Legendre (1752—1833), Fourier (1768—1830), Ampere (1775—1836), Poisson (1781—1840), Fresnel (1788— 1827), Cauchy (1789—1857). J. C. F. Sturm (1803—1855), whose theorem was to have such an importance for Sylvester, was eleven years his senior ; Hermite's life extended from 1822 to 1901. In Germany there were Gauss (1777 — 1855), whose Disquisitiones Arithmeticae is dated 1801, Steiner (1796—1863), von Staudt (1798—1867), Jacobi (1804—1851), W. Weber (1804—1891), Dirichlet (1805—1859), Kummer (1810—1893), while Weier- strass was born in 1815 ; and then there were Helmholtz (1821 — 1894), Kirchhoff (1824—1886), Riemann (1826—1866), and Clebsch (1833—1872). In Italy Brioschi, who took part in the development of the theory of in- variants, was born in 1824 and died in 1897 ; and the name of Abel (1802 — 1829) cannot be omitted. All these, and many others, went to form the atmosphere in which Sylvester's life was spent.
Until Sylvester was fifteen years of age he was educated in London — from the age of six to the age of twelve with Mr Neumegen, at Highgate, subsequently, for a year and a half, with Mr Daniell at Islington, then, for five months, at the University of London (afterwards University College), where apparently he met Professor De Morgan, who (except from 1831 to 1835) taught at this institution fi"om 1828 to 1867 ; for Sylvester speaks in 1840 (l 53) of having been a pupil of De Morgan's. His gift for Mathe- matics seems undoubtedly to have been apparent at this time ; for Mr Neumegen sent him at the age of eleven to be examined in Algebra by Dr Olinthus Gregory, at the Royal Military Academy, Woolwich, and it is recorded that this gentleman was writing to Sylvester's father two years later to enquire for him, with a view to testing his progress in the interval. 1829 In 1829, at the age of fifteen, Sylvester went to Liverpool ; here he attended the school of the Royal Institution, residing with aunts. The Institution, it appears, was founded in 1814, largely by the exertions of William Roscoe (1753 — 1831), and its school in 1819 ; it must not be confounded with the Liverpool Institute, which grew out of the Mechanics Institute, founded in
Biographical Notice xvii
1825, by Mr Huskisson. The Head-master at this time was the Rev. T. W. Peile, afterwards Head-master of Repton, and the mathematical master was Mr Marratt. A contemporary at the school was Sir William Leece Drinkwater, afterwards First Deemster, Isle of Man. At this school Sylvester remained less than two years. In February 1830 he was awarded the first prize in the Mathematical School, and was so far beyond the other scholars that he could not be included in any class. While here, also, he was awarded a prize of 500 dollars for solving a question in arrangements, to the great satisfaction of the Contractors of Lotteries in the United States, the question being referred to him by the intervention of his elder brother in New York. At this early period of his life, too, he seems to have suffered for his Jewish faith at the hands of his young contemporaries ; possibly this may account for the episode recorded, of his running away from school and sailing to Dublin. Here, with only a few shillings in his pocket, he was accidentally accosted by the Right Hon. R. Keatinge, Judge of the Prerogative Court of Ireland, who, having discovered him to be a first cousin of his wife, entertained him, and sent him back to Liverpool.
The indications were by now sufficient to encourage him to a mathe- 1831 matical career. After reading for a short time with the Rev. Dr Richard Wilson, sometime Fellow of St John's College, Cambridge, afterwards Head- master of St Peter's Collegiate School, Eaton Square, London, Sylvester was entered* at St John's College on 7 July, as a Sizar, commencing residence on 6 October 1831, when just over seventeen, his tutor being Mr Gwatkin. He resided continuously till the end of the Michaelmas Term, 1833, though he seems to have been seriously ill in June of this year. For two years from the beginning of 1834 his name does not appear as a member of the College, and apparently he was at home on account of illness. In January 1836 he was readmitted, this time as a Pensioner, and resided during the Lent and Michaelmas Terms, being also incapacitated in the intervening term. In January 1837 he underwent his final University examination, the Mathe- matical Tripos, and was placed second on the list. The first six names of that year were Griffin, St John's ; Sylvester, St John's; Brumell, St John's; Green, Gonville and Caius ; Gregory, Trinity, and Ellis, Trinity. Of these, George Green, bom at Sneinton, near Nottingham, in 1793, was already the author of the famous paper, " An essay on the application of Mathematical Analysis to the theories of Electricity and Magnetism," which was published at Nottingham, by subscription, in 1828. He died in 1841, more than fifty years before Sylvester.
Of the general impression which Sylvester produced upon his con- temporaries at Cambridge, it is difficult to judge. It is recorded that he attended the lectures of J. Gumming, Professor of Chemistry in the
* The Eagle, the College Magazine, «x (1897), p. 603. A list of Sylvegter's scientific dis- tinctions is given in this place (p. 600).
xviii Biographical Notice
University from 1815 to 1861, and, as required by College regulations, the Classical lectures of Bushby. We know how keen was his interest in Chemistry many years later in Baltimore (cf. his paper on The New Atomic Theory, lii 148) : and his writings furnish evidence of the pleasure he took in introducing a Classical allusion. When he became Editor of the Quarterly Journal of Mathematics ia 1855 he secured the printing of a Greek motto on its title-page :
o Ti ovaia rrphs yfve(Tiv, fTrumffif) irpos niariv Ka\ hiavoia npos fiKa<riav eort ;
later on, the American Journal under his care also had (iv 298) a Greek motto :
npayfiaTav (\ty)(Os oi ^XfTro/ifvav ;
in his older age the reading and translation of Classical authors was one of his resources.
He was, in later life at least, well acquainted with French, German and Italian, and rejoices (ii 563) because these with Latin and English "may happily at the present day be regarded as the common property and inherit- ance of mathematical Europe." He was also much interested in Music. We are told that at one time he took lessons in singing from Gounod, and was known to sing at entertainments given to working men. " May not Music," he asks (ll 419), "be described as the Mathematic of sense, Mathematic as Music of the reason ?..." Or again (ill 128), " It seems to me that the whole of aesthetic (...) may be regarded as a scheme having four centres, ..., namely Epic, Music, Plastic and Mathematic " ; and he advocated " a new method of learning to read on the pianoforte " (ill 8).
Of his interest in general literature, and his keen relish for a striking phrase, no reader of his papers needs to be reminded. To his first long paper on Syzygetic Relations, published in the Philosophical Transactions of the Royal Society (i 429), he prefixes the words
How charming is divine philosophy !
Not harsh and crabbed as dull fools suppose,
But musical as is Apollo's lute
And a perpetual feast of uectar'd sweets.
Where no crude surfeit reigus I
In his paper on Newton's rule, also in the publications of the Royal Society (II 380), he quotes
Turns them to shapes and gives to airy nothing A local habitation and a name.
In his Constructive Theory of Partitions (iv 1) he leads off with
seeming parted, But yet a union in partition ;
Biographical Notice Tax.
the Second Act, in which the Partitions are transformed by cunning opera- tions performed on the diagrams which represent them, is introduced by
Naturelly, by composiciouns Of anglis, and slie reflexiouns ;
as the plot thickens he begins to feel more need of apology, and Act III
begins with
mazes intricate, Eccentric, intervolved, yet regular Then most, when most irregular they seem ;
while, when he comes to the Exodion, and feels that, after fifty-eight pages, direct appeal may have lost its power, he takes refuge in Spenser's fairyland with the lines
At which he wondred much and gan enquere What stately building durst so high extend Her lofty towres, unto the starry sphere.
Of his clever sayings we all remember many : " Symmetry, like the grace of an Eastern robe, has not unfrequently to be purchased at the expense of some sacrifice of freedom and rapidity of action " (l 309) ; or again, in support of the contention, that to say that a proposition is little to the point is not to be taken as demurring to its truth (ll 725), " I should not hesitate to say, if some amiable youth wished to entertain his partner in a quadrille with agree- able conversation, that it would be little to the point, according to the German proverb, to regale her with such information as how
Long are the days of summer-tide And tall the towers of Strasburg's fane,
but should be surprised to have it imputed to me on that account that I demurred to the proposition of the length of the days in summer, or the height of Strasburg's towers." More direct still (ill 9), disclaiming the idea that the simplicity of Peaucellier's linkwork should discredit the difiiculty of its discovery, " The idea of the facility of the result, by a natural mental illusion, gets transferred to the process of conception, as if a healthy babe were to be accepted as proof of an easy act of parturition." Some others will be found referred to in the index.
It is also recorded that among the friends of his earlier life was H. T. Buckle, author of the History of Civilisation, with whom, in addition to more serious reasons for sympathy, chess playing was a link of friendship.
Whether the many sides of Sylvester's character, indicated by these gleanings from his later life, were much in evidence at Cambridge, we do not know. The intellectual atmosphere of the place at the time was extremely vigorous in some ways. The Philosophical Society was founded in 1819, largely on the initiative of Adam Sedgwick and J. S. Henslow, and obtained a Charter in 1832; its early volumes are evidence of the great
XX Biographical Notice
width and alertness of scientific interest in Cambridge at this time ; papers of George Green were read at the Society in 1832, 1833, 1837 and 1839 ; James Gumming, whose chemical lectures Sylvester attended, Sir John Herschel, De Morgan, and Whewell are aiiiong the early contributors. Sir John Herschel's Preliminary Discourse on the Study of Natural Philo- sophy is dated 1831. The third meeting of the British Association was in Cambridge, on 24 June 1833. Whewell's History of the Inductive Sciences was published at Cambridge in 1837, the Philosophy of the Inductive Sciences in 1840. But we find* that in 1818 Sedgwick gave up his assistant tutor- ship, whose duties were mainly those of teaching the mathematical students of Trinity College, on the ground that "as far as the improvement of the mind is considered, I am at this moment doing nothing....! am... very sensibly approximating to that state of fatuity to which we must all come if we remain here long enough." This was before Sylvester's student time, and while mathematics at Cambridge was still suffering, partly from the long consequences of the controversy in regard to Leibniz and Newton, and more immediately from the loss of communication with the mathematicians of the Continent due to the war. Yet Sir John Herschelf, writing in 1833, feels compelled to speak very decidedly of the long-subsisting superiority of foreign mathematics to our own, as he phrases it, and there seems to be no doubt that mathematics, as distinct from physics, was then at a very low ebb in Cambridge, notwithstanding the success of the struggle, about a quarter of a century before, to introduce the analytical methods then in use on the Continent. C. Babbage, in his amusing Passages from the Life of a Philo- sopher, describes how he went (about 1812) to his public tutor to ask the solution of one of his mathematical difficulties and received the answer that it would not be asked in the Senate House, and was of no sort of con- sequence, with the advice to get up the earlier subjects of the university studies ; and how, after two further attempts and similar replies from other teachers, he acquired a distaste for the routine of the place. His connexion with the translation of Lacroix's Elementary Differential Calculus (1816), and his association with George Peacock, Sir John Herschel and others in the Analytical Society, is well known ; the title proposed by him for a volume of their Transactions, " The principles of pure D-ism in opposition to the Dot-age of the University," has often been quoted.
In addition to the better known accounts, there is an echo of what is usually said about Cambridge in this connexion in an Eloge on Sir John Herschel, read at the Royal Astronomical Society, 9 February 1872, by a writer who compares the work of Lagrange on the theory of equations with that of Waring, who was born in the same year, and was Senior Wrangler at Cambridge in 1757. We may add to this the bare titles of two continental
* Life of Adam Sedgwick, by J. W. Clark, i, p. 154. t Collected Essays, Longmans, 1857, pp. SO — 39.
Biographical Notice xxi
publications of 1837, the year of Sylvester's Tripos Examination : — C. Lejeune Dirichlet, Beweis des Satzes, doss jede unbegrenzte arithmetische Progression, deren erstes Glied und Differem game Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthdlt ; E. Kummer, De aequatione a?*- + y^ = z^ per numeros integros resolvenda. Augustus De Morgan, who was fourth Wrangler in 1827, speaking in 1865, at the inaugural meeting of the London Mathematical Society, pronounces that "The Cambridge Examination is nothing but a hard trial of what we must call problems — since they call them so — between the Senior Wrangler that is to be of this present January, and the Senior Wrangler of some three or four years ago. The whole object seems to be to produce problems — or, as I should prefer to call them, hard ten-minute conundrums.... It is impossible in such an examination to propose a matter that would take a competent mathematician two or three hours to solve, and for the consideration of which it would be necessary for him to draw his materials from different sources, and see how he can put together his previous knowledge, so as to bring it to bear most effectually on this particular subject." This is the mathematician's criticism of the system then, and, to a large extent, still in vogue. A criticism from another point of view is found in a letter* of Sir Frederick Pollock, written in 1869, to De Morgan : " I believe the most valuable qualities for practical life cannot be got at by any examination — such as steadiness and perse- verance....! think a Cambridge education has for its object to make good members of society — not to extend science and make profound mathema- ticians " These criticisms appear to agree in one implication, the dominance
of the examination in the training offered by the University ; and they are necessary to a right appreciation of Sylvester's university life and subsequent work. Accordingly, we do not hear, as frequently we do in the case of young students at continental universities, of Sylvester being led to study for himself the great masters in Mathematics. We find him, in 1839 (i 39), disclaiming a first-hand knowledge of Gauss's works ; there is no anecdote, known to me, to put with that he himself tells of Riemann. In a sheet of verses issued by himself, in February 1896 — one of many such sheets, I believe — there is a footnote containing the following : " ...the hotel on the river at Nuremberg, where I conversed outside with a Berlin bookseller, bound, like myself, for Prague.... He told me he was formerly a fellow pupil of Riemann, at the University, and that, one day, after receipt of some numbers of the Comptes rendus from Paris, the latter shut himself up for some weeks, and when he returned to the society of his friends, said (referring to newly-published papers of Cauchy), 'This is a new mathematic.'" We find Sylvester, how- ever, writing in 1839 of " the reflexions which Sturm's memorable theorem had originally excited " (I 44), and we know how much of his subsequent thought was given to this matter. Whether he read Sturm's paper of • W. W. B. Ball, HUtory of Mathematict at Cambridge, 1889, p. 113.
xxii Biographical Notice
23 May 1829 {Bulletin de Firmsac, xi, 1829, p. 419 ; Mimoires par divers Savans, Vl, 1835, pp. 273 — 318), or in what way he learnt of the theorem, there seems to be no record. It is not referred to in the Report on Analysis by George Peacock, Cambridge British Association Report, 1833, pp. 185 — 352, which deals at length with Fourier's method. Sylvester records (ii 655 — 6) that Sturm told him that the theorem originated in the theory of compound pendulums, but he makes no reference to Sturm's recognition of the applica- tion of his principles to certain differential equations of the second order.
Another aspect of Sylvester's time at Cambridge must be referred to. At this time, and indeed until 1871, it was necessary, in order to obtain the Cambridge degree, to subscribe to the Articles of the Church of England ; one of the attempts, in 1834, to remove the restriction, is recorded in the Life of Adam Sedgwick, already referred to (i 418 ; Sedgwick writes a letter to the Times, 8 April 1834). Sylvester was, in his own subsequent bitter phrase (ill 81), one of the first holding "the faith in which the Founder of Christianity was educated " to compete for high honours in the Mathematical Tripos ; not only could he not obtain a degree, but he was excluded from the examination for Dr Smith's mathematical prizes, which, founded in 1769, was usually taken by those who had been most successful in the Mathematical Tripos. Most probably, too, had the facts been otherwise, he would have been shortly elected to a Fellowship at St John's College. To obtain a degree he removed to Trinity College, Dublin, from which, it appears, he received in turn the B.A. and the M.A. (1841). He finally received the B.A. degree at Cambridge, 29 February 1872, the M.A. (honoris caiisa) following 25 May of the same year. 1838 In the year succeeding his Tripos examination at Cambridge, he was elected to the Professorship of Natural Philosophy at (what is now) University College, London, and so became a colleague of Professor De Morgan. The list of the supporters of his candidature includes the names of Dr Olinthus Gregory, who had examined him in Algebra when a schoolboy of eleven, of Dr Richard Wilson, who had taught him before his entrance at St John's College, of the Senior Moderator and Senior Examiner in his Tripos examina- tion, of Philip Kelland, of Queens' College, Senior Wrangler in 1834, after- wards Professor at Edinburgh, and of J. W. Colenso, afterwards Bishop of Natal ; the two last had been private tutors of Sylvester at some portions of his career at Cambridge. He held the post of Professor of Natural Philosophy for a few years only; Professor G. B. Halsted (Science, 11 April 1897) makes a statement suggesting that the examination papers set by him during his tenure of the office are of a nature to indicate that he did not find his subject congenial. During these years he was elected a Fellow of the Royal Society (25 April 1839), at the early age of twenty-five. About this time also an oil- painting of him was made by Patten, of the Royal Scottish Academy, from the recorded description of which it appears that he had dark curly hair and
Biographical Notice xxiii
wore spectacles. It has been said that he took his Tripos examination in January 1837 ; he at once began to publish, in the Philosophical Magazine of 1837 — 38. The first four of his papers are on the analytical develop- ment of Fresnel's optical theory of crystals, and on the motion and rest of fluids and rigid bodies ; but the papers immediately following contain the dialytic method of elimination, and the expression of Sturm's functions in terms of the roots of the equation, as well as many results afterwards included in the considerable memoir on the theory of the syzygetic relations of two polynomials, publi-shed in the Philosophical Transactions of 1853.
Leaving University College in the session of 1840 — 41, he proceeded 1841 as Professor of Mathematics across the Atlantic, to the University of Virginia, founded in 1824 at Charlottesville, Albemarle Co., where* his colleague, Key, of University College, had previously occupied the chair of Mathematics. Such a considerable change deserved a better fate than befell ; in Virginia at this time the question of slavery was a subject of bitter con- tention, and Sylvester had a horror of slavery. The outcome was his almost immediate return ; apparently he had intervened vigorously in a quarrel between two of his students.
On his return from America Sylvester seems to have abandoned mathe- 1844 matics for a time. In 1844 he accepted the post of Actuary to the Legal and Equitable Life Assurance Company, and threw himself into the work with great energy. He did not accept another teaching post for ten years, until 1854, but seems to have given some private instruction, as it is related f that he had, what was unusual at that time, a lady among his pupils — whose name was afterwards famous — Miss Florence Nightingale. He entered at the Inner Temple 29 July 1846, and was called to the Bar 22 November 1850. He also founded the Law Reversionary Interest Society. It was in 1846 184& that Cayley, who had been Senior Wrangler in 1842, left Cambridge and became a pupil of the famous conveyancer, Mr Christie, entering at Lincoln's Inn. He was already an author, and had in fact entered upon one of the main activities of his life; for in 1845 he had published his fundamental paper "On the Theory of Linear Transformations," in which he discusses Boole's discovery of the invariance of a discriminant. To us, knowing how pregnant with consequences the meeting was, it would be interesting to have some details of the introduction of Cayley and Sylvester; the latter lived, then or soon after, in Lincoln's Inn Fields, and we are told+ that during the following years they might often be found walking together round the Courts of Lincoln's Inn, discussing no doubt many things but among them assuredly the Theory of Invariants. Perhaps it was particularly of this time that Sylvester was thinking when he described Cayley (l 376) as " habitually
* J. J. Walker, Proe. Land. Math. Soc. «viu (1896—97), p. 582.
+ The EagU, «x (1897), p. .597.
J Biographical notice ol Arthur Cayley, Cayley's Collected Papers, Volume viii.
8. IV. c
xxiv Biographical Notice
1846 discoursing pearls and rubies," or when, much later (iv 300), he spoke of " Cayley, who, though younger than myself, is my spiritual progenitor — who first opened my eyes and purged them of dross so that they could see and accept the higher mysteries of our common mathematical faith." It is in a paper published in 1851 (i 246) that we find him saying, "The theorem above enunciated was in part suggested in the course of a conversation with Mr Cayley (to whom I am indebted for my restoration to the enjoyment of mathematical life) " ; and Sylvester's productiveness during the latter part of this period is remarkable. In particular there are seven papers whose date of publication is 1850, including the paper on the intersections, contacts and other correlations of two conies, wherein he was on the way to establish the properties of the invariant factors of a determinant, afterwards recog- nised by Weierstrass; and there are thirteen papers whose date is 1851, including the sketch of a memoir on elimination, transformation and canonical forms, in which the remarkable expression of a cubic surface by five cubes is given, the essay on Canonical Forms, and the paper on the relation between the minor determinants of linearly equivalent quadratic functions, in which the notion of invariant factors is implicit ; while in 1852 is dated the first of the papers " On the principles of the Calculus of Forms." Dr Noether remarks* how important for the history of mathematics these years were in other respects ; Kummer's memoir, " Ueber die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren," appeared in 1847 {Crelle, xxxv); Weierstrass's " Beitrag zur Theorie der Abel'schen Integrale " (Beilage zum Jahresbericht iiber das Gymnasium zu Braunsberg) is dated 1849 ; Riemann's Inaugural-dissertation, " Grundlagen fur eine allgemeine Theorie der Functionen eiuer veranderlichen complexen Grosse," is dated 1851. Referring to the discovery of the Canonical Forms in order to enforce the statement that observation, induction, invention and experi- mental verification all play a part in mathematical discovery (ii 714), Sylvester tells an anecdote which has a personal interest : " I discovered and developed the whole theory of canonical binary forms for odd degrees, and, as far as yet made out, for even degrees too, at one evening sitting, with a decanter of port wine to sustain nature's flagging energies, in a back office in Lincoln's Inn Fields. The work was done, and well done, but at the usual cost of racking thought — a brain on fire, and feet feeling, or feelingless, as if plunged in an ice-pail. That night we slept no more."
To Englishmen, in whose minds the modern developments of physical mathematics are associated with many familiar names, who recall Thomas Young, Faraday, Herschel, George Green, Stokes, Adams, Kelvin, Maxwell, the activity of Cayley and Sylvester may at first sight seem very natural. But in fact the aim of such men as those first named was primarily the coordination of the phenomena of Nature, not the development of any * Charles Hermite, Math. Annalen, LV, p. 343.
Biographical Notice xxv
mathematical theory. And if we think of such names as those of De Morgan, 1846 Warren, Peacock, their interest perhaps was either systematic or didactic; their endeavours were necessarily largely directed to criticising, and expounding to their countrymen, the proposals of continental mathematicians. But Cayley and Sylvester were in a ditferent position at the time of which we are speaking ; neither of them had any official duties as teacher of mathematics ; to Cayley, as he afterwards said (in 1883) to the British Association, mathe- matics was "a tract of beautiful country seen at first in the distance, but which will bear to be rambled through and studied in every detail of hillside and valley, stream, rock, wood, and flower." To him and to Sylvester, Pure Mathematics was an opportunity for unceasing exploration; or, in another figure, a challenge to carve from the rough block a face whose beauty should for all time tell of the joy there was in the making of it ; or again, it was the discernment and identification of high peaks of which the climbing might be in the years to come the task of those to whom strenuous labour is a delight and fine air an intoxication. And this spirit was a new one in England at this time, of which we may easily miss the significance. It may therefore help if we quote, without expressing any opinion as to its proportionate justice, the impression of an American observer, Dr Fabian Franklin, who succeeded Sylvester as Professor at Baltimore. Speaking* at the memorial meeting held immediately after Sylvester's death, 2 May 1897, he says of Sylvester, " His influence upon the development of mathematical science rests chiefly, of course, upon his work in the Theory qf Invariants. Apart from Sir William Rowan Hamilton's invention and development of Qua- ternions, this theory is the one great contribution made by British thought to the progress of Pure Mathematics in the present century, or indeed since the days of the contemporaries of Newton. From about the middle of the eighteenth century, until near the middle of the nineteenth, English mathematics was in a condition of something like torpor.... And, accordingly, it proved to be the case that in the magnificent extension of the bounds of mathematics which was effected by the continental mathematicians during the first four decades of the present century, England had no share. It is almost literally correct to say that the history of mathematics for about a hundred years might be written without serious defect with English mathe- matics left entirely out of account.
" That a like statement cannot be made in regard to the past fifty years is due pre-eminently to the genius and labours of three men : Hamilton, Cayley and Sylvester.... Not only did other English mathematicians join in the work, but Hermite in France, Aronhold and Clebsch in Germany, Brioschi in Italy, and other continental mathematicians, seized upon the new ideas, and the theory of invariants was for three decades one of the leading objects of mathematical research throughout Europe. It is impossible to apportion • Johiu Hopkiru University Circulari, June 1897.
02
xxvi Biographical Notice
between Cayley and Sylvester the honour of the series of brilliant discoveries which marked the early years of the theory of invariants...."
It would not be right to omit reference to another factor in the mathe- matical life of the time we are dealing with — the influence of George Salmon. At what time Sylvester first became acquainted with him, I have not ascer- tained ; but we know that the theory of the straight lines lying upon a cubic surface was worked out in a correspondence between Cayley and Salmon in 1849. Readers of Salmon are aware of the intimate way in which he followed Sylvester's work, while Sylvester, in his papers, makes frequent reference to Salmon's books. There is a personal letter* from Salmon to Sylvester, of date 1 May 1861, which exhibits the relations of the two men in an interesting light, "...I should be very glad if there was any chance of your preparing an edition of your opuscula. There have been, of course, occasional little statements in your papers requiring verification. Written, as they were, in the very heat of discovery, they are rather to be compared to the hurried bulletins written by a general on the field of battle than to the cool details 'of the historian. Honestly, however, I don't think there is the least chance of your going back to these former studies. I shall be content to let you off some of these if you will do justice to what you have done on the subject of partitions. I wish you would seriously consider whether it is not a duty everyone owes to Society, when one brings a child into the world, to look to the decent rearing of it. I must say that you have to a reprehensible degree, a cuckoo-like fashion of dropping eggs and not seeming to care what becomes of them. Your procreative instincts ought to be more evenly balanced by such instincts as would inspire greater care of your offspring and more attention to providing for them in life, and producing them to the world in a presentable form.
" Hoping you will meditate on this homily and be the better for it, I remain, yours sincerely, Geo. Salmon."
Salmon himself did a great deal for the rearing of many of Sylvester's offspring, and I suppose it would be hard to estimate how much of Sylvester's and Cayley 's reputation in their lifetime was due to his large-minded and genial exposition.
Sylvester himself, in a paper of 1863 (ii 337), supplies some answer to such criticisms as this of Salmon's : " in consequence of the large arrears of algebraical and arithmetical speculations waiting in his mind their turn to be called into outward existence, he [the author] is driven to the alternative of leaving the fruits of his meditations to perish... or venturing to produce from time to time such imperfect sketches as the present, calculated to
evoke the mental cooperation of his readers "
1854 It was not until 10 June 1863 that Cayley returned to Cambridge, as Sadlerian Professor of Pure Mathematics. In 1854, Sylvester was a • Printed in the Eagle, the Magazine of St John's College, xxix (1908), p. 380.
Biographical Notice xxvii
candidate for the Professorship of Mathematics at the Royal Military Academy, Woolwich. At this time he had published the papers now reprinted in Volume i, the Theory of Invariants had an existence firmly established, and Sylvester had an European reputation. But his candidature was unsuccessful. This was in August of 1854. In December of the same year he gave his Probationary lecture on Geometry before the Electors to the Professorship of Geometry in Gresham College, London (ii 2). In this he was also unsuccessful. Professor G. B. Halsted has recorded that Sylvester often deplored the time he had lost "fighting the world," and he would feel these disappointments keenly. However, the successful candidate at Woolwich died a few months after being appointed, and Sylvester was again a candidate. A letter on his behalf by Lord Brougham, of date 28 August 1855, speaks of him as my "learned and excellent friend and brother mathematician Mr Sylvester." This time he was elected. He took up the appointment on 15 September 1855, being, for a year, lecturer in Natural Philosophy as well as Professor of Mathematics. There is record of the exact emoluments of the post, a salary of £550, a Government Residence (K Quarters, Woolwich Common), medical attendance and right of pasturage on the Common. The honse was a pleasant one, with a good garden, in which he could enjoy the shade of his own walnut tree, we are told, and he was able to entertain his scientific friends. The conver- sations with Cayley still went on ; we hear of them walking to meet one another, Cayley from 2 Stone Buildings and he from his home, their meeting point falling near Lewisham. Sylvester retained this post until July 1870, sometimes justifying, we are led to believe, the original hesitation of the electors in regard to his efiBciency as an elementary teacher ; there are stories such as that of his housekeeper pursuing him from home carrying his collar and necktie. His publications during this time are, approximately, those reprinted in Volume II.
Sylvester gave seven lectures on the Theory of Partitions at King's College, London, in 1859 (ll 119), not published until 1897, and then only from outlines privately circulated at the time of delivery ; Capt. (now Sir Andrew) Noble collaborated with him in an important degree in his work on the Theory of Partitions. He wrote the paper on the involution of lines in space considered as axes of rotation (il 236). The long paper on Newton's rule and the invariantive discrimination of the roots of a quintic was published in the Philosophical Transactions, 18(54 (li 376). His work on the proof of Newton's rule made its appeal in various directions — Todhunter remarks in his Theory of Equations, " If we consider the intrinsic beauty of the theorem, the interest which belongs to the rule associated with the great name of Newton, and the long lapse of years during which the reason and extent of that rule remained undiscovered by mathematicians — among whom Maclaurin, Waring and Euler are explicitly included — we must regard
xxviii Biographical Notice
Professor Sylvester's investigations as among the most important contribu- tions made to the Theory of Equations in modem times, justly to be ranked with those of Fourier, Sturm and Cauchy."
1865 Sylvester's outward life also contained points to be remarked. In April 1855 appeared the first number of the QuaHerly Journal of Pure and Applied Mathematics, edited by J. J. Sylvester, M.A., F.R.S. and N. M. Ferrers, M.A.; this replaced the Cambridge and Dublin Mathematical Journal which had first been edited by W. Thomson, M.A. (the late Lord Kelvin) and then by W. Thomson, M.A. and N. M. Ferrers, M.A. In the Preface, the plea is put forward that a more ambitious journal was necessary in view of the growing state of the subject, and might render British mathematicians less dependent on the courtesy of the editors of Foreign journals. Assisted by Stokes, Cayley and Hermite, this joint editorship continued unchanged until June 1877.
1856 In 1856 Sylvester was elected* to the Athenaeum Club, under the special Rule II. The fact is worth recording. Sylvester was never married, and in subsequent years this was the address he frequently appended to his writings.
1859 In 1859 he delivered seven lectures on the Partition of Numbers, at King's College, London, as noted above.
1861 In 1861 he was awarded a Royal Medal by the Royal Society, Cayley having received that honour in 1859.
1863 On 7 Decf 1863 he was chosen correspondent in mathematics by the French Academy of Sciences, in place of the great geometer Steiner, who had died in the preceding April. We notice that he had just commenced (in 1861) what was to be a long series of communications to the Academy, and his paper on Involutions of lines in space had been presented to the Academy by M. Chasles (il 236). His closely following paper on the Double Sixes of lines on a Cubic surface (ii 242) he himself afterwards (ii 451) notes as being an unconscious plagiarism from a paper of Schlafli, which he had read as editor before its publication in the Quarterly Journal (Vol. Ii (1858), p. 116).
1864 His memoir in the Phil. Trans, on Newton's rule is of date 1864 (li 376). In 1865 he delivered a lecture on the subject at King's College, London (ll 498). A syllabus of this lecture forms the first mathematical paper published by the London Mathematical Society. This Society was inaugurated by a speech of Professor De Morgan 16 Jan. 1865, with "the great aim of the cultivation of pure Mathematics and their most immediate applica- tions." The Society consisted at its formation of twenty-seven members, nearly all of whom were members of University College. Sylvester was elected the second President at the Annual General Meeting held at Burlington
* As I have been able to verify by the courtesy of the Secretary. t J. J. Walker, Proc. Land. Math. Soc. xxviii (1896—97), p. 585.
Biographical Notice xxix
House on 8 November 1866 (in the rooms of the Chemical Society), and held oflSce until November 1868. He served on the Council for many years.
In 1869 Sylvester was President of the Mathematical and Physical 1869 Section of the British Association at Exeter. He took as the subject of his Presidential address the charge that Huxley had brought against Mathe- matics, of being the study that knew nothing of observation or induction (II 650), nothing of experiment or causation. An incidental reference in this address to Kant's doctrine of space and time led to a lively controversy in the pages of Sature, in which Sylvester's trenchant style and wide range of intellectual alertness may be well seen (ii Appendix). Characteristically enough Sylvester reprinted the address, with annotations, and the cor- respondence in regard to Kant, as an Appendix to his volume on the Laws of Verse (Longmans, 1870) — a volume which should be consulted for an appreciation of a side of Sylvester's activity which occupied him to the end of his life.
In 1870 Sylvester retired from his post at Woolwicli, in consequence 1870 of what he regarded as an unfair change in the regulations. As may be seen in the article of G. B. Halsted, above quoted. Science, 11 April 1897, and in the Leading Article which appeared in the Times, 17 August 1871 (see also Sylvester's own letter to the Times, 24 August 1871, and Nature, Vol. IV (1871), pp. 324, 326), there was much bitterness as to the question of pension, which was however finally secured to him, if not on the scale desired. For the next few years Sylvester resided near the Athenaeum Club, apparently somewhat undecided as to his course in life. We hear of him as reciting and singing at Penny Readings (cf. his remarks on the utility of these in the Laws of Verse, p. 70), and as being a candidate for the London School Board*, and, in The Gentleman's Magazine for February 1871, there appears "The Ballad of Sir John de Courcy," translated from the German by " Syzygeticus."
In 1874 Sylvester gave a Friday evening discourse at the Royal Insti- 1874 tution on Peaucellier's link bar motion. He was then sixty years old, yet, even in the abstract of the lecture which remains (in 7), the vivacity with which he dealt with the matter is very striking. His enthusiasm evoked a wide interest in the subject.
In 1875 the Johns Hopkins University was founded at Baltimore. A 1875 letter to Sylvester from the celebrated Joseph Henry, of date 25 August 1875, seems to indicate that Sylvester had expressed at least a willingness to share in forming the tone of the young university; the authorities seem to have felt that a Professor of Mathematics and a Professor of Classics could inaugurate the work of an University without expensive buildings or
* Sylvester's election address as candidate for the London School Board for Marylebone in the place of Professor Hnzle;, with a list of his scientific supporters, is found in Nature, 21 March 1872, p. 410.
XXX Biographical Notice
elaborate apparatus. It was finally agreed that Sylvester should go, securing, besides his travelling expenses, an annual stipend of 5000 dollars " paid in gold." And so, at the age of sixty-one, still full of fire and enthusiasm, as appears abundantly from the work he devoted to the papers here reprinted in Volume III, he again crossed the Atlantic, and did not relinquish the post for eight years, until 1883. It was an experiment in educational method ; Sylvester was free to teach whatever he wished in the way he thought best ; so far as one can judge from the records, if the object of an University be to light a fire of intellectual interests, it was a triumphant success. His foibles no doubt caused amusement, his faults as a systematic lecturer must have been a sore grief to the students who hoped to carry away note-books of balanced records for future use ; but the moral effect of such earnestness as we see him shewing for instance in the papers 19 — 21 of Volume ill (on the true number of irreducible concomitants for the cubic and biquadratic), and in paper 34 (on the system for two cubics), must have been enormous. " His first pupil, his first class," was Professor George Bruce Halsted ; he it was who, as recorded in the Commemoration-day Address (in 76) " would have the New Algebra." How the consequence was that Sylvester's brain " took fire," is recorded in the pages of the American Journal of Mathematics. Othere have left records of his influence and methods. Major MacMahon quotes the impressions of Dr E. W. Davis, Mr A. S. Hathaway and Dr W. P. Durfee. Professor Halsted's Article in Science has already been quoted. From Dr Fabian Franklin's long commemorative address*, already referred to, another paragraph may be given: "One of the most striking of Sylvester's achievements was his demonstration and extension of Newton's improved rule concerning the number of the imaginary roots of an algebraic equation. ...We who knew him well in later years can find no diflBculty in understanding the hold this problem had upon him. It was the good fortune of his early hearers in this University to be present when he came into the lecture-room, flushed with the achievement of a somewhat similar task. A certain fundamental theorem in the Theory of Invariants (in 117, 232), which had formed the basis of an important section of Cayley's work, had never been completely demonstrated. The lack of this demonstration had always been, to Sylvester's mind, a most serious blemish in the structure. He had, however, he told us, years ago given up the attempt to find the proof, as hopeless. But, upon coming fresh to the subject in connection with his Baltimore Lectures, he again grappled with the problem, and by a fortunate inspiration, succeeded in solving it. It was with a thrill of sympathetic pleasure that his young hearers thus found themselves in some measure associated with an intel- lectual feat, by which had been overcome a difficulty that had successfully resisted assault for a quarter of a century."
* Johni Hopkins University Circulars, June 1897.
Biographical Notice xxxi
The same writer gives an anecdote illustrating another side of the picture, which may be repeated here. " The reading of the Rosalind poem at the Peabody Institute was the occasion of an amusing exhibition of absence of mind. The poem consisted of no less than 400 lines, all rhyming with the name Rosalind (the long and short sound of i both being allowed). The audience quite filled the hall, and expected to find much interest or amusement in listening to this unique experiment in verse. But Professor Sylvester had found it necessarj' to write a large number of explanatory footnotes, and he announced that in order not to interrupt the poem he would read the footnotes in a body, first. Nearly every footnote suggested some additional extempore remark, and the reader was so interested in each one that he was not in the least aware of the flight of time, or of the amuse- ment of the audience. When he had dispatched the last of the notes, he looked up at the clock, and was horrified to find that he had kept the audience an hour and a half before beginning to read the poem they had come to hear. The astonishment on his face was answered by a buret of good-humoured laughter from the audience ; and then, after begging all his hearers to feel at perfect liberty to leave if they had engagements, he read the Rosalind poem." It may be noted here that it was at Baltimore he wrote " Spring's Debut, a Town Idyll," two centuries of lines all rhyming with "Winn." (January 1880.)
Sylvester's own account of his life at Baltimore, and many other matters, are sufficiently given in the Commemoration-day Address, 22 February 1877 (ill 72) ; it is not necessary to dwell on this further here.
In 1878 appeared the first volume of the Avierican Journal of Mathe- 1878 matics established by the University under Sylvester's care. His first paper is a long account of the application of the new atomic theory to the graphical representation of the concomitants of binary quantics (in 148).
In 1880 he was awarded by the Royal Society the highest honour 1880 possible, the Copley Medal; on 11 June 1880, he was elected Honorary Fellow of his old College of St John at Cambridge, Benjamin Hall Kennedy, the famous schoolmaster and Greek scholar, being elected on the same day. Their portraits are now both preserved in the College.
It is to this period of his life we must refer also the beginning of his investigations in regard to matrices, especially binary matrices. He says (IV 209) " my memoir on TcliebycheflF's method concerning the totality of prime numbers within certain limits, was the indirect cause of turning my attention to the subject, as (through the systems of difference equations therein employed to contract Tchebycheff's limits) I was led to the discovery of the properties of the latent roots of matrices, and had made considerable progress in developing the theory of matrices considered as quantities, when on writing to Professor Cayley upon the subject he referred me to [his own] memoir." Here also, in the interesting communications to the Mathematical
xxxii Biographical Notice
Club reprinted in the Johns Hopkins University Circulars, arose a new interest in developing the Theory of Partitions, which issued in the Con- structive Theory of Partitions (iv 1 — 83) printed in the American Journal (1883). In the course of the year 1883 the University of Oxford conferred upon Sylvester the honorary degree of D.C.L. ; and in Decent) ber of that year, soon after his sixty-ninth birthday, his great distinction was recognised further in the same University by his election to succeed the illustrious H. J. S. Smith as occupant of the chair of Savilian Professor of Geometry. The Professorship had beeu founded in 1619 by Sir Henry Savile, Warden of Merton College, the first professor being obtained by promoting Henry Briggs from the post which Sylvester had vainly sought in 1854, that of Gresham Professor of Geometry in London, so that, as Mr Rouse Ball remarks, Briggs held in succession the two earliest chairs of mathematics that were founded in England — the college founded by Sir Thomas Gresham having been opened in 1596. Other holders of the Savilian chair were John Wallis, 1649, and Edmund Halley, 1704. The companion chair at Oxford, of Savilian Professorship of Astronomy, was held from 1870 to 1893 by the Rev. Charles Pritchard, who was also an alumnus at St John's College, Cambridge. These two were now to be again members of the same house, as Fellows of New College.
The election of Sylvester to Oxford was a matter of importance at Balti- more. On 20 December 1883, a goodbye meeting was held in Hopkins' Hall, Baltimore, by invitation of the President, the guests including Mr Matthew Arnold, Professor Newcomb and others. The following address was agreed to, in Professor Sylvester's presence*.
"The teachers of the Johns Hopkins University, in bidding farewell to their illustrious colleague, Professor Sylvester, desire to give united expression to their appreciation of the eminent services he has rendered the University from the beginning of its actual work. To the new foundation he brought the assured renown of one of the great mathematical names of our day, and by his presence alone made Baltimore a great center of mathematical research.
"To the work of his own department he brought an energy and a devotion that have quickened and informed mathematical study not only in America, but all over the world ; to the workers of the University, whether within his own field or without, the example of reverent love of truth and of knowledge for its own sake, the example of a life consecrated to the highest intellectual aims. To the presence, the work, the example of such a master as Professor Sylvester, the teachers of the Johns Hopkins University all owe, each in his own measure, guidance, help, inspiration ; and in grateful recognition of all that he has done for them and through them for the University, they wish for him a long and happy continuance of his work in his native land, for * Johm Hopkint University Circulars, January 1884, p. 31.
Biographical Notice xxxiii
tliemselves the power of transmitting to others that reverence for the ideal which he has done so much to make the dominant characteristic of this University."
And thus at length, crowned with the gratitude of his American colleagues, 1884 Sylvester was acknowledged in one of the two ancient English Universities, though not his own. And to this, at the age of seventy years, he did not come without something new to say ! On 12 December 1885, he delivered an Inaugural lecture. On the Method of Reciprocants (iv 278), that is of functions of differential coefiScients whose form is unaltered by certain linear transformations of the variables This he followed up by a course of lectures which, as finally edited, extend over more than two hundred pages of the present Reprint. The matter evidently appealed to him as a general- isation of the theory of concomitants, and he worked hard and enthusiastically at the relations of the two theories, gathering round him a school of advanced students. This was the last great continent of thought to be won by him, though he wrote, in 1886, for the centenary volume of "the leading Matlie- matical Journal in the world," Crelle's Journal, a paper on the so-called Tschirnhausen Transformation, which he ascribed to the inspiration of Bring (1786) (IV 531), and a paper ou a funicular solution of Bufifon's " problem of the needle" in 1890 (iv 663), besides other papers. In the Theory of Reciprocants he had been anticipated in detail by Halphen (Thhe, 1878), as he acknowledges. The general idea of differential invariants had been already formulated by Sophus Lie (see his paper on Differential Invariants, Math. Ann. xxiv (1884) in which he states that his investigations go back to 1869 — 72), as an application of his theory of Continuous Groups ; to this Sylvester paid but scant attention. On the other hand it may be recalled that Sylvester had himself in cooperation with Cayley long before stated and frequently employed the principle of infinitesimal trans- formations, and in his first paper on Schwarzian Derivatives (iv 252) he employs the idea of " extended " infinitesimal variations without remark.
One striking note in his Inaugural address at Oxford is the fulness of his references to his colleagues in mathematical work — and of these, what he said about Hammond, fully borne out as it was by the help he gave in the Theory of Reciprocants, seems worthy of being recalled : " I should not do justice to my feelings if I did not acknowledge my deep obligations to Mr Hammond for the assistance which he has rendered me, not only in pre- paring this lecture which you have listened to with such exemplary patience, but in developing the theory ;... saving only our Cayley (...) there is no one I can think of with whom I ever have conversed, from my intercourse with whom I have derived more benefit..." (iv 300)*.
* Another worker to whom he referred in warm terms was Arthur Buchheim. It was his melanchol; duty a few years later to write an Obituary Notice of this distinguished young mathematician, who died at the age of twenty-nine. Nature, 27 September 1888, p. 515.
xxxiv Biographical Notice
1887 In 1887 the Council of the London Mathematical Society made the second award of the De Morgan medal to Sylvester, the first award (in 1884) having been made to Cayley.
1889 In 1889, at the request of a few College friends at Cambridge and elsewhere, he sat to A. E. Emslie for an oil-painting, now hanging in the Hall of St John's College, which was exhibited in the Academy of that year*. It is stated to be a good portrait, though, as he himself writes {Eagle, Vol. XIX, 1897, p. 604), "I was in much trouble at that time... and could scarcely keep awake from the effect of the light on my wearied eyes." This portrait is reproduced at the commencement of the present volume. A copy of it is at New College, Oxford. An oil-painting by Patten, made when he was twenty-six, has already been referred to. An engraving by G. J. Stodart, from a photograph by Messrs I. Stilliard & Co., Oxford, appeared in Nature, accompanying an appreciation by Cayley {Nature, Vol. xxxix, 1889 ; Cayley 's Collected Papers, Xlll, p. 43 gives the appreciation) ; he himself is said to have much prized a particular photograph taken at Venice. On the occasion of his leaving Baltimore a medal was struck in liis honour, of which an exemplar is in the library of St John's College, Cambridge, giving in profile an idea of powerful features. Another medal, struck shortly after his death, is now awarded triennially by the Royal Society of London, for the encouragement of Mathematical Research. This also is a profile with the same impression of strength. It is one side of this medal which is reproduced at the beginning of this Notice (p. xv).
1890 On 10 June 1890 he was awarded the Honorary Degree of Sc.D. by the University of Cambridge. Honorary degrees were conferred on this occasion upon Benjamin Jowett, Henry Parry Liddon, Andrew Clark, Jonathan Hutchinson, George Richmond, John Evans, James Joseph Sylvester and Alexander John Ellis. The speech delivered upon Sylvester by the Public Orator, with his own footnotes, is as follows {Orationes et Epistolae Canta- hrigienses (1876—1909), Macmillan, 1910, p. 83):
" Plus quam tres et quinquaginta anni interfuerunt, ex quo Academiae nostrae inter silvas adulescens quidem errabat, populi sacri antiquissima stirpe oriundus, cuius maiores ultimi, primum Chaldaeorum in campis, deinde Palestinae in collibus, caeli nocturni Stellas innumerabiles, prolis futurae velut imaginem referentesf, non sine reverentia quadam suspiciebant. Ipse numerorum peritia praeclarus, primum inter Londinienses Academiae nostrae studia praecipua ingenii sui lumine illustrabat. Postea trans aequor Atlan- ticum plus quam semel honorifice vocatus, fratribus nostris transmarinis doctrinae mathematicae facem praeferebat;):. Nuper professoris insignis in locum electus, et Britanniae non sine laude redditus, in Academia Oxouiensi
* Graves' Catalogue of the Royal Academy, 1769—1904.
t Genesis, xv. 5.
X University of Virginia, 1841—45 ; Johns Hopkins University, 1877—83.
Biographical Notice xxxv
scientiae flammam indies clariorem excitat*. Ubicunique incedit, exemplo suo nova studia semper accendit. Sive numerorum detopiav explicat, sive Geometriae recentioris terminos extendit, sive regni sui velut in puro caelo regiones prius inexploratas pererrat, scientiae suae inter principes ubique conspicitur. Nonnulla quae Newtonus noster, quae Fresnelius, lacobius, Stunnius, alii, imperfecta reliquerunt, Sylvester noster aut elegantiiis expli- cavit, aut argumentis veris coraprobavit. Quam parvis ab initiis argumenta quam magna evolvit ; quotiens res prius abditas exprimere conatus, sermonem nostrum ditavit, et nova rerum nomina audacter protulitf ! Arte quali numerorum leges non modo poetis antiquis interpretandis sed etiam carmini- bus novis pangendis accommodate ! Neque surdis canit, sed 'respondent omnia silvae§,' si quando, inter rerum graviorum curas, aevi prioris pastores aemulatus,
' Silvestrem tenui musam meditatur avena||.'
Duco ad vos CoUegii Divi loannis Socium, trium simul Academiarum Senatorem, quattuor deinceps Academiarum Professorem, lacobum losephum Sylvester."
During his residence at Oxford he founded the Oxford Mathematical Society. " Members of that Society, even more perhaps than the attendants at his formal lectures, have been impressed and excited to emulation as they have seen his always commanding face grow handsome with enthusiasm, and his eyes flash out irresistible fascination, while he expounded his latest dis- covery or brilliant anticipation," writes the Oxford Magazine (5 May 1897). From the same source we gather that, " despondent over his lecturing work he undoubtedly was, and the feeling of discouragement grew upon him." In 1893 bis eyesight began to be a serious trouble to him, and in 1894 he applied 1893 for leave to resign the active duties of his chair. After that he lived mainly in London or at Tunbridge Wells, sad and dejected because his mathematical power was failing. About August 1896 a revival of energy took place and 1896 he worked at the theory of Compound Partitions, and the Goldbach-Euler con- jecture of the expression of every even number as a sum of two primes. He was present at a meeting of the London Mathematical Society on 14 January 1897, and spoke at some length of his work, answering questions put to him in regard to it. On 12 February he sent a paper, on the number of fractions in their lowest terms that can be formed with limited integers, to the editor of the Messenger of Mathematics, and corrected the proofs about the end of the month (iv 742). At the beginning of March, he had a paralytic seizure 1897 while working in his rooms at Hertford Street, Mayfair. He never spoke again, and died 15 March 1897. He was buried with simple ceremonial at
* Succeeded H. J. S. Smith aa Savilian Professor, 1883—97. t Prof. Cayley in Nature, 3 Jan. 1889.
* The Lam of Verie, 1870 ; Eagle, xiv 251, xv 251, xix 601 f., 604. g Virgil, Jiel. x 8. || ib. Eel. i 2.
xxxvi Biographical Notice
the Jewish Cemetery at Dalston on March 19, the Royal Society, the London Mathematical Society, and New College, Oxford, being represented {Nature, 25 March 1897).
One rises from the task of editing Sylvester's mathematical writings for the Press, with a feeling that here was a great personality as well as a remarkable mathematician, wide and accurate in thought, deep and sensitive in feeling, and inspired with a great faith in things spiritual. " ...is a very great genius," he is reported to have said when pressed on one occasion, " I only wish he would stick to mathematics, instead of talking atheism."
Of the detailed relations of his work with that of contemporary writers, especially for the Theory of Equations, Dr M. Noether has written a masterly and easily accessible account {Math. Annalen, Bd L, 1898). In his Presi- dential address to the London Mathematical Society {Proceedings, xxvill, 1896 — 97) Major MacMahon has given an appreciation of his work on the Theory of Partitions, which should be consulted. Sylvester's long devotion to the Theory of Invariants, in conjunction with Cayley, transforming the whole analysis of Projective Geometry, has left an ineffaceable mark on Mathematics ; but in all questions of algebraical form, working more often by divination than by computation, he is wonderful — his theorems in regard to Sturm's Functions, Canonical Forms, and Determinants suggest themselves at once. So general are some of his results that even the recognition of other theorems as particular cases of them may sometimes be difficult, as very distinguished writers have found.
But another aspect of his mathematical work must, I think, be referred to, if only to place in due proportion what has been said already. It would seem that the multiplicity of the ideas which pressed upon Sylvester's mind left him little leisure to read, more than cursorily, the writings of other mathematicians. He gives a proof of the theorem for six points lying upon a conic section, known as Pascal's theorem, by the method of indeterminate coordinates, and no theorem of analytical geometry seems strange to him, but he makes no reference to the philosophical interest of Poncelet's imaginary elements at infinity. He deals with von Staudt's formulae for the mensuration of pyramids, but von Staudt's scheme for dispensing with the notion of length in geometrical theory does not attract him. The ferment and broad con- clusions as to the foundations of geometry, surely one of the most important of nineteenth century speculations, stir no echo in his pages. Again, he gives remarkable formulae in the Theory of Numbers, but Kummer's investi- gations in regard to ideal numbers, and the vast new regions opened by them, even Gauss's consideration of complex integers, he does not speak of. His silence as to Lie's theory of continuous groups has already been remarked ; he is also silent as to the theory of systems of linear partial differential equations ; and though he gives important results as to the permutations of
Biographical Notice xxxvii
an assigned number of elements, he does not refer to the question of the algebraic solution of the quintic equation, and writes nothing as to the abstract theory of groups. Most remarkable of all, though he gives, and evidently values, an evaluation of an elliptic integral, and proves, in a wonderful way, by partitions, formulae of theta-functions, the majesty of the new world which we associate with such names as those of Cauchy, Abel and Jacobi, of Riemann and Weierstrass and others, does not greatly stir his longing, so far as his writings declare. Indeed the abstract notion of a function whether for a real, or a complex variable, never occurs in his papers ; such a definite instance as Fourier's use of trigonometric series in the Theory of Heat, of 1822, fails to draw him from his combinatorial standpoint ; to him the solution of a differential equation is its solution in explicit form ; and his formula for the quotity of a partition is an isolated result. For an ordinary man, trained in a country where the importance attached to time examinations tends to discourage the study of all mathematical doctrine, this might be easy to understand ; but in Sylvester's case it is very notice- able, and should not be passed over without mention.
Sylvester's position however is secure. As the physicist glories in the interest of his contact with concrete things, so Sylvester loved to mark his progress with definite formulae. He was however before all an abstract thinker, his admiration was ever for intellectual triumphs, his constant worship was of the things of the mind. This it was which seems to have most impressed those who knew him personally. And because of this, his work will endure, according to its value, — mingling with the stream fed by the toil of innumerable men, — of which the issue is as the source. He is of those to whom it is given to renew in us the sanity which is called faith.
H. F. BAKER.
1.
A CONSTRUCTIVE THEORY OF PARTITIONS, ARRANGED IN THREE ACTS, AN INTERACT AND AN EXODION.
[American Journal of Mathematics, v. (1882), pp. 251 — 330; VI. (1884), pp. 334—336.]
Act I. On Partitions Regarded as Entities.
seeming parted, But yet a union in partition.
Twelfth-niffht.
(1) In the new method of partitions it is essential to consider a par- tition as a definite thing, which end is attained by regularization of the snccession of its parts according to some prescribed law. The simplest law for the purpose is that the arrangement of the parts shall be according to their order of magnitude. A leading idea of the method is that of corre- spondence between different complete systems of partitions regularized in the manner aforesaid. The perception of the correspondence is in many eases greatly facilitated by means of a graphical method of representation, which also serves per se as an instrument of transformation.
(2) The most obvious mode of graphically representing a partition is by neans of a network or web formed by two systems of parallel lines or Slaments. Any continuous portion of such web will serve to represent a
irtition, as for example the graph
will represent the partition 3 5 5 4 3 of 20 by reading off the successive numbers of nodes parallel to the horizontal lines of the web. This, however, is not a regularized partition ; the partition will be represented in its regularized form by such a graph as the following :
8 IV.
2 A Constructive theory of Partitions, arranged in [1
which corresponds to the order 5 5 4 3 3, but it may be represented much more advantageously by the figure
which is a portion of the web bounded by lines of nodes belonging to the two systems of parallel filaments. Any such portion becomes then subject to the important condition that the two transverse parallel readings will each give a regularized partition, one being in the present example 5 5 4 3 3, and the other 5 5 5 3 2. Any such graph as this will be termed a regular partition- graph, and the two partitions which it represents will be said to be conjugate to one another. The mere conception of a regular graph serves at once by eflfecting an interchange (so to say) between the warp and the woof, through the principle of correspondence, to establish a well-known fundamental theorem of reciprocity. In the last figure, the extent* of (meaning the number of nodes contained by) the uppermost horizontal line or filament is the maximum magnitude of any element (or part) of the partition, and the extent of the first vertical line is the number of the parts. Hence, every regularized partition beginning with i and containing j parts is conjugate to another beginning with j and containing i parts. The content of the graph (that is, the sum of the parts) of the partition is the same in both cases (it will sometimes be convenient to speak of the partible number as the content of the elements of the partition). From the above correspondence it is clear that if two complete partition-systems be formed with the same content in one of which the largest part is i and the number of parts j, and in the other the largest part is j and the number of the parts i, the order (that is, the number of partitions) of the first system will be identical with the order of the second : so that calling the content n, it follows that n — i may be decom- posed in as many ways into j— I parts as n —j into i—1 parts.
(3) This, however, is not the usual nor the more convenient mode of expressing the reciprocity in question. We may, for the two partition systems spoken of, substitute two others of larger inclusion, taking for the first, all partitions of n in which no one part is greater than i, and the number of parts is not greater than j (that is, is j or fewer), and for the second system, one subject to the same conditions as just stated, but with i and j (as before) interchanged : it is obvious that each regularized partition
* Extent may be used to denote the number of nodes on a line or column or angle of a graph ; content the number of nodes in the graph itself; but I have by inadvertence in what follows frequently applied content alike to designate areal and linear numerosity.
1]
three Acts, an Interact and an Exodion
of one system will be conjugate to one regularized partition of the other system, and accordingly the order of the two systems will be the same*.
(4) When t = oc it follows from the general theorem of reciprocity last established, that the number of partitions of n into j parts or fewer will be the same as the number of ways of composing n with the integers 1, 2, ... j, and is therefore the coefficient of a;" in the expansion of
1
1 -X. \-3? ... 1 —xi'
Thus, then, we can at once find the general term in
(l-a)(l-aa;)(l-aa:")...'
expanded according to ascending powers of a ; for, if the above fraction be regarded as the product of an infinite number of infinite series arising from the expansion of the several factors
1 1 1
1— o' \—ax' 1— aa? ' '"
it will readily be seen that the coefficient of x"a^ will be the number of ways in which n can be resolved into j parts or fewer, that is, by what has been just shown is the coefficient of a^ in
1
1-x.l-x'... 1-xi'
and this being true for all values of n, it follows that the entire coefficient of a^ is the firaction last written developed in ascending powers of x; so that
{l-a)(l-ax){l-ax'). = 1 + as is well known.
The general term in
is also well known to be
a +
l-cc 1-x.l-a^
a» +
l-a;.l-a;».l-a;»
|
1 |
|
|
(l-a)il-ax).. |
.. (1 - ax*) |
|
l-a^+'.l- x*+-' . |
..1 -«••+> |
l — x.l—x'...\—x^
a^,
• The above proof of the theorem of reciprocity is due to Dr Ferrers, the present head of Gonville and Caius College, Cambridge. It possesses the double merit of having set the first example of graphical construttion and of putting into salient relief the principle of correspond- ence, applied to the theory of partitions. It was never made public by its author, but first promulgated by myself in the Land, and Edin. Phil. Mag. for 1853. [Vol. i. of this Keprmt, p. 597.]
1—2
4 A Constructive theory of Partitions, arranged in [1
or in other words, the number of ways of resolving n into j parts none greater than t is the coefficient of a;" in the fraction l-x<+'.l-a:'+'... l-x^-^ \-x.\-a? ...l-xi ' which [denoting 1 — a^ by (g)] is the same as
(l)(2)...(t+j)
(l)(2)...(i).(l)(2)...0-)' and furnishes, if I am not mistaken, Euler's proof of the theorem of reci- procity already established by means of the correspondence of conjugate partitions.
(.5) [It may be as well to advert here to the practical method of obtain- ing the conjugate to a given partition. For this purpose it is only necessary to call Ui the number of parts in the given partition not less than i; a,, Oj, ttj, ... Ui ... continued to infinity (or which comes to the same thing until i is equal to the maximum part), will be the required conjugate.]
(6) The following very beautiful method of obtaining the general term in question by the constructive method is due to Mr F. Franklin of the Johns Hopkins University* :
He, as it were, interpolates between the theorem to be established in general and the theorem for t = oo , and attaches a definite meaning to the above fraction regarded as a generating function when the factors in the numerator are limited to the first q of them, q being any number not exceed- ing i, so that in fact the theorem to be proved, according to this view, is only the extreme case of (the last link in the chain to) a new and more general one with which he has enriched the theory of partitions. The method will be most easily understood by means of an example or two : the proof and use to be made of the construction will be given towards the end of the Act.
Let n = 10, i = o,j = 4.
Write down the indefinite partitions of 10 into 4 or fewer parts, or say rather into 4 parts, among which zeros are admissible : they will be
|
(1) |
10.0.0.0 |
5.5.0.0 |
|
(1) |
9.1.0.0 |
5.4.1.0 |
|
(1) |
8.2.0.0 |
5.3.2.0 |
|
(1) |
8.1.1.0 |
5.3.1.1 |
|
(2) |
7.3.0.0 |
5.2.2.1 |
|
(2) |
7.2.1.0 |
4.4.2.0 |
|
(1) |
7.1.1.1 |
4.4.1.1 |
|
(2) |
6.4.0.0 |
4.3.3.0 |
|
(3) |
6.3.1.0 |
4.3.2.1 |
|
(3) |
6.2.2.0 |
4.2.2.2 |
|
(4) |
6.2.1.1 |
3.3.3.1 3.3.2.2 |
* For a vindication of the oonstractive method applied to this and an allied theorem, see p. [18] et eeq.
1]
three Acts, an Interact and an Exodion
The partitions to which (1) is prefixed are those in which the first excess, that is, the excess of the first (the dominant) part over the next is too great (meaning greater than i, here 5); those to which (2) is prefixed are those in which after the batch marked with (1) are removed, the second excess, that is, the excess of the first over the third element is " too great " ; those to which (3) is prefixed are those in which after the batches marked (1) and (2) are removed, the third excess is " too great," and lastly those (only one as it happens) marked with j (here 4) are those in which, so to say, the absolute excess of the dominant, that is its actual value, is " too great," that is, exceed- ing i (here 5); the partitions that are left over will be the partitions of n (here 10) into 4 parts, none exceeding i (here 5) in magnitude.
It is easy to see from this how to construct the partitions which are to be eliminated from the indefinite partitions of the n (10) into 4 (j) parts so as to obtain the quaternary partitions in which no part superior to 5 (i) appears. To obtain the first batch we must subtract i + 1 (6) from n (10) and form the system of indefinite partitions of 4 into four parts, namely :
4.0.0.0
|
3 |
.1 |
.0 |
.0 |
|
2 |
.2, |
.0 |
.0 |
|
2 |
.1 |
.1 |
.0 |
|
1, |
,1 |
.1 |
.1 |
and adding to each of these 6.0.0.0 (term-to-tenn addition) batch (1) will be obtained.
To obtain the second batch, form the quaternary partitions of n — (i + 2), that is, 3, namely :
3.0.0.0
2.1.0.0
1.1.1.0 [but omit those in which the first excess is "too great" (greater than i); here there are none such to be omitted] and bring the second element into the Irst place ; thus we shall obtain the system
0 3 0 0 12 0 0 1110
Phe augments of those obtained by adding 6 . 1 . 0 . 0 to each of them will eproduce batch (2).
Again, form the quaternary partition-system of n — (i + S}, rejecting all those (here there are none such) in which the second excess is " too great." 7e thus obtain
2 0 0 0 110 0
6 A Constructive theory of Partitions, arranged in [1
and now bringing the third element in each of these into the first place so as to obtain
0 2 0 0
0 110
The augments of these last partitions obtained by adding 6.1.1.0 to each of them will give the third batch, and finally taking the quaternary partition- system to n - {i+j), that is, 1, rejecting (if there should be any such) those in which the third excess is " too great," we obtain 1.0.0.0, and bringing the fourth element to the first place so as to get 0.1.0.0, and adding 6.1.1.1, the fourth batch 6.2.1.1 is reconstructed.
As another example take n = 15, i = 3, j = 3.
The indefinite ternary partitions of 15 are
|
15.0.0 |
(1) |
9.4.2 |
(1)1 |
|
14.1.0 |
(1) |
9.3.3 |
(1) |
|
13.2.0 |
(1) |
8.7.0 |
(2) |
|
13.1.1 |
(1) |
8.6.1 |
(2) |
|
12.3.0 |
(1) |
8.5.2 |
(2) |
|
12.2.1 |
(1) |
8.4.3 |
(1) |
|
11.4.0 |
(1) |
7.7.1 |
(2) |
|
11.3.1 |
(1) |
7.6.2 |
(2) |
|
11.2.2 |
(1) |
7.5.3 |
(2) |
|
10.5.0 |
(1) |
7.4.4 |
(3) |
|
10.4.1 |
(1) |
6.6.3 |
(3) |
|
10.3.2 |
(1) |
6.5.4 |
(3) |
|
9.6.0 |
(2) |
5.5.5 |
(3) |
|
9.5.1 |
(1) |
There are, of course, no partitions left in which no part exceeds 3, as the maxi- mum content subject to that condition would be only 9.
The partitions marked (1) (2) (3) are those in which the first, second and absolute excess respectively exceed 3.
Firstly, the indefinite ternary partitions of 15-4 or 11 augmented by 4.0.0 will obviously reproduce the system of partitions marked (1).
Secondly, taking the indefinite ternary partitions of 10 in which the first excess, and those of 9 in which the second excess, does not exceed 3, we shall obtain
|
6.4.0 |
and 5 . 2 . 2 |
|
6.3.1 |
4.4.1 |
|
5.5.0 |
4.3.2 |
|
5.4.1 |
3.3.3 |
|
5.3.2 |
|
|
4.4.2 |
|
|
4.3.3 |
1] three Acts, an Interact and an Exodion
|
by metastasis become |
|
|
4.6.0 |
2.5.2 |
|
3.6.1 |
1.4.4 |
|
5.5.0 |
2.4.3 |
|
4.5.1 |
3.3.3 |
|
3.5.2 |
|
|
4.4.2 |
|
|
3.4.3 |
and adding to each term of these two groups 4.1.0 and 4.1.1 respectively, the .systems of partitions marked (2) and (3) respectively result.
(7) It may, I think, be desirable to give here my own construction for the case of repeated partitions, which, having regard to its features of resemblance to the one preceding, it is proper to state preceded it in the date of its discovery and promulgation. The problem which I propose to myself is to construct a system of partitions of a given number into parts limited in number and magnitude, by means of partitions of itself and other numbers into parts limited in number but not in magnitude.
As before, let i be the limit of magnitude, j the number of parts (zeros admissible), and n the partible number ; form a square matrix of the jth order in which the diagonal elements are all t + l, the elements below the diagonal all of them unity, and those above the diagonal all of them zero, say Ml.
From this matrix construct if,, if,, ifj, ... Mj, such that the lines in Mq (q being any integer from 1 to j inclusive) are the sums of those in ifj, added (term-to-term) q and q together.
Let (r, q) be the rth line in if, and [r, q] the sum of the numbers which it contains.
Form the complete system of the partitions of n — [r, q] into j parts, and to each such add (term-to-term) (r, q).
In this way, by giving r all possible values we shall obtain a system of partitions of n into _;' parts corresponding to Mq, which may be called P,. I say that P, — P2 + P3 ... -t-(— )^~'Pj will be the complete system of partitions of n into j parts in which one element at least exceeds i ; where it is to be observed that having regard to the effect of the — and + signs (which are used here to indicate the addition and subtraction, or say rather the ad- duction and sub-duction not of numbers but of things), each such partition will occur once and once only; so that calling P the complete system of indefinite partitions of n into _;" parts, the complete system of partitions of n into J parts in which no part exceeds i in magnitude will be
p-p,+p,...+{-yPj\
' It mast, however, be understood that the same partition is liable to appear in more than one, and not exclusively in its regularized phase, or as it may be expressed, the regularized partition undergoes Tnetaatasit.
8 A Constrttdive theory of Partitions, arranged in [1
(8) This construction, which I will illustrate by two examples, proceeds upon the fact which, although confirmed by a multitude of instances, remains to be proved, that if ki, k^, ... kj be any partition of n into j parts and the number of unequal parts greater than i be /it, then the number of times in which this partition, in its regular or any other phase, appears in P, is
^-^ '" — ^ (interpreted to mean 1 when 5=0), and consequently
its total number of appearances in P — P, + P^ . . . is (1 — 1)", that is, is 0.
From this it follows that the total number of partitions of n into j parts none exceeding i in magnitude will be C — C, + Oa — . . ., where (7, is the sum of the number of ways in which the various numbers n^, n^, n,... can be decom- posed into J parts, the numbers n,, n^, n,, ... being n diminished by the sums of the quantities i +1, i + 2 i +j added q and q together ; Cg is therefore
the coefficient of «" in r— — -~ — — — ^; and consequently the number
of partitions of n into j parts none exceeding i in magnitude will be the
coefficient of «" in — '-- '^ -. — as was to be shown.
\-x.\-i^ ... \-xi
(9) As a first example let i = 2, j = 3, 7i=12, the matrices and the partitions corresponding to their several lines will be as underwritten ; the indefinite partitions of the reduced contents, n— [»-, q\, are written opposite to the respective matrix lines to which they correspond, and their augments, found by adding the line to this partition system, are written immediately under them. The zeros are omitted for the sake of brevity.
9 8.1 7.2 7.1.1 6.3 6.2.1 5.4 5.3.1 5.2.2 4.4.1 4.3.2 3.3.3
12 11.1 10.2 10.1.1 9.3 9.2.1 8.4 8.3.1 8.2.2 7.4.1 7.3.2 6.3.3
8 7.1 6.2 6.1.1 5.3 5.2.1 4.4 4.3.1 4.2.2 3.3.2
9.3 8.4 7.5 7.4.1 6.6 6.5.1 5.7 5.6.1 5.5.2 4.6.2
7 6.1 5.2 5.1.1 4.3 4.2.1 3.3.1 3.2.2
8.1.3 7.2.3 6.3.3 6.2.4 5.4.3 5.3.4 4.4.4 4.3.5
5 4.1 3.2 3.1.1 2.2.1
9.3 8.4 7.5 7.4.1 6.5.1
4 3.1 2.2 2.1.1
8.1.3 7.2.3 6.3.3 6.2.4
3 2.1 1.1.1
5.4.3 4.5.3 3.5.4
3.0.0 1.3.0 1.1.3
4.3.0 4.1.3 2.4.3
0 5-4-3|5.4.3
In 6.3.3 there are two unlike elements greater than 2; accordingly 6.3.3 occurs 2 times in P, and 1 time in Pj.
1] three Acts, an Interact and an Exodion 9
In 7.3.2 there are again two unlike elements greater than 2, and 7.3.2, 7.2.3 (the metastatic equivalent to the former) are found in P, and 7.2.3 inPj.
Again, in 5.4.3 there are 3 unlike elements greater than 2, and we find
5.4.3 5.3.4 4.3.5 in Pi
5.4.3 4.5.3 3.5.4 „ P„
5.4.3 „ P,.
But such terms as 11 . 1 10.1.1 9.2.1 8.2.2 in which there is only one distinct element greater than 2 are found 1 time only in P, and not at all in P.i or P3.
As another example let »! = 12, t = 4, J = 3, then a similarly constructed table to the foregoing will be as follows, in which, however, all matrices or lines of matrices which have a sum too large to give rise to partition systems are omitted.
|
5.0.0 |
7 |
6.1 |
5.2 |
5.1.1 |
4.3 |
4.2.1 |
3.3.1 |
3.2.2 |
|
|
12 |
11.1 |
10.2 |
10.1.1 |
9.3 |
9.2.1 |
8.3.1 |
8.2.2 |
||
|
1.5.0 |
6 |
5.1 |
4.2 |
4.1.1 |
3.3 |
3.2.1 |
2.2.2 |
||
|
7.5 |
6.6 |
5.7 |
5.6.1 |
4.8 |
4.7.1 |
3.7.2 |
|||
|
1.1.5 |
5 |
4.1 |
3.2 |
3.1.1 |
2.2.1 |
||||
|
6 |
1.5 |
5.2.5 |
4.3.5 |
4.2.6 |
3.3.6 |
||||
|
6.5.0 |
1 7.5 |
||||||||
|
6.1.5 |
6 |
0 1.5 |
7 . 5 and 6.5.1 are the only two partitions of 12 into 3 parts in which there are two unlike parts greater than 4 ; each of these accordingly is found twice (in one or another phase) in P, and once in P^. Every other partition of 12 into 3 parts in which one of them at least is greater than 4 will be found exclusively and only once in P,.
(10) The two expansions for {\ — ax) {\ — aa?) . . . {\ — ax^) and its reciprocal may readily be obtained from one another by the method of correspondence.
The coefficient of x^a^ in the former is the number of partitions of n into j unequal, and in the latter into j eqital or unequal parts none greater than i or less than unity. The correspondence to be established has been given by Euler for the case of i = x (Comm. Arith., 1849, Tom. i. p. 88), and is probably known for the general case, but as coming strictly within the pur- view of the essay, seems to deserve mention here.
10 A Constructive Uteory of Partitions, arranged in [1
If ifc,, ifcj, i„ ..., kj be a partition of n into j equal or unequal parts written in ascending order, none exceeding i, on adding to it 0, 1, 2 ... (j— 1),
it becomes a partition of n +•' „ into j parts none exceeding i +j — 1, and
conversely, if \j, X, X,- be a partition of n+-^-H-=^ into j unequal parts none
exceeding i-\-j—\, written in ascending order, on subtracting from it 0, 1, 2 ... {j— 1), it becomes a partition of n into equal or unequal (say rela- tively independent) parts none exceeding i.
Hence the complete system of partitions of n into j unlike parts none exceeding i has a one-to-one correspondence with the complete system of the
partitions of w —•^^-g-^ into J parts none exceeding i— j-f 1. Consequently
the coefficient of a-* in the expansion of (1 — ax) ... (1 — iix^) may be found from that of a) in the expansion of its reciprocal by changing i into i —j + 1
and introducing the factor a; ^ .
(11) The expansion of the reciprocal may of course be found algebrai- cally from the multiplication of the expansion which has been given of
—y r ... by (1 — a), or immediately by the correspondence
(l-a)(l-cu;)...(l-cw;*) ^ ^ ^' " ^ »'
between partitions into an exact number j of parts limited not to exceed t, and partitions into j or fewer parts so limited.
By subtracting a unit from each term of k^, k^, ..., kj, a partition of n where no k exceeds i, results a partition q^, q^, ... qj, a partition of n^j where no q exceeds i — 1. Hence the coefficient of ai in
1
1 — aw.l — aa? ... 1 —ax^
may be found from that in
\—a.l — (ix... 1 — cue*
by introducing the factor a;J and changing i into t— 1, so that choosing for the latter the alternative form
l-x-'+'.l-a;J'+''...l-a:^+< \-x.\-a? ...\-x' ' the former becomes
1 - xi+^ ■ 1 - a;i+° ... 1 - xJ+'-^ ^ l-x.l-a^...l-x^^ ^'
and consequently the coefficient of a-* in 1 — oa; . 1 — ax' ... 1 — cur* will be
l-x^+'.l-x^+K.. l-x' •'^ l-x.l-ai'...l-x^j ^ '
1] three Acts, an Interact and an Exodion 11
(12) Before quitting this part of the subject it is desirable to make mention of Dr F. Franklin's remarkable method of proving Euler's celebrated expansion of {\ — x){\— a?){l — a?) ... ad inf. by the method of correspond- ence. This has been given by Dr Franklin himself in the Comptes Rendus of the Institut (1880), and by myself in some detail in the last February Number of the J. H. U. Circular*. The method is in its essence absolutely independent of graphical considerations, but as it becomes somewhat easier to apprehend by means of graphical description and nomenclature, I shall avail myself here of graphical terminology to express it.
If a regular graph represent a partition with unequal elements, the lines of magnitude must continually increase or decrease. Let the annexed figures be such graphs written in ascending order from above downwards.
i.A)
(5) * • * .... ,(A
In A and B the graphs may be transformed without altering their con- tent or regularity by removing the nodes at the summit and substituting for them a new slope line at the base. In G the slope line at the base may be removed and made to form a new summit; the graphs so transformed will be as follows :
{A')
(F) . . . .... (C)
A' and R may be said to be derived from A, B hy a, process of contrac- tion, and C from G by one of protraction.
Contraction could not now be applied to A' and E , nor protraction to C' without destroying the regularity of the graph ; but the inverse processes may of course be applied, namely, of protraction to A' and B' and contraction to C", so as to bring back the original graph A, B, G.
In general (but as will be seen not universally), it is obvious that when
the number of nodes in the summit is inferior or equal to the number in the
base-slope, contraction may be applied, and when superior to that number,
protraction : each process alike will alter the number of parts from even to
[• Vol. in. of this Beprint, p. 664.]
12 A Constructive theory of Partitions, arranged in [1
odd or from odd to even, so that barring the exceptional cases which remain to be considered where neither protraction nor contraction is feasible, there will be a one-to-one correspondence between the partitions of n into an odd number and the partitions of n into an even number of unrepealed parts; the exceptional cases are those shown below where the summit meets the base- slope line, and contains either the same number or one more than the number of nodes in that line ; in which case neither protraction nor contraction will be possible, as seen in the annexed figures which are written in regular order of succession, but may be indefinitely continued :
for the protraction process which ought, for example, according to the general rule, to be applicable to the last of the above graphs, cannot be applied to it, because on removing the nodes in the slope line and laying them on the summit, in the very act of so doing the summit undergoes the loss of a node and is thereby incapacitated to be surmounted by the nodes in the slope, which will have not now a less, but the same number of nodes as itself; and in like manner, in the last graph but one, the nodes in the summit cannot be removed and a slope line be added on containing the same number of nodes without the transformed graph ceasing to be regular, in fact it would take the form
and so the last graph transformed according to rule [by protraction] would become :
which, although regular, would cease to represent a partition into unlike numbers.
The excepted cases then or unconjugate partitions are those where the number of parts being j, the successive parts form one or the other of the two arithmetical series
j,j + \,j + 2, ... 2j-l or j -1-1, j -1-2, ... 2j,
in which cases the contents are •' •' and ' '' respectively, and consequently
1]
three Acts, an Interact and an Exodion
13
since in the product of \—x.\—oc^.\ — a^... the coeflScient of a;" is the number of ways of composing n with an even less the number of ways of composing it with an odd number of parts, the product will be completely
represented by S {—^x ^
(13) It has been well remarked by Prof Cayley that barring the uncon- jugate partitions, the rest really constitute 4 classes, which using c and x to signify contractile and extensile and e and o to signify of-an-even or of-an-odd order, may be denoted by
c.e c .0
x.e X .0.
Hence as each c . e is conjugate to an a; o and vice versa, and each c . o to an a; . e and vice versa, the theorem established really splits up into two, one affirming that the number of contractile partitions of an odd order is the same as the number of extensile ones of an even order, the other that the number of contractiles of an eveu is equal to the number of extensiles of an odd order. It might possibly be worth while to investigate the difference between the number of partitions which each set of one couple and the number of partitions which each set of the sub-contrary couple contain : the sets which belong to the same couple and contain the same number of partitions being those both of whose characters are dissimilar.
(14) There are one or two other simple cases of correspondence which are interesting, inasmuch as the construction employed to effect the corre- spondence involves the operations of division and multiplication, which have not occurred previously.
If fx = (\ - x)(\ - a*){\- a?)(\ -x'){\ -a?) ...
aod «^x = (H-a;)(l-f-a;»)(l+a^)(l-|-ar')(l +a^)...
fx.^x=\,
from which we obtain ^ = Ijfx and l/<f>x=/x.
The first of these equations has been noticed by Euler as involving the elegant theorem that a number may be partitioned in as many ways into only-once-occurring odd-or-even integers as into any-number-of-times-occur- ring only-odd integers.
* ADother proof of this theorem, dedaced as an immediate algebraical consequence of a more general one, obtained b; graphical dissection, will be given in Act 2; and in the Exodion I famish a purely arithmetical proof by the method of correspondence of Jacobi's series for
(l±j;»-'»)(l±j:«+m)(l_x2«){l±X-'»»-'»)(li:z»"-^") (l-l*») ...
(which inclndes Enler'a theorem as a particular case). I prove this theorem in a more extended sense than was probably intended by its immortal author, inasmuch as I regard m and n as absolutely general symbols.
14 A Constnictive theory of Partitions, arranged in [1
The second, which I think he does not dwell upon, expresses that the difference between the number of partitions with an even number of parts and that of partitions with an odd number of parts of the same number n is the same as the number of partitions of n into exclusively odd [unrepeated] numbers (such difference being in favour of the partitions of even or of odd order, according as the partible number is even or odd).
This latter theorem brings out a point of analogy between repetitional and non-repetitional partition systems which appears to me worthy of notice.
Any one of the former contains a class of what may be termed singular partitions, in the sense that they are their own associates, or more briefly, self-conjugate in respect to the Ferrers transformation. Any one system of the latter may also be said to contain a set of singular partitions (0 or 1 in number) in the sense of being unconjugate in respect to the Franklin process of transformation. Since then in this case the difference between the number of partitions of an odd and those of an even order of the same number is equal to the number (1 or 0) of singular partitions of that number, so we might anticipate as not improbable that the like difference for the repetitional partitions of a number should be equal to the number of singular partitions of that number — and such is actually the case; for it will be shown in a future section that the number of self-conjugate partitions of a number is the same as the number of ways in which it can be composed with odd integers.
(15) The correspondence indicated by the equation <^x = 1/fx can be established as follows :
Let 2* .1, 2'^. m, 2" .n, ... be any partition of unrepeated general numbers, where l,m,n... are any odd integers not exceeding unity ; and let U^^ in general denote q parts k, then without changing its content the above parti- tion can be converted into V-'^\ td^\ d^''\ . . . which consists exclusively of odd numbers.
It will of course be understood that the original partition may contain any the same odd number as I multiplied by different powers 2\ 2^', 2^" ... of 2, with the sole restriction that the \, X', \", ... must be all unequal.
Conversely, any such partitions as i^"', m^''\ n^"' may be converted back into one and only one partition of the former kind. For there will be one and but one way of resolving a into the sum of powers of 2 (the zero power not excluded), and supposing o- to be equal to 2^ -(- 2^' -f 2^" -I- ..., /W may be replaced by 2H, 2'^'l, 2*'7, and the same process of conversion may be simul- taneously applied to each of the other products mW, nt"), —
Hence each partition of either one kind is conjugate to one of the other, and the number of partitions iu the two systems will be the same, as was to be shown.
1] three Acts, an Interact and an Exodion 15
(16) But we have here another example of the fact that the theory of correspondence reaches far deeper than that of mere numerical congruity with which it is associated as the substance with the shadow. For a corre- spondence exists of a much more refined nature than that above demonstrated between the two systems, and which, moreover (it is important to notice) does not bring the same individuals into correlation as does the former method.
The partition system made up of unrepeated general numbers may be divided into groups of the first, second, ... ith ... class respectively, those of the t'th class containing i distinct sequences of consecutive numbers having no term in common, with the understanding that no two sequences must form part of a single sequence (so that the largest term of one sequence and the smallest one of the next sequence must differ by more than a single unit), and that a single number unpreceded and unfoUowed by a consecutive number is to count as a sequence.
The partition system, made up of repeatable odd numbers may, in like manner, be resolved into groups of the 1st, 2nd, ... ith, ... class respectively, those of the ith class containing i distinct numbers ; and the new theorem of correspondence is that there is a correlation between the numbers of the I'th class of one system and the I'th class of the other ; so that the number of partitions in a class of the same name must be the same to whichever system it belongs; and thus Euler's theorem becomes a corollary to this deeper- reaching one, obtained from it by adding together the number of partitions in all the several classes in the one system and in the other.
(17) As regards the first class, the theorem amounts to the statement that the number of single sequences of consecutive numbers into which n may be resolved is equal to the number of odd factora which n contains ; so that if iV=2'.P.m''.n'... where I, m,n, ... are odd numbers, N can be represented by (X. + 1) (/* + !)(«' + 1) ... such sequences; thus, for example, ifi\r=15=3.5 we have
1-1-2 + 3 + 4+5 = 4-1-5 + 0 = 7 + 8 = 15.
So 30 = 4 + 5 + 6 + 7 + 8 = 6 + 7 + 8 + 9 = 9+10 + 11,
27 = 2 + 3 + 4 + 5 + 6+7 = 8 + 9 + 10 = 13 + 14,
45 = 1+2 + 3 +...+9 = 5 + 6 + 7 + 8 + 9 + 10
= 7 + 8 + 9 + 10 + 11=14+15+16 = 22 + 23.
So too if iV is a prime number it can only be resolved into the two sequences
N—1 N+1
— 1 ^ — and N. More generally N can be resolved into as many
dififerent sets of i distinct sequences as there are solutions in positive integers
16 A Constructive theory of Partitions, arranged in [1
of the equation 2 (a:,yi + x^y^ + . . . + a;,yj) + Xj + iTj + . . . + a-j = N, of the truth of which remarkable theorem, in its general form, I have for the present only obtained empirical evidence, but may possibly be able to discover the proof in time to annex it in the form of a note at the end, so as not to keep the press waiting*.
(18) The proof for the case of the first class and the mode of establish- ing the correspondence between the partitions of this class of the two kinds is not far to seek. I use as previously a*' to signify a repeated b times.
Consider then any sequence of consecutive numbers for the cases where the number of terms is odd and where it is even separately, calling s the sum of the first and last terms, and i the number of terms ; where i is odd, so
(-) that s is even, the sequence may be replaced by i^' ', and where i is even (so
(-) that 8 is odd) by s ' . Hence each partition of the first class of the first
kind may be transformed into one of the first class of the second kind.
It is necessary to show the converse of this, which may be done as follows : Let X" be any partition of the second kind so that X is necessarily odd. I say that this must be transformable into one or the other (but not into both) of two sequences, namely, one of X terms of which the sum of the first and last is 2/x, the other of which the sum of the first and last terms is X and the number of terms 2/i. The former supposition is admissible if 2/^ is equal to or greater than X + 1, inadmissible if 2^ is less than X + 1. The second supposition is admissible if X is equal to or greater than 2fi + 1, inadmissible if X is less than 2^ + 1.
The two conditions of admissibility coexisting would imply that 2/x, is equal to or greater than 2/t + 2 ; the two conditions of inadmissibility the one that 2/u. is equal to or less than X— 1, the other that X is equal to or less than 2/i— 1, that is, X— 1 equal to or less than 2/Lt — 2, which are inconsistent. Hence one of the two transformations is always possible and the other impossible to be effected ; which proves the con-elation that was to be established. A single example will serve to show that this correspondence is entirely different from that offered by the first and (so to say) grosser method ; suppose N = 15, then 1.2.3.4.5 will be a partition of the first kind and will be converted by the new rule into 5.5.5, whereas, by the former rule, it would be inverted into 1.1.1.3.1.1.1.1.5, that is, into 1' . 3 . 5 belonging to the third class instead of to the first.
(19) I will now pass on to the conjugate theorem corresponding to
fx = \l<^x.
* A complete proof of the general theorem will be giveo in the 3rd Act.
b
1] three Acts, an Interact and an Exodion 17
It may be well here to recall that this identity essentially depends upon the identity 1 — a: = 1/(1 + a;) (1 + a;*) (1 + a^) . . . which, interpreted *, signifies that any number greater than unity may be made up in as many ways with an odd as with an even number of numbers restricted to the geometrical
progression 1, 2, 4, 8 This may be called, for brevity, a geometric
partition. The correspondeuce to which this points is itself worthy of notice ; one mode of establishing it would be to proceed to decompose N into such parts in regular dictionary order — it would easily be seen that each pair of partitions thus deduced would be of contrary parities, but it would not be easy, or at all events evident, how to determine at once the conjugate to a given partition by reference to this principle ; but if we observe that it is possible to pass from the geometric partitions of n immediately to those of n + 1 by the addition of a unit to each of the former, and consequently to
those of n + 2 from the partitions oi E ^, E , E — — , ... 2, 1, by an
obvious process of doubling and adding complementary units, another rule or law of correspondence, which proves itself as soon as stated (and is not identical in effect with that supplied by the dictionary-order method), looms into the field of vision, than which nothing can be simpler. Hence we may derive a transcendental equation in dififerences for m„, the number of geo- metric partitions (with ratlix 2) to n, namely, to find the conjugate of any geometric partition, look at its greatest part — if it is repeated add two of them together: if it is unrepealed split it into two equal parts; these processes are obviously reversible, just as in Dr Franklin's method of correspondence for the pentagonal-series-theorem ; and the method is equally open to the remark made thereon by Prof. Cayley ; that is to say, there will be four classes, extensile even, extensile odd, contractile even and con- tractile odd, and the number of partitions in any class will be the same as in the class in which both the characters are reversed.
The application of this transformation to the construction indicated by the equation fx = 1/0* will be obvious. Let any partition containing only unrepealed numbers consist of odd numbers p, q, r, ... t, each multiplied by one or more powers of 2 ; form batches of these terms which have the same greatest odd divisor (p, q, r, ... t), and arrange those batches in a line according to the order of magnitude of p, q, r, ... t. Then we may agree to proceed either from left to right or from right to left in reading off the batches, and that convention being established once for all, as soon as a batch Ls reached which does not consist of a single odd term, if it contain one term larger than all the rest that term is to be split into two equal parts, but if it contain two terms not le.ss than any
* Jast to the equation 1I{1 -z) = {l + x){l+x'){l+i*) ... teaches that there is one and only one way of effecting the nnrepetitional geometric partition of any number — a theorem which has been applied in the preceding theory.
8. IV. 2
18 A Constructive theory of Partitions, arranged in [1
others in the batch, those two are to be amalgamated into one. In this way the order of a partition consisting of terms not all of them distinct odd numbers, will have its parity (quality of being odd or even) reversed, and it is obvious that if A has been under the operation of the rule converted into B, B by the operation of the same rule will be converted back into A. Hence it follows that (making abstraction of the partitions consisting exclusively of unrepeated odd numbers) all the rest will be separable into as many contractile of an odd as into extensile of an even order, and into as many extensile of an odd as into contractile of an even order, so that the difference between the entire number of the partitions of N into an odd and those of an even order of repeatable numbers (odd or even) will be the number of partitions of N into unrepeated odd numbers, and those of an odd or of an even order will be in the majority according as N itself is odd or even*.
It will be convenient to interpolate here Dr F. Franklin's constructive proof of the theorems referred to in p. [4] of what precedes, as there will be frequent occasion to refer to them in what follows. The theory is thus made completely self-contained. I give the proofs in the author's own words, which I think cannot be bettered.
(20) Constructive Proof of the Formula for Partitions into Repeatable Parts, limited in Number and Magnitude. The partitions herein spoken of are always partitions into a fixed number, j, of parts, written in descending order.
Take any partition of w in which the first excessf is greater than i; subtracting t'+l from the first part we get a partition of w — {i+\); and conversely if to the first part in a partition of w — (i + 1) we add i+\ we get a partition of lu in which the first excess is greater than i. Hence the number of partitions of w in which the first excess is greater than i is equal to the whole number of partitions of w — {i+\); so that if the generating
* Dr F. Franklin has remarked that "the theorem admits of the following extensions," which the method employed in the text naturally suggests, and "which are very easily obtained either by the constructive proof or by generating functions " :
1. The number of ways in which w can be made up of any number of odd and A; distinct even parts is equal to the number of ways in which it can be made up of any number of unrepeated and k distinct repeated parts.
2. The number of ways in which w can be made up of parts not divisible by »i is equal to the number of ways in which it can be made up of parts not occurring as many as m times.
3. The number of ways in which w can be made up of an infinite number of parts not divisible by m, together with k parts divisible by m, is equal to the number of ways in which it can be made up of an indefinite number of parts occurring less than m times, together with k parts occurring m or more times. (3) of course comprehends (1) and (2) as special cases.
Dr Franklin adds, " another extension is naturally contained in the mode of proof, which it is perhaps not worth while to state." See Johns Hopkins Circular for March, 1883.
t The first excess signifies the excess of the largest part over the next largest ; the second excess the excess of the largest over the next part but one, and so on.
1] three Acts, an Interact and an Exodion 19
function for the partitions of w is /(a?), that for those partitions in which the first excess is not greater than i is (1 — x^+^)f{x). Confining ourselves now to this class of partitions, consider any one of them in which the second excess is greater than i; subtracting i + I from the first part and 1 from the next, and putting the reduced first part into the second place we have a partition of w — (i+ 2) in which the first excess is not greater than i; and conversely if in any partition of w — (i + 2) in which the first excess is not greater than i, we add i + 1 to the second part and 1 to the first part and transfer the augmented second part to the first place, we get a partition of w in which the first excess is not greater than i and the second excess is greater than i. Hence the generating function for those partitions in which the second excess is not greater than i is (1 — a;'+')(l — a^'^'') f (x). Considering now exclusively the partitions last mentioned, any one of them in which the third excess is greater than i may be converted into a partition of w — {i + 3) in which the second excess is not greater than i, by subtracting i + 1 from the first part, 1 from the second part, and 1 from the third part, and removing the reduced first part to the third place, and, as before, by the reverse operation, the latter class of partitions are converted into the former. Hence the generating function for the partitions in which the third excess is not greater than i is
(1 -a:'+')(l -a*+»)(l _ a;«+«)/(a;).
So in like manner, the generating function for the partitions in which the k-th excess is not greater than i is
(1 - «•■+') (1 - a^^) (1 - a^») ... (1 - a;*+*)/(a;) ;
and for the partitions in which the _;-th or absolute excess is not greater than i, that is in which the greatest part does not exceed i, the generating function is
(1 - «*+") (1 - a*-^) (1 - «'+») ... (1 - x'+J)/(x).
(21) Constructive Proof of the Formula for Partitions into Unrepeated Palis, limited in Number and Magnitude. All the partitions to be con- sidered consist of a fixed number, j, of unrepeated parts, written in descending order.
Take any partition of w in which the first excess is greater than i + 1 ; subtracting i+1 from the first part we get a partition of w — (I'+l); conversely, if to the first part in any partition oi w — {i + 1) we add i + 1, we get a partition of w in which the first excess is greater than t + 1 ; hence the number of partitions of w in which the first excess is greater than i+1 is equal to the whole number of partitions of «; — (i+l); so that, if the generating function for all the partitions is <^(a;), the generating function for partitions whose first excess is not greater than i+1 is (1 — x^+') <f) (x).
2—2
20 A Constructive theory of Partitions, arranged in [1
Considering now only partitions subject to this condition, if in any such partition of w the second excess is greater than i + 2, we obtain by subtract- ing i + 2 from the first part and removing the part so diminished to the second place a partition of w — (i4-2) subject to the condition; and con- versely from any partition of «; — (i + 2) in which the first excess is not greater than i + 1, we obtain, by adding i + 2 to the second part and removing the augmented part to the first place, a partition of w, in which the first excess is not greater than i + 1 and the second excess is greater than i+2; hence the generating function for the partitions in which the second excess is not greater than i + 2 (which restriction includes the con- dition that the first excess is not greater than t+ 1) is
(1 -*•■+') (1 -»•■+') <^ (a;).
Confining ourselves now to this class of partitions, and taking any partition of w in which the third excess is greater than i + S, we obtain, by subtracting i + 3 from the first part and removing the diminished part to the third place, a partition of w — (i + 3) belonging to the class now under consideration ; and reversely. Hence the number of partitions in which the third excess is not greater than i + S is given by the generating function (1 - a;'+i) (1 - a;'+0 (1 - «'+») </> (a).
Proceeding in this manner, we have finally that the generating function giving the number of partitions into j unrepeated parts, in which the absolute excess, that is the magnitude of the greatest part, is not greater than i+j, is
(1 - «'•+') (1 - «•+'') (1 - a;i+=') ... (1 - x^+J) <f> (x).
For example, if w = 18, j = S,i = 4, the partitions 15, 2, 1 14, 3, 1 13, 4, 1 1.3, 3, 2 12, 5, 1 12, 4, 2 11, 5, 2 11, 4, 3
in which the first excess is greater than 5, become by subtraction of 5 from their first part,
10, 2, 1 9, 3, 1 8, 4, 1 8 3, 2 7, 5, 1 7, 4, 2 6, 5, 2 6, 4, 3 which are all the partitions of 13 ; the partitions
11, 6, 1 10, 7, 1 10, 6, 2 10, 5, 3 9, 8, 1 9, 7, 2 in which the first excess is not greater than 5, but the second excess is greater than 6 become, by the subtraction of 6 from the first part and its removal to the second place,
6, 5, 1 7, 4, 1 6, 4, 2 5, 4, 3 8, 3, 1 7, 3, 2
which are all the partitions of 12 whose first excess is not greater than 5 ; the partitions
9, 6, 3 9, 5, 4 8, 7, 3 8, 6, 4
in which the second excess is not greater than 6, but the third excess (the
1] three Acts, an Interact and an Exodion 21
greatest part) is greater than 7, become, by the subtraction of 7 from the first part and its removal to the last place,
6, 3, 2 5, 4, 2 7, 3, 1 6, 4, 1
which are all partitions of 11 whose second excess is not greater than 6. The only remaining partition of 18 is 7, 6, 5.
Interact.
Notes on certain Generating Functions and their Properties.
(22) (A) It may be as well to reproduce here (so as to keep the whole subject together) the entire proof of the well-known expansions of
1 + aa; . 1 -I- aa^ . 1 + cur* . . . 1 + a«*, and of the reciprocal of
1 —a.l — ax.l — aa^ . 1 — aa? ... 1 — ax*,
which appeared in part in the Johns Hopkins Circular for February* last. This is, I think, distinguishable from the ordinary proofs as being, so to say, classical in form (using the word in an algebraical sense), inasmuch as it establishes the identity of two rational integral functions, one explicitly, the other implicitly given, by comparison of their zeros.
Let the coefficient of a^' in the expansion of
(1 +cw;)(l + aar')...(l+ow;'), say in the expansion of F{x, a), be called Jx, and
I - a^ . 1 - a:*-' ... 1 - a*-J+'
1 — a;.l — a;* ... 1 — a^' be called Xj.
Jx being the sum of the j'-ary combinations of x, a^, ... x^ will necessarily
contain ar^+'+'+i, that is a; * , and will be of the degree
i + (i-l) + ...-l-(»-j+l)
in X, and therefore of the same degree as XjX * .
All the linear factors of Xj are obviously of the form x — p, where x — p is a primitive factor of some binomial expression af — 1 : the number of times
that any x — p occurs in Xj will obviously be equal to E — £^ — E — =^
which is either 1 or 0. Now consider F(p, a), the value of F{x, a) when x
becomes p. Let i = kr + S, where 8 <r; then F(p, a) = (1 + a')* multiplied
[• Vol. in. of this Beprint, p. 677.]
22 A Constructive theory of Partitions, arranged in [1
by h linear functions of a, and consequently if ^ = A;V + 8', where S'<r, Jx vanishes when h' > S, in which case
r r r
Hence any linear factor x — p of Xj possesses the two- fold property of being unrepeated and of being contained in J^. Hence Jx must contain
XjX * , and being of the same degree as it is in x must bear to it a constant
ratio, which, by making x=\, is seen to be that of the coefficient of a^ in
i(i — \\(i—'2.') . (t — 7+1)
(1 + a)', that is of -^^ 'A. — L'. — to the product of the fractions
L. 1.6 ...J
in their vanishing state
1-a;' 1 - X'-' 1 - a;'-J+'
r^' l-x' "■■' 1-xi '
that is, is a ratio of equality, so that Jx = XjX ^ . Q.E.D.
(23) Again let Xj and J^ now stand respectively for
l-a;*+M -x^-^...\-x^} \-x.\-a? ...l-xJ
and the coefficient of a^ in the reciprocal of 1 — a. 1 — cw;... 1 — cue' (say F{x,a)); this latter is the sum of homogeneous products of the _;th order of 1, X, x'^,...x^, and is therefore of the degree ij which is also the degree (as is obvious) of Xj in x. For like reason as in what precedes x — p, any linear factor oi x^ — 1, is contained 1 or 0 times in Xj according as
E^-^-E--Ei=l
or 0.
Let the minimum negative residue of i + 1 to modulus r be — S ; F{p, a) may be expressed as the product of 8 linear functions of a, divided by a power of 1 — a'', and the only power of a (say a*) which appears in its development will accordingly be those for which the residue of 0 in respect to r is 0, 1, 2, ... 8, and consequently if a* appears in the development
E^-±^-Ei-E^- = 0, r r r
or conversely if a; — p is a factor of Xj so that
Ei^^-Ei-E'- = l, r r r
Jx vanishes. Hence Jx contains each linear factor of Xj, and these being simple, contains Xj itself, and on account of their degrees in x being the same must bear to it a ratio independent of x, which, by making x=\.
1] three Acta, an Interact and an Exodion 23
1
so
that the things to be compared are the coefficient of a-' in ^t^^ and
( 1 — a)'"*"
the product of the vanishing fractions — -, -^ —,..., — j, is
readily seen to be a ratio of equality, so that J^ = ^y Q- E. D.
(24) (B) On the General Term in the Generating Function to Partitions into parts limited in number and magnitude, by Dr F. Franklin.
To prove that the coefficient of a^ in the development of
1 • . ( 1 - a;^"^') ( 1 - ^•'"^'') ■ • . (1 - xi**)
{l-d){l-aa;){l-aaf)...(l-aaii) {l-x)(l-x') ...(l-x') '
I showed that the number of partitions of w into i or fewer parts, subject to the condition that the first excess (the excess of the first part over the second) is not greater than j, is the coefficient of ic" in
(l-a:)(l-a!»)...(l-a!')'
and in general that the number of partitions in which the rth excess (the excess of the first part over the (r — l)th) is not greater than j, is the coefficient in
(1 - xi+')(l - »>+') ... (1 - xJ^)
If we look at the question reversely, namely, the coefficient of aJ in
1
(1 - o)(l - ax){l - ax') ... (1 - (M*)
being known to be
(1 - xi+') (1 - xi+^) ... (1 - gJ+O (1 -ar)(l -«»)... (1-a;*) '
if we ask what is the significance of the fractions
l-tef+' (l-x}+')(l-xi^) ...{1- xi-^)
(l-ar)(l-a?)...(l-x«y" (1 -«)(1 -<t») ... (1 -«<) '
the answer is immediately given by the generating function itself For
l-xi+'
(l-a;)(l-<c')...(l-a:')
1 l-xi+^
~ {l-x^)il-x') ...(1-x')' 1-x
= (i-a^)(i-l')...(i-^) (''°- °^ "' ^° (r-oKi^
= '=°- "^"^ ^" (l-a)(l-a^)(l-^)(l-^)...(l-^)-
24 A Constructive theory of Partitions, arranged in [1
But the coefficient of a^a?' in the last written fraction is obviously the number of ways in which w can be composed of the numbers 1, 2, 3, ...i, using not more than j Vs. And the number of I's in a given partition is equal to the excess of the first part over the second part in its conjugate. In like manner
(1 - a;^')(l - a;-'+') ... (1 - a^+Q {\-x)(\-x')...{\-x')
^
= CO. of a^' in
(l-a)(l -aa;)...(l-aa;'-)(l -af+'). ..(!-«<)'
and the coefficient of a^a;'" ia the fraction on the right is the number of ways in which w can be composed of the parts 1, 2, 3, ... i, not more than _;' of the parts being as small as r. But the number of I's in a given partition is equal to the excess of the first part over the second in its conjugate ; the number of 2's to the excess of the second part over the third, and so on. Hence the number of I's plus the number of 2's ... plus the number of r's in a given partition is equal to the excess of the first part over the rth part in its conjugate ; and we have thus proved that the coefficient of it^ in the development of
(1 - a;J+') (1 - a:i+^) ... (1 - aj-'+Q {l-x){l-x')...{l-u^)
may be indifferently regarded as the number of partitions of w into parts none greater than i and not more than j of them as small as r or as the number of partitions of w into j or fewer parts, the excess of the first part over the ?'th part being as small as j. These results may obviously be ex- tended by introducing the a in non-consecutive factors of the product
{l-x)(l-af)...(l-x^).
(25) (C) On the theorem of one-to-one and class-to-class correspondence between partitions of n into uneven and its partitions into unequal parts, by Dr F. Franklin.
The number of partitions of w into k distinct odd numbers, each repeated an indefinite number of times, is evidently the coefficient of a**" in the development of
('-r^J('-T^)(>-r^
«°,
It is not easy to form the generating function for the number of partitions containing k sequences, but it is plain that the number of partitions of w containing one sequence is the coefficient of x"' in
S,-\-8,-\-S,-\-...,
I
1]
where
three Acts, an Interact and an Exodion
25
and in general
Si = x +x- +0^ +3^ +af' +.. Si = a? +a^ +aF +a? +a;"+..
S, = a;" + r» + a^ +a^ +a^ + ... =
g — a;l+S+J+...+r ^ ^+3+4+...+ (r+l) + ... =
X
l-X
X"
a* x">
l-X"
X"
l-X''
So much of Prof. Sylvester's theorem as relates to a single sequence follows from inspection of the above scheme. For S, = = ; adding to S,
X ■"" X
x' the first term of S,, we get r- — —; adding to S, the first term of 8t and the
L *~ X^
x' second term of S^, we get ; adding to Stm+i the first term of S^m , the
second term of Sum^^, the third term of S,(m_<8, .... and the mth term of (S,,
*® 8®* T — Zmk+i' ^^^^ ^^® proposition is proved. The fact is made more
evident to the eye if we write the scheme as follows:
Si = x +af +ie' +x* +0^ + ... S^ = x' + .i:^ +3^ +ai' +x^ + ...
St=af +«» + «" + a;" + ar" + ...
Si = x"+x'+a^ + x^ + ai^+...
S7-a:"+«« + a;*' + a;* + ar»+... - S, = a;*'+ar« + a:" + a:" + a:" + ...
Here :; — , for instance, is obtained by adding the fourth column on the
I — or
\ right to the fifth row on the left.
It may be noted that we have thus found that
|
s.= |
a;'»+aJ*+a;"+a^+... |
|
-sr,= |
x"+aF + af*+ ... |
|
Ss = |
a^ + x**+... |
|
S,o = |
«»+... |
X a^ of
+ , -.+ i -.+ ••• +
X*"'"*''
1 -X l-X* ' l-x"
l_a^+.
+ ...
X a? a"
+ ; ~. + -
\-x l-a? l-a^
j,Jn(n + l)
26 A Constructive theory of Partitions, arranged in [1
(26) [Compare Jacobi's theorem contained in the last-but-one two lines of the last but one page of the Fundamenta Nova, which may be easily reduced to the form
X a? a? X a? a?
\-\-x \-\-a? \-\-of'" 1+a; \-ira? \-Va?
J. J. S.]
Act II. On the Graphical Conversion of Continued Products
INTO Series.
Naturelly, by composiciouns Of anglis, and slie reflexiouns.
The Squieres Tale.
(27) The method about to be explained of representing the elements of partitions by means of a succession of angles fitting into one another arose out of an investigation (instituted for the purpose of facilitating the arrangement of tables of symmetric functions)* as to the number of par- titions of n which are their own conjugates. The ordinary graphs to such partitions must obviously be symmetrical in respect to the two nodal boundaries, as seen below.
Let the above figure be any such graph ; it may be dissected into a square (which may contain one or any greater square number) of say i^ nodes,
and two perfectly similar appended graphs, each having the content — ^— ,
and subject to the sole condition that the number of its lines (or columns), that is that the number (or magnitude) of the parts in the partition which
n—i*
it represents, shall be i or less ; such number is the coefficient of x '^ in
:; r = ., which is the same as that of a;"""" in
I -x.l—x' ...l — x"
l-xW-x^.-.l-x''
x^
or of «" in -. .
\-a?.\-!i^ ...\-x^
* By Mr Durfee, of California (Fellow of the Johns Hopkins University), to whom I suggested the desirability of investigating more completely than had been done the method of arrangement of such tables founded upon the notion of self-conjugate partitions, which M. Fail de Bruno had the honour of initiating. The very valuable results of Mr Durfee's inquiries, embodying, system- atising and completing the theory of arrangement originated by Professor Cayley, and further illustrated by the labours of Professors Betti and De Bruno, will probably appear in the next number of the Journal.
1] three Acts, an Interact and an Exodion 27
Hence giving i all possible values we see that the coefficient of a;" in the infinite series
is the number of self-conjugate partitions of n, or which is the same thing of symmetrical groups whose content is n.
(28) But any such graph, in which there is a square of i* nodes with its two appendices, may be dissected in another manner into i angles or bends, each containing any continually decreasing odd number of nodes, and vice versa, any set of equilateral angles of nodes continually decreasing in number (which condition is necessary in order that the lower lines and posterior columns may not protrude beyond the upper lines and anterior columns) when fitted into one another in the order of their magnitudes will form a regular graph. Thus the actual figure (where there is a square of 9 nodes) formed by the intersections of the lines and columns may be dissected into 3 angles containing respectively 13, 11, 3 nodes ; and so in general the number of ways in which n can be made up of odd and unrepeated parts will be the
same as the number of ways in which — ^^ can be partitioned into not more than j parts ; hence we see that the coefficients of as^a^ in (1 + ax){l +aaf)...(l + aai'^') ...
and in ,
l-x'.l-a*...l-ai'i
are the same, so that the continued product above written is equal to
as is well known.
(29) In like manner if the expansion in a series of ascending powers of a of the finite continued product
(1 + ax)(l+ ax>) ... (1 + cu~»-') be required, the coefficient of jC" in the coefficient of aJ will be the number of ways in which n can be made up with j of the unrepeated numbers 1, 3, ... 2i — l, and as 2i — 1 is the number of nodes in an equilateral angle whose sides contain i nodes, it follows that this coefficient will be the number
of ways in which •' can be composed with parts none exceeding i—j in
magnitude, and will therefore be the same as the coefficient of a; ^ in
1 — a'-^+' . 1 — a;'-^+' ... 1 — a;' i-a;.l-a!'... 1-a:^ '
and consequently the finite continued product above written is equal to
1 + . . . H ;; X™a' + ...
28 A Constructive theory of Partitions, arranged in [1
(30) If it be required to ascertain how many self-conjugate partitions of n there are containing exactly i parts, this may be found by giving j all
possible values and making pj equal to the number of ways in which
can be composed with j or fewer parts the greatest of which is i —j, that is (n—f + 2j — 2i)/2 with j — 1 or fewer parts none greater than i —j, so that Pj will be the coefficient of x'-"-^~'+'^-^^'^ in
1 - x'-}+' . 1 - a<^+'' ■ ■ . 1 - a;'-' I - x.l-ai' ... 1-xJ-' or of of* in
1 _ afi-ij+i . 1 _ x''-'->+* ... 1 - ar"-2
l-x'.l-a^ ... l-af^-
x?-^+^ ■
the sum of the values of pj for all values of j will be the number required : this number, therefore, writing lo for 2i — 1, will be the coefficient of a;" in
1— a;^' ,, 1 - a^-' . 1 - iT-' ^,
af+ -r«^ +—, :T-i -r^ ^""^ + etc.;
1 — ar* 1 — ar". 1 —oc*
the outstanding factor in the 5'th term in this series being a;"+f9-»'' we may suppose q the least integer number not less than 1 + \/{n— m) and then the subsequent term to the (q + l)th being inoperative may be neglected.
(31) In order to see how any self-conjugate graph may be recovered, so to say, from the corresponding partition consisting of unrepeated odd numbers, consider the diagrammatic case of the partition 17, 9, 5, 1 represented by the angles of the graph below written
The number of angles is the number of the given parts, that is 4, and the first four lines of the graph will be obtained by adding 0, 1, 2, 3 to the major half (meaning the integer next above the half) of 17, 9, 5, 1, that is will be 9, 6, 5, 4, the total number of lines will be the major half of the highest term (17) and the remaining lines will have the same contents, namely 3, 2, 1, 1, 1, as the columns of the graph found by subtracting 4 (the number of the parts) from the numbers last found, that is will be the lines of the graph which is conjugate to 5, 2, 1. And so in general the self-conjugate graph corresponding to any partition of unrepeated odd numbers qi, q^, ... qj will be found by the following rule:
1] three Acts, an Interact and an Exoclion 29
Let P be the system of partitions i, , hj, ... kj, in which any term kg is the major half of qg augmented by 6 — \, and P' another system of k^, k.^, ... kj', obtained by subtracting j from each term in P, then P and the conjugate to P' will be the self-conjugate partition corresponding to the given q partition. Thus as an example, 19, 11, 7, 5 being given, P, P' will be 10, 7, 6, 6 ; 6, 3, 2, 2 respectively, and the self-conjugate system required will be 10, 7, 6, 6, 4, 4, 2, 1, 1, 1. Of course P' might also be obtained by taking the minor halves of the given parts in inverse (ascending) order and subtracting from them the numbers 0, 1, 2, ... respectively.
To pass from a given self-conjugate to the corresponding unrepeated odd numbers-partition is a much simpler process, the rule being to take the numbers in descending order and from their doubles subtract the successive odd numbers in the natural scale until the point is reached at which the difference is about to become negative ; thus the partition 6 6 5 4 3 2 is self-conjugate, and the correspondent to it is 11 9 5 1.
(32) The expansion of the reciprocal to (1 — ax) (1 — ax') ... (1 — aa^~') may be read off with the same facility as the direct product. In this case we are concerned with partitions of odd numbers capable of being repeated in the same partition ; now, therefore, if we use the same method of equilateral angles as before, and fit them into one another in regular order of magnitude, it will no longer be the case that their sum will form a regular graph, for if there be 6 parts alike, each line and column which ranges with either side of any (but the first one) of these will jut out one step beyond the anterior line and column (respectively), so that the line joining the extremities of the lines or columns will be parallel to the axis of symmetry. The figure then corre- sponding to i odd parts can no longer be dissected into a square of nodes and two equal regular graphs, but it may be dissected into a line of nodes lying in the axis of symmetry, and two regular graphs one of which has for its boundaries one of the original boundaries and a line of nodes parallel to the axis of symmetry, and the other one the other original boundary and a line of nodes parallel to the same axis, as seen in the annexed figure, where the axial points are distinguished by being made larger than the rest.
• #••••♦
• •*♦••••
• •»#•»»
• •••#•
• ••••#•
The graph read off in angles represents the partition 1111117 3 3. On removing the six diagonal nodes it breaks up into two regular graphs, of
30
A Constructive theory of Partitions, arranged in [1
which one is 5 5 5 3 1 1, and the other the conjugate thereto ; hence the coefficient of a;" in the coefficient of a-* in the expansion of the reciprocal of 1 — ax .1 — aaf 1 — oar^'"' in ascending powers of a is the number of ways
in which —5-^ can be resolved into_; parts limited not to exceed i— 1, which
is the coefficient of x '^ in
\-x^
1
1 - «•'+>-'
or of a;" in
l — x.i —a? ... \—x? 1 - a^ . 1 -a?^ ... 1 -a;»'+«-
-xi.
\.-x^ .\-a* ...\-3?i
(33) Although I shall not require any intermediate expansion whatever in order to obtain the transcendant %-fis product in the form of a series, I will give another of those which are sometimes employed together in combination (see Cayley, Elliptic Functions, pp. 296 — 7) to obtain this result : thus to prove that the continued product of the reciprocal of
(1 -ax){l- aa^) (1 - ax') ... is identical with
a a;*
1 +
g'
1 — x' 1 — xa 1 — X . 1 — x^' 1 — xa . 1 — x'a
of
+
\ — x.].—x'.\—a?'\—xa.\- x^a . 1 — oc^a
+ etc.
if n is partitioned into j parts, the regular graph which represents the result of any such partition must consist either of 1, 2, 3, ...j—1 or of not less than j columns, and its graph may accordingly in these several cases be dis- sected into a square of 1, 4, 9, ...j' nodes ; suppose that such square consists of 6 parts, then there will be n. — ^'' nodes remaining over subject to distribu- tion into two groups limited by the condition as to one of the groups that it may contain an unlimited number of parts none exceeding 0 in magnitude, and as to the other that it must contain j—d parts none exceeding 0 in magnitude, as seen in the following diagrams:
1] three Acts, an Interact and an Exodion 31
in all of which the partible number is 26, and j and d are 7 and 3 respectively. Now the number of such distributions is the coefficient of a;"~*^a-'~* in
1 1
l—x.\—a^... \— a^' \ — ax.\— aa? ... 1 — cut^ that is of a^ai in
\—x.i—jb-... i — a^'\ — ax .\ — aaf ... 1 — aa^ '
and consequently giving 0 all values from 1 to oo , the proposed equation is verified.
(34) It may be desired to apply the same method to obtain a similar development for the reciprocal of the limited product
(1 -aa;)(l-ax')...(l- ax') ;
the construction will be the same as in the last case ; the distribution into two groups can be made as before ; the second group will remain subject to the same condition as in the preceding case (seeing that the number of parts being less than j — 8, will necessarily be less than i— 0, {orj cannot exceed i), but the first group will be subject to the condition of being partitioned not now into an unlimited but into i — 0 (or fewer) parts none exceeding 0 in magnitude, and the number of such distributions into the two groups will accordingly become the coefficient of x^~^a^~' in
l-a:*-»+'.l-a:*-»+'... 1 -a* 1
I —x.\ — x' ...1 — a^ ' 1 — ax.l — ax' ... 1 — aa,^
or of x^a' in the last written fraction multiplied by a^.a^, so that the re- quired expansion will be
1— «* xa 1— a:*.!— a:*"' ar*a'
1 + -; . :; +
I —x' 1 —ax 1— a;.l— «' '1— tw;.l— ax'
1 - ar*. 1 -«*-'. 1 - x'-* ^^
l — x.l—a^.l — x' '1—ax. I— an? . 1 — aa?
(35) It is interesting to investigate what will be the form of the mixed development resulting from an application of the same method to the direct product
1 + ax.l +03^ ...1 + ax'.
For greater clearness I shall first suppose i indefinitely great. Consider the diagram :
32 A Constructive theory of Partitions, arranged in [1
In the above graph j and 0 used in the same sense as ante are 5 and 8 respectively, so that there is a square of 9 points ; an appendage to the right of and another appendage below the square, which I shall call the lateral and subjacent appendages respectively. The content of the graph being 25, there are 16 points to be distributed between these two appendages. What now are the conditions of the distribution of the n — 6^ points between them ?
I say that there will be two sorts of such distribution — one in which the lateral appendage will consist of 6 unrepeated parts, none of them zero, as in the graph above, and the subjacent appendage of j— 0 unrepeated parts, limited not to exceed 8 in magnitude, and another sort as in the graph below written,
in which the ^'th line of the lateral appendage is missing, and consequently the subjacent graph will consist oi j — 6 unrepeated parts limited not to exceed ^ — 1 in magnitude, for there could not be a part so great as 6 with- out the last line of the square having the same content as the first line of the subjacent appendage.
It should be observed that only the last admissible line of the lateral appendage can be wanting, for if more than this were wanting, two lines of the square would belong to the graph, and consequently there would be two equal parts 6.
Hence there are two kinds of association of the appendages, one leading to a distribution of n — 0'' between one group of 6 unrepeated but unlimited parts, and another oi j—d unrepeated parts limited not to exceed 6; the other to a distribution of n — 6" between one group of ^ — 1 unrepeated but unlimited parts, and another oi j — d unrepeated parts limited not to exceed
The number of distributions of the first kind is the coefficient of a.""*'. a^~* in
^-^-y^-^— J— p . (1 + o^) (1 + o^-) . . . (1 + a;rO,
the other of x^~^' . a-'""* in
j-^-^^^— ^— ^.(l+a^)(l + (w;»)...(l+ax«'->);
1] three Acts, an Interact and an Exodion 33
hence the sum of the distributions of the two kinds is the coefficient of the same argument in
--^— ^ [a^ (1 + aa^) + (1 - a;*)} (1 + cue . 1 + oar' ... 1 + aa^%
that is of a;" a-' in
r 2
, f\-¥ax.\ +aa? ...\+ oa;^-' 1 + aaP^^
land consequently we obtain the equation
I ^l+oar"
I + ^j xa
\ — X
l+ax.l + aa? ...\+axi-W+aa^ ^
l+aa;.l+aar.l + aar . . . = 1 + ^; axi + , ^ „ a:" a" +
1 — X 1 —x.l —or'
1 - X .1 - ai' ... I - xJ-^l - xJ
a^ +
and thus by a very unexpected route we arrive at a proof of Euler's
celebrated pentagonal-number theorem ; for on making a = — 1 the above
equation becomes
tP-i l-a:.l-a^.l-a;»... = l-(l+a;)a; + (l+a^)aH'...+(-y(l+a;^)a; « +....
Such is one of the fruits among a multitude arising out of Mr Durfee's ever-memorable example of the dissection of a graph (in the case of a symmetrical one) into a square, and two regular graph appendages.
Even the trifling algebraical operation above employed to arrive at the result might have been spared by expressing the continued product as the sum of the two series (which flow immediately from the graphical dissection process), left uncombined, namely,
1 + ax l+ax.l+aa^l+ax.l+ax'.l+aa^ „,.
\-x 1— a;.l-a^ \— x.\ —a^ .\ — a?
together with
\—x \—x.l — a?
which for a = — 1 unite into the single series
1-x — a?' + a^ + a;'-a:>'-a:" etc.
(36) I will now proceed to find the expression in a mixed series of the limited product
1 -f aa; . 1 -f aa? ...\ + ax'.
In each of the two systems of distribution (as shown already in the theory of the reciprocal of such product) the second group will remain unaffected by the new limitation, but the first group will now consist of partitions (limited in number as before), but in magnitude instead of being unlimited, limited
8. IV 3
34 A Constructive theory of Partitions, arranged in [1
not to exceed (i — 6), so that we will have to take the coefficient of a;""*' . a-'"' in the sum of
^* 1 - a;*-* . 1 - a;'-»-' ... 1 - «'-'"'+■
.(1 +aa:){\ +aa?)...{l + aafi)
1 -a;. 1 -*•»...! -afi and
This will be the same as the coefficient of x"a^ in
X {1 - a^ + (1 - a;«-29+') (a;» + aa^»)}, where the quantity within the final bracket is equal to
1 — «'+' a — a:'~*+' + x'' a. Hence the required series is
1 - a;' 1 - a;'-M - a;*-=
\l + -. ooa+^ T— (H-aa;)a^a'
( I — X 1— a;.l— a;^
l-a;.l-a;M-a;»
.1 + ax.l + aa? . x^^a? +
+ 1 -1 «^a + — i \ ^ (1+ oa;) al>a?
\^\— x \ — X .\ — a?
1 - a;»-' . 1 - a;'-M - a;*-' , , , , ,, . , 1
1— a;.l— a;".!— a^ )
the indices in the outstanding powers of x being the pentagonal numbers in the first, and the triangular numbers trebled, in the second of the above series.
In obtaining in the preceding articles mixed series for continued products, it will be noticed that the graphical method has been employed, Qot to exhibit correspondence, but as an instrument of transformation. The graphs are virtually segregated into classes, and the number of them contained in each class separately determined. (The magnitude of the square in the Durfee-dissection serves as the basis of the classification.)
(37) Now let us consider the famous double product- of
(1 +aa;)(l + aa;')(l + aa;») ...
by (1 +a--ia;)(l+a->a^)(l + a-'a^)....
Here it will be expedient to introduce a new term and to explain the mean- ing of a bi-partition and a system of parallel bi-partitions of a number. The former indicates that the elements are to be distributed into two groups, say into a left and right-hand group : the latter that the number of the elements
1] three Acts, an Interact and an Exoclion 35
(on one, say) on the left-hand side of each bi-partition of the system is to be equal to or exceed by a constant difiference the number (on the other, say) on the right-hand side of the same bi-partition. If we use dots, regularly spaced, to represent the elements (themselves numbers and not units), we get a figure or pair of figures such as the following :
for which the coiTCsponding lines of the contour are respectively parallel — hence the name. When the numbers of elements on the two sides are identical, I call the system an equi-bi-partition-system — in the general case, a parallel bi-partition-system to a constant difference j, where j is the excess of the number of elements in the left-hand over that in the right-hand part of any of the bi-partitions.
(38) Consider now the given double product — it is obvious that it may be expanded in terms of paired powers a^ + a~^ of a, and the coefficient of »" in the term not involving a will evidently be the number of equi-bi-partitions of n that can be formed with unrepeated odd numbers ; and so the coefficient of a;" associated with a-' or a~^ will be the number of parallel bi-partitions of n to the constant difference j that can be so formed.
For the equi-bi-partitions; suppose li, l^.-.U, X,, X^.-.X^ is an equi- bi-partition, all the elements being odd and unrepeated ; take successive angles whose (say horizontal and vertical) sides are the major halves of Zj, X,; Z„ Xj . . . ; li, \i ; these angles will fit on to one another so as to form a regular graph by reason of the relations
k>k+\, k>k+'^---li-x>h+'i; X,>X,4-1, X,>X,-|-1 ...Xi_i>X,-(-l. Conversely any regular graph may be resolved into angles whose horizontal sides shall be the major halves of one set of odd numbers, and their vertical sides the major halves of another set of as many odd numbers, and these two sets of odd numbers will each form a decreasing series ; hence there is a one-to-one conjugate correspondence between any bi-partition of n written in
regular order, and the totality of regular graphs whose content is ^ , so that
it
n
the number of the equi-bi-partitions of n will be the coefficient of ar* in
1
that is of x" in
1 -x.\-a?.\-a?...' 1
which fraction is therefore equal to the totality of the terms not involving a.
3—2
36 A Constructive theorrj of Partitions, arranged in [1
(39) Next for the coefficient of a-*.
Let li, Zj, ... Ij, Ij+i, Zj+2, ... Ij^g; \,, Xj, ... \j be an equi-parallel bi-partition to the difference j (with the elements on each side written in descending order); with the equi-bi-partition Ij^i, lj+2, ■■■ Ij+e ', \,^,---^e, form a graph, as in the preceding case ; say, for distinctness, with major halves of the I series horizontal and of the X series vertical; over the highest horizontal line the successive quantities*
Ij-l Ij-, ~ 3 (,_, - 5 k - (2j - 1) 2 ' 2 ' 2 '■■■ 2
may be laid so as to form a regular graph of which the content will be — —-.
1, — iS
Conversely every regular graph whose content is — -— will correspond to
a parallel bi-partition of unrepeated odd numbers to a difference _;' ; to obtain the bi-partition the first j lines of the graph must be abstracted -f, and the graph thus diminished resolved into angles ; the doubles of the contents of each vertical side of these angles diminished by unity will constitute the right-hand side of the bi-partition, and the doubles of the contents of each horizontal side preceded by the doubles of the lines of the abstracted portion of the graph increased by 1, 3, 5, ...2^'— 1 respectively, will form the left- hand portion. Hence the number of such bi-partitions will be the number
of ways of resolving "^ into unrestricted parts, that is, will be the coefficient of a;" in
x'A -a^.l -a? .
and this being true for all values of n and j, we see that the double product in question will be identical with the infinite series
1
\-a?.\-a^.\-a^ ..
j 14- a; (a + a"') -f ar* (a^ -t- or'') ->r a? {a? -{■ ar^) + ,
(40) To expand the limited double product
(1 -f- aa;)(l -I- aa::») ... (1 -f oar^"')
into (1 -Fa-'a;)(l -f- ar^af)... (1 -|- a-'a~^->)
the procedure and reasoning will be precisely the same as in the extreme case of i infinite, the only difference being that the elements of the bi- partition instead of being unlimited odd numbers will be limited not to exceed 2i— 1. In the case of J = 0 the equi-bi-partition will furnish a series of nodal angles in which neither side can exceed the major half of 2i — 1,
* Any number of these quantities may happen to become zero.
t If the actual number of horizontal lines in the graph is less than j, it must be made to count as ;', by understandmg lines of zero content to be supplied underneath the graph.
1] three Acts, an Interact and an Exodion 37
that is i, and the coeflScient of «" in the term not containing any power of a will consequently be the number of wa3's in which n can be divided into parts limited as well in number as in magnitude not to exceed i, and will therefore be the same as the coefficient of a;^" in the development of
l-a!'+'.l-a;'+»...l-a^ \-x.\-a? ...\-x^ '
or, which is the same thing, of a;" in the development of
l-ar^-^^.l-ar^+^.-.l-a;*' l-ai'.l-a* ... l-ar^~'
and when the bi-partition system has a constant difference j, the correspond- ing graph will be of the same form, except that it will be overlaid with j lines, obtained as in the preceding case by subtracting 1, 3, ... 2; — 1 from the first j left-hand elements, and taking the halves of the remainders ; the graphs thus formed will be subject to the condition of having a content
— =-^ , and parts limited not to exceed i —j in magnitude nor i +j in number
[i —j in magnitude because the topmost line cannot exceed ~ —
in content; i+j in number because without reckoning the j superimposed . lines the subjacent portion of the graph cannot contain more than i lines]. The converse that out of every regular graph fulfilling these conditions may be spelled out a parallel bi-partition with a difference j, and containing only unrepealed odd numbers limited not to exceed 2i— 1 in magnitude may be shown as in the preceding case. Hence the coefficient of af* in the coefficient
of o^ + a~^ in the expansion, is the number of ways of resolving "^ into
parts none exceeding i —j in magnitude nor i -f- j in number, that is, is the coefficient of «" in
1 _ a;S»+«+> . 1 _ 0^+2)+* . . . 1 _ x*'
Hence by the process of reasoning, which has been so often applied, we see that the finite double product
l+ax.l+a^ ...1+ aa^-^
into l+a-^x.l+a-^x'...l+a-^x^^
1 -«'.l-a!«...l-a^ t
1 _ a*'+' . 1 - x^+* . 1 - x^+'"
Compare Hermite, Note sur les foncticms elliptiques, p. 35, where Cauchy's method is given of arriving at this and the preceding identity.
38 A Constructive theory of Partitions, arranged in [1
Act III. On the One-to-one and Class-to-class Correspondence BETWEEN Partitions into Uneven and Partitions into Unequal Parts,
. . . mazes intricate, Eccentric, intervolved, yet regular Then most, when most irregular they seem.
Paradise Lost, v. 622.
(41) It has been already shown that any partition of n into unequal parts may be converted into a partition consisting of odd numbers equal or unequal by, first, expressing any even part by its longest odd divisor, say its nucleus and a power of 2, and, second, adding together the powers of 2 belong- ing to the same nucleus, so that there will result a sum of odd nuclei, each occurring one or more times ; a like process is obviously applicable to convert a partition in which any number occurs 1, 2,,.. or (r — 1) times into one in which only numbers not divisible by r occur with unrestricted liberty of recurrence. The nuclei will here be numbers not divisible by r multiplied by powers of r, and by adding together the powers of r belonging to the same nucleus there results a series of nuclei, each occurring one or more times. Conversely when the nuclei and the number of occurrences of each
■ are given, there being only one way in which any such number can be expressed in the scale whose radix is r, it follows that there is but one partition of the previous kind in which one of the latter kind can originate, and there is thus a one-to-one correspondence, and consequently equality of content between the two systems of partitions.
(42) To return to the case of r = 2, with which alone we shall be here occupied, we see that the number of parts in the unequal partition which corresponds after this fashion with a partition made up of given odd numbers depends on the sum of the places occupied when the number of occurrences of each of the odd numbers is expressed in the notation of dual arithmetic. Such correspondence then is eminently arithmetical and transcendental in its nature, depending as it does on the forms of the numbers of repetitions of each different integer with reference to the number 2.
Very different is the kind of correspondence which we are now about to consider between the self-same two systems, as well in its nature, which is essentially graphical, as in its operation, which is to bring into correspond- ence the two systems, not as wholes but as separated each of them into distinct classes; and it is a striking fact that the pairs arithmetically and graphically associated will be entirely different, thus evidencing that cor- respondence is rather a creation of the mind than a property inherent in the things associated*.
* Just so it is possible for two triangles to stand in a treble perspective relation to each other, as I have had previous occasion to notice in this Journal.
1] three Acts, an Interact and an Exodion 39
(43) I shall call the totality of the partitions of n consisting of odd numbers the U, and that consisting of unequal numbers the V system.
I say that any U may be converted into a V" by the following rule : Let each part of the given U be converted into an equilateral bend, and these bends fitted into one another as was done in the problem of converting the reciprocal of
(\-ax){\-ao^){\-aa?)...
into an infinite series, considered in the preceding section. We thus form what may be called a bent graph. Then, as there shown, such graph may be dissected into a diagonal line of points and two precisely similar regular graphs. The graph compounded of the diagonal and one of these, it is obvious, will also be regular, and I shall call it the major component of the bent graph ; the remaining portion may be called the minor component. Each of these graphs will be bounded by lines inclined to each other at an angle one-half of that contained between the original bounding lines, and each may be regarded as made up of bends fitting into one another. The contents of these bends taken in alternate succes.sion, commencing with the major graph, will form a series of continually decreasing numbers, that is to say, a V partition. As an example let 11 11 9 5 5 5 be the given U partition ; this gives rise to the graph
A D
#••*•♦
A'
M «
C
• • * •
*
Reading off the bends on the major and minor graphs alternately, com- mencing with BAD, GA'E respectively, there results the regularized partition into unequal numbers
11 10 9 8 6 2.
(44) The application of the rule is facilitated to the eye by at once constructing a graph, the number of points in whose horizontal lines are the major halves of the given parts, and construing this to signify two graphs, one the graph actually written down, the other the same graph with its first column omitted ; for instance in the case before us the graph will be*
* This may be regarded as a parallel-rnler form of diBlocation of the figure produced by making the portion to the right of the diagonal of larger asterisks revolve about that diagonal
40 A Constructive theory of Partitions, arranged in [1
If we call the Hues and columns in the directions of the lines and columns of the Durfee-square appurtenant to the graph aia^...ai, o,a^...ai [t (here 3) being the extent of the side of the square], the pai-tition given by the rule will be
ai + Mi — 1, ai + a.i — 2, as + Oj — 3, a, + aj — 4, Oj + as— 5, ...
...[a,-_, + ai_,-(2i-3)], [at., + en - (2i - 2)], [a. + ai-(2i- 1)], [a.-t],
and inasmuch as
a, = or > Oj = or > a, . . . and Oj = or > otj = or >a,...
the above series is necessarily made up of continually decreasing numbers, at all events until the last term is reached. But this term will form no excep- tion, for the fact of i being the content of the side of the square belonging to the transverse graph Oj, a^..., Oj, Of+i ... implies that a,= or > i, hence
[ui + a,— (2i - 1)] - (tti -i) = ai-i+l>0.
In the above example the side of the square nucleus in the original total graph was supposed to be the same for the major and minor graphs of which it is composed. If we suppose that graph to contain only i nodes in the t'th line, then the side of the square to the minor graph which it contains will be i — 1, and the number of parts given by the angular readings of the two graphs combined will be 2ii — 1 instead of 2i, as for example if the 3rd line in the graph above written be 3 instead of 5, the resulting partition will be 11 10 9 8 2, but we may, if we please, regard this as 11 109820 and the last term will then still be at — i, and the general expression will remain unchanged from what it was before.
Next I proceed to the converse of what has been established, namely, that every U may be transformed by the rule into a V, and shall show that any V may be derived from some one (and only one) U.
Whether the number of effective parts in the given V be odd or even, we may always suppose it to be even by supplying a zero part if necessary, and may call the parts i,, Xj, 4, ^ ••■ ■ ^i. Xj. Suppose that it is capable of being derived from a certain U : form with the parts of U a graph expressed in the usual way by equilateral bends or elbows, then the side of the square appurte- nant to the regular graph formed by the major half of this, say 0, must have for content the given number i.
until it coincides with the portion to the left of the diagonal ; the graph thus formed (merely as a matter of conTenience to the eye) may be then made to revolve about an axis perpendicular to the plane, so as to bring the diagonal out of its oblique into the more usual horizontal position. All this trouble of description might have been saved by beginning not with a bent graph but with a graph formed with straight lines of points written symmetrically under each other, which is made possible by the fact of there being an odd number of points in each line. The graph so formed then resolves itself naturally into a major and minor regular graph.
1] three Acts, an Interact and an Exodion 41
Let Oi, Oj-.-a,-, «!, ou...a.i be the contents of the first i rows and first t columns respectively of G, then the equations to be satisfied are ai + a,-l = Zj, a,+ a2 — 3 = ^2, as + a^—o =1^ ..., aj + ai — (2i- 1) = ^;, Oi + aj — 2 = X,, Oo + Oj— 4 = X,, Oj + O4 — 6 = Xj..., a,- — i = \i.
Hence
Oi — a^ = \, — Zj — 1 o^ — a, = X^ — Zj — 1 . . .
Of-i — a,- = \,_i — Z, — 1 Of = \i + i, flj — a^ = Zj — \, — 1 02 — fls = Z, — Xj — 1 . . .
a,-i — a,- = Z,_, — \i_i — 1 «i = Zf — X,- + i - 1, and for all values of 6,
*9 > ^« > '#+1 ■
Hence o,, o, ... a,- are all positive, and a,, Uj ... a,- are all at least equal to i. There will therefore be one and only one graph G satisfying the required conditions, namely a graph the contents of whose lines are
a,, a2,...a,-, A^, A^,... Aa^-i [where Ai, A^, ... Aa, — i is the conjugate partition to Ui — i, a^ — i, ... o,- — t] ; the partition U will be found by subtracting unity from the doubles of each of those parts. Thus then it has been shown that every U will give rise to some one V, and every V be derived from a determinate U; hence there must exist a one-to-one correspondence between the U and V systems. In a certain sense it is a work of supererogation to show that there is a, U cor- responding to each V; it would have been sufficient to infer from the linear form of the equations that there could not be more than one U transformable into a V; for each U being associated with a distinct V it would follow that there could be no V's not associated with a U, since otherwise there would be more V's than D'a, which we know aliunde is impossible.
As an example of what precedes let the partible number be 12. The U system computed exhaustively will be 11.1 9.3 9.P 7.5 7.3.1' 7 . P 5M» 5.3.1*
5.3M 5.r 3* 3M» 3M» 3.1« 1"
Underneath of these partitions I will write the major component graph, and underneath this again the corresponding V; we shall thus have the table
11.1 9.3 9.1» 7.5 7.3.1' 7.1'
7.5 G.5,1 8.4 5.4.2.1 7.4.1 9.3
42 A Constructive theory of Partitions, arranged in [1
5M' 5.3.1* 5.3M o.T 3* 3M' 3M« 3.1" 1" - ' - - - (.)"
(.)'
* • • «
. . (.)»
(.)'
(.)'
\6.3.2.1 8.3.1 6.4.2 10.2 5.4.3 7.3.2 9.2.1 11.1 12 Thus we obtain for the V system :
7.5 6.5.1 8.4 5.4.2.1 7.4.1 9.3 6.3.2.1 8.3.1
6.4.2 10.2 5.4.3 7.3.2 9.2.1 11.1 12
which are all the ways in which 12 can be broken up into unequal parts*.
The U's, corresponding to those given by the arithmetical method of effecting correspondence would be:
1.3^5 1'
P. 7
3.9 1».5=
P. 3' 3.1*
1°.3 P. 3' ,5 P. 3. 7
P.3» 11.1 3*
instead of 11.1 9.3
9.P
SM" 3 . l" 1"
.5 7.3.P 7.P 5M= 5.3.P 5.3M 5.P 3* 3M'
so that there is absolutely not a single pair the same in the two methods of conjugation.
(45) The object, however, of instituting the graphical correspondence is not to exhibit this variation, however interesting to contemplate, but to find a correspondence between the two systems which shall resolve itself into correspondences between the classes into which each may be subdivided.
Thus we may call Ui that class of IPs in which there are i distinct odd numbers, and Vi that class of Vs in which there are i sequences with a gap between each two successive ones : the theorem now to be established is that the V corresponding to any Ui is a Vi, so that class corresponds with class, and as a corollary, that the number of ways in which n can be made up by a series of ascending numbers constituting i distinct sequences is the same as the number of ways in which it can be composed with any { distinct odd numbers each occurring any number of times. This part of the investigation which I will presently enter upon is purely graphical. A few remarks and illustrations may usefully precede.
In the example above worked out it will be observed that there are three classes of U's, namely, 1'= 3*: 11.1 9.3 9.P 7.5 7 . 1" 5\ P
SM" 3^1" 3.P: 7.3.P 5.3.P 5.3M
* In Note D, Interact, Paxt 2, I show how this transformation can be accomplished by the continual doubling of a string on itself.
11 three Acts, an Interact and an Exodion 43
and three classes of Vb agreeing with those above in the number of parti- tions in each, namely,
12 3.4.5: 11.1 9.3 10.2 8.4 7.5 9.2.1
7.3.2 6.5.1 5.4.2.1: 8.3.1 7.4.1 6.4.2.
So again for n = 16 there will be found to be eleven partitions into odd parts of the third class, which, with their quasi-graphs and corresponding partitions into unequal parts are exhibited below :
11. 3. P 9.5.1» 9.3M 9.3.1* 7.5.P
9.6.1 8.5.2.1 8.6.2 10.5.1 9.4.2.1
7.3.1' 7.3M' 5^3.P 5.3M» 5.3M» 5.3. P
• -»•« •••« «•• ••• •*« «*«
*« ** *•« *• •« *•
(#)' •• .. •• •• (•)'
(.)» (.)» . . (.)»
11.4.1 9.5.2 8.4.3.1 8.5.3 10.4.2 12.3.1
The transformed partitions above written are all of them of the third class (that is consist of three distinct sequences) and comprise all that exist of that class. 16 will correspond to 1" and 1.3,5.7 to itself. All the other partitions of each of the two systems will be of the second cla.ss, and will necessarily have a one-to-one gi-aphical correspondence inasmuch as the entire systems have been proved to have such correspondence.
It is worthy of preliminary remark that the association of the first classes of U's and Vs given in the previous section will be identical with the association furnished by the graphical method — but whereas in converting V into U by the antecedent process, the two cases of the sequence being of an odd or even order had to be separately considered, the graphical method is uniform in its operation.
Thus 9 8 7 6a sequence of an even order will be given graphically by
corresponding to 1.5', and 9 8 7 6 5a sequence of an odd order will be given graphically by
44 A Constructive theory of Partitions, arranged in [1
corresponding to 5', whereas it will be observed that 15' = (9 + 6)* and
9+5
5' = 5 « .
It may be noticed that when the major component is an oblate rectangle it gives rise to a sequence of an even order, and when a quadrate or prolate rectangle to one of an odd order.
I subjoin an example of the algorithm by means of which a given V can be transformed into its corresponding U, taking as a first example F= 10 9 8 5 4 1.
The process of finding U is exhibited below :
|
3 3 5 5 (9) |
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2 2 3 3 (8) |
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4 4 2 (7) |
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13 3 (6) |
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10 8 4 (1) |
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9 5 1 (2) |
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111 (3) |
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4 4 4 (4) |
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7 7 7 (5) |
|
|
3^ . 5» . 7' will be the U required. |
|
|
As a second example let 7"= 12 10 9 8 5 4 1 ; |
the algorithm will be |
|
as shown below: |
|
|
1 (9) |
|
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1 (8) |
|
|
1 0 0 0 (7) |
|
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2 1 1 1 (6) |
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12 9 5 1 (1) |
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10 8 4 0 (2) |
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1 3 3 0 (3) |
|
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8 8 6 4 (4) |
|
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15 15 11 7 (5) |
1 7 11 15 15 will be the U required. Lines (1) and (2) are the parts of the given F written alternately in the upper and lower line; lines (3) and (6) arc obtained by oblique and direct subtraction performed between (1) and (2) ; line (4) is obtained from (3) by adding the number of terms in (1) to the last term in (3) which gives the last term in (4) and then adding in successively the other terms in (3) each diminished by one unit; (7) is derived from (6) by diminishing each term in the latter by a unit and taking the continued sum of the terms thus diminished ; (8) is found by the usual
1] three Acta, an Interact and an Exodion 45
rule of "calling"* from its conjugate (7): and finally (5) and (9) are obtained by subtracting a unit from the doubles of the several terms in (4) and (8).
It thus becomes apparent that the passage back from a F to a ?7 is a much more complicated operation than that of making the passage from a f7 to a V, so much more so that it would seemingly have been labour in vain to have attacked the problem of transformation by beginning from the V end.
(46) I now proceed to the main business, which is to show that any U containing i distinct odd numbers will, by the method described, be graphically converted into a V containing i distinct sequences.
Let G be any regular graph ; H what G becomes when the first column of G is removed; a,b,c,d ... the contents of the angles of G, H taken in succession.
Also let i be the number of lines of unequal content in G,j the number of distinct sequences in a, h, c, d, e, ... .
The two first lines of G, say L, L, and also the two first columns, say K, K', may be equal or unequal +.
If i = Z' and ir= if', a - 1 = 6, 6 - 1 = c.
li L = L and K>K', a-l = b, b-l>c.
1{ L > L' and K = K', a - 1 >b. b-l = c.
I( L>L' &Qd K>K', a-l>b, b-l>c. Let G', H' represent what G, H become on removing the first bend, that is the first line and the first column, and let i',j' be the values of i,j for G', H', so that / is the number of sequences in c, d, e ... .
It is obvious from what precedes that in the four cases considered / =j, j'=j — l, j'=j-l, j'=j-^ respectively. But in these four cases i=i, t' = i — 1, i'=i — 1, i' = t — 2 respectively.
Hence on each supposition i —j = i' —j, and continuing the process by removing each bend in succession, i —j must for any number of bends have the same values as it has for one bend ; but in that case if h and k are the contents of the line and column of the bend, the reading of the corresponding G, G' will he h + k—l, h — l,ao that for that case _;' will be 1 or 2 according as h and k are not or are both greater than 1, that is according as i is 1 or 2^.
* I borrow this tenn from the vemacular of the American Stock Exchange.
f For brevity I one line and column to signify the extent of (that is, the namber of nodes in) either.
i The final graph after denudation pushed as far as it will go mast be either a single bend, a column, a line or a single node. In the first case i = 2,j = 2, in each of the remaining three cases i = l,j = l.
46 A Constructive theory of Partitions, arranged in [1
Hence i —j is always equal to zero, consequently a. U of the I'th class will be transformed by the graphical process into a F of the tth class, as was to be proved.
(47) I have previously noticed [p. 25 above] that the simplest case of i =j = 1 leads to the formula
1-q^l-q'^l-^ l-q'^'" I -q 1 - q^^ I - ^ 1 -q* '
which is a sort of pendant to Jacobi's formula
q ^ q° q' _ 9 g" g" 9'° »
r+^ ~i + 5»'*'i+3»~ r+^'"*' ■■■ ~ r+~q ~ i+q''^i+^~i+q*^'" ■
These formulae may be derived from one another or both obtained simul- taneously as follows : From addition of the left-hand sides of the two equations there results the double of
and from addition of the right-hand sides of the same there results the double of
Consequently in order by the operation of addition of the two equations to deduce one from the other we must be able to show that these expressions are identical : observing then that 4i — 3 and 8i — 2 are odd and even respectively for all values of i, but i{2i — l) and i(2i + 3) odd or even, according as for i, 2i — 1 or 2i be written, it has to be shown that
and S ^ . =S(-^ ^ +^ . 1 . (B)
QO 2 qI— 1.81—6 00 q\
{A) is equivalent to 2 y"~' ^_.- =2
or
1 - (f-* 7 1 - (f-*
00 1 _ /jt(a»+2) 00 „»»— i.4i+l
1 ^ 1 - ^'+» 7 1 - J"'-"
Hence if i signify any number from 1 to oo and k signify any number from 0 to i — 1, it has to be shown that (4i + l){2k+ 1) contains the same integers and each taken the same number of times as (2m — 1) (4m -I- 1 -I- 4n), where m is any number from 1 to oo and n is any number from 0 to x . But the (4t -I- 1) (2A; -I- 1) is the same as (2^- +\)[^{k+l-\-\) + \\ where k and I
* My formula is what Jacobi's becomes when every middle minus sign in it is changed into plut and every inferior plus sign into viinus.
1] three Acts, an Interact and an Exodion 47
each extend from 0 to oo , and the (2m — l)(4w+ 47i + 1) is the same as (2to+ l){4(w+ n + l) + Ij where m and n each extend from 0 to oo , and the two latter expressions on writing k = m, l = n become identical.
Again (B) is equivalent to
Hence we have to show that (8i— 2)(1+^') when i=2, 3, ... oo and j = 0, 1, 2, ... , (i - 2), or say (8t + 6)(1 +j), where i = 1, 2, ... oo and j = 0, 1, 2, ...(t — 1) is identical with l{8l+6 + 8m), where 1=1, 2, ... x and m = 0, 1, 2, ... 00 ; the former of these is identical with
(l+j){8(j + k + l) + 6}, where j = 0, 1, ... oo ; k=0, 1, ... oo , and the latter is identical with
(l + 0{8(/ + m+l) + 6}, where 1 = 0, 1,... oo ; m = 0, 1,... oo , consequently the two expressions are coextensive, which proves (B), and (A) has been already proved. Hence we see that either of the two original equations can be deduced from the other from the fact that their sum leads to an identity.
In like manner subtraction performed between the two allied equations leads to the fissiparous equation
2-
gfi+i a;**+» ) " C^ (i+2) (»i+i) ^■+i.ai+s]
1 - a*+» 1 -a;»'+«j o | 1 - x*'+' l-x*'+*] ' which gives birth to the pair
00 .^i+l 00 r~»°+:.4>-l-l ~«'+2.4i+»'\
and 1^Jl^^^^\^^-^ + ''^^[, (D)
0 1—0:"*+' 0 (!—«*+• 1— a;»*+»J ^ '
{(J) is equivalent to
00 /|4i+3 /J g.i-^\ . al+t\ 00 ^21+1 . Ji+J
7 1 - ««•+• "" 7 l-a;»<+« '
which is an identity by virtue of the equivalence of
(4i + 3)[1 + 2[j< (i + l)j] that is (4; + 4i + 3) (1 + 2j) to (2\ + 1) (4\ + 3 + 4^) where J, k, \, /j, each extend from zero to infinity, and (2)) is equivalent to
00 ^+1 (\ ^(gi+t)\ oo ^+J.4i+6
0 l-a*+' "7l-a*''+»' which is an identity by virtue of the equivalence of
(8i + 2) {1 + (j < i)} that is {8 (j + k + l) + 2} (1 +;') to (2\ + 2) (4\ + 5 + ifi), each symbol j, k, fj. having as before the same range, namely from zero to infinity. Thus then the difference of the two allied equations (as previously \heir sum) is reduced to an identity which establishes tlie validity of each of them.
48 A Constructive theory of Partitions, arranged in [1
Interact, Part 2.
With notes of many a wandering bout, Of linkM sweetness long drawn out.
L' Allegro.
(48) D. Transformation of Partitions by the Cord Rule. — The figures below are designed to show how it is possible by means of the continuous doubling of a string upon itself to pass from an arrangement of groups of repetitions of r distinct odd integers to the corresponding one with like sum, made up of r distinct sequences. Each of the two figures duplicated by rotation about its upper horizontal boundary of nodes through two right angles will represent an arrangement of repeated odd numbers, the parts being represented by the contents of the vertical lines in the figures so duplicated.
Fig. 1. Fig. 2.
-A A' -1 1 1 1 i 1 1 1 1 1 iB
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Ui- |
{ |
i |
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HL |
K |
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NF |
< |
!) |
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Rr |
s |
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V |
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1 |
^ |
J |
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i |
i ^ |
1 |
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, |
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t— • |
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|
E |
FD |
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K |
LH |
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O |
PN |
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S |
^R |
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Q |
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M |
||||||||||
|
G |
The first duplicated figure represents the arrangement 33, 29', 23, 21, 9', 7, 6S 3, 1 whose sum is 183 ; its correspondent will be the contents of the lengths of * ABC, CDS, EFG, OHK, KLM, MNO, OPQ, QRS, STU. UV, namely the arrangement 29, 27, 24 (22, 21,), 18, 14, 12, 10, 6 which is the same number 183 partitioned into (ten parts but) nine sequences : the second duplicated figure represents the arrangement 2.5, 23, 17, 15, 9', 7', 5", 1*, whose sum is 130 ; its correspondent is represented by the lengths of ABC, CDE, DEF, FGH. HKL, LMN, NOP, PQR, RST, TU, which is the same number 130 partitioned into the (nine parts but) eight sequences 25, 22 (20, 19,), 15, 12, 10, 6, 1.
* A line containing t units of length represents (t' + l) nodes.
three Acts, an Interact and an Exodion
49
approximately S 1
(49) E. On Graphical Dissection. — It may be not unworthy of notice that there is a sort of potential anticipation of Mr Durfee's dissection of a symmetrical graph, iu a method which, whether it is generally known or not I cannot say, but is substantially identical with Dirichlet's for finding
- and other such like series (a bracketed quantity being
used to signify that quantity's integer part). Constructing the hyperbola oey = n, drawing its ordinates to the abscissas 1, 2, 3, ... «, and in each of them planting nodes to mark the distances 1, 2, 3, ... from its foot, there results a symmetrical graph included between one branch of the curve, its two asymptotes, and lines parallel to and cutting each of them at the distance n from the original. Its content will be the sum in question. The Durfee-square to it will be limited by the square whose side is [%/n], and this added to the original area gives twice over the area in which the
number of nodes is S - , and consequently neglecting magnitudes of the
1 L''J order V,
n
t] = 2«'f i-i» = n{logn + 2C-l)
and as a corollary
V"
f]"-["]} = «(C-2C+l) = (l-(7)n,
where C is Euler's number '57721, so that 1 — C for large values of n will be the average value of the fractional part of n divided by an inferior number. Furthermore a similar graph, but with xy=2n diminished by the portion contained between a branch of the new curve, one of its asymptotes and two parallel ordinates cutting that asymptote at distances n and 2m from the origin (which portion obviously contains (2n — n) that is n nodes) will
represent 2
!["]■
and consequently the sum 2
2m i
-22
(see Berl. Abhand. 1849, p. 75) the number of times that - —
that is
eq
uals
exceeds ^, as i progresses from 1 to n (within the same limits of precision as previously) = 2/i(log 2n+ 2C— 1) — n less 2n (log n + 2(7— 1), that is = (log 4 — 1)71, so that the probability of the fractional part of n divided by an inferior number not falling under ^ is log 4—1*.
* What precedes I recall as having been orally communicated to me manj years ago by the late ever to be regretted Prof. Henry Smith, so untimely snatched away when in the very zenith of his powers, and so to say, in the hour of victory, at the moment when )iis iutellectunl eminence was just beginning to be appreciated at its true value, by the outside world. I was nnder the impression until lately that he was quoting literally from Dirichlet when so communi- cating with me, bat as the geometrical presentation given in the text is not to be found in the
L
50 A Constructive theory of Partitions, arranged in [1
(50) F. Mr Ely's method of finding the asymptotic value of the number of improper fractions with a very large given numerator which are nearer to the integer below than to the integer above*.
"Let a number n be divided by all the numbers from 1 to n; then a value is required for the number of residues which are equal to or greater than J. An example will make evident a method by which we may obtain limits to the value sought. If n be 100 the residues = > ^ are
(I) ^1^17 46454443424140393837363534
^ ^ 51 52 53 54 65 56 57 58 59 60 6l 62 63 64 65 66
32 30 28 26 24 22 20
^ ' 34 35 36 37 38 39 40
(3) (4) (5) (6) (a)
memoir cited from the Berlin Traiuactiom, I infer that it originated with himself. In compar- ing Mertens' memoir, Crelle, 1874, with Dirichlet's (1849), upon which it is a decided step in advance, one cannot fail to be struck with surprise that the point to which the closer drawing of the limits to the values of certain transcendental arithmetical functions achieved by the former is owing, should have escaped the notice of so profound and keen an intellect as Dirichlet's, and those who came after him in the following quarter of a century. The point I refer to is the almost self-evident fact that if in the cases under consideration
Z(p (Fi . x) = <f/x then <px = Z/j. (i) ^ {Fi . x) where fi (t) means 0, if i contains any repeated prime factors, but otherwise 1 or 1 according as the number of prime factors in t is even or odd. Dirichlet works with a function given implicitly by an equation, Meitcns with the same function expressed in a series, wherein exclusively lies the secret of his success.
* It is proper to state that what follows in the text was handed in to me by Mr Ely on the morning after I had proposed to my class to think of some "common sense method" to explain the somewhat startling fact brought to light by Dirichlet, of more than three-fifths of the
residues of n in regard to i = 1, 2, 3, ... n being less than - . Mr Ely's method shows at once, in a very common sense manner, why the proportion must be considerably greater than the half, inasmuch as whilst the terms in the first few harmonic ranges are approximately :j — 5,5—5 , 5 — ■,, etc., in number, the number of them which employed as denominators to n give fractional parts greater than J, instead of being the halves of these are only - — - , i — - , - — - , etc. The mean value in both methods to quantities of the order of y^/n inclusive, turns out to be the same, whichever method is employed, but the margin of unascertained error by the use of Mr Ely's method (as compared with Dirichlet's) is reduced in the proportion of 1 : lH-^2, that is, nearly 2:5.
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22 |
19 |
16 |
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26 |
27 |
28 |
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16 |
12 |
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21 |
22 |
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15 |
10 |
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17 |
18 |
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10 |
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15 |
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4 |
4 |
9 |
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0 |
8 |
13 |
1] three Acts, an Interact and an Exodion 61
In which it will be observed that the residues = > i occur in batches. Let X be the whole number, and Xi the number in batch i. In batch i the numerators decrease by i and the denominators increase by 1. (Those marked (a) of which the denominators are less than ^200 are left out of account for the present.) It is evident for the general case we have approximately
|
n |
- ixi |
|
f+1 |
|
|
n |
+ Xi |
|
b'+lj |
or accurately
**= L(i + l)(2i+l)J *"■ [(i+])(2i+l)J + ^*"
Mr Ely is then able to show that by limiting the calculation of Xi to the values of i which do not exceed [Vn/2], so that roughly speaking the character of V2w of the remainders is left undetermined (and no account taken of them in finding the value of X), and giving to Xi its approximate value
-r. — - — y^r- — T\ , and then extending the series k— s + s— ? + i—a beyond the (i + l)(2n-l) * 2.3 3.5 4.7 •'
[Vn/2]th term, where it ought to stop, to infinity, the errors arising from
each of these three sourcesf and therefore their combined efifect will be
of the order ^n, so that the asymptotic value of X will be
/111 \
l2T3 + 3:5 + iT7 + -j"'
which is (2 log 2 — l)n, with an uncertainty of the order \/n, as was to be shown.
(.51) It may be seen that Mr Ely's method consists in distributing the n numbers from n to 1 into what I have elsewhere termed harmonic ranges and determining what portions of the several ranges employed as denomi- nators to 71 give fractional parts, greater or less than ^. It may assist in forming a more vivid idea of this kind of distribution, if the reader takes a definite case, say of n = 121, the first (10) harmonic ranges will then comprise
* I find by aa exact calcnlation that if R is the remainder of n in regard to (t + 1) (2t + 1) and JJ = X(i" + !) + /», where X<2j + 1 and /t<t + l, then for X = 2«-l or 26, X( = \ - — — ^- — rl+1 if/i = «-l or i-2 ... or «- #, and Jt<= .-: — rr— p — - for all other values of m- Heuce it follows
thatootof (2t' + 3i + l)sncce88iveTalae8 of h, (I' + i) and (i* + 2« + 1) will be the respective numbers of the cases for which the one or the other of these two values of z,- is employed, so that for larger values of i the chances for the two values are nearly the same, but with a slight prepon- derance in favour of the smaller value. See p. [-54].
t The error from the first cause makes the determination of X too small by an unknown amount, that from the third cause too large by a known amount, and that from the second too large or too small (as it may happen) by an unknown amount.
4—2
52 A Constritctive theory of Partitiotis, arranged in [1
all the numbers from 121 to 12 inclusive, and the remaining 111 harmonic ranges will comprise the remaining 11 numbers from 11 to 1 ; that is to say 11 of them will contain a single number, and the remaining 100 ranges be vacant of content.
So again if n = 20 the first four ranges will contain all the numbers from 20 to 5 inclusive ; the 5th, 6th, 9th and 20th range will consist of the sole numbers 4, 3, 2, 1, and the remaining 12 ranges will be vacant. I shall proceed to compare the precision of Mr Ely's result with that of Dirichlet's — for this purpose it will be enough to determine the asymptotic value of the uncertainty and to take no account of quantities of a lower order than */n.
Let us then suppose that \/{kn) ranges are preserved, and consequently
I t) fractions left out {k being an arbitrary constant which will eventually
be determined so as to make the uncertainty a minimum).
The first cause of error necessitates a correction of which the limits are 0
and » /(t) ; the second cause a correction of which the limits are \J{kn) and
— hj{kn) ; and the third, namely the overreckoning of
". n
^/G
(i + l)(2j + l)"(j + 2)(2j + 3) where J = \/{kn), a correction of which the value is — ^. or — ^ a / (x) • Hence making (log 4 — 1) w = U, the superior limit of X is
and the inferior limit U —-^ / i-r] \/{kn). Consequently X = CT + /wi^ where
p< \Jk+ -^ I ( r) , of which the minimum value is found by making A; = ^ , so that /3< V2 and the uncertainty is V2 • i^ . Adopting Mertens' asymptotic
value of the uncertainty of 2
« L* J
, namely V", and using Dirichlet's formula.
, X has the same mean value as above, but the uncertainty
becomes (\/2 + 2) n- which is nearly two and a half times as great as that given by the direct method employed by Mr Ely.
I use the word uncertainty, it will be noticed, in a different sense from error ; the latter is objective, referring to fact, the former subjective, referring to knowledge. Both methods in the case here presented give the same mean value, and therefore the error is the same, but the uncertainty is widely
i|
IJ three Acts, an Interact and an Exodion 53
different according to the method made use of. Of two formulae referring to the same fact one might very well give the smaller error and the other the smaller uncertainty.
I have shown above that for considerable values of i, the average value
n 1 .
of Xi is -. — ,, ,_. — 7-r + j; ; if then it may be assumed (and there seems no (i + l)(2i + l) 2' ^ ^
reason for suspecting the contrary) that for i = 1, 2, ... , \/2n, the mean value
71 I « 1 J ^^
of - — I - is s . f^ will not only be the mean value of the known limits of I LiJ 2 •'
X but also the mean value of X itself. The value found for k shows that the
most advantageous mode of employing Mr Ely's method is to make the
series 5— s + «— ^ + ... t-. — ttto^ — r\'^ ••• ^''^P ** ^°® °^ the terms which
A.O 0.0 (1*4*1^ {Z% "T" \.J
is approximately equal to unity.
(52) It is not without interest to consider the exact law for the extent of a harmonic range of a given denomination, say i : this it is easily seen will
be always equal to j^^.j or |^^^J + 1.
I shall regard i as given and determine the values of n which correspond to the one or the other of the two formulae : this will depend not on the absolute value of n but on its remainder in respect to the modulus I' + t. To fix the ideas, let t = 4 so that i" + i = 20, and let n take in successively all values from 40 to 59 inclusive.
Then corresponding to n equal to
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40 44 48 |
52 |
56 |
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41 45 49 |
53 |
57 |
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42 46 50 |
54 |
58 |
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43 47 51 |
55 |
59 |
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12, 11, 10 |
13. 12, |
11 |
14, |
13, |
12 |
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12, 11, 10 |
13, 12, |
11 |
14, |
13, |
12 |
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12, 11 |
13, 12, |
11 |
14, |
13, |
12 |
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12, 11 |
13, 12 |
14, |
13, |
12 |
the fourth range will be 10, 9 11, 10, 9 10, 9 11, 10 10, 9 11, 10 10, 9 11, 10
that is in half the terms of the period -^ — : and in the other half
^— -. + 1 gives the extent of the range.
So in general, if n=k{i? + i) + Xr + /*, where X = 0, 1 , 2, . . . t, and yit = 0, 1, 2, ... {i— 1), when the remainder of n to modulus (t' + i) is of the form
54 A Constructive theory of Partitions, arranged in [1
X. (t' + i) + {0, 1, 2, . . . (\ - 1)} that is in
%^A-l
cases the extent of the zth
harmonic range to n is i^ — = + 1, and when of the form
\(i* + i)+[W + \,...(,i-\)],
that is in the remaining
I' + i
cases it is
i» + i
As the sum of the harmonic ranges to n is n. itself, and
n
n n n
rT2"''2T3"'" ■■■'^«(« + i)
n + 1'
it follows that if we separate all the numbers from 1 to « into two classes, say t"s and j's, i being any number for which n is of the form
k (i? + i) + \i + 0, 1, 2, ... (\- 1), and j any other number within the prescribed limits, then
1 t 1
= number of i's —
n+1'
and consequently the number of the i terms has {I —C)n for its asymptotic value.
(53) In like manner the law previously stated in a footnote, p. [51], for
giving the extent of that portion of the I'th range for which - contains
z
a fractional part not less than ^ may be verified. Thus let i = 3 then (t+l)(2i+l) = 28, let n = 56, 57, ... 83. Then for the values of n
28 32 36 40 44 48 52
29 33 37 41 45 49 53
30 34 38 42 46 50 54
31 35 39 43 47 51 55
the portion of the third range having the required character will contain the numbers
8 8 8 8
9 9 9 10, 9
10 10 10 11, 10
11 11
12, 11 12, 11
12 12
12, 11
13, 12
13
14, 13 14, 13
14, 13
14
15, 14 15, 14 15, 14
so that there are 2(1 + 2 + 3), that is 3 . 4 forms of n out of 7 . 4 for which the formula 7 — p + 1 has to be employed, and so in general if R is the residue of n in respect to (i+l)(2i+l), there are i'+i cases where the
^"•■"""'^ L(t + i)(2t + i)J + ^ ^""^ ^' + ^^' *^^^" *^" ^"'■"'"^^ l(^ri)(2i^)\
has to be employed.
1] three Acts, an Interact and an Exodion 55
G. On Farey Series.
(54) This note is a natural sequel to and has grown out of the two which precede ; it has also a collateral affinity with the subject-matter of the Acts, inasmuch as a graph affords the most simple mode of viewing and stating the fundamental property of an ordinary Farey series, and any series ejusdem generis. For instance, let ^, 5, C be a reticulation in the form of an equilateral triangle, where fi is a right angle, and n the number of nodes in the base or height of the triangle ; if the hypothenuse be made to revolve in the plane of the triangle about (either end say about) A, the triangle formed by joining A with any two consecutive nodes of greatest proximity to the centre of rotation traversed by the rotating line will be equal in area to the minimum triangle which has any three nodes for its apices, that is its double will be equal to unity. This law of uniform description of areas, (say of equal areas in eqxml jerks) is identical with the characteristic law of an ordinary Farey series which deals with terms whose number is the sum-totient to: but it will also hold good if the triangle be scalene instead of equilateral, which corresponds to Glaisher's extension of a Farey series, to the case where the numerator and denominator of each term has its own separate limit (Phil. Mag. 1879), or again, when the rotation takes place about the right angle B as centre, which gives rise to a Farey series of a totally different species, defined by the inequality ax + by<n, or again when the hypothenuse is replaced by the quadrant of a circle or ellipse, and in an infinite variety of other cases, as for example when the graph is contained between a branch of an equilateral hyperbola and the asymptotes, which case corresponds to the subject-matter of the theory of Dirichlet {Berl. Abhand. 1844) concerning the sum of the number of ways in which all integers up to n can be resolved into the product of two relative primes, which is the same thing as the half of the number of divisors (containing no repeated prime factors) which enter into the several integers up to n, or as the entire number of solutions in rela- tive primes of the inequality xy = or < n. The law of equal description of areas (pq' — p'q^ ± 1), Mr Glaisher has shown very acutely, is an immediate inference (by an obvious induction) from the well-known fact that between a fraction and its two nearest convergents (namely the one ordinarily so called and that which is obtained by substituting 8 — 1 and 1 for the last partial quotient), no other fraction can be interposed whose denominator is not greater than that of the one first named.
From the areal-law obviously follows the equation f-j,= -~, — - (where
n p' p"
- , —,, ^ are any three consecutive terms of the series), so that in order to
construct explicitly such a series from the two first terms, all we have to do is to give to X at each step the highest value it can assume, consistent with
66 A Constructive theory of Partitions, arranged in [1
the imposed limit or limits. Thus for example I have found by this method when the limiting inequality is a; + y = or < 15, the series
1 15 14 13 12 11 10 9 8 7 6 11 5
9 4 11 7 10 3 11 8 5 7 9 2 1 and the complements in respect to unity of the several terms which precede
5 taken in reverse order, and again for xy = or < 15 the series (which might
be called the Dirichlet-Farey series)
9. L L 1 } L L I I \ I I 1 15 14 13 12 11 10 9 8 7 6 5
121213231 47352534 1'
In general if we agree to understand respectively by the deceTnent and the secernent to x, the number of divisors without restriction, and the number of divisors restricted to contain no square number, that go into x, ^ud denote the sum-secernent and sum-decernent of n by 8n and Dn respectively, Dirichlet's mode of looking at the question leads immediately to the equation
1,8 -7- = Dn. Mertens' equation 1 *"
somewhat more difficult process is in point of fact merely that equation
reverted. On pointing out to Mr F. Franklin this elegant passage in
Dirichlet's memoir, he remarked to me to the effect that it was an example,
which might admit of wide generalization, of a concept resembling that
inherent in the subject-matter of the ordinary Farey series; which excellent
and keen-witted observation led me to look into the subject from the point
of view herein explained. The present theory diverges from the ordinary
one in quite another and more natural direction (I imagine) than that pursued
by M. Darboux. whose article on the subject of quasi-Farey series {Bulletin
de la SocUte Mathimatique de France, tome vi.) I have not been able to obtain
sight of, and can only conjecture its purport through the reference made to it
in a subsequent article which I have been able to procure in the same journal
by M. Edouard Lucas.
* It is advisable for the purpose of securing generality in reasoning upon Farey series not to omit the initial and final terms ^, \ which seem generally to have been lost sight of by previous writers on the subject Even then the series is only half complete, for after I should follow the reciprocals of the preceding terms until I is reached. Thus a complete ordinary Farey series beginning with | and ending with J consists of two symmetrical branches with \ as their point of junction, each made up of two symmetrical sub-branches meeting respectively in the terms ^ and ^, and such that the sum of a corresponding pair of fractions on the one side of | and of their reciprocals on the other side is equal to unity : whereas in the two complete branches the product of each corresponding pair is unity.
. n Sn = 1,u.iD — ,
1 *'
obtained by a longer and
1] three Acts, an Interact and an Exodion 57
(55) I prove the persistency of the fundamental property of ordinary Farey series for such series generalized in the manner supposed above, as follows.
Let us use 0. F. Si to denote an ordinary Farey series for which the limit is i, and G. F. S. a Farey series in which, calling the numerator and denominator of any term x, y, <f> (x, y) < = i, <f> {x, y) meaning a rational function which increases when either x or y increases. If in an 0. F. St any
two consecutive terms be 7 , -j, and in an 0. F. jS,+, - intervenes between
CE C
r, J we know, p being greater than b and d, the two nearest convergents to - must be contained in 0. F. Si, and consequently must be r, t them-
(X c
selves, so that p = a + c, q = b + d, and as a corollary if t , t he consecutive terms in any 0. F. S., and " be any one of the terms which subsequently
intervene between r , j . we must have p = or > a + c, ^ = or > 6 + rf. In
order to fix the ideas let us suppose <f> (x, y) to represent x + y, so that x + y <=n.
For the values 2, 3, 4, 5, 6, 7, 8, 9 ... of n, the 0. F. S. will be 11' \ \2) V 1 V3/ 2 1' 1 W 3 2 Uy 1' 1 W 4 3 2 3 1' 1 U; 5 4 3 U/ 2 3 W 1 ' 1 uJ 6 5 4 3 5 2 U/ 3 4 1 '
1 W 7 6 5 4 VtJ 3 5 2 5 3 4 Uj 1 ' •"•
where the terms in parenthesi.s are the new terms which intervene as n increases from any value to the next following integer, and where it will be
noticed that if - be any such parenthesised fraction lying between r and
/J
J , p = a + c and q — h + d, just as in the successive form of an 0. F. S.
The theorem to be proved may be made to depend on the following lemma.
If for any given value of n every two consecutive terms in a G. F. S. appear as consecutive terms in an 0. F. S. for the same or any smaller value of »; this will continue to be true for all superior values of n.
The proof is immediate, for 'et r , -j be any two consecutive terms in the 0. F. Sj which are also consecutive terms in 0. F. Si where i— or < ;.
58 A Constructive theory of Partitions, arranged in [1
If a term - intervene between t , j in G. F. <?,>,, p = or > a + c, q= or
>b + d, by virtue of the remark made. But if p> a + c and q>b + d, <f>{a + c,b + d)<^ {p, q) <j + 1,
but -. ; is intermediate in value between r, -, , hence i ; must have
b+d b d b+d
appeared in a 0. F. 8^, where /< J, which is contrary to hypothesis.
Hence r. . j will have been consecutive terms in some 0. F. S., b q d
and in like manner any two consecutive terms in G. F. S. either remain con- secutive in G. F. Sj+i or admit a new term between them which is consecutive to each of them in some 0. F. S., so that the supposed relation if it holds
good for _;' is true for all superior values of _;'; but -, - will in any of
the supposed cases be a G.F.S.; consequently in all these cases no two terms are consecutive in any G. F. S. which are not so in some 0. F. S., and there- fore the law of equal description of areas will apply to them equally as to the 0. F. S., as was to be proved.
The theory may be extended to G. F. S., defined by several concurrent limiting equations. Thus for example Mr Glaisher has proved this for the case of X < = m, y < = n : I have not had time as yet to consider what are the restrictions to which the limiting functions may be subject, but the theorem is obviously an extremely elastic one, and the above proof suffices for all the special cases which I have enumerated*.
(56) I am indebted to Mr Ely for the following additional examples of Farey series, in the enlarged sense, which may interest some of my readers.
Ex. (1). x + y=or<20
19 ■"9 17 8 15 7 13 6 17 n 16
|
1 |
3 |
2 |
3 |
1 |
4 |
3 |
2 |
3 |
4 |
1 |
5 |
4 |
3 |
5 |
2 |
|
5 |
14 |
9 |
13 |
4 |
15 |
11 |
7 |
10 |
13 |
3 |
14 |
11 |
8 |
13 |
5 |
|
5 |
3 |
4 |
5 |
6 |
1 |
7 |
6 |
5 |
1 |
7 |
3 5 |
7 |
2 |
7 |
5 |
|
12 7 9 11 13 2 13 11 |
9 7 12 5 8 11 3 10 7 |
|
8 3 7 4 9 |
5 6 7 8 9 |
|
11 4 9 5 11 |
6 7 8 9 10' |
|
Ex. (2). x^ + y= or <20 |
|
|
1 112 12 12 12 |
1323 1323123 |
|
19 ■■'9 8 15 7 13 6 11 5 9 |
4 11 7 10 3857234 |
* Since the above was in type I have discovered the tme principle of Farey series, for which see Note H following the Exodion.
1] three Acts, an Interact and an Exodion 59
Ex. (3). y - Va: = or < 15
16 ■■• 8 15 7 13 6 11 16 5 14 9 13 17 4 15 11 7 17 To 13 17 3 17 14 11 8 13 5 17 12 7 17 0 11 13
AAlAA_LA^A^_L12?liA^l.
l5172 17 15l3TT9l6712T75 18 13 8 11
14 17 3 16 13 10 17 7 18 11 15 4 17 13 9 14
4 13^1451611613715817^10111213
5 16 11 17 6 19 13 7 15 8 17 9 19 10 11 12 13 14
14 15 16 17 18 1
15 16 17 18 19 1"
Exodion. On the Correspondence between certain Arrangements of Complex Numbers.
At which he wondred much and gan enquere What stately building durst so high extend Her lofty towres, unto the starry sphere.
Faerie Qtteene l. x. 56.
(57) Starting from the expansion in a series of &iX, multiplying in the usual notation both sides of the equation by
(l-9')(l-9*)(l-9«).... and intercalating the factors of tliis product between those of
(l-qz){l-q'z)...{l- qz-') (1 - 9'^-) - taken in alternate order, there results the equation
(1 - g2-')(l - qz){l - 9')(1 - 9*^')(1 - 9'^)(1 - (t) - = "S" {-yq''z\
and writing 9" in place of q and making ^ = + 9", Jacobi (Crelle, Vol. xxxil. p. 166) derives the identity
(1 ±9"-")(l ±9'»+^)(l-9»»)(l +9»»-™)(l-9»»+"')(l-g«»)...= S (+)'?"'■'+"'*.
— 00
3 1
From this equation, using the lower sign and making n = s . "* = s >
he observes, may be deduced Euler's expression in a series for
(l-q)(l-q^)(l-^)...,
and using the upper sign and making « = h > "' = s > another known series
"given by Gauss in the first volume of the Gottingen Commentaries for the years 1808-11."
60 A Constructive theory of Partitions, arranged in [1
It is not without interest, I think, to observe that by making w = ^ ,
m = 5 + e (where e is an infinitesimal), and using the lower sign we may
immediately deduce Jacobi's own celebrated postscript (so to say) to Euler's equation, namely,
(1 - 9)» (1 - q^r {I -(ff ... = "£ (-)< q~^^" -^ (1 - r')
— QO
= l-3g + 5r^- 1q^..., the general term being
which is (-y(2i+l)q ^ .
(58) It is obvious, that by the same right and within the same limits of legitimacy as the equation involving q, n, m (or if we please to say so in q, m) has been derived from the equation in {q, z), the equation in q, z may be recovered from the equation in q and to, if this latter can be shown to be true, morphologically interpreted for general values of m. I shall show that regarding m and n as absolutely general symbols, such as V(~ 1) '^'^ V^ or p or the quaternion units, or any other heterogeneous or homogeneous units we please, the equation in question which I shall write under the equivalent form
(1 + g«)(l + 3»)(1 - 50(1 + 5»+0(l + 5''+0(l -?=0- ='"2"(+)'9^''^^^''"*'
i = — ■»
[where c=a-\rh, and a, 6 are absolutely general symbols or species of units entirely independent of one another] does hold good as a morphological identity *. Thus interpreted, it amounts to a theorem in complex quantities, dealing with arrangements of three sorts of elements which I shall call Cs, £'s, ^'s respectively, meaning by a (7 any non-negative integer (that is zero or any positive integer) multiple of c, by a 5 such multiple augmented by a single h, and by an A such multiple augmented by a single a.
The C"s, the J5's and the ^'s in any such arrangement will be regarded as three separate series, the terms in each of which flow from left to right in descending order, that is the multiples of c which represent totally or with the exception of a single 6 or a single a, the terms in each such series taken in severalty are to form a continually decreasing series.
* This theorem is less transcendental than Newton's binomial theorem when the same latitude is given to the meaning of the symbols in either case: for (l + a;)"' = l + 7nx+ — = — x2+ ... does not admit of direct interpretation when m is a general symbol. The passage from numerical proximate equality to absolute identity, prepared but not perfected nor capable of being explained by infinitesimal gradation, brings to mind the analogous transfiguration of sensibility into sensation, or of sensation into consciousness, or of consciousness into thought.
1] three Acts, an Interact and an Exodion 61
The total number of elements and the number of Cs will be called the major and minor parameters respectively — the relation to the modulus 2 (that is the parity or imparity) of either one of them its character : and for brevity, the terms major and minor character will be used to signify the character of the major or minor parameter. The totality of all arrange- ments whatever of id's, B's, Cs iu which no element is repeated, will constitute the sphere of the investigation, limited only by the absence of what I term the exceptional or isolated arrangements, consisting exclusively of a series of consecutive B's ending in b, or of consecutive .4's ending in a. Within the prescribed sphere I shall prove that a process may be instituted for transforming any arrangement which shall satisfy the five following conditions :
(1) That it shall be capable of acting on every licit and unexceptional arrangement.
(2) That it shall transform it into another such arrangement.
(3) That operating once upon an arrangement, and then again upon the operate, it brings back the original arrangement.
(4) That it leaves the sum of the elements in the arrangement un- altered.
(.5) That it 'reverses each of its two characters*.
From (3) it will follow that all the arrangements within the prescribed sphere are associated in pairs, and from (1) that the sum of the elements in each such pair is the same. This being so, it is obvious from the fact of the parity of the total number of elements being opposite for any pair of associated arrangements, that in the development in a series of
(l-j«)(l-9»)(l-5')(l-g»+«)-,
no term will appear in which the index of q is other than the sum of the terms in one of the exceptional (we may now call them unconjugated or unconjugable) arrangements, and from the fact of the parity of the number of the Cs being opposite in any pair, the same will be true of the develop- ment in a series of
(1 -H 9«)(1 + 5»)(1 - y«)(l -t- r/»+«) ... .
As regards the coefficient in this latter series of any term whose index is
* It will presently be seen that all the licit and unexceptional arrangements will be divided into 3 classes and a specific operator be found for each class capable of acting on each arrange- ment of that class and converting it into another of the same class, and which will satisfy also the 3rd, 4th and 5'b of the ennmerated conditions. The total operator contemplated in the text may then be regarded as the sam of these specific ones, each of which, within its own sphere, will have to fulfil the five conditions of Catholicity, Homoeogenesis, Mutuality, Inertia and Enantiotropy (the last a word used in the school of Heraclitus to signify " the conversion of the primeval being into its opposite "). See Kant's Critique of Pure Reason by Max Miiller, Vol. I., p. 18.
62 A Constructive theory of Partitions, arranged in [1
the sum of the elements in an unconjugate arrangement it will manifestly be the number of ways in which the same complex number can be thrown under the form of a sum of the arithmetical series
a, a + c a + (i— l)c,
which is — - — c + ia,
that is o c + 5 (a — 6),
or of h, h + c, ... , h + {i—\)c,
which is , -jr c — jr (a — 6).
If
then —^r-a-\ ^r~ h = ^-~^ a -V
I z I i,
which necessitates i = j, and if
then -j-~ a + -^ b = ^'-^ a + =^-y^ b,
so that i' + i — {i' — i) = (j^ — j) — (j' +j) or i = —j.
Hence the general term is 9^ ^ , where i is an integer stretching from zero to infinity, and in like manner, and for the same reason, the
general term in the former series will be (— )'g^ * with the like in-
terpretation : or which is the same thing, comprising both cases in one and interpreting i to be integer stretching from — x to +00, the general term
wiUbeC+y^^"^^'""*'.
(59) The task before us then is to show the possibility of instituting, by actually instituting, a law of operation which shall satisfy the five preliminary conditions of catholicity, homoeogeriesis, reciprocity, reversal of characters and conservation of sum.
The following notation will be found greatly to conduce to clearness in effecting the needful separation into classes or species. A capital letter with a point above, as X, will be used to signify the greatest value, and with a point below, as X, the least value of any term in a series which that letter is used to denote. Z = 0, Z>0, X +Y = 0, X + Y>0 will signify respec- tively that there are no X's, that there are X's, that there are no A"s and
1] three Acts, an Interact and an Exodion 63
no F's, that there are either X's or F's or both in any arrangement under consideration. B's will be separated into 'B and B"s, or as we may write it B = 'BB', where 'B is the general name for all the B's, which beginning with the highest term B form an arithmetical series of which c is the common difference. If there is a gap of more than one c between B and the next lowest B, 'B is of course the single term B: B' ia any B which is not a 'B.
So again, Ai is any A which belongs to a series of A's forming an arith- metical series whose constant difference is c and lowest term a, so that unless 4= a, ^i = 0: any other A will be designated by lA. The signs of accent and point may of course be separate or combined : thus for example C will mean the smallest C in any given arrangement, B will mean the greatest B, A will mean the lowest A, ^A will mean the lowest of the lA's and A I the highest of the Ai's. Every 'B is necessarily greater than any B', and every jA than any 4,. If 'B — 6 = 0, this will indicate that all the B"s will form a consecutive series of terms (that is having a constant difference c) and ending in b, so that here B' = 0, that is there are no B's except those that belong to the regular arithmetical progression ending in b. If ,4 = 0, all the A'a will form an arithmetical progression ending in a. Thus we see that the arrangements belonging to the 1st terms (those that I have called exceptional) will consist of two species denoted respectively by
^A+B + C = 0 and {'B-b) + A + G = 0.
It may sometimes be found convenient to use a point to the left centre of a quantitative letter to signify that the quantity denoted is t« be increased, and a point to the right centre to signify that the quantity denoted is to be diminished, bye. Thus B- will mean ^ — c, and Ai will mean 4i + c, the first signifying the greatest B diminished by and the second the smallest 4, increased by c. When any general letter, say X, is wanting as indicated by the equation X = 0, X must be understood to mean zero. So for instance if -4 = 0, and consequently ,A = 0 and 4, = 0, i4 = 0. Again, when there is a gap between the highest B and the one that follows it in any arrangement, the arithmetical progression of 'B'a reduces as above remarked to a single term and there results 'B = 'B. It may be noticed also that always 'B = B, and 4i = A-
The arrangements which are comprised under the forms (a) A, A — c, A — 2c, ... , a, (y9) B. B-c, B-2c,..., b, may be regarded as belonging to what I shall term the first genus.
The second genus, namely that consisting of unexceptional combinations of unrepeated A'a, B'a, C'a, may then be divided into the following three species, the conditions by which they are severally distinguished being attached to each in its proper place.
64 A Constructive theory of Partitions, arranged in [1
1st Species. Conditions (7) 'B — b>0,
or (7') '^-6 = 0, C>0, C-c< = 'B-b. 2nd Species. (5) 'B -b = 0, A + C >0, C = 0 or C-o'B-b,
or (8') B = 0, C>0, A=0. or ,4 - a = > C 3rd Species, (e) B = 0, A>0, ,A+C>0, 0 = 0, or (J>,A-a.
Where it is to be understood that the conditions set out in the same line are simultaneous conditions. Thus for example the conditions of an arrange- ment being of the second species are when all the conditions of the upper or else all the conditions of the lower of the two lines written under that species are fulfilled: the conditions of the upper line (be it noticed) are that 'B is b, and that there are either some 4 's or some C'a, and that if there are some O's, C — o'B — b, and of the lower line, that there are no B's and some C's, and that if there are ^'s, A —(t=>G, and so for the interpretation of the conditions of the existence of each of the other two species.
To these (7) systems of conditions o, /3, 7, 7', S, S', e may be joined the trivial system (a) A = 0, B = 0, C = 0*; the (8) systems thus constituted will easily be seen to be mutually exclusive and between them to comprehend the entire sphere of possibility, leaving no space vacant to be occupied by any other hypothesis. I will now proceed to assign the operators <f}, -v/r, ^ appro- priate to the three species of the second genus.
Office of the Operator </>. ^ = '(f> + </>'.
When in Genus 2, Species 1, 0=0 or C — o'i — 'B, '(j) is to be per- formed, meaning that for each 'B, 'B- is to be substituted, and the inertia kept constant by forming a new C with the sum of the c's thus abstracted. In the contrary case <^' is to be performed, meaning that C is to be resolved into simple c's and as many of the 'B's, commencing with 'B and taken in regular order to be converted into '-B as are required to maintain the inertia constant, that is c is to be added to each B in succession, until all the c's which together make up C are absorbed.
Oj^ce of the Operator yfr. ^fr = 'y}r + yfr'.
When in Genus 2, Species 2, C = 0 or C > 'B + A, '"^l^ is to be performed, meaning that for 'B and A their sum is to be substituted, producing a C [which, on the second hypothesis, will be a new C]. In the contrary case -v/r' is to be performed, meaning that for C is to be substituted '-B (which will form a new 'B) and C—'-B which will form a new Ai-
* It would be perfectly logical, and indeed is necessary to regard the trivial case as belonging to the cases of exception, and then we might say that there are two genera, each containing three species, those of the first genus solitary, and those of the second, each of them comprising two sub-species, namely the sub-species subject to the action of the left-accented and that subject to the operation of the right-accented operators. The trivial species of the first genns consists of a single individual.
1] three Acts, an Interact and an ExocKon 65
Office of the Operate ^. ^ = '^ + ^'.
When C > 0 and C + Ai< ,4, '^ is to be performed, meaning that for C and Ai their sum is to be substituted, producing a new i4. In the contrary case ^' is to be performed, meaning that for i4, -4.1 forming a new 4, and i4 — 4i forming a new G are to be substituted.
(60) It will be seen that every species of the second genus consists of two contrary sub-species having opposite charactere, and it will presently appear that any arrangement belonging to one of these sub-species under the eflfect of its appropriate operator passes over into the other, which operated upon in its turn by its appropriate operator becomes identical with the original one, so that any two contrary sub-species may be said to be of equal extent : in fact if the sum of the parts is supposed to be given there will be as many arrangements in any sub-species as in its opposite, for each one will be conjugated with some one of the others.
It may not be amiss to call attention here to the fact that the scheme of classification adopted is, in a certain sense, artificial. Thus, for instance, it proceeds upon an arbitrary choice between which shall be regarded as the 4 and which as the B series, so that by an interchange of these letters a totally different correspondence would be brought about between the arrange- ments of the second genus, those of the first genus remaining unaltered. Nor is there any reason for supposing that these are the only two correspondences capable of being instituted between the arrangements of the second genus — in particular there is gieat reason to suspect that a symmetrical mode of procedure might be adopted, remaining unaffected by the interchange between 4 and B. As a simple example of the effect of interchange, applying the method here given, suppose 4 = 0, iJ = 0, a case belonging to the second species and that sub-species thereof to which •^' is applicable, and imagine further that the C series is monomial. Then C will be associated according to the scheme here given with b,G — b, but in the correlative scheme it would be associated with a, C — a.
(61) I need hardly say that so highly organized a scheme, although for the sake of brevity presented in a synthetical form, has not issued from the mind of its composer in a single gush, but is the result of an analytical process of continued residuation or successive heaping of exception upon exception in a manner dictated at each point in its deveIo.pment by the nature of the process and the resistance, so to say, of its subject-matter. The initial step (that applicable to species 7) is akin to the procedure applied by Mr F. Frdnklin to the pentagonal-number theorem of Euler, of which I shall have more to say presently. It will facilitate the comprehension of the scheme to take as an example the particular case where a and b represent actual and real quantities, say, to fix the ideas, 6 = 1, a = 2. Nothing, it will
8. IV. 6
66 A Constructive theory of Partitions, arranged in [1
be noticed, turns upon the fact of this specialization, which is adopted solely for the purpose of greater concision and to afford more ready insight into the modus operandi.
To illustrate the classes and laws of transformation consider (with 6 = 1, a = 2*, c = a + 6 = 3) all the arrangements, the sum of whose parts is 12, namely 12, 11.1. 10.2, 9.2.1, 8.4, 8.3.1, 7.5, 7.4.1, 7.3.2, 6.5.1, 6.4.2, 5.4.3, 5.4.2.1.
One of these, 7.4.1, belongs to the exceptional genus. The rest will be conjugated and fall into species in the manner shown below, where the first species means where the conditions (7) or (7'), the second that where (S) or (8'), and the third where the conditions (e) are satisfied. The C"s, B'b, A's are now numbers whose residues are 0, 1 or 2 in respect to the modulus 3. For greater clearness in each arrangement, numbers belonging to the same series are kept together, the law of descent only applying in this theory to elements belonging to the same series.
Species 1. 10. 2 3.7.2; 4.8 3.1.8; 7.5; 3.4.5; 6.4.2 6.3.1.2; 5.7 3.2.7.
Species 2. 9.1.2 9.3; 6.1.5 4.1.5.2;
Species 3. Caret.
Or again let the collection of arrangements be one in which the sum is 18. The partitions of 18 are 18 17.1 16.2 15.3 15.2.1 14.4 14.3.1 13.5 13.4.1 13.3.2 12.6 12.5.1 12.4.2 12.3.2.1 11.7 11.6.1 11.5.2 11.4.3 11.4.2.1 10.8 10.7.1 10.6.2 10.5.3 10.5.2.1 10.4.3.1 9.8.1 9.7.2 9.6.3 9.6.2.1 9.5.4 9.5.3.1 9.4.3.2 8.7.3 8.7.2.1 8.6.4 8.6.3.1 8.5.4.1 8.5.3.2 8.4.3.2.1 7.6.5 7.6.4.1 7.6.3.2 7.5.4.2 7.5.3.2.1 6.5.4.3 6.5.4.2.1. In this case there are no exceptional arrangements.
1st Species. 16.2 3.13.2; 4.14 3.1.14; 13.5 3.10.5; 13.4.1 3.10.4.1; 7.11 3.4.11; 10.8 3.7.8; 12.4.2 12.3.1.2; 10.7.1 6.7.4.1; 6.10.2 6.3.7.2; 10.1.5.2 3.7.1.5.2; 9.4.5 9.3.1.5; 6.7.5 6.3.4.5; 7.1.8.2 3.4.1.8.2; 6.4.8 6.3.1.8; 7.4.5.2 6.4.1.5.2;
2nd Species. 18 17.1; 15.3 15.1.2; 12.6 12.5.1; 6.1.11 4.1.11.2; 9.1.8 4.1.8.5; 9.7.2 9.3.4.2; 9.6^3 9.6.1.2; 11.5.2 3.8.5.2.
3rd Species. Caret.
If the partible number is 11, of which the partitions are 11 10 . 1 9.2 8.3 8.2.1 7.4 7.3.1 6.5 6.4.1 6.3.2 5.4.2 5.3.2.1, there will be no exceptional arrangements and the pairs of unexceptional ones will be as below.
* No use it will be seen is made of the accidental relation a=h + h.
1] three Acts, an Interact and an Exodion 67
1st Species. 10.1 3.7.1; 7.4 6.4.1; 4.5.2 3.1.5.2.
2nd Species. 3.81.8.2.
3rd Species. 11 9.2; 6.5 6.3.2.
By interchanging a and b, that is making a = 1, 6 = 2, the correspondence changes into the following :
1st Species. 11, 3 . 8 ; 6 . 3 . 2, 6 . 5 ; 8.2.1,3.5.2.1; 7 . 4, 6 . 4 . 1.
2nd Species. Caret.
3rd Species. 10. 1, 6 . 4.1 ; 7. 4, 3. 7 .1.
According to Mr Franklin's process the correspondence takes a form quite distinct from either of the above, namely 11, 10.1; 9.2, 8.2.1; 8.3, 7.3.1; 7 . 4, 6 . 4 . 1 ; 6.5, 5.4.2; 6.3.2, 5 . 3 . 2 . 1, all these arrange- ments constituting one single species.
A careful study of the preceding examples will suflBciently explain to the reader the ground of the divisions into species with their appropriate rules of transformation, and might almost supersede the necessity of a formal proof of the operator supplying the conditions of catholicity, homoeogenesis and mutuality ; from their very definition they are seen to comply with the other two essential conditions of inertia and enantiotropy.
Signifying by fi the total operator ^ + i/r + ^, it has been already remarked that n will in the general case have two values which only come together when a = h, or which is the same thing, each of them is 1 ; a special case of the special case when the complex reduces to simple numbers, namely, it is the case indicated in the well-known equation
■ — ao
But besides the two correspondences given by the two values of fl, if we take the actual (no longer a diagrammatic case) 6 = 2, a = l, we revert to Euler's theorem concerning the partitions of all pentagonal and non- pentagonal numbers, and can obtain by Dr Franklin's process, given in Art. (12), a totally different distribution into genera and species, namely the first genus instead of containing arrangements of the species
1, 4, 7,...3i-2; 2, 5, 8,...3i-l
will, as previously shown, consist of the very different arrangements (giving the same infinite series of numbers as those for other sums)
i, i-l-l,i-l-2,... 2i-l; i -f- 1, i -f- 2, i-(-3 ... ; 2t.
The character of each arrangement in the new solution depends in part on the relation to the modulus 2 of the whole number of parts and of the number of parts which are divisible by 3, so that we may divide the conjugate arrange-
5—2
68 A Constructive theory of Partitions, arranged in [1
ments into four groups* designated respectively by Oo, Oe ; Eo, Ee, using the capital letters to signify the oddness or evenness of the whole set of parts, and the small letters the same for the parts divisible by 3. There will thus be a cross classification of the arrangements of the second genus into groups over and above that into species, each species in fact consisting of four groups, which may be denoted as above, and of which Oo and Ee are one associative couple, and Oe, Eo the otheri".
(62) The following elegant investigation has been handed in to me by Arthur S. Hathaway, fellow and one of my hearers at the Johns Hopkins University, to which, although it does not exactly strike at the object of the constructive theory here expounded, I gladly give hospitality in these pages.
" The theorem to be proved is as follows :
1 + ea^ . 1 + ea^+* . 1 + €«;"+'* ... X 1+ ea^ . 1 + €0;*+'' . 1 + ea/'+^ ...
xl-aJ'.\-a?^.\-a?^...= 2 e«.a! « « ,
3= —00
where e' = 1 and h = a + b, a and b being any quantities whatever.
" The general term contains, say i exponents of a; selected from the first \ine,j from the second line, and k from the third line, namely
a + HoA, ...a + Oi_, A,
b + ^oh....b + ^j^^h,
'yih,...yth,
where a<|...ai_i, /9o'"/9j-i. yi---yk are respectively sets of i, j, k unequal integers arranged in ascending order, none representing a less integer than its subscript. This term is (remembering that h = a + b)
where
m = [(Oo + 1) + ... (ai-i+ 1)] + [/3„+ ... /3j_0 + [7, + ... 7^ (1)
n = K + ...ai_J + [(y8„+l) + ...(^,_, + l)] + [7,+ ...7i] (2)
* It will be seen later on that there is a division into sixteen groups analogous to the division into four groups first noticed by Prof. Cayley arising under the Franklin process.
+ The Oe and Eo conjugation has a very striking analogue in nature (as I am informed) in the existence of dissimilar hermaphrodite characters in two sorts of the wild English primrose and the American flower Spring-beauty or Quaker-lady — it being the law of nature that only those of different sorts can fertilize one another. Possibly the double symbolic character of Oo and Ee will justify or suggest the inquiry whether there may not be a latent duality in the unisexual specimens of such flowers as those just mentioned, where male and female are found codomiciled with the bisexual florets. There is also, it seems, a trace of analogy to the sparsely distributed unconjugate individuals of my first genus in Darwin's " complemental males."
1] three Acts, an Interact and an Exodion 69
In addition to these we obtain by subtraction
m — n = i— ^' = i+j mod 2. (3)
Whence (since e= = 1) e'-y = e™-".
"Thus all the above general terms having the same m and the same n divide themselves into positive and negative groups (corresponding to even and odd values of k), a term from one group cancelling a term from the other group. I propose to prove that the number of terms in each of these groups are equal, except when a certain relation exists between m and n, namely
(m-n)(m-n+l) . , ... .
m — ^ -^ = 0, (or m = 0 if m = n),
corresponding to which there is but one general term having the same m and the same n which falls into the positive group (^' = 0). This establishes the theorem in question, as we see by putting m — n = S.
" It is sufficient to consider (1) in connection with (3). In the first place the first two partitions in (1) may be converted by a (1 : 1) correspondence into an indefinite partition (bearing in mind (3)) with a decrease (vi — n> 0) in the sum or content of the integers by ^(m — n)(m— n+ 1), as follows: extend a, + 1 in a horizontal line of dots, and under the first dot extend /So in a vertical line of dots, thus fonning an elbow; in a similar manner form elbows out of o, + 1, /8i &c. until one of the partitions is exhausted ; this will be according to (3), the first or the second, according as wi< or > n, leaving in the inexhausted partition m — n integers ; place these elbows successively one without the other, and place on top (/« — n>0) horizontal lines of dots corresponding to the successive unmatched integers decreased respectively by 0, 1, ...(n — OT— 1) or 1, 2, ...(m — n), according as m< or >n; in either case the total decrease is ^ (m — n) (m— n+ 1). In other words, the above tri- partition of m has a (1 : 1) correspondence with a bi-partition of
(m — n)(m-n + l), .„ ,
m — ^ -, (or m if m = n),
consisting of an indefinite partition on one side and a partition of unrepeated integers on the other (71,... 7*). Such a bi-partition (on removing the line of demarcation) is an indefinite partition ; and, conversely, every indefinite partition involving 0 different integers gives rise as follows to (1 + 1)* such bi-partitions, the number of those involving even and odd values of k being respectively the positive and negative parts of the expansion of (1 — 1)', which are equal: namely, first, the indefinite partition itself (A = 0); second, the 6 bi-partitions obtained by placing each of the 0 integers successively on the k side (i = 1) ; third, the i^ 6{d — 1) bi-partitions obtained by placing the \0{0 — l) pairs of the 0 integers successively on the k side (k = 2), and so on.
70 A Constrttctive theory of Partitions, arranged in [1
The only exception to this equality of the number of partitions for even and odd values of k is when the partible number,
(m — ra) (m — n + 1 ) m — -^^ or m,
is zero, for which case there is but one bi-partition [0] + [0] {k = 0). Q.E.D. The tri-partition of in corresponding to the celibate case reduces to the natural sequence above subtracted whose content is
(m-»)(m-n + l)^^^Q^^
it
which is the second or the first partition (according as m < or > w), the others being wanting."
(63) The same infinitesimal method which applied to the expansion of %^x gives rise as was shown to the expression for the cubes of the successive rational binomial functions may be applied to the development of
(1 + aa;) (1 + cw^) (1 -^aa?)...
given in Art. (35), but will not lead to any new result. Making a — — a;~'~*, where e is infinitesimal, we obtain from the general theorem
{\- af){\-x){\- a?){\- og^) ...
, 1 — «• 1— a;'.l-a; . \— af.\— x.Y-a? „ \-x \-x.\-ii? \-x.\-x^.\-a?
1-a;' . X—al'.X—x „
or (l-^)(i_^)(i-^)... = i-^^-^ + ^...
= 1 - a;(l + a;) + ar^ (1 + ar') . .. ,
the same equation as results from writing a = — 1.
To arrive at any new result it would be necessary to have recourse to processes of differentiation ; the above calculation serves, however, as a verification if any were needed of the accuracy of the theorem to which it refers.
(64) Since sending what precedes to press I have thought it would be desirable in the interest of sound logic to set out the marks or conditions of the several species of the arrangements of unrepeated A, B, C's, somewhat more fully and explicitly than before. And first, I may observe that since it has been convenient to understand that when there are no X terms X shall signify zero, the quantitative equation X —0 dispenses with the necessity of
i
1] three Acts, an Interact and an Exodion 71
using the symbolical one X = 0, and in like manner X > 0 supersedes the symbolical inequality X > 0, and, of course, the same remark extends to the equality or inequality X + F = or > 0.
We have then for what I shall term the first, second and third species of genus 1, the conditions
C' + 5 + 4=0, CVB^A=h, (7 + 5 + 41 = 0
respectively — the first, the trivial case of vacuous content ; the second, of only a complete natural B progression, that is, one ending with h (the minimum value of B), and the third, the same for A similarly ending with the minimum a. In what follows the conditions in each separate line are to be understood to be not disjunctive but simultaneous or accumulative ; they of course refer to the species of the second genus. Marks of species (1) (a) ^ — 6 > 0,
or (yS) S-6 = 0, 'B-'B = >C-c, C>0. „ (2) (a) 5-6 = 0, C-c>'6-'B,
or (/9) B-h = 0, (7=0 [4 >0],
or (7) B = 0, A-a = >C, C>0.
or (B) B = 0, 4 = 0 [(7>0].
„ (3) (a) 5 = 0, C>A-a, 4>0, or (/9) 5 = 0, (7=0[,4-a>0].
The three inequalities included in brackets are only required in order to exclude arrangements belonging to the first genus. Leaving these out of account for the moment, merely for the sake of greater concision of state- ment, it is easy to see by mere inspection of the above table that the three species are mutually exclusive and share between them the total sphere of possibility, for (1) a exhausts the hypothesis of there being other 5s besides those forming a complete natural progression, (1) /3 and (2) a of the 5's forming such progression when there are existent C'b, and (2) /3 when there are not. Also ((2) 7, (2) 8), (3) a exhaust between them the hypothesis of there being no 5's when there are some existent C'a, and (3) y3 of neither 5*8 nor C's appearing in an arrangement.
Thus all unexceptional arrangements must bear the marks occurring in one or the other of the first four lines of the table, and all those where no 5*8 occur, either of the last line when there are neither 5's nor C's, and of the three preceding ones when there are no 5'8 but some C's, and the total sum of these hypotheses plus the hypothesis of the first genus together make up necessity, as was to be shown.
The convention X = 0 when an arrangement contains no X with the consequent reduction of the conditions to a purely quantitative form has lent
72 A Constructive theory of Partitions, arranged in [1
itself very advantageously to the above bird's-eye view of the completeness of the scheme (as covering the whole ground of possibility) ; it also will be found to simplify the expression of the proof. I did not employ it until the necessity for so doing forced itself upon my notice, for a very obvious reason, namely that X is a J9 (or an A), which is defined to be congruous to 6 (or a) [mod c], which zero is not: there is thus an apparent paralogism in ad- mitting that any X of these two where there is a B (or when there is an ^) is congruent to b (or to a), but that when there is no B (or no ^) then the conventional least B (or ^) is zero. It will be seen, however, ex post facto, that no inconvenience in working the scheme results from this extended definition which constitutes an important gain to the perfect evolution of the method. It is usually in the form of some apparent contradiction or paradox that a scientific advance makes its first appearance.
(65) Aided by this clearer and fuller expression of the definitions of the genera and species, I will now set out a logical proof that the respective operators fulfil the three additional necessary conditions. I may observe preliminarily that the Greek letterings a, /9 ; o, /S, 7, S ; a, ^, do not express sub-species, for one distinguishing mark of species (or sub-species) may be taken to be that conjugation cannot take place except between individuals of the same species or sub-species, but it will be presently seen that individuals belonging to the differently lettered divisions of the above species are susceptible of mutual conjugation — and are therefore in conformity with biological precedent to be regarded as mere varieties. Besides these varieties of each of the species there is another entirely different principle of cross classification applicable to each of them, namely in general an arrangement must belong to one of sixteen groups designated by combining together one out of each of the four pairs of opposite symbols X, G ; x, c; 0, E ; o, e, where the large 0, E refer to the oddness or evenness of the major, and the small 0, e to the same for the minor parameter ; and in like manner the large X and large G to the result of the operation appropriate to any arrangement, being to extend or contract the major, and x, c to extend or contract the minor parameter. There are thus eight pairs of groups, and conjugation can only take place between individuals belonging to the same pair.
The pairs are as follows :
fXxOo\ fXxOe\ fXxEo\ fXxEe
\GcEeJ' [GcEoJ
fXxEo\ /XxEe\ \GcOe)' [CcOoJ'
, fXcOo\ fXcOe\ fXcEo\ (XcEe\
KcxEel' [CxEeJ' [CxOeJ' \GxOo)'
Species (1) and species (3) it will be seen may each be separately divided
into four sub-species denoted by the upper four, and species (2) into the four
sub-species denoted by the lower four paii"s of combined characters, so that
there will be in all twelve (and not as might at first be supposed twenty-four)
1] three Acts, an Interact and an Exodion 73
sub-species of conjugable arrangements. The different sub-species of the same species do not admit of cross-conjugation ; it is the property which they have in common of being subject to the same law of transformation when passage is made from an individual to its conjugate, which binds them together into a single species. In the arrangements peculiar to Euler's problem, we see that there was no division of the second genus at the outset, but that a separation would be made of it into two pairs of groups with con- jugation possible only between individuals belonging to the same pair, and consequently there may be said in this case to be two species of the second genus, analogous, however, not to the species but the sub-species in the more general theory. The final separation of a pair of groups into its component elements has nothing to do with the concept of species, sub-species or variety, but may be regarded as similar to the separation of the sexes.
In what follows, a bracket enclosing a letter will be used to denote that it belongs to an arrangement after it has been operated upon by its appropriate operator, or what may be called its operate.
Species (1). When B -h>0,\i C -o'B -'B or C =0,'4> may be per- formed, giving [fT] ='B — 'B+C <C so that the law of descending magnitude is maintained ; we have then [B] — ['B] = or > B — 'B = > [C] — c ; hence <^' has to be performed and will obviously restore the original arrangement. Again if in the original arrangement B — 'B = >C—c and C>0, ^' has to be applied ; a resolution of C can take place into c's and the Q/c first 'B'a, and will each be increased by c and [B] — '[B] = C— c, so that either [C] = 0 or [6']-c<C'— c< [£] — '[/i], and '<f> being applicable to the new arrange- ment will convert it back to the original one.
First Species (/8). When 5-6=0 and B-'B=>C-c and (7>0, </>' can be performed, and the new arrangement as before may be operated upon by <j>' and so brought back to its original value. If (7 = 0 or C—oB — 'B, '(f> could not be performed, for then B = b and has no c to part with to help make up [C].
These two hypotheses belong to Species (2), which we will now proceed to consider throughout its full extent. When B-b=0, then 'B = b, and I shall first suppose [(a) and (/3)] that (7=0 or Q-oB-h. When (7 = 0 or B-\-4>C, then > will be applicable, making [C] = B+4; if now [B]>0 and [S]>0,[B]+[4]=>{B-c)+i4+c) = >B + 4 = >[C], and
[C]- c= 6 + 4 -c = [B] + 4 >[B]-b.
Hence we are still within Species 2 and have fallen upon the case to which the reversing operator yjr' has to be applied. If [£] = 0, [4] =0 we must have B [C] > 0, inasmuch as the original content (or inertia) is originally greater than zero and is kept constant, and this is a case which still belongs to Species 2 and falls under the operation of y^'.
74 A Constructive theory of Partitions, arranged in [1
If [B] = 0 so that j& = £ = 6 and [4] > 0, then
[4]-a = >4+c-a=>4+j6 = >C', which also falls within the second species and is amenable to the reversing operator i/r'.
Finally, if {E\ > 0, that is 5 - 6 = 0 and [4] = 0, [(7]-c = 5 + 4-c = >[^]-6, that is = > [E] — 'B, and we are still within Species (2) and in the case amenable to the reversing operator i^'.
If now on the other hand we begin with an arrangement of the second species in the case amenable to t/t' we must suppose either 5 = 0 or 4 = 0, or else C>0 and C<=B + A.
Take first this last supposition. The operation of i/r' gives \C] = > (7 + c, [B] = B-\-c and [A^^G - c- B>B-h-B>-h = >c-h=> a. And [B1 + [4.] = B-\-G-B=C<[Gl
[C]- c = >{C- c) + c=> B -h + c = >[B]-[B]. Hence the operate is licit, belongs to the second species and is amenable to the reversing operator 'y^.
If ^=0 and 4 = 0, [B] = [E\ = h and [4] = (7-6 and [C] = Oox>G. If [0] = 0 since [4] > 0, the operate is included in variety (/3) of the second species and amenable to the reversing operator 'i/r, and if
[(7]>C'[C'-c]>C7-c>0, that is > [E] — B which belongs to variety (a) of the second species ; and since [(7] > (7 > [-B] + [4] is amenable to the reversing operator 'i/r.
If jB > 0 and 4 = 0, then (7 > 0 [otherwise it would be an arrangement in Genus 1, Species 2] [(7] = 0 or > (7, [S] = 5 + c,
[4] = (7-[-B]>(c + £-6)-(c + £) = >a, and either [(7] = 0 and [4]>0 or
[G]-c>{G-c)+c>B+c-b>[B]-'B and [4] + [-B] = (7 > [(7]. Hence in either hypothesis the operate is still in Species (2) and amenable to the reversing operator 'i/r.
Lastly, if£ = 0, 4 — a = >C7 and (7 > 0, the arrangement is amenable to the operator -^^ which will make [5] = 6, [4] = (7— 6 < C + a < 4- We have then [£] -6 = 0 and [(7] = 0, and consequently also 4 > 0 or
[G^-OC-OO, that is > [B] — '[£], and the result is still contained within Species (2) and is amenable to the reversing operator 'i/r.
(66) The following are examples of paired arrangements belonging to" the first species, adapted to the case of a = 2, 6 = 1. The G and B terms are
1]
three Acts, an Interact and an Exodion
75
expressed ; the A line is the same for each of any pair of this species, and may be filled in at will.
"^'lie.is.io.FpiiD.ie.is.Fj
where X, Y represent any licit series of Cs and B'a respectively. jZ.9 ] jZ.9.6. \ ,,(Z.9 Ui^-
116.13.7.FJ (13.10.7. F( ^ ] 16. 13. 10. 4 J 119.16.13.4, ^,|X.9 ) fZ
'4>
,4.1
*1:
.} = {
10.7.4 16.7.4
) = {:
={7.4.1}
10.7.4
The following are examples of paired arrangements of the second species with a = 2 and 6 = 1 as usual.
fZ.12.| /Z.12.9.) (Z.12.
». >-! 7.4.1.^ = -! 4.1 \ V-17.4.1.
F.2
>{V%'1-(^J-)
z
r
We come now to the third species. Here, I think, the reader will find it a great relief to the strain upon his attention if I invite him before attacking the demonstration to consider the annexed diagrammatic cases accommo- dated to the supposition a = 2, 6 = 1. The 5*8 it will be remembered in this species do not exist, and the action neither of '^ nor ^' introduces any B into the transformed arrangement. In the examples given below the G and A terms occupy the higher and lower lines respectively — the comma is used in the latter to mark off the ,.4's from the il/s.
J 9.6. 1 9.6.3. ,f 6.3. 1 6.
'[14.11.8.5, J 14.11.8,2 |l4.11.8, 2) 14.11.8.5,
') = 17.'8;2 '^("•»-«') = 17.'8:2
^(l7.8.5.2) = ^jj\-.^ '^(l7.14,8.5.2) = ^,_j^%^ 2 9. ^, f 12.9.3. ) 12.9.
-^
(n.
8.5
' ,2 ^ 1 , 11.8.5.2j~14, f 9.6.3. ) 9.6
1 , 11.8.5. 2J 14, 8. 5, 2"
8.5.2
76 A Constructive theory of Partitions, arranged in [1
The left-hand accent is used here as elsewhere to signify that phase of the operator which brings about an increase and the right-hand one a decrease in the number of Cs. It will readily be seen that the action of the operator in each of the above examples prepares the arrangement for the action of the contrary one which will restore it to its original value. It is worthy of notice that in any two associated arrangements above, an a (here 2) may appear in each and mvLst appear in one of them. I will now proceed to the general demonstration.
(67) Let us first suppose 4i = 0, then i4 > 0, otherwise we shall be dealing with the antecedent species and '^ will be applicable, making [4] = [-^i] = a [(7] — 4-—0'<Q and > (4 — «)■ Thus the generated arrange- ment is licit and belongs still to the third species; but now [(7] + [4,] = 4 and [i4] = 0 > 4- Hence the reversing operator ^' is applicable to the new arrangement ; the remaining cases to consider (in which 4 = a for the arrangement as well before as after being operated upon) may be separated into those where C. > 0, and at the same time either 0 -(- 4j < ,4 or i4 = 0, which are amenable to the operator ^' and the complementary cases which are amenable to '^.
>
In the cases first considered [4,]=4i — c, [i4] = (7 = 41 ^[C*] + 0 or C (and d, fortiori >0), consequently the new arrangement is licit and still belongs to the third species, and since either [C] = 0 or else
and [,4] > 0, it is one of the complementary cases and is subject to the reversing operator '^.
Again, any arrangement for which 4 = a belonging to the complementary cases is defined by the conditions i4 > 0 and (7 + 4i = > i4 and is by hypothesis to be subjected to the operator '^ which will make [4i] = 4, -1- c, [i4] = 0 or > i4 [(7] = i4 - ^1 - c, and since (7 = > i4 — 4i , [(7] < (7, so that the operation leads to a licit new arrangement.
Also [C] + [4i] = i4. and consequently either [,4] = 0 or [C + 4,] < [i4], which is a condition belonging to the first considered class of cases, subject to the reversing operator ^', and thus for the third as for both the antecedent species of the second genus, it has been proved that each designated operator prior to any arrangement being performed does not take away its licit character nor carry it out of the species to which it belongs, and on being repeated brings it back to its original form, and that the eflFect of any single operation is to maintain the content (or inertia) of the arrangement constant but to reverse each of its characters. This is the thing that was to be proved and brings my wearisome but indispensable task to an end.
1] three Acts, an Interact and an Exodion 77
(68) Another and perhaps somewhat clearer image of the classification of the numbers of the second Genus may be presented as follows : The com- binations of the characters XGOExcoe give rise to eight pairs of groups, say eight classes. Of these classes four belong to Species 2, and may be repre- sented by four indefinite vertical parallelograms, set side to side, and sub- divided each of them into four, (say) black, white, grey and tawny stripes, corresponding to the four varieties of the second species. The other four classes may be similarly represented by four such parallelograms as before, but separated by a transverse horizontal line into eight sub-classes, four corresponding to the first species and four to the second. The upper parallelograms may then be each divided into blue and green, the lower into yellow and red stripes to represent the respective couples of varieties of the first and third species. There will thus be in all thirty-two stripes, namely four blue, green, yellow and red, and four black, white, grey and tawny, each of which is bifid, representing two groups of opposite sexual characters, which may be fittingly represented by the upper and under sides of the sixteen unlimited single-coloured stripes of the first and the eight unlimited double-coloured stripes of the second set of parallelograms.
The above logical scheme is not intended to convey any notion of the relative frequency of the three species. The general case is that of the first species. The second is conditioned by 'i?= 6 or -S =0, and the third by 5= 0. When 'B = b it is about an even chance whether the arrangement is of the second or first species, and when 5 = 0 of the second or third. Either equality is a particularization of the B series, the latter signifying that there are no B'a in the arrangement, the former that there are B'a descending in rational progression down to b : this supposition is apparently infinitely more general than the former, because there is no limit to the number of terms in the progression, and the case of a natural progression of B's of the kind men- tioned with any given number of terms as regards the probability of its occurring in an arrangement seems to be on a par with the case of the jB's being all wanting. Hence the first species is infinitely more frequent than the second, and the second than the third. According to Prof. Max MuUer's theory of the relation of thought to language (if I interpret it rightly) I ought to have thought out my divi.sions and schemes of operation in language, but I certainly had formed in my mind a dim abstract of them before I had found the language that was competent to give them expression.
In conclusion, I may remark that whilst the experience of the past indi- cated the probability that there did exist (if one could find it) a method of di.stributing the arrangements of the second genus into pairs, in sUch a way that in each pair the total or partial character should be reversed in passing from the one to the other, there was nothing to induce a reasonable degree of assurance that both those characters should be found simultaneously reversed
78 A Constructive theory of Partitions, arranged in [1
in one and the same distribution ; for aught that could have been foreseen to the contrary, it might very well have happened that one mode of distribution might have been needed to prove Jacobi's theorem for the case of only negative signs appearing in the factors on the left-hand side of the equation, and a dififerent one for the other case where only every third factor contains such sign — indeed upon the principle of divide et impera or doing one thing at a time (as invaluable a maxim to the algebraist as to the politician) I had completed the proof for the former case without thinking of the latter, and only when on the point of attacking it was agreeably surprised to find that there was nothing left to be done, for that the proof found for the one extended to the other — in familiar phrase, I had hit two birds with one stone. We may now ask whether this was a happily found chance solution or was predestined by the nature of things, and that simple necessarily implies double enantiotropy of conjugation. Probably I think not, and if so, a question arises as to the number of solutions for each of the two sorts of enantiotropy and whether the number of each kind of simply-enantiotropic conjugations is the same.
Viewed merely as a question of direct multiplication, I think it must be allowed that what I have here called Jacobi's theorem (including Euler's marvellous one, as the ocean a drop of water) is the most surprising revela- tion that has been made in elementary algebra since the discovery of the general binomial theorem, and that the space devoted to its independent, and so to say, materialistic proof in these pages, although considerable, is not out of proportion to its intrinsic importance.
H. Intuitional Exegesis of Generalized Farey Series*.
(69) The demands of the press will only admit of a rapid sketch of what appears to me to be the true underlying principles of the theory initiated by Farey, honoured by the notice of Cauchy, and to a certain extent generalized by Mr Glaisher, whose inductive method in the cases treated by him finds its full development in the method of continuous change of boundary, explained in the course of what follows. Let us start from the conception of an infinite cross-grating formed by two orthogonal systems of parallel lines in a plane, the distance between any two parallels being made equal to unity. The intersections of any two lines of the grating may, as heretofore, be termed nodes. A triangle which has nodes at its apices and at no other point on or within its periphery, may be termed an elementary triangle, and the double of the area of any such triangle will be unity. If any finite aggregate of nodes be given it must be possible to pick out a certain number of them which may be formed together by right lines so as to form a sort of ring- fence, within which all the rest are included : the area thus formed, if it * Continued from note G, Interact, Part 2.
1] three Acts, an Interact and an Exodion 79
admits of being mapped out into elementary triangles, may be termed a complete nodal aggregate. Any other contour consisting of lines of any form (curved or straight) drawn outside of this ring-fence in such a manner that no nodes occur between the two, may be termed a regular contour.
If any node 0 be taken as origin and any nodal lines through 0 as axes of coordinates, and if 'A, A' are the nearest nodes to 0 in the radial lines on which they lie, and if no nodes of the given aggregate are passed over as an indefinite line rotating round 0, passes from one of these radial lines to the other, '.40.4 is an elementary triangle, and if 'p, 'q ; p, q be the coordinates of 'A, A respectively, 'pq — p'q = e where e is + 1 or — 1 but is fixed in sign when the direction of the rotation is given.
When the aggregate is complete, if the values of the coordinates of the successive points passed over by the rotating line be called . . . "p, "q ; 'p, 'q ; p, q ; p', q', p", q" ; ... ,vte shall have a Farey series formed by the successive coifples p, q, that is p" 'q — p'q" = e; p'q—pq' = e; pq' —p'q = e ... . Thus we see that the Farey property is invariantive in the sense of being inde- pendent of the position of the origin.
Next I say, that if any contour to a given aggregate is regular, every contour similar thereto in respect to any node of the aggregate regarded as the centre of similitude is also regular, provided the boundary is simple; meaning that there are no interior limiting lines giving rise to holes or perforations in aggregate, and no loops formed by the boundary cutting itself.
In the above figure 'BOB is any triangle whose sides are bisected in 'A, A, A'. Suppose 0 to be the origin, 'A, A' two nodes of greatest proximity to 0 successively passed over by the rotating line for a given
80 A Constructive tJieory of Partitions, arranged in [1
contour. As this contour expands uniformly in all directions through 0, the line 'AA' remains parallel to itself. Since 'AOA' is an elementary triangle so also must the similar triangles 'AAA', A'AB', 'AA'B be all elementary, consequently A will be the first new node intervening between 'A, A' brought into the enlarged aggregate as 'AA' moves continuously parallel to itself, and AOA, AOA' will be elementary triangles ; it may be noticed in order to bring this method into relation with that indicated by Mr Glaisher, that the coordinates of this new node A are the sums of the coordinates of its neighbours 'A, A'. If the contour were not supposed to be simple, this condition could not be drawn ; for if there were a hole round the middle point of 'AA' the node A would be missing in the enlarged aggregate, and if the first node to intervene as the contour went on enlarging be called (4), 'AO{A) or {A)OA' or each of them would be a multiple of the elementary triangle, so that the constancy of the value of the successive determinants would no longer hold. In like manner it will be seen that on the same supposition as above made, if in consequence of the contour contracting about 0 as the centre of similitude, two points 'A, A' which originally are non- contiguous, at any moment become contiguous, at the moment previous to this taking place A (and no other point) must have intervened, and after A has disappeared from the reduced aggregate, no other point can make its appearance between 'A, A'.
(70) Hence we may contract at pleasure the given contour about any node as origin, and if the contour so contracted contains at least one node besides the origin, it will suffice to determine whether the given contour is or is not regular.
Thus for example in the case of a triangle limited by the axes and by the right line a; + y = n, we may make w = 1 and the trial series will then become
:r Y K which possesses the Farey property. Hence this will hold good for
a triangular boundary of any size and wherever the origin is situated : this includes the case of the ordinary Farey series when the origin is taken at either extremity of the hypothenuse. So again for the area contained within the axes and the hyperbola xy = n, we may take xy = \ and the trial series is the same as before.
(71) It is easy to form unperforated areas of any magnitude which shall
not satisfy the Farey law : for example we may as in the annexed figure
draw a curve passing through the origin, the point (0, 1), and the point (2, 3),
0 2
-, - does not satisfy the Farey law, and consequently no similar contour
X o
obtained by treating any one of the three nodes which it contains as a centre of similitude will be a " complete contour," and the successive values of (p, q)
1] three Acts, an Interact and an Exodion 81
obtained by the rotation of a line round the origin in such contour will not constitute a Farey series.
The theory will, I believe, admit of being extended to solid reticulations, formed by the intersections of three systems of equidistant parallel planes, determinants of the third order between the three coordinates of successive points, replacing the pq —p'q of the plane theory. The chief difference will consist in the introduction of a new element in the multiplicity of the " normal orders " in which a given set (of points in a plane or) of radii in solido may be taken. (Points in a plane arranged in any order of sequence, such that the successive determinants formed by their trilinear coordinates are of uniform sign, are said to be in a normal order. Rays of a conical pencil arranged in any order of sequence, such that their intersections by a plane satisfy the above condition, are also said to be in a normal order : see privately printed syllabus* of my lectures on Partitions, 1859, or M. Halphen's theory of Aspects.) But as far as I can see this will in no way militate against the existence of the laws of invariance and similitude established for the case of a plane reticulation, but will only introduce a further principle of invariance, namely that the law of unit-determinants if satisfied by one normal arrangement of the points of the solid reticulation will be satisfied by every other.
APPENDIXt.
LIST OF CORRECTIONS SUGGESTED BY M. JENKINS TO PROFESSOR SYLVESTER'S CONSTRUCTIVE THEORY OF PARTITIONS.
Page 5, 5 lines from end, 2« — (t + 3) should be n — (t + 3). „ 6, between 2nd and 3rd rows of sinister table insert 13.2.0. 7th and 8th „ „ „ „ 11.2.2.
„ in 6th row of dexter table, for 8 . 4 . 3 (2) write 8.4.3 (1). „ 11, line 8 from the end, interchange protraction and contraction so as to read "contraction could not now be applied to A' and B' nor protraction to C." „ 13, line 25. If /(a;) = (l-a;)(l -a:»)(l-a»)(l -a;')(l -a*), for the second a* read «*.
[* Vol. IX. of this Reprint, p. 119.]
[t These corrections have been included in those made in the text preceding.]
8. IV. 6
82 A Constructive theory of Partitions, arranged in [1
Page 13, line 29, for " latter " read " former," „ 15, line 11 from end, for l' read l\ „ 20, line 4, for 1 + 2 read i + 2. „ „ line 5, for 1 + 2 read i + 2.
„ 22, line 11, for XjX « read XjX « .
„ „ line 20, for " the minimum negative residue of i — \ " read i + 1.
„ 25, line 7, for j-^n^ ""^^ i_^ •
„ „ line 4 from the end, for " to the 5th now " read " to the 5th row
now." „ 27, line 15, for 15, 7, 3 read 13,. 11, 3. „ „ line 19, for (1 + ax) (1 — aa^) (1 —ax^) ... read (1 +ax)(l + aaf)...(l+ aa^').
„ „ line 22, for ^ a read , a.
1—x l—x"
„ „ line' 30, for " angle whose nodes contain i nodes " read whose sides.
„ 28, line 5, for " with j — i or fewer parts " read j — 1.
„ line 12, for 1 + — af + — —— ic-+i etc.
1 — ar \ — X .i. — ap
, ^ l-a^-' ^,, l-a;"-'.!-**"-' ^^
read of + -, - x"+^ + — — ; — «»+* + etc.
\—a? 1— ar'.l— a^
If in the expression in line 9, namely in
1 _a;^-!!?+2 . 1 _ a^"-!!?+< ... 1 _ aJit-a
\-3? . 1 -a^... \-x'i-''
we put J = 3 we^obtain
g.fl--i + 1X
l-ar-.l-a:* 1-a^.l-ar* '
l-a;»-M-«"-'
\-a?.\-a^ ' since to = 2i— 1, and similarly for other terms when we put.;" = 2 and j = 1.
The correction which I offer seems to me to be right, and the expression in the paper to give a wrong result in the cai^e when n happens to be equal to w + 2 ; for then the number of parts being supposed to be exactly i, the first bend contains 1i — 1 or to nodes, and there is then no way of placing the remaining 2 nodes so as to make the partition a conjugate partition — sup- posing I have not misunderstood the article. Page 29, line 8, for 19, 7, 6, 6 read 10, 7, 6, 6. „ „ figure, either insert a node at junction of 5th column and 7th row
or remove a node from junction of 7th column and 5th row. „ „ lines 7 and 8 from the bottom, if we remove a node from the figure no change is required in these two lines; but if we
1] three Acts, an Interact and an Exodion 83
insert a node in the figure, then 111111733 should be 11 11 11 7 5 3 and 5 5 5 3 1 1 should be 5 5 5 3 2 1.
Page 31, line 15 from end, after ; ; -r insert "or of
° \—ax.\—aa?...\ — aa?
x" ai'.' „ 34, line 7, for a) read a*. „ „ line 8, for (x» + aa;'«)} read (a? + x^).
I., 36. line 8. for ^' ^^ 7 -^> read ^-(|-^). „ 37, line 4, for a:" read a;^, „ line 7, forar^+' read 0,'^+^ „ 40, line 6, a,— i is, I believe, the right final term; but it appears as if it were the first of a pair instead of the last of a pair,
»ai~i being a quantity which may vanish. If the pair of expressions which in the text precede a<— i, if definitely expressed and not left to be understood, should be
[ai_, + «(_,- (2i - 3)], [tti-, + a._, - (2i - 2)], and not as in the text
[a.-_, + ai_, - (2i - 1)], [a.-_, + o.- - 2i], the factor which should precede Oi—i is [aj + o, — (2i — 1)].
I do not quite follow lines 9 — 13 of p. 40, possibly from the oversight in the subscripts I do not see what is intended. But it seems to me the following proof would be right :
The expressions of the same form succeeding a, + a, — 1 and Ui + 0.^—2 must be continued so long as they are positive, and must be rejected when they become negative.
Now from the fact of i being the content of the side of the square belong- ing to the transverse graph a,- = or > i, a, ■= or > i, therefore Oj + a< — (2t — 1 ) is positive and is therefore one of the terms of the series. Also aj+, = or < i and tti+i = or <i, therefore af+, + Oj^., — (2i + 1) is negative and must conse- quently be rejected.
The intermediate expression is a. + Oi+i — 2i; and for this we may in all eases put o^ — i as the last term of the series for the following reason :
If the extreme inside bend have more than one node in the row, then «,+, = i and Oj + a,+, — 2i is = aj — i, which is not negative since a,- = or > i. If the extreme inside bend degenerate, so that it consists only of a vertical I line or of a single point, then 04 = 1; and since a,+, <i in this case, therefore a, + o,+, — 2i is negative and inadmissible as a term in the series; but since Oj — 1 = 0 there is no harm in putting it as the final term in the series.
Page 601, Vol. in. of this Reprint, line 6 from the end, for 3100 read 3110.
6—2
2.
SUR LES NOMBRES DE FRACTIONS ORDINAIRES IN^GALES QU'ON PEUT EXPRIMER EN SE SERVANT DE CHIFFRES QUI N'EXCEDENT PAS UN NOMBRE DONN^^.
[Comptes Rendus, xcvi. (1883), pp. 409—413.]
Dans le Philosophical Magazine, 1881, p. 175, M. Airy, associ^ etranger de rinstitut, annonce qu'il a calculi, pour I'usage de I'lnstitution of civil Engineers, k Londres, les valeurs logarithmiques de toutes les fractions
ordinaires — , dans lesquelles m et n ne contiennent nul facteur commiin et
n'excedent pas 100, arrangees dans Tordre de leurs grandeurs, et que le nombre de ces fractions est 3043.
Je vais montrer qu'on peut appliquer la methode dont M. Tchebycheff' s'est servi dans sa thdorie cdlfebre sur les nombres premiers, avec I'addition que j'y ai faite*, pour trouver des limites supt?rieures et inf^rieures au nombre d'un systfeme pareil de fonctions quand la limite des valeurs de m et de n est un nombre quelconque donnd
1. Je dis que si Ti signifie le nombre de nombres inf^rieurs et premiers k i, nombre entier (ce que nous nommons, a Baltimore, le totient de i), on aura I'identit^
r=.x\ r J 2
C'est une consequence du th^oreme plus g^ndral que "si aj, a^, ..., a,- sont des nombres entiers quelconques, et si Ton nomme le nombre des a qui contiennent r la frequence de r par rapport au systeme des a, et qu'on prenne le produit de la frequence de r par son totient, la somme de ces produits (quand r prend toutes les valeurs de 1 jusqu'a I'infini) sera la somme des a."
* Voir American Journal of Mathematics. [Vol. iii. of this Eeprint, pp. 530, (iOo, 672.]
2]
Fractions orcUiiaires inegales
85
2. Nommons Jx la somme-totient de x, c'est-a-dire la somme des totients de tous les nombres qui n'excedent pas la valeur de Ex (la partie entiere de x).
Je me servirai ddsormais de ( - j pour signifier la partie entifere de - . Or ecrivons les suites successives
x-l,
(!)-■■
(i)-^^ (i) ©-'^ (I)
(^r-i)-' (f,)-'^ (£)
©-'■
(1)+'^
+ 1
2?
+ 1
5 augmentant ad libitum.
Je dis que, " si r est un nombre entier quelconque qui se trouve dans les suites d'ordre impair, c'est-k-dire commen^ant avec x, f^j, (^j, ..., et si
j = 2t ou 2t + 1, on aura
E
-2^© = 1,
\XJ \rj
et que, si r appartient k une suite quelconque d'ordre pair, on aura
^(i)-2^(i) = o."
Consequemment, en appliquant le th^oreme precedent, on aura
i(i+l)_2^:(\+i)=^. + ^, + ...4-s.
;+•..,
oil <S^_, est la somme des totients des nombres qui sont en m^me temps 6gaux ou inferieurs k E "^ et plus grands que E~, c'est-k-dire
Si done on dcrit
ex = jx-jl+jl-jl+jl-jl + ...,
on aura, quand a; = un nombre entier pair (soit 2i),
a? 6x = (2i» + 1) - (i" + i) = i^= -r,
4
et, quand a;= un nombre entier impair (soit 2i + 1),
9x={i+l)(2i+ 1) -(i' + i) = ^^^.
86 Fractions ordinaires indgales [2
Avec I'aide de ces ^galitds, si x est un nombre positif quelconque entier ou fractionnel, on obtient facilemenb les in^galitfe
a?-2x
Ox — on > ^a; = ou <
4
ar'+2a;4-l
4
En appliquant a ces deux in^galites la m^thode d'approximation successive que j'ai appliqu^e, dans* le Memoire cit6, aux in^galitfe auxquelles est assujettie la fonction ■^(a;) (?;oiV Senet, Alghhre supdrieure, Edition de 1879, t. II. p. 233), je parviens facilement et rigoureusemont a d^montrer que, etant donn^e une quantite e aussi petite qu'on veut, on peut trouver une limite supdrieure L et une limite inf^rieure A a Jx, ou
L=^(— + Tl\a^-Ax + R(\ogx)
A = ('i-
T)'\x'-A'x + R'(\ogx),
ou R (log x), ii'(log x) sont tous les deux fonctions rationnelles et entieres de log« d'un degr^ fini, dont les coefficients aussi bien que A et A' restent toujours finis et ou r], rj' sont tous les deux plus petits que e.
II s'ensuit que la fraction \^ possede une valeur asymptotique —
(ce qui n'est pas demontr^ pour la fraction analogue ^-- , dans la th^orie
Jx parallels de M. Tchebycheff ) et que la valeur de — approche indefiniment
3 prfes quand x est pris suffisamment grand de — , c'est-a-dire de •30396....
II est facile de voir que la quantite Jx diminu^e de I'unit^ n'est autr
chose que le nombre des fractions dans les Tables pareilles a celles de
M. Airy. Ainsi, pour le cas de a; = 100 selon M. Airy, Ja;=3044. Poui
o ce cas —ar'= 3039-6.
Avec I'aide de ces limites on peut calculer la probabilite que deux
nombres dont la limite sup^rieure est tres grande soient premiers autre
eux. Car si cette limite est x, le nombre total des cas qui peuvent arriver
est of, et le nombre des cas pour lesquels les nombres choisis sont premiers
entre eux sera 2/a; — 1. Consdquemment, la probability en question
6 sera — ..
M. Franklin, I'auteur de la belle demonstration, ins^ree dans les Gomptes rendus, du thdorfeme d'Euler sur le produit (1 — x) {\ — a?) {\ — a?) ... , a bien
[* Vol. in. of this Eeprint, p. 532.]
2] Fractions ordinaires in^gales 87
voulu m'adresser la remarque que cette conclusion pent etre au moins con- firmee, peut-etre meme absolument d^montr^e, de la maniere suivante :
X ^tant pris trfes grand, la probability que deux nombres inf^rienrs h, X, pris au hasard, ne contiennent pas tons les deux le nombre premier p,
sera 1 — ^ . Done, la probability cherch^e sera
(•4)(-J.)(-|.)('4.)-.
qui est la r^ciproque de
1 1,1.1,1 1
2' ^ 3' 4' 5' 6* "^ 7» "*" ■"'
c'est-k-dire est ^gal a — - . ° it'
II y a une suite doublement infinie d' Equations fonctionnelles exactes qu'on pent former avec les J{x). En particulier, il y a une serie simplement infinie de telles fonctions ou les signes sont altemativement positifs et n^gatifs, et consdquemment peuvent servir chacun a donner une suite infinie de limites k Jx.
Ainsi, si Ton ^crit
|
dx=Jx- |
-'I |
|
-f/3- |
-Jl |
|
+ /?- 0 |
■'1 |
|
+ |
|
|
+ |
e,x = 3Jt -4>J-,+3Jt-*Jt+^J'^-J
3 '" 4 ■ - 6 *^ 8 + '*'' 9 "^ 12
+
+
on aura toujours, quand
X ^{le' + k) i, 6tx =
2(/fc'' + A;)'
{x+\f
et quand x = {k'-^k)i-l, etx=^-^^ ^ ,
et, quel que soit le r^sidu de x par rapport au module k^ + k,on pent calculer
la valeur de O^x. Enfin, si x est une quantity positive quelconque, on trouvera
rt a? —x . cc' + 1x ->r\
^*^ = °"> 2(FTT)' ^*^=""< 2-(ifc'T)fc)' •
3.
NOTE SUE LE TH^ORjfcME DE LEGENDRE CIT^ DANS UNE NOTE INS6r6e dans LES COMPTES RENDU S.
[Comptes Rmdus, xcvi. (1883), pp. 463—465.]
Le thdorbine de Legendre, cit^ par MM. de Jonquiferes et Lipschitz, est une consequence immediate d'un theorfeme logique bien connu, lequel, mis sous forme sensible, equivaut a dire que, si A, B, C, ... sont des corps avec la faculty de s'entrecouper, contenus dans un vase d'eau, et si a, ab, abc, ... reprdsentent symboliquement les volumes de .4, de la partie commune k A et k B, de la partie commune a A, B, C, ..., alors le volume du liquide d^place par la totality des corps sera
Sa — l,ab + tahc — ....
Consdquemment, ce th^orfeme admet une generalisation infinie dent je donnerai un seul exemple.
Nommons les nombres premiers qui n'excedent pas 7i, nombres premiers subordonnds k n, et distinguons entre eux ceux qui sont plus grands que \/n comme sup^rieurs,
Le theor^me de Legendre Equivaut k dire que, si p^, p^ pi sont
les nombres premiers subordonn^s a >Jn, le nombre des nombres premiers subordonnds ^ n du genre sup^rieur augmente de Tuuit^ est egal k
\pj \PipJ \PiPiPJ Or, repr^sentons la fonctiou ^x(x + l) par Ax; alors on aura le th^oreme que la somme des nombres premiers subordonn^s a « du genre superieur augment^ de I'unite sera ^gale k
An-%pA{^)^tp.p.^{-^^)-....
Par exemple, si » = 11, les nombres premiers subordonn6s a 11 du genre superieur seront 5, 7, 11, et les nombres premiers subordonn^s a V'* sont 2, 3.
3] Sur im thS&reme de Legendre 89
On doit done trouver, et en effet on trouve
(11 . 12) - 2 (5 . 6) - 3 (3 . 4) + 6 (1 . 2) = 2 (1 + 5 + 7 + 11).
Je saisis cette occasion pour dire que j'ai fait calculer la valeur de J (n),
" somme-totient de n," pour toutes les valeurs entieres de n jusqua 500, et je
3 trouve que sans aucune exception J(n) est toujours plus grand que —^(n")
3 et plus petit que — (n + 1)^
TT"
II reste a demontrer que ces limites sont d'application universelle pour un nombre entier quelconque n.
On peut faire une extension illimitee du th^oreme donne dans le numero prec^ent des Comptes rendus sur les sommes-totients, tout a fait analogue a I'extension ci-dessus donn^e au theoreme de Legendre sur les nombres premiers. Nommons, par exemple, u (j) la somme de tous les nombres premiers et inf^rieurs a j, et Uj la somme
u(l) + u(2) + ... + u(J). On ^tablit facilement* I'identit^
£A(^^)u(i) = iJ(j+l)0-+2),
oil Ax signifie le nombre triangulaire iix(x + 1), et avec ce th^or^me, en se servant, comme dans la th^orie des sommes-totients, du principef de la division harmonique et en ^crivant
vj = Uj -2u^^+3u i-wi+ 5ui- ...,
on en d^duit facilement ^ = fn — o q»and j est pair,
( J -)- 1 )» J -f- 1 Vj = —Ta h quand j est impair, etc.
Dans ma Note J Sur le nombre des fractions ordinaires inigahs, etc., j'ai omis de dire que I'^quation
lEiTr=^-^ r r 2
peut etre dcrite sous la forme
^; + ^i + ./| + /|+...=-^^. (1)
[•With u(r) = Jrr(r), ti(l) = i, T{r) being the totient of r, we have
2 2 A(E^^tt(r) = J»(t + l)(2i + l).] [t Vol. rn. of this Reprint, p. 678.] [J p. 84 above.]
90 Sur un thioreme de Legendre
De mfime, I'^quation
[3
<j ..J . J0'+l)0' + 2)
r r 0
equivaut a I'^quation '
c:;-+2[ri+3frU4f7-U...=-i<i±lMi±2)
4
6
(2)
II est facile de d^montrer, avec I'aide des Equations (1) et (2), que les
valeurs asymptotiques de ^ et ^ pour _; ind^fiDiment grand sont — et —
respectivement.
Cauchy, MM. Halphen et Lucas out ^crit sur les suites de Farey. II est done bon de faire remarquer que Jj est le nombre des fractions et Uj la somme des numerateurs des fractions dans une telle suite pour laquelle la limite donn^e est _;'.
[• For iJ(i + l)(j + 2) read xViU + l) (2/ + 1)-]
4.
SUR LE PRODUIT INDl^FINI 1 -x .1-a^.l -x' ....
[Comptes Rendm, xcvi. (1883), p. 674]
Dans le Johns Hopkins Circular, numero de fevrier*, on trouvera I'ex- plication d'une m^thode graphique pour convertir les produits continus en series. J'ai applique cette m^thode pour obtenir la formule connue (Cayley, Elliptic Functions, p. 296)
1
I— ax. 1 — ax'. \ — aa?...
xa aW
= 1 + .
1 —X. i. —ax i — x.i—x'.i — ax.i — ax'
a^a*
+ ...
l—x.l—ai'.l—a^.l—ax.l —ae^.l — aa?
Je me suia demand^ quelle serait I'expression obtenue en appliquant la meme construction (ou dissection) graphique (qui fournit la formule cit^e en haut), au produit \ + ax .\+a^ .1 ■^ax' ..., et j'ai trouvd sans aucune difficulte I'expression suivante :
1+aa^ . . 1 +aar. 1 +ax*
\-\-xa-z, +a;»a'-j r — +...
1 —X 1 — a; .1 — ar
'-^ ^l + ax.l+ax' ...1+ axJ-' 1+ax'i ■"""^ " l-x.l-x'...l-xJ-' • 1-xi ■^•••-
En faisant a — — 1, on obtient
l-x.l-x'.l-x'...
= l-a!(l+a;) + a^(l + a»)+... + (-ya: * {l+a^)+....
C'est le th^orfeme bien connu d'Euler, lequel, sous ce point de vue, n'est qu'un coroUaire d'un theor^me plus g^n^ral.
Par la meme m^thode, j'obtiens la s6rie pour les thita fonctions et d'autres series beaucoup plus gen^rales, sans calcul alg^brique aucun, [• Vol. m. of this Reprint, pp. 669, 686 ; and above pp. 30, 33.]
5.
SUR UN TH]60REME DE PARTITIONS.
{Comptes Rendm, xcvi. (1883), pp. 674, 675.]
SoiENT Si, Si, ..., Si des suites de nombres consdcutifs, telles que le plus petit terme dans aucune d'elles n'excede de plus de I'unite le plus grand terme dans la suite qui precede ; bien entendu que t pent devenir I'unite et qu'une suite quelconque pent se reduire k un seul terme. On peut envisager ce systeme de suites comme une partition de la somme des nombres contenus dans leur totality : alors on aura le theoreme suivant :
Le nombre de systemes de i suites de nomhres consecutifs dont la somme est N est le meme que le nombre de paHitions de N qu'on peut former avec les repetitions de i nombres impairs. Comme exemple, en faisant N=\Q et i= 1, 2, 3 successivement, on aura d'un cotd les divers groupes de partitions
1, 3, 6
|
10 |
9, 1 |
1, 2, 7 |
|
1, 2, 3, 4 |
8, 2 |
2, 3, 5 |
|
7, 3 |
1, 4, 5 |
|
|
6, 4 |
et de I'autre (en se servant d'un indice superieur pour signifier le nombre des reflexions de sa base),
3^ 1 \\ 3, 5
3S 1* 3, 1'
110
9,
7, 7, 5,
En ajoutant ensemble les equations qui, pour la mfime valeur de N, r^pondent a toutes les valeurs possibles de i, on retombe sur le th^orfeme bien connu d'Euler que le nombre des partitions de N, en excluant seulement les repetitions, est le meme que le nombre de ses partitions en excluant seulement les nombres pairs. Ainsi, on peut envisager ce dernier theoreme comme un corollaire d'un theoreme bien autrement profond et qui n'est pas du tout facile a demontrer, sinon pour le cas le plus simple, c'est-a-dire quand il n'y a qu'une seule suite. Pour ce cas, le theoreme peut s'exprimer en disant que le nombre de suites de nombres consecutifs dont la somme est N est 6gal au nombre de diviseurs impairs de N.
I
6.
PREUVE GRAPHIQUE* DU THEORfeME D'EULER SUR LA PARTITION DES NOMBRES PENTAGONAUX.
[Comptes Rmdus, xcvi. (1883), pp. 743—745.]
Une partition quelconque de n peut etre representee par un assemblage de points uniformdment distribuds sur un plan et limites par deux lignes droites. Ainsi, par exemple, I'arrangement suivant:
sera la representation graphique de la partition du nombre 22 dans les parties
7, 5, 5, 3, 2.
Mais, de plus, un tel arrangement de points peut 6tie distribu^ dans un carr6 et deux groupes que je nommerai lateral et in/^rieur. Ainsi, I'arrangement ecrit ci-dessu8 peut etre decompose dans un carrd de neuf points, dans un groupe lateral de buit et dans un groupe infi^rieur de cinq points.
Considerons les partitions de n dans j parties inegales. Tous les arrange- ments de points qui correspondent k ces partitions peuvent etre classifies selon la valeur du cot^ du carr^ qui y correspond et que je nommerai 0. Alors, pour une valeur donn^e de 0, le groupe lateral contiendra n^ces- sairement ou ^ ou ^ — 1 lignes de points, car autrement il y aurait des parties egales dans I'arrangement. Dans le premier cas, le nombre de colonnes dans ce groupe inf^rieur peut ^tre un nombre quelconque, mais pas plus grand que 0 ; dans le second cas, pas plus grand que ^ — 1. Done, en se rappelant que le nombre de partitions de v en 6 parties inegales est le coefficient de x dans le d^veloppement de
et que le nombre de partitions de p dans j — 0 parties inegales et pas plus grandes que 0 est le coefficient de x'a^'^ dans le developpement de
(l + aa;)(l + oaf) . . . {1 + ax'), on voit que, quand le nombre de lignes dans le groupe lateral est 0, le nombre
[* See p. 32 above.]
94 Sur la partition des nombres pentagonaux [6
total d'arrangements de n dans j parties indgales qui correspondent k cette espece de distribution sera le coefficient de a;"~*'a-^~* dans le developpement de l+ax.\+aa?...\+(ud> ^ 1 — a; . 1 — a;" . . . 1 — a^ De mfime, le nombre des partitions qui correspondent k la seconde hypothese sera le coefficient de af*~^cJ~^ dans le developpement de
l + ax.l+aa? ...l + aa^-' ~
a; " .
l-x.l-a:!'...l-x*-' En donnant a 6 toutes les valeurs depuis 1 jusqu'a I'infini, on obtiendra toutes les partitions de n dans j parties inegales. Les cas oh 6 excede j n'offrent rien d'exceptionnel, car, pour ces cas, le coefficient de a^~* dans les deux fonctions generatrices sera nul.
Or le coefficient de a;"~*'a-'~* dans chacune de ces deux fonctions est le meme que le coefficient de a:"a^ dans les produits qui resultent de leur multiplication par «*"«*.
En comparant les coefficients de x^a^ pour toute valeur de n et i, on trouve done
(!+««) (1 +x'a){l+a^a)+ ...
1 + ax „ l+ax.l + aa? . „
1—x 1—x.l — a^
1 + ax. 1 + ax'. ..l + aa^ ^ e
1 + ax . „
+ xa + ~ af'a^ + ...
1 — x
l+ax.l+aa?...l + ax^-'^ ?^ „
H ^ 5 z a_. X ^ a» + ....
1— x.l—oc' ... 1— af^^
En mettant a = — 1, on obtient ainsi
l-x.l-a?.l-a?- ... = l-x-a?-... + {-f\x ^ +x « ) + ..., ce qui est le theoreme d'Euler.
En r^unissant les deux series dans une seule, on obtient, pour le cas general,
{l+xa){l + a?a){l+a?a)+ ...
l + aar* 1 + ax .1 +ax* ^ „ H-air.l+ oar* . 1 + aa^
1 — a; 1 — a;.l— ar* 1 —x.l — x' .1 — x'
c'est-ii-dire I'equation que j'ai donnde dans la Note prec^dente [p. 91].
Je dois dire que c'est M. Durfee, dtudiant a Baltimore, qui, le premier (dans un tout autre probleme), a fait usage du genre de decomposition d'une assemblee reguliere de poiuts dans un carr^ et deux groupes supple- men taires dont j'ai profite dans I'analyse precedente (voir Johns Hopkins Circular, [Vol. Iil. of this Reprint, pp. 661 fF.]).
7.
DEMONSTRATION GRAPHIQUE* D'UN TH^OREME D'EULER CONCERNANT LES PARTITIONS DES NOMBRES.
[Comptes Rendus, xcvi. (1883), pp. 1110—1112.]
CoMME confirmation de la puissance de la methode graphique appliqude a la theorie des partitions, la preuve suivante d'un thdoreme que je crois etre nouveau ne sera pas, je I'espfere, tout h fait depourvue d'int^ret pour les g^ometres; car il serait, il me semble, assez difficile d'en trouver une preuve directe analytique au moyen de la comparaison de fonctions generatrices, comme on le fait ordinairement pour des theoremes de ce genre.
Euler a trouve facilement, par une comparaison de telles fonctions, que le nombre de partitions de n en nombres impairs est le meme que le nombre de partitions de n en nombres inegaux ; je precise ce theoreme en ajoutant que le nombre de partitions de n en nombres impairs, qui se divisent en i groupes de nombres distincts, est egal au nombre de partitions de n en i suites tout a fait distinctes de nombres cons^cutifs.
Nommons U une partition en nombres impairs et V une partition en nombres inegaux.
Je dis qu'on peut passer de U k V par la methode suivante. Supposons, par exemple, que U soit la partition 11.11.7.7.7.5.
Je forme deux assemblages r^guliers de points en prenant dans I'un
11+1 11 + ] 7 + 1
d'eux, sur cbaque ligne, un nombre de points ^gal k 7+1 7 + 1 5+1
2 ' 2 ' 2 ' , et I'antre assemblage en diminuant de I'unit^ chacun
2 ' 2 ' 2
de ces nombres de points. On forme ainsi ces deux assemblages : 1. 2.
et, en comptant le nombre de points dans les angles successifs de chaque figure, on obtient, dans I'un, 11, 9, 5, 2, et, dans I'autre, 10, 8, 3 ; en les r^unissant, on obtient la partition
11.10.9.8.5.3.2, qui est un V.
[* See p. 39 above.]
96 Le» partitions des nomhres [7
Or il est facile de voir que dans cette m^thode de transformation U devient V, et Ton d^montre (en construisant un certain systeme d'equations lindaires) que, pour un V quelconque donne, on pent trouver un et un seul U qui se transformera dans ce V, de sorte qu'il y a correspondance un k un entre la totalite des U et la totality des V, ce qui sert k d^niontrer le theoreme original d'Euler. Mais si tel ^tait le but de cette recherche, cette ra^thode de transformation serait peine perdue, car il existe una tout autre m^thode, infiniment plus simple, d'^tablir une telle correspondance : on la trouvera expliquee dans le cahier de V American Journal of Mathe- matics qui va paraitre. L'utilite de cette m^thode spe'ciale de cr^er la correspondance consiste en ceci : que le V ainsi conjugue avec un U contiendra le meme nombre de suites distinctes de nombres consecutifs que le ?7contient de nombres impairs distincts: cela veut dire que le nombre des lignes in^gales (disons i) dans I'un ou I'autre assemblage de points est toujours dgal k j, nombre de suites distinctes obtenu en operant de la maniere expliquee ci-dessus. La preuve en est facile ; car si Ton enleve Tangle ext^rieur a I'un et a I'autre des assemblages, on verra facilement que quatre cas se pr^senteront : pour un de ces cas, j ne change pas de valeur, a cause du changement opdr6 dans les deux assemblages ; dans un autre cas, j subira une diminution de deux unites, et dans les deux cas intermediaires d'une seule unite. Ces cas correspondent aux quatre suppositions qui resultent de la combinaison des hypotheses que les deux premieres lignes ou les deux premieres colonnes dans I'un ou I'autre des assemblages sent ou ne sont pas dgales entre elles : de sorte qu'on verra facilement que le j et le i seront toujours diminu^s de la meme quantity, ou 0, ou 1 ou 2, et cons^quemment on aura i — j constant; si Ton enleve I'un apres I'autre les angles des deux assemblages jusqu'a ce qu'on arrive a un assemblage qui sera de I'une ou I'autre des quatre formes suivantes :
4.
pour lesquels cas i=2, j' = 2 ; i = \, j=\\ i=l, j=l; t' = l, 7=1; re- spectivement on aura toujours ainsi i =j, de sorte qu'il y a correspondance une k une entre les partitions du mSme nombre n qui contiennent justement i nombres impairs r^p^t^s (ou non) k volont^, et celles qui contiennent justement i suites distinctes de nombres consecutifs, et cons^quemment il y aura le meme nombre des unes et des autres : ce qui est le theoreme que j'ai voulu d^montrer.
8.
SUR UN THEOREME DE PARTITIONS* DENOMBRES COMPLEXES CONTENU DANS UN THEOREME DE JACOBI.
[Gomptes Rendus, XCVI. (1883), pp. 1276—1280.]
Dans le Journal de Crelle, t. xxxii. p. 166, Jacobi fait la remarque que le developpement en s^rie de SiX donne lieu k un thdoreme que j'expriine de la maniere suivante.
Soient a etb deux quantites c=a + h; alors le produit infini
+00 fle+Ha-b)
{lTg')(l + q>')(l-q')(lT^+'){l+q'^')(l-r)--='^ i+Yq *
— oo
Ce theoreme etant vrai pour un nombre infini de valeurs de t sera, par
sa forme meme, n^cessairement vrai quand a et b sont de symboles absolument arbitraires, et Ton voit facilement que, pour le montier dans ce sens universel, il suflBra d'^noncer un certain theoreme sur les nombres complexes dont voici I'^nonc^ :
Designons par C, B, A des nombres complexes de la forme fc, fc + b, fc + a, oil / est ou ziro ou un nombre entier et positif quelconque.
Considerons un arrangement compose avec des C, des B et des A nan rip4tis ou avec des C, B, A pris seuls ou combines deux d deux, en excluant les arrangements (que je nomme exceptionnels) qui ne contiennent que des B formant vne sirie arithmetique dont b est le dernier terme et c la difference constante, ou des A formant une sirie semblable dont a est le dernier terme.
Par le caractire majeur et le caract^re mineur d'un tel arrangement, je disigne la parite ou Vimpariti du nombre total des termes et du nombre des G qu'il contient. Je dis qu'd chaque arrangement {non exceptionnel) on pent en associer un autre pareil dont la somme totale des dements (les A, B,G) sera la meme, mats dont les caracteres seront tous les deux opposds.
La demonstration deviendra plus claire en se servant de la notation suivante. En d^signant par X un symbole d'une s^rie de termes, je me eervirai de Z et de X pour signifier le terme le plus haut et le terme le plus
[* See above, p. S9 ff.] 8. IV. 7
98 8ur un tMorertie de partitions [8
bas de la sdrie, et en me servant de Y on Z pour signifier un symbole ou simple ou affect^ de marques quelconques, j'emploie les notations
F=0, Y+Z=^0, F>0, Y+Z>0,
pour signifier que les Y manquent, que les Y et les Z manquent tous les deux, que les Y ne manquent pas, que les Y et les Z ne manquent pas tous les deux.
Je divise les B (d'un arrangement quelconque) en deux especes, 'B et B', dont 'B repr^sente un B appartenant a la s6rie arithmetique (la plus grande qu'on puisse former) commen^ant avec le plus grand B, et B' les autres B qui se trouvent dans I'arrangement.
Ainsi je divise les A en ^A et en A^ ; A^ signifie un A appartenant a la s^rie arithmetique la plus grande qu'on puisse former, dont a est le terme minimum (de sorte que, si I'arrangement ne contient pas un a, A^ manque) et ^A signifie les autres A de I'arrangement.
Finalement un point au centre d'un symbole a droite ou k gauche signifiera ce symbole diminu^ ou augmente respectivement de c.
On voit que dans cette notation les arrangements exceptionnels seront exprim^s ainsi: ceux qui appartiennent k I'une des deux classes par lea conditions 'B — b = 0 avec A + C=0, et les autres par les conditions B = 0 avec ,A + C=0.
Je divise les arrangements non exceptionnels en trois classes, dont les conditions seront respectivement les suivantes:
Premiere classe :
'B-b>0 ou ('B-b = 0 avec C-c&'B-h).
Deuxifeme classe :
'B-h = 0 avec (C-o'B-b ou (7=0, mais A + G>0),
ou B = 0 avec {A = 0 ou A —a > G).
Troisieme classe :
5 = 0 avec A>0 et A-a<C et ^+G>0.
Toutes les hypotheses possibles se trouvent comprises dans ces tableaux des arrangements exceptionnels et non exceptionnels.
A chacune des trois classes des derniers je vais assigner un ope'rateur qui pent etre appliqu^ a chaque arrangement de cette classe et qui le trans- formera dans un autre arrangement appartenant a la nieme classe; cette disposition, appliqu^e deux fois successivement, reproduira I'arrangement sur lequel on opere, lequel ne changera pas la somme des elements, mais changera chacun des deux caracteres en sens oppos^ : c'est-a-dire que chacun des trois op^rateurs que je vais d^finir, et que je nommerai </>, yjr, ^, doit
8] de nomhres complexes 99
satisfaire h. cinq conditions qu'on pent nommer catholicite, homoeoghihse, mutualite, inertie et enantiotropie.
1. <f> signifie que, si C = 0 ou C — o'B — 'B, on doit former un nouveau C, en substituant, pour chaque 'B, 'B- (c'est-a-dire sa valeur diminu^e de c), et reconstituer I'inertie originale en ajoutant ensemble les c ainsi soustraits pour former un nouveau G, et que, dans le cas contraire, C doit etre decompose en simples c, dont on ajoutera un au premier 'B (le B le plus grand), un au second 'B, etc., jusqu'i ce que tous les c dont on a a disposer soient ^puises.
2. i/r signifie que, si B>0 ou C = 0, ou C>'B-\-A, on doit former un nouveau C en substituant k'B et A leur somme et que, dans le cas contraire, C doit etre decompose en 'B et .d si £ > 0 et en 6 et J. si £ = 0.
3. ^ signifie que, si C=0 ou (7 + ^4, = >4, il faut decomposer ^ en •A, et C7 ou en a et G, selon que 4, = ou > 0, et que, dans le cas contraire, pour G et A,, il faut substituer leur somme. On sera satisfait en ^tudiant les conditions des trois classes que les ^, i/r, ^ possedent tous les trois cinq attributs voulus: la preuve en est facilitee en supposant que, dans chaque s^rie des G, des B et des A, prise s^parement, on suit un ordre regulier de grandeur dans I'arrangement de ces termes respectivement au multiple de c qui entre dans chacun d'eux.
Si Ton donne a a et a 6 des valeurs quantitatives (ce qui est toujours permis), et en particulier les valeurs 1 et 2 respectivement, on retombe 8ur le theoreme d'Euler, mais (chose a noter) la correspondance donnee par le proc^d6 general applique k ce cas ne sera nuUement identique h la correspondance donnee par le proced^ de Franklin. En effet, les arrange- ments exceptionnels ne seront pas les memes dans les deux m^thodes : selon le proc^d^ de Franklin, les arrangements non conjugables sont de la forme
i, i+1, ..., 2t-l ou i+1, i + 2, ..., 2i,
tandis que la methode actuelle donnera, comme non conjugues, les arrange- ments de la forme
1, 4 3t-2 ou 2, 5 3i-l.
La methode employee ici fournira elle-meme toujotirs deux systfemes de correspondance absolument distincts, dont on obtient I'un, qui n'est pas exprim^, en ^hangeant entre eux les o, A et les 6, B, car la methode n'est pas sym^trique dans son operation sur ces deux systfemes de lettres.
Ce cas est analogue k celui de la correspondance perspective entre deux triangles, laquelle pent ^tre simple ou triple, comme je I'ai raontre ailleurs. Jacobi, dans I'endroit cite, a fait la remarque que, pour a = l, 6 = 2, en se servant du signe sup«5rieur (?) dans son th^orfeme, on retombe sur le
7—2
100 Sur un tMoreme de partitions de nombres complexes [8
th^or^me d'Euler et que, pour le cas de a = 1, 6 = 1, en se- servant du signe infdrieur, sur un thdoreme donnd (il y a longtemps par Gauss). On peut ajouter que, si avec cette supposition on se sert du signe superieur, on obtient 0 = 0, mais si Ton &rit a = 1 — e, 6=1, en faisant e infinit«^simal, on tombe (chose singuli^re) sur I'^quation de Jacobi elle-mSme,
(1 - qy(l-qj{l -q>f+ ... = 1 -<iq+ og^ -1q' + ....
Puisque j'ai introduit le nom de I'auteur des Fundamenta nova, qu'on me permette la remarque que, dans les deux avant-derniferes lignes de I'avant-derniere page de cet immortel Ouvrage, on trouve un th^oreme qui ^quivaut k I'^quation
q q' ^ _ q 5'+' 5'+'+'
l + ~q~T+'^'^l+q'~ r+^~ l+q''^l+q'~'"'
or, le. premier cas du thdoreme intitule : Sur un theor^me d' Eider, contenu dans une Note precedente des Comptes rendus*, affirme que le|nombre des series arithni^tiques avec lesquelles on peut exprimer n est ^gal au nombre des diviseurs impairs de n, laquelle consideration mene imni^diatement a une consequence qu'on ne pourrait manquer d'observer (mais que M. Franklin, effectivement, a remarqu^e le premier) et qui s'exprime par I'equation
Equation tres ressemblante a I'autre et qui peut ^tre combinde avec elle de maniere a donner naissance a quatre autres Equations de la meme esp^ce.
On n'a pas besoin de dire que le theoreme qui constitue la matiere principale de cette Note, en faisant a = 1 et en considerant h corame une quantity arbitraire, contient ou au moins conduit immediatement au de- veloppement de ©ja; dont Jacobi I'a traits comme consequence.
[* p. 95 above. Cf. p. 25 above.]
9.
ON THE NUMBER OF FRACTIONS CONTAINED IN ANY "FAREY SERIES" OF WHICH THE LIMITING NUMBER IS GIVEN.
[Philosophical Magazine, xv. (1883), pp. 251—257 ; xvi. (1883),
pp. 230—233.]
A Farey series (" suite de Farey ") is a system of all the unequal vulgar fractions arranged in order of magnitude, the numerator and denominator of which do not exceed a given number.
The first scientific notice of these series appeared in the Philosophical Magazine, Vol. XLVii. (1816), pp. 385, 386. In 1879 Mr Glaisher published in the Philosophical Magazine (pp. 321 — 336) a paper on the same subject containing a proof of their known properties, an important extension of the subject to series in which the numerators and denomiuators are subject to distinct limits, and a bibliography of Mr Goodwyn's tables of such series. Finally, in 1881 Sir George Airy contributed a paper also to the Philosophical Magazine of that year, in which he refers to a table calculated by him "some years ago," and printed in the Selected Papers of the Transactions of the Institution of Civil Engineers, which is in fact a Farey table with the logarithms of the fractions appended to each of them. Previous tables had only given the decimal values of such fractions. The drift of this paper is to point out a caution which it is necessary to observe in the use of such tables, and which limits their practical utility: this arises from the fact of the ditferences receiving a very large augmentation in the immediate neighbour- hood of the fractions which are a small aliquot part of unity — a fact which may be inferred d priori from the well-known law discovered by Farey applicable to those diflFerences, but to which the author of the paper makes no allusion.
In addition to the tables of Farey series by Goodwyn, Wucherer, an anonymous author mentioned in the Babbage Catalogue, and Gauss, referred to by Mr Glaisher in his Report to the Bradford Meeting of the British Association (1873), may be mentioned one contained in Herzer's Tabellen
102 On the Number of Fractions contained in any [9
(Basle, 1864) with the limit 57, and another in Hrabak's Tabellen-Werk (Leipsic, 1876), in which the limit is taken at 50.
The writers on the theory are : — Cauchy (as mentioned by Mr Glaisher), who inserted a communication relating to it in the Bulletin des Sciences par la Socidtd Philomathique de Paris, republished in his Exercices de Mathe- matiques ; Mr Glaisher himself (loc. cit.) ; M. Halphen, in a recent volume of the Proceedings of the Mathematical Society of France ; and M. Lucas, in the next following volume of the same collection. I am indebted to my friend and associate Dr Story for these later references.
For theoretical purposes it is desirable to count \ as one of the fractions in a Farey series. The number of such fractions for the limit j then becomes identical with the sum of the totients of all the natural numbers up to _;' inclusive — a totient to x (which I denote by tx) meaning the number
x = l
of numbers less than x and prime to it. Such sum, that is, S tx, I denote
by Tj. My attention was called to the subject by this number Tj expressing the number of terms in a function whose residue (in Cauchy's sense) is the generating function to any given simple denumerant (see Amencan Journal of Mathematics, [Vol. III. of this Reprint, p. 605]) ; and 1 became curious to know something about the value of Tj. I had no difficulty in finding a functional equation which serves to determine its limits (see Johns Hoplcins University Circular, Jan. and Feb. 1883*). The most simple form of that equation (omitted to be given in the Circular) is
Ti 4- t1- -I- T-L 4- T-i 4- T^ 4- _.Li? ^J^-^ 2^ 3^ 4^ 5^'"~ 2 '
(where, when a; is a fraction, Tx is to be understood to mean Tj,j being the
integer next below x) ; and from this it is not difficult to deduce by strict
demonstration that Tjjj^, when j increases indefinitely, approximates
indefinitely near to S/tt^
I have subsequently found that if ua; be used to denote the sum of all
x=l
the numbers inferior and prime to x, and Uj = "^ ux, then f
uj+2u{ + 3ui + iui + ...J^J+^l^^+^^
(where Ux, when a; is a fraction, means the U of the integer next inferior to x). From this equation it is also possible to prove that Uj/j', when j becomes indefinitely great, approximates to I/tt*. Uj, it may be well to notice, is the sum of all the numerators of the fractions in a Farey series whose limit is j, just as Tj is the number of these fractions.
In the annexed Table the value of tx (the totient), of Tx (the sum-totient), and of S/tt^ . x' is calculated for all the values of x from 1 to 1000 ; and the [* See pp. 84, 89 above.] [t The right side should be ^j (j + 1) (2/ + 1).]
9] "Farey Series" of which the Limiting Number is given 103
remarkable fact is brought to light that Tx is always greater than S/Tr" . a? (the number opposite to it), and less than S/tt^ . (a; + ] )", the number which comes after the following one in the same table.
I have calculated in my head the first few values of Ux, and find (if I have made no mistake) that it obeys an analogous law, namely is always intermediate between I/tt". a? and I/tt*. (a; + 1)*.
It may also be noticed that when « is a prime number, Tn is always nearer, and usually very much nearer, to the superior than to the inferior limit — as might have been anticipated from the circumstance that, when this is the case, in passing from n — 1 to n the T receives an augmentation of
3
n — 1, whereas its average augmentation is only — (2n — 1).
in
In like manner and for a similar reason, when n contains several small factors Tn is nearer to the inferior than to the superior limit. For instance, when n = 210, Tn = 13414 and S/tt" . n' = 13404-79.
Table of Totients, of Sum-totients, and ofS/ir' into the Squares of all the Numbers from 1 to 1000 inclusive.
[~ = -sosgessol.
|
n |
r(«) |
T{n) |
i- |
n |
r(n) |
T(n) |
i- |
n |
r(n) |
T{n) |
> |
|
1 |
1 |
1 |
■30 |
27 |
18 |
230 |
221-59 |
63 |
62 |
882 |
863-83 |
|
2 |
1 |
2 |
1-22 |
28 |
12 |
242 |
238-31 |
54 |
18 |
900 |
886-36 |
|
3 |
2 |
4 |
2-74 |
29 |
28 |
270 |
255-63 |
55 |
40 |
940 |
919-49 |
|
4 |
2 |
6 |
4-86 |
30 |
8 |
278 |
273-56 |
56 |
24 |
964 |
953-23 |
|
5 |
4 |
10 |
7-60 |
31 |
30 |
308 |
292-11 |
57 |
36 |
1000 |
987-58 |
|
6 |
2 |
12 |
10-94; |
32 |
16 |
324 |
311-26 |
58 |
28 |
1028 |
1022-64 |
|
7 |
6 |
18 |
14-90 i |
33 |
20 |
344 |
331-01 |
59 |
58 |
1086 |
1058-10 |
|
8 |
4 |
22 |
19-46 |
.34 |
16 |
360 |
.351-38 |
60 16 |
1102 |
1094-27 |
|
|
9 |
6 |
28 |
24-62 i |
3a |
24 |
384 |
372-35 |
61 |
60 |
1162 |
11.31-06 |
|
10 |
4 |
32 |
.30-40 |
36 |
12 |
396 |
393-93 |
62 |
30 |
1192 |
1168-44 |
|
11 |
10 |
42 |
.36-78 |
37 |
36 |
432 |
416-12 |
63 |
36 |
1228 |
1206-43 |
|
12 |
4 |
46 |
43-7"! |
.38 |
18 |
450 |
438-92 |
64 |
32 |
1260 |
1245-03 |
|
13 |
12 |
58 |
51-37 |
39 |
24 |
474 |
462-32 |
65 |
48 |
1308 |
1284-26 |
|
14 |
6 |
64 |
69-58 |
40 |
16 |
490 |
486-.34 |
66 |
20 |
1.328 |
132407 |
|
15 |
8 |
72 |
68-39 |
41 |
40 |
530 |
510-96 |
67 |
66 |
1394 |
1364-49 |
|
16 |
8 |
80 |
77-81 |
42 |
12 |
542 |
.536-19 |
68 |
32 |
1426 |
1405-63 |
|
17 |
16 |
96 |
87-84 |
43 |
42 |
.584 |
662-02 |
69 |
44 |
1470 |
1447-17 |
|
18 |
6 |
102 |
98-481 |
44 |
20 |
604 |
688-47 |
70 |
24 |
1494 |
1489-42 |
|
19 |
18 |
120 |
109-73 |
45 |
24 |
628 |
615-62 |
71 |
70 |
1564 |
15.32-28 |
|
20 |
8 |
128 |
121 -.58 |
46 |
22 |
650 |
64.3-19 |
72 |
24 |
1588 |
1575-76 |
|
21 |
12 |
140 |
1.34-06 ! |
47 |
46 |
696 |
671-46 |
73 |
72 |
1660 |
1619-82 |
|
22 |
10 |
150 |
147-12 |
48 |
16 |
712 |
700-33 |
74 |
36 |
169C |
1664-51 |
|
23 |
22 |
172 |
160-79 |
49 |
42 |
754 |
729-82 |
75 |
40 |
17.36 |
1709-80 |
|
24 |
8 |
180 |
175-08 |
50 |
20 |
774 |
759-91 |
76 |
36 |
1772 |
1755-69 |
|
25 |
20 |
200 |
189-98 |
61 |
32 |
806 |
790-61 |
77 |
60 |
1832 |
1802-20 |
|
26 |
12 |
212 |
205-48 |
52 |
24 |
830 |
821-92 |
78 |
24 |
1866 |
1849-31 |
104 On the Number of Fractions contained in amj
Table {continued).
[9
|
n |
T(n) |
T(n) |
!."• |
n |
T(n) |
T(n) |
i- |
n |
T(n) |
T(n) |
!."■ |
|
79 |
78 |
1934 |
1897-04 |
134 |
66 |
5498 |
5457-97 |
189 |
108 |
10904 |
10857-88 |
|
80 |
32 |
1966 |
1945-37 |
135 |
72 |
5570 |
5539-74 |
190 |
72 |
10976 |
10973-09 |
|
81 |
54 |
2020 |
1994-31 |
136 |
64 |
5634 |
5622-11 |
191 |
190 |
11166 |
11088-90 |
|
82 |
40 |
2060 |
2043-85 |
137 |
136 |
5770 |
5705-09 |
192 |
64 |
11230 |
11205-31 |
|
83 |
82 |
2142 |
2094-01 |
138 |
44 |
5814 |
5788-68 |
193 |
192 |
11422 |
11322-34 |
|
84 |
24 |
2166 |
2144-77 |
139 |
138 |
5952 |
5872-88 |
194 |
96 |
11518 |
11439-97 |
|
85 |
64 |
2230 |
2196-14 |
140 |
48 |
6000 |
5957-69 |
195 |
96 |
11614 |
11558-21 |
|
86 |
42 |
2272 |
2248-12 |
141 |
92 |
6092 |
6043-10 |
196 |
84 |
11698 |
11677-06 |
|
87 |
56 |
2328 |
2300-70 |
142 |
70 |
6162 |
6129-12 |
197 |
196 |
11894 |
11796-52 |
|
88 |
40 |
2368 |
2353-90 |
143 |
120 |
6282 |
6215-75 |
198 |
60 |
11954 |
11916-59 |
|
89 |
88 |
2456 |
2407-70 |
144 |
48 |
6330 |
6302-99 |
199 |
198 |
12152 |
12037-26 |
|
90 |
24 |
2480 |
2462-10 |
145 |
112 |
6442 |
6390-83 |
200 |
80 |
12232 |
12158-54 |
|
91 |
72 |
2552 |
2517-12 |
146 |
72 |
6514 |
6479-29 |
201 |
132 |
12364 |
12280-43 |
|
92 |
44 |
2596 |
2572-75 |
147 |
84 |
6598 |
6568-35 |
202 |
100 |
12464 |
12402-93 |
|
93 |
60 |
2656 |
2628-98 |
148 |
72 |
6670 |
6658-02 |
203 |
168 |
12632 |
12526-03 |
|
94 |
46 |
2702 |
2685-82 |
149 |
148 |
6818 |
6748-29 |
204 |
64 |
12696 |
12649-75 |
|
95 |
72 |
2774 |
2743-27 |
150 |
40 |
6858 |
6839-18 |
205 |
160 |
12856 |
12774-07 |
|
96 |
32 |
2806 |
2801-33 |
151 |
150 |
7008 |
6930-67 |
206 |
102 |
12958 |
12899-00 |
|
97 |
96 |
2902 |
2860-00 |
152 |
72 |
7080 |
7022-77 |
207 |
132 |
13090 |
13024-54 |
|
98 |
42 |
2944 |
2919-27 |
153 |
96 |
7176 |
7115-48 |
208 |
96 |
13186 |
13150-68 |
|
99 |
60 |
3004 |
2979-15 |
154 |
60 |
7236 |
7208-80 |
209 |
180 |
13366 |
13277-43 |
|
100 |
40 |
3044 |
3039-64 |
155 |
120 |
7356 |
7302-72 |
210 |
48 |
13414 |
13404-79 |
|
101 |
100 |
3144 |
3100-73 |
156 |
48 |
7404 |
7397-26 |
211 |
210 |
13624 |
13532-76 |
|
102 |
32 |
3176 |
3162-44 |
157 |
156 |
7560 |
7492-40 |
212 |
104 |
137-28 |
13661-34 |
|
103 |
102 |
3278 |
3224-75 |
158 |
78 |
7638 |
7588-15 |
213 |
140 |
13868 |
13790-52 |
|
104 |
48 |
3326 |
3287-67 |
159 |
104 |
7742 |
7684-51 |
214 |
106 |
13974 |
13920-32 |
|
105 |
48 |
3374 |
3351-20 |
160 |
64 |
7806 |
7781-47 |
215 |
168 |
14142 |
14050-72 |
|
106 |
52 |
3426 |
3415-34 |
161 |
132 |
7938 |
7879-04 |
216 |
72 |
14214 |
14181-73 |
|
107 |
106 |
3532 |
3480-08 |
162 |
54 |
7992 |
7977-22 |
217 |
180 |
14394 |
14313-34 |
|
108 |
36 |
3568 |
3545-44 |
163 |
162 |
8154 |
8076-01 |
218 |
108 |
14502 |
14445-57 |
|
109 |
108 |
3676 |
3611-40 |
164 |
80 |
8234 |
8175-41 |
219 |
144 |
14646 |
14578-40 |
|
110 |
40 |
3716 |
3677-96 |
165 |
80 |
8314 |
8275-41 |
220 |
80 |
14726 |
14711-84 |
|
111 |
72 |
3788 |
3745-14 |
166 |
82 |
8396 |
8376-02 |
221 |
192 |
14918 |
14845-89 |
|
112 |
48 |
3836 |
3812-92 |
167 |
166 |
8562 |
8477-24 |
222 |
72 |
14990 |
14980-54 |
|
113 |
112 |
3948 |
3881-31 |
168 |
48 |
8610 |
8579-07 |
223 |
222 |
15212 |
15115-81 |
|
114 |
36 |
3984 |
3950-31 |
169 |
156 |
8766 |
8681-50 |
224 |
96 |
15308 |
15251-68 |
|
115 |
88 |
4072 |
4019-92 |
170 |
64 |
8830 |
8784-55 |
225 |
120 |
15428 |
15388-16 |
|
116 |
56 |
4128 |
4090-14 |
171 |
108 |
8938 |
8888-20 |
226 |
112 |
15540 |
15525-25 |
|
117 |
72 |
4200 |
4160-96 |
172 |
84 |
9022 |
8992-46 |
227 |
226 |
15766 |
15662-94 |
|
118 |
58 |
4258 |
4232-39 |
173 |
172 |
9194 |
9097-33 |
228 |
72 |
15838 |
15801-24 |
|
119 |
96 |
4354 |
4304-43 |
174 |
56 |
9250 |
9202-80 |
229 |
228 |
16066 |
15940-15 |
|
120 |
32 |
4386 |
4377-08 |
175 |
120 |
9370 |
9308-88 |
230 |
88 |
16154 |
16079-67 |
|
121 |
no |
4496 |
4450-33 |
176 |
80 |
9450 |
9415-57 |
231 |
1-20 |
16274 |
16219-80 |
|
122 |
60 |
4556 |
4524-19 |
177 |
116 |
9566 |
9522-87 |
232 |
112 |
16386 |
16360-53 |
|
123 |
80 |
4636 |
4598-66 |
178 |
88 |
9654 |
9630-78 |
233 |
232 |
16618 |
16501-87 |
|
124 |
60 |
4696 |
4673-74 |
179 |
178 |
9832 |
9739-29 |
234 |
72 |
16690 |
16643-82 |
|
125 |
100 |
4796 |
4794-43 |
180 |
48 |
9880 |
9848-42 |
235 |
184 |
16874 |
16786-38 |
|
126 |
36 |
4832 |
4825-72 |
181 |
180 |
10060 |
9958-15 |
236 |
116 |
16990 |
16929-55 |
|
127 |
126 |
4958 |
4902-63 |
182 |
72 |
10132 |
10068-49 |
237 |
156 |
17146 |
17073-32 |
|
128 |
64 |
5022 |
4980-14 |
183 |
120 |
10252 |
10179-44 |
238 |
96 |
17242 |
17217-70 |
|
129 |
84 |
5106 |
5058-26 |
184 |
88 |
10340 |
10290-99 |
239 |
238 |
17480 |
17362-70 |
|
130 |
48 |
5154 |
5136-98 |
185 |
144 |
10484 |
10403-15 |
240 |
64 |
17544 |
17508-30 |
|
131 |
130 |
5284 |
5216-32 |
186 |
60 |
10544 |
10515-92 |
241 |
240 |
17784 |
17654-51 |
|
132 |
40 |
5324 |
5296-26 |
187 |
160 |
10704 |
10629-30 |
242 |
110 |
17894 |
17801-32 |
|
133 |
108 |
5432 |
5376-81 |
188 |
92 |
10796 |
10743-29 |
243 |
162 |
18056 |
17948-74 |
9] "Farey Series" of which the Limiting Number is given 105
Table (continued).
|
n |
r{n) |
T{n) |
^- |
n |
r{n) |
T{n) |
5.. |
n |
r{n) |
T{n) |
i- |
|
244 |
120 |
18176 |
18096-77 |
299 |
264 |
27318 |
27174-65 |
354 |
116 |
38174 |
38091-50 |
|
245 |
168 |
18344 |
18245-41 |
300 |
80 |
27398 |
27356-72 |
355 |
280 |
38454 |
38307-01 |
|
246 |
80 |
18424 |
18394-66 |
301 |
252 |
27650 |
27539-40 |
356 |
176 |
38630 |
38523-12 |
|
247 |
216 |
18640 |
18544-51 |
302 |
150 |
27800 |
27722-69 |
357 |
192 |
38822 |
38739-85 |
|
248 |
120 |
18760 |
18694-97 |
303 |
200 |
28000 |
27906-59 |
358 |
178 |
39000 |
38957-18 |
|
249 |
164 |
18924 |
18846-04 |
304 |
144 |
28144 |
28091-10 |
359 |
358 |
39358 |
3917513 |
|
250 |
100 |
19024 |
18997-72 |
305 |
240 |
28384 |
28276-21 |
360 |
96 |
39454 |
39393-68 |
|
251 |
250 |
19274 |
19150-01 |
306 |
96 |
28480 |
28461-93 |
361 |
342 |
39796 |
39612-83 |
|
252 |
72 |
19346 |
1930290 |
307 |
306 |
28786 |
28648-26 |
362 |
180 |
39976 |
39832-60 |
|
253 |
220 |
19566 |
19456-40 |
308 |
120 |
28906 |
28a35-20 |
3&3 |
220 |
40196 |
40052-97 |
|
254 |
126 |
19692 |
19610-51 |
309 |
204 |
29110 |
29022-75 |
364 |
144 |
40340 |
40273-95 |
|
255 |
128 |
19820 |
19765-23 |
310 |
120 |
29230 |
29210-90 |
365 |
288 |
40628 |
40495-54 |
|
256 |
128 |
19948 |
19920-56 |
311 |
310 |
29540 |
29399-66 |
366 |
120 |
40748 |
40717-74 |
|
257 |
256 |
20204 |
20076-49 |
312 |
96 |
29636 |
2958903 |
367 |
366 |
41114 |
40940-55 |
|
258 |
84 |
20288 |
20233-03 |
313 |
312 |
29948 |
29779-01 |
368 |
176 |
41290 |
41163-96 |
|
259 |
216 |
20504 |
20390-18 |
314 |
156 |
30104 |
29969-59 |
369 |
240 |
41530 |
41387-98 |
|
260 |
96 |
20600 |
20547-94 |
315 |
144 |
30248 |
30160-79 |
370 |
144 |
41674 |
41612-61 |
|
261 |
168 |
20768 |
20706-30 |
316 |
156 |
30404 |
30352-59 |
371 |
312 |
41986 |
41837-85 |
|
262 |
130 |
20898 |
20865-28 |
317 |
316 |
30720 |
30545-00 |
372 |
120 |
42106 |
42063-69 |
|
263 |
262 |
21160 |
21024-86 |
318 |
104 |
30824 |
30738-01 |
373 |
372 |
42478 |
42290-15 |
|
264 |
80 |
21240 |
21185-05 |
319 |
280 |
31104 |
30931-64 |
374 |
160 |
42638 |
42517-21 |
|
265 |
208 |
21448 |
21345-84 |
320 |
128 |
31232 |
31125-87 |
375 |
200 |
42838 |
42744-87 |
|
266 |
106 |
21556 |
21507-25 |
321 |
212 |
31444 |
31320-71 |
376 |
184 |
43022 |
42973-15 |
|
267 |
176 |
21732 |
21669-26 |
322 |
132 |
31576 |
31516-16 |
377 |
336 |
43358 |
43202-04 |
|
268 |
132 |
21864 |
21831-88 |
323 |
288 |
31864 |
31712-22 |
378 |
108 |
43466 |
43431-53 |
|
269 |
268 |
22132 |
2199511 |
324 |
108 |
31972 |
31908-88 |
379 |
378 |
43844 |
43661-63 |
|
270 |
72 |
22204 |
22158-95 |
325 |
240 |
32212 |
3210615 |
380 |
144 |
43988 |
43892-34 |
|
271 |
270 |
22474 |
22323-39 |
326 |
162 |
32374 |
32304 03 |
381 |
252 |
44240 |
44123-65 |
|
272 |
128 |
22002 |
22488-44 |
327 |
216 |
32590 |
32502-52 |
382 |
190 |
44430 |
44355-58 |
|
273 |
144 |
22746 |
22654-10 |
328 |
160 |
32750 |
32701-62 |
383 |
382 |
44812 |
44588-11 |
|
274 |
136 |
22882 |
22820-37 |
329 |
276 |
33026 |
32901-32 |
384 |
128 |
44940 |
44821-25 |
|
275 |
200 |
23082 |
22987-25 |
330 |
80 |
33106 |
33101-63 |
385 |
240 |
45180 |
4")055-00 |
|
276 |
88 |
23170 |
23154-73 |
331 |
330 |
33436 |
33302-55 |
386 |
192 |
45372 |
45289-35 |
|
277 |
276 |
23446 |
23322-82 |
332 |
164 |
33600 |
33504-08 |
387 |
252 |
45624 |
45524-32 |
|
278 |
138 |
23584 |
23491-52 |
333 |
216 |
33816 |
33706-22 |
388 |
192 |
45816 |
45759-89 |
|
279 |
180 |
23764 |
23660-83 |
334 |
166 |
33982 |
33908-96 |
389 |
388 |
46204 |
45996-07 |
|
280 |
96 |
23860 |
23830-75 |
335 |
264 |
34246 |
34112-31 |
390 |
96 |
46300 |
46232-86 |
|
281 |
280 |
24140 |
24001-27 |
336 |
96 |
34342 |
34316-27 |
391 |
352 |
46652 |
46470-25 |
|
282 |
92 |
24232 |
24172-40 |
337 |
336 |
34678 |
34520-84 |
392 |
168 |
46820 |
46708-25 |
|
283 |
282 |
24514 |
24344-14 |
338 |
156 |
34834 |
34726-01 |
393 |
260 |
47080 |
46946-87 |
|
284 |
140 |
24654 |
24516-49 |
339 |
224 |
a5058 |
34931-80 |
394 |
196 |
47276 |
4718609 |
|
285 |
144 |
24798 |
24689-44 |
340 |
128 |
35186 |
35138-19 |
395 |
312 |
47588 |
47425-91 |
|
286 |
120 |
24918 |
24863-00 |
341 |
300 |
35486 |
35345-19 |
396 |
120 |
47708 |
47666-35 |
|
287 |
240 |
25158 |
•25037-18 |
342 |
108 |
35594 |
35552-80 |
397 |
396 |
48104 |
47907-39 |
|
288 |
96 |
25254 |
25211-96 |
343 |
294 |
35888 |
3576101 |
398 |
198 |
48302 |
48149-04 |
|
289 |
272 |
25526 |
25387-34 |
344 |
168 |
36056 |
35969-83 |
399 |
216 |
48518 |
48391-30 |
|
290 |
112 |
25638 |
25563-34 |
345 |
176 |
36232 |
36179-26 |
400 |
160 |
48678 |
48634-17 |
|
291 |
192 |
25830 |
25739-94 |
346 |
172 |
36404 |
36389-30 |
401 |
400 |
49078 |
48877-64 |
|
292 |
144 |
25974 |
25917-15 |
347 |
346 |
367.W |
36599-95 |
402 |
132 |
49210 |
49121-73 |
|
293 |
292 |
26266 |
26094-97 |
348 |
112 |
36862 |
36811-21 |
403 |
360 |
49570 |
49366-42 |
|
294 |
84 |
263r.0 |
26273-40 |
349 |
348 |
37210 1 37023-07 | |
404 |
200 |
49770 |
49611-72 |
|
|
295 |
232 |
26582 |
26452-43 |
350 |
120 |
37330 |
37235-54 |
405 |
216 |
49986 |
49857-62 |
|
296 |
144 |
26726 |
2663207 |
351 |
216 |
37546 |
37448-61 |
406 |
168 |
50154 |
50104-14 |
|
297 |
180 |
26!MJ6 |
26812-32 |
352 |
160 |
37706 |
37662-30 |
407 |
360 |
50514 |
50351-26 |
|
298 |
148 |
27054 |
26993-18 |
353 |
352 |
38058 |
37876-59 |
408 |
128 |
50642 |
50598-99 |
106 On the Number of Fractions contained in any
Table {continued).
[9
|
n |
r(n) |
T(n) |
i." |
n |
r(n) |
T{n) |
> |
n |
r(n) |
T(n) |
>■ |
|
409 |
408 |
51050 |
50847-33 |
464 |
224 |
65630 |
65442-14 |
519 |
344 |
82028 |
81875-93 |
|
410 |
160 |
51210 |
51096-27 |
465 |
240 |
65870 |
65724-52 |
520 |
192 |
82220 |
82191-75 |
|
411 |
272 |
51482 |
51345-83 |
466 |
232 |
66102 |
66007-51 |
521 |
520 |
82740 |
82508-18 |
|
412 |
204 |
51686 |
51595-99 |
467 |
466 |
66568 |
66291-11 |
522 |
168 |
82908 |
82825-21 |
|
413 |
348 |
52034 |
51846-76 |
468 |
144 |
66712 |
66575-31 |
523 |
522 |
83430 |
83142-85 |
|
414 |
132 |
52166 |
52098-14 |
469 |
396 |
67108 |
66860-13 |
524 |
260 |
83690 |
83461-10 |
|
415 |
328 |
52494 |
52350-12 |
470 |
184 |
67292 |
67145-55 |
525 |
240 |
83930 |
83779-95 |
|
416 |
192 |
52686 |
52602-72 |
471 |
312 |
67604 |
67431-58 |
526 |
262 |
84192 |
84099-42 |
|
417 |
276 |
52962 |
52855-92 |
472 |
232 |
67836 |
67718-22 |
527 |
480 |
84672 |
84419-49 |
|
418 |
180 |
53142 |
53109-73 |
473 |
420 |
68256 |
68005-46 |
528 |
160 |
84832 |
84740-17 |
|
419 |
418 |
53560 |
5336415 |
474 |
156 |
68412 |
68293-32 |
529 |
506 |
85338 |
85061-46 |
|
420 |
96 |
53656 |
53619-17 |
475 |
360 |
68772 |
68581-78 |
530 |
208 |
85546 |
85383-36 |
|
421 |
420 |
54076 |
53874-80 |
476 |
192 |
68964 |
68870-85 |
531 |
348 |
85894 |
85705-87 |
|
422 |
210 |
54286 |
54131-04 |
477 |
312 |
69276 |
69160-52 |
532 |
216 |
86110 |
86028-98 |
|
423 |
276 |
54562 |
54387-89 |
478 |
238 |
69514 |
69450-81 |
533 |
480 |
86590 |
86352-70 |
|
424 |
208 |
54770 |
54645-35 |
479 |
478 |
69992 |
69741-70 |
534 |
176 |
86766 |
86677-03 |
|
425 |
320 |
55090 |
54903-42, |
480 |
128 |
70120 |
70033-20 |
535 |
424 |
87190 |
87001-97 |
|
426 |
140 |
55230 |
55162-09 |
481 |
432 |
70552 |
70325-31 |
536 |
264 |
87454 |
87327-51 |
|
427 |
360 |
55590 |
55421-39 |
482 |
240 |
70792 |
70618-03 |
537 |
356 |
87810 |
87653-66 |
|
428 |
212 |
55802 |
55681-26 |
483 |
264 |
71056 |
70911-35 |
538 |
268 |
88078 |
87980-42 |
|
429 |
240 |
56042 |
55941-76 |
484 |
220 |
71276 |
71205-29 |
539 |
420 |
88498 |
88307-79 |
|
430 |
168 |
56210 |
56202-86 |
485 |
384 |
71660 |
71499-83 |
540 |
144 |
88642 |
88635-77 |
|
431 |
430 |
56640 |
56464-57 |
486 |
162 |
71822 |
71794-98 |
541 |
540 |
89182 |
88964-35 |
|
432 |
144 |
56784 |
56726-89 |
487 |
486 |
72308 |
72090-73 |
542 |
270 |
89452 |
89293-54 |
|
433 |
432 |
57216 |
66989-82 |
488 |
240 |
72548 |
72387-10 |
543 |
360 |
89812 |
89623-34 |
|
434 |
180 |
57396 |
57253-36 |
489 |
324 |
72872 |
72684-07 |
544 |
256 |
90068 |
89953-75 |
|
435 |
224 |
57620 |
57517-50 |
490 |
168 |
73040 |
72981-65 |
545 |
432 |
90500 |
90284-77 |
|
436 |
216 |
57836 |
57782-26 |
491 |
490 |
73530 |
73279-84 |
546 |
144 |
90644 |
90616-39 |
|
437 |
396 |
58232 |
58047-62 |
492 |
160 |
73690 |
73578-63 |
547 |
546 |
91190 |
90948-62 |
|
438 |
144 |
58376 |
58313-58 |
493 |
448 |
74138 |
73878-04 |
548 |
272 |
91462 |
91281-46 |
|
439 |
438 |
58814 |
58580-.16 |
494 |
216 |
74354 |
74178-05 |
549 |
360 |
91822 |
91614-91 |
|
440 |
160 |
58974 |
58847-34 |
495 |
240 |
74594 |
74478-67 |
550 |
200 |
92022 |
91948-97 |
|
441 |
252 |
59226 |
59115-14 |
496 |
240 |
74834 |
74779-90 |
551 |
504 |
92526 |
92283-64 |
|
442 |
192 |
59418 |
59383-54 |
497 |
420 |
75254 |
75081-73 |
552 |
176 |
92702 |
92618-91 |
|
443 |
442 |
59860 |
59652-54 |
498 |
164 |
75418 |
75384-18 |
553 |
468 |
93170 |
92954-79 |
|
444 |
144 |
60004 |
59922-16 |
499 |
498 |
75916 |
75687-23 |
554 |
276 |
93446 |
93291-28 |
|
445 |
352 |
60356 |
60192-38 |
500 |
200 |
76116 |
75990-89 |
555 |
288 |
93734 |
93628-38 |
|
446 |
222 |
60578 |
60463-22 |
501 |
332 |
76448 |
76295-15 |
556 |
276 |
94010 |
9396608 |
|
447 |
296 |
60874 |
60734-66 |
502 |
250 |
76698 |
76600-03 |
557 |
556 |
94566 |
94304-39 |
|
448 |
192 |
61066 |
61006-70 |
503 |
502 |
77200 |
76905-52 |
558 |
180 |
94746 |
94643-31 |
|
449 |
448 |
61514 |
61279-36 |
504 |
144 |
77344 |
77211-61 |
559 |
504 |
95250 |
94982-84 |
|
450 |
120 |
61634 |
61552-62 |
505 |
400 |
77744 |
77518-31 |
560 |
192 |
95442 |
95322-98 |
|
451 |
400 |
62034 |
61826-49 |
506 |
220 |
77964 |
77825-62 |
561 |
320 |
95762 |
95663-72 |
|
452 |
224 |
62258 |
62100-97 |
507 |
312 |
78276 |
78133-54 |
562 |
280 |
96042 |
96005-07 |
|
453 |
300 |
62558 |
62376-06 |
508 |
252 |
78528 |
78442-06 |
563 |
562 |
96604 |
96347-03 |
|
454 |
226 |
62784 |
62651-75 |
509 |
508 |
79036 |
78751-19 |
564 |
184 |
96788 |
96689-60 |
|
455 |
288 |
63072 |
62928-05 |
510 |
128 |
79164 |
79060-93 |
565 |
448 |
97236 |
97032-77 |
|
456 |
144 |
63216 |
63204-97 |
511 |
432 |
79596 |
79371-28 |
566 |
282 |
97518 |
97376-55 |
|
457 |
456 |
63672 |
63482-48 |
512 |
256 |
79852 |
79682-23 |
567 |
324 |
97842 |
97720-94 |
|
458 |
228 |
63900 |
63760-61 |
513 |
324 |
80176 |
79993-79 |
568 |
280 |
98122 |
98065-94 |
|
459 |
288 |
64188 |
64039-35 |
514 |
256 |
80432 |
80305-96 |
569 |
568 |
98690 |
98411-55 |
|
460 |
176 |
64364 |
64318-69 |
515 |
408 |
80840 |
80618-74 |
570 |
144 |
98834 |
98757-76 |
|
461 |
460 |
64824 |
64598-64 |
516 |
168 |
81008 |
80932-13 |
571 |
570 |
99404 |
99104-58 |
|
462 |
120 |
64944 |
64879-20 |
517 |
460 |
81468 |
81246-12 |
572 |
240 |
99644 |
99452-01 |
|
463 |
462 |
65406 |
65160-36 |
518 |
216 |
81684 |
81560-72 |
573 |
380 |
100024 |
99800-05 |
9] "Farey Series" of which the Limiting Number is given 107 P Tablk (continued).
|
n |
T(«) |
T{n) |
5- |
n |
r(n) |
r(n) |
3 2 |
n |
r{n) |
T(«) |
3 „ —„ n- 7r- |
|
574 |
240 |
100264 |
100148-70 |
629 |
576 |
120544 |
120260-45 |
684 |
216 |
142380 |
142211-17 |
|
575 |
440 |
100704 |
100497-95 |
630 |
144 |
120688 |
120643-14 |
685 |
544 |
142924 |
142627-30 |
|
576 |
192 |
100896 |
100847-81 |
631 |
630 |
121318 |
121026-44 |
686 |
294 |
143218 |
143044 03 |
|
577 |
576 |
101472 |
101198-28 |
632 |
312 |
121630 |
121410-35 |
687 |
456 |
143674 |
143461-37 |
|
578 |
272 |
101744 |
101549-36 |
633 |
420 |
122050 |
121794-86 |
688 |
536 |
144010 |
143879-32 |
|
579 |
384 |
102128 |
101901-05 |
634 |
316 |
122366 |
122179-98 |
689 |
624 |
144634 |
144297-88 |
|
580 |
224 |
102352 |
102253-34 |
635 |
504 |
122870 |
122565-71 |
690 |
176 |
144810 |
144717-05 |
|
581 |
492 |
102844 |
102606-24 |
636 |
208 |
123078 |
122952-05 |
691 |
690 |
145500 |
145136-82 |
|
582 |
192 |
103a36 |
102959-75 |
637 |
504 |
123582 |
123338-00 |
692 |
344 |
145844 |
145557-20 |
|
583 |
520 |
103556 |
103313-87 |
638 |
280 |
123862 |
123726-55 |
693 |
360 |
146204 |
14597819 |
|
584 |
288 |
103844 |
103668-60 |
639 |
420 |
124282 |
124114-71 |
694 |
346 |
146550 |
146399-79 |
|
585 |
288 |
104132 |
104023-93 |
640 |
256 |
124538 |
124503-48 |
695 |
552 |
147102 |
146821-99 |
|
586 |
292 |
104424 |
104379-87 |
641 |
640 |
125178 |
124892-86 |
696 |
224 |
147326 |
147244-80 |
|
587 |
586 |
105010 |
104736-42 |
642 |
212 |
125390 |
125282-85 |
697 |
640 |
147966 |
147668-22 |
|
588 |
168 |
105178 |
105093-58 |
643 |
642 |
126032 |
125673-44 |
698 |
348 |
148314 |
148092-25 |
|
589 |
540 |
105718 |
105451-35 |
644 |
264 |
126296 |
126064-64 |
699 |
464 |
148778 |
148516-89 |
|
590 |
232 |
105950 |
105809-72 |
645 |
336 |
126632 |
126456-45 |
700 |
240 |
149018 |
148942-14 |
|
591 |
392 |
106342 |
106168-70 |
646 |
288 |
126920 |
126848-87 |
701 |
700 |
149718 |
149367-99 |
|
592 |
288 |
106630 |
106628-29 |
647 |
646 |
127566 |
127241-89 |
702 |
216 |
149934 |
149794-45 |
|
593 |
592 |
107222 |
106888-49 |
648 |
216 |
127782 |
127635-52 |
703 |
648 |
150582 |
150221-52 |
|
594 |
180 |
107402 |
107249-29 |
649 |
580 |
128362 |
128029-76 |
704 |
320 |
150902 |
150649-20 |
|
595 |
384 |
107786 |
107610-70 |
650 |
240 |
128602 |
128424-60 |
705 |
368 |
151270 |
151077-48 |
|
596 |
296 |
108082 |
107972-72 |
651 |
360 |
128962 |
128820-06 |
706 |
352 |
151622 |
151506-37 |
|
597 |
396 |
108478 |
108335-35 |
652 |
324 |
129286 |
12921612 |
707 |
600 |
152222 |
151935-87 |
|
598 |
264 |
108742 |
108698-59 |
653 |
652 |
129938 |
129612-79 |
708 |
232 |
152454 |
152365-98 |
|
599 |
598 |
109340 |
109062-43 |
654 |
216 |
130154 |
130010-07 |
709 |
708 |
153162 |
152796-70 |
|
600 |
160 |
109500 |
109426-88 |
655 |
520 |
130674 |
130407-96 |
710 |
280 |
153442 |
153228-02 |
|
601 |
600 |
110100 |
109791-94 |
656 |
320 |
130994 |
130806-46 |
711 |
468 |
153910 |
153659-95 |
|
602 |
252 |
110352 |
110157-61 |
657 |
432 |
131426 |
131205-56 |
712 |
352 |
154262 |
154092-49 |
|
6a3 |
396 |
110748 |
110523-89 |
658 |
276 |
131702 |
131605-27 |
713 |
660 |
154922 |
154525-64 |
|
604 |
300 |
111048 |
110890-77 |
659 |
658 |
132360 |
132005-59 |
714 |
192 |
155114 |
154959-40 |
|
605 |
440 |
111488 |
111258-26 |
660 |
160 |
132520 |
132406-52 |
715 |
480 |
155594 |
155393-76 |
|
606 |
200 |
111688 |
111626-36 |
661 |
660 |
133180 |
132808-06 |
716 |
356 |
155950 |
155828-73 |
|
607 |
606 |
112294 |
111995-07 |
662 |
330 |
133510 |
133210-20 |
717 |
476 |
156426 |
156264-31 |
|
608 |
288 |
112582 |
112364-39 |
663 |
384 |
133894 |
133612-95 |
718 |
358 |
156784 |
156700-50 |
|
609 |
336 |
112918 |
112734-31 |
664 |
328 |
134222 |
134016-31 |
719 |
718 |
157502 |
157137-30 |
|
610 |
240 |
113158 |
113104-84 |
665 |
432 |
134654 |
134420-28 |
720 |
192 |
157694 |
157574-70 |
|
611 |
552 |
113710 |
113475-98 |
666 |
216 |
134870 |
134824-86 |
721 |
612 |
158306 |
158012-71 |
|
612 |
192 |
113902 |
113847-73 |
667 |
616 |
135486 |
135230-04 |
722 |
342 |
158648 |
158451-33 |
|
613 |
612 |
1 14514 |
114220-09 |
668 |
332 |
135818 |
135635-83 |
723 |
480 |
159128 |
158890-56 |
|
614 |
306 |
114820 |
114593-05 |
669 |
444 |
136262 |
136042-23 |
724 |
360 |
159488 |
159330-40 |
|
615 |
320 |
115140 |
114966-62 |
670 |
264 |
136526 |
136449-24 |
725 |
560 |
160048 |
159770-84 |
|
616 |
240 |
115380 |
115340-80 |
671 |
600 |
137126 |
136856-86 |
726 |
220 |
160268 |
160211-89 |
|
617 |
616 |
115996 |
115715-59 |
672 |
192 |
137318 |
137265-08 |
727 |
726 |
160994 |
160653-55 |
|
618 |
204 |
116200 |
116090-99 |
673 |
672 |
137990 |
137673-91 |
728 |
288 |
161282 |
161095-82 |
|
619 |
618 |
116818 |
1 16466-99 |
674 |
336 |
138326 |
138083-35 |
729 |
486 |
161768 |
161538-69 |
|
620 |
240 |
117058 |
116843-60 |
675 |
360 |
138686 |
138493-40 |
730 |
288 |
162056 |
161982-17 |
|
621 |
396 |
117454 |
117220-82 |
676 |
312 |
138998 |
138904-05 |
731 |
672 |
162728 |
162426-26 |
|
622 |
310 |
117764 |
117598-65 |
677 |
676 |
139674 |
139315-31 |
732 |
240 |
162968 |
162870-96 |
|
623 |
528 |
118292 |
117977-08 |
678 |
224 |
139898 |
139727-18 |
733 |
732 |
163700 |
163316-27 |
|
624 |
192 |
118484 |
118356-12 |
679 |
576 |
140474 |
140139-66 |
734 |
366 |
164066 |
163762-18 |
|
625 |
500 |
118984 |
118735-77 |
680 |
256 |
140730 |
140552-75 |
735 |
336 |
164402 |
164208-70 |
|
626 |
312 |
119296 |
119116-03 |
681 |
452 |
141182 |
140966-44 |
736 |
362 |
164754 |
164655-83 |
|
627 |
360 |
119656 119496-90 |
682 |
300 |
141482 |
141380-74 |
737 |
660 |
165414 |
165103-57 |
|
|
628 |
312 |
119968 119878-37 |
683 |
682 |
142164 |
141795-65 |
738 |
240 |
165654 |
165551-92 |
108 On the Number of Fractions contained in any
. Table (continued).
[9
|
n |
T(n) |
T(n) |
i- |
n |
T(n) |
T(n) |
> |
n |
r(n) |
T(n) |
h" |
|
739 |
738 |
166392 |
166000-87 |
794 |
396 |
191870 |
191629-56 |
849 |
564 |
219340 |
219097-23 |
|
740 |
288 |
166680 |
166450-43 |
795 |
416 |
192286 |
192112-56 |
850 |
320 |
219660 |
219613-66 |
|
741 |
432 |
167112 |
166900-60 |
796 |
396 |
192682 |
192596-17 |
851 |
792 |
220452 |
220130-71 |
|
742 |
312 |
167424 |
167351-38 |
797 |
796 |
193478 |
193080-39 |
852 |
280 |
220732 |
220648-36 |
|
743 |
742 |
168166 |
167802-77 |
798 |
216 |
193694 |
193565-21 |
853 |
852 |
221584 |
221166-62 |
|
744 |
240 |
168406 |
168254-76 |
799 |
736 |
194430 |
194050-64 |
854 |
360 |
221944 |
221685-48 |
|
745 |
592 |
168998 |
168707-36 |
800 |
320 |
194750 |
194536-67 |
855 |
432 |
222376 |
222204-96 |
|
746 |
372 |
169370 |
169160-57 |
801 |
528 |
195278 |
195023-32 |
856 |
424 |
222800 |
22272504 |
|
747 |
492 |
169862 |
169614-39 |
802 |
400 |
195678 |
195510-57 |
857 |
856 |
223656 |
223245-73 |
|
748 |
320 |
170182 |
170068-82 |
803 |
720 |
196398 |
195998-43 |
858 |
240 |
223896 |
223767-03 |
|
749 |
636 |
170818 |
170523-85 |
804 |
264 |
196662 |
196486-90 |
859 |
858 |
224754 |
224288-93 |
|
750 |
200 |
171018 |
170979-50 |
805 |
528 |
197190 |
196975-98 |
860 |
336 |
225090 |
224811-44 |
|
751 |
750 |
171768 |
171435-75 |
806 |
360 |
197550 |
197465-66 |
861 |
480 |
225570 |
225334-56 |
|
752 |
368 |
172136 |
171892-61 |
807 |
536 |
198086 |
197955-96 |
862 |
430 |
226000 |
225858-29 |
|
753 |
500 |
172636 |
172350-07 |
808 |
400 |
198486 |
198446-86 |
863 |
862 |
226862 |
226382-62 |
|
754 |
336 |
172972 |
172808-14 |
809 |
808 |
199294 |
198938-37 |
864 |
288 |
227150 |
226907-57 |
|
755 |
600 |
173572 |
173266-82 |
810 |
216 |
199510 |
199430-48 |
865 |
688 |
227838 |
227433-12 |
|
756 |
216 |
173788 |
173726-11 |
811 |
810 |
200320 |
199923-21 |
866 |
432 |
228270 |
227959-28 |
|
757 |
756 |
174544 |
174186-01 |
812 |
336 |
200656 |
200416-54 |
867 |
544 |
228814 |
228486-05 |
|
758 |
378 |
174922 |
174646-52 |
813 |
540 |
201196 |
200910-48 |
868 |
360 |
229174 |
229012-43 |
|
759 |
440 |
175362 |
175107-63 |
814 |
360 |
201556 |
201405-03 |
869 |
780 |
229954 |
229541-41 |
|
760 |
288 |
175650 |
175569-35 |
815 |
648 |
202204 |
201900-19 |
870 |
224 |
230178 |
230070-01 |
|
761 |
760 |
176410 |
176031-68 |
816 |
256 |
202460 |
202395-95 |
871 |
792 |
230970 |
230599-21 |
|
762 |
252 |
176662 |
176494-62 |
817 |
756 |
203216 |
202892-32 |
872 |
432 |
231402 |
231129-02 |
|
763 |
648 |
177310 |
176958-16 |
818 |
408 |
203624 |
203389-30 |
873 |
576 |
231978 |
231659-43 |
|
764 |
380 |
177690 |
177422-31 |
819 |
432 |
204056 |
203886-89 |
874 |
396 |
232374 |
232190-46 |
|
765 |
384 |
178074 |
177887-07 |
820 |
320 |
204376 |
204385-09 |
875 |
600 |
232974 |
232722-09 |
|
766 |
382 |
178456 |
178352-44 |
821 |
820 |
205196 |
204883-89 |
876 |
288 |
233262 |
233254-33 |
|
767 |
696 |
179152 |
178818-42 |
822 |
272 |
205468 |
205383-30 |
877 |
876 |
234138 |
232787-18 |
|
768 |
256 |
179408 |
179285-00 |
823 |
822 |
206290 |
205883-32 |
878 |
438 |
234576 |
234320-64 |
|
769 |
768 |
180176 |
179752-19 |
824 |
408 |
206698 |
206383-95 |
879 |
584 |
235160 |
234854-70 |
|
770 |
240 |
180416 |
180219-99 |
825 |
400 |
207098 |
206885-19 |
880 |
320 |
235480 |
235389-37 |
|
771 |
512 |
180928 |
180688-40 |
826 |
348 |
207446 |
207387-03 |
881 |
880 |
236360 |
235924-65 |
|
772 |
384 |
181312 |
181157-42 |
827 |
826 |
208272 |
207889-48 |
882 |
252 |
236612 |
236460-54 |
|
773 |
772 |
182084 |
181627-04 |
828 |
264 |
208536 |
208392-54 |
883 |
882 |
237494 |
236997-04 |
|
774 |
252 |
182336 |
182097-27 |
829 |
828 |
209364 |
208896-21 |
884 |
384 |
237878 |
237534-14 |
|
775 |
600 |
182936 |
182568-11 |
830 |
328 |
209692 |
206400-49 |
885 |
464 |
238342 |
238071-85 |
|
776 |
384 |
183320 |
183039-56 |
831 |
552 |
210244 |
209905-37 |
886 |
442 |
238784 |
238610-17 |
|
777 |
432 |
183752 |
183511-61 |
832 |
384 |
210628 |
210410-86 |
887 |
886 |
239670 |
239149-10 |
|
778 |
388 |
184140 |
183984-28 |
833 |
672 |
211300 |
210916-96 |
888 |
288 |
239958 |
239688-64 |
|
779 |
720 |
184860 |
184457-55 |
834 |
276 |
211576 |
211423-67 |
889 |
756 |
240714 |
240228-78 |
|
780 |
192 |
185052 |
184931-43 |
835 |
664 |
212240 |
211930-98 |
890 |
352 |
241066 |
240769-53 |
|
781 |
700 |
185752 |
185405-92 |
836 |
360 |
212600 |
212438-91 |
891 |
540 |
241606 |
241310-89 |
|
782 |
352 |
186104 |
185881-01 |
837 |
540 |
213140 |
212947-44 |
892 |
444 |
242050 |
241852-86 |
|
783 |
504 |
186608 |
186356-71 |
838 |
418 |
213558 |
213456-58 |
893 |
828 |
242878 |
242395-43 |
|
784 |
336 |
186944 |
186833-02 |
839 |
838 |
214396 |
213966-32 |
894 |
296 |
243174 |
242938-62 |
|
785 |
624 |
187568 |
187309-94 |
840 |
192 |
214588 |
214476-68 |
895 |
712 |
243886 |
243482-41 |
|
786 |
260 |
187828 |
187787-47 |
841 |
812 |
215400 |
214987-64 |
896 |
384 |
244270 |
244026-81 |
|
787 |
786 |
188614 |
188265-60 |
842 |
420 |
215820 |
215499-21 |
897 |
528 |
244798 |
244571-81 |
|
788 |
392 |
189006 |
188744-34 |
843 |
560 |
216380 |
216011-39 |
898 |
448 |
245246 |
245117-43 |
|
789 |
524 |
189530 |
189223-69 |
844 |
420 |
216800 |
216524-18 |
899 |
840 |
246086 |
245663-65 |
|
790 |
312 |
189842 |
189703-65 |
845 |
624 |
217424 |
217037-57 |
900 |
240 |
246326 |
246210-48 |
|
791 |
672 |
190514 |
190184-22 |
846 |
276 |
217700 |
217551-58 |
901 |
832 |
247158 |
246757-91 |
|
792 |
240 |
190754 |
190665-39 |
847 |
660 |
218360 |
218066-19 |
902 |
400 |
247558 |
247;i05-96 |
|
793 |
720 |
191474 |
191147-17 |
848 |
416 |
218776 |
218581-40 |
903 |
504 |
248062 |
247854-61 |
9] "Farey Series" of ichich the Limiting Number is given 109
Table* (continued).
|
n |
T(n) |
Tin) |
1- |
n |
r(n) |
T(n) |
!>'■ |
n |
x(«) |
T{n) |
i- |
|
904 |
448 |
248510 |
248403-88 |
937 |
936 |
267256 |
266870-57 |
1 1 970 1 384 |
286076 |
285999-30 |
|
|
905 720 '249230 24895375 |
938 |
396 |
267652 |
267440-51 |
971 1 970 287046 |
286589-30 |
|||||
|
'906 300 249530 249504-22 |
939 |
624 |
268276 |
268011-05 |
972 1 324 287370 |
287179-90 |
|||||
|
907 906 250436 250055-31 |
940 |
368 |
268644 |
268582-19 |
973 828 288198 |
287771-11 |
|||||
|
908 , 452 250888 250607 00 |
941 |
940 |
269584 |
269153-95 |
974 i 486 288684 |
288362-92 |
|||||
|
909 1 600 251488 251159-31 |
942 |
312 |
269896 |
269726-31 |
975 1 480 , 289164 |
288955-35 |
|||||
|
910 288 251776 25171222 |
943 |
880 |
270776 |
270299-28 1 |
976 i 480 1 289644 289548-39 |
||||||
|
911 ,910 |
252686 252265-73 |
944 |
464 |
271240; 270872-861 |
977 976 290620 ! 290142-03 |
||||||
|
912 288 |
252974 252819-86 |
945 |
432 |
271672 |
271447-05 |
978 ! 324 1 290944 |
290736-28 |
||||
|
913 820 |
253794 ) 253374-59 |
946 |
420 |
272092 |
272021-84 |
979 880 1 291824 |
291331-13 |
||||
|
914 456 |
254250 253929-93 |
947 |
946 |
273038 |
272597-25 |
980 336 292160 |
291926-60 |
||||
|
915 480 |
254730 254485-88 |
948 |
312 |
273350 |
273173-26 |
981 648 |
292808 |
292522-67 |
|||
|
916 456 |
255186 ( 255042-44 |
949 |
864 |
274214 |
273749-88 |
982 490 |
293298 |
293119-35 |
|||
|
917 780 |
255966 255599-61 |
950 |
360 |
274574 |
274327-10 |
983 ; 982 |
294280 |
293716-64 |
|||
|
918 i 288 |
256254 256157-38 |
951 |
632 |
275206 |
274905-94 |
984 320 |
294600 |
294314-54 |
|||
|
919 918 |
257172 |
256715-76 |
952 |
384 |
275590 |
275483-38 |
985 784 |
295384 |
29491304 |
||
|
920 352 |
257524 |
257274-75 |
953 |
952 |
276542 276062-43 |
986 448 |
295832 |
295512-15 |
|||
|
921 |
612 |
258136 |
257834-34 |
954 |
312 |
276854 276642-09 |
987 552 |
296384 |
296111-87 |
||
|
922 |
460 |
258596 |
258394-55 |
955 |
760 |
277614 277222-36 |
988 432 |
296816 |
296712-20 |
||
|
923 |
840 |
259436 |
258955-36 |
956 |
476 |
278090 277803-23 |
989 924)297740 |
297313-14 |
|||
|
924 |
240 |
259676 |
259516-78 |
957 |
560 |
278650 278384-71 |
990 |
240 |
297980 |
297914-68 |
|
|
925 |
720 |
260396 |
260078-81 |
958 478 |
279128 j 278966-80 |
991 |
990 |
298970 |
298516-83 |
||
|
926 |
462 |
260858 |
260641-45 |
959 816 |
279944 279549-50 |
992 |
480 |
299450 |
299119-59 |
||
|
927 |
612 |
261470 |
261204-69 |
960 256 |
280200 280132-81 |
993 |
660 |
300110 i 299722-96 |
|||
|
f»28 |
448 |
261918 |
261768-55 |
961 ; 930 |
281130 |
280716-72 |
994 |
420 |
300530 1 300326-94 |
||
|
.•J9 |
928 |
262846 |
262333-01 |
962 i 432 |
281562 |
281301-24 |
995 |
792 1 301322 |
300931-52 |
||
|
j:jo |
240 |
263086 |
262898-07 |
963 , 636 |
282198 |
281886-37 |
996 |
328 |
301650 |
301536-71 |
|
|
931 |
756 |
263842 |
263463-75 |
964 |
480 |
282678 |
28247211 |
997 |
996 |
302646 |
302142-51 |
|
932 |
464 |
264306 |
264030-03 |
965 |
768 |
283446 |
283058-46 |
998 |
498 |
303144 |
302748-92 |
|
933 |
620 |
264926 |
264596-93 |
966 |
264 |
283710 |
283645-41 |
999 |
648 1 3a3792 |
3a3355-93 |
|
|
934 |
466 |
265392 |
265164-43 |
967 |
966 |
284676 |
( 284232-97 |
1000 |
400 1 304192 |
303963-55 |
|
|
935 |
640 |
266032 |
265732-53 |
968 |
440 |
285116 |
t 284821-14 |
||||
|
936 |
288 |
266320 |
266301-25 |
969 |
576 |
285692 |
1 285409-92 |
* Id the extended as well as in the original Table it will be seen that the som-totient is always intermediate between 3/t' . n" and 3/t' . (n + 1)'.
The formula of verification applied at every tenth step to the T colnmn preclndes the possibility of the existence of other than typographical errors or errors of transcription. Accnmolative errors are rendered impossible.
10.
ON THE EQUATION TO THE SECULAR INEQUALITIES IN THE PLANETARY THEORY.
[Philosophical Magazine, xvi. (1883), pp. 267 — 269.]
A VERY long time ago I gave, in this Magazine*, a proof of the reality of the roots in the above equation, in which I employed a certain property of the square of a-symmetrical matrix which was left without demonstration. I will now state a more general theorem concerning the product of any two matrices of which that theorem is a particular case. In what follows it is of course to be understood that the product of two matrices means the matrix corre- sponding to the combination of two substitutions which those matrices represent.
It will be convenient to introduce here a notion (which plays a conspicuous part in my new theory of multiple algebra), namely that of the latent roots of a matrix — latent in a somewhat similar sense as vapour may be said to be latent in water or smoke in a tobacco-leaf If from each term in the diagonal of a given matrix, X be subtracted, the determinant to the matrix so modified will be a rational integer function of X ; the roots of that function are the latent roots of the matrix ; and there results the important theorem that the latent roots of any function of a matrix are respectively the same functions of the latent roots of the matrix itself: for example, the latent roots of the square of a matrix are the squares of its latent roots.
The latent roots of the product of two matrices, it may be added, are the same in whichever order the factors be taken. If, now, m and n be any two matrices, and M=mn or nm, I am able to show that the sum of the products of the latent roots of M taken i together in every possible way is equal to the sum of the products obtained by multiplying every minor determinant of the ith order in one of the two matrices m, n by its altruistic opposite in the other : the reflected image of any such determinant, in respect to the principal diagonal of the matrix to which it belongs, is its proper opposite, and the corresponding determinant to this in the other matrix is its altruistic opposite.
[• Vol. I. of this Keprint, p. 378.]
10] On the Equation to the Secular Inequalities, etc. Ill
The proof of this theorem will be given in my large forthcoming memoir on Multiple Algebra designed for the American Journal of Mathematics.
Suppose, now, that m and n are transverse to one another, that is, that the lines in the one are identical with the columns in the other, and vice versd, then any determinant in m becomes identical with its altruistic opposite in n ; and furthermore, if in be a symmetrical matrix, it is its own transverse. Consequently we have the theorem (the one referred to at the outset of this paper) that the sum of the i-ary products of the latent roots of the square of a symmetrical matrix (that is, of the squares of the roots of the matrix itself) is equal to the sum of the squares of all the minor determinants of the order i in the matrix ; whence it follows, from Descartes's theorem, that when all the terms of a symmetrical matrix are real, none of its latent roots can be pure imaginaries, and, as an easy inference, cannot be any kind of imaginaries ; or, in other words, all the latent roots of a symmetrical matrix are real, which is Laplace's theorem.
I may take this opportunity of stating the important theorem that if \,, \,, ... Xi are the latent roots of any matrix m, then
_ (m-\,)(m-X,)...(m-X,)^^
^"-^(\-x,)(x.-x.)...(x.-x,)'''^
This theorem of course presupposes the rule first stated by Prof. Cayley {Phil. Trans. 1857) for the addition of matrices.
When any of the latent roots are equal, the formula must be replaced by
another obtained from it by the usual method of infinitesimal variation. If 1
(fun = wi", it gives the expression for the wth root of the matrix ; and we see
that the number of such roots is <<>•, where i is the order of the matrix.
When, however, the matrix is unitary, that is, all its terms except the
diagonal ones are zeros, or zeroidal, that is, when all its terms are zeros, this
conclusion is no longer applicable, and a certain definite number of arbitrary
quantities enter into the general expressions for the roots.
The case of the extraction of any root of a unitary matrix of the second order was first considered and successfully treated by the late Mr Babbage ; it reappears in M. Serret's Cours d'Algebre superieure. This problem is of
course the same as that of finding a function j of any given order of
periodicity. My memoir will give the solution of the corresponding problem for a matrix of any order. Of the many unexpected results which I have obtained by my new method, not the least striking is the rapprochement which it establishes between the theory of Matrices and that of Invariants. The theory of invariance relative to associated Matrices includes and transcends that relative to algebraical functions.
11.
ON THE INVOLUTION AND EVOLUTION OF QUATERNIONS. [Philosophical Magazine, xvi. (1883), pp. 894 — -396.]
The subject-matter of quaternions is really nothing more nor less than that of substitutions of the second order, such as occur in the familiar theory of quadratic forms. A linear substitution of the second order is in essence identical with a square matrix of the second order, the law of multiplication between one such matrix and another being understood to be the same as that of the composition of one substitution with another, and therefore depending on the order of the factors ; but as regards the multiplication of three or more matrices, subject to the same associative law as in ordinary algebraical multiplication.
Every matrix of the second order may be regarded as representing a quaternion, and vice versa; in fact if, using i to denote V(— 1), we write a matrix m of the second order under the form
we have by definition,
1 0 ^ t 0 0 1 . 0 z
where a = ^ ^, ^ = o_., 7 = _i o' ^=^0"
Now a- = a, yS'-' = 7^ = g2 _ _ jK^
a/3 = /3a = /3, ay = 7a = 7, aS = 8a = S,
^7 = _7^ = a, 7S = -S7 = ^, 8;8 = -^S = 7;
so that we may for a, /8, 7, S, substitute 1, /(, k, I, four symbols subject to the same laws of self-operation and mutual interaction as unity and the three Hamiltonian symbols. Now I have given the universal formula for expressing any given function of a matrix of any order as a rational function of that matrix and its latent roots; and consequently the 5th power or root of any
|
a + hi, |
c + |
di, |
|
|
- c -f di. |
a- |
hi, |
|
|
m |
= aa -f 6^ -f- C7 -f dh. |
||
|
/3 = |
i 0 "0 -i' |
7 = |
0 1 -1 0' |
11] On the Involution and Evolution of Quaternions 113
quadratic matrix, and therefore of any quaternion, is known. As far as I am informed, only the square root of a quaternion has been given in the text- books on quaternions, notably by Hamilton in his Lectures on Quaternions.
The latent roots of m are the roots of the quadratic equation
X'' -2a\ + a? +11' + cr + d" = 0. The general formula
where i is the order of the matrix m, when i = 2 and ^m = mi, becomes
A-x — A,2 Aj — Aj
where Xj, X, are the roots of the above equation. If fi, is the modulus of the quaternion, namely is V(«' + 6* + C + d"), and /a cos ^ = a, the latent roots \, , Xj assume the form
fi (cos 6 ±i sin 6).
When the modulus is zero the two latent roots are equal to one another, and to o, the scalar of the quaternion ; so that in this case the ordinary theory of vanishing fractions shows that
\a q J
In the general case there are q' roots of the gth order to a quaternion. Calling
1
77
- = w, and writing m» = Am + B,
^ cos f - + 2^•w j + i sin f - + 2k<o\ - cos ( - + 2i'<a j + i sin [- + 2k' oA fi 2i sin 6
(^- 0+ 2k<o^ + i sin (^ — ^ 0 + 2ka))
cos [
9
1 -cos(^l^0 + 2k'<o) + i3in(^!^0 + 2k'co] B = -ai L2 I L2 l.
'^ 2i sin 0
For the q system of values k^k" =1, 2,3 ...q, the coefficients A and jB will be real, for the other q' — q systems of values imaginary ; so that there are q quaternion-proper i/th roots of a quaternion-proper in Hamilton's sense, and g*— 9 of the sort which, by a most regrettable piece of nomenclature, he terms
bi-quateruions. The real or proper-quaternion values of mi are
1 sin 0
8 IV.
jsin (- -1- 2k<o^ - + sin (^—^ 0 + 2Aa))[ ,
114 On the Involution and Evolution of Quaternions [11
111 meaning the or (when there is an alternative) either real value of the gth root of the modulus.
In the qth root (or power) of a quaternion m, the form Am + B shows that the vector-part remains constant to an ordinary algebraical factor pres ; and we know a priori from the geometrical point of view that this ought to be the case. When the vector disappears a porism starts into being; and besides the values of the roots given by the general formula, there are others involving arbitrary parameters. Babbage's famous investigation of the form
of the homographic function of -^ of x, which has a periodicity of any
given degree q, is in fact (surprising as such a statement would have appeared to Babbage and Hamilton) one and the same thing as to find the yth root of unity under the form of a quaternion !
It is but justice to the eminent President of the British Association to draw attention to the fact that the substance of the results here set forth (although arrived at from an independent and more elevated order of ideas) may be regarded as a statement (reduced to the explicit and most simple form) of results capable of being extracted from his memoir on the Theory of Matrices, Phil. Trans. Vol. CXLViii. (1858) {vide pp. 32 — 34, arts. 44 — 49).
12.
ON THE INVOLUTION OF TWO MATRICES OF THE SECOND ORDER.
[British Association Report, Southport (1883), pp. 430 — 432.]
If ot, » be two matrices of any order i, then, taking the determinant of the matrix z + i/n + xm, there results a ternary quantic in the variables X, y, z, which may be termed the quantic of the corpus to, n.
In what follows I confine myself almost exclusively to the case of a corpus of the second order ; the quantic may be written
z^ + ^hzx + Icyz + da^ + lexy +ff :
it is then easy to establish the identical relations
to' - Ihm + d = 0,
TOW + nm — Ihn — 2cni + 2e = 0,
n' - 2cn +/= 0.
It hence easily appears that any given function of to, n can, by aid of the five parameters b, c, d, e,/, be expressed in the form A + Bm + On + Dmn.
This form containing four arbitrary constants, it follows that in general any given matrix of the second order can be expressed as a function of m and n ; for there will be four linear equations between A, B, C, D and the four elements of the given matrix. But this statement is subject to two cases of exception.
The first of these is when n and m are functions of one another : for in this case A + Bm + Cn + Dmn is reducible to the form P + Qm, and there will be only two disjHJsable constants wherewith to satisfy the four linear equations.
The second case is when the determinant of the fourth order formed by
the elements of the four matrices
m, n = \
1, TO
n, vm
vanishes; writing
8—2
116 On the Involution of Two MatHces of [12
respectively, it is not difficult to show that the value of this determinant is
«2T, - T^^s)'' + {(«, - U) Tj - (t, - T,) «,} ((«, - O T, - (t, - T4) U].
This expression is a function of the five parameters h, c, d, e,/, as may be shown in a variety of ways.
Thus it is susceptible of easy proof that if /x, , Mj are the roots of the equation fi' — 2bfj, + d = 0, and i/,, v^ the roots of the equation v' — 2cv +f= 0, then, the two matrices being related as above, we must have
(m — fii) (n — r,) = 0, (n - i/j) (m - /ia) = 0, and consequently, by virtue of the middle one of the three identities,
/i, r, + fi^Vi— 2e = 0. Writing this in the form
(/ill/, + /j-^Vi — 2e) {fiiVi + fiiVi — 2e) = 0, this is 4e= - 2e . 46c + {fi^ + ^if) v^v^ + (j^i^ + vi) /ii/xj = 0,
which gives e' - 26ce + iPf +d'd-df=0;
the function on the left hand is the invariant (discriminant) of the ternary quantic appurtenant to the corpus, and we have this invariant =0 as the necessary and sufficient condition of the involution of the elements of the corpus ; the invariant in question is for this reason called the involutant.
Expressing the values of the coefficients in terms of the elements of the two matrices, namely
26 = A + ti, 2c = Ti + T4,
d^titi—tit^, 2e=«iT4 + Ti^4— tjTs— isTo, /=TjT4 — T2T3,
it at once appears that the two expressions for the involutant are, to a numerical factor prhs, identical.
It can be shown d priori that the involutant of a corpus of the second order must be expressible in terms of the coefficients of the function ; and therefore, being obviously invariantive in regard to linear substitutions . impressed on m, n, it must be also invariantive for linear substitutions impressed on z, x, y, and must therefore be the invariant of the function. The corresponding theorem is not true, it should be observed, for the involutant of a corpus beyond the second order ; for such involutant cannot in general be expressed in terms of the coefficients of the function.
The expression for the involutant in terms of the <'s and t's may also be obtained directly from the equation (m — /li,) (n — i/,) = 0. To this end it is only necessary to single out any term of the matrix represented by the left- hand side of the equation and equate it to zero : the resulting equation rationalised will be found to reproduce the expression in question.
12] the Second Order 117
I have thus indicated four methods of obtaining the involutant to a matrix -corpus of the second order ; but there is yet a fifth, the simplest of all, and the most suggestive of the course to be pursued in investigating the higher order of involutants.
I observe that for a corpus of any order the function mn—nm is invarian- tive for any linear substitution impressed on in and n. Its determinant will therefore be an invariant for any substitution impressed on m and n. When VI and n are of the second order, reducing each term of {mn — nniy, that is mnmn — mnhn — nm^n + nmnm, and of mn — nm, by means of the three identical equations, to the form of a linear function of nm, m, n, 1, it will be found without diflBculty that there results the identical equation
(mn — nmy + 1=0,
the coefficient of mn — nm vanishing. Consequently the determinant of the matrix mn — nm is equal to /, which on calculation will be found to be identical with the invariant of the ternary quadric function.
It is obvious from the three identical equations that if m, n are in involution — that is, if their involutant is zero — every rational and integral function of m, n will be in involution with every other rational and integral function of m, n. Hence follows this new and striking theorem concerning matrices of the second order : If /(j«, n) and <f> (m, n) are any rational functions whatever of m, n, the determinant to the matrix m,n — mn is contained as a factor in the determinant to the matrix f<f> — <^/.
It may be noticed that /, <f> need not be integer functions by stipulation, because any linear function of mn, m, n, 1, divided anteriorly or posteriorly by a second like function, can it-self be expressed as a linear function of the same four terms.
As a very simple example of the theorem, observe that the determinant of m^n — mnm will contain as a factor the determinant of mn — nm.
13.
SUR LES QUANTIT^S FORMANT UN GROUPE DE NONIONS ANALOGUES AUX QUATERNIONS DE HAMILTON.
[Comptes Rendus, xcvii. (1883), pp. 1336—1340,]
On sait qu'on peut tout a fait (et tres avantageusetnent) changer la base de la theorie des quaternions en consid^rant les trois symboles i, j, k de Hamilton comme des matrices binaires.
Si h, j sont des matrices binaires qui satisfont a I'^quation hj = —jh, on deraontre faeilement que, en ecartant le cas ou hj =jh = 0, h^ et ^■^ seront de la forme
c 0 7 0
0 c' 0 7
o'est-k-dire cu, yu, ou u est I'unit^ binaire
1 0 0 1
On peut ajouter, si Ton veut, les deux conditions c^ = l, 7- = !; alors, en
supprimant, pour plus de brievete, le u, qui jouit de propri^tes tout a fait
analogues k celles de I'unite ordinaire, on obtient faeilement les Equations
connues _ _ _
/i,2=i^ j^=l_ k^ = l^
hj = —jh = k, jk = — kj = i, ki = — ik =j.
De plus, en supposant que {i, j) soit un systfeme particulier qui satisfait a r^quation ij = —ji, on peut deduire les valeurs universelles de /, J qui satisfont a I'^quation IJ = — JI en termes de i, j. En efifet, on d^montre rigoureusement que, en dcartant toujours la solution mn = nm = 0, on aura
I = ai + hj + cij,
J=m + ^j + rij,
avec la seule condition aa + b^ + cy = 0. De plus, si Ton suppose t' =j' = u et aussi I^ = J' = u, on aura
a» + 6^ + c» = l, a» + ^' + 7*=l,
13] Sitr les qiiantites formant un groupe de nonions, etc. 119
de sorte que, en ecrivant ij = k, IJ = K et K = Ai + Bj+ Ck, la matrice
a b c
a /3 7
ABC
formera une matrice orthogonale. Une solution, parnii les plus simples, des equations ij = —ji, i' = u.,j- = u, est la suivante :
1 =
et cons^quemment
0 -0
J =
0 -1
1 0
k = ij =
0 -0
-0 0 ou^=V'(-l)-
En dcrivant une quantity binormale quelconque (c'est-i-dire une matrice binaire) sous la forme
a + b0, - c - d0,
c -d0, a- b0, on voit qu'elle pent ^tre mise sous la forme au + bi + cj + dk, oil il est souveut commode de supprimer (c'est-k-dire de sous-entendre) sans ^crire I'unite binaire u.
On peut construire d'une maniere tout k fait analogue un systeme de nonions en considerant I'^quation m — pn, ou to, n sont des matrices ternaires et p une racine cubique primitive de I'unitd (voir* la Circular du Johns Hopkins University qui va prochainement paraitre), en prenant pour les nonions foudameutaux u (I'unite ternaire)
10 0
0 10
0 0 1 et les huit matrices m, m'; n, n'; m'n, mn'; mn, m'n" construites avee les valeurs les plus simples de m, n qui satisfont aux equations
Les valeurs
nm = pmn, nr
m'
|
1 |
0 |
0 |
|
|
0 |
p |
0 |
et n = |
|
0 |
0 |
p' |
n' = u.
0 1 0
0 0 /}
p» 0 0
peuvent Stre prises pour les valeurs basiques du systfeme de nonions.
Une quantity ternaire (c'est-a-dire une matrice) quelconque s'exprime alors sous la forme
o + 6to + /9to' + cn + Tw' + dm'w + hmn^ + emn + em?n^ ; [* VoL m. of this Reprint, p. 647. Also below, p. 122.]
120 Sur les qtiantit^s formant un groupe de nonions [13
mais, quand cette matrice M est capable de s'associer avec une autre N dans r^uation NM = pMN, alors il devient n^cessaire que
a = 0, 6/3 + C7 + dS + ee = 0.
Je n'entrerai pas ici dans les details de la m^thode d'associer la solution gdndrale de I'equation NM = pMN avec une solution quelconque particuliere de cette Equation, mais je me bornerai k expliquer quelles sont les conditions auxquelles les Elements de M Qt Aq N doivent satisfaire afin que cette Equation ait lieu.
M. Cayley a resolu la question analogue pour les matrices binaires dans le beau M6moire, qu'il a publid dans les Transactions of the Royal Society de 1858. En supposant que m et n sont les matrices
ah a' b'
c d c' d'
il trouve que, afin que nm = — mn, il faut avoir
a + d = 0, a' + d' = 0, aa' + be' + cb' + dd' = 0.
Au lieu de cette troisieme Equation (en la combinant avec les deux precddentes), on pent ecrire
ad' + a'd - be' - b'c = 0.
Alors ces trois conditions equivalent a dire que le determinant de la matrice osu + my + nz {u etant I'unite binaire), qui, en general, est de la forme
ar^ + -IBxy + 2Cxz + Dy^ + 2Eyz + Fz^,
se r^duira k la forme
a? + By'' + J^^^
car, dans le determinant de arit + my + nz, c'est-^-dire de
a; + 03/ + a!z by + b'z
cy + c'z x + dy + d'z
les coefficients de ocy, xz, yz seront ^videmment
a + d, a' + d', ad' + a'd - be' - b'c respectivement.
Passons au cas de m et n, matrices ternaires qui satisfont k I'equation
nm = pmn. Formons le determinant de a^ + ym + zn, ou u represente I'unite ternaire
10 0
0 1 0 0
13]
aiialogues mix quaternions de Hamilton
121
Ce determinant sera de la forme
a? + Wa?y + ^Ca?z + SBxf + QExyz + ^Fxz'- + Gf + 2Hfz + SKyz^ + Lz',
et je trouve que, dans le cas suppose, il faut que les sept conditions souscrites soient satisfaites; B = 0, C=0, D = 0, E = 0, F=0 H=0, K = 0, de sorte que la fonction en x, y, z devient une somme de trois cubes, mais ces sept conditions, qu'on pourrait nommer conditions paramdtriques, quoique necessaires, ne sont pas suflBsantes; il faut y ajouter une huitieme condition que je nommerai Q = 0.
Pour former Q, voici la manifere de proc^der : En supposant que
a b c a! y
et n<
»»= d e f
g h h
on ecrit, au lieu de m, son transversal
I a' d' b' e'
a d' 9
e h'
c
r k'
h'
et Ton forme neuf produits en multipliant chaque d(?terrainant mineur du second ordre contenu dans m avec le determinant mineur semblablement pos^ dans le transversal de n : la somme de ces neuf produits est Q.
Ces huit conditions que je d^montre sont suffisantes et necessaires (en ecartant corame auparavant le cas oil nm = mn = 0) pour que nm<=pmn.
On pourrait tres bien se demander ce qui arrive dans le cas oil les sept conditions parametriques sont satisfaites, mais non pas la huitifeme condition supplementaire.
Dans ce cas, je trouve* que mn et nm restent fonctions I'une et I'autre et qu'on aura
nm = A + B^mn + G(mny, mn = — A + B^nm + G (nmf, oil 5,, 5, sont les racines de I'equation alg^brique
5'' + 5+1 =0,
A, G etant deux quantit^s arbitraires et ind^pendantes, sauf que I'une d'elles ne pent pas s'evanouir sans I'autre, les deux s'^vanouissant ensemble pour le cas (et seulemeut pour le cas) ou Q (qui foumit la condition supplementaire) s'^vanouit.
[* See footnote [f], p. 154 below.]
14.
ON QUATERNIONS, NONIONS, SEDENIONS, ETC.
[Johns Hopkins University Circulars, ill. (1884), pp. 7 — 9.]
(1) Suppose that m and n are two matrices of the second order.
Then if we call the determinant of the matrix a; + my + nz,
x^ + 2bxy + 2cxz + dy" + 2eyz +fz-,
the necessary and sufficient conditions for the subsistence of the equation nm = — mn is that b = 0, c = 0, e = 0, and if we superadd the equations ?n^ 4- 1 = 0, n" + 1 = 0, then d=l and /= 1, or in other words in order to satisfy the equations mn = — nm, m^ = — 1, ?i- = — 1, where it will of course be understood that in these (as in the equations m- + 1 = 0, n'' + 1 = 0) 1 is
the abbreviated form of the matrix and 1 of* the form -^ , the necessary
and sufficient condition is that the determinant of a; + my + nz shall be equal
to X- + y* + z".
i 0 The simplest mode of satisfying this condition is to write 111= .,
w= A ' * meanmg v(— 1). which gives mn= . and nm= . .
-I \j "~~ % yj % \j
It is easy to express any matrix of the second order as a linear function
of 1 ( meaning \ m,n,p, where p stands for mn.
For if ' , be any such matrix it is only necessary to write a =/+ ig, h = —h— ki, d=f—ig, c = — h + ki, and then ' j =/+ ffin + /'w + kp.
The most general solution of the equations MN =■ — NM, M^ = N^ = —l, must contain three arbitrary constants, namely, the difference between the number of terms in m and n, and the number of conditions 6 = 0, c = 0, e = 0, ci!=l, /=!, which are to be satisfied.
[* I denotes - 1.]
I
14] On Quaternions, Nonions, Sedenions, etc. 123
Suppose M, N to be the most general solution fulfilling these conditions; we may write
M= f + gm + hn + kp,
N =/' + g'm + h'n + k'p,
where m, n is any particular solution and p = mn, and we shall have inas- much as M^ = 1,
/■-g'- h- -!<? + %fgm + 2fhn + 2/^•p = the matrix 1,
and consequently g- + h- + k" = 1 +/ ^
fg = 0, /h = 0, fk = 0.
Hence /= 0 and g-+h' + k'= 1.
Similarly /' = 0 and g'' + A'" + ^■'^ = 1,
and also inasmuch as MN = — NM,
gg" + hh' + kk' = 0,
and since the equations M" =1, iV» = 1, MN = — NM imply if we make MX=P that P' = -l, and MP=-PM, and NP = -PN, it follows that M, N, P, are connected with ni, n, p, in the same way as the coordinates of a point referred to one set of rectangular coordinates in space are connected with the coordinates of the same point referred to any other set of the same*.
Herein lies the ground of the geometrical interpretation to which quaternions lend themselves and it is hardly necessary to do more than advert to the fact that the theory of Quaternions is one and the same thing as that of Matrices of the second order viewed under a particular aspect f.
(2) Let m, n now denote matrices of the third order.
We might propose to solve the equation mn = — nm.
The result of the investigation is that we must have m* = n', m' = 0, n' = 0, and writing mn=p, Tn' = n' = q, there results a set of quinions, 1, m, n, p, q, for which the multiplication is that marked {a^) p. 144 of the late Prof Peirce's invaluable memoir in Vol. iv, of the American Journal of Mathematics.
* There is another Bolntion possible, obtained by writing
bnt this leads to a linear relation between m and n, so that mn = nm and consequently mn = nm=0 which is not the kind of solation proposed in the question.
+ See my article in the Land, and Edin. Phil. Mag. on "Involution and Evolution of Qnatcmiong," November, 1883. [Above, p. 112.]
124 On Quaternions, Nonions, Sedenions, etc. [14
But instead of this let us propose the equation mn = pnm, where p is one of the imaginary roots of unity ; if now we write the determinant of X + my + nz under the form
a^ + Zba?y + ^ca^z + Zdxz- + Qexyz + Zfyz"^ + 9^ + ^hy'z + Skyz' + Iz*,
it may be shown [cf p. 126, below] that we must have
6 = 0, c = 0, d = 0, e = 0, /=0, h = 0, k = 0,
and if we superadd the conditions m'=l, n' = l, we must also have g=l, 1 = 1, or in other words the determinant to x + my + nz must take the form x' + y^ + z'; but this condition (or system of conditions) although necessary is not sufficient (a point which I omitted to notice in my article entitled "A Word on Nonions" inserted* in a previous Circular).
It is obviously necessarj' that we must have (mn)' = 1.
Now if the identical equation to mn be written under the form
{mnf - SB {mnf + Wmn - E=0,
B may be shown to be a linear homogeneous function of b, c, and e ; also E = gl = \; but D is not a function of h, c, d, e, /, g, h, k, I, and will not in general vanish (as it is here required to do) when b, c, d, e, f, h, k vanish. Its value is the sum of the products obtained on multiplying each quadratic minor of m by its altruistic opposite in n : (the proper opposite to a minor of m means the minor which is the reflected image of such minor viewed in the Principal Diagonal oi m regarded as a mirror; and the altruistic opposite is the minor which occupies in n a position precisely similar to that of the proper opposite in m). There are, therefore, 10 equations in all to be satisfied between the coefficients of m and n when m' = «' = 1 and nm = pmn.
These ten conditions I have demonstrated are sufficient as well as necessary. There remains then 18 — 10 or 8 arbitrary constants in the general solution. If m, n is a particular solution we may take for M, N (the matrices of the general solution),
M=a + ^m+ ym" + a'n + ^'mn + <^'m^n + a"n= + ^"mn"^ + i'ni'n^,
iV = «! + /3, m + 7i/tt2 ^ 3^'„ ^ ^^rnn + 7, Ww + a"n^ + /S/'wm' + '^"m'^n-,
and 10 relations between the 18 coefficients m/ust be sufficient to enable to be satisfied the equations M' = iV' = 1, NM=pMN: but what these relations are and how they may most simply be expressed I am not at present in a condition to state f.
[* Vol. III. of this Keprint, p. fi47.]
f The solution of this problem would seem to involve some unknown expansion of the idea of orthogonalism. Unless MN=NM=0, a solution to be neglected, it may be proved that 0 = 0, ai = 0.
14] On Quaternions, Nonioiu, Seclenions, etc. 125
I showed in "A Word on Nonions" that the 9 first conditions are satisfied by taking
10 0 0 0 1
m=0 p 0 n = p 0 0
0 0 p\ 0 p^ 0.
The 10th condition is also satisfied ; for the only quadratic minors (not
.. j.-.v- lOpOlO, ,..
having a zero determinant) in m are - , ^, ,, „ „; the altruistic
V p V p^ {) p-
,.,. OpOp^'OO,,
opposites to which in n are _ j; , (. f^ , ■. ^ , the determinants to each
of which are zeros, and accordingly we find
10 0
|
m' |
= n» |
= 0 0 |
1 0 |
0 1, |
|||||
|
0 |
0 |
P' |
0 |
0 |
I |
||||
|
nm = p |
0 |
0 |
mn — p^ |
0 |
0 |
||||
|
0 |
1 |
0, |
0 |
P |
0 |
so that win = pnm and m» = /i' = 1 as required.
I subjoin an outline proof of the fundamental portion of the theory of Quaternions and Nonions above stated as it will serve to throw much light upon the nature of the processes employed in that new world of thought to which I gave the name of Universal Algebra or the Algebra of multiple quantity: a fuller explanation will be found in the long memoir which I am preparing on the entire subject for the American Journal of Mathematics.
(1) As regards the equation 7im = — mn, where m, n are matrices of the second order.
As before let the determinant of (x + ym + zn) be a? + 2bxy + 2cxz + dy'' + 2eyz +fz''.
I may observe here parenthetically that the Invariant of the above Quantic is equal to the determinant of m.n — nm, and that when it vanishes 1, m, n, mn, as also 1, n, m, nm are linearly related — or, as I express it, this Invariant is the Involutant of the system m, n or n, m. When m, n are of higher than the second order, the Involutant of m, n, say /, is that function whose vanishing implies that the 9 matrices (1, m, m'j^l, n, n^) are linearly related, and the Involutant of n, m, say J, that function whose vanishing implies that the 9 quantities (1, n, n'Jl, m, m^) are so related (/, J being two distinct functions), and so for matrices of any order higher than the second.
126 On Quaternions, JVonions, Sedenions, etc. [14
By virtue of a general theorem for any two matrices m, n of the second order, the following identities are satisfied : m" - 26m + d = 0, mn + nm — 2bn — 2cm + 2e = 0, n' - 2cn +/= 0. If then mn + nm = 0, since vi and n cannot be functions of one another (for then mn = nm), the second equation shows that 6 = 0, c = 0, e = 0, and conversely if 6 = 0, c = 0, e = 0, mn + nm = 0, and m^ + d = Q, n^+f=0, where, if we please, we may make d = l, f= 1.
(2) Let m, n be matrices of the third order, and write as before, Det. {x+ym + Z7i) = x^ + Sbx^y + Scx'z + Sdxy"
+ Qexyz + Sfxz^ + gf + ^hyH + Skyz^ + Iz".
Then by virtue of the general theorem last referred to* there exist the identical equations
m» - 36m'' + 3dm -g = 0,
m^n + mum + nm' — 36 {mn + nm) — 3cm' + Sdn + Gem — 3A = 0,
mn^ + nmn + n^m — 3c {inn + nm) — Sbn- + Sfm + Qen — Sk = 0,
n^-3c7i' + ^fn-l = 0.
Let now nm = pmn, where p is either imaginary cube root of unity, then
(1) m^ n + mnm + nm" = 0 and (2) mn- + ?im?i + n-m = 0;
for greater simplicity suppose also that m^ = n' = 1, where 1 means the matrix
10 0
0 1 0.
0 0 1
From the 1st and 2nd of the four identical equations combined it may be
proved that b = 0, d = 0 ; I do not produce the proof here because to
make it rigorous, the theory of Nullity would have to be gone into which
would occupy too much space ; and in like manner from the 3rd and 4th
it may be shown that c = 0, /=0t. Hence returning to the two middle
equations it follows that e = 0, A = 0, A; = 0, and from the two extremes that
^ = 1,^ = 1.
If then nm = pmn, m^ = 1, and n' = 1, it is necessary that
6 = 0, c = 0, d = 0, e = 0, f=0, g = l, /i = 0, ^- = 0, 1=1. But these equations although necessary are manifestly insufficient ; for they lead to the equations m' -1 = 0, ?i' — 1 = 0, and
(1) 7ii''n + mnm + nm' = 0; (2) mn" + nmn + n-7n = 0,
r* By Cayley's theorem, if in Det. {x + ym + z7i) we replace x by -ym-zn, the result vanishes identically in regard to y and z.]
t Except when m, n are functions of one another, so that mn and nm are identical and consequently are each of them zero.
14] On Quaternions, Nmiions, Sedenions, etc. 127
but not necessarily to nm = pmn. In fact the supposed equations between m and n involve as a consequence the equation (mny = 1. Now the general identical equation to (mn) is
{mny - SB (mny + SD (mn) -F=0,
where B is the sum of each term in m by its altruistic opposite in n = 36c — 2e = 0, F = gl = \, and D is the sum of each first minor in m by its altruistic opposite in n which sum does not necessarily vanish when b, c, d, e, f, h, k, all vanish. Hence there is a 10th condition necessary not involved in the other 9, namely, D = 0. These 10 conditions I shall show are sufiBcient as well as necessary. For when they are satisfied since (innf = 1, mn . mn = n'm".
Hence from (1) m'n' + n^m' + nm^n = 0,
and from (2) m'n^ + n'm'' + mn^vi = 0.
Hence nm . mn = mn . nm *, and consequently nm is a function of mn [cf p. 149, below]. Hence we may write
nm = A + Bmn + C (mny.
But the latent roots of mn and nm (which are always identical) are 1, p, p', hence
A+B + C, A+Bp + Cp', A+Bp' + Cp,
must be equal to 1, p, p', each to each taken in some one of the 6 orders in which these quantities can be writtenf.
Solving these 6 systems of linear equations there results :
A=0, B = 0, C=l, p OT p'
or ^ = 0, B = l, p or p\ 0=^ 0.
Hence nm = dmn, or 6 (mny where 6 =\, p, p'.
If nm = 6 {mny, nmmn = 6 (mn)" = 6.
Hence m' = On' .en' = e^n\
and m'n + mnm + nm' = S6m* = Mm = 0,
so that m = 0, and m' = 0 = 1 ; and again if nm =■ mn,
m'n + mnm + nm' = 2m'n + mnm = Sm'n = 0,
* This eqnation is independent of the equation (mny= 1 ; for
nm'n - mn'm = (ffi*n + mnm + nm')n- m (mn' + nmn + n^m) = 0
by virtue of equations (1) and (2) above : accordingly these equations taken alone imply the
equations
nm = A + Bi mn + C (mn)-, mn = - A + £,nm - C (nm)'
AC where B, , D^ are the roots of B'' + B + l--i- = 0; A, C being arbitrary and independent except
that each vanishes when and only when the cube of mn and (as a consequence) of mn, is a scalar matrix. [See below, p. 154. Footnote [t].]
t By virtue of the general theorem that the latent roots of any function of a matrix are the like functions of the latent roots of the original matrix.
128 On Quaternions, Nonions, Sedenions, etc. [14
so that m'n = 0, n = 0, and n' = 0 = 1 as before, where it should be noticed
0 0 0 10 0
that 0 = 1 means that 0 0 0 is identical with 010.
0 0 0 0 0 1
Heace the only available hypothesis remaining is the equation nm = v . mn, where v is one of the imaginary cube-roots of unity as was to be proved.
(3) It remains to say a few words on the general equation nm = kmn, where m, n are matrices of any order a>. To avoid prolixity I shall confine my remarks to the general case, which is, that where the determinants (or as I am used to say the contents) of m and n are each of them finite ; with this restriction, the proposed equation is impossible for general values of A; as will be at once obvious from the fact that the totalities of the latent roots of mn and of nm are always identical, but the individual latent roots are by virtue of the proposed equation in the ratio to one another of 1 : k, which, since by hypothesis no root is zero, is only possible when fr* = 1.
When the above equation is satisfied the w' equations arising from the identification of nvi with kmn cease to be incompatible and (as is necessary or at all events usual in such a contingency) become mutually involved. Thus, for example, when w = l and k = \, the number of independent equations is 0, that is, 1 — 1, when w = 2 and A; = — 1 the number is 3, that is, 4 — 1, when w = 3 and k = p or p' the number is 8, that is, 9 — 1 ; it is fair therefore to presume (although the assertion requires proof) that for any value of m when A is a primitive wth root of unity the number of conditions to be satisfied when nm = kmii is w' — 1. Of these the condition that the content of a; + my + m shall be of the form x" + cy" + c'z" will supply
C'" +!)(<" +1) ■ 3, that is, ^'^4^- 2,
and there will therefore be
'-3w , (i»-l)(a)-2) h 1 or ^^ ~ -
'I to be supplied from some other source.
When A; is a non-primitive wth root of unity, the number of equations of
condition is no longer the same. Thus when k=\ we know that n may be
of the form
A + Bm + (7m= -F . . . + Xm"-',
where A, B, ... L, and all the <»' terras in m are arbitrary, and consequently the number of conditions for that case is 2(o^ — (co' + w) or <u' — w. It seems then very probable that if A; is a qth power of a primitive toth root of unity the number of conditions required to satisfy nm = kmn is &>-— B where 8 is
14] On Quaternions, Nonions, Sedenions, etc. 129
the greatest common measure of q and a> : but, of course, this assertion awaits confirmation.
When &) = 4 besides the case of nm = mn, that is, of n being a function of m of which the solution is known, there will be two other cases to be considered, namely, nm — — mn and nm = imn : the former probably requiring 14 and the latter 15 conditions to be satisfied between the coefficients of m, the coefficients of n and the two sets of coefficients combined.
It is worthy of notice that the conditions resulting from the content of X + my + nz becoming a sum of 3 powers are incompatible with the equation nm = vmn when v is other than a primitive wth root of unity (a> being of course the order of m or n).
Thus suppose w = 4 ; the conditions in question applied to the middle one of the 5 identical equations give
m'n' + n'm' + mn'm + nm^n + mnmn + nmnm = 0 ; when nm = imn the left-hand side of this equation becomes
(1 + 1* + i* + 1» + i + 1") m'n*, that is, is zero, but when nm = — mn, the value is
(1 + 1-1 -l-l-l)m'7i' which is not zero, and bo in general. Thus the pure power form of the
content of x + my + nz is a condition applicable to the case of — being a
mn
primitive root of unity and to no other.
The case of nm being a primitive root of ordinary unity is therefore the one which it is most interesting to thrash out.
There are in this case, we have seen, ^(o»' + 3a) — 4) simple conditions expressible by the vanishing of that number of coefficients in the content of x + my + nz and ^(w — l)(w — 2) supplemental ones. What are these last ? I think their constitution may be guessed at with a high degree of probability. For revert to the case of a» = 3 in which there is one such found by equating to zero the second coefficient in the identical equation
(mnf - 3B (mny + SDmn -G = 0.
Suppose now (m*ny - 3B' {m*n*y + ZD'm'ri' -G' = 0
is the identical equation to m^n\ By virtue of the 8 conditions supposed to be satisfied we know that nm = pmn as well as m' = 1, w' » 1, and consequently that (m»ri»)'= 1. Hence fi'= 0, D' = 0, by virtue of the 7 parameters in the oft-quoted content and of D being all zero, and thus the evanescence oi B' or D' imports no new condition.
8 IV. 9
130 On Quatemiotis, Nonions, Sedenions, etc. [14
Now suppose o) = 4, and that
{mn)* - 45 (mn)' + QD (mny - 4,Gmn + M = 0, {nv'n')* - iB'{ni'ny + 6D'{m''ny - 4:G'm,^n' + M' = 0. Here we know that B vanishes by virtue of b, c and e vanishing, but D = 0, G = 0, which must be satisfied if nm = imn, will be two new conditions not implied in those which precede. It seems then, although not certain, highly probable that B' = 0, D' = 0, will be implied in the satisfaction of the antecedent conditions but that G' = 0 will be an independent condition, so that Z) = 0, G = 0, G' = 0, .will be the three supplemental conditions: and again when to = 5 forming the identical equations to mn, ni'n', m^n^, and using an analogous litteration to what precedes, the supplemental conditions will be
J) = 0, G =0, M =0,
M" = 0, and so in general for any value of as.
The functions D, G, M, etc., above equated to zero are known from the following theorem of which the proof will be given in the forthcoming memoir*.
If (mn)'" + ki (rmi)--'^ + . . . + ij (mnf-' + . . . = 0
is the identical equation to mn, then ki is equal to the sum of the product of each minor of order i in m multiplied by its altruistic opposite in n.
The annexed example will serve to illustrate in the case of « = 3 that unless the supplemental condition is satisfied we cannot have nni = pmn.
Write m = l 0 0, n= 0 c k,
0 p 0, k 0 cp,
0 0 p\ cp' k 0,
then the determinant to a; + my + nz will be easily found to be
ic' + ^ + (C + ^)^; but B becomes — Spck, and does not vanish unless c = 0 or k = 0, and accordingly we find
n7u =0 pc p'k, mn =0 c k, k 0 c, pk 0 p^c,
p'c pk 0, pc p-k 0.
When k = 0 mn = p'nm, when c = 0 nm = p^mn, but on no other supposition
will — be a primitive cube root of unity. mn ^ ■'
* This theorem furnishes as a Corollary the principle employed to prove the stability of the Solar System. (See Land, and Edin. Phil. Mag., October, 1883.) [Above, p. 110.]
14] On Quaternions, Nmiions, Sedenions, etc. 131
Addendum.
Referring to the equation MN = — NM, and to the eight equations expressing M and N in terms of the combinations of the powers of m with those of n, in which it is to be understood that M and N are non-vacuoiis, we know that the sums of the latent roots of M and of N must each vanish and consequently, as may be proved, that a = 0, a' = 0, leaving 8 — 2 or 6 conditions to be satisfied. If we further stipulate that Jlf' = l, i\r'=], there will be 8 relations connecting the coeflScients h, c, ... k and V, c', ... k', so that the 64 coefficients in the 8 equations connecting M, M'; N, N'; MN, M'N'; WN, MN", or say rather M, M^; N, N"; p^N, p'M^N'; pM^N, pMN'^, with like combinations or multiples of combinations of powei-s of m, n * will be connected together by 56 equations ; the coefficients in the expression for any one of the above 8 terms may then be arranged in pairs /i,yi'; gi, Qi \ hi, V> ^i, ki ; and in the expression for its fellow by Fi, Fi ; Gi, Gi ; Hi, HI; Ki, Kl; so that the Matrix is resolved as it were into 4 sets of paired columns and 4 sets of paired lines : the 4 different sets of paired lines being found by writing successively i = 1, 2, 3, 4.
It is then easy to see that there will be 4 equations of the form
2(/.(?.'+/.G.')=l. and 6 quaternary groups (that is, 24 equations) of the form
S (/.G^' + /.(?/) = 0,
with liberty to change / into F or G into g or each into each : together
then the above are 28 of the 56 conditions required. But inasmuch as the
8 [m, n] arguments may be interchanged with the 8 \M, N'\ ones, we may
transform the above equations by substituting for each letter/ its conjugate
d log A
— -P. — (where A is the content of the Matrix) and thus obtain 28 others,
giving in all (if the two sets as presumably is the case are independent) the required 56 conditions : the latter 28, however, may be replaced by others of much simpler formf.
* It is easy to see that the sam of the latent roots of 'MSiii mast be zero for all values of i, j 80 that it is a homogeneous linear function of the 8 quantities m, m^, ..., mn, m^v?,
t I am still engaged in studying this matrix, which possesses remarkable properties. Is it orthogonal? I rather think not, but that it is allied to a system of 4 pairs of somethings drawn in four mutually perpendicular hyperplanes in space of 4 dimensions. In the general case of ilN=pNM where p is a primitive uth root of unity, there will be an analogous matrix of the order u'-l where each line and each column will consist of u+l groups of w-1 associated terms.
The value of the cube of any one of the 8 matrices M, W; ... ; MN, M^N^ may be expressed as follows : It is JF* into ternary unity. Such a quantity may be termed by analogy a Scalar. To find P(_,- I imagine the 8 letters corresponding to M'N' (but without powers of p attached) to be set over 8 of the 9 points of inflexion to any cabic curve, the paired letters being made suitably
9—2
132
On Quaternions, Nonioiu, Sedenions, etc.
[14
To me it seems that this vast new science of multiple quantity soars as high above ordinary or quaternion Algebra as the M^canique Celeste above the " Dynamics of a Particle " or a pair of particles, (if a new Tait and Steele should arise to write on the Dynamics of such pair,) and is as well entitled to the name of Universal Algebra as the Algebra of the past to the name of Universal Arithmetic.
coUinear with the missing 9th point. Then among themselves the 8 letters may be taken in 8 different ways to form collinear triads and the product of the letters in each triad may be called a collinear product ; P,_y (which is identical with the Determinant to M*N') will be the sum of the cubes of the 8 letters less 3 times the sum of their 8 collinear products, and its 8 values will be analogous to the 3 values of the sum of 3 squares in the Quaternion Theory. Each of these 8 values is assumed equal to unity.
It may be not amiss to add that the product of four squares by four is representable rationally as a sum of four squares, so if we place (not now 8 specially related but) nine perfectly arbitrary letters over the nine points of inflexion of a cubic curve the sum of their 9 cubes less three times their 12 collinear products multiplied by a similar function of 9 other letters may be expressed by a similar function of 9 quantities lineo-linear functions of the two preceding sets of 9 terms.
By the 8 letters of any set as, for example, 6, ..., h' being " specialized," I mean that they are subject to the condition bb' + dd' +//' + hh' = 0. When this equation is satisfied, and not otherwise, M^ will be a Scalar, and it must be satisfied when MN=:pNM.
15.
ON INVOLUTANTS AND OTHER ALLIED SPECIES OF INVARIANTS TO MATRIX SYSTEMS.
[Johns Hopkins University Circulars, in. (1884), pp. 9 — 12, 34, 8.5.]
To make what follows intelligible I must premise the meaning and laws of vacuity and nullity.
A matrix is said to be vacuous when its content (the determinant of the matrix) is zero, but it may have various degrees of vacuity from 0 up to w the order of the matrix.
If from each term in the principal diagonal of a matrix X be subtracted, the content of the resulting matrix is a function of degree o) in \ ; the to values of X which make this content vanish are called its latent roots, and if i of these roots are zero, the vacuity (treated as a number) is said to be i. This comes to the same thing as saying that the vacuity is i when the determinant, and the sums of the determinants of the principal minors of the orders w — 1, w — 2, ... (« — I'+l) are each zero. A principal minor of course means one which is divided into 2 [equal] triangles by the principal diagonal of the parent matrix.
Again the nullity is said to be i when every minor of the order (w — i+l), and consequently of each superior order, is zero. It follows therefore that it means the same thing to predicate a vacuity 1 and a nullity 1 of any matrix, but for any value of i greater than 1, a nullity i implies a vacuity i but not vice versd ; the vacuity may be i, whilst the nullity may have any value from 1 up to i inclusive.
The law of nullity which I am about to enunciate is one of paramount importance in the theory of matrices*.
* The three cardinal laws or landmarks in the science of mnltiple quantity are (1) the law of nullity, (2) the law of latency, namely, that if X,, X,, ... X„ are the latent roots of m, then /X]> /\< ... /Xu are those ot fm, inclading as a consequence that
■^ -''■'*' (X|-X,)(Xi-X3)-(V.-M' and (3) the law of identity, namely, that the powers and combinations of powers of two matrices m, n of the order u are connected together by (u + 1) equations whose coefficients are all included among the coefficients of the determinant to the Matrix
. x+ym + zn.
134
On Involutants and other allied species of
[15
I
The law is that the nullity of the product of two (and therefore of any number of) matrices cannot be less than the nullity of any factor nor greater than the sum of the nullities of the several factors which make up the product.
Suppose now that X,, X™, ... X„ are the latent roots of any matrix with unequal latent roots of the order w. It is obvious that any such term as 7/1 — Xi will have the nullity 1, for its latent roots will be 0, X, — X,, X3 — Xi, ... X„ — Xi, and consequently its vacuity is 1.
Moreover we know from Cayley's famous identical equation that the nullity of the product of all the w factors is w. -
Hence it follows that if Mi contains i, and Mj the remaining w—i of these factors (so that i +j = <u), the nullity of Mi must be exactly i and of Mj exactly j.
For the theorem above stated shows that Mi cannot have a nullity greater than i, nor Mj a nullity greater than j.
Hence if the nullity of the one were less than i or of the other less than j, the nullity of MiMj would be less than i +j, that is, less than m, whereas its nullity is w ; hence the two nullities are respectively i and j as was to be shown.
Furthermore we know that the latent roots of (m — Xi)' are (X, — X,)' ; (Xj — Xi)° ; ... (X„ — Xi)".
Hence if the latent roots of m are all distinct, the nullity of {m — X,)" is unity and consequently by the same reasoning as that above employed it follows that the nullity of
(m — Xi)"' . (m — X-j)"' . . . (m — Xi)"* is exactly i.
I will now explain what is meant by the Involutant or Involutants of a system of two matrices of like order.
It will be convenient here to introduce the term " topical resultant " of a system of w" matrices each of order a.
We may denote any matrix say
(h.i
ai.2
by the linear form
*<u, 1
T ^w, 1 f «, 1 T ^to>, 2 'i(», 2 T • • ' "r" ^W, « 'w
15] Invariants to Matrix Systems 135
where the t system is the same for all matrices of the order to. If, then, we have a^ such matrices, their topical resultant is the Resultant in the ordinary sense of the w- linear forms above written, proper to each of them re- spectively.
Suppose now that m, n are two independent matrices of the order «, we may form ox' matrices by taking each power of m from 0 to a> — 1 as an antecedent factor, and can combine it with similar powers of n as a con- sequent factor, and in this way obtain tu' matrices, of which the first will be the ta-ary unity, that is, a matrix of the order w in which the principal diagonal terms are all units and the other terms all zero. The topical resultant of these co- matrices I shall for brevity denote as the Involutant to m, n.
In like manner, inverting the position of the powers of m and of n so as to make the latter precede instead of following the former in the w" products above referred to, we shall obtain another topical resultant which may be termed the Involutant to n, m.
The reason why I speak of these topical resultants as involutants to m, n or n, mi is the following :
In general if m, n are two independent matrices, any other matrix p, by means of solving w' linear equations, may obviously be expressed as a linear function of the «' products
(1, TO, m', ... , TO"~')(1, 11,71", ... , n"~').
There are, however, exceptions to this fact.
The most obvious exception is that which takes place when n is a function of m ; for then any m of the to' products will be linearly related, and there will be substantially only co disposable quantities to solve w^ equations.
Another exception is when the m, n Involutant, that is, the topical resultant of the w' matrices, is zero ; in which case the general values of the «' disposable quantities each becomes infinite. So that m, n may be said to be in a kind of mutual involution with one another. So, again, p may in general be expressed as a linear function of the &>' matrices
(1, n, n», ... , n--») (1, to, m', ... , to*^'),
but when the i», to Involutant vanishes this is no longer possible.
When o) = 2 the two involutants, considered as definite determinants, are absolutely equal in magnitude and iu Algebraical sign, but when to exceeds 2 this is no longer the case; the two Involutants are then entirely distinct functions of the elements of m and n.
136 On Involutants and other allied species of [15
10 0 Q p k
Thus to take a simple example : if m = 0 p 0 and n = k 0 p' it will
0 0 p» 1 A 0
be found by direct calculation of two topical resultants of the 9th order, that the two involutants will be
81 {p - p^) (A* - pY and 81 (p" -p)(k'- p»)»
respectively. The reason why the two involutants coincide in the case of ft) = 2 is not far to seek. It depends upon the fact of the existence of the mixed identical equation
mn + nm — 2bn — 2cm + 2e = 0 ; from which it is obvious that the topical resultant of 1, m, n, mn is the negative of that of 1, m, n, nm or identical with that of 1, n, m, nm.
By direct calculation it will be found that the Involutant m, n, or n, m, where m = {^ n^{,(,is
-(9h'-g'kf+[{f-k)g'-{f'-k')g][{f-k)h'-{f'-k')h],
which is the same thing as the content of the matrix {mn — nm). It may also be shown d priori or by direct comparison to be identical (to a numerical factor prh) with the Discriminant of the Determinant to the matrix (x + ym + zn) which is a ternary quantic of the second order. Its actual value is 4 times that discriminant.
Let us consider the analogous case of Mechanical Involution of lines in a plane or in space. There are two questions to be solved. The one is to find the condition that the Involution may exist, that is, that a set of equilibrating forces admit of being found to act along the lines ; the second, to determine the relative magnitudes of the forces when the involution exists, and this is the simpler question of the two.
In like manner we may consider two questions in the case of m, n being in either of the two kinds of involution; the one being to find what the condition is of such involution existing, the other what are the coefiicients of the (iP coefficients in the equation which connects the co" products, when the involution exists.
This latter part of the question (surprising as the assertion may appeal- and is) admits of a very simple and absolutely general direct and almost instantaneous solution by means of the Law of Nullity, above referred to, as I will proceed to show.
The determination of the Involutants, or at all events of their product, will then be seen to follow as an immediate consequence from this prior determination of the form of the equations which express the involutions of the two kinds respectively.
15] Invariants to Matrix Systems 137
But first it may be well to explain why and in what sense I refer in the title to Involutants as belonging to a class of invariants. I say, then, that universally involutants are invariants in this sense, that if for m and for n, any function of m, or any function of n be substituted, the ratio of the two Involutants, say / and J, remains unaltered. By virtue of the Identical Equation (m)' will be of the form of
Ai + Bi + CiVi' + ... + Liin"-^
and as a consequence it is easy to see that when m» is substituted for m, / and J will become respectively PI, PJ where P is the Q>th power of the determinant to the matrix formed by writing under one another the (&>— 1) lines of terms, of which the line £,-, C,-, ... ; Z,- is the general expression.
Moreover, in the particular case where w = 2 and I=J*, besides being an Invariant in this modified sense, / will be an invariant in a sense including but transcending the more ordinary conception of an Invariant ; for if when, for m and n, f{m, n) and <f> (m, n) are substituted, / becomes /', then /' will contain 7 as a factor ; this is a consequence of the fact that when m and n are in involution f{m, n) and <^{m, n) will also be in involution, for in consequence of the identical equation
mn + nm — 2bn — 2cm + 2e = 0 /and ^ andy^ will each be reducible to the form
A + Bm + (7n + Dmn and it is obvious from the ordinary theory of the determinants that the
topical resultant of 1, (meaning ^ ,). and three linear functions of 1, m,
n, nm, will contain as a factor the topical resultant of 1, m, n, mn.
Nor must it be supposed that Involutants are the only species of invariants in the modified sen.se first described which appertain to the
* I for some time had imagined, and indeed thought I had proved, that the two involutants were always identical. When crossing the Atlantic last month on board the "Arizona," having hit npon a pair of matrices of the third order, for which the two topical resultants admitted of easy calculation, I found, to my surprise, that they were perfectly distinct. The cause of the tailnre of the supposed proof constitutes a paradox which will form the subject of a communication to a future meeting of the Johns Hopkins Mathematical Society.
I will here only premise that the seeming contradiction between the logical conclusion and the facts of the case takes its rise in a sort of mirage with which invariantistg are familiar, namely : the apparent a priori establishment of algebraical forms as the result of perfectly valid processes, which forms have no more real existence in nature than the Corona of the Sun under oar Or Hastings' scrutinizing gaze : the contradiction between the logical inference and the truth being accounted for by the circumstance that any such supposed form on actual per- formance of the operations indicated, turns out to be a congeries of terms, each affected with a nnll coefficient ; we are thus taught the lesson that all a priori reasoning until submitted to the test of experience, is liable to be fallacious, and it is impossible to prove that a proof may not be erroneous by any other method than that of actoal trial of the results which it is supposed to yield.
138 On Involutants and other allied species of [15
system m and n. Thus, for example, when w = 2 it is not only true that the determinant of the matrix mn — nni is such a kind of Invariant (which for greater clearness it may be desirable to denote by the term Perpetuitant*), but each element of that matrix will also be a perpetuitant, and these 4 per- petuitants, when for m, n pm, <f)n are substituted, will be in an invariable ratio to one another and to either square root of the Involutant.
In like manner it will eventually be seen that for two matrices m, n of any order w, it is possible to form a matrix of the order w analogous to mn — nm
I which be it observed may be regarded as the Determinant of the matrix 1
each of whose w" terms will be in a constant ratio to each other and to any ftjth root of / and of J.
I will now return to the problem of finding what is the form of the equation which connects the <»' matrices denoted by
(1, m, m=, ... m"-^ (1, n, 7^^ ... n"-') when such an equation admits of being formed, that is, 7=0.
To fix the ideas let us suppose that m, n are matrices of the 3rd order of perfectly general form so that the m, n involution necessitates the satisfaction of one single condition, 1=0.
Let A+Bn + Cn^=0 be the equation whose form is to be determined where A, B, C, are each of them quadratic functions of m. I say that neither A, B, nor G, can contain a non-vacuous linear factor. For suppose that any one of them as A should contain the non- vacuous factor m + q, and that
A = {m + q) (am + p). Then we may multiply the equation by (m + q)~^ and thus obtain the equation
(am+p) + B'n + CV = 0,
that is, we have an equation in which not all 9 but only 8 of the terms signified by (1, m, m') (1, n, n') = 0 are linearly related. But this obviously implies, contrary to the hypothesis, the existence of two equations of condition instead of one.
Hence then A must be of the form c (m — \) (m — X') where \, \' are each of them a latent root of m ; whether the same or different remains to be determined.
In like manner it may be shown that B is of the form Ci{m — Xj)(m — \') and C of the form Cj (?n — X2) (m — V)- But now I say further that
(m — \) (m — X'), (m - X,) (m — X,'), (?« — Xj) (m — X2') must be identical.
* Perpetuitant formed from perpetuity by analogy to Annuitant from Annuity. Perpetnaut would have been better, but that it has already been applied by myself in the theory of Invariants in a sense recognized and adopted by Cayley, Hammond, and MacMahon.
15] Invariants to Matrix Systems 139
For, firstly, suppose that any one pair of the X's, say X, X', are distinct. If any other pair, say X-., Xj', is not identical with this pair, on multiplying the equation by m — X", where \" is the 3rd latent root of M, the term containing the term A{\ ...\") will vanish, but B(K...\") will not vanish and conse- quently there will be an equation, if C(\...\"} does not vanish, between 6 only, and if C(\ ... X") does vanish, between 3 only of the 9 terms denoted by (1, m, »i-)(l, n, n°), contrary to hypothesis.
The only remaining supposition is that A, B, G are each perfect squares. Suppose, then, that any one of them as ^ is a multiple of (?ft — X)'; unless B, C are each of them also multiples of the same, on multiplying the equation by (»i — X') (m — X"), one of the three coefficients of 1, n, n" will vanish but one at least of the other two will not vanish, which is impossible for the same reason as before. Hence the left-hand side of the equation of involution must contain (m — X)(ot— X') as a sinister factor where X, X' (whether the same or different) are latent roots of X. And in like manner precisely, by arranging the equation of involution under the form A' + mB' + m'C' where A', B', C are quadratic functions of n, it may be found that the same function must contain (n — fi.)(n~ fi) where /a, /i' are latent roots of n as a dexter &ctor.
Hence the form of the equation must be
(m - X) (m - X') (n -fx.)(n- fi.') = 0.
It is easy to see that we cannot have X and X' the same latent root of m and at the same time fi, ft the same latent root of n, for then the above product would have at most the nullity 2 whereas it is an absolute null, that is, has the nullity 3.
But I will now show that X, X' and fi, ft' must each consist of unlike roots. Let t be any term of the matrix
(m — X) (m — X*) (n — fi.)(n — fj,'),
where t will be a known function of the elements of m, n, of X, X' entering gymmetrically, and of /i, fi also entering symmetrically : this is the same thing as saying that t will be a function of the elements of m and n, of X", /*", and of the coefficients of the equations which contain the 3 latent roots of X and /i respectively.
Consequently the product of the 9 values of t found by writing X", X', X for X", and fi", ft, fi for /t", will be a rational integer function of the elements of m, n which vanishes when the Involutant / vanishes and must conse- quently contain / as a factor. If then, in any single instance, the matrix
{m-\f{n-fi'){n-ix")
does not vanish for some one value of X and /t when / vanishes, it cannot be the form, or one of two conceivably possible coexisting forms, of the
140
On Involutants and other allied species of [15
left-hand side of the general equation of involution. A similar remark of
course applies to
(m - X,) (m - Xj) (n - fj^y.
0
Let now
m-
9 0
0 0,
0 n = k 1 p\ and of n are 0, p6, p^6, where
9 0
k
9- 0
-p'k
11"= p'
-k 9
) 0 ) 3p ) 0
1
h? . — pk
0
0, 3p»
9' k^
9-
k'
+
-k + p6- p + p-dk
-p'k + p^ff' p^ +pek k" + pd
e
k? +p'0
-k+p"-e^
1
k^
+pek + e
p +p6k -pk + p'&' + p''ek + p6
1 0 0 u p
The latent roots of m are 1, 6= ^{\+k?); we have also
10 0
m= = 0 p» 0,
0 0 p
The three values of (m — V) {m — \") are 3 0 0 0 0 0 0 0 0, 0 3p> 0, 0 0 0 0 0 3p and the three values of (w — p-i) (n — /x,) are p^k + 6^ k' +pd 1 +ek + ek -k+ ff^ k^ +p^0
+ e p +ek -pk+ &'
p^k + p^d k
+ p-'ek + p'e
The general value of
(vi — \,) (m — X,) (n — /li) (n — /ju) will (to a numerical factor pres) be a matrix consisting of a single column accompanied by two columns of zeros, the non-zero column being some one of the 9 columns found in the above 3 matrices.
Now by direct calculation we know that the n, m Involutaut in this case is a numerical multiple of (P — p'^y and vanishes when k^ = p-, which gives 6 = ^{\ + p-), that is, — p = 6^, and if we please k=-6^.
Hence not merely one but three of the products of (m - X') (m - X") (n - jji') {n - p,") will in this case vanish, for the above equations will cause the 2nd, 4th and 9th columns all to become columns of nulls.
If now instead of the factor (m — X') (m — X") we substitute the factor (m — X)', the three values of (m — X)' will become
00 0 -3 00 -3 00
0-8/9 0 00 0 0-3/3 0
0 0 -3/D^ 0 0 -3p» 0 0 0
1
k'
-pk + pd'
15] Invariants to Matrix Systems 141
so that if {m - Xf (n - /x') (n - fj.")
is to vanish, it will readily be seen that each of two columns of one or the other of the two matrices representing (n — /uf) (n — jx") will have to vanish simultaneously, and that this cannot be brought to pass when 6' = — p and p = p- =. $^ whether we make A; = ^ or — ^^ or ^.
Hence {m-Xf{n-^l){n-^^.") = 0
is not an admissible general involution form of equation. Similarly by interchanging the above special values assigned to m and n, it may be shown that
(m - V) (to - X") (n - fj.y = 0
is not an admissible form, and consequently that the one universal form of the involution equation is expressed by saying that
(to - X') (m - X") (n - fj.') (n - m")
is an absolute null. If no connexion exists between the elements of to and n, we know from the law of nullity that the above matrix has a nullity 2, that is, that all its minors except the elements themselves have zero contents. The effect of the vanishing of / is to make the elements themselves one and all vanish when the two sets of latent roots are duly selected.
So in general if
jP = X- - ^,X--' + ^,X"-» - ^sX--' . .. = 0, and (? = /!-- 5,M"-' + jB.M""' - B.^l'^'' . . . = 0,
are the two equations to the latent roots of to, n matrices of order w, and if
M= m"^' - (Ai - X)m"-» + (A^ - ^,X + X')to"-» ... and i\r = 71— ' - (5. - /i) n"-* + (5, - B,X + X») w^' ... ,
MN = 0 for some value of X and of /* is the one equation of involution, and NM = 0 for some value of X and some value of /* ia the other such equation.
I will now show how to deduce from the above statement the following marvellous theorem.
Let H represent the sum of the product of each term in the matrix M by its altruistic opposite in N (so that H is a. function of X and /j. and of degree B - 1 in each of them) then will the ordinary Algebraical Resultant of* F, G, H* be exactly equal (in magnitude as well as form) to the product of the two involutants to the corpus m, nf.
* The system of equations whose resultant expresses the undifiFerentiated condition of tnTolation, may be written under the form (x,y)" = 0; {z,t)" — 0; (x, y)"-'=0. Qu(Bre whether CDch a resultant may not be written nnder the form of a determinant by an application of the Dialytie Method?
+ If / and J be the two involutants, /=0 will be the condition of left-handed involution of m, n or right-handed of n, m, and J=0 of right-handed involution of 711, n or left-handed of n, 711, for Involution, like light, " has sides." But IJ = 0 will be the condition of cme or the other kind, or so to say of undifferentiated Involution.
142 On Involutants and other allied species of [15
By the theorem proved at the beginning of this note, the nullity of M and that of N are each w— 1, hence the nullity of MN and consequently a fortiori its vacuity cannot be less than « — 1, and accordingly the identical equation to MN may be written under the form
{MN)''-H{MN)'^' = Q,
where H is the sum of the product of each element in the Matrix M or the Matrix N multiplied by its altruistic opposite in the other. Suppose now that 7=0 then for some one system of X, /x out of the w' systems given by the equations F=0, G = 0, H must vanish (for the nullity and a fortiori the vacuity of MN in that case becomes w) ; hence the double norm of H, that is, the product of the w^ values of H, or, which comes to the same thing, the resultant of F, G, H, must vanish when / vanishes and must therefore contain 7; in like manner because the nullity of NM and cb fortiori its vacuity is to when 7=0, it follows that the same resultant, say R, must contain also J\ R will therefore contain IJ, from which it may readily be concluded that it can differ from IJ, if it differ at all, only by a numerical factor.
I need hardly pause to defend the assumption that 7, J have no common factor, and that it is the first and not necessarily any higher power of R which contains 77; the single instance, when
10 0 0 p k
m = 0 p 0, n = k 0 p", 0 0 p' 1 k 0
of 7, J being respectively (to a numerical factor prh) the cubes of ^ — p and k' — p^ which have no common factor, settles the first part of this assumption at all events for the case of o) = 3, and as regards the second, it is only necessary to show that neither 7 nor J is equal to, or contains a square or higher power of a function of the letters in m and n as may be done easily enough when w = 3 by another simple instance*. We may then at once proceed to compare the dimensions of R with those of 7 and J.
* Limiting ourselves to the case of matrices of the third order, if we take for m, n the matrices 0 fc 0 0 B 0
d 0 f, D 0 F, it may be shown by direct computation that one of the Involutants 0 ft 0 0 H 0 becomes
(bH-hBf (fD - dFf (bd+fh) {BD - FH) {dB -fH) .{(hF+ bDf - (bd +fh) (BD + FH)\, and consequently if there were any square factor in either involutant such factor would contain the elements belonging to the two sets indecomposably blended, but on the other hand, if we
10 0 Q f F take for m, n the matrices 0 p 0, O 0 g, either involutant to m, n may easily be shown
0 0 p» ft il 0 (also by direct computation) to be made up of three factors, each of which is an indecomposable cubic function of /, g, ft, F, G, H. Hence it follows that neither Involutant can in its general
15] Invariants to Matrix Systems 143
R being the product of tu- values of \"~' ^"^^ + etc., where \, ^ are codimensional with the elements in m and n respectively, is obviously of the degree &>-. (tu — 1) in regard to each set of elements, that is, of the degree 2<B-((u — 1) in regard to the two sets taken together.
Consider now the degree of/; this is the topical resultant of «»" matrices of the form m' . n', where
1 = 0, 1, 2, ... tu-1, j = 0, 1, 2, ... m-l,
so that each term in I will consist of a combination of cd^ elements selected
respectively from these m' matrices. If m is even, there will be -^ pairs
of matrices, one of any such pair of the form m'n-', the other of form m"~'~* . n"~'""^, and the combination of elements taken from any such pair will be of the collective degree 2 (a) — 1) in the two sets of elements, so that
the total degree of the Involutant will be ^.2(w — 1) or «=(« — 1). If
again (o is odd, there will be ^(a)'+ 1) such pairs, and one factor (unpaired)
belonging to the matrix to * . n * of the collective degree (w — 1). Hence the degree of the involutant will be
(a»»-l)(a)-l) + (w-l) or <o'(<»-l) as before.
Hence the product of IJ is of the degree 2(u*(o) — 1), or the same as R, and consequently (at all events to a numerical factor prht) R and IJ coincide, which is the essential thing to be proved.
N.B. As regards ta = 3, the above proof is exact ; for higher values of u> to make it valid, it must be demonstrated as a Lemma that the two general twin involutants (even were they decomposable forms, which they un- doubtedly are not) could not have any common factor, nor either of them contain any square factor. The Resultant of F, G, H may be compared to a cradle just large enough to contain the twin forms in question, so as to give assurance that no other form is mixed up with them ; and the proof given above shows that this must be the case if neither twin is doubled
form contain any square factor. As a matter of fact, not only for ternary matrices but for mAtrices of any order, there can be no reasonable doubt whatever in any sane mind that every Involutant is abiolutely indecomposable. One must try, however, to obtain a strict proof of this npon the general principle of crushing every logical difficulty regarded as a challenge to the human reason, which falls in our way; it is in overcoming the difficulties attendant upon the proof of negative propositions that the mind acquires new strength and accumulates the materials for future and more significant conquests. To prove that involutants in their general form are indecomposable may possibly, I imagine, prove to be a hard nut to crack, or it may be exceedingly euy.
144 On Tnvolutants and other allied species of [15
up upon itself, and if the two do not grow into one another, but like such creatures each possesses a perfectly distinct organization.
A single instance will serve to establish the fact that the Resultant of F, G, H is the very product IJ itself, without any numerical multiplier. I have made this verification for binary and ternary matrices, and as the point is not one of an essential importance need not dwell here further upon it.
To pass to a much more important subject, I am inclined to anticipate as the result of a long and interesting investigation into the relations of the involutants of a certain particular corpus of the third order that the sum of the two involutants of any corpus admits of being represented by means of invariants similar in kind to that which expresses the single involutant to a binary corpus (m, n), namely, the content of (that is, the determinant to) the matrix mn — nin, which itself (as previously observed) may be written as the
{Til, 71 1 ^ , or say (m, n\ ; and in some similar way
it is, I think, not unlikely that the product also of the two involutants (the resultant of F, 0, H) is capable of being expressed; but I must for the present content myself with exhibiting the bare fact of the existence of invariants of the kind referred to for matrices of any order.
Suppose then that m, n is a corpus of the third order. Form the deter- minant
771 71 m' nr m n m' n" m n m? n" m n tn? n'
say (»n, n, ni", n'^.
The number of terms, half of them positive and half of them negative, in such determinant is 24 ; but of these, all but 8 will obviously appear as pairs of equal terms affected with opposite signs and so cancel one another : the 8 excepted ones are those in which no m and n come together, to wit :
mnm^n' + nmn^ni' + ni'n'mn + n^m^nm — rn'nmn' — nui'n^m — mn'm^n — n'mnni^.
The determinant to this matrix will be of the total degree 18 in the two sets of elements belonging to ?/i and n respectively, that is, of the degree 9 in respect to eacli set of elements per se. And so in general if m, n be of the order w the determinant
(m, m', ... m"~S n, »i", ... n"~^)^ will contain only 2 (tto))' effective terms, of which half will bear the positive and the others the negative sign.
15] Invariants to Matrix Systems 145
The determinant to this matrix will be of the order
a>[2{l + 2+...+(w-l)}], that is, (a)-l)a)>,
in regard to the combined elements in m, and n, that is, equi-dimensional with either involutant to the corpus m, n.
Whatever else may be its properties (on which I do not dare yet to pronounce), it is certain that such determinant (and over and above that, every term in the matrix of which it is the content) will be an Invariant to the corptis in the same sense in which either Involutant has been previously shown to be entitled to bear that name. And here for the present it becomes necessary for me to break oflF, bidding au revoir to any reader who may peruse this sketch, and trusting to meet him again in the broader field of the American Journal of Mathematics, where I hope to be spared to set out this portion of the theory with more certainty, and the whole doctrine of multiple quantity with much greater completeness and in more ample detail than is possible within the limits of the Circulars and in the short interval re- maining between the present time and the date of my intended departure for Europe.
8 IV.
10
16.
ON THE THREE LAWS OF MOTION IN THE WORLD OF UNIVERSAL ALGEBRA.
[Johns Hopkins University Circulars, III. (1884), pp. 33, 34, 57.]
In the preceding Circular allusion was made to tbe three cardinal prin- ciples or conspicuous landmarks in Universal Algebra ; these may be called, it seems to me (without impropriety), its Laws of Motion, on the ground that as motion is operation in the world of pure space, so operation is motion in the world of pure order, and without claiming any exact analogy between these and Newton's laws, it will be seen that there is an element in each of the former which matches with a similar element in the latter, so that there is no difficulty in pairing off the two sets of laws and determining which in one set is to be regarded as related by affinity with which in the other. They may be termed the law of concomitance or congruity, the law of consentaneity and the law of mutuality or community.
The law of congruity is that which affirms that the latent roots of a matrix follow the march of any functional operation performed upon the matrix, not involving the action of any foreign matrix ; it is the law which asserts that any function of a latent root to a matrix is a latent root to that same function of the matrix ; in so far as it regards a matrix per se, or with reference solely to its environment, it obviously pairs off with Newton's first law.
The law of consentaneity, which is an immediate inference from the rule for combining or multiplying substitutions or matrices, is that which affirms that a given line (or parallel of latitude) can be followed out in the matrices resulting from the continued action of a matrix upon a fixed matrix of the same order, that is, in the series M, mM, m^M, m'M, ... (which may be regarded as so many modified states of the original matrix) without reference to any other of the lines or parallels of latitude in the series, or again any column or parallel of longitude in the correlated series M, Mm, Mm'', ... without reference to any other such column or parallel of longitude.
16] On the Three Laws of Motion, etc. 147
An immediate consequence of this obvious fact (a direct consequence for the rule of multiplication) obtained by dealing at will with either of the systems of parallels referred to, is that a system of simultaneous linear equations in differences may be formed for finding each term in any given line or in any given column at any point in the series, and the integration of these equations leads at once to the conclusion that any term of given latitude and longitude in the I'th term of either series is a syzygetic function of the tth powers of the latent roots of m.
If, then, M be made equal to multinomial unity, this at once shows that supposing &) to be the order of m, on substituting rti for the carrier (or latent variable) in the latent function to m, and multiplying the last term by the proper multinomial unit, the matrix so formed is an absolute null, which proves the proposition concerning the " identical equation " first enunciated by Professor Cayley in his great paper on Matrices in the Philosophical Transactions for 1858.
This proposition admits of augmentation, (1), from within, as shown in a former note, by applying to it the limiting law of the nullity of a product (a branch of the 3rd law), which leads to the very important conclusion that the nullity of any factor of the function of a matrix which is an absolute null, or more generally of any product of powers of its linear factors, is exactly equal to the number of distinct linear factors which such factor or product contains, at all events, in the general case where the latent roots are all unequal ; and (2), from without, by substituting for m,m-'ren where n is any second matrix whatever and e is an infinitesimal. This leads to the catena of identities, to which allusion has been made in the preceding Circular. Then, again, the endogenous growth of the theorem (that which determines the exact nullity of any factor of the left-hand side of the identical equation) in its turn seems to lead to a remarkable theorem concerning the form of the general term of any power of m into M.
Observe that every such term is expressed as a syzygetic function of powers of the <a latent roots, and contains, therefore, a> constants, so that the total number of syzygetic multipliers is w'; but the number of variables in m and M together is 26)' ; and, consequently, apart from the to arbitrary latent roots the number of independent constants in m'M should be 2«i)' — w. The ft)' syzygetic multipliers ought then to contain only ft)(2ft) — 1) arbitrary constants, and such will be found to be the case by virtue of the following hypothetical theorem : Calling \ any one of the latent roots, the multipliers of X' in m'M will form a square of w' quantities; the theorem in question* is that every minor of the second order in such square is zero, so that the o)' terms in the square is given when the bounding angle containing
* I have not had leisure of mind, being mach occupied in preparing for my departure, to reduce thia theorem to apodictic certainty. I state it therefore with all due reserve.
10—2
148 On the Three Laws of Motion in [16
2ft) — 1 terms is given ; and the same being true for the multipliers of each latent root (which resolve themselves into o> squares) the number of arbitrary quantities in all is m{2m — 1) as has to be shown.
The law of consentaneity in so far as it relates to the decomposition of the motion of a matrix into a set of parallel motions, has an evident affinity with Newton's second law*.
Remains the law of mutuality, which is concerned with the effect of the mutual action upon one another of two matrices, and so claims kindred with Newton's third law.
This law branches off into two, one of which may be termed the law of reversibility, the other that of co-occupancy or permeability.
The law of reversibility affirms that the latent function of the product of two matrices is independent of the sense in which either of them operates upon the other, that is, is the same for mn as for nm, just as the kinetic energy developed by the mutual action of two bodies is not affected by their being supposed to change places.
As regards the second branch of the third law, the word co-occupancy refers to the fact that although the space occupied by two similarly shaped figures (say two spheres) is not absolutely determined (in the absence of other data) by the spaces occupied by them each separately (for they may intersect or one of them coincide with or contain the other), a superior as well as an inferior limit to such joint occupation is so determined ; the inferior limit being the space occupied by either such figure, that is, the dominant of these two given spaces, and the superior limit their arithmetical sum. So the nullity resulting from the action in either sense of two matrices upon one another is not given when their separate nullities are assigned, but has for an inferior limit the dominant of these two nullities and for a superior limit their sum ; the nullities of the two component matrices may also be conceived under the figure of two gases or other fluids which are mutually permeable and capable of occupying each other's pores.
Although the limits spoken of are independent of the sense in which the two matrices act on one another, it must not however be supposed that the actual resultant nullity is unaffected by that circumstance ; thus, for example, if the latent roots of a ternary matrix m are X, \', X.", the nullity resulting from {m — \) (m — \') acting sinistrally upon {m — \") n, that is, of (m — X) (m — X') (m — X") n is 3, but from the same acting dextrally upon the same, that is, of (7?i — X") n (m — X) (m — X'), need not necessarily exceed 2.
* For another and closer bond of affinity between the two laws see concluding paragraph of this note.
16] the World of Universal Algebra 149
Snch then are the three primary Laws of Algebraical Motion ; but as Conservation of areas, Vis viva, D'Alembert's Principle, the principle of Synchronous Vibrations, of Least action, and various other general laws may be deduced from Newton's three ground laws, so, of course, various subordinate but very general laws may be deduced from the interaction of the above stated three ground laws, namely, the law of Congruity, the law of Consentaneity, and the law of Mutuality.
The deduction of the catena of identical equations connecting two matrices m and n from the second and third laws combined, atfords an instance of such derivative general laws. Another instance of the same is the theorem that when the product resulting from the action upon one another of two matrices, is the same in whichever of the two senses the action takes place, the matrices must be functionally related, unless one of them is a scalar, that is, a multiple of multinomial unity, at all events when neither m nor n possesses a pair of equal latent roots.
This very important and almost fundamental law (seemingly so simple and yet so hard to prove) may be obtained as an immediate inference from that identical equation in the catena of such equations connecting the matrices m and n, in which one of the two enters only singly at most in any term. As for example if m and n are of the 3rd order, beside the identical equation w»* — 36to* + Sdm — gr = 0 we have* the identity
nihi + mnm + nm* — 36 (mn + nm) — 3cm' + 2dn + 6em — 3/t = 0.
But if nm = win then mnm = m'», nm* = mnm — m'n, so that this equation becomes
mhi — 2bmn + dn = m'c — 2em + h, or n = — ; — —. ; f,
m' — 26m + d '
unless m* — 26m + d is vacuous.
The first branch of the third law, namely, the law of reversibility, is an almost immediate inference from the rule for the multiplication of matrices, and becomes intuitively evident when the process of multiplication in each of the two senses between m and n is actually set out. The second branch, namely, the law of co-occupancy or permeability, as it is the most far-reaching 80 it is the most deep seated (the most cach^ of all the primary laws of
[• See p. 126 above.]
t Whence it follows that n mast be a fnnction of m convertible into an integral polynomial form, unless the namerator and denominator of the fraction to which n is equated vanish simul- taneously, which is what happens when m is scalar. If the numerator exactly contains the denominator n becomes a scalar. Seeing that a constant c is a specialized case of a function of • variable x althongh the converse is not true, we may say that whenever nm = rim, one at least of the two matrices m and n is a function of the other, and that each is a function of the other aniess that other is a scalar. Compare Clifford's " Fragment on Matrices '' in the posthumous edition of his collected works.
150 On the Three Laim of Motion in [16
motion. I found my proof of it upon the fact that the value of any minor determinant, say of the ith order, in either product of m and n (two matrices of the order m) may be expressed as the quantitative product of a certain couple of rectangular matrices (in Cauchy's sense of the term), of which one is formed by i columns and the other by t lines in the two given matrices respectively. Such rectangle as shown by Cauchy (and as may be intuitively demonstrated by the simplest of my umbral theorems on compound deter- minants) is the sum of the
7r(<a) ^
TT (<i> — i) TTl
complete determinants of the one rectangle multiplied respectively by the corresponding complete determinants of the other rectangle.
This shows at once the truth of the proposition in so far as relates to the lower limit, that is, that if mn=p, and m, n have the nullities e, f, and p the nullity 0, then d must be at least as great as e and at least as great as ^. As regards the superior limit the proof is also founded on the theorem in deter- minants already cited, and the form of it is as follows. If e be any number r, it may be shown that ^ must be at least as great as 9 — r; hence giving r all values successively from 0 to f — 1, it follows that e+ ^ cannot be less than 6, that is, that 0 cannot be greater than e + f .
The proof of the first law, that of concomitance or congruity, I ought to have stated antecedently, is a deduction from the theory of resultants and the well-known fact that the determinant of a product of matrices is the product of their determinants. Thus each of the three laws of motion is deduced independently of the two others.
As another example of a derivative law of motion, I may quote the very notable one which results from the interaction of the first and second funda- mental laws upon one another, and which gives the general expression for any function whatever of a matrix in the form of a rational polynomial function of the same and of its latent roots, to wit, the magnificent theorem that whatever the form of the functional symbol <^, and whether it be a single or many valued function, if Xj, Xj, ... \„ be the latent roots of 7n,
•^"^ = ^"^^^ (X,-X,)(X,-X3)...(X.-XJ • p p As for example if <pm = m', m' will have q" roots which are completely determined by the above formula.
The first law, as already stated, regards a single body or matrix, un- influenced by the action of any external force. The second law regards the effect upon a single matrix, subject to external impulses, taking their rise in an external source; whilst the third law has regard to the mutual
16] the World of Universal Algebra 151
action or joint effect of two bodies or matrices simultaneously operating upon one another.
Note. Making [in p. 149] m' - 36m= + 3dm -g = F (m), we found (F'm) n = cm?- 2em + g.
When two of the latent roots of m are equal, it is easy to prove that F'm is vacuous, and conversely, that when F'm is vacuous, two of the latent roots of m are equal ; but when F'm is vacuous it is no longer permissible to drive it out of the equation, and accordingly the true statement of the theorem in question is that when m. n are two matrices of (any) the same order, such that mn = nm, n must in general be a function of m, but that this ceases to be true, when and only when m has two equal roots. The theorem requires further investigation in order to make out what happens when, or how it can happen that, two of the latent roots of one and only one of the two convertible matrices are equal ; for supposing this to happen it would seem to lead to the conclusion that n may be a function of m, but m not a function of n ; which, however, is not quite so paradoxical as it looks, inasmuch as in ordinary algebra a constant may be regarded as a specialized function of a variable, whilst a variable in no sense can be regarded as a function of a constant. The following example of two matrices not functions of one another, but forming commutabie products, has recently occurred to me in practice, and led to the discovery of the oversight I had committed in stating the theorem in question in too absolute terms.
Opp* Oil lix=lOl,y = pOp' where p»+ p + 1 = 0, it will be found that xy = yx, p'pO pp^O but that neither x nor y is a function of the other; this may easily be deduced from the fact that x'-p'x-2p = 0, so that if y were any function of X, it would be reducible to the form of a linear function thereof, and con- sequently (on account of the zeros in the two matrices) y must be a multiple of JB, which is absurd.
In like manner it will be found that y» - p^y - 2/> = 0, and that conse- quently X cannot be a function of y.
17.
EQUATIONS IN MATRICES. [Johns Hopkins University Circulars, III. (1884), p. 122.]
I HAVE been lately considering the subject of equations in matrices. Sir William Hamilton in his Lectures on Quaternions has treated the case of what I call unilateral equations of the form x'+px + q = 0, or x' + xp + q^O, where we may, if we please, regard x, p, q as general matrices of the second order. He has found there are six solutions, which may be obtained by the solution of an ordinary cubic equation. In a paper now in print and which will probably appear in the May number of the Philosophical Magazine, I have discussed by my own methods the general unilateral equation, say
af+paf^^ + qocf-'' + ...+l = 0,
where x,p, q ...I, are quaternions or matrices of the second order, and have shown, by a method satisfactory if not absolutely rigorous, that the number of solutions is w' — o)' + to, that is to say, the nearest superior integer to the general maximum number of roots {m*) divided by the augmented degree (a, + 1).
But after I had done this it occurred to me that there were multitudinous failing cases of which neither Hamilton nor myself had taken account, as for example x" + px = 0, besides the solutions x = 0, x = — p, will admit of a solution containing an arbitrary constant, I think ; but that is a matter which I shall have to look further into before committing myself to a positive assertion about it. I have only had time to pass in review the more elementary case of a unilateral simple equation, say px = q, where p, q are matrices of any order to.
If p is non-vacuous there is one solution, namely, x = p~^q; but suppose p is vacuous : what is the condition that the equation may be soluble ?
(1) Suppose q — O,]) being vacuous has for its identical equation pP = 0, and consequently we may make x = \P where \ is an arbitrary constant.
(2) Suppose q is finite and that a; = r is one solution, then obviously the general solution is a; = r + XP.
17] Equations in Matrices 153
We have now to inquire what is the condition that r may exist. I find
from the mere fact of x being indeterminate (and confirm the result by
another order of considerations) that the determinant of ? + Xp must vanish
h' c identically ; so that for instance when p, q are of the second order and , .
are the parameters to the coi'pus (p, q), we must have when d = 0, which is implied in the vacuity o{p,f=0 and e = 0. The first of these conditions is known d priori immediately from my third law of motion ; but not so, without introducing a slight intervening step, the intermediate one (I mean the con- nective to d and /, namely) e = 0.
So in general in order that px + q = 0 may be soluble, that is, in order that p~^q where p is simply vacuous may be Actual and not Ideal, q must satisfy as many conditions as there are units in the order of p or q, all implied in the fact that the determinant to p+\q, where \ is an arbitrary constant, vanishes identically. When these conditions are satisfied p~'q becomes actual but indeterminate. (This, by the way, shows the disadvantage of calling a vacuotis matrix indeterminate, as was done in the infancy of the theory by Cayley and Clifford — for we want this word as you see to signify a combination of the inverse of a vacuous matrix with another which takes the combination out of the ideal sphere and makes it actual.)
So in general in order that p~^q where jo is a null of the ith order (that is where all the (i + l)th but not all the tth minors of p are zero) shall be an actual (although indeterminate) matrix, it is necessary and sufficient that p + \q, where \ is arbitrary, shall be a null of the .same (ith) order. What will be the degree of indeterminateness in p~^q, that is, how many arbitrary constants are contained in the value of x which satisfies the equation px = 0 remains to be considered.
The law as to the conditions is an immediate corollary to my third law of motion, for if px = q then p + X^ = p (1 + \x) ; consequently p + \q, what- ever X may be, must have at least as high a degree of nullity as p. q.e.d.
18.
SUR LES QUANTITES FORMANT UN GROUPE DE NONIONS ANALOGUES AUX QUATERNIONS DE HAMILTON.
[Comptes Rendus, xcvm. (1884), pp. 273—276, 471—475.]
Dans une Note prec^dente*, j'ai fait allusion au cas oil le determinant de X + ym + zn devient une fonction lineaire de a?, y, 2? sans que la quantity noram^e Q s'^vanouisse. . Dans ce cas, on aura
(r)inf + Q{mn)-R=Q, (1)
R ^tant le determinant de mn. C'est bien la peine, comme on va le voir, de donner plus de precision aux Equations qui lient ensemble mn et nm pour ce cas.
En suivant la meme marche que pour le cas particulier oil Q = 0, on trouvera sans difficult^ les r^sultats suivants :
«m = - -^ {mny - ^^— mn--^, (2)
3Q, „ ?-9iJ 2Q^ ,„,
mn= -^ {nmy - ^ nm + -^, (3)
f ^tant le produit des differences des racines de la fonction \' + Q\ — iJ, de sorte que ^ = - (4Q» + 27i?»).
Cons^querament on peut ^crire
nm= A{mnf + Bmn+G, (4)
mn=-A {nmf + B'nm - G, (5)
ou A et 0 peuvent etre tons les deux z4to, ou tous les deux des quantity finies quelconques, mais non pas I'un d'entre eux une quantity finie et I'autre zero, et B, B' les deux racines par rapport a B de I'^quation
An
B' + B+l + ^ = Oi. (6)
* Comptes rendus, t. xcvii. p. 1336.
[tit follows from n {mn -f- 9) = {nm + 6) n that M, =mn and N, = nm both satisfy equation (1) ; further MN=NM (footnote * p. 127 above), so that (p. 149 above) there exists an equation N=pM' + qM + r; from (1), if \M-N\*0, follows M^ + MN + N^ + Q = 0. Hence (2), (3) can be deduced.]
18] Sur les quantites formant uu groupe de nonions 155
On peut verifier, comme je I'ai fait, par un calcul alg^brique direct, que les eqiiations (4) et (5), en vertu des Equations (1) et (6), sont compatibles.
Or une chose digne de remarqiie, c'est ce qui arrive quand ?=0, car cela servira a reveler un phenomeue d'Algebre universelle d'un genre que personne n'avait encore ineme soup9onne.
Dans ce cas, les deux equations (4) et (5) changent leur caractere et deviennent
Qimnf + ^RmnJrlQ'^O,
Q (nm)"- + SRnm + ^Q' = 0,
de sorte que mn et nm cessent d'etre fonctions I'un de I'autre.
Nommons, pour le moment, mn = u, nm = v; on aura, comme auparavant, uv = vu, sans que v et u soient fonctionnellement li^s ensemble. Dans le Johns Hopkins Circular de Janvier 1884 (dans I'article intitult^ Ox the three laws of motion in the world of universal Algebra, [above p. 146]), on trouvera le moyeii d'etablir qu'en general cette Equation amene k la conclusion que ou
COO u doit etre un scalar, c'est-i-dire de la forme 0 C 0, ou bien v un scalar, ou
0 0 C sinon que nm, mn doivent etre fonctions I'un de I'autre ; mais on remarquera (ce qui m'avait alors ^chapp^) que, si Fu = 0 est I'^quation identique en u et que la deriv«?e fonctionnelle F'u est une matrice vide (vacuous), c'est-a-dire dont le determinant est z^ro, le raisonnement est en d^faut ; cette vacuity a lieu dans le cas, et seulement dans le cas, ou deux des racines latentes (lambdaiques) de m sont egales. On peut g^ndraliser celte conclusion et r^tendre a deux matrices u et v d'un ordre quelconqne au-dessus du deuxieme ; c'est-^-dire qtiand les racines latentes de u (ou bien de v) ne sont pas toutes in^gales, il est des cas ob. uv = vu, sans que u ou v soient des scalars et sans que v et w soient fonctions I'un de I'autre. Par exemple, si Ton fait
u =
|
0 p |
P' |
|
1 0 |
1 |
|
p" p |
0 |
|
- p |
|
|
uv = |
p |
v =
|
0 |
1 |
1 |
|
p |
0 |
p' |
|
p |
p' |
0 |
on trouvera
P 1 I — p 1 ' = tm.
P' P' -p\
Mais on d^montrera sans difficult^ que v ne peut pas s'exprimer comme sorame de puissances de u, ni vice versa v comme somme de puissances de u.
On n'a pas besoin de remarquer que la seule condition de I'existence de racines latentes Egales en u ou en v ne peut pas suffire en elle-mSme pour
156 Siir lea quantiUs formant un groupe de nonions [18
assurer que m = vu, mais il faut r^server pour une autre occasion la pleine discussion de la totality des solutions de cette equation importante.
J'ajouterai seulement cette remarque, qui est essentielle. En supposant I'existence des Equations
TO*n + mnm + nm* = 0,
n'm + nmn + mn* = 0, (m»)» + Qmn - i2 = 0, {nmf + Qnm - R =0, qui ont lieu necessairement quand le determinant de x-\-ym-\-zn devient une fonction lineaire de a?, f, s?, et en regardant nm comme fonction de mn (en vertu de I'^quation mn . nm = nm . mn), alors, en additionnant aux deux valeurs de nm (exprimd comme fonction de mn) donn^es ci-dessus, qui corre- spondent aux deux valeurs de f, c'est-^-dire V-(4Q'+ 27^),on a k consid^rer quatre autres valeurs, le nombre total en ^tant six. Car si Ton suppose nm = A (mny + Bmn + C et si \,\^, X, sont les trois racines de X' + Q\ - iJ = 0, les valeurs d^ A, B, G sont determin^es en mettant
^Xi» + -BX. + C=X.-,
A\^^ + B\, + C = \k,
oh i,j, k sont respectivement
13 2 2 3 1
1 2 3 ou ou bien 3 2 1
3 12
2 13
Les valeurs de A,B,C donn^es ci-dessus correspondent au deuxifeme de ces groupes de valeurs de i,j, k.
Si Ton 6crit i = 1, j = 2, k = 3, on trouvera nm = mn. Si Ton ecrit i =l,j = 3, k = 2, en faisant X, = A, on trouvera
2Aimny-Qmn + 2AQ 3A' + Q Dans le cas critique ou f = 0, de sorte que 3A^ + Q = 0, I'equation devient (mny + Amn - 2A» = 0, comme dans le cas d^ja traite. Quand on suppose Q egal k z^ro et R (c'est-^-dire le determinant de mn) fini, les seules solutions possibles avec ces conditions sont celles fournies en ^crivant i,j, A = 2, 3, 1, ou 3, 1, 2 ; mais, pour le cas gdn^ral, il n'y a pas de raison (au moins trh Mdente) pour exclure aucune des trois classes de solution. Si Ton admet la Mgitimit^ des solutions de la troisieme classe, en dcrivant
nm = A (mny + Bmn + C,
A r
on trouvera B^ + B + — - = 0
18] mmlogues aiix quaternions de Hamilton 157
au lieu de 1 equation
AC 5^ + 5 + 1+^ = 0,
qui est applicable aux solutions de la deuxieme classe.
Avant de coiisiderer I'equation xy = yx, il importe d'avoir une id^e nette d'une certaine classe de matrices que je nomme priviUgUes ou derogatoires, en tant qu'elles d^rogent a la loi g^nerale que toute matrice est assujettie a satisfaire a une Equation identique dent le degre ne pent pas etre moindre que I'ordre de la matrice.
Les matrices derogatoires sont justement celles qui satisfont a une Equation d'un ordre inf^rieur k leur ordre propre; on pent les nommer simplement, doublement, triplement, . . . derogatoires, selon que le degre de I'equation identique a laquelle elles satisfont differe par une, deux, trois, ... unites du degre minimum ordinaire.
Pour le cas des matrices du deuxifeme ordre, il n'y a que les scalars „
0 a qui soient derogatoires.
Pour le cas des matrices du troisifeme ordre, en dcartant les scalars de la a 0 0 forme 0 a 0, toute matrice x derogatoii-e peut 6tre ramen^e ou k la forme 0 0a
a + 6(6 + €»),
oh. e est une matrice qui satisfait a I'equation e* = 1, c'est-i-dire une matrice dont les racines latentes sont 1, p, p', ou a la forme
a + 6(l + e + r')?,
oil 6»=1, f=l et ^€ = pe^,
p signifiant une racine cubique primitive de I'unit^ Dans le premier cas,
ar" - (2a + 6) a; + (a» + o6 - 26») = 0, ' et dans le second
sd'-2ax + d' = 0,
car on trouvera facilement que
(l + e + e')C(l+€ + €»)?=0.
Pour le cas du quatrifeme ordre, en ^cartant les scalars et en se bornant au cas ou I'equation identique d^rog^e (vue pour le moment comme une equation ordinaire en x) ne contient pas des racines egales, toute matrice X peut 6tre ramen^e k I'une ou k I'autre des deux formes suivantes :
a + b{U+U',) oubien a + 6 (tr+ ^^ f7^ + /fcf/'
158 Sur leg qvantit^a formant un grmipe de nonions [18
oh U est une matrice du qiiatri^me ordre telle que U*+l =0; a,b,k sont des scalars arbitraires et i est une racine primitive biquadratique de I'unitd ; quand, pour la seconde forme A;= 1, on trouvera qu'il y aura une derogation double de I'ordre de I'equation satisfaite par x, I'^quation identique pour x ne sera que du deuxieme degre.
En r^servant les details du calcul, voici le resultat g^ndral que j'ai demontr^ rigourensement (en m'aidant de la notation des nonions) pour les matrices du troisieme degr^ qui satisfont a I'equation xy = yx.
A moins que x ne soit une matrice privil^giee ou derogatoire, y sera toujours une fonction rationnelle et entiere quadratique de x, et de meme, a moins que y ne soit privil^giee, x sera une fonction pareille de y.
II est bien entendu que le caractfere derogatoire d'une seule des deux matrices n'emp^che pas quelle ne soit une fonction entiere et rationnelle quadratique de I'autre. Dans le cas oil a; et y sont tous les deux d^rogatoires, ni I'un ni I'autre ne peut etre exprim^ comme fonction explicite I'un de I'autre, mais ils seront li^s ensemble par une Equation lineo-lineaire.
II parait peu douteux qu'une regie semblable doive #tre applicable a requation xy = yx, quel que soit I'ordre des matrices x et y, sauf quand I'equation qui lie ensemble x et y pourra etre d'un degr^ moindre que I'ordre de chacune d'elles.
II est bon de remarquer que nulle matrice ne peut etre derogatoire, sauf pour le cas ou il existe des egalites entre ses racines latentes; mais ces egalites peuvent parfaitement subsister sans que la matrice a laquelle elles appartiennent soit derogatoire. En general, si x = a + by + cy*, on peut, par une formule generale que j'ai d4jk donnee, exprimer y sous la fonne
a + ^x + yx' ;
avec I'aide des racines latentes de x, cette formule ne cesse pas en general d'etre valable, meme pour le cas ou x contient des racines egales, en regardant leur difference comme une quantite infinitesimale ; seulement le nombre des racines finies subira dans ce cas une diminution ; mais, dans le cas ou I'equation xy = yx (x etant derogatoire) menerait a I'equation
x = a + hy + cy'',
on trouverait que nulle fonction explicite de x avec des coefficients finis ne peut exprimer le y cherche.
II est a peine necessaire d'ajouter que rien n'empeche, dans le cas ou I'un ou I'autre de x et y ou tous les deux sont derogatoires, qu'on puisse satisfaire k xy = yx, en supposant que x et y soient des fonctions explicites chacune I'une de I'autre : tout ce qu'on affirme, c'est que, dans le cas admis, cette supposition cesse d'etre obligatoire ; c'est un cas tres semblable a ce qui arrive dans le cas de defaut (failing case) du theoreme de Maclaurin : c'est
I
18] analogues aux quaternions de Hamilton 159
celui ou une variable est une fonction sans pouvoir etre ddvelopp^e dans une serie de puissances d'line autre variable.
Dans ce qui precede, on a vu un example du fait general que, m. ^tant ime matrice donnee, I'^quation ^ (a;, m) = 0, pour certaines valeurs de m, cesse Ld'admettre la solution ordinaire x = Fm.
Mais il existe encore une classe assez etendue d'equations entre x ei m pour lesquelles, quand m prend certaines valeurs, x ii'a aucune existence actuelle ; par exeniple, m dtant une matrice vide d'un ordre quelconque, si mx = 1, la matrice x devient inexprimable et n'a, pour ainsi dire, qu'une existence ideale.
Je citerai encore I'exemple a? = m, m 6tant une matrice du deuxieme
ordre ; si les racines latentes de in sont inegales, on trouvera, par la formule
I g^n^rale, quatre valeurs de x. Si les deux racines latentes sont ^gales et
finies, ces quatre valeurs se r^uisent a deux ; mais, si les deux racines sont
' toutes les deux egales a zero, il n'y aura aucune valeur de x qui satisfasse k
a [I'^quation donnee, c'est-^-dire si m= k; I'^quatiota devient absolument
ka —a
[insoluble, ou, si Ton peut s'exprimer ainsi, les quatre racines carrees de m [sont toutes ideales.
Dans le cas suppose, on v^rifiera ais^ment que m' = 0 et, vice versa, toute
a racine carrde du zero binomial est de la forme k , de sorte que Ton peut
ka — a dire qu'une racine carree quelconque du zero binomial ne poss^de pas elle- meme des racines aig^briques quelconques, ou, en d'autres termes, une racine alg^brique quelconque du quaternion i + V(— l)i est puremeut ideale et n'adniet pas d'etre representee sous la forme d'un quaternion. Finalement je remarque que toute matrice est d'un certain ordre et d'une certaine classe; I'ordre, c'est le nombre total de ses racines latentes ; la classe, c'est le degre minimum de I'equation latente (c'est-k-dire de I'^quation identique k laquelle la matrice satisfait), lequel ne peut etre plus petit que le nombre des racines latentes in«?gales.
Je dois ajouter (ce que j'aurais Ah dire auparavant) que, quand x est une matrice temaire ddrogatoire dont toutes les racines latentes sont Egales, r^quation xy = yx peut subsister sans que ni a; ni y ne soit une fonction explicite Tun de I'autre, meme quand y n'est pas une matrice privile'git^e ; c'est le cas ou, e et f fai.sant partie d'un groupe de nonions ^l^meataires, on a x= a + b{l + e+ f')^. Les caiculs sont un peu compliques pour ce cas special, mais je crois ne pas me tromper en faisant cette correction. Le champ de la theorie de la quantity multiple est tellement nouveau et inex- ploite que, sans les plus grandes precautions, on est toujours en danger de se beurter centre quelque cause impr^vue d'incertitude ou meme d'erreur.
19.
SUR UNE NOTE R^CENTE DE M. D. ANDRE*.
[Comptes Rendus, xcviii. (1884), pp. 550, 551.]
Le the'oreme de M. Andr^ est une consequence immediate de la gene- ralisation que j'ai donnee du thdorfeme de Newton {Arithmitique universelle, 1" Partie, Ch. ll.) sur les racines imaginaires des Equations.
On verra, eu consultant mon travail f sur ce sujet (Proceedings of the London Mathematical Society, No. 2), que si Mo, w,, «,,... «,„ sont les co- efficients d'une Equation du degrd m et si
Gr = rUr' - (r -h 1) 7r M,_, M^+l
V + r—l ou 7»- = ; .
7r ^tant une quantity r^elle quelconque qui n'est pas interm^diaire entre 0 et — m, I'dquation aura n^cessairement au moins autant de racines imaginaires qu'il y a de variations de signes dans la s^rie G^, Gi, G.^ G,„.
En faisant i) -= — m, on a le th^oreme de Newton ; en faisant t; = 1, on voit qu'on peut prendre G, = m^'' — Mr-i^^+i. Cons^quemment le theoreme de M. Andr^ subsiste, quel que soit le signe de la quantity qu'il nomme a et quels que soient les signes des quantit^s qu'il nomme u^, v^, ..., «„.
De plus, le theoreme subsistera encore quand, outre ces modifications, au lieu de I'^quation
M„ = aM„_i-|-/3u„_2,
on ecrit i)„ = o[u„_i + /3y„_2
ou ^0, fi, V.2, ... , Vm,
identiques avec
M„
Ml Wo
m' 1
^(m.m-l) ^--^-^Tn(m-l)(in-2)
* Comptes rendus, stance du 18 Kvrier 1884. [+ Vol. II. of this Eeprint, pp. 501, 507.]
19]
Sur line Note recente de M. D. Andre
161
II y a encore une autre extension importante a ajouter, en considdrant r^quation
M„_, !/„+, -Un' = Aa'' + B/a"^ + Crf,
dont j'ai donn^ une solution particuliere dans Y American Mathematical Journal, Vol. IV. [Vol. lii. of this Reprint, pp. 546, 633.]
II est peut-^tre digne de remarque que si, dans la formule dtablie pour 7,, on fait V infini, la regie calquee sur celle de Newton (mais plus gen^rale) enseigne que, quels que soient o, b, c ou m, I'^quation
f, , , a? a? a;"" \
"V^'^^ir^^r^r^-'-'TY-rn)
+H'-" + T-r:2T3 + -±r2r:^) + ^=o
ne peut jamais avoir plus de deux racines reelles.
IT.
11
20.
SUR LA SOLUTION D'UNE CLASSE TRfiS ETENDUE D'^QUATIONS EN QUATERNIONS.
[Comptes Rendus, xcviii. (1884), pp. 651, 652.]
L'^QUATION parfaitement gen^rale du deuxifeme degre en quaternions sera de la forme
2 {axhxc + dxe) +/ = 0
et admettra seize solutions, qu'on pourrait obtenir d'une manifere directe au moyen de quatre Equations, chacune du deuxifeme degr6, contenant les quatre elements de x comme inconnus. De meme, I'^quation en quaternions ou en matrices du deuxifeme ordre du degre a admettra m* solutions. Parmi ces formes g^n^rales, on peut distinguer celles dans lesquelles tous les quaternions donnes se trouvent du meme c6te du quaternion cherch^, par exemple aa? + 6a; + c = 0. On peut nommer de telles Equations equations uni- laterales. Hamilton a considdre le seul cas de I'^quation quadratique {voir Lectures on Quaternions, art. 636, pp. 631 — 2), et a determine le nombre (6) des racines.
Or, je trouve que ma m^thode g^n^rale de traiter les matrices amfene directement k la solution d'une equation unilat^rale d'un ordre quelconque (o (c'est-a-dire la fait d^pendre de la solution d'une equation alg^brique ordinaire) et donne sans la moindre difficult^ et sans aucun eflfort d'in- vention le nombre des racines. Ce nombre est exprim6 par la fonction ft)» — 0)2 4- o), de sorte que le nombre des racines, pour ainsi dire ^vanouies par suite de I'unilateralisme de la forme, est w* — «' + w' — <», c'est-a-dire (o)'— o)) (o)" -f 1). On comprend bien qu'en certains cas le nombre des racines subit une reduction ; par exemple, le nombre des racines de a;" + i = 0 est co" et celui de x" + kx + l = 0 est 2a)' — to. II semble que le nombre, pour I'^quation
doit etre (0 + l)(o^— Ow, lequel, quand 0 = co — l, devient le nombre g^n^ral to'— ay' + CO. Les details de ce petit travail seront donnes dans un prochain num^ro du London and Edinburgh Philosophical Magazine.
21.
SUR LA CORRESPONDANCE ENTRE DEUX ESPfeCES DIF- F^RENTES DE FONCTIONS DE DEUX SYST^MES DE QUANTITES, CORR^LATIFS ET ^GALEMENT NOMBREUX.
[Comptes Rendus, xcviii. (1884), pp. 779—781.]
Voici le theor^me a demontrer, dans lequel, par somme-puissance, on sous-entend une somme de puissances de quantit^s donndes:
A i quantites on pent en associer i autres telles, que chaque fonction syme- trique (qui est une fonction des differences) des premieres sera une fonction des sommes-puissances du 2', du 3', ... , du i**"" ordre des demieres.
Faisons, pour plus de clart^, t = 3.
Soient r,, r^, r, les racines de I'^uation
fr = aT^ + br' + cr+d = 0.
En prenant b,c,d; ri,r,,r, com me deux systfemes correlatifs de variables ind^pendants, on trouve
Done ZaZb + 26S, + cS^ = - S S,,
oBi, + bBc + cSa= dt -^ Sr. rj r
Soient a = a, 6 = 3/3, c = 3 . 2 . 7, d = 3 . 2 . 1 . 8, et soient p,, p,, /», les racines de lequation
ap'+/3p'' + 7p+5=0. Alors, si 2 8, <^ = 0, on aura (a.1^ + /38^ + 784) <^ = 0. C.Q.F.D.
L'integrale g^n^rale de la premiere ^uation est
^ = S (''i - n, n - n),
et celle de la demifere est
</> = JF. (/>>' + />.' + P*', Pi* + pi + />.')•
11—2
164 Sur la correspondaiice entre deux especes [21
Ces deux int^grales sont done identiques, et, le raisonnement ^tant g^n^ral pour une valeur quelconque de i, on voit que chaque fonction des difiPSrencea des r doit pouvoir s'exprimer comrae une fonction de i— 1 sommes-puissances cons^cutives des p (commen9ant avec la seconde), les r et les p dtant li^s ensemble par les Equations
ar^ + 6r*-' + ct^-* + dr^' + ... = 0,
api+ ~.p'-'+ .,.^ ..p^-'+ ... .f,. „■/)<-'+ ■■■ = 0, I "^ t(t — 1) i(i— l)(t — 2) '^
et cons^quemment une fonction symAtrique des differences des r sera une fonction rationnelle et entifere des i — 1 puissances cons^cutives (dont on a d^ja fait mention) des p.
En prenant i = oo , on voit que le th^orfeme ^quivaut h, dire que tous les sous-invariants, sources des covariants de (a, h, c\x, y)', (a, h, c, d^x, yf, ... (a I'infini), seront des fonctions des sommes-puissances prises a I'infini, avec la seule exception de la somme lineaire, des racines de I'equation
a + bx + -^^x'+ ^ , „ af+...(k I'infini).
Tel est le thdorfeme capital d^couvert par M. le capita! ne Mac-Mahon, de rArtillerie royale anglaise, dont il a fait le plus heureux usage en d^veloppant la th^orie des perp^tuants (voir America7i Journal of Mathematics). II est Evident que le meme principe peut etre appliqu^ aux invariants de toute espfece, de sorte que, grace a la belle d^converte de M. Mac-Mahon, avec la generalisation (qui en sort presque intuitivement) que j'ai donnee, on est aujourd'hui en etat de traiter les parties les plus difficiles et les plus essentielles de la thdorie des formes alg^briques, comme M. Schubert I'a fait avec sa Zahl-Geometrie pour les figures dans I'espace, en faisant abs- traction, pour ainsi dire, de toute question de substance (de matiere contenue dans les formes), et en se bornant a un calcul purement arithmetique.
Je dois avertir que le theorfeme de correspondance, tel que M. Mac-Mahon I'a donne, a paru dans YAvierican Journal of Mathematics (Vol. VI. p. 131). M. Mac-Mahon affirme (mais sans aucune preuve) que, si (o, /3, 7, ... etant des nombres entiers plus grands chacun que I'unite) </> est de la forme l,r^s^P, ... ,ou r,s, t, ... sont les racines de I'equation
(«»'"- 0'nf:3'-)^^'^>"=«'
alors (ttoSa, -f- a,Saj -f- a^Sa^ ■+ ...) <^ = 0,
et il donne Et ^ le nom de fonction symetrique nan unitaire des racines. Ce theoreme est vrai seulement pour le cas ou n est infini (ce que M. Mac-
n
21] differ entes de fonctions 165
Mahon a oublid de dire), et dans ce cas il conduit k la consequence que les differentiants (c'est-a-dire les sous-invariants) de
sent Ae^ fonctions symetriques non unitaires des racines de I'equation
«„ + Oi a;- + ^ x-^ + j-|^ a;-» + ... = 0
et vice versa. Or il est Evident que chaque fonction symdtrique non unitaire d'un nombre infini de quantites n'est autre chose qu'une fonction des sommes • de toutes les puissances de ces quantites au del^ de la premiere. Voil^ pourquoi j'ai attribud k M. Mac-Mahon, dans ce qui pr^cfede (pour le cas d'une Equation dont le degre est infini), la connaissance du th^oreme que j'ai d^montre dans toute sa gendralit4
22.
SUR LE THEORJ^ME DE M. BRIOSCHI, RELATIF AUX FONCTIONS SYMETRIQUES.
[Comptes Rendus, xcviii. (1884), pp. 858—862.]
Dans la demonstration du th^oreme sur une correspondance alg^brique, ins^rd dans les Comptes rendus de la semaine dernifere [p. 163 above], j'ai eu occasion de consid^rer I'int^graie de I'^quation
( d _ d d \ ^ .
Je me suis aper5u depuis que cette int^grale pent se d^duire imm^dia-
tement du beau theoreme de M. Brioschi, sur les fonctions syndtriques, k savoir que :
d^ d^ d<j) d^ _
dSr dUr dUr+i " dOn
On en tire cette consequence immediate que, si <f> est une fonction des n premiferes sommes-puissances des racines de I'equation
ao a;" + a, «"-' + ... =0,
avec exclusion de la puissance r'^"", on aura
dd> dd> _
aar aa„
et consdquemment F(si, s^, ... , Sr-i, Sr+i, ... , s„) sera I'^quivalent complet de I'expression
f d , d . d\-' f.
V aar aar+i (ia„/
Dans le cas que j'ai consider^, r = 1, et nous avons trouv6
On pent trouver aussi facilement I'integrale complete de I'equation / d d d \*\ .
22] Sur le theoreme de M. Brioschi 169"
ou I'asterisque signifie qu'on doit prendre le produit complet de Taction de la forme lineaire agissant i — \ fois sur elle-meme. Ainai, par exemple,
('^ i + ^ zT''^'''^' "' [S + ^"^ 11+^^ (I)
» d
+ a
dc '
On trouvera sans difficulte que la valeur de cette int^grale est F+s,F, + s,^F, + ... + s,'-' Fi^^ , oil chaque F est une fonction exclusivement de s^, Sj, ... ,s„.
Consequemment le t**"* coefficient d'un covariant quelconque de (a„a,, ...,a„){x,yy* pent etre mis sous cette forme, si Ton se sert de s„ pour exprimer la somme des o)''""** puissances des racines de
«" + a,a;»- + ~ a;"-» + f-f-g*""' + ••• = 0.
En efifet, en ecrivant — = », tout covariant de degre arbitraire v apparte- nant a ce quantic sera de la forme
[Mo, (m«, Mi$S, 1), (Mo. «i . Wal*, l)*, ("o. "i, "2. Mj"5», 1)'. • ••] (.'"< !/)',
oh, en general,
dug rf", dt/g
v„ ^tant une fonction exclusivement de (o,ii; Sj, s,, ...,«„ du poids « + 1. J'ajoute encore cette observation que tout differentiant (c'est-^-dire sous- invariant ou seminvariant) d'un systeme de i quantics des degr^s m, fi, ... , M sera fonction exclusivement de s^, a,, .... Sm', a-,, a,, ..., a-^, ..., 82,83, ...,Sx et de I — 1 fonctions lin^aires ind*?pendantes de la forme
Isi + Xo-i + . . . + LSi ,
soumises a la condition que l + \+...+L = 0.
Je ne sais s'il vaut la peine de dire, comme conclusion, qu'en combinant le theoreme de M. Brioschi avec le mien sur les puissances (avec astdrisque) on trouve, pour I'equation
(ou le i est sans asterisque), I'int^grale partielle
4> = F+F,s, + F,s,' + ...+ Fi^, sr\ oil chaque F est une fonction arbitraire de s,.j.,, «i+s, ..., s„.
Ed effet, cette expression est I'integrale complete du systfeme form^ par I'equation supposee conjointe avec les Equations
168 Sur le tMoreme de M. Brioschi [22
On voit aussi facilement que I'int^grale de
est ^ - JJ, + UtSr + U^Sr' + ... + Ur-i8r*-\
oil chaque U est une fonction arbitraire de Si, s„ ... , »,_i, «r+ii ... s».
On peut former un nombre infini de syst^mes construits au moyen des op^rateurs (flo 3 — H ••• ) dont on connaitra d'avance les int^grales ; ainsi, par exemple, le systfeme de r Equations
aura pour int^grale complfete
<!>= [/o + «,£/,+«j''I/j+...+Sj*-'Cr,_,,
oil chaque U repr^sente une fonction arbitraire de (sjSjSj ... Sji-iSj,- ... s„), en omettant celles des quantit^s s,, S3, ..., Sji_i dont les sous-indices excfedent n.
Pour indiquer le moyen de justifier ces ^nonc^s, prenons comme exemple le cas des Equations simultan^es
(ooSai + . . . + a„_i Sanf (f) = 0, ou Ei'tf) = 0,
(ucBtti + . . . + a„_a5a„) <f> = 0, ou E^tf) = 0,
(aoSa,-!- ... +a„_sSa„) ^ = 0, ou Es<f> = 0.
On trouvera facilement qu'en general ^i' = ^V— 2£'V^a + -^j. de sorte que le syst^me donn^ ^quivaut au systfeme
EV<f) = 0, E^4> = 0, E,<l> = 0.
Pour que ces equations soient satisfaites s^parement, il faut et il suffit que <f> soit respectivement de la forme
F («aSjS4 . . . S„) + SiFi (S2S3S4 • • • «n) + «i'-f2 (SsSjSi • ■ • ««), G (S1S3S4 . . . S„), H (SxSaSi . . . S„).
Cons^quemment, afin que les trois Equations soient toutes satisfaites simultan^ment, la condition suflSsante et n^cessaire sera que <f) soit de la forme
F(Si... S„) + S: F^ (S, . . . Sn) + si'F., (s^... S„),
laquelle est cons^quemment I'integrale complete du systeme donn^. De mSme, on ddraontre facilement que I'integrale complete des equations (ao So, + . . . + a„_i 8a„)' <^ = 0,
(OoSoj + . . . 4- ffln-aSctn) <f> = 0,
(aoSaj+... + an-iBa„y <f) = 0
sera
<f) = F{s,SiS, . . . Sn) + Si Fi (SjSsSj . . . s„).
23.
SUR UNE EXTENSION DE LA LOI DE HARRIOT RELATIVE AUX EQUATIONS ALGEBRIQUES.
[Comptes Rendus, xcvill. (1884), pp. 1026—1030.]
On peut envisager la loi de Harriot comme une loi qui affirme la possibility de decomposer d'une seule manifere un polyn6me en x dans un produit de facteurs lin^aires composes avec les differences entre x et les racines du polyn6me. En r^fl^chissant sur la cause de cette possibility et la mani^re de la demon trer, on voit facilement que le meme principe doit, avec une certaine modification, s'appliquer a toute equation en matrices d'un ordre quelconque dont les coefiBcients sont transitifs entre eux-memes, c'est-^-dire qui agissent les uns sur les autres exactement comme les quantites de I'Algfebre ordinaire, si chaque coefficient, par exemple, est une fonction rationnelle de la meme matrice. On peut nommer les ^uations dont les coefficients satisfont k cette condition Equations manothStiques : on remarquera que de teiles Equations forment une classe spdciale des equations que j'ai nomm^es unilatSrales dans une Note pr^c^dente.
Pour fixer les id^es, prenons comme exemple une Equation monoth^tique du second degr^ en matrices binaires, laquelle peut toujours ^tre ramen^e k la forme
ar" - 2px + Ap + B = 0.
En supposant que p!' — {a + fi)p + o/3 = 0 soit Vdquation identique de p, on aura
Faisons ^^ V(a' -A<x-B) = u, ^^ ^<^-A0-B) = v. a — p p — a.
Alors les quatre racines de p seront
p + u + v, p — u — v; p + u — v, p — u + v.
Disons r,, Tj, r,, r4.
170 Sur une extension de la loi de Harriot [23
On trouve
(p - /3)' = (;) - /8) (p - a) + (a - ^) (;) - /3) = (a - ^) (p - ^),
et de meme {p - a)' = (/9 - a) (|> - o),
de sorte que
M' + r' = ^^(a'-^o-5) + §^(/3'-^/9-5) a — p /3— a
= (a + fi)p-a^-Ap-B=f-Ap-B.
On a aussi uv = 0 et cons^quetnment (u + vy=u'' + v'' = (u — v)\ Done
(a; — r,) (« - r,) = (a; - 1))» - (« + v)" = a,-' - 2px + Ap + B,
{x - rj) (a; - r^) = (a; -p)''-(M -vy = a^- 2px + Ap + B.
Or consid^rons le cas g^nc^ral d'une Equation monothetique du degr^ n en matrices de I'ordre w.
Cette Equation (que j'6crirai fx = 0), en vertu de ce que j'ai nomm^ la seconde loi de mouvement alg^brique (c'est-a-dire la formule
.„ _ y (m - h) (m - c)...{m-l) {a — b){a — c) ...{a-l) ^
oi a,h,c,...,l sent les racines latentes de la matrice m), aura n" racines qu'on peut repr^senter par les symboles composes
'*l > '*2 1 • • • > ^01 )
oil chaque r parcourt les valeurs 1, 2, 3 n.
En refl^chissant sur la manifere de d^montrer le principe de Harriot, on arrivera facilement a la conclusion suivante : en prenaut une combinaison
quelconque de n symboles r^.r^ r„, de telle mani^re que chaque r
parcoure toutes ses n valeurs, R^, R^, ...,Rn, on aura
/x = {x- J?,) (x -R,)...{x- A.).
Ainsi on arrive au theoreme suivant :
Toute fonction monothetique rationnelle et entiere de x du degre n en matrices de I'ordre a petit etre reprhentde de (1 . 2 . 3 ... n)"~' manieres differentes comme un prodxdt de n facteurs lineaires dont chacun sera la difference entre x et une des racines de la fonction donnee.
Telle est la loi de Harriot, ^tendue au cas des quantit^s multiirration- nelles.
Dans le cas de I'Algfebre ordinaire, w = 1, et le nombre des decompositions de/x en facteurs, selon la formule, devient unique, comme il doit etre.
De mSrae, pour les quaternions, le nombre des decompositions d'une ' fonction monothetique du degr6 n en facteurs lineaires sera im. Par
41
I
23F] relative aiix dquations algebriques 171
exemple, si w = 3, les racines de fx peuvent etre exprimees par les neuf symboles
0.0 0.1 0.2
1.0 1.1 1.2
I 2.0 2.1 2.2
La fonction (comme on le demontrera facilement) pent §tre mise sous la forme a; — 0 . 0 multipliee par une fonction quadratique dont les racines seront des racines de fx, et consequemment, par raison de sym^trie, seront les quatre racines
1.1 1.2,
2.1 2.2;
done la fonction quadratique dont j'ai parl^ sera ^gale h,
(a;-l.l)(a;-2.2)
eti (a;-1.2)(a;-2.1).
Ainsi il y aura deux decompositions de fx qui correspondent aux deux diagonalea 0 . 0, 1 . 1, 2 . 2 ; 0. 0, 1 . 2, 2 . 1, et de meme il y aura des decom- positions qui r^pondent aux diagonalea 0.1,1.2, 2.0; 0.1,1.0, 2.2; 0.2, 1 . 0, 2 . 1 ; 0.2, 1 . 1, 2 . 0, de sorte que le nombre total est 6gal &, 1 . 2 . 3.
De mSme, quand fx est monothetique et matrice da troisieme ordre, on peut prendre les diagonales d'un cube. Par exemple, les racines de I'^quation monothetique du second degre en matrices du troisieme ordre peuvent etre representees par
0.0.0 0.0.1 0.1.0 0.1.1
1.1.1 1.1.0 1.0.1 1.0.0
et Ton aura les quatre decompositions
{x-0.(\.O^x-\.\.\); {x-0.0.\\x-\.\.Qi);
(a; -0.1. 05a; -1.0.1); {x -0 A A\x-\ .0 .0); et de m^me, en general, pour le degr^ n, le nombre des diagonales (en se servant de ce mot dans le sens analytique, bien entendu) sera
(1.2. 3. ..«)». C'est ainsi qu'on trouve I'expression g^nerale que j'ai donn^e (ttti)""' pour le nombre des decompositions quand le degre est n et que I'ordre des matrices est 10.
En multipliant ensemble toutes les Equations de decomposition, et en nommant v chacune des n" racines, on parvient k 1' equation
^(^x- «)'("-■>'•-' = ifxy"" ; done, quoiqu'on ne pui-sse pas en general conclure que, si X' = F* (X et Y
172
Sur une extension de la loi de Harriot
[23
6tant des matrices), X est n^essaireraent ^gal k Y,'\\ y & toute raison de croire qu'on pourra d^montrer que, dans le cas actuel, on aura
,r(a;-t;) = (/x)»— .
Ainsi la regie de Harriot se reproduira de nouveau sous la forme trfes peu modifi^e qu'un polyn6me (monothdtique) en x (dlev^ a une puissance convenable) est ^gal au produit des differences eutre x et toutes les racines en succession de ce polyndme.
On aura remarqu^, dans ce qui precede, qu'en appliquant la seconde des trois lois du mouvement alg^brique aux Equations monoth^tiques, on a trouv6 que le nombre des racines est n", et cons^quemment est w' dans le cas des quaternions, tandis que le nombre des racines pour la classe des Equations en quaternions unilatdrales (h laquelle les formes monoth^tiques appartiennent) est en g^n^ral n^ — n^ + n {voir le num^ro d'avril 1884 du London and Edinburgh Phil. Mag.), de sorte qu'il y a une Elimination n(n — 1)' de racines en passant du cas general au cas particulier.
II reste k examiner s'il n'est pas possible d etendre la loi de Harriot aux Equations unilat^rales polyth^tiques. C'est ce que je vais Etudier, mais sans cela, et en me bornant au cas monoth^tique, il me semble qu'en attribuant aux elements des matrices des valeurs entiferes (simples ou complexes), comme le fait M. le professeur Lipschitz pour les quaternions, on voit s'ouvrir un nouveau champ immense de recherches arithm^tiques fondles sur la loi fondamentale de Harriot gen^ralis^e de la mani^re indiqude dans ce qui pr^cfede.
24.
SUR LEs Equations monoth^tiques.
[Gomptes Rendus, xcix. (1884), pp. 13 — 15.]
Dans une Note pr^cddente sur une extension de la loi de Harriot, j'ai eu occasion de consid^rer les Equations dites monothetiques dont tous les coeffi- cients sont des fouctions d'une seule matrice. Or il y a une circonstance tres int^ressante et importante relative aux Equations de cette forme qu'il est essential de faire connaitre ; car, k defaut d'une telle explication, le lecteur de la Note cit^e pourrait facilement ^tre induit dans une erreur tres grave. Voici en quoi consiste I'addition k faire.
Supposons que tous les coefficients d'une equation donn^e soient des fonctioDS d'une seule matrice m. En appelant x I'inconnue, on pent r&oudre r^quation en regardant x comme fonction de m, et Ton trouvera ainsi n" racines, en supposant que n soit le degr^ de I'^quation et (o I'ordre de m. Ces racines seront parfaitement d6termin6es: mais on n'a nullement le droit de supposer qu'il n'y a pas d'autres racines qui ne sont pas des fonctions de m, qu'on peut nommer racines aberrantes, et un exemple, des plus simples qu'on puisse imaginer, suffira a d^montrer que de telles racines, en effet, existent ; je me servirai, pour cet objet, de I'dquation en quaternions (ou matrices binaires) se' — px=0.
En effet, on connait d^jk, a priori, la possibility de I'existence des racines aberrantes, car I'&juation en matrices a^ + q = 0, quand q est une matrice
/ . f^ ^ ^
scalar I comme si, par exemple, q^iO q 0 I, possede, on le sait, bien des
\ [O 0 q/
racines qui ne sont pas scalars et consequemment ne sont pas des fonctions
de q, et, de plus, ces racines contiennent des constantes arbitraires. Comme
ou va le voir, c'est aussi le cas pour I'^quation a? —px = 0, qui possfede une
seule constante.
Si Ton veut trouver ses racines normales (ou non aberrantes), on n'a qu'k r^oudre cette equation comme une Equation ordinaire, et Ton trouve ainsi
174 Sur les ^qtiations monolMtiqiiea [24
En nommant r et s les racines latentes de p, on obtient par ma formule d'interpolation (pour ainsi dire), r^cemment cit^e par M. Weyr,
1 / .p-s ^p-r \
IT i ti ^ s\ S i Ti ~~ 7* I
c'est-Jl-dire a; = 0, p, —^ , — — , et il n'y a pas d'autres racines de
^ r — 8 8 — r ■' ^
ce caractfere. Mais sortons de cette restriction arbitraire (produit de la
paresse de I'esprit humain, qui se fatigue enfin en voyant sans cesse se
reproduire des horizons nouveaux et inattendus), et posons hardiment
a B a b
70 ^ c a
oil a, /9, 7, 8 sont les quantit^s a determiner.
Puisqu'on fait abstraction des solutions x = 0, x=p, on sent, en vertu de la troisieme lot du mouvement algebrique, que x et x—p auront chacun un degr^ de nullity (car leur produit possede deux degr^s) ; ainsi, si a + S = 0, on aura
a!'=0,
done aussi px = 0,
et p sera aussi une matrice vide, c'est-k-dire qu'on aura
ad — he — 0.
La solution pour ce cas (dont, dans ce qui suit, je veux faire abstraction)
8er»
, {ac — a? x = X\ ,
(a' — ac
\ ^tant arbitraire.
Dans tout autre cas, en ^galant la raison du second au troisifeme membre de a? avec la meme pour px, on trouve sans difficulte que x sera de la forme
-\{d-r) \b
fiC —fi{a — r)
oil r et « sont les racines latentes de p, c'est-^-dire les racines de I'^quation
i^-{a + d)r + ad-bc = 0.
Alors, en calculant a^ et px, et en les ^galant terrae a terme, on obtient les quatre Equations suivantes :
\(d- r)' +fibc =bc-a{d- r),
b [\ {d - r) + n (a - r)]= - br,
c [\ (d — r) + /x (a — r)] = — cr,
\bc + /M(a — ry =bc — d(a — r).
24] &ur les equations monothetiques 175
En dcartant le cas special pour lequel 6 = 0 et c = 0, on voit (et c'est M. Franklin, de Baltimore, qui le premier s'est aper9u de cette conclusion capitale) que toutes ces Equations seront satisfaites avec la seule supposition
\ (d - r) + /* (a - r) + r = 0,
de sorte qu'une constante reste parfaitement libre dans la solution aberrante de I'equation iv'—px = 0.
Dans le cas ou p = . , on trouvera facilemeut les deux solutions deter- ■^ 0 a
minees
a 0 , 0 0
^=0 0 '' ^ = 0 d-
Dans ses Lectures sur les quaternions, Hamilton n'a pas mis le doigt sur les cas v^ritablement singuliers des Equations quadratiques unilat^rales. La condition de singularite, c'est-^-dire de la presence de I'un ou de I'autre des cas oil une ou plusieurs des trois paires de racines de I'equation pa^+qx+r=Q disparaissent ou deviennent indetermiu6es (c'est-a-dire affect^es de constantes arbitraires), peut se r^sumer dans la seule Equation 1=0, oil I est Tin variant quartique temaire quadratique (en u, v, w) qui exprime le determinant d'une matrice up-\-vq-\- wr.
25.
SUR L':6QUATI0N en matrices px^xq. [Comptes Rendua, xcix. (1884), pp. 67—71 ; 115, 116.]
SoiENT p et q deux matrices de I'ordre a>.
Pour r^soudre I'^quation px = xq, on obtiendra m^ Equations homogfenes lin^aires entre les w" elements de I'inconnue x et les ^l^ments de p et de q, de sorte que, afin que I'^quation donn^e soit r&oluble, les dl^ments de p et de q doivent etre lies ensemble par une et une seule equation.
Mais, si Veqiuition identique en p est dcrite sous la forme p" + Bp"-' + Gp"-^ + . . . + i = 0, on aura apparemment, en vertu de I'^quation p = xqx~^,
x(fx-^ + Exq^-^x-' + Cxq-'-'x-^ + ...+L = 0 ou bien ^ + Bq"~^+ Cq-^^ + ... + L = 0;
done les m racines de q seront identiques avec celles de p et, au lieu d'une seule Equation, on aura en apparence (au mains) a equations entre les i\i- ments dep et de q.
Pour faire disparaitre ce paradoxe, il n'y a qu'une seule supposition k faire : c'est que x, sous les suppositions faites, devient une matrice vide, car alors x~^ n'a plus une existence actuelle, et I'^quation p=xqx~^ n'aura pas lieu ; c'est ce qu'on va voir arriver dans le cos gdndral, ou px = xq.
Pour fixer les idees, supposons « = 1 et faisons
p:
a b
c d
1 =
a B \ \ X a
I \ V IT
En 6galant px k xq, on obtient les quatre Equations simultandes et homo- g^nes entre X, ^, v, tr suivantes :
(a - a) \ + c/i - /Sv + Ott = 0,
iX + (d - a) /i + 0 J/ - /Stt = 0,
-yX + Ofj, + {a-S)v+ C7r = 0,
0X + yfi + bv + {d - B) -TT = 0,
25] Sur V Equation en matrices px = xq 177
et consequemment on aura*
6»c» + ^-'f - 2hc^-f - 2abcd - 2a^yB + (be + ^y) (a + d}(a + B) - be {oi' + B') - ^y (a' + d') + aS (a' + #) + ad (a= + 8') + 2adaS + a^d^ + a=8' - (a + d) (a + S) {ad + aB) = 0, ou, en dcrivant a + d= B, ad—bc=D, a + 8 = C, aS — ^y = F,
(^D-Ff + iB- C){BF- CD) = 0; c'est-^-dire, si R est le resultant de X- - Bx + D, X''-Cx + F, R = 0 sera la condition g^nerale de la possibility de satisfaire k I'equation px = xq.
II est facile de faire voir que ce r^sultat peut etre etendu au cas general
lOU p et q sont des matrices de I'ordre w : on n'a qu'a demontrer que si une
ies racines latentes de p est egale k une de q, I'equation px = xq est resoluble;
let de plus, sans que cette condition soit satisfaite, I'equation est irr^soluble.
Solent done X,, X,, ..., X„ Ies racines latentes de p et /Xj, (i^, ..., /t„ de q et
Bupposons que Xj = /^ , alors
{p-\-^)x = x{q- ixi), Bt Ton peut satisfaire k cette equation en ecrivant
x={p-X,){p-\,)...(p-\^){q-^l^){q-|M3)...{q-^i„). Consequemment, si Ies racines latentes de p et de g^ sont Ies racines des deux formes alg^briques X" + BX"'^ + ... +L, X" -h GX"-^ + ... + M, quand R (le resultant de ces deux formes) s'evanouit, le resultant des oj" Equations homogenes lineaires obtenues en ^galant px = xq s'^vanouira ; mais R est indecomposable et du mSme degr6 (&>') que ce dernier resultant dans Ies Elements de p et q. Consequemment Ies deux resultants (a un facteur num^rique pres) sont identiques : ce qui d^montre que la condition R = 0 est non pas seulement n^cessaire, mais de plus suffisante afin que px = xq soit resoluble.
Pour ce qui regarde la valeur de x, posons x= UV, oil
U=(p-\)(p-X,)...{p-X„); V = {q - fj^) {q - fi^) ... {q - fj,J, le seul fait que x contient U comme facteur ou que x contient Fcomme facteur suffit a constater que x n'est pas seulement vide, mais de plus possede au moins ta — 1 degrds de nullity, c'est-a-dire que tous ses determinants mineurs du second ordre sont des z^ros.
Cela est la consequence d'un th^oreme que j'ai d^montr^ dans le Johns Hopkins Circular^ relatif au degr6 de nullity des combinaisons des facteurs latents d'une matrice, dont le th^orfeme relatif k I'equation dite identique de Cayley ou de Hamilton n'est qu'un cas particulier, ou pour mieux dire le cas extreme; seulement il faut y ajouter un theoreme qui fait partie de ma troisieme loi de mouvement algebrique, c'est-a-dire que le degr6 de nullity d'un facteur ne peut jamais exc^der le degre de nullity du produit auquel il appartient.
[' The expressions for p, q in line 7 from the bottom of p. 176 should be interchanged; in the last line of p. 176, for +yfi read -7/1.] [t p. 134 above.]
8. IV. 12
178 Sur r^qitation en matrices px = xq [25
Nous avoDS douc compl^tement r^solu le paradoxe qui ^tait k expliquer. Mais, sur-le-champ, une nouvelle contradiction surgit, car il semble que nous avons d^montr^ que, dans tout cas sans exception, si px = xq, x est ndcessaire- ment une matrice vide, ce qui est evidemment faux, car on sait bien que, si, CD dtant de I'ordre de p et de q, q=^ yfi\)p, alors, afin que I'^quation px = xq soit resoluble, il n'est jamais n^cessaire que x soit vide. Ainsi, par exemple, pour les matrices binaires, I'equation qx = xq est satisfaite quand x est une fonction quelconque de q, et I'equation qx——xq est resoluble, pourvu que g' soit scalar, en imposant deux conditions (dont une que son carre soit scalar) sur x. Pour lever cette contradiction, revenoas au cas oil w = 2 et aux Equations fondamentales
(a — a) \ + c/x — /9i' = 0,
6\ + (d-a)/x-/S7r = 0,
— 7\ + (a — S) 1/ + CTT = 0,
- 7/i + ti/ + (d - S) TT = 0.
Certes, si ces Equations donnent des valours d^termiii^es aux rapports X, /t, V, v, le raisonnement precedent rend certain que x doit etre vide, c'est-k-dire que Xir — fiv = 0, mais cette conclusion devient fausse aussitdt que p et q sont pris tels que ces rapports deviennent ind^termin^s, ce qui arrive quand tous les premiers determinants mineurs de la matrice (a -a) c -/3 0
b (d-a) 0 -/3
-7 0 (a-B) c
0 -ry b (d-B)
s'^vanouissent simultan^ment.
Dans ce cas, quoique la solution gendrale qui donne x vide tienne bon, rien n'empeche qu'il n'existe d'autres valeurs de x, c'est-k-dire de ^ , pour lequels cela n'est pas vrai.
La matrice ecrite en haut doit posseder et possede, en effet, la propri^te remarquable que, en supprimant une ligne horizontale quelconque et en nommant A, B, C, D les quatre determinants mineurs de la matrice rect- angulaire qui survient, affectes de signes convenables, la quantity AD — BG contiendra le determinant complet comme facteur. II sera peut-etre utile, avant de conclure, de donner un exemple d'un genre nouveau de subsistance de I'equation pa; = a;g avec une valeur finie du determinant de a;. Faisons done
a - 8 = 0, d-a = 0, be- fiy = 0,
on aura (a — c?) \ + c/t — /Sv = 0,
6\ - /Stt = 0, -yX + CTr = 0, — 7/x + Jv + (d — a) TT = 0,
i
25] Sur r equation en matrices px = xq 179
Equations qui n'equivalent qu'k deux,
6\ - /Stt = 0, {a-d)\ + (cfj, - ^v) = 0, et le determinant de x, c'est-a-dire Xir — /xp, aura en g^n^ral une valeur finie.
Dans la demifere Note (insdr^e dans les Comptes rendus*) qui roule sur r^quation en matrices binaires or' —px = 0, j'ai remarqu^ qu'en addition aux solutions normales
p-s p-r
x=<j, x = p, x = r- , x = s'
•^ r —s s — r
(ou r, s sont les racines latentes de p), on a la solution indt^termin^e (due en grande partie k la sagacity de M. Franklin)
x =
(-\{d-r) \6 1
( fic -ti{a-r)]
avec la condition \{d — r)+fi(a — r) + r = 0. Eviderament on a aussi la solution tout a fait distincte
-fiia-s)]
(-\(d-s)
avec la condition \(d — s) + fi (a — a) + s = 0 ; mais on doit noter que, quand
on prend \=fi, on reprend les deux valeurs normales x — r- , x=8- ;
'^ r — 8 s — r
le fait curieux que, quand 6 = 0 et c = 0, les deux solutions aberrantes
forment un troisi^me couple tout a fait d^termin^ a ^te d^j^ not^, et Ton
peut y ajouter la remarque que si, en addition ^ 6 = 0, c = 0, on a aussi
a-d = 0,
alors I'ind^termination reparait k pas redouble, la solution entiere ^tant dans ce cas extra-spdcialement constitute par une paire de solutions dont I'une et I'autre contiennent deux constantes arbitraires au lieu d'une seule.
Je dois ajouter que, dans le cas oil i racines de p (X,,, X,, ..., X,) sont identiques avec i de q (fij, ^, ..., fti), I'^quation
px = xq,
qui amene a p'x = xq', ..., p'x = xq'
et, par consequent, k
(p-X,)--- (P - 'W) a; = a; (gr - ^i,) ... (q-/j^),
sera satisfaite si Ton fait x= UV, oh
U= (p -Xi+i) ...(p-K), r=((jr -/i<+.) ... (g - fiJ),
[* p. 174 above.] % 12—2
180
Sur rSquation en matrices px = xq
[25
de sorte que x (en vertu du theor^me d^ja cit6) aura au moins a> — 6 degr^s de nullity, c'est-k-dire tons ses determinants mineurs de I'ordre ^ + 1 s'dva- nouiront. Mais on sait, pour le cas oh 6 = a> (et Ton a toute raison de croire pour le cas oil ^ a une valeur quelconque au-dessus de I'unit^), qu'il existe pour des valeurs sp^ciales de /) et de q des solutions singulieres de I'^qua- tion px=:xq, lesquelles (comme dans le cas de I'equation de Riccati) sont bien autrement int^ressantes et beaucoup plus importantes que la solution g^n^rale.
On remarquera que, quand d — o>, la solution g^n^rale disparait, tandis que les solutions singulieres pour des valeurs particulieres de p et de q, ayant tontes les racines latentes de I'un identiques avec celles de I'autre, forment la base de la presentation des matrices sous la forme de quaternions, nonions, etc.
n
i
26.
SUR LA SOLUTION DU CAS LE PLUS GENERAL DES EQUA- TIONS LINEAIRES EN QUANTITES BINAIRES, C'EST-A- DIRE EN QUATERNIONS OU EN MATRICES DU SECOND ORDRE.
[Comptes Rendus, xcix. (1884), pp. 117, 118.]
SoiENT p, q deux matrices d'un ordre donnd et servons-nous du symbole p( )q pour signifier I'operateur, lequel, applique a nne autre matrice x du mSme ordre, donne pscq.
Alors, si Ton pose
pA )qi+pA )q2+ ■■■+Pn{ )qn = 4>,
<f>x sera une matrice dont chaque element sera une fonction lin^aire des ^l^ments de x ; cons^uemment, en supposant que les matrices p, q sont de I'ordre co, on parvient ainsi k une matrice de I'ordre to', et cons^quemment (f> sera asaujetti a une Equation identique de I'ordre to'^; disons F=0.
Je vais donner la valeur de F pour le cas ou w = 2, c'est-a-dire oii F sera une fonction du quatri^me degrd Suppo.sons que P et P' sont deux quantics du second ordre dans les deux systemesde variables a;i,a:2, ...,a;„; fi, fa, ...,fn contragredients. Alors, si Ton represente par P' ce que devient P' quand on
^crit Sx,, 8«,, ..., S^ au lieu de fi, fa fn. (-P*)'- P* sera un invariant du
systfeme donn^ pour toute valeur de i.
Considerons le cas ou P = ax' + bxy + cy- et P' = af + /Jf?; + 77;^. Dans ce cas, on trouvera que \ \jJ^Y ^ ~ ^{^ • -^ W sera identique avec le r^sultat de ax' + bxy + cy'', '^a? — ^xy + ay^, de sorte qu'on peut le nommer le contra- risultant des formes (a, b, c), (a, /3, 7). Je nommerai done, en g^n^ral, I'invariant ^ [(P')' P* — 4 (P' . P)'] le quasi contra-rdsultant des deux formes P, P quand elles contiennent un nombre quelconque de variables.
Or, en revenant k I'expression </>, nommons P le determinant de
U^pi + U^p.i + . .. + UnPn + 4>-'U
et Q le determinant de
"igi + Ma?'^ + . . . + u„qn - V,
182 Le cos le plus gSn^ral des equations lin^aires [26
oh. ^, pour le moment, est traits comme une quantity ordinaire. J'ai trouv^ que le quasi contra-resultant de P, Q, quand <^ appartient k des matrices du second ordre (lequel sera une fonction biquadratique de ^), dgal^ k zero, est r^quation identique cherchde en ^.
II est probable, mais je n'en suis pas encore absolument convaincu, qu'une m^thode analogue donnera I'^quation identique de <^ pour des matrices d'un ordre quelconque.
Si Ton suppose que les p et les q sont des quaternions, rien ne change avec I'exception que P et Q seront d^finis comme dtant les modules (les tensors carr^s) au lieu d'etre les determinants de ^v + 2p«, — v + 'S.qu respectivement.
Connaissant ainsi I'^quation identique de <f), on peut r^soudre immediate- ment I'equation
lipxq) = T,
car, en ^rivant p( )q = (f>, on a I'dquation connue
<f>* + B^' + Cif}' + J)<f> + E = 0,
et, consequemment, en exceptant toujours le cas oil E=0 (dans lequel cas I'equation devient on impossible ou indetermin^e), on trouve
Par exemple, si I'equation donn^e est pxq + rxs = T,
<i>T = pTq + rTs,
(fy'T = p^Tq^ + prTsq + rpTqs + r'Ts',
4>^T = fTf -irp^Tsq^ + prpTqsq
+ rp^Tq^s + pi^'Ts'^q + rprTsqs + r'pTqs' + r»2V,
et, ^ventuellement, en ne se servant que des coeflScients qui entreat dans les fonctions P et Q par le moyen de formules connues, on r^duit x a une somme de multiples de termes de la forme
pT, rT, prT; pTq, rTq, prTq; pTqs, rTqs, prTqs,
et ainsi en g^n^ral. Done le problfeme de la resolution des Equations lin^aires est completement r^solu ; seulement il reste k traiter en detail le cas singulier oil la matrice appartenant k <f> est vide.
27.
SUR LES DEUX METHODES, CELLE DE HAMILTON ET CELLE DE L'AUTEUR, POUR RESOUDRE L':6qUATI0N LIN^AIRE EN QUATERNIONS.
[Comptes Rendus, xcix. (1884), pp. 473—476, 502—505.]
Un celebre quaternioniste m'ayant demande de lui expliquer la portee de ma solution de I'^quation lineaire en matrices sur la solution du meme probleme en quaternions, il me semble desirable de donner explicitement le moyen de passer d'une solution a I'autre. Prealablement, il sera bon cependant de remarquer que, faute d'un examen suffisamment attentif de la forme du resultat obtenu ou plutot indique par Hamilton {Lectures on Quaternions, pp. 559 — 561), on pourrait attribuer a sa solution une pro- pridt^ qu'elle ne poss^de pas, celle de fournir le moyen de trouver la solution de I'^quation lineaire en quaternions sous une forme reduite semblable a celle que fournit ma m^thode : mais, en effet, I'examen d'un seul terme de m (voir au bas de la page 561), par exemple Srj7'^, suffit k montrer que le d^nominateur to de Hamilton est du douzieme degr^ dans lea Elements des quaternions (b et a) de son Equation 'S.bqa = c (p. 559), tandis que le degr6 pour la forme rdduite n'est que huit. II s'ensuit que le num^rateur (si Ton avait la patience de le deduire des formules de Hamilton), aussi bien que le denominateur obtenu par ce moyen, serait affectd d'un facteur Stranger k la question, du quatrieme degr^, dans les elements nommes.
J'ajoute qu'il est parfaitement possible de donner la valeur de x dans I'equation ^pxp = T comme fonction seulement des p et p' et des coefficients des deux formes assoaides sans aucune irrationnalite. Car le determinant du nivellateur '2p{ )p', disons N, etant obtenu sous la forme il^ + \/(^4). le determinant du nivellateur
-10 NO
^P()P'+ ()
0-1 ON
(disons FN) sera aussi exprim^ sous une forme semblable k celle-la, disons
*a + s/{<t>,).
184 Resolution de Equation lin^aire en quaternions [27
Or, au lieu de I'^quation identique FN = 0, on peut se servir d'un multiple quelconque de cette Equation pour obtenir I'inverse de N comme fonction de puissances positives de JV^ Ainsi Ton peut, dans ce but, se servir de I'equation *j' — 4>4* = 0, au lieu de FN= 0, et, avec I'aide de cette Equation, on obtiendra x exprim^ en fonction des p et p' et de fonctions rationnelles des coefficients des deux formes associ^es ; mais alors, au lieu d'etre obtenu sous sa forme la plus simple, son num^rateur et son d^nominateur con- tiendront un facteur commun qui sera une fonction du huitifeme degr^ des ^l^ments des p et des p'.
Je passe k la regie pour traduire ma solution de I'equation en matrices ^pscp' = T en solution de cette m6me Equation quand les p, les p' et le T, au lieu d'etre matrices, sont donnas comme quaternions. Evidemment tout ce qui est n^cessaire, c'est de connaitre I'equation qui serait identique pour ^P( )p'; J6 v^'is donner la rfegle pour I'obtenir.
Sous le signe S, je suppose compris p, q, r, ..., p, q', r'
Ecrivons la forme symbolique [Nx + {p)y + (q)z + ...]', disons X; les co- efficients de xy, xz, ..., symboliquement ecrits, sont
2(p)N, 2iq)N, ...; a (p), (q), ... il /aut svhstituer Sp, Sq, ... ; le coefficient de y^ est (p)' auquel il faut substituer Tp'; finalement le coefficient de yz est (p)(q), auquel il faut substituer S(Vp. Vq)*.
De meme, on construit et Ton interprete la forme
[-x' + (p')y+(q') / + ...]' (disons X').
On calculef la valeur de J^'^X- — ^ {XXy. Ce rdsultat (une fonction du quatrieme degrd en N) (disons UN) sera une partie de la fonction qui doit Stre identiquement zero. Le reste de cette fonction (disons 64fl,iV) sera
[■2S{VpVqVr) S{Vp'Vq'Vr')] N-lSpSp' S(VpVqVr) S (Vp Vq' Vr'),
etjedisque niV+64n,iV = 0
sera I'equation identique en N, et servira pour trouver la valeur de x, c'est- ^-dire N-^T comme fonction du quaternion T, des quaternions p, q, .... p', q, ... et des symboles S, V, T; de plus la valeur ainsi obtenue sera x sous sa forme reduite.
II y a encore une petite observation a ajouter k mes remarques sur la solution de Hamilton de I'equation ^bqa = c (Lectures, p. 559). II divise q en deux parties, le scalar w et le vecteur p.
C'est cette derniere quantite (p) qu'il exprime sous la forme — ; alors
1^ ~ — ^CY/ t\ . de sorte que, k d^faut d'avoir recours k des reductions 2,0 (ao)
[* See first note on p. 191 below.] [+ See p. 181 above and p. 202 below.]
27] Resolution cle T equation lineaire en quaternions 185
ulterieures, le d^nominateur de q contiendra, non sevdement le facteur etranger du quatrieme degre dans les elements des a et des b dont j'ai d^ja parld, mais encore le facteur etranger XS{ab).
On remarquera que, dans cette solution, on aura des combinaisons des b avec des a et des fonctions quatemionistiques de ces combinaisons, tandis que, dans la solution infiniment plus simple que je donne du probleme, il ne se trouve nulle part des melanges de cette nature, mais seulement des fonctions quatemionistiques de combinaisons des a entre eux-memes et des b entre eux-memes. Le vice fondameutal de la m^thode de Hamilton, c'est la reduction du problfeme donne a un autre, ou, au lieu de q, il n'entre que sa partie vectorielle. Neanmoins le travail de Hamilton (quoique sa raison d'etre ne subsiste plus) ra^ritera toujours d'etre regard^ comme un monument du g^nie de son grand et admirable auteur.
C'est la, pour la premiere fois dans I'histoire des Math^matiques, qu'on rencontre la conception de i'^quation identique (voir Lectures, pp. 566, 567) qui est la base de tout ce qu'on a fait depuis et de tout ce qui reste k faire dans revolution de la Science vivante et remuante de la quantity multiple, c'est-a-dire I'Algebre universelle, nee a peu pres 250 ans apres I'organisation definitive de sa sceur ain^e I'Arithmetique universelle, dans le Memoire de M. Cayley sur les matrice-s, daus les Philosophical Transactions, vol. 148.
Dans une Note pr^c^dente, on a vu que dans la rioiivelle et seule bonne m^thode pour r^soudre, par rapport a x, I'equation en quaternions
pxp' + qxq' -(- rxr' + sxs' + ...=V,
on fait trois operations. La premiere, k laquelle on peut donner le nom de nivellation, consiste a trouver le nivellant, c'est-a-dire le determinant de la matrice du quatrieme ordre appartenant k un nivellateur donne du second ordre. La seconde, qu'on peut appeler deduction, consiste a obtenir I'dquation identique, k laquelle un nivellateur correspond au moyen d'un autre nivel- lateur qu'on obtient du nivellateur donn6 en y adjoignant un couple de plus de la forme— i\r( )S«, ou,ce qui revientau mdme.le couple v'(—-^)( )V(— -^). ou N est consid^rd comme un scalar. Finalement, on arrive h. la demi^re operation, que je nommerai substitution et reduction, et qui consiste k sub- stituer a I'inverse du nivellateur sa valeur en fonction rationnelle du troisieme ordre de lui-m6me, puis k faire des reductions dont je parlerai tout k I'heure.
Au moyen de ces operations, on arrive a la valeur de I'inconnue de I'equation sous sa forme reduite la plus simple qu'elle puisse prendre.
Pour obtenir la forme de I'dquation identique, voici ce que j'ai trouv^ en appliquant la methode indiqu^e dans la Note pr^cedente.
186 absolution de Viquation lin4aire en quaternions [27
Pour plus de simplicite, je me sers de la notation suivante, qui s'applique k des lettres quelconques, accentu^es ou non, repr^sentant des quaternions.
Je pose
Sp = ip), Tf=p,, S(VpVq) = (j>q), S{VpVqVr) = (pqr).
Alors, en ^crivant
P( )p' + q( )q' + r{ )r' + s( )s' + ...=N, on aura
N* - 4S (p)(p')N' + 2 [HpYp, + 4 {pjp, - 2p,p\] N"
- S (4 {p){p')p^p\
+ 8 [.ip){q'){n)-p'^ + ip'Kq}p'q -Pi] - ^{p){p')q.q\
+ 4 [{q){p')p.q\ + (p) {q')p\q.] - 8pp'(qr) {q'r) + 8 \.<.P){q)iqr)p'r + (p'KqKq'r'Kpr)] + St (pqr}(p'qV)] N + t[p^'p\''-2p^p\.q^q\
+ 4 [p^qi (p'q'y + p'2q\ (pqY] - ip^p'tipq ■ p'q
+ 4p,p\qr . qV + 8 [p, (qr)(pq)(p'r') + p\ (q'r'){pq)(pr)]
+ 8[pq.rs. p'r . qs +p'q' . r's' .pr.qs\-8 {p){p'){qrs){q'r's')] = 0,
oh. le dernier terme de la partie fonctionnelle de I'^quation est le nivellant deK
Quant a la substitution, si, dans I'dquation prdcedente
on remplace N~^ F par la fraction
NT - AN'T + BNr - Cr D
tons les termes du numdrateur de cette fraction seront des multiples connus de la forme FTP', ou P est de I'une des formes suivantes : p' ; p''q, pqp, qp? ; P', pq; p; ..., et oil de meme P' a des types semblables avec des lettres accentuees. II ne reste plus qu a reduire chaque P a sa forme la plus simple, c'est-a-dire a I'exprimer comme fonction lindaire de ^,p, q,pq —qp, et de meme pour P'. Alors le num^rateur de a; ne contiendra plus que des termes dont les arguments seront- tous d'un des types suivants (je remplace la moiti^ de pq — qp par [pq]) :
r, pV, Tp, pTp', pTq\
[pq-\v, np'q'l pnp'q^ ipqWp, [pqWlv'q'V,
il faut y ajouter le type pqrTr'q'p', qui est d^jk sous sa forme la plus simple et n'exige aucune formule de reduction.
• D est le determinant de la matrice qui appartient au nivellement N. Qnand D = 0, la solution de I'^quation Nx = T devient ou idiale (ce qui a lieu en g^n^ral), ou (ce qui a lieu pour des cas particuliers) actuelle, mais iuditermin^e.
H
27] Resolution de T equation lin^aire en quaternions 187
Je n'entreprendrai pas pour le moment de calculer les coefficients de ces guments, mais j'indiquerai du moins les formules de reduction qui seules nt necessaires pour effectuer ce calcul. Ce travail, bien digne d'attirer I'attention de quelque jeune gdomfetre, peut tres probablement amener k des resultats qui, a I'aide d'une notation symbolique, pourront Stre presentds sous une forme d'une simplicity tout a fait inattendue et pour ainsi dire pro- videntielle. J'en ai eu I'experience pareille dans d'autres recherches du meme genre, dans la solution de certains cas d'^quations quaternionistiques du second degre.
Voici toutes les formules de reduction dont on aura besoin : p' = 2(p)p-p„ p> = [4,{py-p,]p-2{p)p„ pq = [pq] + (p) ? + (3) p - ( pq), qp = - [pq] + {p)q + (q)p - (pq), p'q = 2 (p)[pq] + 2 (p)(q)p + (2p» -p,) q-2 ip)(pq),
pqp = *(p)[pq] + [8 (p)(q) - 2 ipq)]p
- [4 (py +p.]q- [2 (q) p. + 4 {p)(pq)] ; dans les formules on peut, au lieu de [pq], ecrire V(VpVq).
Remarque. — Quand un nivellateur devient symetrique, c'est-a-dire quand p = p', q = q', ... , alors les deux formes associ^es coincident en une seule dont le nivellant devient un invariant orthogonal.
Qu'il me soit permis, avant de conclure, d'ajouter encore une petite reflexion sur I'importance de la question traitde ici. Elle constitue, pour ainsi dire, un canal qui, comme celui de Panama, sert a unir deux grands oceans, celui de la theorie des invariants et celui des quantity complexes ou multiples : dans I'une de ces theories, en efifet, on considere Taction des substitutions sur elles-m^mes, et dans I'autre, leur action sur les formes; de plus, on voit que la theorie analytique dea quaternions, ^tant un cas particulier de celle des matrices, cesse d'exister comme une science ind^- pendante ; ainsi, de trois branches d'analyse autrefois regardees comme ^nt ind^pendantes, en voila une abolie ou absorbee, et les deux autres i^anies en une seule de substitution alg^brique.
28.
SUE LA SOLUTION EXPLICITE DE L'^QUATION QUADRATIQUE DE HAMILTON EN QUATERNIONS OU EN MATRICES DU SECOND ORDRE.
[Comptes Rendus, xcix. (1884), pp. 555—558, 621—631.]
Hamilton, dans ses Lectures on quaternions (p. 632), a fourni un moyen de resoudre I'^quation (en quaternions ou en matrices binaires) de la forme
of - 2px + q = 0;
mais les circonstances les plus int^ressantes de la solution ne se font pas voir dans sa m^thode de traiter la question. Voici la maniere analytique directe que nous employons pour obteuir x sous sa forme explicite.
On suppose a.- — 2Bx + D = 0
r^quation identique pour x, oh B et D sont des scalars k trouver.
En combinant ces deux Equations en x, on obtient
2x=(p-B)-\q-D),
et, en supposant que la forme associSe a [1], p, q, c'est-^-dire le determinant de X + /ji,p + vq, soit
X- + 26X/i, + IcX-v + dfj? + 2eii,v -\-fv-, on aura*
4>{d- 2hB + &) x,^ - A>{e-bD - cB -^ BD)x,J^f - 2cD + D-- = Q.
Cons6quemment, en ecrivant u=B — h,v = D—c,
d-b^ = a, e-bc = ^, f-c'=y,
et, en comparant cette Equation avec I'equation donnee, on voit qu'on pent " 6crire
M^ + a = \, uv + ^= 2X(i(, + b), v- + y = 4X (v + c).
De plus, puisque p^ — 2bp + d = 0, on aura
_ {p — b + u)(q — c — v) _ _ {p — b + u)(q — c — v)
^~ 2(b'-d-u') ~ 2\ •
[* The determinant of 2Bx^ -D- 2x,y + q being zero, if Xg is a latent root of x.]
i
28] Solution de V equation quadratique de Hamilton 189
En ^liminant m, v entre les trois equations qui les lient avec h, c, a, yS, y, on trouvera I'equation bien remarquable
e^(-«c-«d' .1 = 0,
ou I est le discriminant de la forme associ^e donnde plus haut, c'est-^-dire
16c
1= b d e =d/+2bce-dd'-e^-fb-,
c e f
de sorte que la quantity exponentielle symbolique repr^sente une fonction cubique et donne lieu a une equation cubique en \.
A chaque valeur de X correspondent les deux valeurs + i\/{\ —a) de w
et k chaque valeur de u (autre que u = 0) correspondra la seule valeur
„^ , {2\ + c)b-e ,
2X + ^^ de V,
u
Quand m = 0, X = a = d — 6^ et I'equation
»» - 4Xi; + 7 - 4\c = 0
a ses deux racines finies. Done, quand u = 0, il faut que
prenne la forme ^, et i cette valeur de n (qu'on peut envisager comme
deux valeurs de u r^unies en une) correspondront pour v les deux valeurs donn^es par I'equation quadratique ci-dessus.
Ainsi Ton voit qu'en general x a trois paires de valeurs d^termin^es et qa'aucune de ces valeuis ne cesse d'etre actuelle et determinde que pour le seul cas ou I'une des trois valeurs de \ est 6gale a z^ro, c'est-k-dire ou /, Tin variant de \&pleine* forme associ^e h, (p, q), s'^vanouit.
Cela revient k dire que / est le crit^rium de la normality de I'equation doDD^e.
Si Ton regarde p et q comme des quaternions, on aura
b=Vp, c=Vq, d = Tp\ e = SpSq-S{VpVq), f=Tq\
II est bien digne de remarque que 4/ est identique avec i^pq — qp^.
On peut d^montrer que, si p et g sont des matrices d'un ordre quelconque, les racines de I'equation a? — 2px + 3 = 0 seront toujours (comme icl) associe'es en paires ; car, si Ton ^crit x + x, = 2p, on aura
x^-2a:ip + q = 0,
et consequemment, si p" — (obp"^^ + ... = 0 est I'equation identique connue en p et x"— a)Baf~^ + . . . = 0 I'equation identique k trouver en x, k chaque valeur
* Noug avoDg diji d^fini la formt ataoeUe au corps p, q, r, .... Par la pleine forme, on peat sons-entendre oe que devient la forme aB8ocite quand on adjoint an corps one matrice nnitaire.
190 Solution de Tdqimtion [28
de 5 — 6 correspondra une valeur ^gale de b — B, c'est-k-dire que I'dquation pour trouver B sera de la forme F{B — b)* = 0.
En se servant de I'^quation conjugu^e (c'est-k-dire en a;,) dont la somme des raciues sera ^videmment la meme que pour I'equation en x, on obtient immediatement, dans le cas oil p et q sont dii second ordre, par le moyen de la formula
(p + b-v)iq-c-v) * 2k
et de I'dquation en \, la valeur de 2a;*.
Cette valeur sera 6[p + (28c — Sd) -^^1, de sorte que la valeur moyenne d'une racine de I'dquation a;" — 2px + q = 0 est j) (la valeur moyenne pour le cas ou p et g' sont scalars), augment^e de (2Be — Bg)I^, ou I^ doit avoir le signe qui le rend dgal k ^{pq — qp). De mSme on trouve
2a;' = 2j32a; - 6q, et ainsi la valeur moyenne de a;* sera
2p'-q + {iB,-2Sa)I^p, et Ton peut trouver successivement, par la meme m^thode, la valeur moyenne d'une puissance quelconque de x. Les details du calcul pr^c^dent, et encore d'autres propriet^s de I'equation en x, seront donnas prochainement dans le Quarterly mathematical Journal ou quelque autre recueil math^matique. Ici on n'a voulu que produire les rdsultats principaux obtenus par notre m^thode.
L'^quation de Hamilton en quaternions ou en matrices binaires est celle que nous avons traitee dans une Note precedente. C'est I'dquation
ar" + 2g'a! + r = 0.
Nous avons trouv^ que la solution de cette Equation depend d'une equation cubique ordinaire en X, a chaque valeur de laquelle correspondent deux valours de x, et qu'elle est normale ou rt^guliere quand le dernier terme de cette Equation differe de zero. L'^quation est dite reguliere ou normale quand sa solution depend du nombre maximum de racines determinees, c'est- k-dire de trois paires de racines d^termint^es ; chaque paire est alors connue comme fonction de \, q, r et des parametres b, c, d, e,f qui dependent de q
On aura 2x= -2 'P-'' + "^^^-^-'').
H
On retranche une Equation de I'autre, on substitue pour 2 — sa valeur tir^e de I'equation
A
cubique en X, et on 6orit pq -qp='2I',
i\
28] qiiadratique de Hamilton 191
et r et sont d^finis au moyen du determinant de u-\-vq-\- wr* qu'on a sup- pose etre mis sous la forme
u- + 2lmv + 2ciiw + dv'' + 2evw + fw", d'oh
b = Sq, c = Sr, d=Tq\ f=Tr^e = SqSr- S{Vq.Vr)*.
Dans ce cas, on pent dire que la solution elle-meme est r^gulifere.
En nommant / I'invariant de la forme ternaire, ^crite plus haut, c'est-a-
dire en posant
I = df+ 2bce - l^f-cFd - e^
nous avons trouv^ que I'^quation en \ pent etre mise sous la forme
oii n = 2S, - Srf,
c'est-^-dire qu'on aura
4V + (4c-4d)X» + (46e-4cd+c--/)X-L/=0.
Ainsi, afin que la solution soit rdgulifere, il faut et il suffit que / difffere de z6ro+.
De la il suit que, dans le cas d'une equation r^guli^re, deux x ne peuvent dtre egaux, h. moins qu'ils n'appartiennent k la meme paire ou bien que deux \ ne deviennent dgaux ; car x pent Itre exprimd comme une fonction lineaire
de qr, q, r, 1, dans laquelle le coefficient de qr est — ^ .
Done, si deux des x sont egaux sans que deux \ le soient, une Equation lineaire subsistera entre pq, p, q, 1, mais dans ce cas nous avons trouvd ailleurs que / = 0, et la solution cesse d'etre rdguli^re.
Nous allons pour le moment nous borner au cas ou I'^quation est r^- guli^re, et cons^quemment nous n'aurons qu'^ consid^rer les cas oti il y a ^galit^ ou entre deux racines de \ ou bien entre deux valours de a; qui corre- spondent a la m^rae valeur de X.
Si Ton suppose que deux valeurs de \ soient egales, il en rdsultera que deux des paires de valeurs de x deviendront identiques, de sorte qu'une seule condition suffira k r^duire le nombre des racines distinctes de 6 a 4, c'est-k-
* Par an oubli trAs regrettable nous avons pris, dang une Note pr^c^ente, pour le coeflBcient de 2xi/ dans la forme associ^e &
S(Vp Vq) an lien de sa vraie valenr, Sp Sq - S {VpVq),
et de mtime ponr les autres coefficients des temies mixtes, de sorte que le calcol du determinant dn nivellateuT "Lp ( )p' dans la Note snr I'aofadvement de la solution de I'^quation lineaire en qnatemions est erron^ et a besoin d'etre fait de nouveau.
t Consiquemment, quand I'^qnation est r^guli&re, ni 7 ni u ne pent devenir z<5ro ; car, dans I'an et I'antre de ces denz cas, 1=^; aussi, pour la mdme raison, r ne peut pas Stre une fonction deg.
192 Solution de Tiqiiation [28
dire que lea valeurs de x, qui, en general, sont de la forme m, m; n, n; p,p', deviendrorit de la forme m, vi'; n, n'; n, n.
Au lieu de calculer directement le discriminant de I'^quation en X, qui donnera un r^t^ultat trfes compliqu^ nous allons montrer qu'on pent substituer le discriminant de la forme tr^s simple biquadratique
c+2d
l.h,'^,e.f)(r,8r.
Mais pr^alablement il sera utile d'op^rer une transformation lindaire sur r^quation en \.
Ecrivons \ = fi + d; I'i^quation en fi sera
V + 4 (c + 2d)/x2 + [(c + Uf + 46e - /]/i + 26 (c + 2d)e - 6=/ - e' = 0.
On voit done que le discriminant qu'on veut calculer est une fonction complete de 6, c + 2d, e, f.
Nous avons trouv6 m'' = \ — d + 6", c'est-^-dire /a + fr*. On aura done
4M» + 4(c + 2d-3&')u*
+ [126* - 8 (c + 2d)}P + (c + 2df + (46e -/)]«»
-[26^-6(c + 2ci) + e]"-».
Dans r^quation donn^e, substituons x + e, oil e est un infinitesimal
. e 0 . {scalar si Ton parle de quaternions ou repr^sentant la matrice si Ton
0 e parle de matrices) ; alors p sera augmente par e et g par 26j9, et ainsi
(\ + /ip + vq) deviendra (X + e/ii) + (/i + 2ev)p + vq, de sorte qu'en designant le discriminant cherch^ par D, I'accroissement de D est nul quand \ et ft deviennent \ + e/i, jx + ev simultandment, c'est-k-dire quand la forme ternaire en u, V, w devient
m" + 2 (6 + e) Mt) + 2 (c + 2e6) mv + {d + 2eh) v'
+ (2e + 26C + 4ed) m) + (f + 4-ee) w'. Done [aBb + 2bS, + 2cSa + (c + 2d) 3, + 4e8/]Z) = 0.
Ecrivons c+2d= Sm. On sait que B est une fonction complete de b, m, e,f, de sorte que, par rapport k D (comme op^rande), Bc+Ba = Bm', aiusi, en I ^crivant 1 = a, on aura
(aBb + 2bB,n + 2mB, + 4eSy) D = 0.
D sera done ou un invariant ou un sous-invariant de la forme biqua- dratique (a, b, m, e, f).
* u sera la partie scalar de x si I'^quation est donn^e sons la forme quaternionique, ou bien la moiti6 de la somme du premier et du quatrieme element de x ei I'^quation est donnce entre des matrices. Hamilton a trouv6 I'equation ^quivalente k celle donnfie pour « dans le texte; mais, dans sa formule, les coefficients sont exprim^s sous une forme compliqu^e et assez difficile k d^brouiller.
28] quadratique de Hamilton 193
Mais, en faisant attention a I'equation en fi, on voit que D sera de I'ordre 6 dans les coefficients et du poids 12; il est done un invariant et une fonction lineaire de s' et <' (ou s et t sont les deux invariants irr^ductibles) de la forme biquadratique.
En nommant A le discriminant de cette forme, on a
A = s'-27<^
dont une partie sera /' — 27b*/' ;
mais on voit, par I'examen de I'equation en /*, qu'une partie de D sera
et, consequemment, i) = — ^ A.
II s'ensuit que la condition n^cessaire et suffisante pour I'^galite de deux des racines de I'equation donnde avec deux autres est tout simplement A = 0, comme nous I'avons d^ja ^nonce.
Cherchons la condition pour laquelle les trois paires coincideront toutes dans une seule paire ; alors les trois racines de /jl deviennent toutes ^gales, et Ton a noa seulement
A = 0,
mais encore (1 2m') — (9m' + 46e — /) = 0,
e'est-a-dire /— 46e + 3w' = 0 ou « = 0.
Done les conditions necessaires et suffisantes, pour qu'il n'y ait que deux
racines distinctes chacune, prises trois fois dans la solution de I'equation
donn6e, seront
s=0, <=0.
On pent aussi demander quelle est la condition ou plutdt quelles sont les Equations de condition pour que deux racines de la meme paire soient ^gales.
Dans ce cas, nous avons trouv^ que m = 0; cela exige que le dernier terme dans I'equation k u' devienne z^ro. On aura done, en vertu de I'equa- tion en u\
ae - Sbm + 26^ = 0,
c'est-i-dire que le sous-invariant gauche ou bien le premier coefficient du Hessien k la forme biquadratique s'^vanouit. Mais cela ne suffit pas pour que les deux x d'une paire deviennent parfaitement identiques. II faut aussi que les deux valeurs de v, qui correspondent k la valeur z^ro de u, ou que les deux racines de I'equation
■t^-*\{v + c) + y= 0,
oji \ = a = d - 6',
deviennent egales, c'est-k-dire que
7+c»-(2a-)-c») = 0, a IV. 13
194 Solution de liquation [28
ou bien, puisque y=f — c^, que
f-{Sm-2b'y = 0;
k cette Equation il faut joindre I'^quation d4jk trouv^
ae-3bm + 2b' = 0;
le systfeme de ces deux Equations exprime la condition de la coincidence des deux a; d'une paire. Quoique/— (3wi — 26*)* = 0 ne soit pas en elle-meme un sous-invariant, les deux Equations ci-dessus constituent (comme elles doivent le faire) un plextis sous-invariantif ; car on trouvera
(aSft + 26S,„ + 3mSe + ^eB/) [af- (3am - 26*)'] = 4 (a« - 36m + 26») = 0. En effet, puisque / - (3»» - 2b-)- ne difffere de / - 9m» + 2a6e + 66*m (le second coefficient du Hessien) que par — 26 (ae — 36m. + 26'), on peut sub- stituer, pour le plexus ^crit plus haut, le plexus H^^O, Hi = 0, oh Hi, H^ sont le premier et le second coefficient du Hessien de la forme quadratique.
Or il est facile de demontrer que, quand dans la forme (a, 6, m, e, / ) {x, y) a n'est pas zdro, mais que les deux premiers coefficients du covariant irr^- ductible gauche le sont, le covariant s'evanouit compl^tement*, et la forme biquadratique a deux paires de racines egales.
On sait aussi que, quand les deux invariants irr^ductibles s'^vanouissent, il y a trois racines egales, et, quand en meme temps les deux invariants et le covariant gauche s'^vanouissent, toutes les racines de la biquadratique sont Egales.
Ainsi on voit que les seuls cas d'^galite possibles entre les racines de r^quation quadratique donn^e, quand sa solution est regulifere, correspondent aux quatre cas d'^galit^ entre les racines de la biquadratique ordinaire qui s'y est associee.
En prenant les quatre cas : 1° ou la quadratique a deux racines Egales ; 2° ou elle a deux paires de racines egales ; 3° trois racines Egales ; 4° toutes ses racines egales ; alors la quadratique donn^e aura, dans le premier cas, deux paires de racines Egales ; dans le deuxieme, quatre racines Egales ; dans le troisifeme, trois paires de racines Egales, et dans le dernier cas toutes ses racines seront Egales.
Quant au rapport de la biquadratique binaire a la forme ternaire quadra- tique, on passe de la seconde a la premiere, en se servant de la substitution dont s'est servi notre trfes honor^ collegue, M. Darboux, dans sa belle Note sur la resolution de I'equation biquadratique {Journal de Liouville, t. xviil. p. 220). On n'a qu a faire x = «*, y = 2uv, z = x^, et la forme ternaire passe dans la forme binaire biquadratique. On voit ainsi que les genres de solutions r^gulieres de I'equation en quaternions donn^e dependent ex-
* Quand les deux premiers coefficients du covariant irreductible gauche d'une biquadratique binaire s'^vanouissent, le discriminant s'evanouit a6cessairement : nous avons trouv^ que ce discriminant pris ndgativement ^gale 16 fois le produit des coefficients extrSmes, moins le produit du second et I'avant-dernier coefficient du covariant gauche.
1
28] quadratique de Hamilton 196
clusivement de la relation entre la conique qui s'y est associee avec la conique absolue y"^ — 4r^. Dans le cas le plus g^n^ral, les deux courbes se coupent en quatre points ; dans les quatre autres cas, il y aura I'une ou I'autre des quatre especes de contact entre les deux coniques.
Mais, de plus, on voit ^videmment que cette id^e des deux coniques pent etre dtendue h, I'^quation de Hamilton, meme pour le cas ou la solution devient irreguliere.
Dans ce cas, la forme ternaire, associee h. I'^quation a? +qx + r, perdra sa forme de conique et deviendra un systeme de deux lignes droites qui se croisent ou de deux lignes coincidentes. Dans la premiere supposition, il y aura le cas ou les deux droites toutes les deux coupent et les cas ou I'une ou toutes les deux touchent la conique fixe ; il y aura aussi les cas ou la conique fixe passe par le point d'intersection des deux droites en les coupant
» toutes les deux ou en touchant une. Dans la seconde supposition, il y aura les deux cas oil les droites coltncidentes coupent ou touchent la conique fixe. Ainsi done il nous parait qu'on pent aflSrmer avec pleine confiance que, dans I'equation de Hamilton*, il y a exactement douze cas, ou au moins douze cas principaux, k consid^rerf. Nous devons cette m^thode si simple
* Qaant k I'equation plus gin^rale pa;^ + gx + r = 0, dans le cas oil le discriminant ou le tensenr de p devient z^ro et que, par consequent, la forme ne rentre pas dans celle de Hamilton (puisqu'on ne pent plus diviser I'equation par p), il pent se presenter encore un grand nombre de cas singuliers que nous n'avons pas encore studies a fond.
t Cela donne lieu k une reflexion curiense. Si Ton consid^re tous les genres de rapports qui peavent avoir lieu entre une vraie conique et une conique variable et capable de d^gen^rer en n'excluant pas les deux cas oti la conique variable coincide avec I'autre ou s'^vanouit tout H fait, le nombre de ces genres sera 14, qui est le nombre de doubles decompositiona du nombre 4, Bavoir:
4: 3,1: 2,2: 2,1,1: 1,1,1,1: 3:1 2,1:1 1,1,1:1 2:2 1,1:2 1,1:1,1 2:1:1 1, 1:1:1 1:1:1:1.
De m^rae on trouvera faeilement que, pour le cas de formes binaires, le nombre de genres semblables sera 6, car, ayant sur une ligne droite deux points fixes et deux points variables, ces demiers peuvent itre distincts entre eux-mSmes en coiucidant avec un ou tous les deux ou avec ni I'un ni I'autre des deux premiers, on bien ils peuvent 6tre r^unis dans un seul point qui peut coincider ou ne pas co'incider aveo un des points fixes, et finalement ils peuvent disparaitre ; or le nombre de decompositions doubles du nombre 3, c'est-i-dire
3: 2,1: 1,1,1: 2:1 1,1:1 1:1:1, est aussi 6.
Mais nous avons demontri autrefois, dana le Philotophical Magazine, que pour le cas de deux formes qnadratiqnes de n variables dont chacane restc g^nerale, c'est-i-dire n'a pas le dis- criminant zdro, le nombre des genres de rapport est exactement le nombre de doubles decom- positions du nombre n. C'est une question qui merite d'etre examinee, si cette identite entre le Dombre de genres pour n variables dans le second cas avec celni pour le nombre n-1 dans le premier, reste vraie pour toute valeur de n. Une consideration qui s'y oppose, c'est que, dans le premier cas, qnand (n - 1 = 1) le nombre de genres, au lieu d'etre 3 (le nombre de decompositions doubles de 2), n'est que 2, mais il peut arriver que pour ce cas (le cas d'une seule variable), la forme generale etant la mSme que la forme de coincidence parfaite, ce genre doit compter pour deux, et ainsi la loi se maintiendra.
13—2
|96 Solution de Vequation [28
de d^Dombrement k la connaissance que nous avons acquiae du M^moire ci- dessus cit^ de M. Darboux*.
Mais ce qui plus est, on peut beaucoup simplifier, corame on va voir, la solution de I'^quation quadratique fx^pa^ + qx + r = 0.
En regardant pour le moment x comme une quantity ordinaire, soient Fx le determinant de la matrice a?p + ceq+r ot <^ un quelconque des six facteurs quadratiques de Fx ; alors ^ = 0 sera I'equation identique d'une des racines de fx — O, et ces deux Equations, en ^liminant ar", donneront la valeur precise de cette racine^f". De meme nous ferons voir qu'en g^n^ral, quel que soit le degr^ (n) de fx (fonction rationnelle entiere et unilat6rale de x), lequel, comme aussi chaque coefficient, est une matrice d'un ordre donnd (o)) quelconque, en prenant le determinant Fx defx (od pour le moment on regarde x comme une quantity ordinaire), chaque facteur du degr^ <o de Fx sera la fonction identiquement zero d'une des racines (prise negativement) de I'equation /a; = 0, et r^ciproquement.
Ce beau theoremej, pulcherrima regula, repose sur les considerations suivantes :
Soit (fix. le determinant de X. + «; alors on peut ddmontrer facilement que «f>x = 0 sera I'equation identique de x.
Or soit fx = 0, alors y(— \) =/(— \) — f{x) et consequemment contiendra le facteur x + \. Done le determinant de /(— \) contiendra le determinant de (\ + x), c'est-^-dire contiendra <f>X, oix <f)x = 0 est I'equation identique.
Ainsi <j)x (la fonction de x qui est identiquement zero) ne peut qu'etre un facteur du determinant de f(— x) pris comme si x etait une quantite ordinaire. De plus, puisqu'en general ce determinant sera une fonction ir- reductible de x, de sorte qu'on ne peut plus distinguer une racine d'avec une autre, tout facteur qu'il contient dont le degre est egal a I'ordre de x sera la fonction identiquement nulle d'une des racines de I'equation /a; = 0.
* On doit remarqner que le discriminant de I'eqoation en \ ou m ou «' est le m£me que celui de la biquadratique associ^e a I'equation donn^e ; en e£fet, I'equation en /t a pour racines
* — ^-!^ — ', ^ '-—- , 5 !^ — 11, ou o, /3, 7, S sent les racines de cette biquadratique ;
ainsi on peut dire que les six racines cherch^es sent associees respectivement anx six cot^s du quadrangle complet forme par les quatre points d'intereection de la conique appartenant aux co- ef&cients de r^quatiou donnee avec la conique absolue y^ - ixz.
On comprend que la forme appartenant ^ p, q, r vent dire le determinant de la matrice xp + yq + zr qui est une courbe dont I'ordre sera toujours celui des matrices p, q, r.
t Ainsi on poss^de une m^tbode immediate, et qui s'applique a tous les cas qui peuvent se presenter pour r^soudre I'equation de Hamilton. L'aualyse pr^cedente suflBt pour en donner une demonstration qui a ete passee dans le texte.
X On peut donner 4 cet enonce une autre forme, a savoir : Toute racine latente de cluique racine de fx (fonction rationnelle entiere et uuilaterale par rapport a x) est une racine (prise negativement) du diterminant de fx (oil x est traite comme une quantite ordinaire) el reciproque- ment chaque racine aitisi prise de ce diterminant est une racine latente d^une des racines de fx.
28] quadratique de Hamilton 197
II parait done (s'il n'y a aucune erreur dans ce dernier raisonnement) que le nombre des racines de fx sera le nombre exact de combinaisons de n<o choses prises co a, co ensemble, oil n est le degr6 de fx en a; et w I'ordre des matrices qui paraissent la-dedans; cons^querament le nombre des racines sera
irnm ^ _
IT {n — 1) a . irco ' ainsi, par exemple, le nombre des racines dans le cas d'une Equation du degr6 n en quaternions sera 2n^ — nf.
Pour trouver ces racines, on n'a qu'a combiner les deux Equations yic = 0 qui ne change pas, avec ^x = 0, qui varie avec chaque combinaison des racines de Fx [c'est-a-dire le determinant de /(— x)], et, en 61iminant les puissances superieures de x, on trouvera une Equation lineaire qui sert k donner x sous la forme d'une fraction : par des proced^s qui ne presentent nulle difficult^, cette fraction pent ^tre ramen^e (au moins pour le cas des matrices binaires) k la forme d'une autre fraction dont le d^nominateur sera une fonction ex- clusivement des coefficients de lei forme associde a I'ensemble des coefficients de I'equation donnee dont nous nous proposons d'essayer de trouver la valeur g^n^rale. Ce d^nominateur sera toujours (comme dans le cas que nous avons traite en detail dans ce qui prdcfede) le criterium de la rSgularit^ de I'equation donnee. Quand ce criterium s'dvanouit (et pas autrement), quelques-unes des racines vont k I'infini, c'est-a-dire cessent d'etre actuelles et deviennent pure- ment conceptuelles.
En general, pour r^soudre I'equation unilat^rale du degre n et I'ordre (o, on n'aura besoin que de resoudre une equation ordinaire du degr^ n<o. Si une racine de I'equation donnee est connue, on n'aura qu'i resoudre deux equations ordinaires des degres la et {n—l)<o respectivement. Dans le cas d'une equation quadratique, quand une racine est donnee, on pent trouver immediatement I'equation identique d'une seule autre qui y est associee, et consequemment en determiner la valeur sans resoudre une equation d'un degre superieur au premier. Quand deux racines de I'equation resolvante
(celle du degre na>) sent egales, on a — -. — — — ^r- i- -^ paires de
TT (&) — 1) . TT L(n — 1) a) — IJ
racines Egales dans I'equation du degre n qui est k resoudre.
* Dan8 le cas le plas g^n^ral d'ane Equation en x da degr^ n et de I'ordre u par rapport auz matrices, on pent sapposer nn nombre ind^fini de termes dans I'equation. Chacun de ces termes sera compost d'un nombre pas plus grand que n dea x dont chacun sera saivi et pr^c^d^ par une matrice multiplicatrice. En appliquant la m^thode algebrique directe pour resoudre cette Equation, on sera amen^ k un syst^me de to' Equations du degr^ n chacune. Ainsi le nombre des racines sera en general n .
+ Cela dimontre que le nombre 21 que nous avions trouv4 pour le cas de n = 3 dans le Philosophical Magazine, (mai 1884) [p. 229 below] et la formula g^nerale que nous avons bas^e 14-de68us sont erron^s ; la raison en est ^videmment que Tordre apparent da syst^me d'iquations qui nous a fourni ce r^sultat surpasse I'ordre actuel de 6 unites.
Nous n'avions pas discut^ en detail ces Equations, et ainsi cet abaissement da degr^ nous a ichapp^. C'est un point curieuz qui reste & discuter.
198 Solution de T Equation quadratique de Hamilton [28
Prenons comme exemple de I'application de la m^thode I'^quation en quaternions
?««* + gaa^ + qiic + ^0 = 0.
La fonction r^solvante sera
(3 . 3)a;« + (3 . 2K + (3 . 1+ 2 . 2) a;* (3 . 0 + 2 . l)a;»
+ (2.0 + l.l)ar'(1.0)a; + (0.0) = 0,
oh en g^n^ral i . i et i .j signifient
Tq^, HSq,qj-S{Vq,Vqj)] respectivement.
Les quinze facteurs quadratiques de cette fonction 6gales k z6to don- neront chacun une Equation quadratique a laquelle doit satisfaire une des quinze racines de I'^quation donn^e, et, en combinant s^par^ment chacune de ces Equations avec la cubique donnee, on pent 61iminer a;* et a;" et obtenir ainsi quinze equations lin^aires pour determiner les quinze racines voulues.
4
29.
SUR LA RESOLUTION G^N^RALE DE L'^QUATION LIN^AIRE EN MATRICES D'UN ORDRE QUELCONQUE.
[Comptes Bendus, xcix. (1884), pp. 409—412, 432—436.]
Ce qui interesse le plus dans les r^sultats nouvellement acquis que j'ai I'honneur de presenter a I'Acad^mie, c'est I'union ou bien I'anastomose dont lis oflFrent un exemple frappant et tout a fait inattendu entre les deux grandes theories de YAlgebre moderne et de VAlgebre nouvelle, dont I'une s'occupe des transformations lin^aires, et I'autre de la quantite g^neralisde, de sorte qu'au mSme titre que Newton d^finit I'Algebre ordinaire comrae dtant I'Arithm^tique universelle, on pourrait trfes bien caract^riser cette Algfebre-ci comme dtant I'Algebre universelle, ou au moins une de ses branches les plus importantes.
En general, un invariant de deux formes signifie une fonction de deux systemes de coefficients qui reste invariable, k un facteur pres, quand les deux systfemes des variables sont ou identiques ou assujettis k des substitu- tions semblables; mais rien n'empeche qu'on n'applique ce meme mot au cas ou les substitutions sont reciproques : ainsi, sans parler du cas de deux formes mixtes, on aura des invariants de deux formes donnees k mouvement semblable et des invariants k mouvement contraire; on peut tr^s bien noramer ces demiers (comme titre distinctif) contrariants. C'est a une classe sp^ciale de contrariants que nous aurons affaire dans la solution de I'^quation generale lin^aire en matrices d'un ordre quelconque.
En supposant que chaque p et p' soit une matrice de I'ordre cd, I'operateur qui contient i couples
Pii )p'i + pA )p'2+--+Pi( )p'i peut fitre noram^ provisoirement un nivellateur de I'ordre a> et de I'dtendue i, et on peut le caract^ri.ser par le symbole fl„j. Servons-nous toujours du symbole 0 pour signifier une matrice dont tons les 616ments sont des z^ros, et d^signons par 1 (ou bien par v indiff^remment) une matrice dont tons les
I.
200 Sur la resolution g6n4rale de I'dquation linSaire [29
dl^ments sont z^ro, k I'exception des ^l^ments de la diagonale qui seront des unites: ce sont les matrices nommdes matrice nulle et matrice unitaire respectivement.
J'ai d^j^ expliqu^ comment un nivellateur g^n^ral, de I'ordre w, donne naissance k one matrice de I'ordre cu': je nomme le determinant de cette matrice le determinant du nivellateur*. Ces determinants poss^dent des propri^tds tout k fait analogues k celles des determinants des matrices simples; ainsi, par exemple, je demontre la propriety dont je me suis servi avee grand avantage dans les recherches actuelles, que le determinant du produit de deux nivellateurs est egal au produit de leurs determinants separes, et que le determinant d'une fonction rationnelle d'un nivellateur, disons FH, est egal au resultant (par rapport k fl regarde comme une quantite ordinaire) de F£l et IH, oh. lil = 0 represente I'equation identique du degre a>^ k laquelle fl est assujetti.
En general, a un systerae ou corps de matrices pi, p.^, ...,pi de I'ordre to correspond un quantic de I'ordre m, c'est-a-dire le determinant de
<«iP\ + iHiPi + ... + a;ipi. Je nomme les coefficients de ce quantic les parametres du corps. Ces paramfetres doivent etre regardes comme des quantites connues. Ainsi, par exemple, si au corps p, q (deux matrices binaires) on adjoint la matrice unitaire v, et qu'on forme le determinant de la matrice x + yp + zq, on obtiendra un quantic
a^ + Bxy + Cxz + Dy'^ + Eyz + Fz'^,
ovi, si Ton regarde p, q comme des quaternions, on aura, dans le langage du grand Hamilton,
5 = Sp, G = Sq, D=T'p, F=T% E^SiVp.Vq).
II resulte de cette definition qu'k chaque nivellateur ri„,,- appartiennent deux quantics de I'ordre w et avec i variables, dont I'un appartient au corps Pi, Pa, •••,Pi et I'autre au corps p\, p\ p'i.
Si Ton connait I'equation identique /Q = 0 a laquelle le nivellateur fl obeit, on peut immediatement, comme je I'ai dejk montre, resoudre I'equation nx = T.
Mais il est tres facile de voir que lil n'est autre chose que le determinant du nivellateur il — \v{ )v, quand dans ce resultat on substitue XI k X. Done la question de la solution lineaire la plus generale est raraenee k ce seul probleme:
Exprimer le determinant d'un nivellateur en termes de quantites connues.
Or la premiere conclusion et la plus difficile k etablir dans cette recherche, mais que j'ai enfin reussi a demontrer, c'est que ce determinant est toujours * Quelqnefois ce determinant sera nomm^ un nivelUint.
I
29] en matrices d^un ordre quelconque 201
une fonction entiere, mais pas ndcessairement rationnelle, des coefficients des deux qtiantics qui sont associ^s au nivellateur.
Cela ^tant convenu, on demontre avec une extreme facility que ce determinant est un contrariant du degr^ <» dans chaque syst^me de coeffi- cients des deux quantics associ6s.
Cela ne suffit pas ou pent ue pas suffire en soi-meme k d^finir com- pletement le contrariant cherche; nommons, en general, ce contrariant le nivellant des deux quantics.
Supposons que Nx, y, ... z, t soit le nivellant ponr deux quantics d'un ordre donne «d, et representons par N^, y, ... z, o ce que ce nivellant devient quand on r^duit a ze'ro tons les coefficients qui appartiennent aux termes dans les deux quantics qui contiennent t; alors il est facile de voir que
■"X, y, ..., z, 0 '^ ■" X, y, ... z-
Cette propriete seule est suffisante (avec I'aide d'un quelconque des op^- rateurs differentiels qui servent pour annuler un contrariant) pour preciser le contrariant (nivellant) dans le cas de deux quantics du second ordre, et c'est ainsi que j'ai obtenu la solution de I'^quation lindaire pour le cas des matrices binaires donn^ dans la Note prec^dente. Or il est bieu con- cevable que cette loi ne pent pas suffire ^ determiner les parametres arbitraires qui entreut dans le contrariant d'ordre (w, ai) appartenant k deux quantics de I'ordre &>.
Mais il y a encore une autre loi (constituant par elle-ineme un trfes beau th^oreme) qui doit suffire surabondamment a cette fin.
Cest une loi qui ^tablit une liaison entre les nivellants de deux systfemes de quantics contenant chacun le meme nombre de variables, mais dont I'un est d'un ordre plus grand par unite que I'ordre de I'autre.
Supposons que N soit le nivellant de deux quantics de I'ordre w,
F{x,y 2) et 0{x,y,...,z);
soit N' ce que devient N quand
F{x,y,...,z) = {lx + my+...->rnz)Fi{x,y,...,z)
et Q(,x,y z) = {\x + fiy+ ...+vz)Qiix, y,...,z);
alors je dis que, quand
l\ + ni/i, + ... +nv = 0,
le nivellant de (Fi, 6i) sera contenu comme tacteur dans le nivellant modifie N'.
A I'aide de ces principes, je me propose de calculer les nivellants pour les degi-es sup^rieurs au second. On voit par ce qui precede que la solution de I'equation lineaire 'S.pxp' = T sera alors connue en termes des p, des p', de T et des paramfetres des deux corps Pi, pa, .... Pi, p'l, p'n, ..., p'i,&\igment6s I'un et I'autre d'une matrice unitaire.
202 Sur la resolution gSnSrale de V^quation linSaire [29
C'est dans les Lectures, publi^es en 1844, que pour la premiere fois a paru la belle conception de I'equation identique appliqu^e aux matrices du troisifeme ordre, envelopp^e dans un langage propre k Hamilton, apres lui mise k nu par M. Cayley dans un tres important M^moire sur les matrices dans les Philosophical Transactions pour 1857 ou 1858, et ^tendue par lui aux matrices d'un ordre quelconque, mais sans demonstration; cette demonstra- tion a 6t^ donn6e plus tard per feu M. Clifford {voir ses ceuvres posthumes), par M. Buchheim dans le Mathematical Messenger (marchant, comme il I'avoue, sur les traces de M. Tait, d'Edimbourg), par M. Ed. Weyr, par nous- meme, et probablement par d'autres; mais les quatre m^thodes cities plus haut paraissent Stre tout k fait distinctes I'une de I'autre.
Par le moyen d'une chaine de matrices coupleos (disons N), operant non pas sur une matrice gen^rale, mais sur une niatrice x (disons du degre a>) d'une forme sp^ciale suivie par un autre op^rateur V qui aura I'effet de reduire la matrice du degr^ to de Nx (dont les elements sont des fonctions lindaires des ^l^ments de x) a une forme identique a celle de x, il est facile de voir qu'k I'op^rateur compose VN on peut faire correspondre une matrice d'un ordre quelconque non sup^rieur k to", et c'est ainsi virtuellement que Hamilton, a cause d'une transformation qu'il effectue sur I'equation lin^aire g^nerdle, est tombe dans ses Lectures sur la matrice du troisieme ordre, et ce n'est que dans les Elements publics en 1866 (apres sa mort) qu'on trouve quelque allusion k I'equation identique pour les matrices du quatrieme ordre.
On pourrait nommer I'operateur composd VN, pour lequel I'equation identique est d'un degr^ moindre que ay', nivellateur qualifie, mais il est essentiel de remarquer que ces operateurs ne poss^deront pas les propriet^s analogues k celles des matrices que possfedent ces nivellateurs purs dont il est question dans ma m^thode. Comme exemple d'un nivellateur qualifid, on pourrait admettre que le x (matrice du deuxifeme ordre), sur lequel opfere le N, aura son quatrieme element zdro, et que I'effet du V sera d'abolir le quatrieme Element dans Nx, ou. Ton peut supposer (et cette supposition est, dans son essence, a peu pres identique a la mdthode des vecteurs de Hamilton) que le premier et le quatrieme Element de x sont ^gaux, mais de signes contraires, et que I'effet de V est de substituer dans la matrice du second ordre N (x) la moitie de la difference entre le premier et le quatrieme element au lieu du premier et, au lieu du quatrifeme, cette meme quantity avec le signe algebrique contraire.
Evidemment un tel operateur donnera naissance a une matrice et sera assiijetti k une equation identique du troisieme ordre. Avant de conclure, pour convaincre de la justesse de la formule importante
i[{Pyp'-i{P'.py]-^^(i.ir,
* Pour rendre intelligible cette formule, il est nicessaire de dire que I'expression
29] en matrices d^un otdre quekonque 203
applicable au cas d'un nivellateur du second ordre a quatre couples de matrices, il sera bon d'en donner une demonstration parfaite a posteriori, ce qu'une transformation legitime rend trfes facile a faire. Remarquons que le
determinant du nivellateur du second ordre 2 , ( ) \> est le determinant
c d^ ^7 0
de la matrice suivante:
|
2aa |
2ca |
2a/3 |
2c/3 |
|
26a |
%dcL |
26/8 |
2d/8 |
|
2a7 |
Icy |
2a8 |
2c8 |
|
267 |
2^7 |
26S |
Xdh |
laquelle contiendra dans le cas suppose 144 termes, puisque chaque 2 com- prend 4 produits : mais, sans perdre en gdneralite, on peut prendre une forme de nivellateur dont le determinant ne comprendra pas plus de 24 termes ; car il est facile de demontrer que, si aux 4 matrices de gauche on substitue 4 fonctions lin^aires quelconques, pourvu que sur les 4 de droite on opere une substitution contragrediente k la substitution pr^ce- dente, la valeur du determinant ne subira nul changement. On peut done supposer que les 4 matrices de gauche sent
10 01 00 00
00 00 10 01
respectivement, et, si la formula est verifiee dans cette supposition (vu que les contravariants des deux quantics associes ne sent pas afifectes par les sub- stitutions contragredientes operees sur les deux s^-stemes de matrices), elle
donn^ dans la Note da 21 joiilet [pp. 181, 184 above], a besoin d'nne correction (dont je pensaia avoir fait mention dans le texte) : il fait lui ajoater la racine carrfe d'un contrariant eonnue da qnatridme degr£ (appartenant aax denx Jormea aifocUea), laqaelle sera ane fonction rationnelle des ^l^ments des matrices da nivellatear. Poor le cas d'nn nivellateur k quatre couples de matrices, c'est la racine carr^e du prodoit de / et /', les discriminants des denx formes a88oci^es prises s^par^ment ; en nommant les quatre matrices a gauche a b a' b' c d c' d' la racine carr^ de / sera igale an determinant
a b
a' V
a" b"
a'" b"'
qn'on peut nommer le developpant de ces quatre matrices ; de mgme la racine carrce de I' sera igale au developpant des quatre matrices correspondantes i, droite, de sorte que le terme irration- nel dans la formale poor le nivellant k quatre couples de matrices est ^gal an produit de ces deux d^veloppants ; dans le cas g^n^ral, la partie relativement irrationnelle de la formule pour an nivellant sera ^gale a la somme de tons les produits de d^veloppants accoupl^s qu'on peut former en combinant qaatre k quatre, ensemble, les couples de matrices qui en dependent. Dans le cas o£i le nivellateur contient moins de qnatre couples, la racine carrce disparalt enti^re- ment de la formale pour le nivellant. Je nommerai P . P et (P')'P', 3-, et ^j respectivement.
|
a" |
b" |
a'" |
b'" |
|
c" |
d" |
c'" |
d!" |
|
c |
d |
||
|
e |
d' |
||
|
c" |
d" |
• |
|
|
e'" |
d'" |
204 Sur la rholution ginfyrale de V^uation linfyiire [29
sera noD pas seulement verifide, mais absolument dimontrie pour les valeurs parfaitement g^ndrales des deux syst^mes.
Avec ces valeurs des matrices gauches, la matrice ^crite plus haut, en prenant
a /8 a' /S' 78 7' S' pour les matrices k droite, devieut
a «! a' a 7 7i
7' 7 dont je nommerai le determinant Q.
De plus, le quantic k gauche deviendra xt — yz, et le quantic k droite (aS - ^7) ^ + (5 S - yS 7) «' + (a'8' - ySV) f + («i8> - A7.) ^'
+ (1 . 2)xy + (3 . 4)z« + (1 . 3)a;2 + (2 . 4) y< + (1 . 4) ««+ (2 . 3) yz, oh (1.2) = oS'+Sa'-^7'-/3'7, (3 . 4) = a,S + ^.S - /9,7 - 7,^8,
|
a. A |
ayS |
|
7i Si |
7S |
|
iS A |
|
|
/3' y8 |
|
|
S 8. |
|
|
s' a |
Done
a. = («S + fiS - /37 - /87) - (o'S, + a,S' - ;8'7i - ^W). \% = (aS + aS - ^7 _ ;S7)^+ (a'8, + a,8' - /3'7i - /8.7')'
+ 2 (aS - ^7) {al - 0y) + 2(a'B' - /3 '7') (a:8, - ^i7:)
- (aS' + Sa' - ,87' - /3'7) (a J + g^a - ^7 - /871)
- (aS, + Sa. - y37i -_A7) (a'S + B'a - ^'7 - ^y')
- (aS + fiS - /37 - /87) (a'S. + a,S' - /3'7i - ^i7').
et V(-^ • ^') (pris avec le signe convenable) sera le determinant de la matrice
a ^ y B
a /3' y B'
tti ;8, 7, Si
a y9 7 S. En faisant les multiplications necessaires, on trouvera que
i^,-V-V(/-/') = 2Q.
ce qui d^montre I'exactitude de la formule donn^e pour un nivellateur du deuxieme ordre k quatre couples de matrices.
D'ici a pea de temps, j'espere avoir I'honneur de soumettre k I'Aca- d^mie la valeur du determinant du nivellateur du troisifeme ordre a trois couples de matrices. Pour presenter I'expression g^n^rale de ce determi- nant pour une matrice d'un ordre et d'une etendue quelconques*, il faudrait avoir une connaissance des proprietes des formes qui va beaucoup au deli • C'est-i-dire pour r^soudre I'^quation lin^aire en matrices dans toute sa g6n^ralit6.
29] en matrices (Tun ordre quelconque 205
des limites des facultes humaines, telles qu'elles ne sont manifestoes jusqu'au temps actuel et qui, dans mon jugement, ne pent appartenir qu'a I'intelli- gence supreme.
Post-scriptum. — Qu'on me peimette d'ajouter une petite observation qui fournit, il me semble, une raison suffisante a priori pour le signe ambigu du terme v'(-^ ■ ^ ) qui entre dans la formule donnee pour un nivel- lant (c'est-a-dire determinant d'un nivellateur) du deuxieme ordre.
Les determinants d'un nivellateur et de son conjugue ^tant identiques en signe alg^brique tout autant qu'en grandeur, ce n'est pas dans cette direction qu'on peut chercher I'origine de I'ambiguitd
Mais, si, en se bornant aux matrices correspondantes d'un nivellateur de la menie esphce, c'est-a-dire a main droite ou a main gauche du symbole ( ), on ^change entre eux, dans chacune de ces matrices, le premier terme avec le quatrieme et le deuxieme avec le troisieme, on verra f'acilement que le nivellant et en meme temps les deux qualities associes restent absolument sans alteration; mais, si Ton execute I'une ou I'autre de ces substitutions s^par^ment, alors, tandis que les deux quantics associes restent constants, le nivellant (quand son nivellateur possede plus de trois couples) subira un changement de valeur (et, pour I'une et I'autre substitution, le vieme change- ment), de sorte que pour les qnatre positions qu'on peut assignor simul- tan^ment aux elements des matrices de la meme espece sans changer en rien les quantics associes, le nivellant aura deux valeurs distinctes. Voila, il me semble, I'explication suffisante et la veritable origine de I'ambiguitO dont il est question.
A peine est-il n^cessaire de remarquer qu'on peut faire 4 autres dis- positions semblables et simultan^es des matrices a I'un ou I'autre c6t6 du symbole ( ), dispositions qui donneront naissance k des nivellants identi- ques en valeur avec les deux dont j'ai parlO (c'est-k-dire deux k une valeur et deux a I'autre), et pour lesquelles les deux quantics associes seront sans autre changement que celui du signe alg^brique.
En combinant les 24 dispositions semblables des matrices d'un c6t6 d'un nivellateur donne avec les 24 de I'autrd c6te, on obtiendra un systfeme de 576 nivellateurs correlatifs dont les determinants ne prendront que 3 paires le valeurs; de plus, les deux valeurs d'une quelconque de ces paires seront es racines d'une Equation quadratique dont les coefficients seront des con- trariants rationnels et entiers d'une des trois paires de formes quadratiques; mais le discriminant de ces trois Equations sera le meme certainement quand les nivellateurs du systeme seront formes avec quatre couples de matrices et probablement quel que soit le nombre de ces couples. Quand ce nombre est moindre que 4, le discriminant de ces trois quadratiques devient nul pour toutes les trois.
le
30.
SUE L'jfeQUATION LIN^AIRE TRINOME EN MATRICES D'UN ORDRE QUELCONQUE.
[Coniptes Rendus, xcix. (1884), pp. 527 — 529.]
Pour r^soudre I'^quation trindme pa^' + qxq' + r = 0 (oil toutes les lettres li^signent des matrices du mStne ordre to) sous sa forme sym^trique, on a besoin de coimaitre I'^quation identique k un nivellateur de cet ordre k deux couples de matrices, ce qui equivaut virtuellement k connaitre le determinant d'un nivellateur k trois de ces couples. Mais, sans avoir recours k cette mdthode g^n^rale, il existe, comme on va le voir, un raoyen plus court et plus direct pour r^soudre I'equation et exprimer x sous la forme essentiellement bonne d'une fraction reduite, si Ton est d'accord k se dispenser de la condition que le numdrateur soit sym^trique.
A cet effet, on pent multiplier I'equation, a volont^, ou par q~^ (, )p'~^ ou par j3~'( )q'~^. Choisissons le premier de ces deux multiplicateurs et ^crivons q~^p=<f>, q'p'~^ = — yfr, — </"' rp'~' = /li ; alors on obtient I'equation <f>a; — ooyjr = /ji (mais d^ja avec une breclie de sym^trie, par la raison du choix d'une entre deux choses pareilles). En multipliant cette Equation par le nivellateur 0'( ) + (^'-'( )-«/r + ^'-2( )^2 + ...+( )-^< (disons Ui) et en dcrivant Uifi = /u.i+i, on obtient la suite d'^quations
<f>X — xyfr = fi, <j>*x — x-\Jr^ = fj,3, (fy^x — X'y^ = fji^, . . . , <f>'-'x— xyfr'' = /i„.
Soient B^, £,,..., 5„ et Go, Cj, ...,(7„ les coefficients des deux formes associ^es aux deux systemes p, q et p', q' respectivement; alors, en vertu d'un theor^me general en matrices*, on aura
C'^t" + C„-it""' + . . . + Co = 0, 5„ - A0 + . . . + (-yB^cfy = 0. Avec I'aide de ces deux Equations et de la suite prec^dente, on pent deduire une equation de I'une ou de I'autre des deux formes Mx = N ou xM=N. Faisons le choix (qui amfene encore une fois une brfeche de sym^trie) de la premifere.
On aura (C„^" + C„_i^"-i + . . . + C,^ + C„) a; = C„/4„ + C„_,/i„_i + . . . + C,yn.
Or, selon la th^orie ordinaire d'elimination, on pent determiner 'is et H deux fonctions chacune du degr^ (w — 1) en <^ (traitd comme une quantity ordi- naire), telles que
^ [£o - ^a</> + • • • + {-YB^r] + SiC.<f>" + C„-i <^"-' + . . . + Co)
* Ainsi, par exemple, si p, q sont des qaaternious, on a
2>2 (p-i q)^-2S{rpVq){p-^q) + Tq^ = 0.
I
30] Sur Tequation lineaire trindme en matrices 207
sera egal k R, le contre-rdsultant des deux formes associees a (p, q)et (p', q')* respectivement, et Ton aura
_CiHiJ. + C^Hfi^ + ... + G„Hfi„ '^~ R '
et ainsi x sera determine.
Si ft est zero, alors, afin que x ne soit pas zero, le R doit devenir zdro, comme nous avons deja trouvd dans une Note pr^c^dente. En g^n^ral, si R (le contre-r^sultant des deux formes adjointes a |), 5' et p', 5' dans Tequation pxp' + qx(i + r = 0) s'^vanouit, Tequation ne peut pas admettre une solution en ineme temps actuelle et ddterminee; sans autres conditions, la solution deviendra ideale; avec conditions convenables, elle peut redevenir actuelle, mais contiendra (selon les circonstances) une ou plusieurs constantes arbitraires.
Hamilton, dans ses Lectures, a considere I'equation trindme pour les quaternions, mais il n'en a pas pousse la solution, e'est-^-dire la valeur de I'inconnue, a sa forme finale dans laquelle le d^nominateur doit 6tre un scalar (je dis doit etre), parce que, ici comme dans toutes les Equations en matrices, c'est le denominateur de I'iuconnue convenablement exprim6 dont I'evanouissement est le cntinuvi pour distinguer le cas 011 la solution est actuelle et ddterminee d'avec les cas ou elle devient ou ideale ou inde- terminde.
En combinant le rdsultat ici obtenu avec celui de notre Note prec^dente, on voit qu'on est entre en pleine possession de la solution de I'equation Nx = r dans les deux cas ovi le nivellateur N est de I'ordre 2 et d'une ^tendue quelconque ou bien de I'^tendue 2 et d'un ordre quelconque.
Remarque. — On peut objecter que le num«5rateur de I'expression trouv6e pour X dans I'equation trin6me contient des combinaisons de q'^-p, q'p'~^, q~^rp'~^ et que, cons^quemment, x pourrait devenir id^al k. cause de I'eva- nouissement du determinant de p' ou de q sans que le contre-resultant R s'evanouisse. Pour r^pondre k cette objection, soient D', A les determinants de p' et de g~'; alors, en se servant des equations identiques k p' et k q, on peut substituer pour leurs inverses des fonctions rationnelles de I'un et de Tautre divisees respectivement par Z)' et A, et alors le numerateur de x sera une quantite incapable de devenir iniinie, tandis que son denominateur sera R multiplie par des puissances de Z)' et de A ; mais, vu qu'on peut repre- senter x tout aussi bien par une autre fraction dont le numerateur sera aussi incapable de devenir infini et dont le denominateur sera R multiplie par des puissances de D' et de A (les determinants de p et de q'), il est evident que ces deux fractions doivent toutes les deux admettre d'etre simplifiees et que dans leurs formes reduites le denominateur sera tout simplement R et qu'ainsi ce contre-resultant est le seul criterium pour distinguer le cas de Tactuel et determine d'avec le cas de I'ideal ou indetermine.
* C'e8t-a-dire le r£>ultant dea fonctions multipli^es par ^ et if ci-dessus.
31.
LECTURES ON THE PRINCIPLES OF UNIVERSAL ALGEBRA.
[American Journal of Mathematics, vi. (1884), pp. 270 — 286.]
LECTURE I. Preliminary Conceptions and Definitions.
Apotheosis of Algebraical Quantity.
A MATRIX of a quadrate form historically takes its rise in the notion of a linear substitution performed upon a system of variables or carriers; regarded apart from the determinant which it may be and at one time was almost exclusively used to represent, it becomes an empty schema of operation, but in conformity with Hegel's principle that the Negative is the course through which thought arrives at another and a fuller positive, only for a moment loses the attribute of quantity to emerge again as quantity, if it be allowed that that term is properly applied to whatever is the subject of functional operation, of a higher and unthought of kind, and so to say, in a glorified shape, — as an organism composed of discrete parts, but having an essential and undivisible unity as a whole of its own. Naturam, expellas furcd, tamen tisque recurret*. The conception of multiple quantity thus rises upon the field of vision.
At first undifferentiated from their content, matrices came to be regarded as susceptible of being multiplied together; the word multiplication, strictly applicable at that stage of evolution to the content alone, getting transferred by a fortunate confusion of language to the schema, and superseding, to some extent, the use of the more appropriate word composition applied to the reiteration of substitution in the Theory of Numbers. Thus there came into view a process of multiplication which the mind, almost at a glance, is able to recognize must be subject to the associative law of ordinary
* Choitez le naturel, il revient au galop, a familiar quotation which I thought was from Boilean, but my friend Prof. Babillon informs me is from a comedy of Destonches (bom in 1680, died 1754).
31] Lectures on the Principles of Universal Algebra 209
multiplication, although not so to the commutative law; but the full signi- ficance of this fact lay hidden until the subject-matter of such operations had dropped its provisional mantle, its aspect as a mere schema, and stood revealed as bona-fide multiple quantity subject to all the affections and lending itself to all the operations of ordinary numerical quantity. This revolution was effected by a forcible injection into the subject of the concept of addition, that is, by choosing to regard matrices as susceptible of being added to one another ; a notion, as it seems to me, quite foreign to the idea of substitution, the nidus in which that of multiple quantity was laid, hatched and reared. This step was, as far as I know, first made by Cayley in his Memoir on Matrices, in the Phil. Trans. 1858, wherein he may be said to have laid the foundation-stone of the science of multiple quantity. That memoir indeed (it seems to me) may with truth be affirmed to have ushered in the reign of Algebra the 2nd; just as Algebra the 1st, in its character, not as mere art or mystery, but as a science and philosophy, took its rise in Harriot's Artis Analyticae Praxis, published in 1631, ten years after his death, and exactly 250 years before I gave the first course of lectures ever delivered on Multinomial Quantity, in 1881, at the Johns Hopkin.s University. Much as I owe in the way of fruitful suggestion to Cayley's immortal memoir, the idea of subjecting matrices to the additive process and of their consequent amenability to the laws of functional operation was not taken from it, but occurred to me independently before I had seen the .memoir or was acquainted with its contents; and indeed forced itself upon my attention as a means of giving simplicity and gene- rality to my formula for the powers or roots of matrices, published in the Comptes Rendus of the Institute for 1882 (Vol. xciv. pp. 55, 396). My memoir on Tchebycheff's method concerning the totality of prime numbers within certain limits, was the indirect cause of turning my attention to the subject, as (through the systems of difference-equations therein employed to contract Tchebycheff's limits) I was led to the discovery of the properties of the latent roots of matrices, and had made considerable progress in developing the theory of matrices considered as quantities, when on writing to Prof. Cayley upon the subject he referred me to the memoir in question: all this only proves how far the discovery of the quantitative nature of matrices is removed from being artificial or factitious, but, on the contrary, was bound to be evolved, in the fulness of time, as a necessary sequel to previously acquired cognitions.
Already in Quaternions (which, as will presently be seen, are but the simplest order of matrices viewed under a particular aspect) the example had been given of Algebra released from the yoke of the commutative principle of multiplication — an emancipation somewhat akin to Lobat- chewsky's of Geometry from Euclid's noted empirical axiom; and later on, 8. IV. 14
210 Lectures on the Principles [31
the Peirces, father and son (but subsequently to 1858) had prefigured the universalization of Hamilton's theory, and had emitted an opinion to the effect that probably all systems of algebraical symbols subject to the associative law of multiplication would be eventually found to be identical with lineai' transformations of schemata susceptible of matricular representation.
That such must be the case it would be rash to assert; but it is very difficult to conceive how the contrary can be true, or where to seek, outside of the concept of substitution, for matter affording pabulum to the principle of free consociation of successive actions or operations.
Multiplication of Matrices.
A matrix written in the usual form may be regarded as made up of parallels of latitude and of longitude, so that to every term in one matrix corresponds a term of the same latitude and longitude in any other of the same order.
Every matrix possesses a principal axis, namely, the diagonal drawn from the intersection of the first two parallels to the intersection of the last two of latitude and longitude ; and bj' a symmetrical matrix is always to be understood one in which the principal diagonal is the axis of symmetry. If there were ever occasion to consider a symmetrical matrix in which this coincidence does not exist, it might be called improperly symmetrical. This designation might and probably ought to be extended to matrices symmetrical, not merely in regard to the second visible diagonal, but to all the (ft) — 1) rational diagonals of a matrix of the order &>, a rational diagonal being understood to mean any line straight or broken, drawn through <o elements, of which no two have the same latitude or longitude.
The composition of substitutions directly leads to the following rule for the multiplication of matrices. If m, n, be matrices corresponding to substitutions in which m is the antecedent or passive, and n the consequent or active, their product may be denoted by inn (that is, m multiplied by n), and then any term in the product of the two matrices will be equal to its parallel of latitude taken in the antecedent or passive and multiplied by its parallel of longitude taken in the consequent or active matrix. Cauchy has taught us what is to be understood by the product of one rectangular array or matrix by another of the same length and breadth, and we have only to consider the case of rectangles degenerating each to a single line and column respectively, to understand what is meant by the product of the multipli- cation of the two parallels spoken of above. It may, however, be sometimes convenient to speak of the disjunctive product of two sets of the same number of elements, meaning by this the sum of the products of each element in the
31] of Universal Algebra 211
one by the corresponding element in the other. Thus (XZ) mn denoting the term in mn of latitude \ and longitude I, we have the equation
(X/) mn = Xto X In,
where, of course, Xm means the Xth parallel of latitude, and In the fth parallel of longitude, in m and n respectively. This notation may be extended so as to express the value of any minor determinant of mn ; such minor may obviously be denoted by
Xjti, X,(2, . . ■ Xji,-,
Xa'i) Xgt.j, , , , Xgti,
Xi(,, Xifj, . . . X,i,-,
and its value will be the product of the two rectangles (in Cauchy's sense) formed respectively by the X,, Xj,...X^ parallels of latitude in m, and the li,l„...li parallels of longitude in n.
Any other definition of multiplication of matrices, such as the rule for multiplying lines by lines, or columns by columns, sins against good method, as being incompatible with the law of consociation, and ought to be in- exorably banished from the text-books of the future. It is almost unnecessary to add that by a ^th power of a matrix m is to be understood the result of multiplying/) m's together; and by the ^th root of m, a matrix which multi- plied by itself q times produces m : hence we can attach a clear idea to any positive integral or fractional power. The complete extension of the ordi- Dary theory of surds to multinomial quantity will appear a little further on. But it is well at this point to draw attention to the fact that at all events, if if, M' are positive integer powers of the same matrix m, the factors M, M' are convertible, that is, MM' = M'M, this commutative law being an imme- diate consequence (too obvious to insist upon) of the associative law of multiplication.
On Zero and Nullity.
The absolute zero for matrices of any order is the matrix all of whose elements are zero. It possesses so far as regards multiplication (and as will presently be evident as regards addition also) the distinguishing property of the ordinary zero, namely, that when entering into composition with any other matrix, either actively or passively, the product of such composition is itself over again; so that it may be said to absorb into itself any foreign matrix (of its own order) with which it is combined. This is the highest degree of nullity which any matrix can possess, and (regarded as an integer) ■will be called m, the order of the matrix. On the other hand, if the matrix has finite content, its nullity will be regarded as zero. Between these two
U— 2
212 Lectures on the Principles [31
limits the nullity may have any integer value ; thus, if its content, that is, its determinant, vanishes without any other special relation existing between its elements, the nullity will be called 1; if all the first minors vanish, 2; and, in general and more precisely, if all the minors of order w — t + l vanish, but the minors of order a—i do not all vanish, the nullity will be said to be t: as an example, if the elements are not all zero, but every minor of the second order vanishes, the nullity is eo — 1.
In general, a substitution impressed on a set of variables may be reversed, and the problem of reversal is perfectly determinate; but when the matrix — the schema of the substitution — is affected with any degree of nullity, such reversal becomes indeterminate. Hence the use of the word indeterminate employed by Cayley to characterize matrices affected with any degree of nullity, in which he has been followed by Clifford, who goes a step further in distinguishing the several degrees of indeterminateness from one another.
On Addition and Monomial Multiplication of Matrices.
The sum of two matrices of like order is the matrix of which each element is the sum of the elements of the same latitude and longitude as its own in the component matrices; thus, as stated by anticipation in what precedes, the addition of a zero matrix to any matrix of like order leaves the latter entirely unchanged.
Addition of matrices obviously will be subject to the same two associative and commutative laws as the addition of monomial quantities. This seems to me a sufficient ground for declining to accept associative as the dis- tinguishing name of the algebra of multinomial quantity; for the emphasis thereby laid on association would seem to imply the entire absence of the commutative principle from the theory, whereas, although not having a place in multinomial multiplication, it flourishes in full vigour in the not less important, and, so to say, collateral process of multinomial addition. If k is any positive integer, the addition of the same matrix taken k times obviously leads to a matrix of which each element is k times the corre- sponding element of the given one; and if p times one matrix is q times
another, the elements of the first are obviously - into the corresponding
ones of the other: hence, if k is any positive monomial quantity, k times a given matrix, by a legitimate use of language, should and will be taken to mean the matrix obtained by multiplying each element in the given one by k. And as the negative of a given matrix ought to mean the matrix which added to the given one should produce the zero-matrix previously defined, the meaning of multiplying a matrix by k may be extended, with the certainty of leading to no contradiction, to the case of any commen- surable value of k positive or negative, and consequently, by the usual and
■ ^
tr
31] of Universal Algebra 213
valid course of inference, to the case of k being any monomial symbol what- ever, whether possessing arithmetical content or not.
On the Multinomial Unit and Scalar Matrix.
On subjecting a matrix of any order to to a resolution similar to that by which one of the second order may be resolved into a scalar and a vector, it will be shown hereafter that the a>^ components separate into a group of ft)' — 1 terms analogous to the vector and to a single term analogous to the scalar of a quaternion. This outstanding single term is of an invariable form, namely, its principal diagonal consists of elements having the same value, which may be called its parameter, and all the other elements are zeros.
A matrix of such form I shall call a scalar. When the parameter is unity
it may be termed a multinomial unity and denoted by T *, or in place of m we may write to dots over T, or for greater simplicity when desirable write simply T. Any scalar, by virtue of what precedes, is a mere monomial multiplier of some such T.
Let kT be any scalar of order m. It will readily be seen, by applying the laws of multiplication and addition previously laid down, that
<fi{kT) = <f>(k).T, and that kT.m==m .kT = km.
Thus a scalar possesses all the essential properties of a monomial quantity, and a multinomial unity of ordinary unity; in particular, the faculty of being absorbed in any other coordinate matrix with which it comes in contact. A scalar whose parameter vanishes of course becomes a zero-matrix.
The properties stated of a scalar ^T serve to show that in all operations into which it enters the T may be dropped, and supplied or understood to be supplied at the end of the operations when needed to give homogeneity to
expression. Thus, for example,
(m + AT) (wi + kT) = rn' + (h + k) Tm + hkV = m^+(h + k)m + hkT;
but this result may be obtained by the multiplication of (m+ h)(m-{-k), and supplying T (or imagining it to be supplied) to the final term in order to preserve the homogeneity of the form. In like manner, 0^ or 0 with « points over it may be used to denote the absolute zero of the order a ; but it will be more convenient to use the ordinary 0, having only recourse to the addi- tional notation when thought necessary or desirable in order to make obvious the homogeneity of the terms in any equation or expression. Thus, for example, such an expression as ni' + 2bm + d = 0, where to is a matrix, say of the 2nd
* Perhaps more advantageously by !„. I shall hold myself at liberty in what follows to use whichever of these two notations may appear most convenient in any case as it arises.
214 Lectures on the Principles [31
order, and h and d monomials, set out in full would read m^ + 26m + dT = 0, meaning m.m + 2 6?n+,= .
On the Inverse and Negative Powers of a Matrix.
The inverse of a matrix, denoted by m~S means the matrix which multi- plied by m on either side produces multinomial unity. It is a matter of demonstration that when a matrix is non-vacuous (that is, has a finite content or determinant appertaining to it), an inverse to it fulfilling this double condition can always be found, and that if the product of mn is unity, so also must be that of nm.
It is a well-known fact, proved in the ordinary theory of determinants, that if every element in the first of two matrices is the logarithmic dif- ferential derivative, in respect to its correspondent in the second, of the content of that second, so conversely, every element of the second is the logarithmic derivative, in respect to its correspondent in the first, of the content of the first.
But two such matrices multiplied together in either sense would not give for their product multinomial unity; to obtain this product either matrix must be multiplied indifferently into or by the transverse of the other (meaning by the transverse of a matrix, the new matrix obtained by rotating the original one through 180° about its principal diagonal). In other words, if m be a given matrix and n be obtained from it by substituting for each element the logarithmic derivatives of its content in respect to its opposite,
then mn = 'T and nin = T, where w means (as will always be the case throughout these lectures) the order of the matrices concerned. The n which satisfies these two equations (and it cannot satisfy the one without satisfying the other) will be called the inverse of m and be denoted by m~K
For brevity and suggestiveness it will be advantageous to write in
general 1 for T as we write 0 for 0„, so that mn = 1 will imply nm = 1 = mn
and n = m~^.
We may define in general (as in monomial algebra) m~* to mean the inverse of m*, that is, (m')~^ We shall then have (m~')' = m~*, for mn . mn = 1 implies m . mn . n — mn = 1 or mV = 1. Hence n'' = m~'\ that is, (»ft~')' = m~'. Also since mVi' = 1, m'n' = m,n = 1 or w' = m~', that is, (mr^f = m~', and so in general for all positive integer values of i, (m~')' = wi~*. And, as in monomial algebra, it may now be proved and taken as proved that, for all real values of i and j, whether positive or negative, m* . W = m'+-', and the same relation may be assumed to continue when i, j become general quantities. The elements in the inverse to any matrix m all involving the reciprocal of the
(
31] of Universal Algebra 215
determinant to m, if D be the content of m we may write mr^ = j^ fj,,
where ft, is a, matrix all of whose elements are always finite. Hence we come to the important conclusion that for vacuous matrices inverses only exist in idea and are incapable of being realized so as to have an actual existence. In the sequel it will be shown that the inverse is only a single instance of an infinite class of matrices which exist ideally as functions of actual matrices, but are incapable of realization.
Suppose now that M, N are any two matrices such that MN = 0 or that NM=0; multiplying each side of the equation by if-' if such expression ha-s an actual existence (that is, if M is non- vacuous), we obtain, from the known properties of zero, iV^ = 0, but if M is vacuous no such conclusion can be drawn. So further if »n' = 0 (i being any positive integer), it will be seen under the third law of motion that m is necessarily vacuous. Hence from this equation it cannot be inferred that any lower power than the ith of m is necessarily zero.
On the Latent Roots and Different Degrees of Vacuity of Matrices.
If m be any matrix, the augmented matrix m — \T or m — \ . 1„ or m — \ will be found simply by subtracting \ from each element in the principal diagonal of m. The content of this matrix or the same multiplied by — 1 or any other constant, I term the latent function to m, which will be an algebraical function of the degree &> in \ (which may be termed the latent variable or carrier); and the w roots of this function (that is, the to values of the carrier which annihilate the latent function) I call the latent roots of the unaugmented matrix m. It is obvious from this definition that if \i be any latent root of m, the content of m — \i will vanish, that is, m — \, will be vacuous, and conversely that if rre — X, is vacuous, \, must be one of the latent roots to m. Thu.s if m is vacuous, one of the latent roots must be zero ; if only one of them is zero I call m simply vacuous and say that its vacuity is 1 : thus zero vacuity and simple vacuity mean the same thing as zero nullity and simple nullity respectively. More generally if any number i, but not i+1, of the latent roots of m are all of them zero, m will be said to have the vacuity i.
By a principal minor determinant to any matrix I mean any minor deter- minant whose matrix is divided by the f)rincipal diagonal into two triangles. It will then easily be seen that if «< means in general the sum of the principal tth minors to nt, and «„ means the complete determinant, the assertion of m having the vacuity i is exactly coextensive with the a.ssertion that
s„ = 0, «! = 0, s, = 0, . . . s,_, = 0. If the nullity of m is t, every gth minor of m is zero when q < i. Hence the vacuity cannot fell short of the nullity, but the converse is not true.
216
Lectures on the Principles
[31
A matrix may not have any vacuity up to <o inclusive without the nullity being greater than 1. It will hereafter be shown, under the 2nd law of motion, that if X,, Xj, ... \„ are the to latent roots of m, then
(m — \,)(hi- X,)... (m — X„) = 0 or say M=0. But it will be interesting even at this early stage to show that a theorem closely approaching this may be deduced from the distinction drawn between vacuous and non-vacuous matrices as regards their possession of real inverses.
I propose to prove instantaneously by this means that at all events 3f"-i = 0. It is obvious from any single instance of multiplication that mn and 717)1 are not in general coincident. But if n could be expressed as a linear function of powers of m (including m' or 1„ among such powers), mn and nm must be coincident. If now we take the <a' matrices
1, VI, m', ... m"'~*, n at first blush one would say ought to be expressible as a linear function of these «' quantities determinable by means of the solution of m' linear equations, and can only escape being so expressible in consequence of the fact that these w' powers of vi are linearly related. Hence we must have an identical equation of the form
Am'^-^ + Bm'''-^ + Gm."'-^ ... + Om + H=0„ or say Fm = 0. If now Fm were supposed to contain any factor other than
m-X,, wi — X,, ... 7ft — X„,
such factors being non-vacuous may be expelled from Fm; consequently the equation in question must be of the form
(m - X,)"' (m - X,)*" {m - X„)«" = 0, and as the coefficients of the equation in m are necessarily rational we must have a, = ct, ou = a. Hence ma = Hi + Oj 4- ...«„< a)', and conse- quently a < o).
Hence, at all events (since M''~^~' = 0 on multiplication by ilf* gives Jlf'-i = 0),
{(m -X,)(m - Xj) ...(m-X^)}"-^ = M"-^ = 0. Q.KD.
LECTURE II.
On Reduction.
It follows from what has been already shown in Lecture I, when m is a matrix of the second order (a> — I being here unity) that (m — X,) (m — X,) = 0.
Understanding by m the matrix
«.,
ti — K,
the latent equation to w is
'J.
Tj— X
= 0,
31] of Universal Algebra 217
that is, >.'-(«, + T,) X + ((jTj - t,T,) = 0,
so that ni' - (<, + r.) m + (ttT, — t^Tj) = 0,
or, using the literation applied to the parametric triangle,
m'-2bm + d = 0; (1)
for since the content of x+ym + zn is supposed to be
a? + "ibxy + 2cxz + dy^ + 2eyz +fz\ that of - \ + m will be found by making z = 0,x = — \,y=\. The variation of equation (1) obtained by taking en for the increment of m (remembering that the variation of m' is {m + €n){m + en) - in", that is, e{mn + nm)) gives rise to the identical equation
win + nm - 2bn — 2cm + 2e = 0, (2)
and the variation of this again gives
n» + n' - 2cn - 2cn + 2/= 0, or n' — 2cn+/=0, as of course will be obtained immediately from (1) by substituting n, c, f in place of m, 6, d.
The parameters c, /, if n represents ' ' are the sum of the principal
diagonal elements and the content of u, just a&b,d are such sum and content in respect to m.
The parameter e (the connective to d and /) or rather its double 2e is obviously the emanant of d in respect to the operator
or, if we please, of/" in respect to the inverse operator
that is, tiVt + WiTj — Uvx — MjT,.
With the aid of the catena of equations in m, in m and n, and in n, any combination of functions of vi and n may be reduced to the standard form
Amn+Bm + Cn + D.
For, in the first place,
^ = P (m' — 26m + d) + rm + « «■ rm + «, and similarly ■^n = pn + a.
Hence the most general combination referred to is expressible as the product of alternating linear functions of m and n, and may therefore be reduced to a sum of terms of which each is a product of alternate powers of m and of n, each of which powers may again be reduced to the form of linear functions, and this process admits of being continually repeated.
Suppose then, at any stage of it, that the greatest number of occurrences of linear functions of jn and n in the aggregate of terms is i; then at the
218 Lectures on the Principles [31
next stage of the process the new aggregate will consist of monomial multi- pliers of one or more simple successions of m and n, and of terms in which the number of alternating linear functions never exceeds i — 1; hence, eventually we must arrive at a stage when the aggregate will be reduced to a sum of monomial multipliers of simple successions of m and n, every such succession being of the form
(7n»)« or w~' (?«»)« or (mn)''n~^ or m~' (mn)^n~\ But {mnf = TO . nm ,n = —m (mn — 2bn — 2cm + 2c) n
= — m'n' + 2bmn' + 2cm?n — 2evin
= - (26m - d) (2cM -/) + 26m {2cn -/) + 2c (26m -d)n- 2emn
= — (2e — 46c) mn - df.
Hence {mnf + 2 (e - 26c) mn + df= 0.
Hence (mn)« = P {{mnf + 2 (e - 26c) mn + df}-\-A mn + B=Amn + B,
where A and B are known functions of (e — 26c) and/ ; and therefore
,/ x» ^ r> , A B 2Bb
m~' (mn)« = An + Bm~^ = An ? m H j- .
d d
_,..,, y ^„ , J -S 2Bc
Similarly (mn)'^n~^ = Am — ^n H — ^ ,
^ ' 2e 46c\
and ' m~' (mn)in-^ =A+B (mn)~' = — -rj.mn + (A—B — -j^ — ] .
And this being true (mutatis mutandis) for all values q, it follows that the function expressed by any succession of products of functions of m and n is reducible to the form of a linear expression in m, n, m?i, in which the 4 monomial coefficients are known or determinable functions of the parameters to the corpus m, n.
The latent function to any such linear expression, say Amn + Bm + Gn + D,
may be found in the same way as the latent function to mn has been found, namely, as follows:
(Amn + Bm + On + Dy = A^ (mnf + AB (mnm + mmn) + AC (mnn + nmn) + 2ADmn + B'm^ + BG (mn + nm) + (7»n» + 2BDm + 2GDn + Z>^ = A'(-2e+ 46c) mn - AHf+ ABm (2bn + 2cm - 2e) + AG(2bn + 2cm -2e)n + 2ADmn + B^m^ + BG(2bn + 2cm - 2e) + C7W + 2BDm + 2GDn + J>. Let (Amn + Bm + Gn + Df - 2P(Amn + Bm+Gn + B) + Q=0
be the identical equation to Amn + Bm + Gn+D.
The coefficient of mn in the development of the first term being (46c - 2e) A^ + 2bAB + 2c.4a + 2AD,
m
31]
of Universal Algebra
219
and m?, n" being reducible to linear functions of m, n respectively, it follows that P = A{nc-e)^Bh + Gc + D.
To find Q it is only needful to fasten the attention upon the constant terms in the before named development reduced to the standard form. These will be
- AHf- 2ABcd - 2ACbf- BH - 2BCe - G'f+ D\ say K, and the constant part in —2P{Amn+Bm+Cn+D) being— 2Z>P, it follows that Q = 2AD{2hc -e) + 2BDb + 2GDc + IP - K = A'df+ 2ABcd + 2ACbf+ 2AD (26c - e) + B^d + 2BCe + C^/+ 2BDb + 2CDc,
and consequently the latent function A'' — 2PA + Q, of which the algebraical roots are the latent roots of Amn + Bm + Cn + D, is completely determined. Thus, for example, if the latent function ofm + n is required, making A=D = 0, £ = (7=1, its value will be seen to be A''-2 (6 +c) A +d + 2e +/=0, so that the roots will be 6 + c ± '/{{b + cf -{d + 2e +/)}.
On Involution.
In general, if m and n be two given binary matrices, and p any third matrix, say
m= ... n= ' % p =
nm =
10
U U ' Tj T4 ' jT g J4
p may be expressed as a linear function of T, m, n, mn or of T, m, n, nm. For in order that p may be expressible under the form A + Bm + On + Dnm, observing that
^1X3 + ^3X4 ^jTj + ^4X4
and that T = ^ " it is only necessary to write
A+Bt^ + Cr^ + D (<,T, + <3T,) = Ti,
|| jB<,+ CT,+ 2)(<,Ti + <4T,)=r„
5«, + Ct, + Z> («, T, + <,T4) = ^3 ,
f ! and then A, B, C, D may be found by the solution of these four linear
equations : and this solution must always be capable of being effected unless
Lthe determinant 1, U, Ti, <iT, + <3T, 0, t), T2, (jT, + <4Tj 0, <3. T3, «,T, + <jT4 1. h, T4, <2T3 + <4T4 van .
vanishes.
220 Lectures on the Principles [31
When this is the case the matrices m, n, in the order in which they are written, will be said to be in sinistral involution. In like manner, if 1, n, m, mn are linearly related, w, n may be said to be in dextral involution. But it is very easy to see from the identical equation (2) that in this case these two involutions are really identical, for, since A + Bm + Gn-^ Dmn = 0, by subtraction
A + Bm + Cn- Dmn + 2Dcm + 2Dlm - We = 0, that is, {A- 2eD) + ( J9 + 2cl>) m + (0 + 2bD) n - Dnm = 0.
The above determinant then will be called the involutant to m, n or n, m, indifferently, for it will be seen, and indeed may be shown, a priori, that its value remains absolutely unaltered (not merely to a numerical factor pres, but in sign and in arithmetical magnitude as well) when the Latin and Greek letters, or which is the same thing, when the matrices m and n are interchanged.
On the lAnearform or Summatory Representation of Matrices, and the Multiplication Table to which it gives rise.
This method by which a matrix is robbed as it were of its areal dimensions and represented as a linear sum, first came under my notice incidentally in a communication made some time in the course of the last two years to the Mathematical Society of the Johns Hopkins University, by Mr C. S. Peirce, who, I presume, had been long familiar with its use. Each element of a matrix in this method is regarded as composed of an ordinary quantity and a symbol denoting its place, just as 1883 may be read
1^ + 8A + 8« + 3m, where 6, h, t, u, mean thousands, hundreds, tens, units, or rather, the places occupied by thousands, hundreds, tens, units, respectively.
Take as an example matrices of the second order, as
a /3 a b 78 c d. These may be denoted respectively by
a\ + 0lj.+ yv + Btt, aX + bfi + cv + d-rr; their product by
(aa + c/3) \ + (6a + d/3) fi + (ay + cS) p ■¥ (by + dS) tt, which therefore must be capable of being made identical with
aa\' + a/3X/i + arfKv + aSkir + boifi\ +b^fi'' +byfj,p +bSfnr + cavX + c^v/j. + cyv^ + c^mr + dcL-rrX + d^irix + dyirv + dhir'.
%
31]
of Universal Algebra
221
when a proper system of relations is established between the quadric com- binations and the simple powers of \.
The arguments of like coeflScients in the two sums being equated together, there result the equations
V = X, Xl' = V, fjX = /J,, flV = TT, Vfl = \, VTT^V, ■Trfl = H, 77^ = TT,
and again, the arguments to the 8 coefficients in the second sum which are not included among the coeflScients of the first, being equated to zero, there result the equations
X/i = 0, XtT = 0, ft- = 0, /iTT = 0,
i,\ = 0, 1/2 = 0, 7rX = 0, ■7rv = 0.
These 16 equalities may be brought under a single coup d'oeil by the follow- ing multiplication table :
|
\ V fl IT |
|
|
X |
X I. 0 0 |
|
1^ |
0 0 X v |
|
/» |
fl n 0 0 |
|
TT |
0 0 /i »r |
a b c
In like manner it will be found that any rnatrix of the 3rd order as ci e f,
g h k regarded as a quantity, may be expressed linearformly by the sum
aX -f i^ -f- cv + dTT -I- e/a +/<7 + gfT + Aw + i<^,
where the topical symbols are subject to the multiplication table below written :
|
X |
IT |
r |
p- |
p |
t; |
V |
a <f) |
|
|
X |
X |
V |
r |
0 |
0 |
0 |
0 |
0 0 |
|
»r |
0 |
0 |
0 |
X |
TT |
r |
0 |
0 0 |
|
r |
0 |
0 |
0 |
0 |
0 |
0 |
X |
jr T |
|
M |
M |
p |
V |
0 |
0 |
0 |
0 |
0 0 |
|
P |
0 |
0 |
0 |
V- |
p |
V |
0 |
0 0 |
|
V |
0 |
0 |
0 |
0 |
0 |
0 |
p |
P " |
|
V |
V |
<T |
•^ |
0 |
0 |
0 |
0 |
0 0 |
|
a- |
0 |
0 |
0 |
V |
(T |
<> |
0 |
0 0 |
|
<P |
0 |
0 |
0 |
0 |
0 |
0 |
V |
tr (p |
And, in like manner, matrices of any order « may be expressed linearformly as the sum of oj' terms, each consisting of a monomial multiplier of a topical
222 Lectures on the Principles [31
symbol, the entire w" symbols being subject to a multiplication table con- taining (u* places, of which w' will be occupied by the <u' simple symbols, each appearing a> times, and the remaining &>* — <o' places by the ordinary zero.
This conception applied to quadratic matrices might have served to establish the connection between them and Hamilton's quaternions, regarded as homogeneous functions of 1, i, j, k, themselves linear functions of the topical symbols \, fi, v, tt; but the same result may be arrived at somewhat more simply by a method given in a subsequent lecture.
On the Corpus formed by two Iiidependent Matrices of the same order, and the Simple Parameters of such Corpus.
By the latent function of a corpus (m, n) we may understand the content or any numerical multiplier of the content of (that is, the determinant to) the matrix x + ym+zn, where x, y, z are monomial carriers. This function will be a quantic of the order m in x, y, z, and in the standard form the coetficient of of may be supposed to be unity, so that it will contain ^(eo'+So)) coefficients, which may be termed the parameters of the corpus.
To fix the ideas, suppose eo = 3 and let the latent function to
|
b |
c |
a. |
/« |
7 |
|
y |
c' |
a' |
/3' |
7 |
|
b" |
0", |
a" |
/S" |
7 |
be called F, where
F = a? + Zba?y + ^cx'z + ^dxy^ + Qeaoyz + S/c^" +gf + Shy'z + Skyz^ + ^2*.
Let m become m + en, where e is a monomial infinitesimal. Then the function to the corpus becomes the content of
x + y{m + en) + zn, that is, x + ym + (z + ey) n,
and consequently the variation of the function to (m, n) is eyB^F. If then the rate of variation of any of the parameters, when n is the rate of variation of m, be denoted by prefixing to such parameter the symbol E, we shall find
Eb = c\ M = 2e; Ee=f; Eg = Zh; Eh = 2k; Ek = l;
and similarly, if ^, preceding a parameter, be used to indicate its rate of variation corresponding to n's rate of variation being m, then
a'c=^; a/=2e; 3e = d; 3l = 3k; 3k = 2h; 3h=g;
and the variations of c, /, I, as regards E, and of b, d, g, as regards g[, are of course zero.
a
31] of Universal Algebra 223
By forming the triangle of parameters
1
h c
def
g h k I
p q r s t
the law of variations of the parameters of the function to {ra, n) (expressed in
the ordinary manner by a ternary quantic affected with the proper numerical
multipliers) becomes evident, whatever may be the order of the corpus (that
is, of the matrices m and n, of which it is constituted): thus, for example,
when oj = 4, in addition to the previous expressions we shall find
Ep=^, Eq=Sr, Er = -2s, Es = t, Et = 0,
St = 4,8, 3S = 3r, sr = 1q, 3q=p, ap = 0.
By means of the above relations, any identical equation, into which enters one or more matrices, admits of being varied, so as to give rise to an identical equation connecting one additional number of the same.
Scholium. — In what precedes it will have been observed that the matter under consideration has always regard to matrices, or, as we may say, quantities of a fixed order «, combined exclusively with one another and with ordinary monomial quantities. Every such combination forms as it were a clausum or world of its own, lying completely outside and having no relations with any other. It is, however, possible, and even probable, that as the theory is fiirther evolved, this barrier may be found to give way and the worlds of all the various orders of quantity be brought into relation and intercommunion with one another.
LECTURE III.
On Quantity of the Second Order.
The theory of matrices of the second order seems to me to deserve a special eliminary investigation on various grounds. First, as affording a facile and natural introduction to the general theory (as the study of Conic Sections is usually made to precede that of universal Geometry); secondly, because it presents certain very special features distinguishing it from all other kinds of quantity, such as the coincidence of the two involutants (reminding one of the single image in the case of ordinary refraction as contrasted with the double image seen through iceland spar), or, again, the rational relation between the products of matrices of the second order, in whatever order the factors are introduced in the performance of the multiplication ; and thirdly,
224 Lectures on the Princijths of Universal Algebra [31
because the theory of this kind of quantity has already been extensively studied and developed under the name or aspect of Quaternions. Hence it may not be out of place to make the remark that, as it surely would not be logical to seek for the origin of the conception included in the symbol V( — 1) in geometrical considerations, however important its application to geometrical exegesis, so now that an independent algebraical foundation has been dis- covered for the introduction and use of the symbols employed in Hamilton's theory, it would (it seems to me) be exceedingly illogical and contrary to good method to build the pure theory of the same upon space conceptions; the more so, as it will hereafter be shown that quantities of every order admit of being represented in a mode strictly analogous to that in which quantity of the second order is represented by quaternions, namely, if the order is <o, by «i)"-ions, or as I shall in future say, by Ions, of which the geometrical interpre- tation, although there is little doubt that it exists, is not yet discovered, and it must, it is certain, draw upon the resources of inconceivable space before it can be effected.
4
I
32.
ON THE SOLUTION OF A CLASS OF EQUATIONS IN QUATERNIONS.
[Philosophical Magazine, xvii. (1884), pp. 392 — 397.]
The general equation of the degree w in Quaternions or Binary Matrices is obviously <o*, but in certain cases some of these roots evaporate and go off to infinity. The only equation considered by Sir William Hamilton in his Lectures is the Quadratic Equation of a form which I call unilateral, because the quaternion coefficients in it are supposed all to lie on the same side of the unknown quantity. I propose here to show how Hamilton's equation, and indeed a unilateral one of any order, may be solved by a general algebraical method and the number of its roots determined.
It will be convenient to begin by setting out certain general equations relating to any two binary matrices m, n.
Writing the determinant of a; + ym + zn under the form
a? + 2bxy + 2cxz + dy^ + 2eyz +fz''
(6, c, d, e,f, thus constituting what I call the parameters of the corpus m, n), we have universally
»m^-2bm+d=0, w»-2cn+/=0, d(m-'n)^ - 2e(m-'n)+/= 0. Moreover if m, n receive the scalar increments fj.,v;d, e,/ become respectively d — 2/jJ) + fi?, e — fw — vb + fiv, f— 2vc + v^. Let us begin with Hamilton's form, say ar'-2pa; + 5=0, and suppose ar'-2fia; + D = 0,
where B, D are scalars to be determined.
Let h, c, d, e,fhe the five known parameters of the corpus p, q. Then, ». since I! {p-B)-^(q-D) = 2x,
8. IV. 16
226 On the Solution of a Class of [32
we shall have [cf. p. 188 above]
4i(d-2bB + B*)a^-i{e-bD - cB+BD)a! + f- 2cD + 2)' = 0. Hence, writing B — b = u, D — c = v,
d-b' = a, e-bc = fi, f-c^ = y, we have «*+a = \, mw + /8 = 2X (m + 6), i/* + y = 4-\(v + c).
From the last two equations, eliminating v, there results
(2\u - 2bX - /3)' - 4\ (2Xm - 26X - yS) m + (7 - 4cX) «' = 0. Hence substituting X — a for «',
(4X' + 4cX.- 7) (X - o) - (26X - /9)» = 0. We have thus six values of u, namely
+ V(X - «) (where X has three values), to which correspond six values of v, namely
and, finally, 2a; = (p — m — b)~^(q — ?; + c)
= {(p - by - u»}-'(p -b + u){q-c-v),
__pq-(c + v)p-(b-u)q + (b-u)(c + v)^ °' '^ 2(6»-d-M») '
which equation gives six values for x, and shows that ten have evaporated.
It is easy to account d priori for the solution depending only upon a cubic in u'.
For a? — 2px + q = 0 is the same as y* — 2yp + q = 0, where y = — x + 2p. But obviously, from the nature of the process for determining them, B and C are independent of the side of the unknown on which the first coefficient lies. Hence the actual B will be associated with B', B' being what B becomes when X becomes —x + 2p, which is obviously —B + 2b.
Hence with any value of B — b, which is u, is associated a corresponding B — b, which is — u.
I will now proceed to apply a similar or the same method to the trinomial cubic equation in quaternions (or binary quantity) a^ +jua;— 51 = 0, with a view to ascertain the number of its roots.
Retaining the same notation as before, and still supposing
a^ - 2Bx + D = 0,
32] Equations in Quaternions 227
we obtain a:» + (Z)'-4£')a; + 2jSi> = 0,
and , = Jl+25^,.
Hence [{^B^ - Df - 2h {'^B' - D)-\- d] x"
-2 [ii^B" - D) BD - dA.B'- D)- 2bBD + e} X
+ 4,&]>-2cBD+f=0.
Hence we may write
('iB' - Dy - 2b{'iB' - D) + d = \,
2{'tB'-D)BD-c{'iB'-D)-2bBD + e = \B,
4,B'D'-2cBD+f=\D;
from which equations B and D are to be determined. Eliminating \ between the first and second and between the first and third of these equations, we obtain two equations, of which the arguments are
If; B'D',1)'; B*D, BB, BD, D ; 1
for the one,
Biy; B'D.BD.D; B',B',B',B; 1
for the other.
Eliminating D by the Dialytic method between these two equations, we shall have (using points to signify unexpressed coefficients) the following three linear equations in B', D, 1, namely:
■BI> +(-5' + &c.)Z)+(-5» + &c.) = 0,
■ B'D' + (■ B> + &c.) D + (-B' + &c.) = 0,
• B'D' + (-B' + &;c.)D +(-B> + &c.) = 0.
Hence in the final equation B rises to the 15th power; and by com- bining any two of the above equations, D is given linearly in terms of B; and, finally, a; is known from the equation
_:(p + D-'iB'-2b)(q + 2BD) * - _(45i_ />)2_ 2(4/^ - i>) + d' and has 15 values.
A like process may be extended to a unilateral equation (of the Jerrardian form) of any degree, say af + qx + r = 0.
Introducing the auxiliary equation with scalar coefficients as before, namely
x'-2Bx-\-D = 0,
X may be expressed as a function of q, r, B, D; and the term containing the
• I ose^ and -v^ to signify M~^L and LM~^ Tespectively.
15—2
228 On the Solution of a CUm of [32
highest power of B in the equation for determining B (of which D is a one- valued function), when a, = 4, will be found to be the determinant
-B B' •£" •B'
.£• .£» .£' 'B>
.B" •B' ■B> -B"
.£' •£• -B" -B"*
and a similar determinant will fix the degree of B in the resolving equation
for any value of to. Hence the number of solutions of the unilateral equation
in quaternions of the Jerrardian form of the degree o) is o) (2a, - 1) or 2a,' - <o,
and the evaporation will accordingly be to* - 2a>' + «■>, or
{(O" -to) (to' + CO- 1).
Moreover the same method with a slight addition will serve to de- termine the roots of the general unilateral equation in quaternions, the number of which will be a cubic function of «, as I propose to show and to give its precise value in some future communication, either in this Journal, or at all events in the memoir on Universal Algebra now in the course of publication, under the form of lectures, in the American Jourmil of Mathematics^.
I very much question whether the old method of Hamilton, as taught by its most consummate masters, Tait in this country, or the late Prof. Benjamin Peirce in America, would be found sufficiently plastic to deal effectually with an analytical investigation in quaternions of this degree of complexity, so as to lead to the formula for the number of solutions of the unilateral equation of the Jerrardian form above given.
I invite my much esteemed and most capable former colleague and former pupil DrStory.of the Johns Hopkins, and Prof Stringham, of the University of California, who carry on the traditions of the Harvard School, to put the power of the old method as compared with the new to this practical test.
Postscript— U a^-5pa?+5qa;-r = 0,
(where p, q, r are perfectly general matrices of the second order which satisfy the general equations
q'-2bq + d = 0. qr+rq-2bq-2b,q + 2e = 0, r'-2b,q + d, = 0,
pq + qj)-2bp-2fiq-^2e = 0, p»-2^p + S = 0,
pr + rp- 2bip - 2^r + 2ei = 0),
• It may readily be seen that the highest term in the equation for finding B is identical with the resultant of ^^ _ ^^^^ ^ ^^^^^ ^^^ ^^^ _ ^^^,^, ^ g^,^ _ g^^^,
that is will be 2'8.8.7.19iJ=«; and that the last term (at aU events to the sign prlt) will be b*S\ which is of 4 . 3 + 2 . 2 . 4 {that is of 28) dimensions in *, and is therefore codimensional (as it ought to be) with B^.
t It is given in the Postscript below.
I
32]
Equations in Quaternions
229
: r + .SDp px= —
2Bx + D = 0, BD
and if we write a? ■
and I find by perfectly easy and straightforward work that B, D may be
determined by means of the following equations:
(B'-BY B^ — D
^ — 9—^+2 {h - PB) ~^~ + {d- 2eB + 485=) = 9\,
B'D — BD' B^ — D
g + ih + 3/3i)) -^ + (e-e,B + 3eD- 6BBD) = 3B\,
B'D^ - 2 (6, + 3/3i)) BD + d, + QDe, + 981)' = D\. The order (by which I mean the number of solutions of this system of equations) is readily seen to be the same as that of
D' +B'D+B*D = 0 ■BBp+B^D+R =0; that is, is the same as the degree in 5 of fi' (£»)' . R, where R is the resultant of
• D^ + ■ 5» + ■ 5^ and • D^ + ■ Br-D + ■ B\
Hence* the number of solutions is 3 + 10 + 8, that is, is 21.
Practically, therefore, we have now suflBcient data to determine the number of solutions of a unilateral equation in quaternions of any order w; for it is morally certain that such number is a rational function of m ; and as it cannot but be of a lower order than &>*, we have only to determine a cubic function of o) whose values for w = 0, 1, 2, 3 are 0, 1, 6, 21, which is easily found to be (o' — (o' + m; so that the evaporation is tu* — o)* + w' — <a, that is
(a>>+l)(a>'-a>).
Practically also we can solve (subject to hardly needful verification) the number of roots of a unilateral equation of the special form
sr + q^a^ + qt-ia^~' + ... + qo = 0. For when ^ = &>, we know the number is a)'; and when ^ = 1, the number is «*' + <»* — <»; consequently if the second differences of the function of (&>, 6) which ejtpresses the number of roots are constant, the value of this function when ^= &) — 1 is ft)' — a>' + ft), which we have found to be the actual number; and consequently, if the second differences are not constant, they must be sometimes positive and sometimes negative, which is in the highest degree improbable. Hence in all probability it will be found that the required number of solutions in the form supposed is (1 + 6)0}' — 6m.
I need hardly add that the nine quantities 26, 26,, 2/3; 2e, 26,, 26; d.h.d^, which occur in the discussion above given of the general unilateral cubic, or, say, rather the ten quantities obtained by adding on to these unity, are the
[* See footnote t p. 197 above.]
230 On, the Solution of a Class of Equations, etc. [32
ten coefficients of the determinant to the binary matrix {x ■\- py -^^ qz + rt), which of course there is not the slightest difficulty in expressing in terras of scalar and vector affections of p, q, r and their combinations, if any one chooses to regard them as given in quaternion form.
Scholium. In what precedes it is very requisite to notice that only general cases are considered; and that there are multitudinous others which escape the direct application of this method, and do not conform to the rule which assigns the number of solutions. Thus, for example, the equation a^ + px = 0, besides the solutions x = 0,x = —p, will have two others which will require the method of the text to be modified in order to determine. Or take the most elementary case of all, the simple equation pa; = q. If p is not vacuous (that is, if its determinant when regarded as a matrix, or its modulus when regarded as a quaternion, is finite), there is the one solution x=p~^q. But if p is vacuous, then, unless q is also vacuous, the equation is insoluble. If g = 0, there will be two solutions ; one of theni x = 0, the other x = con- jugate of p in quaternion terminology; or
— d; b , a; b
x= , when «= ' ,
c; — a ^ c; a
in the language of matrices. If, p still remaining vacuous, q is vacuous but not zero, a further condition must be satisfied, namely, if
a; b , o; /9 p- , and q= j, ^ c; a ^ y; 6
the condition is aB + ad — by — c^ = 0;
or if p'=a + bi + cj + dk and q=a + ^i+yj + Sk,
the condition is aa + b^ + cy + dB= 0.
When this condition (besides that of q being vacuous) is satisfied, the equation px = q is soluble, and p~^q becomes finite but indeterminate, containing two arbitrary constants*.
* So in general if p, q be two simply vscnons matrices of any order, the condition that the equation px=q may be soluble, or, in other words, that p~'g (a combination of an ideal with a vacuous matrix) may be non-ideal, may be shown to be that the determinant to the matrix \p + nq (where X, fi are scalar quantities) shall vanish identically— which (p being supposed already to be vacuous) involves just as many additional conditions as there are units in the order of the matrix.
4
33.
[. ON HAMILTON'S QUADRATIC EQUATION AND THE GENERAL UNILATERAL EQUATION IN MATRICES.
[Philosophical Magazine, xviii. (1884), pp. 454 — 458.]
In the Philosophical Magazine of May last I gave a purely algebraical method of solving Hamilton's equation in Quaternions, but did not carry out the calculations to the full extent that I have since found is desirable. The completed solution presents some such very beautiful features, that I think no apology will be required for occupying a short space of the Magazine with a succinct account of it.
Hamilton was led to this equation as a means of calculating a continued fraction in quaternions, and there is every reason for believing that the Gaussian theory of Quadratic Forms in the theory of numbers may be extended to quaternions or binary matrices, in which case the properties of the equation with which I am about to deal will form an essential part of such extended theory*. Let us take a form slightly more general than that before considered, namely, the form
pa^ + qx + r = 0, with the understanding that the determinant of p (if we are dealing with matrices), or its tensor if with quaternions, dififers from zero. Let us construct the ternary quadratic
aw' + 2buv + 2cuw + dir" + 2ei>«; +fu/', defined as the determinant of up + vq+ wr, on the one supposition, or by means of the equations
a = Tp', d=T<f, f^Tr', b = SpSq- SVpVq, c = SpSr-SVpVr, e = SqSr - SVqVr, on the other supposition.
• I have foond, and stated, I believe, in the form of a question in the Educational Timet some years ago, that any fraction whose terms are real integer qnaternions may be expressed as a finite continaed fraction, the greatest-common-measure process being applicable to its two terms, provided both their Moduli are not odd multiples of an odd power of 2, which can always be guarded against by a previous preparation of the fraction.
232 On Hamilton's Quadratic Equation and the [33
On referring to the article of May [p. 226 above], it will be seen that the solution of the equation may be made to depend on the roots of a cubic equation in the quantity therein called X. When fully worked out, this equation will be found to take the remarkable form e^°./= 0, where / is the invariant of the ternary quadratic above written, and Q. = 2aSc - aS^. It may also be shown that
{p + h-u){q-c-v) '"- ^ •
where « is a two-valued function of X, and v a linear function of u.
I shall suppose that /, the final term in the equation in \ differs from zero: the solution of the given equation in x will then be what may be termed regular, and will consist of three pairs of actual and determinate roots. When 1 = 0, the solution ceases to be regular; some of the roots may disappear from the sphere of actuality, or may remain actual but become indeterminate, or these two states of things may coexist. The first coefficient of the equation in \ is a, the determinant of p (or its squared tensor), which also must not be zero, as in that case one root at least of \ would be infinite. Let us suppose, then, that neither a nor / vanishes. The very interesting question presents itself as to what kind of equalities can arise among the three pairs of roots, and what are the conditions of such arising.
. This equation admits of an extremely interesting and succinct answer as
follows: — Let m represent — ^ — ; the equalities between the roots of the
given equation in x will be completely governed, and are definable by the equalities existing between those of the biquadratic binary form
{a,h,m,e,f){X, Yy*.
* If the equation is regarded as one in quaternions, the determining biquadratic is the
modulus of x'^ + xp + q; from which it follows immediately that, if p, q are real quaternions, all
the four roots, say a, /3, 7, S, are imaginary. It may ba shown that the roots of Hamilton's
determining cubic are
(o + /9)(7 + ») (tt + 7)(^ + 8) (a + «)(|3 + 7) d -^ , d J . d J ,
and these therefore are (as shown also by Hamilton) all of them real. The biquadratic serves to determine the points in which the variable conic associated to the equation px^ + qx + r (that is, the determinant to xp + yq+zr) is intersected by the absolute conic xz-y-. Each root of the given equation corresponds to a side of the complete quadrilateral formed by the four points of intersection of these two conies ; and thus we see that there are five cases to consider when the variable conic is a conic proper, according as it intersects or touches the fixed conic (which can happen in four different ways); and seven other cases where the conic degenerates into two intersecting or two coincident lines (in which cases the solution becomes irregular) ; namely, the intersecting lines may cut or touch in one or two points the fixed one, and may cut or touch the conic at their point of intersection, which gives five cases; and the coincident lines may cut or touch the fixed conic, which gives two more. Hence there are in all twelve principal cases to consider in Hamilton's form of the Quadratic Equation in Quaternions: or rather tliirteen, for the case of the variable and fixed conios coinciding must not be lost sight of.
I
33] genet^al Unilateral Equation in Matrices 233
If the biquadratic has two equal roots, the given quadratic will have two pairs of equal roots.
If the biquadratic has two pairs of equal roots, the given quadratic will have four equal roots.
If the biquadratic has three equal roots, the quadratic will have three pairs of equal roots.
If the biquadratic has all its roots equal, the quadratic will have all its roots equal.
In the first case two of the three pairs of roots of the given quadratic coincide, or merge into a single pair.
In the second case, not only two pairs merge into one pair, but the two roots of that pair coincide with one another.
In the third case the three pairs merge into a single pair.
In the fourth case the two members of that single pair coincide with one another.
So long as the equation in x remains regular, no kind of equalities can exist between the roots other than those above specified.
For instance, let us consider the possibility of two values of x, and no more, becoming equal. First, let us inquire what is the condition to be satisfied in order that the scalar parts of two roots which belong to the same pair shall become equal. It may be shown that the sufficient and necessary condition that this may take place is that the irreducible sub-invariant of degree 3 and weight 6 (that is, the first coefficient of the irreducible skew- covariant of the associated biquadratic form [a, b, m, e,f]) shall vanish.
If, now, the vectors as well as the scalara of the two roots are to be equal, it may be shown that the second as well as the first coefficient of the skew- covariant must vanish. But this cannot happen without the discriminant vanishing*; for it may easily be seen that the discriminant of a binary biquadratic with its sign changed is equal to sixteen times the product of the first and last coefficients, less the product of the second and penultimate coefficients of its irreducible skew-covariant. Hence when two roots belong- ing to the same pair of the given quadratic coincide, two values of \ become equal, and therefore all four roots belonging to two pairs merge into one.
Agam, it is not possible for two roots belonging to two pairs correspond- ing to two different values of X to coincide ; for in such case the expression
The first two coefficients of the skew-covariant vanishing implies the existence of two pairs of equal roots and vice versa. This is on the supposition made that a, the first coefficient of the given qnartic, is not zero.
234 On Hamilton's Quadratv; Equation and the [33
given for x shows that pq, p, q, 1 would be connected by a linear equation. But when this happens (as has been shown by me elsewhere), the invariant of the associated ternary quartic vanishes and the equation ceases to be regular. Thus, then, it appears that it is impossible for a single relation of equality (and no more) to exist between the roots of the given equation when its form is regular. So, again, it may be shown that it is impossible for four, and no more, relations of equality to exist between the roots.
It need hardly be added, that the equation pa? + qx + r = 0 ceases to be regular when q or r vanishes.
The reader may satisfy himself as to the truth of what has been alleged as to the relation of the discriminant of a binary biquadratic to the coefiScients of its skew-covariant by simple verification of the identity
16 (a^d - 3ahc + 26') (e»6 - 3edc + 2(P)
- (a»e + 2abd - 9c'a + 66'c) (e'a + 2edb - Qed' + 6d^c)
= 27 (ace + 2bcd -c^-b^e- ad'')' - (ae - ibd + 3c=)'.
The biquadratic equation in X, Y is what the determinant of Xp+fiq + vr becomes when X', XY, Y^ are substituted therein for \, fi, v; so that we may say that (a, b, m, e,f) (x, 1)* is the determinant of px' + qx + r, when x is regarded as an ordinary quantity. Let <f>x be any quadratic factor of this biquadratic function in a;: I have found that ^ = 0 will be the identical equation to one of the roots of the given equation fx = 0, where
fx = px^ + qx + r.
Between the two equations yic = 0, (f)x=0, a^ may be eliminated and x found in terms of known quantities : <f>x will have six different values, which will give the six roots of/a;=0. It is far from improbable that a similar solution applies to a unilateral equation /ic = 0 of any degree n in matrices of any order «a.
Call Fx the determinant offx when x is regarded as an ordinary quantity; then, if <l>x is an algebraical factor of the degree o) in a; contained in Fx, it would seem to be in all probability true that (f)x = 0 is the identical equation to one of the roots of fx = 0 ; and, vice versa, that the function identically zero of any such root is a factor of Fx. By combining the equations fx = 0, ^ = 0, all the powers of x except the first may be eliminated, and thus every root of X determined. The solution of the given equation will depead upon the solution of an ordinary equation of the degree n<o, and the number of roots will be the number of ways of combining na things to and to together. Thus, for a cubic equatiim in quaternions the number of roots would be ^6 . 5, or 1.5. In the May number of this Magazine [p. 229 above] it was supposed to be shown to be 21 ; but it is quite conceivable that this determination may
m
33]
general Unilateral Equation in Matrices
235
be erroneous, especially as it was deduced from general considerations of the degrees of a certain system of equations without attention being paid to their particular form, which might very well be such as to occasion a fall in the order of the system. I am strongly inclined, with the new light I have gained on the subject, to believe that such must be the case, and that the true number of roots for a unilateral equation in quaternions of the degree n is 2n' — n*; in which case the theorem above stated, and which may be viewed as a marvellous generalization of the already marvellous Hamilton- Cayley Theorem of the identical equation, will be undoubtedly true for all values of n and w. But I can only assert positively at present that it is true for the case of w = 1 whatever &> may be, and for the case of n = 2, w = 2-f-.
* From the number 21 above referred to, now known to be erroneous, the general value was inferred to be n^ -n- + n, whereas it is demonstrably 2n^-n only for the general unilateral equation of degree n in quaternions, as I proved it to be for the Jerrardian form of that equation.
t I have since obtained an easy proof of the truth of the conjectural theorem for all values of n and a; see the Comptes Rendus of the Institute of France for October 20th last [p. 197 above].
34.
NOTE ON CAPTAIN MACMAHON'S TRANSFORMATION OF THE THEORY OF INVARIANTS.
[Messenger of Mathematics, xiii. (1884), pp. 163 — 165.]
The whole question as is well known consists in finding the free forms of
fi~'0, where
il = aoSa, + 2aiSa2 + ... + iof-iSoj;
but, as long ago noticed by me* in the Atti. Math. Journal, il~'0 is only a deformation of V~^0, where
F= ajSai — a^Sa^ + ... ± ai_iBai, n~'0 being deducible from F~'0 by altering the dimensions of the a elements which it contains in known numerical proportions, so that X1~*0 may be said to be V~^0 subjected to a known strainf.
To fix the ideas let i = 3 and call the a's by the names a, b, c, d or, for greater simplicity, 1, b, c, d.
Let b=r + s + t,
c = rs + rt + St, d = rst.
Then the matrix
so that
D{b,c,d) \ ^ rtV \l = S-ift t + r r + s, D(r,s,t)
st tr rs ,
(r-s){r-t) {s-r)(s-t) (t-r)(t-s)' D {r, s, t) r s t
D{b,c,d)~ {r-s){r-t) {s-r){s-t) (t-r)it-s)' 1 1 1
(r-s)(r-t) (s-r){s-t) (t-r){t-s)'
[* Vol. III. of this Reprint, p. 570.]
t In fact the numerical multipliers of the terms in 0 may be taken perfectly arbitrary without prodaoing any effect upon the form Q-'O than what may be represented by a strain.
I
34] Note on Captain MacMahon's Transformation, etc. 237
Consequently
y _^r' — jr -^^ a + t)r + {rs ■¥ rt + st) ^ _ ^ st -
{r-s){r-t) ^'--^(r-sXr-O
In like manner in general for \, a^, a^, ... at we shall find, on writing
ai = ri + r2+ ... +ri,
0-3 = n^j + r^rs + . . . + n^iVi,
ai = rir^...ri,
V= Sui - OiSOi + ... ± Oi^iBoi = 2 , .,"'"'* r Sn.
(ri-r^){r,-r,)...{n-ri)
Hence V-'O = i'(«i, s„ ... Si), ■
where, in general, s^ = r," + rj" . . . + Vi" ;
and consequently the theory of invariants, which endoscopically treated in the ordinary way hinges upon symmetrical functions of the differences of a set of letters, is made to depend upon functions of the simple sums of powers com- mencing with the second power and ending with a power whose index is the order of any given finite quantic, but in the case of perpetuants taking in all the powers except the first.
It goes without saying that the same method applied to the constrained V will show that it is equal to SSr,, so that Fo~' is an arbitrary function of the differences of the r's corresponding to that hypothesis, as we know ought to be the case.
What has been established in the foregoing investigation is a principle of correspondence whose importance as a simplifying agent recalls Ivory's use of such principle in Attractions, namely, the remarkable algebraical law that any symmetrical function of the differences of a set of i quantities is a symmetrical function of the sums of the 2nd, 3rd, ..., ith powers of another equi-numerous set.
By virtue of this principle the numerical part of the Calculus of Invariants is capable of being entirely divorced from all question of algebraical content and a Zahl-Invariant theory comes into being, in its fundamental conception analogous to the Zahl-Geometrie of Schubert.
Further remarks on this subject will be found in the Comptes Rendus de I'Institut presumably for March 31 and April 7 of this year [p. 163 above].
35.
ON THE D'ALEMBERT-CARNOT GEOMETRICAL PARADOX AND ITS RESOLUTION.
{Messenger of Mathematics, xiv. (1885), pp. 92 — 96.]
I WILL presently state the simple geometrical problem which led D'Alem- bert to call into question the validity of the received Cartesian doctrine of positive and negative geometrical magnitudes, and which, according to Camot, furnishes an unanswerable argument against it. See Mouchot, La refwrme Cartesienne, pp. 74, 75.
Against this doctrine, presented in its crude form, the objections of these illustrious impugners of it are unquestionably well founded and unanswerable; but the inference to be drawn from this is not that no such or such-like doctrine reposing on an unassailable logical basis exists or is capable of being established (woe worth the day! when such a conclusion should be admitted), but that the doctrine as usually stated is incomplete and requires a supplement.
This has been anticipatively furnished by me many years ago in this very Journal, and in conjunction with the substitution of positive and negative indefinite rotation in lieu of Euclid's positive and limited angular magnitude, made the basis of a strictly logical deduction (which was before wanting) of the trigonometrical canon.
It consists in the notion of a line having, so to say, sides (returning upon itself at its two semi-points at infinity), or to put the matter in a more practical form, in regarding an Euclidean indefinite straight line as repre- senting two distinct lines locally coincident, but running in contrary directions, and in referring the algebraical sign of any rectilinear segment to the con- currence or discordance of its flow (which is represented by the order in which its two extremities are named or written down) with that of the indefinite line, upon which it is supposed to be carried.
«
35] On the D'Alembert-Camot Geometrical Paradox, etc. 239
Thus, for example, AB taken on the upper side of a line or line-pair will be the negative of AB taken on the same side, but the same as BA taken on the under side.
I will now state the D'Alembert-Caraot problem. "Voici" says Carnot, "un exemple aussi simple que frappant, qui seul suffit pour renverser toute cette doctrine" of positive and negative magnitudes.
"D'un point K, pris hors d'un cercle donn^, soit propose de mener une droite Kmm', telle que la portion mm', intercept^e dans le cercle, soit egale k une droite donnee.
K
"Du point K, et par le centre du cercle menons une droite KAB qui rencontre la circonfdrence en A et B. Supposons KA = a, KB = h, mm' = c, Km = X. On aura done par les propri^tfe du cercle
ab = x{c + x)=:cx + a?
done a? + ex — ab = 0
ou x= — \c±*J{\(i' + ah).
X a deux valeurs: la premiere, qui est positive, satisfait sans difficult^ k la question; mais que signifie la seconde, qui est negative? II parait qu'elle ne peut repondre qu'au point m', qui est le second de ceux oil Km coupe la circonf^rence ; et, en eflFet, si Ton cherche directement Km', en prenant cette droite pour I'inconnue x, on aura
x(jc — c) = ab
on x = \c ± >J{\(? + ah)
dont la valeur positive est pr^is^ment la mSme que celle qui s'dtait pr^sent^e dans le premier ca.s avec le signe n^gatif Done, quoique les deux racines de liquation
soient I'une positive et I'autre negative, elles doivent etre prises toutes les deux dans le mSme sens par rapport au point fixe K. Ainsi, la rfegle qui veut que ces racines soient prises en sens opposes porte k faux. Si au contraire le point fixe K ^tait pris sur le diamfetre meme AB et non sur le prolongement,
240 On the D'Alembert-Carnot Geometrical [35
on trouverait pour x deux valeurs positives et cependant elles devraient dtre prises en sens contraires i'une de I'autre. La regie est done encore fausse pour ce cas.
" Si Ton dit que ce n'est pas ainsi qu'il faut entendre ce principe, que les racines positives et negatives doivent dtre prises en sens opposes, je de- manderai comment il faut I'entendre ? et j'en conclurai par la mfeme qu'il faut une explication pour empecher qu'il ne soit pris dans I'acceptation la plus naturelle. II suit que ce principe est obscur et vague."
The answer has been already given to the question, "comment il faut entendre ce principe," and it will be seen in such a way as to remove all grounds for the charge of its being any longer "obscur et vague."
This is how the problem set out in full ought to be enunciated :
A complete line (that is, a line-pair or two-sided line) drawn from K cuts the circle in the points m, m'; mm' measured on either side of the line (and of course denoted quantitatively by the number of units of given length which it contains) is to be equal to c a given positive or negative number. Required the value of Avi.
(1) Suppose K to be exterior to the circle as in the diagram above.
I distinguish the two sides of the complete line, as the under and upper line, and suppose the flow of the under one to make an acute Euclidean angle with the flow from K to the centre of the circle. In all cases
Km' = Km + mm',
and consequently the equation for finding x remains always sc' + cx = ah, of which the two roots are —hc + n/(\c^ + ab) and — ^c — V(ic'-t- a6).
Adhering to the letters of the diagram, if c is positive the two values of a; will correspond to Am on the under line and Am' on the upper line of the line-pair. If, again, c is negative, the two values of x will correspond to Am on the upper and Am' on the under one.
(2) Suppose K to be within the circle.
It will still be true (paying attention to the signs) that Km' = Km + mm' (that being a universal identity in algebraical geometry), but the algebraical values of KA, KB being contrary, we may regard KA as positive and equal to a, KB as negative and equal to — b, and shall have the equation
x' + cx = — ab, of which the two roots are
-hc + »/(i(^-abj, --^c-VdC-ai). Understand by the two segments Km and Km'.
I
■
It^
35] Paradox and its Resolution 241
We may suppose the indefinite line-pair mKm to swing round K, its under-side in the position of coincidence with the diameter having the same flow as KA\ then, if c is positive, until the swinging line revolving with the sun has described a right angle, the first root will be the in/rd-diametral segment taken on the lower line (or side), and the second root the suprd- diametral segment taken on the upper line (or side) of the line-pair (or complete line); in the next quadrant of rotation the first root will be the «M/w«-diametral segment on the under and the second root the infrd- diametral segment on the upper side of the complete line. When c is negative a similar statement may be made it only the words under and upper are interchanged. In the critical position, when the swinger is at right angles to the diameter, the two roots become equal and undistinguishable ; but throughout and subject to no exception, the complex of the two roots contains the complete solution of the problem, and the complete solution of the problem necessitates the retention of the complex of the two roots.
Thus, then, as in the preceding case, it has been shown that the Cartesian view of the equipollence of positive and negative roots (the latter Descartes influenced by hereditary prepossessions calls radices falsae) is made exact through the intermediation of the conception of sides to a line. D'Alembert and Carnot are entitled to the gratitude of Geometers and all lovers of truth for raising objections so perfectly well founded to the then, and even now, too prevalent interpretation of the meaning of the geometrical positive and negative, but the difficulty which they so justly appreciated and so clearly expressed is overcome and exists no longer.
P.S. I am informed that M. Laguerre has emitted the same view as that I have set forth relative to the sign to be given to geometrical distances, and made use of the same conception of the double or complete line-carrier.
My note on the subject appeared before my exodus across the Atlantic, probably nine or ten years ago. M. Laguerre 's publication must have been many years posterior to this. The references to the reappearance of the theory on the other side of the Channel, obligingly furnished to me by M. Mannheim in Paris, have unfortunately got mislaid. I believe the com- munication containing it was made by M. Laguerre within the last three or four years, but it has already had time to find its way into some of the most esteemed French text-books. Being not only true but the truth, it must eventually find universal acceptance. It is not without interest (it seems
me) that we may regard a double or complete right line as a sort of embryonic embodiment of the idea of a Riemann Surface.
8 IV. 16
36.
SUR UNE NOUVELLE TH^ORIE DE FORMES ALG^BRIQUES*.
[Comptes Rendus, ci. (1885), pp. 104.2—1046, 1110—1111, 1225—1229,
1461—1464.]
Si Ton imagine une fonction de d^rivees diffdrentielles (toutes d'un ordre sup^rieur k I'unit^) de y par rapport k x, qui, sauf I'introduction d'un facteur
multiple num^rique, d'une puissance de t^ , ne change pas sa valeur quand
on remplace x par y ei y par x, il est evident qu'une telle fonction restera
invariable (sauf Tintroduction d'une constante comme facteur) quand pour x
et y on substitue des fonctions lin^aires quelconques, homogenes ou non
homogenes de y et x. Ainsi une telle fonction couduira immMiatement k
la connaissance d'un point singulier d'une courbe d'un degre quelconque. Le
seul exemple d'une telle fonction, traits jusqu'4 ce jour, est la simple fonction
dJ'v
-7^ qui, par cette seule propri^td, sans aucune autre consideration, sert k
d^montrer I'existence d'une propri^t^ projective de courbes dont la condition
est -T^ = 0. II nous parait done tr^s utile de chercher un moyen de produire
toutes les fonctions de cette espece auxquelles nous donnerous le nom de rdciprocants purs ou simplement r6ciprocants. On verra qu'il existe des
r^ciprocants mixtes, c'est-a-dire contenant des puissances dcT^f comme la
dy d'v 3 d^v d''y\ forme bien connue de M. Schwarz, -^ -^ — ^ -~_, -~] qui possfedent la m6me
facultf^ d'invariance par rapport a I'echange de y avec x, comme les rdciprocants purs, mais qui dvidemment ne peuvent pas indiquer I'existence de points singuliers dans les courbes.
Nous ^crirons, au lieu de 8xy, Bx'y, ^xy, Bx*y,..., les lettres t, a, b, c, ..., et
pour leurs r^ciproques B^x, By'x, S/x, ..., t, a, /3, 7, On verra facilement
que, pour que F(t, a, b, c, ...) soit un reciprocant pur, F doit etre d'un degr^ et d'un poids constant dans les lettres de chaque terme ; de plus (pour uu
[* See the lectures, below p. 303.]
36] Sur une nouvelle tMorie de formes algehriques 243
r^ciprocant ^d'une nature quelconque), on aura i^(a ...)/^ (a ...) = (— 1)*<^, oii 6 sera le plus petit nombre des lettres a,b, c, ... dans un terme quelconque de F, et X sera la moyenne arithmetique entre le poids et trois fois le degr^ de F, en
comptant le poids de t, a, b, c, ... comme etant — 1, 0, 1, 2 Cela donne
lieu a une remarque importante par rapport aux re'ciprocants rrdxtes: pour qu'on puisse additionner deux formes mixtes afin de former un nouveau reciprocant, il faut non seulement que le degr6 et le poids soient les memes pour tous les deux, mais aussi le caractire qui depend de la valeur de 6 et que Ton peut qualifier comme caractere pair ou impair selon la parite de 6. Ainsi, par exemple, Ith — 3a' et a? sent tous deux r^ciprocants, mais 2i6 ne le sera pas, parce que les caracteres des deux donnees sont contraires. II est facile de d^montrer que, si R est un reciprocant quelconque,
{2tb - 3a') BaR + {2tc - 4a6) Sji? + {2td - 5ac) S^iJ + . . .
sera aussi un reciprocant de mfeme caractere que R. Ainsi, en commen9ant avec le reciprocant a, on peut obtenir une suite infinie de reciprocants mixtes : ces reciprocants ainsi obtenus ne seront pas en general irreduc- tibles ; mais, sans les reduire, leur forme fait voir imm^diatement que tout
, reciprocant, qu'il soit pur ou mixte, peut etre exprim^ comme une fonction rationnelle et aussi (si Ton regarde t comme unit^) entifere de combinaisons
' legitimes* de ces quantity.
Pour obtenir tous les reciprocants purs de poids, degr^ et ordre (c'est-k- dire nombre de lettres) donnes, lin^airement independants les uns des autres, on peut former une Equation partielle differentielle, lin^aire, oii R est la variable d^pendante, et a, b, c, ... les variables ind^pendantes ; elle exprimera la condition n^cessaire et suffisante pour que R soit un tel reciprocant et foumira un moyen sftr de resoudre le probleme propose. Voici la maniere de demontrer ce theorfeme fondamental.
Si, dans I'equation
F{a.b,c....) = (-l)*t^F(a,ff,y,...),
on donne k y \& variation ea;, on voit que a, i, c, ...,et consequemment F, restent in variables. Les variations de a, /S, 7, ... sont faciles a determiner, et la variation de t est donnee.
Ainsi, apres quelques calculs faciles, en egalant k zero, separement, dans la variation de t^F{a, 0, ...), les termes qui contiennent t et ceux qui ne le contiennent pas, on arrive a deux equations dont I'une sera
Je nomme Ugitime nne combinaison qaelconqae de reciprocants oi Ton evite d' additionner eenx dont le poids, le degr^, I'ordre et le earactire ne sont pas les mSmes pour tous.
16—2
244 Sur une nouvelle theorie [36
qui exprime la valeur num^rique de X, comme fonction du poids et du degr^ de F; I'autre equation, eu ^crivant
F= Sa'Sft + lOaftS, + (15ac + 106') 8^
+ (21arf + 356c) S, + (28ae + 566d + SSc") S/+ .. .,
sera VR=0.
Pour voir la loi des chiffres arithmetiques dans V, formons les suites des coefficients de (1 + «)' en commen^ant avee i= 4 ; divisons chaque coefficient central en deux parties ^gales, et suppriiiions la derniere moiti^ des series num^riques ainsi forraees ; on obtiendra ainsi la Table :
14 3
|
1 |
5 |
10 |
|
|
1 |
6 |
15 |
10 |
|
1 |
7 |
21 |
35 |
|
1 |
8 |
28 |
5G 35 |
En n^gligeant les deux premieres colonnes, on trouve les nombres qui paraissent dans la formule.
On demontre ainsi que Fi? = 0 est une condition n^cessaire pour que R soit un reciprocant. Mais il faut aussi d^montrer que cette condition est suffisante. Soit done D la valeur de i^(a, 6, ...)— t"^^(a, yS, ...), exprim^e comme une fonction de a,h,c,... seulement. D sera done une fonction de la meme forme que F(a, 6, ...).
On suppose que AD = 0 ;
c'est-a-dire que la variation de D produite par la substitution dQ x+ ey k x est dgale a z(5ro, en vertu de I'^quation VR = 0.
Donnons k y une variation arbitraire y + rju; alors, si I) devient £>', la variation de D' sera nuUe, quand on substitue, pour as, x+ey + er)U, et, con- sdquemment, quand on substitue x -\-2y pour x ; on aura done
AD' = 0,
et, en prenant la difference des variations de D et D', on obtient
a(u'^ R + u"~R + u"'^R + ...)=0. \ da do ac I
Done, h. cause de la forme arbitraire de u, il faut que
^l;^-"' '^k"'"- ■■■■■
et, en raisonnant sur ^- B, -r. D, ... comme on a raisonn^ sur D, on voit que da do ^
le A de chacune des deriv^es secondes diff^rentielles de D sera z^ro; en
I
36] de formes algebriqnes 245
poursuivant le meme calcul, on trouve ^videmment que le A d'une d^riv^e de D d'un ordre quelconque par rapport a a, 6, c, ... sera nul.
Done D est nul ; car, dans le cas contraire, s'il contient un terme quel- conque, dent les lettres peuvent etre distinctes ou identiques, en isolant une seule de ces lettres et prenant la d^riv^e de D par rapport k toutes les autres lettres, on aura le A de la lettre isolee, c'est-a-dire de S^y, h^y,..., zero quand on substitue x + ey pour x, ce qui est absurde. Ainsi Ton voit que, quand AD = 0, c'est-^-dire quand VR =0, D = 0, ce qui 6tait a d^montrer.
Soient a>, i,j le poids, le degre et I'ordre d'un r^ciprocant quelconque : de meme que pour les sous-invariants, le nombre de fonnes lineairement inde- pendantes s'exprime par(a); i,j) — (w — 1; i,j), oh, en general, (to; i, j) signifie le nombre de partitions de a en i parties dont nuUe n'excede j; ainsi Ton voit que, en vertu de lequation VR = 0, on aura, pour le nombre des r^ciprocants lineairement independants, la formule
(w; t,j)- (<--!; i+l,j).
Mon long exil en Amerique expliquera, je I'espere, comment j'ai pu ignorer I'identitd des invariants differentiels de M. Halphen avec les formes que j'ai aomm^ea rSciprocants purs. Les travaux vraiment remarquables de M. Halphen n'ont pas besoin de mes ^loges et auront ^te couronnes par I'admiration de toua les gdometres dignes de ce nom.
Je crois cependant qu'il y a assez de difference entre le but et la marche de mes recherches sur ce terrain et ceux de M. Halphen pour justifier I'in- sertion dans les Comptes rendus de ma discussion de la th^orie regardee comme une theorie de formes algebriques. Si je ne me trompe pas, M. Halphen, s'il I'a d^couverte, n'a fait nul usage de I'equation partielle difffirentielle que j'ai donnee et qui sert h. dtablir le parall^lisme mer- veilleux entre les invariants differentiels et les semi-invariants ordinaires.
De plus, il n'a pas eu occasion de faire allusion atix formes que j'appelle rddprocants mixtes orthogonaiuc, qui ne sont point compris dans la definition des invariants differentiels, et qui sont essentiels pour expliquer les singu- larit^s quasi-metriques des courbes.
Nous rappelons que par le mot redprocant (sans qualification) il a ^te convenu de sous-entendre une forme de cette espfece qui ne contient pas t Ic'eat-k-dire j^j et nous avons trouv^ que le nombre de ces reciprocants
lineairement independants, du degr^ i, de I'etendue j (c'est-k-dire contenant j+1 lettres distinctes) et du poids <», s'exprime par la formule
(«; hjy-ioi- 1 ; i + l,j), oh en general {I; m, n) signifie le nombre de partitions de i en m ou un plus
246 Sur une nouvelle thdorie [36
petit Dombre que m de parties dont aucune n'excede n en grandeur ; de sorte que (l; m, n), quand m est plus grand que I, signifie la m^me chose que {I; I, n), car tous les deux sont equivalents ^ (/; oo , n). Cons^quemment
(i; i, i) - (i - 1 ; i + 1, j) = (i; i, j) - (i - 1 ; i. j). lequel sera toujours positif quand i et j sont tous les deux plus grands que I'unit^ ; et, puisque a, qui est du degr^ 1, est un r^ciprocant, il s'ensuit que, pour un degre quelconque donn^, il existe toujours des reciprocants (car on pent faire a> = i), mais en nombre fini, car, en faisant croitre to, (oi — 1 ; t'+l, oo), au dela d'une certaine valeur de to, deviendra n^cessairement plus grand que (w; i, x ). On peut exprimer par (I : m) ce que devient (/; m, n) quand n = oo , et alors (« : t) — (w — 1 : i + 1) exprimera le nombre de reciprocants lin^airement ind^pendants du poids « et du degr^ i sans autre limitation. Ainsi on trouvera que du degr^ 1 il n'existe qu'un seul r^cipro- cant du poids 0; pour le degr6 2, un seul du poids 2 ; pour le degr^ 3, deux qui seront respectivement du poids 3 et du poids 4 ; etc.
On trouvera qu'etant donnd _;' il existe toujours, sauf pour le cas ou j = 1, un r^ciprocant qui contient toutes les j +\ lettres et qui de plus contiendra un terme qui est un produit de la derniere lettre par une puissance de a. Ces formes, qu'on peut nommer les protomorphes, sont les analogues des formes a, ac — b', a'd + ..., ae + ..., qu'on connait dans la th^orie des sous- invariants. Dans le cas des reciprocants, ces protomorphes seront a, ac, ..., a'd a'e, .... a'/, ..., a'g, ..., etc.
Evidemment une fonction rationnelle quelconque des lettres peut, au moyen de substitutions successives, Stre exprim^e comme une fonction rationnelle des protomorphes et de h divis^e par une puissance de a. Soit done R un rdciprocant quelconque ; on aura
a»i2 + P+Q6+... + Ji* = 0, ou P, Q, ..., J" sont eux-memes des reciprocants. En operant i fois sur cette Equation avec notre op^rateur V, on voit qu'on obtient a^J = 0 ; done J est nul, et Ton voit ainsi que tous les termes Q, ..., J disparaissent et que R (en faisant a=l) devient une fonction rationnelle et entifere des protomorphes. Nous allons appiiquer ce principe fondamental, commun aux deux theories des sous-invariants et des reciprocants, pour obtenir les formes irr^ductibles (les Grundformen) des reciprocants pour les ordres 2, 3, 4.
Faisons j = 2,i = 2, a = 2 et supposons que le r^ciprocant R soit Xac + ^; on obtient
VR = (Sa^Bb + lOabBc) R = {6fj.+ lOX) a'b = 0.
Done — \ : /t :: 3 : 5 et nous obtenons le r^ciprocant Sac — 5b'*.
* II est bon de remarquer que Sac - 56^ = 0, c'est-A-dire
3j|g_5(gy=0, ax' ax* \ax'J
indique que le point (x, y), quand cette Equation est satisfaite par telles coordonn^s d'nne courbe
quelconque, est un point supra-parabolique, c'est-i-dire oH une parabole passe par 5 au lieu de 4
points cons^cutifs seulement.
I
36] de foj'nies algebriques 247
PassoDs au cas _;' = 3, i = 3, o) = 3, et posons
R = Xa'd + jMihc + vb\ On aura VR = {^a'hi + 10a68c + 15ac + 106^8^) R
= (3/i + 15\) a^c + (9«' + 10/x + lOX) a-6= = 0. On aura done A' = — 5\, Oj^ = 40X,
de sorte qu'on peut ecrire
R = 9a^d - 45a6c + 406'.
On reconnaitra immediatement que R = 0 est I'^quation diff^rentielle donn^e par Monge et retrouvee par M. Halphen a une conique et que
9 (B^'yf (S^'y) - VoK'yh^'yK*y + 40 {K^y? = 0 exprime la condition que le point {x, y) d'une courbe quelconque sera nn point d'inflexion du second ordre, c'est-a-dire un point oil une conique passe par six points cons^cutifs. Le nombre de ces points peut etre trouv^ en fonction lin^aire de n, ordre d'une courbe donnee, en operant sur cette Equation une transformation analogue k celle au moyen de laquelle on
passe du systfeme y = 0, -j^ =0 au systeme Equivalent, mais dpure, 0 = 0;
^0 = 0*.
Passons au cas ou j = 4, i = 3, m = 4, et ^rivons
R = Xa'e + /xabd + vac' + ■rrb'c. On aura
F=3a=86 + 10a6S, + (15ac + 106")8d + (21ad + 3o6c)S.,
et, en posant RV=0, on obtient, en Egalant s^par^ment k z6to les coefficients de a'd, d'bc, cib', les Equations
21\ + 3m=0, 35\+15/i. + 20i/ + 67r=0, 10/i4-107r = 0,
• Pour le caa d'une cubiqae, le nombre de ces points d'inflexion du second ordre est vingt-sept; on d^montre facilement que ce sont les intersections de la courbe avec son covariant du degri- ordre 12 . 9.
On voit immediatement, an moyen de notre thtorie connne de riiddus giomitriques, que cei vingt-sept points sont les points de la cubique oil elle est rencontr^e par les nenf faisceanx des tangentea qu'on peut mener des nenf points d'inflexion ordinaire. Car un quelconque de ces points doit ^tre tel que sa d^riv6e k I'indice 5 sera coincidente avec le point lui-m6me. On aura done 1, 1 = 1,5, c'est-i-dire 2 = 4, ce qui veut dire que le tangentiel du point est un point d'inflexion ; ce qui ^tait k d^mootrer.
Soit dit, par parentb^se, que la m6me th^orie de r^siduation enseigne que le point fixe Q, oil nne cubiqae donnie sera coup^ par une autre cubique quelconque qui a en coramun avec la premiere 8 points consecutifs k nn point donn^ P, sera le troisi^me tangentiel de P et pent fitre nornm^ son tateUite ; quand le satellite coincide avee son primaire, en se servant pour le moment de la forme canoniqne pour exprimer la cubiqae donnee, et en nommant x, y, z les coordonn^es du primaire, cellea da satellite seront (d'apr^s notre th^orie exposie dans \' American Journal of Malhematict) x, y, z multiplies respectivement par des fonctions rationnelles de x', y', «', cbacune du degr6 21. [Vol. ni. of this Reprint, p. 339.]
C'est nn fait depnis longtemps connu que les points primaires qui coincident avec leurs satellites (en ne tenant pas compte des neuf inflexions) sont en nombre 72.
248 Sur une nouvelle thSorie [36
et ainsi on peut ^rire
R = 5a«e - 'S5abd + lac* + 356'c.
Voici done le systfeme de protomorphes pour tous les ordres jusqu'au quatrieme inchisivement :
«, (1)
3ac - 66», (2)
9a'd-45a6c + 406', (3)
5o'e-35a6d+7ac' + 356''c. (4) En corabinant le cube du deuxieme avec le carr^ du troisieme, et en
divisant par a, on obtient la forme (analogue au discriminant) de la cu- bique, mais d'un degre plus ^levd,
405a'd» - 4050a»6cd + 1728a»c»
+ 1585a6'cr' + 3600a6'rf-180006«c*. f ^^^
En combinant le produit de (2) et de (4), lineairement, avec (5), on obtient
4800a'ce - 8000a6^e - 2835a^d= - 5376ac' ) ,„.
- o'IbOabcd + 308()06'd + \\^Qo¥c\ j ^ '
Si Ton se borne aux lettres a, b, c, d, les formes (1), (2), (3), (5) formeront un systfeme complet de Gh~und/ormen : si on laisse entrer la nouvelle lettre e, (5) n'est plus irr^ductible, et le systeme complet de Grundformen est con- stitud par les formes (1), (2), (3), (4), (6).
Tout cela se passe precisf^ment com me avec les sous-invariants avec les memes lettres : les poids des formes sont les memes pour les deux systfemes, et la seule diffi^rence essentielle entre les deux consiste en ce fait, que les trois derni^res formes subissent chacune une Elevation d'une unit6 de degr^ en passant du systeme des sous-invariants a celui des reciprocants.
II est n^cessaire d'ajouter quelques mots sur les reciprocants mixtes, qui se distinguent en deux espfeces, homog^nes et h^t^rogenes. Comme exemple des premiers, on a la d^riv^e Schwarzienne 2tb — Sa^, laquelle, ^gal^e a z^ro, ne dontie aucune espfece de singularity, mais signifie seulement qu'au point (x, y) on peut mener une conique qui passera par cinq points consecutifs, en ayant ses deux asymptotes paralleles aux axes, ou bien la forme tc — bah. Comme exemple de I'autre classe, on a la forme counue (1 + t^)b — 3to^ dont I'evanouissement (pourvu que x, y soient des coordonn^es rectangulaires) signifie que le point {x, y) est un point de courbure maximum ou minimum.
* Cette fonction, 6gal^e k z^ro, exprime que x, y sont les coordonn^es d'un point par od I'on peut {aire passer une parabole cubique ayant 5 points consecutifs communs k la coorbe dont x, y sont les coordonn6es.
I
36] de formes algebriques 249
Nous avons remarqud, par parenthese, que lequation
{\ + t')b-Ua? = Q
indique I'existence d'une singularite au point dont les coordonn^es sont les X, y sous-entendus dans t, a, b de I'^quation.
Mais, pour que cela soit vrai, 11 faut introduire la restriction que x, y sont des coordonnees rectangulaires.
On peut donner le nom de rAxiprocant orthogonal k tout reciprocant mixte qui jouit de la propri^te de rester invariable (sauf I'introduction d'une puis- sance de i) quaud on opfere sur x et y une transformation lineaire ortho- gonale. Cela etant convenu, on peut demoatrer t'aeileiuent que le coefficient difFerentiel par rapport a t d'un reciprocant est lui-meme un reciprocant ou pur ou mixte. La proposition reciproque est aussi vraie, de sorte qu'on a le , beau theoreme suivant :
Si R et -J- sont tons les deux reciprocants, alors R est un reciprocant ortho- gonal.
Par exemple, le reciprocant que nous avons cit^ plus haut a pour co- efficient difFerentiel par rapport a i la Schwarzienne 2tb — 3a' ; done c'est un reciprocant orthogonal ; et, en effet, il exprime qu'au point (x, y), ou I'^quation 2tb — 3a= = 0 est satisfaite, on peut appliquer un cercle qui aura nn contact du troisieme ordre avec la courbe dont xety sont les coordonnees; au contraire, la Schwarzienne elle-nieme ne correspond pas a une singularity quelconque, car sa derivee par rapport a t, c'est-a-dire 26, n'est pas un reciprocant.
J)e raSme nous avons trouv6 qu'en integrant le reciprocant 2tc — lOab par rapport a t, entre les limites t et — c — 1.5a', la forme resultante
{t''+l)c-l0abt + l5a'
sera un reciprocant et consequemment un reciprocant orthogonal, de sorte que I'equation
(1 +t^)c-lOabt+loa'=0
sera la condition d'une singularity de la courhe f(y, x) =0 qui se rapporte aux points circulaires k I'infini*. Peut-Stre trouvera-t-on que I'int^grale, par rapport k t, d'un reciprocant mixte quelconque, prise entre des limites convenablea, conduira necessairement k un reciprocant orthogonal. Les singularites d'une courbe peuvent etre partagees en trois classes : celles de la premifere classe seront projectives et peuvent etre d^finies indifferemment au moyen de covariants de formes ternaires ou par des reciprocants purs ;
* M. James Hammond, dont on oonnalt les belles et importantes dcSeouvertes dans la th^orie invariantive des formes binaires, a trouvi I'int^grale de cette Equation, que nous avons donnie dans un discours inaugural, prononcfi devant I'Unirersit^ d'Oxford, lequel va 4tre public dans le joamal anglais Nature, [p. 278 below.]
250 Sur une nouvdle thiorie [36
celles de la deuxi^me claase seront non projectives, mais n'aurout affaire qu'avec la ligne h. I'infini ; les singularit^s de cette classe seront exprimables au moyen de r^ciprocants purs, mais non pas au moyen de covariants de formes temaires. Restent celles de la troisifeme classe qui non seulement ne sont pas projectives, inais sont quasi m^triques en caractere, c'est-a-dire ont des rapports avec les points circulaires a I'infini ; les singularites de cette classe sont signalees par I'^vanouissement de reciprocants orthogonaux. Les reciprocants mixtes, qui ne sont ni purs ni orthogonaux, comme celui, par exemple, de M. Schwarz, ne r^pondront a aucime de ces trois espfeces de singularites ; mais, quoique ne servant pas a repr^senter une propriety invariable d'une courbe, ils serviront souvent, peut-dtre toujours, comme bases des reciprocants orthogonaux, c'e8t-&,-dire qu'ils seront les coefficients diff^rentieis par rapport a < de ces derniers.
L'echelle des protomorphes, aussi bien dans la th^orie des reciprocants purs que dans celle des sous-invariants, joue un role si capital, en ce qui conceme la determination des formes irreductibles, qu'il nous semble in- dispensable de donner une demonstration rigoureuse de son existence dans I'une et I'autre theorie.
1° Quant aux sous-invariants, soit j I'ordre (c'est-a-dire j + 1 le nombre des lettres que Ton considere). Si j est pair, on connait les formes inva- rianti ves ac+ ..., ae+ ..., ag + ..., et Ton pent passer au cas ou j est impair. Dans ce cas, le nombre de sous-invariants du poids_; et du degre 3 sera
0';3,i)-(i-l;3,;).
Mais il faut demontrer qu'il existe une forme de ce type, dans laquelle le coefficient du produit de a' et de la derniere lettre n'est pas nul.
Or je dis que le nombre des formes du type suppose, qui ne contiennent pas cette lettre, sera
(j;S,j-l)-(j-l;S,j-l).
Mais 0'-l;3,i) = 0-l;3,i-l)
et, evidemment, {j; 3, j) - (j; 3, j - 1) = 1 ;
car les partitions dont le nombre est (j; 3, _;') contiendront toutes les partitions dont le nombre est (j; S,j — 1) et en plus la partition constituee par j com- bine avec des z^ros.
Consequemment il existe un sous-invariant dont un terme sera le produit de a" par la derniere des lettres que Ton considere.
2° Quant aux reciprocants purs de I'ordre j, nous avons deja demontr^ qu'on pent satisfaire a I'inegalite
(j;*,j)-(i-i;a;-i-i,i)>o
en donnant k x une certaine valeur pas plus grande que j —\\ et, pour de- montrer qu'il y aura un reciprocant pur qui contient actuellement un terme
I
36] de formes algebriques 251
a^~' multiplie par la derniere lettre, on pourrait faire pr^cisement le meme raisonnement que nous avons fait ci-dessus pour le cas precedent, et, puisque
(j;a^.j)-0"-i;« + i.i)
exeede de I'unite la valeur de {j\ x,j — 1) — {j; x + \, j — 1), on conclura avec certitude I'existence d'un protomorphe pour I'ordre j.
On pent, en general, trouver plusieurs valeurs de x qui rendent 0; *. i) — (i — 1 ; ^ + 1. i) positif; parmi ces valeurs, il est commode d'adopter, comme protomorphe par excellence, une quelconque de celles pour lesquelles la valeur de x qui satisfait a cette inegalite est un mini- mum. Quand la lettre la plus avanc^e est inf^rieure a h, il n'y en a qu'un seul qui reponde a cette definition. Ainsi, par exeraple, si _; = 5, I'in^galite
(5:a;)-(4 :a; + l) >1 donne pour x la valeur minimum a; = 4 et, avec I'aide de I'aneantisseur
Sa'Sj +10a68e + (15ac + 106') S^ + (21ad + 356c) 8, + (28ae + h<ohd + 35c=) 8/, on obtient le protomorphe
45a'/- 420a'6e - 42a=cd + 1120a6'd- 315a6c= - 11206" c. Cela servira pour conduire a la connaissance de tous les reciprocants purs de Tordre 5, dont le nombre sera au moins 6gal k celui des Grundformen du quantic binaire.
Dans une Communication qui suivra .celle-ci, nous nous proposons de donner la theorie des reciprocants doubles ou multiples dont ceux de I'espece pure sont prdcis^ment analogues aux invariants ou sous-invariants de syst^mes de formes binaires.
La theorie des doubles reciprocants purs comprend n^cessairement, comma cas particulier, I'dtude des formes qui d^terminent la position des tangentes communes k deux courbes et les points bitangeutiels d'une seule.
Dans la remarque que nous avons faite, dans la premiere Note, sur le meme sujet que la Note actuelle, k propos des reciprocants mixtes de la forme
[(2tb - 3a') Ba + (2te - 4a6) Sj + (2td - 5ac) S„ +...]' a,
nous avons affirme que tout reciprocant pur ou mixte pent etre exprim^ en fonction rationnelle et, de plus (quand on fait t 4ga,\ k I'unitd), entiere de re- ciprocants de cette famille ; nous n'avons pas limite, comme nous aurions dft le faire, cette affirmation au cas de reciprocants homogenes : la proposition a besoin d'une certaine modification si on veut la rendre applicable au cas de reciprocants non homogfenes; mais nous ne croyons pas necessaire d'y insister en ce moment. Seulement, il est bon de remarquer que I'existence d'une equation partielle differentielle lineaire, que nous avons trouvde pour les reciprocants purs, suffit k etablir immediatement que ces reciprocants seront necessairement, et sans exception aucune, ou homogenes ou separables en parties homogenes, dont chacune sera elle-meme un reciprocant.
37.
NOTE ON SCHWARZIAN DERIVATIVES. [Messenger of Mathematics, xv. (1886), pp. 74 — 76.]
Reading with great pleasure and profit Mr Forsyth's masterly treatise on Differential Equations (in my opinion the best written mathematical book extant in the English language), it occurred to me to find an easy proof of the fundamental and striking identity concerning Schwarzian derivatives, from
(dz\' -J- I (y,z),
where one of which is, it may be observed, that (y, x) like y" has the property of remaining a factor of what it becomes when x and y are interchanged ; a persistent factor, so to say, of its altered self. I will return to this point subse- quently, my present concern is to give a natural proof of the above striking identity ; to do this, it will be sufficient to show that (considering y, z, x, the two former as fixed, and the last as a variable function of a common variable)
I ifj '}?) — (2 ^\
- , ' — does not vary when x becomes x + e<f> (x) where e may be
regarded as infinitesimal*. For then this must remain true by successive accumulation when x becomes any function whatever of itself, and accordingly making x = z we obtain (y, z) as the value of the invariable quotient as was to be shown. Calif eh<^x<= ^, then using dashes to denote differentiation qud X, and a parenthesis to signify the augmented value of the derivatives, we obtain
iy') = y'-ey', il
(y") = y"-26y"-0'y',
{y"') = y"'-'3ey"'-2e'y"-e"y'. |
• It is easy to see It priori that if the theorem is true, it can only be so in virtue of {y, x) when X receives an infinitesimal, becoming of the form
{l-2$){y, x) + \e", as is subsequently shown to be the case in the text.
[+ Cf. p. 306 below.]
}
37]
Note oil Schwarzian Derivatives
253
Hence
and Hence
i^y'") = y'y'" - Wy" - ^&y'y" - &'y'\ f(y'") = iy'"-6%"'-80y/,
{{y, a;)) = (1 + -26) {{y, x) -^0 (y, x)} - 6"
= {\-W){2i,x)-e",
((y, x) - {z. x)) = (1 - 20) [{y, x) - (z, x)]. r{y^c)-(z^\ _ (y, x) - (z, x)
Ay, ^) - {^' ^)\
\dx) J [dij
that is, the right-hand expression does not change, when y, z remaining fixed forms of function, x passes from one form of function of the independent variable to another ; as was to be shown.
From what precedes, it appears that if y,z,x be regarded as functions of <^
(dxY -J- J is a constant function in the sense that it remains
unaltered, whatever function x may be of t, or which is the same thing if y and z functions of a- when expressed as functions of a;' (any function of x) are written y', z , then {y , x) — {z', x') is identical with (y, x) — (z, x), save as to a factor which depends only on the form of the substitution of x for x. Hence to all intents and purposes, any function of the differences of the Schwarzian derivatives of any system of functions of the same variable, in respect thereto, is (in a sense comprising, but infinitely transcending the sense in which that word is used in Algebra) a covariant of the system.
Addendum. — Let us for the moment call functions of x, y which either
remain unaltered or only change their sign when x and y are interchanged
self-reciprocating functions.
// ' /// //»
The first case of the kind is — , , the next is '-^ — ,, '^ , and obviously a
y'\ y'
very general one of this sort will be the function
For greater simplicity, let us call the numerator of any such function when expanded and brought to the lowest possible common denominator, a reciprocant, the highest index of differentiation wliich such recipiocant contains its order, and the number of factors in each term its degree. Then in any reciprocant so formed the degree is always just one unit less than the order: but as a matter of fact the function so obtained is in general not irreducible, so that its degree may be depressed, and it becomes a question of much interest to form the scale of degrees of reciprocants of this sort. For the
254 Note on Schwarzian Derivatives [37
orders 2, 3, 4, 5, 6 the degrees in question are respectively 1, 2, 2, 3, 3. Calling the successive derivatives of y, a, b, c, d, ..., they will be found to be
a,
2ac - 36=,
ad — bbc, 2a'e- loacd - lOad' + 356'c, 20"/- 2 laic - Soacd + GOab'd + 1106c',
where each form is obtained by operating upon the preceding one with the operator a (bBc + cSd + dSe + ...)—\b{\ meaning half the weight + the degree of the operand), combining the result of this operation in each alternate case with a legitimate combination of those that precede, and in that case dividing out by a. I have proved that in this way can be obtained an infinite pro- gression of reciprocants, of which the leading terms (substituting numbers for letters), will be alternately of the forms lV(2i + l) and l\(2i+2). Every other reciprocant can be formed algebraically from these primordial forms, as every seminvariant can be obtained from the primordial forms a, ac — b',
a'd — 3a6c + 26', The two theories run in parallel courses, but their
relationship is that which naturalists call homoplazy as distinguished from honiogeny; I propose to give further developments of this new algebraical theory in a subsequent Note.
M
38.
ON RECIPROCANTS. [Messenger of Mathematics, xv. (1886), pp. 88 — 92.]
In a note on Invariant Derivatives in the September number of the Messenger I have given a definition and examples of reciprocants.
If in any of the forms at the end of the postscript to the note we restore to a, b, c, ... their values Sxy, Bx'y, S/y, ... any such function divided by a certain power of S^y will change its sign, but otherwise remain unaltered when X and y are i iterchanged. The index of that power is the degree added to half the weight and will be called the index of the reciprocant. Any product of i of such reciprocants will be a reciprocant of the same kind or contrary kind to those in the table (subsequent to a) according as i is odd or even. In the latter case the interchange of x and y will leave the function absolutely unaltered. Reciprocants which cause a change of sign will be said to be of an odd, those which cause no change of sign of an even character. Any linear function of reciprocants of the same weight, degree, and charactei' will be itself a reciprocant of that character, but reciprocants of opposite characters cannot be combined to form a new reciprocant : those of an odd character may be regarded as analogous to skew, those of an even character to non-skew seminvariants ; the rule against combining forms of opposite characters becomes superfluous in the case of seminvariants, because those that offer themselves for combination as having the same weight and degree must of neces.sity be of like character. Any reciprocant being given there is a simple ex post facto rule for assigning its character without any knowledge of the mode of its genesis, namely its character is odd or even according as the smallest number of letters other than a in any of its terms is odd or even. Thus the character of a reciprocant whose leading term is a'e, or ab'e, or abce is odd ; that of one whose leading term is abe or ahf is even, as is also that of the remarkable reciprocant hd—o(? in which no power of a appears.
A further important distinction between the two theories* is that there are two linear reciprocants a and 6 but only one linear seminvariant. As an illustration of the combinatorial law of like character it will be seen that if we operate upon 2ac — 36^ with the operator
a (bBa + cSb) - 3b, * That is of reciprocants and invariants.
256 On Reciprocants [38
we obtain a new reciprocant
2ad-106c + 96',
of which the character is the same as that of 6', namely both are odd ; we may therefore add — 96" to the latter expression, and then dividing out by 2a there results the reciprocant ad — 56c, but we cannot combine 2ac — 36' with 6' because these two reciprocants are of opposite characters.
Again, remembering that a is of an even and b of an odd character, the three reciprocants
- i^\ 5 {ac - f 6')', 3a6 (ad - 56c)
are all of an even order, hence we may add them together and divide the sura by a", which gives the new reciprocant Sbd — 5c' a form not containing the first letter a.
No seminvariant exists, nor, except the one just given bd — oc", have I been able to discover any other reciprocant in which the first letter does not make its appearance "f.
The infinite progression of odd reciprocants with the leading terms ac, ad, a.a.e, a.a.f, a.a.a.g, a.a.a.h, ...
will easily be seen to exist by virtue of the general theorem that any reci- procant of degree, extent, and weight (say brietly of dew i, j, w) gives birth to two others of the same character as its own, one of dew i+l,j + l, w+ 2, the other of dew { + 1, j + 2, w + 3.
For let ^w-ti = \,
then denoting the operator
bSa + cSb+... by n, and the result of the action of D, upon itself (n*)^ which is in fact Ct' + fia (lij meaning c8„ + d8j+ ...) ; (ali — \6)ii will obviously be a reciprocant of dew i+l,j + l,w + 2, and will give rise to a second reciprocant
{an-{\ + ^)b}{an-Xb)R, which is a' (n*)' - (2X, + i) abUR - XacR + {X^ + |X) b^R;
the last term of this being a reciprocant of the same character as the entire expression may be omitted, and dividing out the residue by a we obtain the
second new reciprocant
(a(n*)''-(2X+i)6-Xc]ii,
which will be of dew i + 1, j + 2, w + 3, as was to be shown.
It is easy to see that every reciprocant must be a rational integral
function of the forms above stated commencing with a, 6, 2ac — 36^ (whose
dew's are alternately of the form i, 2i — 1, 3i — 2 ; i, i — 2, 3t — 1) divided by
some power of a.' For if any reciprocant contains only the letters a, 6, ...
t Since the above went to press I have made the capital discovery that there are an infinite number of such reciprocants, and that all those of a given weight, extent and degree may be obtained by aid of a certain quadratico-Uuear partial differential equation.
38] On Reciprocants 257
h,k, I, it may be expressed as a rational integral function of the protomorph in which I first appears and of the letters «, 6, ...k divided by a power of a, and consequently the reciprocant may be so expressed, and continually repeating this process of substitution it follows that the reciprocant will be a rational integer of the protomorphs exclusively divided by a power of a*: this of course will necessarily be tound only to contain combinations of like character; we already know the converse that the sum of all combinations of like character of the protomorphs is a reciprocant"^. If any homogeneous reciprocant consists of portions of unlike degree (although of the same index) it is obvious that each portion must be itself a reciprocant, for if P, P', P" ... be such portions, P + P' + P" .. . must be identical with 11 -t- II' + 11" + ... when n, n', n" ... are the same functions of a, /3, 7 ... (that is, S^a;, Sy'a;, S^'a; ...) \ that P, P', P" ... are of a, 6, c If then we make
p-a■^Ii=^, P'-o^n' = A'...,
Pwe have A + A' + A" -> ... identically zero.
But P, P" ... being of the same index but diflFerent degrees must be of [diflferent weights, and consequently A, A',... are of different weights. Hence I we must have A = 0, A' = 0, &c., as was to be shown.
It follows from this that every reciprocating function whatever may be [obtained by an algebraical combination of the protomorphs, and consequently by an algebraical combination of the forms
(^S^Jlogy,
* The proof that every seminvariant is a rational integral function of the protomorphs is very similar : any proposed seminvariant is by the method employed in the text shown to be at worst a function of the protomorphs and of b ; but the terms involving any power of b must disappear becaase no identical equation can connect seminvariants with a non-seminvariant b. In the text we see in like manner that any given reciprocant may be reduced to the form H + K, where H and K are protomorphic combinations of opposite character, so that one of them will disappear.
t Another general mode of generating a class of reciprocants would be to express any function of o,6, c, ... say 0(a, 6, c, .,.) under the form ^(a, j3, 7, ...). The product 0 (a, 6,c, .,.)^(a, 6, c,.,.), or its numerator, will then obviously be a reciprocant. To take a simple example.
H
d»i/_ dy_dy^_%l£l_ '-d^- AJTT =-«7 + S
Hence, by the rule laid down, c(ac-SV), that is, ac''-36*c ought to be a reciprocant, which U right, for it is equal to {2ac - 362)» - 9li* divided by a multiple of a. The law that the factors of neminvariants must be seminvariants cannot be extended to the theory of reciprocants. In this case the factors may some or none of them be reciprocants, and the others on reciprocation exchabge forms monocyclioally or polycyclically with one another. / add the remark that this is
not true of pure reciprocants, that is, those in which -^ does not appear. Every factor of a pure reciprocant must be itself a reciprocant.
8. IV. 17
258 On Reciprocanta [38
and that we should gain nothing in generality by operating with successive operators of the form
'^7*^''^'')' (74^'*'')'-
where ^1, ^j, ... are arbitrary functions of y' + —, instead of with the simple
V
operator —= B^ continually repeated.
The results of using the more general operators would only amount to algebraical combinations of the results obtained from the simple forms
(^jS^yiogy,
where i may take all values from zero to infinity*.
As in the case of seminvariants so also reciprocants would in extent contain only a finite number of ground-forms ; but furthermore for re- ciprocants limited in degree the number of ground-forms will also be finite. Whether reciprocants which are irreducible for a given extent ever cease to be so and become reducible when the order is increased, as is the case with seminvariants, remains to be seenf.
In order to facilitate the verification of the results obtained and to be obtained it may be well to express the successive derivatives of a; in regard to y in terms of those of y in regard to x, that is, of o, y8, 7, ... in terms of a,b,c, ... as shown in the following short table.
a = a a",
b = -^ ^.
c = -ay{- 3;8' a',
d=-a^S-fl0a/37- 1.5/3' oJ,
e = - a»6 + 15a=/SS + lOa'^^" - 105a^^7 + 105/3* a',
/ = _ ar"^ + 21a»/3e -I- S5ac>yh - 210a'^/3--8 - 2800^,87= + 1260a/3'7 - 945/3" a",
g = -a.*7) + 28a*/3f -1- 56a'76 + 35a*8^ - SlSa'^-e - 1260a»^78 + 31.50a=^»8| „
- 280a»y -I- 6300a''/S^7= - 17325a/3*7 -I- 10395^ J " '
where a, b, c, d, e,f, ... represent the successive derivatives of y with respect to X ; and a, /3, 7, S, e, f, ... of a; with respect to y.
In any subsequent paper on reciprocants in this Journal, I shall make the absolutely necessary transliteration referred to in a preceding footnote, re- placing the present letters a, b, c, d, ... by the letters t, a,b,c,... or possibly, for reasons which carry great weight, by the expressions
t, 2a, 2.36, 2.3.4c,...
* This is not true of homogeneous reciprocants.
+ I have since found that this is true for reciprocants, as for seminvariants.
39.
NOTE ON CERTAIN ELEMENTARY GEOMETRICAL NOTIONS AND DETERMINATIONS.
[Proceedings of the London Mathematical Society, xvi. (1885),
pp. 201—215.]
A CURVE, as every one knows, may be regarded as a locus of points or as
%u assembly of directions, every point being common to two consecutive
irections of the assembly, and every direction to two consecutive points of
[the locus ; the locus is called the envelop of the a.ssembly (that is part of the
ccepted language of geometry), and, conversely, the assembly may be called
the environment of the locus. So we may regard a surface as an assembly of
mgent planes or as a locus of points standing to each other in the relation
of envelop and environment, and extend these definitions to apace of any
number of dimensions.
By a plasm, waiting a better word, we may understand a figure analogous to a point-pair in a line, a triangle in a plane, a pyramid in space, etc.; and an n-gonal plasm or n-gon will signify a plasm having n vertices and n faces themselves {n— l)-gons.
It is easy and desirable to find the general value of the content of a regular »-gon, say abcde, all whose edges we may call unity.
If h^ = \ab, C'i = lcp, dS = idy...,
it is easily seen by an elementary process of integration that fi,y, B ... are the centres of figure to the successive plasms ab, abc, abed, ..., and, making
ba=pi, c^=p.„ dy=p,...,
each term in p,, /)j, p, ... will be perpendicular to the one which precedes it, 80 that, if F„ is the content of the plasm,
(1, 2, 3 ...n)' Vn=PiPi...pn'
17—2
260 Note on Certain Elementary [39
Moreover, we shall have
of which the general integral is
in the present case, since j^i = 1, (7=0, so that
tr,_ '» + !
(1.2...7i)»2'»
If a, b, c be the angles of a fixed triangle, and A, B, C are proportional to the distances of a variable line from a, b, c, respectively, we may denote the line hy A: B:C; as regards a variable point, it will presently be seen to be advantageous to denote its proportional coordinates, not, as is rather more usually done, by equimultiples of its distances from the three sides, but as equimultiples of these distances multiplied by the sides of the triangle from which they are measured*; so that, calling these coordinates a, b, c, the image t of the line at infinity becomes a + 6 + c.
Consider now the universal mixed concomitant (which it will be convenient to call a mutuant) Aa + Bb + Cc (where a, b,c,A,B,G are used in lieu of the more usual letters x, y, z, f, 17, 0; it will readily be seen that, when a, 6, c vary, and A, B, C are fixed, the mutuant images the line ^ : J8 : C, and that, when A, B, C vary and a, b, c are fixed, the mutuant images the radiant point a:b:c; that is to say, Aa + Bb + Cc=0 is true for every point in the point- containing line A: B:C in the one case, and to every line through the radiant point a : 6 : c in the other.
Supposing, then, that the two kinds of coordinates are chosen in this manner, we see (what would not be the case if the simple distances were taken) that a form F and its " polar-reciprocal " (/> image the self-same curve referred to the self-same fundamental triangle.
These consequences would moreover continue to subsist if, calling the distances of a line from the vertices P, Q, R, and of a point from the sides p, q, r, we took AP -.MQ-.NR.Xp-.nq.vr for the two sets of coordinates, provided only that XAP = /iM G = vNZf ; F, G, H being the distances of the sides from the vertices of the fundamental triangle, in which case the line at infinity would no longer be imaged by a -I- 6 -t- c. I shall, however, adhere in what follows to the convention above laid down. I need hardly add that in like manner, in space taking A:B:C:D (the distances of a plane from the
* Or rather divided by the distances of these sides from the opposite angles of the funda- mental triangle, whose coordinates thus become 1, 0, 0, 0, 1, 0, 0, 0, 1.
t If F=0 is the equation to any locus or atsembly, I call J^ the image, and such locus or assembly the object; to a given image responds in general an absolutely definite object, but, when the object is given, the image is only determined to a constant factor pr'es.
I
39]
Geometrical Notions and Determinations
261
vertices of a fundamental pyramid) as the coordinate-representation of such plane, and a:b:c:d (the contents of the volumes which any variable point makes with the respective faces) as the coordinate-representation of such point, the mutuant aA+bB + cC+dD will be the image of the radiant point a:b:c:d when the capital letters are the variables, and of the plane A : B :C : D when the small letters are the variables, meaning of course that Aa + Bb+ Cc+ Dd = 0 will be true of every point in the plane A: B.C-.D and of every plane through the point a:b:c:d, and, as before, F and ^ polar- reciprocals to each other will image the self-same surface (referred to the self-same fundamental pyramid) viewed as a locus or envelop on the one hand, as an assembly or environment on the other.
If a, b, c, d be used to signify the actual as distinguished from the pro- portional coordinates of a point, a linear function of these is constant, whereas it is a quadratic function o{ A, B, C, D ..., when used to signify the actual distances of a variable line, plane, &c., from the vertices of the funda- mental plasm which is constant ; and it is the principal object of this note to determine the form of this quadratic function, which, as Prof Cayley was the first to show, may be expressed by the determinant to a matrix standing in close relation to the well-known "invertebrate symmetrical matrix," the determinant to which represents a numerical multiple of any plasm in terms of its edges, as, for example :
ab ac ad 1
ba . be bd 1
ca cb . cd 1
da db dc .1
1111.
where ab, ac, be... are used for brevity to signify the measure of absolute distance between a,b, a,c, b,c ..., that is, stand for what in ordinary notation
would be denoted by (abf, (ac)', (bcf This may be quoted as the
mutual-distance matrix ; its determinant, besides representing a numerical multiplier of the squared content of the pyramid when equated to zero, expresses the conditions of the four points a, b, c, d lying in a plane, the former property being a consequence immediately deducible by strict alge- braical reasoning from the latter.
That this determinant does image the condition of the plasm to which the points a, b, c, d ... are the vertices, losing one dimension of space, may be shown in a somewhat striking manner as follows. If for a moment we use X, y, z, the distances of any point in the plane of abc from be, ca, ab as coordinates, the equation to a circle circumscribed about abc will be of the iorm fyz + gzx + hxy, and, calling the sides of the triangle a, b, c respectively,
262
Note on Certain Elementary
[39
cw; + 6y + c^ is constant. Hence, substituting for z its value in terms of x and y, the image of the circle may be put under a form in which fb and ga will be the coefficients of y' and a? respectively ; but, since x and y are proportional to the Cartesian coordinates y and x respectively, the coefficients of x' and y' must be equal. Hence f:g:h::a:b:c, and if now ax, by, cz, instead of a;, y, z, be used as the coordinates of the variable point, the image
to the circumscribing circle becomes 2 -f- , or if we please 'S.a^yz, that is,
'S.bcyz, where 6c stands as convened for (be)'.
Hence, if a, b, c, d be the vertices of a pyramid, labyz will be the image of the circumscribing sphere, for when any coordinate t is made zero the image becomes that of a circle; and so universally for a plasm of any number of dimensions.
Consider the case of a circle, and suppose that
ab ac 1
ac be
ba
ca cb
1 1 1
vanishes ; this means that the line x + y + z touches the circle
abxy + bcyz + cazx.
But, '\ix+y-\-z images the line at infinity, it must cut this (as it cuts any other circle) in two distinct points, namely, the so-called circular points at infinity. Hence x + y +z must, when the above determinant vanishes, cease to be the line at infinity, which can only come to pass by the triangle abc losing a dimension of space, and a, b, c coming into a straight line, in which case x + y + z = Q, instead of being true of a particular line, is true of every point in the plane.
Just in like manner, if
vanishes, unless x-\-y + z + t ceases to image the plane at infinity, this plane would touch the sphere 'Eabxy, that is, would cut it in a pair of straight lines, whereas it intersects it in a circle. Consequently the plasm abed must, as before, lose one dimension, and so in general. The content of a plasm vanishes when the mutual-distance determinant does so, and the latter as
|
• |
ab |
ac |
ad |
|
ba |
. |
be |
bd |
|
ca |
cb |
. |
cd |
|
da |
db |
dc |
. |
|
1 |
1 |
1 |
1 |
39]
Geometrical Notions and Determinations
263
which is easily transfonnable into
|
0 |
1 |
1 |
1 |
|
1 |
0 |
1 |
1 |
|
1 |
1 |
0 |
1 |
|
1 |
1 |
1 |
0 |
|
to 0 |
1 |
1 |
1 |
|
1 |
I |
0 |
0 |
|
1 |
0 |
1 |
0 |
|
1 |
0 |
0 |
I |
well as the former may be expressed rationally in terms of ordinary Cartesian coordinates; but the expression for the content (being linear in each set of coordinates) is obviously indecomposable, and must therefore be a numerical multiple of some power of the mutual-distance determinant ; a comparison of dimensions shows at once that this power is the square root.
I As regards the numerical multiplier, when the plasm has all its edges equal to unity (say a triangle, for example), the mutual-distance determinant becomes
of which the value is - 3 ; and so in general for a regular plasm with (n -I- 1) vertices; that is, in space of n dimensions the mutual-distance determinant, say Z>„, becomes (—)"+'(» -j- 1), whereas the (volume)^, say F„^ has been
8howDtobe^,^^"+ ^^,.
Hence, universally,
i)„ = (_)n+i 2» (1 . 2 . . . n)^ F„».
It may be here noticed that, if p be the perpendicular from any vertex on an opposite face of the plasm whose content is F„_,, we shall have
L 1V,P = «F„.
Consequently, Dn-if = (-)" 2»-' {1 . 2 . . . (n - 1))' FVi f
= (-)» 2»-' (1 . 2 ... nf F„» = - ii)„.
I now pass on to the leading motive of this note, namely, the determin- ation of the connection between the coordinates A, B, C ... drawn from a, b, c ....
It is clear d priori that the form of the condition will be in all cases that a homogeneous quadratic function of the distances must be constant. Thus, for example, when there are four point.s, if A, B, C be assumed, we may describe three spheres with these quantities as radii, and the fourth point will be determined by means of one of the pairs of tangent planes drawn to them, the particular pair depending on the relative signs attributed to
H
264 Note on Certain Elementary [39
A, B, C. Hence, if i''(^, B, C, D)= <x> be the general equation, each of the quantities must enter in the second and no higher degree; moreover, since by transporting the plane from which the distances are measured parallel to itself, A, B, C, D will be all increased by the same quantity, F must express a function of their differences, and consequently, since any two distances may be interchanged, F can contain no terms of the first order in the variables, so that jP=0 must amount to the predication of a homogeneous quadratic function of the distances being constant.
Thus, for example, in the case of three points, we have the well-known equation
2 {ah) (A-C)iB-G) = i (obey.
Suppose now that A, B, C are taken in proportions consistent with making
2(a6)»(^-C)(£-C) = 0.
Let S (aby (A — C)(B—C) = P.Q, where P, Q are two linear functions of A, B, C; then P, Q image two radiant points, each of which will have the property that any of its rays is at an infinite distance from a, b, c, or at all events, if it should pass through one of them, from the other two, and it is easy to anticipate that these two points must be the circular points at infinity. That such is the fact is obvious, because (using Cartesian co- ordinates) the perpendicular distance from any point upon x ± ij—l .y contains zero in its denominator; so that the two points of the absolute may be regarded as the centres of two points of rays, all of them infinitely distant from the finite region.
But these two points are the intersections of the circumscribing circle with the line at infinity, and consequently their collective ecjuation will be found by taking the resultant of "^abxy, Sa;, ^Ax, which is well known to be the determinant of the quadratic function bordered by the coeflScients of the two linear ones. Hence the constant quadratic function \n A, B, G, namely, l.ab(A — B){A — C), ought to be a numerical multiple of the determinant
.ABC.
A . ab ac 1
B ba . be \
G ca cb .1
. 111.
as is the case, the value of this determinant being
-2lab(A-G)(B-G).
The same thing may be shown in a more elementary manner as follows- Combining
X + y + z = 0, abiBy + bcyz + cazx = 0,
I
I
I
39] Geometrical Notioiu and Determinations 265
we have aca^ + (6c + ca — ah) xy + icy' = 0,
at each point of the absolute. And, taking x-^y^z-i, x^y^z^ as the coordinates at these two points, it follows that
a^i^Jj : y,ya : z^z^ : x^y^ + x^yi : y^z^ + y^z^ -.z^x^Jr z^x^
::bc:ca:ab: — bc — ca + ab: — ca — ab + bc: — ab—bc + ca.
And, as the two points will be imaged by
XjA +yiB + ZiG, x^A+y^B + z^G,
respectively, it follows that their collective image will be
^{bcA' + (bc-ab-ac)BC],
which is easily seen to be identical with
26c (A -B){A- C).
The universal algebraical theorem upon which the first method of proof depends is the well-known one that, if Q is a quadratic function and Z,, Zj, ... Li i linear functions of _;' variables, and if Q' (where j is not less than i+ 1) is what Q becomes when i of its variables are expressed in terms of the rest, then the necessary and suflBcient condition of the discriminant of every such Q' vanishing is that the determinant to Q bordered by the coeffi- cients of the i linear functions shall vanish. When j is equal to i + 1, the theorem shows that the resultant of the quadratic and its i attendant linear functions will be the bordered determinant in question. In the above example we bad j = 3, t = 2.
Let us now proceed to apply a similar principle to the case of four points a,b,c,din space.
If we take the case a^ -|- y* + 2' -K' = 0, any tangent plane to it at x', y', z', i! will be
x'x -f y'y -t- zz -H i!t, and , as a;'» + y'^ + /» + P = 0,
it follows that every tangent plane will be at infinite distance from any point external to it; and, as this is true wherever the centre of the cone be placed, and all the cones so obtained have the "circle at infinity" in common, — it follows that every tangent plane to the circle at infinity is infinitely distant from any external point in the finite region, — the infinitely-infinite system of planes thus obtained one may regard, if one pleases, as consisting of sheaves of planes whose axes form the environment to the circle at infinity, and will be the correlative to the infinitely-infinite system of points in the plane at infinity, which are infinitely distant from all external planes in the finite region. We see, then, that the coordinates to each such plane must satisfy the condition that, on making 1x=0 and l.Ax = 0, and expressing any two of the variables x, y, z, t in terms of the two others, the discriminant
U
266
Note on Certain Elementari/
[39
|
A |
B |
C |
D |
|
|
A |
, |
ah |
ac |
ad |
|
B |
ha |
, |
be |
bd |
|
G |
c« |
c6 |
cd |
|
|
D |
da |
dh |
dc |
, |
|
1 |
1 |
1 |
1 |
of the form then assumed by 'S.abxy must vanish, and consequently, as before, the mutual-distance determinant to the points a, b, c, d, bordered with a row and column of units and a row and column consisting of the letters A, B,C, D, will represent to a numerical factor pres the constant quadratic function of distances, that is, this function will be
and obviously a similar algebraical conclusion will continue to apply, what- ever may be the number of points n in a space of n — 1 dimensions.
As regards the value of the constant, in any case, that may be obtained by taking a face of the plasm as the term (line, plane, etc.) from which the distances A, B, G ..., are measured; that is, we may make 5 = 0, (7=0, Z) = 0..., provided we make A equal to the perpendicular from a on the opposite face. The value of the bordered determinant then becomes the negative of the squared perpendicular from a on bed ... multiplied by the mutual-distance determinant to bed...; that is, by virtue of what has previously been shown, will be half of the mutual-distance determinant of abed
Hence the complete relation between A, B,G, D may be exhibited by making
|
-h A |
B |
G |
D |
|
A |
ab |
ac |
ad |
|
B ba |
. |
be |
bd |
|
G ea |
cb |
, |
cd |
|
D da |
db |
dc |
|
|
1 |
1 |
1 |
1 |
0,
and similarly for any number of points.
Professor Cayley has obtained the same result by a more direct but not more instructive process, as follows. Taking, by way of example, three points, A+k, B+ k, C+ k, (where k is infinite,) may be regarded as the distances of a, b, c from a fourth point at an infinite distance, and accordingly we may write
= 0.
|
• |
ab |
ac |
{A + ky |
1 |
|
ba |
. |
be |
(B+ky |
1 |
|
ca |
eb |
. |
(G+ky |
1 |
|
{A + ky |
{B + ky |
{G+ky |
1 |
|
|
1 |
1 |
1 |
1 |
I
n
39]
Geometrical Notions and Determinations
267
For the gnomon bordering the squaxe formed by the small letters and dots, we may substitute
2lcA + A'' 1
2kB + & 1
2kC+C^ 1
2it^ + ^= 2kB + B' -IkC+C -2k' 1
I 1 1 1 1 .
without altering the value of the determinant, which therefore, remembering that k is infinite, is in a ratio of equality to (2A)" multiplied into the deter- minant
ab ac A 1
|
ba |
be |
B 1 |
|
|
ca |
cb |
. |
C 1 |
|
A |
B |
C |
-i . |
|
1 |
1 |
1 |
This last determinant therefore must vanish, agreeing with what has been shown above by a more purely geometrical method*. I will now proceed to develop this determinant deprived of its constant term, expressing it as a function of the dififerences of the capital letters.
It is obvious that it may be expressed as a sum of terms of which each variable part will be of one or the other of these three forms {A -By. iA-B)(A-C), {A-B)(C-D);
and accordingly we may distribute the totality of the terms of the constant function of difference into three families depending on the form of the variable argument.
In general, if we consider any invertebrate symmetrical determinant
» ex pressed by the umbral notation
aa ab ac ... al
^- ba bb be ... bl
II
la lb Ic
• As a corollary, we may infer, from the vanishing of this determinant, that, using the
notation previonsly employed,
n
and consequently that D,= - (2)"(1 . 2 ... n)» F„»,
and that thus the content of a regular plasm with unit edges and (n + 1) vertices is
n + 1
for triangle, pyramid, plu-pyramid, etc.
I 3 - namely, r^, =s
16 ' 72 ' 9 . 2»o ■
268 Note on Certain Elementary [39
where aa=hb = cc — ll...=0 and -pq = qp, we have this simple rule of pro- ceeding :
Divide the letters a . . . Hn every possible manner into cyclical sets, each set containing at least two letters.
Any cycle OjOs ... a< is to be interpreted as meaning
0,0,. OjOs ... ai^xOi .ata-i,
which, by virtue of the supposed condition ab = ba, will be the same in
whichever direction the cycle i.s read, the effect of the inversion of the cycle
being merely to give the same product over again, written under the form
OiOf.OsOi ... OiOi—i.
The cycle of two letters a^a^ must be interpreted to mean (a,a,)'. If now 0,(7, ... Gi are cycles of two letters each, and x>X2 ••• Xi cycles of three or more letters, the total value of the determinant will be
2(-)n+i+J2iC,C,...GiX.X^...Xj- If, the principal diagonal terms remaining zero, the other terms were general, then the expression of the value of the determinant, calling the cycles CiCj ... C^, and making no distinction between the case of their being binary or super-binary, would be 2 (— )"+'' Cj Cj . . . C,; only it would have to be understood that each cycle of two letters, as (ab), would mean (aby, but a cycle of three or more letters, as (abc), would mean ab .be .ca + ac .cb. ba.
This being premised, it is easy to deduce the following rule for the deter- mination of the three different families of terms belonging to the constant determinant of distances, which, to avoid prolixity, must be left to the reader to verify.
Family I. — Omitting any two letters, and forming all possible cyclical products with the remaining (n — 2) letters, if (7, Ca ■ • ■ C, be any set thereof, and v the number of them containing more than two letters, the general term will be ll(-y'+-2'' G^. C^ ...c/(A- By, a, b being the two letters which do not occur in the cycles GyGi...G,.
Family II. — Omitting any one letter, and forming with the remaining n — \ letters, in every possible way, a chain x containing two or more letters, and cycles G^G, ... Gy, then, supposing the chain to be bed ... kl, and under- standing by (x) the product bc.cd ... H, the general term will be
22 (- )»+■' 2-'+' G,G,...Gy(x){A- B) {A - L),
a being the letter which does not appear in the chain or any of the cycles, and v meaning as before the number of the cycles which contain 'at least three elements.
Family III. — Form all the letters in every possible way into two chains (each containing two or more letters) X' %• ^^^ ^^^'^ cycles C,, C„ ... G,;
i
39]
Geometrical Notions and Determinations
269
then, supposing the initial and final letters of ;^ to be a, h, and of ■x^ to be k, I, the general terra of this family will be
22 i-r^^ 2--+' C,C,...C. (x) (%') {{A -K)(H-L) + iA- L) {H - K)].
I subjoin in the following table the types of the coefficients of the several families for all the values of n from 2 up to 7 ; the vacant cycle ( ) of course means unity, and a cycle {ah) means {ahy ; that is, the fourth power of the length ah.
Every cycle enclosed in a parenthesis of three or more letters, will be understood to be affected with a coefficient 2, and for greater brevity the variable part of each term is left to he supplied. A round parenthesis indi- cates a cycle, a square parenthesis a chain.
Number
of Letters
2 3 4
Types
Name
of Family
lat
2Dd
1st
2nd
3rd
Ist
2nd
3rd
1st
2nd
3rd
Ist
() {he)
-{cd)
2 [6cd]
2[a6].[cd]
5 {cde)
-2[6ca!e]:2(6c)[de]
- 2 [a6] [cde]
6 -{cdef):{cd){ef)
- 2 {hcd) [ef] : - 2 (6c) [def] : - 2 [bcdef]
- 2 {ab) [cd] [ef] : [ahc] [def] : [ab] [cdef]
7 {cdefg)-{cd){efg) „ 2 {bcde) [fg] : 2 {bed) [efg] -.-2 {be) {de) [ fg] 2 {be) [defg] : - 2 [hcdefg] 2nd „ 2 {ahc) [de] [fg] : 2 {ab) [cd] [efg] : - 2 [ahc] [defg] : - 2 [ahcdefg] 3rd
Thus, for example, the constant function of distances for three points in a plane is 226c {A — B) {A — C) ; for four points in space is -l,cd{A-By+21bc.ed{A -B){A-D)
+ 2Sa6 . cd \{A -C){B - D) + {A - D){B- C)} ; for five points in hyper-space is
21{cd.de.ec){A - Bf -21 {be .cd . de) {A - B){A - E)
+ 2{hcy{de){A-D){B-E) - 2Sa6 .cd.de.ec{{A- C) {B-E) + {A- E) {B - C)]. The part of the constant function of distances for seven points belonging to the 2iid family of terms will be
426c .cd.de. eh .fg {A -B){A-E) + 4-lhc . cd . db . ef.fg {A -E){A- 0) - 2 {hcf {deffg {A-F){A-G)^2 {bcf {de . ef.fg) {A -D){A- G) -2hc.cd.de.ef.fg{A-B){A-0).
270 Note on Certain Etementarj/ [39
The number of types in each family for n points is easily expressible by a generating function.
Obviously in the Ist family this number is the number of ways of resolving n into parts none less than 2 ; that is, it is the coefficient of ic""' in
1
l-x».l-«».l-a;*...'
In the 2nd family, it is the sum of the number of ways of decomposing
n — 3, n — 4, ... into parts none less than 2; tiiat is, it is the coefficient of
x"~* in
l+x+x' + ... ^, ^ . . 1
, that is, in
(l-x')(l-x>)...' ' (l-x){l-a!')il-af)...'
In the 3rd family, if the number of ways of dividing r into two parts, neither of them less than 2, is called (r), and of dividing (n — r) into any number of parts, none less than 2, is called [n — r], the number of types is 2 (r) [» — r] ; that is, it is the coefficient of x^~* in
that IS, in
(l-ai'){l-x'){l-x*)... ' ' (l-x)(l-x'y{l-ai')(l-x')... '
Hence the total number of types in all three families combined will be the coefficient of x"~' in
(l-x)(l-a^) + x (1 - x^) + a,'' . 1
l-x.l-x^.l-af... ■ ■ *''^' "■ '" l-x.il-x^y.l-a^A-x^... '
Consequently, the indefinite partitions of 0, 1, 2, 8, 4, 5, 6, 7, ... being 1, 1, 2, 3, 5, 7, 11, 15, ..., the series for the type-number will be found by summing all the terms in the odd and even places successively. We thus obtain the series 1, 1, 3, 4, 8, 11, 19, 26, ... for the number of types in the constant-distance function for 2, 3, 4, 5, 6, 7, 8, 9, ... points respectively.
It may be worth while to exhibit the rule for the formation of the constant function of distances under a slightly different aspect.
As before, by the reading of any cycle, understand the product of its successive duads affected with the multiplier — 1 or — 2, according as the number of letters in the cycle is two or more than two.
By a modified reading of a cycle, understand what the reading becomes on substituting for any two duads pq, rs the product (P — Q){R — S), as for instance (.4 - -BJC — B) in lieu of ab . cd, (A — B^B — G) in lieu of ab . be, and (which can only happen in the case of a cycle of two letters), {A -B){B-A), that is, -{A- B)' in lieu of ab . ba.
Then, to find the constant function of distances to any given set of letters, we must begin with distributing the letters in every possible way into cycles containing between them two or more letters. Each such combination of cycles we may call a distribution.
I
L
39] Geometrical Notions and Determinations 271
In each distribution the cycle is to be taken (each in its turn), and the sum of its modified readings is to be multiplied by the product of the read- ings of the remaining cycles, if there are any. The sum of these sums (or the single sum, if there is but one cycle) is the portion of the quadratic function sought, due to the particular distribution dealt with ; and the sum of these double sums, taken for each distribution in succession, is the total value of the function, and will be equal exactly to its representative deter- minant when the number of letters is odd, and to the same with its sign changed when that number is even.
As an example for five letters a, b, c, d, e, there will be ten distributions of the form (ab) (cde), and twelve distributions of the form (abcde).
From any one of the first ten distributions, as (ah) (cde), by modifying first (ab) and then (cde), we obtain
(1) 2(cd.de.ec)(A-B)(B-A),
(2) 2(aby [ce (C-D)(D - E)+dc(B - E)(E-C) + ed(E - C)(C - D)].
And from a distribution of the form (abcde) we obtain, by operating on con- secutive duads,
5 terms of the form —2[cd.de. ea (A —B)(B — G)],
and, by operating on non-consecutive duads,
5 terms of the form -2 [bc.de. ea (A -B)(C- D)} *.
The sum of all the sums of terms due to the twenty-two distributions is the constant function of distances for the five given letters.
In the case of six letters the distributions into cycles will be of four kinds, corresponding to the partitions 6; 4, 2; 3, 3; 2, 2, 2.
The first kind will contain two types of the 3rd family and one of the 2nd family; the second kind will contain one type of each of the three families, and the third and fourth kinds single types of the 2nd and 1st families respectively, thus giving eight distinct types of terms in all, as should be the case according to the rule.
* It will be observed that the distribution {acbde) will give a term -2 {cb . de . ea{A - C) (B - D)], in which the literal part cb.de. ea is equal to the literal part bc.de .fa in the term above expressed. This is bow it comes to pass that the terms of the 3rd family may be grouped in pairs, as stated in the prior mode of arranging the result according to families instead of according to cycles.
40.
ON THE TRINOMIAL UNILATERAL QUADRATIC EQUATION IN MATRICES OF THE SECOND ORDER.
[Quarterly Journal of Mathematics, xx. (1885), pp. 305 — 312.]
In the May number [p. 225 above] of the present year of the London and Edinburgh Philosophical Magazine (disfigured by numerous errors or inaccuracies) I investigated the number of the solutions of an equation in quaternions or matrices of the second order, belonging to what I term the unilateral class, meaning one in which the coefficients of any actual power of the unknown quantity lie on the same side of it ; this number for the Jerrardian Trinomial form I proved strictly is 2t' — i {i being the degree of the equation) and with evidence little short of moral certainty i^ — i'' + i* in the general case where none of the terms are wantingf .
But it must be well borne in mind that these numbers only apply when the coefficients are left general, and that for special relations between them some or all of the roots may become either ideal or indeterminate, or some the one and some the other. In all cases of equations in matrices one principal feature of the investigation is, or should be, to determine the equation of condition between the coefficients, in order that the solution may lose or retain its normal form ; if we wish to avoid being compelled to enter upon a complicated consideration of exceptions piled upon ex- ceptions, it is necessary to presuppose a certain criterion function to be other than zero ; otherwise it is like the opening of Pandora's box, letting loose an almost incalculable train of vexatious inquiries scarcely worth the trouble they give to answer correctly.
* This article was written and sent to the press many mouths ago. I have since shown that the number of roots of a general unilateral equation of degree i in matrices of the order u is the number of combinations of iu things taken u and w together, and consequently for the case of quaternions is 2t^-i for the general and not merely for the Jerrardian form. See [above, pp. 197, 233. Also] Nature, Nov. 13, 1884.
t I made the assumption that the required number is an analytical function of u.
I
I
40] On the Trinomial Unilateral Quadratic Equation, etc. 273
Take as an instance the subject of monothetic equations. I have defined a monothetic equation to be one in which all the coetficients are functions of a single matrix, which may be called the base. In such an equation of the degree i and of the order <b in the matrices, we may suppose the unknown quantity to be a function of the base, and then the general formula for expressing a function of a matrix as a rational and integral function of the matrix with the aid of its latent roots, shows that i" and no more of such roots exist. But this in no manner precludes the possibility of the existence ofother roots which aie not functions of the base. Thus, fur example, in the very simple case of the equation a? -\-px = Q, where x and p are quaternions or matrices of the second order, I have shown in the Comptes Rendus [pp. 174., 179 above] that besides the four determinate ones, all of which (0 included) may be regarded as functions of p, there are two other indeterminate ones, each one containing an arbitrary constant, and neither of them (to use quaternion language) coplanar with the base. Here there is a sort of rever- sion to the normal case of 3 pairs of roots to an unilateral quadratic, with the modification of two of them having become indeterminate. It becomes then of importance to fix accurately the condition of this normal state of things ceasing to exist. Being intent on the Denumeration theory of the roots in the general quadratic, I did not in the paper cited do this explicitly for the unilateral quadratic, although I gave there my own form of solution. Moreover, there are other features of much interest belonging to the question, which, for the same reason, I omitted to notice. These omissions and shortcomings it is the object of this present article to supply.
Starting with the form a? — 2px + q = 0, and for convenience of com- parison with Hamilton's formulae treating p, q indifferently as matrices or as quaternions, and forming the equation a^ — 2Bx + D = 0, where B, D are scalars to be determined, so that B = Sx and D= Ta?, we shall have
2x^{p-BY^{q-D). If now we understand by 6, c, d, e, f
Sp, Sq, Tp\ S{Vp Vq), Tq' respectively, by means of the general formula
r^ . (tt-'x)' - 2S(F7r 7x) (tt-x) + Tx^ = 0», [remembering that
r(p-BY = d'-2bB+B',
T(^q-Df = f'-'2cD + D\
* This formula, which I have not met with in Treatises on Qoatemions, is a particular case only of the general Theorem in Matrices, that if
A\'' + U\'''^^+...+Lpr U the determinant to {\L + iiM), where L and M are two matrices of the order a and \ and /i two ordinary qaantities, then
4(L-ii/)"-iJ(Z,-iJ/)"-'... + (-)"L = 0. 8. IV. 18
274 On the Trinomial Unilateral Quadratic [40
and S{V{p-B)V{q-D)] = e-bD-cB-\-BDl
we shall obtain [see p. 188 above]
4 (d - 265 + £•') a» - 4 (e - 6Z> - cfi + £i)) .T + (/ - 2cZ) + DO = 0.
Hence, writing B-b=u, D-c = v,
d - b- = a, e-bc = 0, /- c" = 7.
and comparing with each other the two quadratic equations in x, we may
write
u'+a = \, Mt) + /3 = 2X(M+6), v^ + y = 'iX{v + c).
Eliminating v from the latter two equations there results
- (2\w + 2bX -^y + 4t\ (2Xm + 26\ - /3) m - (7 - 4cX) u' = 0, and finally writing \- a for «', we obtain
(4V + 4cX. - 7) (\ - a) - (26\ - y3)= = 0. There are thus 3 pairs of roots, for to each of the three values of \ corre- spond two values of v, namely
±{\-d + b')\
and to each value of \ and m one value of v, namely
26\ + bc-e
2\
u
We have also x = ^{{p-b-u)-'(q - D)},
consequently, since p' - 2bp + d = Q,
_ (p - b-\-u)iq-c-v) ^ _ (p-b+u)(q-c-v) '" 2(6»-rf-wO 2X
Thus then we see that x can only cease to have 6 determinate values when \ = 0, and consequently the Criterion of Normality is the last term in the equation to X.
This equation, written out at length, is
4X3 + 4 (c _ 62 _ a) x' + (- 4ca + 46^ - 7) X + 07 - /3- = 0, that is, 4V + 4 (c - d) X^ + (- 4crf + 46e - / + c^ X + (d - b') (/- C) - (e - 6c)». Hence the Criterion in question is (d - ¥) (f - c^ - (e - 60^ or (//- c=rf - 6=/- e' + 2bce, which is the discriminant to the quadratic form
X» + 2b\p + 2cfi,v + dfj? + 2efiv + fv^; this, as I have elsewhere shown, is the Criterion of the matrices p.q* being in involntion-l, that is, of a linear equation existing between the matrices 1, p. q, pq; or ifp, q are regarded as quaternions, it is the condition of the square of
• When p, q are regarded as matrices, then
p^-2bp + d = 0, q'-2cq+f=0, i(pq + qp) -bq-cp + e = 0, ^vhere \^ + 2h>a> + 2cny + d^L* + ie^p■^■fy'
is the determinant toX + fip + vq. [t Above p. 116.]
I
40] Equation in Matrices of the Second Order 275
the sine of the angle between the vectors oip and q vanishing; a condition which of course does not imply the coincidence of the vectors unless accom- panied by the futile limitation of such vectors being real.
It admits of easy demonstration by virtue of the foregoing that in the case of the more general equation
pa? + qx + r = 0, the Criterion of Normality will be the discriminant of the ternary quadratic, which is the determinant of
pu + qv + rw ; this seems to me a very remarkable and noteworthy theorem. When this Criterion does not vanish, the quadratic equation above written must have 3 pairs of determinate roots.
Why they go in pairs and can be found by solving only a cubic instead of [ a sextic is best seen d priori by reverting to the original form a^ — 2px + q = 0.
It follows from the nature of the process for finding B and D that they will be the same for that equation as for the equation y' — 2yp 4-g = 0.
But on writing x + y = 2p these two equations pass into one another.
Hence each value of B, say Bj , will be associated with another value, say E, where Bj + R = 2b*, that is to say, if Ui, namely B — b is one value of u, then b — B, that is, — !<, will be another value of u, so that the equation in u^ ought to be (as it has been shown to be) a cubic.
It might for a moment be supposed that X = a = d — 6" would lead to a
breach of normality on account of the equation i; — 2\ = , where
m'=0.
This, however, is not the case. For the equation
«» + 7 = 4X(t;+ c) becomes, when \= a,
v^ - 4 (d - 6») « + / - c^ - 4cci + 46^c = 0, so that V remains finite ; consequently 26\ + 6c — e, that is, 2bd — 2b'+bc—e, must vanish when \ = d — b', and v — 2X assumes the form - . Obviously then
in this case, to the one value m = 0 will be associated the two values of v, say Vi and v„ given by the above quadratic, and to \ = a will still correspond two values of (m, v), namely (0, v,), (0, v^); where, ideally speaking, the two zeros may be regarded as the same infinitesimal affected with opposite signs.
* In qnatemion phrase, it x + y = 2p, Sx + Sy = 2Sp.
It should be observed, in order to understand what follows in the text, that b-B^ = B' -b, and that the values of B mast obviously be the same in the equation x'-2px + q = 0 as in the equation x^ - 2jj> + q = 0.
18—2
276 On the Trinomial Unilateral Quadratic [40
The equation in X may be made to undergo a useful linear transformation.
Let \= n + a, so that fi = m'.
Then
fj. (V + (8a + 4c) M + 4a« + 4ca - 7} - {2bfi + 2ba - y3)= = 0, that is 4fji' + {4 (c + 2d) - 1 2b'] fj.' + ((c + 2d)' - 8 (c + 2d) b'
+ 126* + ibe-/}fi- {b (c + 2d) - 2i'-e)' = 0, where it is noticeable that the number of parameters is reduced from 5 to 4, c and d only appearing together in the linear combination c + 2d. This is tantamount to the form obtained by Hamilton.
Let us make another linear transformation suggested by the preceding remark. Write c + 2d = ^r, and fi — b' = y = \ — d, the equation becomes
4y + 4:gy^ + (^= + 46e -/) 7 + 2beg -¥f-e' = 0.
But obviously, notwithstanding this reduction of the parameters, X itself is the most natural quantity to employ as the base of the solution, or, so to say, as the independent variable, and this admits of being determined by an equation of extraordinary simplicity.
For, let / be the discriminant of
det. (\ + /xp + 1/9) = 7 = d/" + 2bce - c-d - b'f - ef.
Then it will be seen by actual inspection that the equation found for \ takes the following form
that is
(the terms in the exponential function subsequent to the fourth term adding nothing to the value of the series).
If in the equation of — 2px + q=0, p and 5' be regarded as quaternions, then X = Sx' + Ip' - {Spy, c = 8q, d^Ip', and I=i(pg-qpy, which is a scalar quantity, and is to be regarded as an explicit function of Sp, Sq ; Tp', S{VpVq), Tq^; it is in fact the discriminant of the form
Z' + 2SpZ F + 2SqXZ + Tp" F» + 2S ( Vp Vq) YZ + Tq'Z\
an identity unknown I believe to the geometrical quatemionists.
[As an example of it, let p = i, q =j, then
8p = 0, 8q = 0, S{VpVq) = 0, Tp' 1, Tq'=-\,
\ (P? - iPf = 1 = the discriminant of X'-Y-- Z-.]
With these definitions e^^^*'"*"/ becomes identically zero.
40] Equation in Matrices of the Second Order 277
The equation a? — 2px+q = 0 having six roots it is natural to inquire as to the value of their sum. This may be readily found as follows. We have
found
(p-b + u){q-c-v) '^~ 2\
Also, if x + x'= 1p,
af^ - 2x'p + g = 0, and obviously Sa; = Sa;'.
Hence ^^^_^{p -h ^u^q- o-v) ^
and 12;. - S. = - S (3jlA:it|(Pzi±^) .
3
Therefore Sa; = 6p — S ^ {pq — qp)
where the sign of /4 must be so taken that it shall be equal to \ ipq — qp)-
So again 'Ex^ = 2p'S.x — 67
= I2p' -6q + l2 (28, - Ba)I^-p.
Thus the mean value of each root is e in excess, and that of each square root ep in excess, of what these means would be if p and q were nominal quantities, e denoting (28^ — 8^) I^p. Of course Sa:* may be found by the formula of derivation
lx^+' = 2p'S.x'-9lx'-\
In conclusion it may be observed in regard to the equation a^ — 2px + q = 0, (since in writing a; + ar, = 2p, we have x^^ — 2x,p + g = 0) it follows that (what- ever be the order of the quantities p and q) the roots of either equation must be associated in pairs; because, if the identical equation to p is jj"— a)6p"~' + ... and to a; is af — coBoT'^ + ..., the equation for finding B must be of the form T(B-by = 0.
P.S. — Since the above was sent to press I have discovered the general solution of the unilateral equation of any degree in matrices of any order ; see the Comptes Rendtis of the Institute for Oct. 20, 1884 [pp. 197, 233 above], and Nature for Nov. 13, 1884*.
[• This paper contains the Theorem " Every latent root of every root of a given unilateral function in matrices of any order, is an algebraical root of the determinant of that function taken as if the unknown were an ordinary quantity, and conversely every algebraical root of the determinant so taken is a latent root of one of the roots of the given function."]
41.
INAUGURAL LECTURE at Oxford 12 December 1885.
ON THE METHOD OF RECIPROCANTS AS CONTAINING AN EXHAUSTIVE THEORY OF THE SINGULARITIES OF CURVES*.
[Nature, xxxiii. (1886), pp. 222—231.]
It is now two years and seven days since a message by the Atlantic cable containing the single word " Elected " reached me in Baltimore informing me that I had been appointed Savilian Profes.sor of Geometry in Oxford, so that for three weeks I was in the unique position of filling the post and drawing the pay of Professor of Mathematics in each of two Universities: one, the oldest and most renowned, the other — an infant Hercules — the most active and prolific in the world, and which realises what only existed as a dream in the miad of Bacon — the House of Solomon in the New Atlantis.
To Johns Hopkins, who endowed the latter, and in conjunction with it a great Hospital and Medical School, between which he divided a vast fortune accumulated during a lifetime of integrity and public usefulness, I might address the words familiarly applied to one dear to all Wykeliamists : —
"Qui condis Iseva, condis collegia dextra, Nemo tuanim unam vicit utraque manvi."
The chair which I have the honour to occupy in this University is made illustrious by the names and labours of its munificent and enlightened founder, Sir Henry Saville ; of Thomas Briggs, the second inventor of logarithms; of Dr Wallis, who, like Leibnitz, drove three abreast to the temple of fame — being eminent as a theologian, and as a philologer, in addition to being illus- trious as the discoverer of the theorem connected with the quadrature of the circle named after him, with which every schoolboy is supposed to be familiar, and as the author of the Arithmetica Infinitorum, the precursor of Newton's Fluxions; of Edmund Halley, the trusted friend and counsellor of Newton, whose work marks an epoch in the history of astronomy, the reviver of the study of Greek geometry and discoverer of the proper motions of the so- [* The tables referred to in the text are given pp. 301, 302 below.]
41] On the Method of Reciprocants 279
called fixed stars ; and by one in later times not unworthy to be mentioned in connection with these great names, my immediate predecessor, the mere allusion to whom will, I know, send a sympathetic thrill through the hearts of all here present, to whom he was no less endeared by his lovable nature than an object of admiration for his vast and varied intellectual acquirements, whose untimely removal, at the very moment when his fame was beginning to culminate, cannot but be regarded as a loss, not only to his friends and to the University for which he laboured so strenuously, but to science and the whole world of letters.
As I have mentioned, the first to occupy this chair was that remarkable man Thomas Briggs, concerning whose relation to the great Napier of Merchiston, the fertile nursery of heroes of the pen and the sword, an anecdote, taken from the Life of Lilly, the astrologer, has lately fallen under my eyes, which, with your permission, I will venture to repeat: —
" I will acquaint you (says Lilly) with one memorable story related unto me by John Marr, an excellent mathematician and geometrician, whom I conceive you remember. He was servant to King James and Charles the First. At first, when the lord Napier, or Marchiston, made public his logarithms, Mr Briggs, then reader of the astronomy lectures at Gresham College, in London, was so surprised with admiration of them, that he could have no quietness in himself until he had seen that noble person the lord Marchiston, whose only invention they were : he acquaints John Marr here- with, who went into Scotland before Mr Briggs, purposely to be tliere when those two 80 learned persons should meet. Mr Briggs appoints a certain day when to meet at Edinburgh; but failing thereof, the lord Napier was doubtful he would not come. It happened one day as John Marr and the lord Napier were speaking of Mr Briggs : ' Ah John (said Marchiston), Mr Briggs will not now come.' At the very moment one knocks at the gate; John Marr hastens down, and it proved Mr Briggs to his great contentment. He brings Mr Brigg.s up into my lord's chamber, where almost one quarter of an hour was spent, each beholding other almost with admiration before one word was spoke. At last Mr Briggs began : ' My lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto astronomy, namely, the logarithms ; but, my lord, being by you found out, I wonder nobody else found it out before, when now known it is so easy.' He was nobly entertained by the lord Napier ; and every summer after that, during the lord's being alive, this venerable man Mr Briggs went purposely into Scotland to visit him*."
* A very gimilar story is told of the meeting of Leopardi and Niebnhr in Bome. What Briggs ■aid of logarithms may be said almost in the same words of the subject of this lecture: — " This most excellent help to geometry which, being found out, one wonders nobody else found it ont
280 On the Method of Reciprocants [41
Some apology may be needed, and many valid reasons might be assigned, for the departure, in my case, from the usual course, which is that every professor on his appointment should deliver an inaugural lecture before commencing his regular work of teaching in the University. I hope that my remissness, in this respect, may be condoned if it shall eventually be recog- nised that I have waited, before addressing a public audience, until I felt prompted to do so by the spirit within me craving to find utterance, and by the consciousness of having something of real and more than ordinary weight to impart, so that those who are qualified by a moderate amount of mathe- matical culttire to comprehend the drift of my discourse, may go away with the satisfactory feeling that their mental vision has been extended and their eyes opened, like my own, to the perception of a world of intellectual beauty, of whose existence they were previously unaware.
This is not the first occasion on which I i)ave appeared before a general mathematical audience, as the messenger of good tidings, to announce some important discovery. In the year 1859 I gave a course of seven or eight lectures at King's College, London, at each of which I was honoured by the attendance of ray lamented predecessor, on the subject of " The Partitions of Numbers and the Solution of Simultaneous Equations in Integers," in which it fell to my lot to show how the difficulties might be overcome which had previously baffled the efforts of mathematicians, and especially of one bearing no less venerable a name than tfiat of Leonard Euler, and also laid the basis of a method which has since been carried out to a much greater extent in my "Constructive Theory of Partitions," published in the American Journal of Mathematics, in writing which I received much valuable co-operation and material contributions from many of my own pupils in the Johns Hopkins University*. Several years later, in the same place, I delivered a lecture on the well-known theorem of Newton, which fills a chapter in the Arithmetica Universalis, where it was stated without proof, and of which many celebrated mathematicians, including again the name of Euler, had sought for a proof in vain. In that lecture I supplied the missing demonstration, and owed my success, I believe, chiefly to merging the theorem to be proved, in one of
before ; when, now known, it is so easy." I quite entered into Briggs's feelings at his interview with Napier when I recently paid a visit to Poincare in his airy perch in the Eue Gay-Lussac in Paris (will our grandchildren live to see an Alexander Williamson Street in the north-west quarter of London, or an Arthur Cayley Court in Lincoln's Inn, where he once abode?). In the presence of that mighty reservoir of pent-up intellectual force my tongue at first refused its office, my eyes wandered, and it was not until I had taken some time (it may be two or three minutes) to peruse and absorb as it were the idea of his external youthful lineaments that I found myself in a condition to speak.
* In one of those lectures, two hundred copies of the notes for which were printed off and distributed among my auditors, I founded and developed to a considerable extent the subject since rediscovered by M. Halphen under the name of the Theory of Aspects.
I
41] On the Method of Reciprocants 281
greater scope and generality. In mathematical research, reversing the axiom of Euclid, and converting the proposition of Hesiod, it is a continual matter of experience, as I have found myself over and over again, that the whole is less than its part. On a later occasion, taking my stand on the wonderful discovery of Peaucellier, in which he had realised that exact parallel motion which James Watt had believed to be impossible, and exhausted himself in contrivances to find an imperfect substitute for, in the steam-engine, I think I may venture to say that I brought into being a new branch of mechanico- geometrical science, which has been, since then, carried to a much higher point by the brilliant inventions of Messrs Kempe and Hart. I remember that my late lamented friend, the Lord Almoner's Reader of Arabic in this University, subsequently editor of the Times, Mr Chenery, who was present on that occasion in an unofficial capacity, remarked to me after the lecture, which was delivered before a crowded auditory at the Royal Institution, that when they saw two suspended opposite Peaucellier cells, coupled toe-and-toe together, swing into motion, which would have been impossible had not the two connected moving points each described an accurate straight line, " the house rose at you." (The lecture merely illustrated experimentally two or three simple propositions of Euclid, Book III.)
The matter that I have to bring before your notice this afternoon is one far bigger and greater, and of infinitely more importance to the progress of mathematical science, than any of those to which I have just referred. No subject during the last thirty years has more occupied the minds of mathe- maticians, or lent itself to a greater variety of applications, than the great theory of Invariants. The theory I am about to expound, or whose birth I am about to announce, stands to this in the relation not of a younger sister, but of a brother, who, though of later birth, on the principle that the masculine is more worthy than the feminine, or at all events, according to the regulations of the Salic law, is entitled to take precedence over his elder sister, and exercise supreme sway over their united realms. Metaphor apart, I do not hesitate to say that this theory, minor natu potestnte major, infinitely transcends in the extent of its subject-matter, and in the range of its applications, the allied theory to which it stands in so close a relation. The very same letters of the alphabet which may be employed in the two theories, in the one may be compared to the dried seeds in a botanical cabinet, in the other to buds on the living branch ready to burst out into blossom, flower and fruit, and in their turn supply fresh seed for the main- tenance of a continually self-perpetuating cycle of living forms. In order that I may not be considered to have lost myself in the clouds in making such a statement, let me so far anticipate what I shall have to say on the meaning of Reciprocants and their relation to the ordinary Invariantive or Covariantive forms by taking an instance which happens to be common
282 On the Method of Beciprocants [41
(or at least, by a slight geometrical adjustment, may be made so) to the two theories. I ask you to compare the form
a'd - Sabc + 26* as it is read in the light of the one and in that of the other. In the one case the a, b, c, d stand for the coefficients of a so-called Binary Quantic, and its evanescence serves to express some particular relation between three points lying in a right line. In the other case the letters are interpreted* to mean the successive differential derivatives of the 2nd, 3rd, 4th, 5th orders of one Cartesian co-ordinate of a curve in respect to the other. The equation expressing this evanescence is capable of being integrated, and this integral will serve to denote a relation between the two co-ordinates which furnishes the necessary and sufficient condition in order that the point of the curve of any or no specified order (for it may be transcendental) to which the co- ordinates may refer, may admit of having, at the point where the condition is satisfied, a contact with a conic of a higher order than the common. In the one case the letters employed are dead and inert atoms ; in the other they are germs instinct with motion, life, and energy.
A curious history is attached to the form which I have just cited, one of the simplest in the theory, of which the narrative may not be without interest to many of my hearers, even to those whose mathematical ambition is limited to taking a high place in the schools.
At pp. 19 and 20 of Boole's Differential Equations (edition of 1859) the author cites this form as the left-hand side of an equation which he calls the "Differential Equation of lines of the second order," and attributes it to Monge, adding the words, "But here our powers of geometrical interpretation fail, and results such as this can scarcely be otherwise useful than as a registry of integrable forms." In this vaticination, which was quite uncalled for, the eminent author, now unfortunately deceased, proved himself a false prophet, for the form referred to is among the first that attracts notice in crossing the threshold of the subject of Reciprocants, and is but one of a crowd of similar and much more complicated expressions, no less than it susceptible of geometrical interpretation and of taking their place on the register of integrable forms. A friend, with whom I was in communication on the subject, and whom I see by my side, remarked to me, in reference to this passage : — " I cannot help comparing a certain passage in Boole to Ezekiel's valley of the dry bones: 'The valley was full of bones, and lo, they were very dry.' The answer to the question, ' Can these bones live V is supplied by the advent of the glorious idea of the Reciprocants ; and the grand invocation, ' Come from the four winds, 0 breath, and breathe upon these slain, that they may live,' may well be used here. That they will
L 2!<ijr2' °--i\dxi' '-i\di*' "~6!dx»J
41] On the Method of Reciprocants 283
' live and stand up upon their feet an exceeding great army ' is what we may expect to happen." This, as you will presently see, is just what actually has happened.
Not knowing where to look in Monge for the implied reference, I wrote to an eminent geometer in Paris to give me the desired information ; he replied that the thing could not be in Monge, for that M. Halphen, who had written more than one memoir on the subject of the differential equation of a conic, had made nowhere any allusion to Monge in connection with the subject. Hereupon, as I felt sure that a reference contained in repeated editions of a book in such general use as Boole's Differential Equations was not likely to be erroneous, I addressed myself to M. Halphen himself, and received from him a reply, from which I will read an extract :—
" En premier lieu, c'est une chose nouvelle pour moi que I'equation differentielle des coniques se trouve dans Boole, dont je ne connais pas I'ouvrage. Je vais, bien entendu, le consulter avec curiosite. Ce fait a ^chappe a tout le monde ici, et Ton a cru g^neralement que j'avais le premier donne cette Equation. Nil sub sole novi! II m'est naturellement impossible de vous dire ovi la meme equation est enfouie parmi les ceuvres de Monge. Pour moi, c'est dans Le Journal de Math. (18T6), p. 375, que j'ai eu, je crois, la premiere occasion de developper cette Equation sous la forme meme que vous citez ; et c'est quand je I'ai employee, I'ann^e suivante, pour le probleme sur les his de Kepler (Comptes rendus, 1877, t. Lxxxiv. p. 939), que M. Bertrand I'a remarquee comme neuve. Ce qui vous int^resse plus, c'est de connaitre la forme simplifide sous laquelle j'ai donne plus tard cette equation dans le Bulletin de la Soci^t^ Math^matique. C'est sous cette derniere forme que M. Jordan la donne dans son cours de I'Ecole Poly- technique (t. I. p. 53)."
All my researches to obtain the passage in Monge referred to by Boole have been in vain*.
I will now proceed to endeavour to make clear to you what a Reciprocant means : the above form, which may be called the Mongian, would afford an example by which to illustrate the term ; but I think it desirable to begin with a much easier one. Consider then the simple case of a single term, the second derivative of one variable, y, in respect to another, x. Every tyro in algebraical geometry knows that this, or rather the fact of its evanescence, serves to characterise one or more points in a curve which possess, so to say,
• Search has been made in the collected works of Monge and in manuscripts of his own or Prony in the library of the Institute, but without effect. I have also made application to the Universal Information Society, who undertake to answer " every conceivable question," but nothing has so far come of it. Perhaps until the citation from Monge is verified it will be safer in future to refer to the go-called Mongian as the Boole- Mongian. It may be regarded aa the starting-point of the Differential Invariant Theory, as the Sohwarzian is of the deeper-lying and more comprehensive Beciprocant Theory.
284 On the Method of Eeciprocants [41
a certain indelible and intrinsic character, or what is technically called a singularity ; in this case an inflexion such as exists in a capital letter S, or Hogarth's line of beauty.
If we invert the two variables, exchanging, that is to say, one with the other, the fact of this indelibility draws with it the consequence that in general these two reciprocal functions must vanish together, and as a fact each is the same as the other multiplied or divided by the third ]K)wer of the first derivative of the one variable with respect to the other taken negatively. In this case we are dealing with a single derivative and its reciprocal. The question immediately presents itself whether there may not be a combination of derivatives possessing a similar property. We know that no single derivative except the second does.
Such a combination actually presents itself in a form which occurs in the solution of Differential Equations of the second order, the form
dy cPy _ 3 /d^/V
dx'd^ 2 wj ' which, after the name of its discoverer, Schwarz, we may agree to call a Schwarzian (Cayley's " Schwarzian Derivative*"). If in this expression the x and y be interchanged, its value, barring a factor consisting of a power of the first derivative, remains unaltered, or, to speak more strictly, merely undergoes a change of algebraical sign. We may now arrive at the generalised conception of an algebraical function of the derivatives of one variable in respect to another, which, if we agree to pay no regard to the algebraical sign, or to any power of the first derivative that may appear as a factor, will remain unaltered when the dependent and independent variables are interchanged one with another; and we may agree to call any such function a Reciprocant.
But here an important distinction arises — there are Reciprocants such as the one I first mentioned, -r^, or such as the Mongian to which allusion has
* More strictly speaking this is Cayley's Schwarzian derivative cleared of fractions — it may well be called the Schwarzian (see my note on it in the Mathematical Messenger for September or October past). Prof. Greenhill in regard to the Schwarzian derivative proper writes me as follows : —
"I found the reference in a footnote to p. 74 of Klein's Vorlesumjen ilber dat Ikoiaeder, &c., in which Klein thanks Schwarz for sending him the reference to a paper by Lagrange, ' Sur la construction des cartes g^ographicjues ' in the Nouveaux Memuires de VAcadimie de Berlin, 1779. Compare also Schwarz's paper in Bd. 75 of Borchardt's Journal, where further literary notices are collected together. Klein says further that in the ' Sachsischen (iesellschaft von Januar 1883,' he has considered the inner meaning {innere Bedeutung) of the differential equation
There are two papers by Lagrange, one immediately following the other, "Sur la construction des cartes g^ographiques," but I have not been able to discover the Schwarzian derivative in either of them.
41] On the Method of Reciprocants 285
been made in the letter from M. Halphen, in which the second and higher differential derivatives alone appear, the first differential derivative not figuring in the expression. These may be termed Pure Reciprocants.
Thus I repeat -7^ , and
\da?j ' da?
..<Py d^ tV.AoC^ dx' ' da? ' dx*^ \daJ>.
are pure reciprocants. Those from which the first derivative -— is not
excluded may be called Mixed Reciprocants. An example of such kind of Reciprocants is afforded by the Schwarzian above referred to. This dis- tinction is one of great moment, for a little attention will serve to make it clear that every pure reciprocant expressed in terms of x and y marks an intrinsic feature or singularity in the curve, whatever its nature may be, of which X and y are the co-ordinates ; for if in place of the variables (x, y) any two linear functions of these variables be substituted, a pure reciprocant, by virtue of its reciprocantive character, mu.st remain unaltered save as to the immaterial fact of its acquiring a factor containing merely the constants of substitution *.
The consequence is that every pure reciprocant corresponds to, and indicates, some singularity or characteristic feature of a curve, and vice versa every such singularity of a general nature and of a descriptive (although not necessarily of a projective) kind, points to a pure reci- procant.
Such is not the case with mixed reciprocants. They will not in general remain unaltered when linear substitutions are impressed upon the variables. Is it then neces.sary, it may be asked, to pay any attention to mixed reci- procants; or may they not be formally excluded at the very threshold of the inquiry ? Were I disposed to put the answer to this question on mere personal grounds, I feel that I should be guilty of the blackest ingratitude, that 1 should be kicking down the ladder by which I have risen to my present commanding point of view, if I were to turn my back on these humble mixed reciprocants, to which I have reason to feel so deeply indel)ted ; for it was the putting together of the two facts of the sub- stantial permanence under linear substitutions impressed upon the variables of the Schwarzian form and the simpler one which marks the inflexions of a curve — it was, if I may so say, the collision in my mind of these two facts — that kindled the spark and fired the train which set my imagination in a blaze by the light of which the whole horizon of Reciprocants is now illumined.
* The form as it stands shows that for y a linear function of x and y may be substituted; and the form reciprocated (by the interchanRe of x and y) shows that a similar substitution may be made for x. Hence arbitrary linear substitutions may be simultaneously impressed on x and y vithont inducing any change of form.
286 Oil the Method of Reciprocanta [41
But it is not necessary for me to defend the retention of mixed reciprocants on any such narrow ground of personal predilection. The whole body of Reciprocants, pure and mixed, form one complete system, a single garment without rent or seam, a complex whole in which all the parts are inextricably interwoven with each other. It is a living organism, the action of no part of which can be thoroughly understood if dissevered from connection with the rest.
It was in fact by combining and interweaving mixed reciprocants that I was led to the discovery of the pure binomial reciprocant, which comes im- mediately after the trivial monomial one, — the earliest with which I became acquainted, and of the existence of compeers to which I was for some time in doubt, and only became convinced of the fact after the discovery of the Partial Differential Equation, the master-key to this portion of the subject, which gives the means of producing them ad libitum and ascertaining all that exist of any prescribed type. Of this partial differential equation I shall have occasion hereafter to speak ; but this is not all, for, as we shall presently see, mixed reciprocants are well worthy of study on their own account, and lead to conclusions of the highest moment, whether as regards their applications to geometry or to the theory of transcendental functions and of ordinary differential equations.
The singularities of curves, taking the word in its widest acceptation, may be divided into three classes : those which are independent of homographic deformation and which remain unaltered in any perspective picture of the curve ; those whicii, having an express or tacit reference to the line at infinity, are not indelible under perspective projection, but using the word descriptive with some little latitude may, in so far as they only involve a reference to the line at infinity as a line, be said to be of a purely descriptive character ; and, lastly, those which are neither projective nor purely descriptive, having relation to the points termed, in ordinary parlance, "circular points at infinity" — for which the proper name is "centres of infinitely distant pencils of rays," that is, pencils, every ray of which is infinitely distant from every point external to it. Such, for instance, would be the character of points of maximum or minimum curvative, which, as we shall see, indicate, or are indicated by, that particular class of Mixed to which I give the name of " Orthogonal Reciprocants." All purely descriptive singularities alike, whether projective or non-projective, are indicated by pure reciprocants, and are subject to the same Partial Differential Equation; just as, in the Theory of Binary Quantics, Invariants, although under one aspect they may be regarded as a self-contained special class, admit of being and are most advantageou.sly studied in connection with, and as forming a part of, the whole family of forms commonly known by the name of "semi-, or subinvariants," but which I find it conduces to much
^:
41]
On the Method of Reciprocants
287
I
greater clearness of expression and avoidance of ambiguity or periphrasis to designate as Binariants.
The question may here be asked, How, then, are projective and non- projective pure reciprocants to be discriminated by their external characters?
I believe that I know the answer to this question, which is, that the
former are subject to satisfy a second partial differential equation of a certain
simple and familiar type, but this is a matter upon which it is not necessary
for me to enter on the present occasion*. It is enough for our present
purpose to remark that every projective pure reciprocant must, so to say, be
in essence a masked ternary covariant. For instance, if we take the simplest
dhi of all such, namely, a, that is -r^ , we have, if <^ {x, y) = 0,
daf
fd4\^_ \dy) ~
d^ daf
d^ dxdy
d^ dx
|
d'(l> dxdy |
d<\> dx |
|
d'<t> |
d4> dy |
|
d<i> dy |
• |
Obviously we might instead And now if we write <I> as
<»-)-(|)'
* = 0.
which, for facility of reference, let me call M.
of a = 0 substitute M = 0 to mark an inflexion.
the completed form of <f>, when made homogeneous by the substitution of z
for unity ; and if we suppose it to be of n dimensions in x, y, z, and call its
Hessian H, we shall obtain the syzygy
4.H+\— — _/'-*!5-V \da? ' dy^ \dxdy)
Hence the system «1> = 0, a = 0, will be in effect the same as the system <I> = 0, H = 0, and in this sense a may be said to carry // as it were in its bosom. And so in general every pure projective reciprocant may, in the language of insect transformation, be regarded as passing, so to say, first from the grub to the pupa or chrysalis, and from this again, divested of all superfluous integuments, to the butterfly or imago state.
Non-projective pure reciprocants undergo only one such change. There is no possibility of their ever emerging into the imago — their development being finally arrested at the chrysalis stage.
It would, I think, be an interesting and instructive task to obtain the imago or Hessianised transformation of the Mongian, but I am not aware
* In Paris, from which I correct the proofg, I have succeeded in reducing this conjecture to a certainty and in establishing the marvellous fact that every Projective Reciprocant, or, which is the same thing, every Differential Invariant, is, at the same time, an Ordinary Subinvariant. Thus a di£FereDtial invariant (or projective reciprocant) may be regarded as a single personality clothed with two diitinct natures — that of a reciprocant and that of a subinvariant.
288 On the Method of Reciprocants [41
that anyone has yet done, or thought of doing, this*. It seems to me that by substituting Reciprocants in lieu of Ternary Covariants we are as it were stealing a dimension from space, inasmuch as Reciprocants, that is. Ternary Covariants in their undeveloped state, are closely allied to, and march pari passu with, the fiarailiar forms which appertain to merely binary quantics.
I will now proceed to bring before your notice the general partial differential equation which supplies the necessary and sufiScient condition to which all pure reciprocants are subject.
It is highly convenient to denote the successive derivatives
dy d'y d*y da^' d^' dx"' ■■■ by the simple letters a,b,c,
The first derivative -^ plays so peculiar a part in this theory that it is
necessary to denote it by a letter standing aloof from the rest, and I call it t. This last letter, I need not say, does not make its appearance in any pure reciprocant. This being premised, I invite your attention to the equation in question, in which you will perceive the symbols of operation are separated from the object to be operated upon.
Writing F= 3a% + 10abBc+ (l-5ac + lOb") Ba + and calling any pure
reciprocant R,
VE = 0 is the equation referred to.
I cannot undertake, within the brief limits of time allotted to this lecture, to explain how this operation, or, as it may be termed, this annihilator Fis arrived at. The table of binomial coefficients, or rather half series of binomial coefficients, shown f in Chart 4, will enable you to see what is the law of the numerical coefficients of its several terms. Let the words weight, degree, extent (extent, you will remember, means the number of places by which the most remote letter in the form is separated from the first letter in the alphabet) of a pure reciprocant signify the same things as they would do if the letters a,b,c, ... referred, according to the ordinary notation, to Binariants instead of to Reciprocants. The number of binariants linearly independent of each other whose weight, degree and extent or order are w, i,j is given by the partition formula (w; i,j) -(w- 1; i,j) where in general (w; i,j) means the number of ways of partitioning w into i or fewer parts none greater than j.
* M. Halphen informs me that this has been done by Cayley in the Phil. Trans, for 1865, and subsequently in a somewhat simplified form by Painvin, Coniptes Rendus, 1874. But neither of these authors seems to have had the Boole-Mongian objectively before him, so that a slight supplemental computation is wanting to establish the equation between it and the function which either of them finds to vanish at a sextactic point.
[t p. 302 below.]
41] On the Method of Reciprocants 289
It follows immediately from the mere form of V that the corresponding formula in the case of Reciprocants of a given type w.i.j will be
(w; i,j)-{w-\; t' + l, j) the augmentation of i in the second term of the formula being due to the fact that, whereas in the partial dififerential equation for Binariants it is the letters themselves which appear as coefficients, it is quadratic functions of these in the case of Reciprocants. From the form of V we may also deduce a rigorous demonstration of the existence of Reciprocants strictly analogous to those with which you are familiar in the Binariant Theory, which are pictured in Chart 2, and are now usually designated as Protomorphs, as being the forms by the interweaving of which with one another (or rather by a sort of combined process of mixture and precipitation), all others, even the irreducible ones, are capable of being pro- duced. The corresponding forms for Reciprocants you will see exhibited in the same table. Each series of Protomorphs may of course be indefinitely extended as more and more letters are introduced. In the table I have not thought it necessary to go beyond the letter g. You also know that besides Protomorphs there are other irreducible forms, the organic radicals, so to say, into which every compound form may be resolved, always limited in number, whatever the number of letters or primal elements we may be dealing with. The same thing happens to Reciprocants as you will notice in the comparative table in Chart 2. Without going into particulars, I will ask you to take from me upon faitli the assurance that there is no single feature in the old familiar theory, whether it relates to Protomorphs, to Ground-forms, to Perpetuants, to Factorial constitution, to Generating Functions, or whatever else sets its stamp upon the one, which is not counterfeited by and reproduced in the parallel theory.
So much — for time will not admit of more — concerning pure reciprocants. Let me now say a few words en passant on Mixed Reciprocants. Pure Reciprocants, we have seen, are the analogues of Invariants, or else of the leading terms, for that is what are Semi- or Subin variants, of Covariantive expansions; each is subject to its own proper linear partial differential equation. Mixed Reciprocants are the exact analogues of the coefficients in such expansions other than those of the leading terms. Starting from the leading terms as the unit point, the coefficients of rank a> are subject to a partial differential equation of order to; and just so, mixed reciprocant-s, if involving t up to the power to, are subject to a partial differential equation of that same order.
I have alluded to a peculiar class of mixed under the name of "Orthogonal
Reciprocants." They are distinguished, as I have proved, by the beautiful
property that, if differentiated with respect to t, the result must be itself a
Reciprocant. In Chart 1 you will see this illustrated in the case of a mixed
& IV. 19
I
290 On the Method of Reciprocants [41
reciprocant (1 + <')6 — 3ta*, which serves to indicate the existence of points of maximum and minimum curvature. Its differential coefficient with respect to t is the oft-alluded-to Schwarzian, transliterated into the simpler notation. Proceeding in the inverse order — of Integration instead of Differentiation — I call your attention to a mixed reciprocant, of a very simple character, one which presents itself at the very outset of the theory, namely
tc — bah, which, integrated in respect to t between proper limits, yields the elegant orthogonal reciprocant
{t- + \)c-\Oabt+loa?.
Expressed in the ordinary notation, this, equated to zero, takes the form
\(dy\\Ad^_.Qdy d?y d'y fd^yV _
\\dx)^^]da^ ^"d^-d^-flt^ + ^H^/"
Mr Hammond has integrated this, treated as an ordinary differential equation, and has obtained the complete primitive expressed through the medium of two related Hyper-Elliptic Functions connecting the variables x and y (see* Chart 3). It may possibly turn out to be the case that every mixed reciprocant is either itself an Orthogonal Reciprocant, or by inte- gration, in respect to t, leads to one.
It will of course be understood that, in interpreting equations obtained by equating to zero an Orthogonal Reciprocant, the variables must be regarded as representing not general but rectangular Cartesian co-ordinates.
Here seems to me to be the proper place for pointing out to what extent I have been anticipated by M. Halphen in the discovery of this new world of Algebraical Forms. When the subject first dawned upon my mind, about the end of October or the beginning of November last, I was not aware that it had been approached on any side by any one before me, and believed that I was digging into absolutely virgin soil. It was only when I received M. Halphen's letter, dated November 25, in relation to the Mongian business already referred to, accompanied by a presentation of his memoirs on Differential Invariants, that I became aware of there existing any link of connection between his work and my own. A Differential Invariant, in the sense in which the term is used by M. Halphen, is not what at first blush I supposed it to be, and as in my haste to repair what seemed to me an omission to be without loss of time supplied, I wrote to M. Hermite it was, in a letter which has been or is about to be inserted in the Comptes Rendus of the Institute of France ; it is not, I say, identical with what I have termed a general pure reciprocant, but only with that peculiar species of Pure Reciprocants to which I have in a preceding part of this lecture referred as corresponding and pointing to Projective Singularities. In his
[* p. 302 below.]
I
41] (hi the Method of Reciprocants 291
splendid labours in this field Halphen has had no occasion to construct or concern himself with that new universe of forms viewed as a whole, whether of Pure or Mixed Reciprocants, which it has been the avowed and principal object of this lecture to bring under your notice.
I anticipate deriving much valuable assistance in the vast explorations remaining to be made in my own subject from the new and luminous views of M. Halphen, and possibly he may derive some advantage in his turn from the larger outlook brought within the field of vision by my allied investigations.
Let me return for a moment to that simplest class of pure reciprocants which I have called protomorphs. Each of these will be found (as may be shown either by a direct process of elimination, or by integrating the equations obtained by equating them severally to zero, regarded as ordinary dififerential equations between x and y) each of these, I say, will be found to represent some simple kind of singularity at the point {x,y) of the curve to which these co-ordinates are supposed to refer. Thus, for instance, No. 1 marks a single point of inflexion ; No. 2, points of closest contact with a common parabola ; No. 3, what our Cayley has called sextactic points, referring to a general conic ; No. 4, points of closest contact with a common cubical parabola ; and 80 on. The first and third, it will be noticed, represent projective singularities, and as such, in M. Halphen's language, would take the name of Differential Invariants. The second and fourth, having reference to the line at infinity in the plane of the curve, are of a non-projective character, and as such would not appear in M. Halphen's system of Differential Invariants. It is an interesting fact that every simple p.arabola, meaning one whose equation
m
can be brought under the form y = a;", corresponds to a linear function of a square of the third, and the cube of the second protomorph, and con- sequently will in general be of the sixth degree. In the particular case of the cubical parabola, the numerical parameter of this equation is such that the highest powers of b cancel each other so that the form sinks one degree, and becomes represented by the Quasi- Discriminant, No. 4.
This simple instance will serve to illustrate the intimate connection which exists between the projective and non-projective reciprocants, and the advantage, not to say necessity, of regarding them as parts of one organic whole.
It would take me too far to do more than make the most cursory allusion to an extension of this theory similar to that which happens when in the ordinary theory of invariants we pass from the consideration of a single Quantic to that of two or more. There is no difficulty in finding the partial differential equation to double reciprocants which, as far as I have
19—2
\
292 On the Method of Reciprocants [41
as yet pursued the investigation, appear to be functions of a, b, c, ... ; a', b', c, ...; and of {t — t').
The theory of double reciprocants will then include as a particular case the question of determining the singularities of paired points of two curves at which their tangents are parallel, and consequently the theory of common tangents to two curves and of bi-tangents to a single one.
I think I may venture to say that a general pure multiple reciprocant which marks ofif relative singularities, whether projective or non-projective, of a group of curves, is a function of the second and higher differential derivatives appertaining to the several curves of the group, and of the differences of the. first derivatives, whereas in a mixed multiple reciprocant these last-named differences are replaced by the first derivatives themselves. As a particular case, when the group dwindles to an individual and there is only one t, this letter disappears altogether from the form, for there are no differences of a single quantity.
In the chart (marked No. 2) you will see the table of Protomorphs carried on as far as the letter g inclusive, and will not fail to notice what may be termed the higher organisation of Reciprocantive as compared with ordinary Invariaiitive Protomorphs ; the degrees of the latter oscillate or librate between the numbers 2 and 3, whereas in the former the degree is variable according to a certain transcendental law dependent on the solution of a problem in the Partition of Numbers. Another interesting difference between general Invariants and general Pure Reciprocants consists in the fact that, whilst the number of the former ultimately (that is, when the extent is indefinitely increased) becomes indefinitely great, that of the latter is determinate for any given degree even for an infinite number of letters. ^M
In carrying on the table of protomorphs up to the letter h (see Chart 6) a new phenomenon presents itself, to which, however, there is a perfect parallel in tlie allied theory. An arbitrary constant enters into the form, its general value being a linear function of U and W (for which see Chart 6). But this is not all. If you examine the terms in both U and W (there are in all twelve such) you will find that these twelve do not comprise all of the same type to which they belong. There is a Thirteentii (a banished Judas), equally (I priori entitled to admission to the group, but which does not make its appearance among them, namely, h*d. I rather believe that a similar phenomenon of one or more terms, whose presence might be expected, but which do not appear, presents itself in the allied invariantive theory, but cannot speak with certainty as to this point, as the circumstance has not received, and possibly does not merit, any very particular attention.
41] On the Method of Reciprocants 298
Still, in the case before us, this unexpected absence of a member of the family, whose appearance might have been looked for, made an impression on my mind, and even went to the extent of acting on my emotions. I began to think of it as a sort of lost Pleiad in an Algebraical Constellation, and in the end, brooding over the subject, my feelings found vent, or sought relief, in a rhymed effusion, a jeu de sottise, which, not without some apprehension of appearing singular or extravagant, I will venture to rehearse. It will at least serve as an interlude, and give some relief to the strain upon your attention before I proceed to make my final remarks on the general theory.
TO A MISSING MEMBER Of a Family Group of Terms in an Algebraical Formula.
Lone and discarded one I divorced by fate,
Far from thy wished-for fellows — whither art flown ?
Where lingerest thou in thy bereaved estate,
Like some lost star, or buried meteor stone ?
Thou mindst me much of that pi-esumptuous one
Who loth, aught less than greatest, to be great,
From Heaven's immensity fell headlong down
To live forlorn, self-centred, desolate :
Or who, new Heraklid, hard exile bore,
Now buoyed by hope, now stretched on rack of fear,
Till throned Astraea, wafting to his ear
Words of dim portent through the Atlantic roar,
Bade him "the sanctuary of the Muse revere
And strew with flame the dust of Isis' shore."
Having now refreshed ourselves and bathed the tips of our fingers in the Pierian spring, let us turn back for a few brief moments to a light banquet of the reason, and entertain ourselves as a sort of after-course with some general reflections arising naturally out of the previous matter of my discourse. It seems to me that the discovery of Reciprocants must awaken a feeling of surprise akin to that which was felt when the galvanic current astonished the world previously accustomed only to the phenomena of machine or frictional electricity. The new theory is a ganglionic one : it stands in immediate and central relation to almost every branch of pure mathematics — to Invariants, to DiflFerential Equations, ordinary and partial, to Elliptic and Transcendental Functions, to Partitions of Numbers, to the Calculus of Variations, and above all to Geometry (alike of figures and of complexes), upon whose inmost recesses it throws a new and wholly unexpected light. The geometrical singularities which the present portion of the theory professes to discuss are in fact the distinguishing features of curves; their technical name, if applied to the human countenance, would lead us to call a man's eyes, ears, nose, lips, and chin his singularities ; but
V
294 On the Method of Reciproeants [41
these singularities make up the character and expression, and serve to distinguish one individual from another. And so it is with the so-called singularities of curves.
Comparing the system of ground-forms which it supplies with those of the allied theory, it seems to me clear that some common method, some yet undiscovered, deep-lying, Algebraical principle remains to be discovered, which shall in each case alike serve to demonstrate the finite number of these forms (these organic radicals) for any specified number of letters. The road to it, I believe, lies in the Algebraical Deduction of ground- forms from the Protomorphs*. Gordan's method of demonstration, so difficult and so complicated, requiring the devotion of a whole University semester to master, is inapplicable to reciprocants, which, as far as we can at present see, do not lend themselves to symbolic treatment.
How greatly must we feel indebted to our Cayley, who while he was, to say the least, the joint founder of the symbolic method, set the first, and out of England little if at all followed, example of using as an engine that mightiest instrument of research ever yet invented by the mind of man — a Partial Differential Equation, to define and generate invariantive forms.
With the growth of our knowledge, and higher views now taken of invariantive forms, the old nomenclature has not altogether kept pace, and is in one or two points in need of a reform not difficult to indicate. I think that we ought to give a general name — I propose that of Binariants — to every rational integral form which is nullified by the general operator
\aBb + fibSc + vcBa +... ,
where \, /i, v, ... are arbitrary numbers.
This operator, I think, having regard to the way in which its segments link on to one another, may be called the Vermicular.
Binariants corresponding to unit values of \, /i, v, ... may be termed standard binariants. Those for which these numbers are the terms of the natural arithmetical series 1, 2, 3, ... Invariantive binariants, which may be either complete or incomplete invariants ; these latter are what are usually termed semi- or sub-invariants. I may presently have to speak of a third class of binariants for which the arbitrary multipliers are the numbers 3, 8, 15, 24 ... (the squares of the natural numbers each diminished by unity) which, if the theorem I have in view is supported by the event, will have to be termed Reciprocantive Binariants. But first let me call attention to what seems a breach of the asserted parallelism between the Invariantive and the
* See the section on the Algebraical Deduction of the Ground-forms of the Quintic in my memoir on Subinvariants in the American Journal of Matliematics. [Vol. iii. of this Reprint, p. 580.]
I
41] On the Method of Reciprocants 295
Reciprocantive theories. In the former we have complete and incomplete invariants, but we have drawn no snch distinction between one set of pure reciprocants and another. A parallel distinction does however exist.
If we use IV, i, j to signify the weight, degree, and extent of an invariantive form, w is never less than the half product of ij ; when equal to it the form is complete. In the case of reciprocants certain observed facts seem to indicate that there exists an analogous but less simple inequality. If this
conjecture is verified it is not merely ~ — w, but ~ — (j — 2) — w, which
is never negative : and when this is zero, the form may be said to be complete*. There would then be thus complete forms in each of the two theories ; in the earlier one they take a special name : this is the only difference.
We have spoken of Pure Reciprocants as being either projective or non- projective, but so far have abstained from particularising the external characters by which the former may be distinguished from the latter. I have good reason to suspect that the former are distinguished from the latter by being Binariants; that, in addition to being subject to annihilation by the operator V, they are also subject to annihilation by the Vermicular operator when made special by the use of the numerical multipliers y, 8, 15 ... above alluded to, or in other words (as previotisly mentioned incidentally) are subject to satisfy two simultaneous partial differential equations instead of only onef.
* If thiB should turn out to be true, the "crude generating function" for reciprocants would be almost identical with that of in- and co-variants of the same extent j. The denominators would be absolutely identical ; as regards the numerators, while that for invariantive forms is l-o"ix~' the numerator for reciprocants would be l-a~^x~^. As I write abroad and from memory there is just a chance that the index of a here given may be erroneous.
t As already stated in a previous footnote this conjecture is fully confirmed, my own proof having been corroborated (if it needed corroboration) by another entirely different one invented by M. Halphen, who fully shares my own astonishment at the fact of there being forms (half- horse, half-alligator) at once reciprocants and sub-invariants, and as such satisfying two simultaneous partial differential equations.
If instead of denoting the successive differential derivatives (starting from the second) a,b,c, ... we call them 1.2. a, 1.2.3.b, 1.2.3. 4. c, ... the two Annihilators will be
a3j -I- 2i«, + Scig + ids, + ... a' »nd * 2 ** "*" ***'' + ^ (*" ■'■**'' *'« + '' ("''■*" **) '« ■*■ •••
the latter being my new operator, the Reciprocator V, accommodated to the above-stated change of notatiou for the successive differential derivatives.
Hardly necessary is it for me to point out in explanation of the semi-sums }b^, ... that we may write the MacMahonised V under the form
4a% + o {ab + ba) d^+&(ac + lj' + ca)S^ + T {ad + bc + cb + da)St+ .... It is to be presumed that in addition to mixed reciprocants (the ocean into which flows the sea of pore reciprocants, as into that again empties itself the river of projective reciprocants) there may
^1 dx reciprocants, the most general of all, in which case we must speak of the content of these as the
exist a theory of forms in which w as well as —- will appear, or, so to say, doubly mixed
dx
296 On the Method of Reciprocants [41
Projective Reciprocants we have seen are disguised or maaked Ternary Covariants — Covariants in the grub, the first undeveloped state. Now ternary covariants are capable, it may or may not be generally known, of satisfying 6 reducible to 2 simultaneous Partial Diiferential Equations, and at first sight it might be surmised that nothing would be gained by the substitution of the two new for the two old simultaneous partial differential equations. But the fact is not so, for the old partial differential equations are perfectly unmanageable, or at least have never, as far as I know, been handled by any one, for they have to do with a triangular heap, whereas the new ones are solely concerned with a linear series of coefficients.
I have alluded to there being a particular form common to the two theories. In the one theory it is the Mongian alluded to in the correspondence, which has been read, with M. Halphen. In the other it is the source of the skew covariant to the cubic. If the latter be subjected to a sort of MacMahonic numerical adjustment, it becomes absolutely identical with the former. Let ns imagine that before the invention of Reciprocants an Algebraist happened to have had both forms present to his mind, and had thought of some contrivance for lowering the coefficients of the Mongian written out with the larger coefficients, and had thus stumbled upon this striking fact. It could not have failed to vehemently arouse his curiosity, and he would have set to work to discover, if possible, the cause of this coincidence. He would in all probability have addressed himself to the form which precedes the source alluded to in the natural order of genesis, and have applied a similar adjustment to the much simpler form, ac — b": having done so he would have tried to discover to what singularity it pointed — but his efforts to do so we know must have been fruitless, and he would have felt disposed to throw down his work in despair, for the intermediate ideas necessary to make out the parallelism would not have been present to his mind. So long as we confine ourselves to Differential Invariants, that is, to projective pure reciprocants, we are like men walking on those elevated ridges, those more than Alpine summits, such as I am told* exist in Thibet, where it may be the labour of days for two men who can see and speak to each other to come together. Reciprocants supply the bridge to span the yawning ravine and to bring allied forms into direct proximity.
ocean and of the others as sea, riTer, and brook. Curious is it to reflect that in the theory which as it exists comprises Invariantives, Reciprocants, and Invariantive Beciproeants or Reciprooant Invariantives, the order of discovery was (1) Invariantives (Eisenstein, Boole, <fcc.) ; (2) In- variantive Reciprocants (Monge and Halphen); (3) Reciprocants (Schwarz, the author of this lecture).
* I think my informant was my friend Dr Inglis, of the Athenteum Club, who some time ago undertook a journey in the Himalayas in the hopes of coming upon the traces of a lost religion which he thought he had reason to believe existed among mankind in the pre-GIacial period of the earth's history.
41] On the Method of Reciprocants 297
I have spoken of mixed reciprocants as being subject to satisfy not a linear partial differential equation, but one of a higher order dependent on the intensity, so to say, of its mixedness — the highest power, that is to say, of the first differential derivative which it contains, and it might therefore be supposed that these forms are much more difficult to be obtained than pure reciprocants. But the fact is just the reverse, for as I discovered in the very infancy of the inquiry, and have put on record in the September or October number* of the Mathematical Messenger, mixed reciprocants may be evolved in unlimited profusion by the application of simple and explicit processes of multiplication and differentiation. From any reciprocant whatever, be it mixed or pure, new mixed ones may be educed infinitely infinite in number, inasmuch as at each stage of the process, arbitrary functions of the first differential derivative may be introduced.
The wonderful fertility of this method of generation excited warm interest on the part of one of the greatest of living mathematicians, the expression of which acted as a powerful incentive to me to continue the inquiry. They may be compared with the shower of December meteors shooting out in all directions and covering the heavens with their brilliant trains, all diverging from one or more fixed radiant-points, the radiant-point in the theory before us being the particular form selected to be operated upon.
The new doctrine which I have endeavoured thus imperfectly to adumbrate has taken its local rise in this University, where it has already attracted some votaries to its side, and will, I hope, eventually obtain the cooperation of many more. I have ventured with this view to announce it as the subject of a course of lectures during the ensuing term.
When I lately had the pleasure of attending the new Slade Professor's inaugural discourse, I heard him promise to make his pupils participators in his work, by painting pictures in the presence of his class. I aspire to do more than this — not only to paint before the members of my class, but to induce them to take the palette and brush and contribute with their own hands to the work to be done upon the canvas. Such was the plan I followed at the Johns Hopkins University, during my connection with which I may have published scores of Mathematical articles and memoirs in the journals of America, England, France, and Germany, of which probably there was scarcely one which did not originate in the business of the class- room ; in the composition of many or most of them I derived inestimable advantage from the suggestions or contributions of my auditors. It was frequently a chase, in which I started the fox, in which we all took a common interest, and in which it was a matter of eager emulation between my hearers and myself to try which could be first in at the death.
[* p. 255 above.]
\
298 On the Method of Reciprocanta [41
During the past period of ray professorship here, imperfectly acquainted with the usages and needs of the University, I do not think that my labours have been directed so profitably as they might have been either as regards the prosecution of my own work or the good of my hearers : my attention has been distracted between theories waiting to be ushered into existence and providing for the daily bread of class-teaching. I hope that in future I may be able to bring these two objects into closer harmony and correlation, and I think I shall best discharge my duty to the University by selecting for the material of my work in the class-room any subject on which my thoughts may, for the time being, happen to be concentrated, not too alien to, or remote from, that which I am appointed to teach; and thus, by example, give lessons in the diflRcult art of mathematical thinking and reasoning — how to follow out familiar suggestions of analogy till they broaden and deepen into a fertilising stream of thought — how to discover errors and to repair them, guided by faith in the existence and unity of that intellectual world which exists within us, and is at least as real as that with which we are environed.
The American Mathematical Journal, conducted under the auspices of the Johns Hopkins University, which has gained and retains a leading position among the most important of its class, whether measured by the value of its contents or the estimation in which it is held by the Mathematical world, bears as its motto —
TrpayfiaTtov f\(y)(os oil ^Xenofiivatv.
I have the pleasure of seeing among my audience this day the most distinguished geometer of Holland, Prof. Schoute, who has done me the signal honour of coming over to England to be present at this lecture, who hospitably entertained me at Groningeu (in a vacation visit which I recently paid to his country, the classic soil which has given birth to an Erasmus, a Grotius, a Boerhaave, a Spinoza, a Huyghens, and a Rembrandt), and who was kind enough, in proposing my health at a party where many of his colleagues were present, to say that he felt sure " that I should return to England cheered and invigorated, and would, ere long, light on some discovery which would excite the wonder of the Mathematical world."
I do not venture to affirm, nor to think, that this vaticination has been fulfilled in the terms in which it was uttered, but can most truly say tiiat the discovery, which it has been my good fortune to be made the medium of revealing, has excited my own deepest feelings of ever-increasing wonder rising almost to awe, such as must have come over the revellers who saw the handwriting start out more and more plainly on the wall, or the sciemiati crowding round the blurred palimpsest as they began to be able to decipher
I
41] On the Method of Reciprocants 299
characters and piece together the sentences of the long lost and supposed irrecoverable De Repuhlicci.
When I was at Utrecht, on my way to Groningen, Mr Grinwis, the Professor of Mathematics at that University, showed me au English book on " Differential Equations," which had just appeared, of which he spoke in high terms of praise, and said it contained over 800 examples. I wrote at once for the book to England, and on seeing it on my arrival, forgetting that it had been ordered, mistook it for a present from the author or publisher, and, what is unusual with me, read regularly into it, until I came to the section on Hyper-geometrical series, where the Schwarzian Derivative (so named by Cayley after Prof. Schwarz) is spoken of
Perhaps I ought to blush to own that it was new to me, and my attention was riveted by the property it possesses, in common with the more simple form which points to inflexions on curves, of remaining substantially unaltered, of persisting as a factor at least of its altered self, when the variables which enter it are interchanged. Following out this indication, I at once asked myself the question, " ought there not to exist combinations of derivatives of all orders pos.sessing this property of reciprocation?" That question was soon answered, and the universe of mixed reciprocants stood revealed before me. These mixed reciprocants, by simple proces.ses of combination, led me to the discovery of the first pure reciprocant, 36- — oac : whereupon I again put the question to myself, " are there, or are there not, others of this form, and if so, what are they?"
In an unexpected manner the question was answered, and my curiosity gratified to the utmost by the discovery of the partial differential equation which is the central point of the theory, and at once discloses the parallelism between it and the familiar doctrine of Invariants. Two principal exponents of that doctrine, who have infused new blood into it, and given it a fresh point of departure — Capt. MacMahon and Mr Hammond — I have the pleasure of seeing before me. Mr Kempe, who is also present, has lately entered into and signally distinguished himself in the same field, availing himself in so doing of his profound insight into the subject of linkages, his interest in which I believe I may say received its first impulse from the lecture which he heard me deliver upon it at the Royal Institution in January 1874, on the very night when the Prime Minister for the time being sent round letters to his supporters announcing his intention to dissolve Parliament. Of the two events I have ever regarded the lecture as by far the more important to the permanent interests of society. He has lately applied ideas founded upon linkages to produce a most original and remarkable scheme for explaining the nature of the whole pure body of Mathematical truth, under whatever different forms it may be clothed, in a memoir which has been recommended to be printed in the Transactions of the Royal Society, and which, I think.
300 On the Method of Redprocants [41
cannot fail when published to excite the deepest interest alike in the Mathematical and the Philosophical worlds*.
I also feel greatly honoured by the presence of Prof. Greenhill, who will be known to many in this room from his remarkable contributions to the theory of Hydrodynamics and Vortex Motion, and who has sufficient candour and largeness of mind to be able to appreciate researches of a diflferent character from those in which he has himself gained distinction.
I should not do justice to my feelings if I did not acknowledge my deep obligations to Mr Hammond for the assistance which he has rendered me, not only in preparing this lecture which you have listened to with such exemplary patience, but in developing the theory; I am indebted to him for many valuable suggestions tending to enlarge its bounds, and believe have been saved, by my conversations with him, from falling into some serious errors of omission or ovei-sight. Saving only our Cayley (who, though younger than myself, is my spiritual progenitor — who first opened my eyes and purged them of dross so that they could see and accept the higher mysteries of our common Mathematical faith), there is no one I can think of with whom I ever have conversed, from my intercourse with whom I have derived more benefit. It would be an immense gain to Science, and to the best interests of the University, if something could be done to bring such men as Mr Hammond (and, let me add, Mr Buchheira, who ought never to have been allowed to leave it) to come and live among us. I am sure that with their endeavours added to my own and those of that most able body of teachers and researchers with whom I have the good fortune to be associated — my brother Professors and the Tutorial Staff of the University — we could create such a School of Mathematics as might go some way at least to revive the old scientific renown of Oxford, and to light such a candle in England as, with God's grace, should never be put outf.
* In his memoir for the Phil. Trans. Mr Kempe contends that any whatever mathematical proposition or research is capable of being represented by some sort of simple or compound linkage. One would like to know by what sort of linkage he would represent the substance of tlie memoir itself.
t I have purposely confined myself in my lecture to reciprocants, indicative of properties of plane curves, but had in view to extend the theory to the case of higher dimensions in space leading to reciprocants involving the differential derivatives of any number of variables y, z, .... M. Halphen, with whom I have had the great advantage of being in communication during my stay in Paris, has anticipated me in this part of my plan, and has found that the same method which I have used to obtain the Annihilator V applied to a system of variables leads to an Annihilator of a very similar form to V, and at my request will publish his results in a forth- coming number of the Comptcs Rendm. Thus the dominion of reciprocants is already extended over the whole range of forms unlimited in their own number as well as in that of the variables which they contain.
41]
On the Method of Reciprocants
301
Tables of Singularities and Formula referred to in the Preceding Lecture.
Chart 1.
Inflexion
Node
Cnsp
Pointa of maximum and minimnm cnrvatare
Bitangent
Chart 2. — Protomorphs.
Binarianti. a
ac — l^
ay+6a6e + 2acd+862rf_e6c» ag-ebf+lbce-lOcP
Reciprocants. a
3ac-5l^
9a'd-ibabc + iOlr> ba'e -Zbabd Jr lac^ + Zbb-c iba?f- Alfkfibe -iia'cd + \\20a}^d-Z\bahc*
-112063c a*g - 12a6/- 4b0ace+ 7926% + b68ade^
-27726crf+1925c3
302
On the Method of Reciprocanta
[41
Chart 3. No. 1. a No. 2. Zac-bV* No. 3. 9a2rf - 45a6c + 406'
No. 4. 45aSflP- 450a«ic+ 192aV+400a6Srf+ 165a6V - 4006*c ^ I dt
infe
'-/.
^|« ( 1 - 15<i' + 15<* - <«)+X (3< - lOf + 3<»)} ■*" " V=3a% + l0ab8c+{nac + \0b^)ba + {2lad+35bc)», + (28a« + 566rf + 35c2) 8/+ . . .
ChaKT 4. — COEFFICIRNTS OF AnSIHILATOR V.
|
1 4 |
3 |
|||
|
1 5 |
10 |
|||
|
1 6 |
15 10 |
|||
|
1 7 |
21 35 |
|||
|
1 8 |
28 56 35 |
|||
|
1 9 |
36 84 126 |
|||
|
1 10 |
45 120 210 126 |
|||
|
Chart 5.— Rkciprocant Transformations. |
||||
|
Grub |
Chrytalii |
Imago |
||
|
dh) |
d^(t> d^cji |
d({> |
d^* oP* cP* |
|
|
d^ |
dx^ dvdy |
dx |
dx^ dxdy dxdz |
|
|
CP0 flP<^ |
d<t> |
d^<t> d'* (P* |
||
|
flterfy dy' |
dy |
dxdy dy' dydz |
||
|
d<f> d4> |
d'* d^ cP* |
|||
|
dx dy |
• |
dxdz dydz dz* |
(a)
(M)
(B)
dy d?y 3 /flPw\« . ^, „ u • .u • •.. r 3a« "3^ —^ " o V ^7^ ) ^^ Schwarzian, otherwise written to .
Chart 6. — The H Reciprocantive PROTOMOKrH.
|
u |
ir |
||
|
65a<A |
120aV |
The Vermicular Operator. |
|
|
- 975a36^ -990a3c/ |
- 200a'i2/ -195a3rfe |
\a8^+^b8^ + vcdi+7rd8, + ... |
|
|
+ 6200a262/ |
- \\ba?hce |
Examples. |
|
|
+ 4690a26ce |
+ 1000a63e |
||
|
-1540a6% |
+ 1365a26rf2 |
ahb + bh, + cbi+d&,+ ... |
|
|
- 2730a26ci2 |
- nld^c^d |
a\ + 26S, + Zcbi + idb. + . . . |
|
|
+7161aVrf |
-22260o6W |
3o86+868c+ 15cSi + 24<i8,+ ... |
|
|
+ 3080a6W |
+ 2485a6c3 |
||
|
- 24255a6c3 |
+ 1056^02 |
Ig^ b*d does not appear in either |
|
|
+ 254106^02 |
H^-kU^-MW |
6^ or W. |
I
I
A and M are arbitrary numbers.
I
42.
LECTURES ON THE THEORY OF RECIPROCANTS.
[American Journal of Mathematics, viil, (1886), pp. 196 — 260 ; ix. pp. 1 — 37, 113—161, 297—352 ; x. pp. 1—16. Delivered in Oxford, 1886.]
The lectures here reproduced were delivered, or are still in the course of delivery, before a class of graduates and scholars in the University of Oxford during the present year. They are to be regarded as easy lessons in the new Theory of Reciprocants of which an outline will be found in Nature for January 7, which contains a report of a Public Lecture on the subject delivered before the University of Oxford in December of the preceding year.
They are designed as a practical introduction to an enlarged theory of Algebraical Forms, and as such are not constructed with the rigorous adhesion to logical order which might be properly expected in a systematic treatise. The object of the lecturer was to rouse an interest in the subject, and in pursuit of this end he has not hesitated to state many results, by way of anticipation, which might, with stricter regard to method, have followed at a later period in the course.
There will be found also occasional repetitions and intercalations of allied topics which are to be justified by the same plea, and also by the fact that the lectures were not composed in their entirety previous to delivery, but gradually evolved and written between one lecture and another in the way that seemed most likely to the lecturer to secure the attention of his auditors.
Since the delivery of his public lecture in December last, papers have been contributed on the subject to the Proceedings of the Mathematical Society of London by Messrs Hammond, MacMahon, Elliott, Leudesdorf and Rogers, and one to the Comptes Rendus de I'Institut by M. George Perrin. It may therefore be inferred that the lectures have not altogether failed in attaining the desired end of drawing attention to a subject which, in the opinion of the lecturer, constitutes a very considerable extension of the previous limits of algebraical science.
304 Lectures on the Theory of Redprocants [42
LECTURE I.
A new world of Algebraical forms, susceptible of important geometrical applications, has recently come into existence, of which I gave a general account in a public lecture at the end of last term. I propose in the follow- ing brief course to go more fully into the subject and lay down the leading principles of the theory so far as they are at present known to me. The parallelism between the theory of what may be called pure reciprocants and that of invariants is so remarkable that it will be frequently expedient to pass from one theory to the other or to treat the two simultaneously. It may be as well therefore at once to give notice that the term invariant will hereafter be applied alike to invariants ordinarily so called and to those more general algebraical forms which have been termed sources of covariants, differentiants, seminvariants, or subinvariants. A form which is an invariant in the old sense will be termed, when necessary to specify it, a satisfied invariant, an expression which the chemico-graphical representation of invariants or covariants will serve to explain and justify.
In an elucidatory course of lectures such as the present, it will be advis- able to follow a freer order of treatment than would be suitable to the presentation of it in a systematic memoir. My object is to make the theory known, to excite curiosity regarding it, and to invite co-operation in the task of its development.
By way of introduction to the subject, let us begin with an investigation of the properties of a differential expression involving only the first, second and third differential coefficients of either of two variables in respect to the other. For this purpose let us consider not what I have called the Schwarzian itself, which is an integral rational function of these three quantities, but the fractional expression
d£ _ 3 / cte^M dy •iXdyj dx \dx/
which becomes the Schwarzian when cleared of fractions, and which after Cay ley we may call the Schwarzian Derivative and denote by
{y, «) ; f
(x, y) will then of course mean
d?x /^\'
5p_3[rf7M
dx 2 I ofa- I "
dy \dy/ j
I
42] Lectures on the Theory of Reciprocants
It is easy to establish the identical equation
Using for brevity y', y", y"' to denote, as usnal,
305
(1)
dy d?y d'y dx' dx^' d^'
and x„ «„, a;,,, to denote
dx d^x d'x -dy' 5^" dp'
respectively, the relation to be verified is
Now,
and
y' ^ • a;/
_ 1
""'dy'^'^'^'y'- dx\y') y'''
'" -J" ^ "' y" dx\ y'V y'* y''
dy'
Whence we obtain
^F,.. ^^„ -[ y'^^ y"J y'«
= -±.J2y'y"'-3y"').
and the truth of (1) is manifest. This may be put under the form
2y'y"'-Sy"' 2x^„-3x„
showing that a certain function of the first, second and third derivatives of one variable in respect to another remains unaltered, save as to algebraical sign, when the variables are interchanged. An example/'of a similar kind with which we are all familiar is presented by the well-known function dhf /dy\i .... , ^ d'x /dx\i
di^^(dx) ' ""^''^ '' "^"^' '""-df^Kdy) •■
We are thus led to inquire whether there may not be an infinite number of algebraical functions of differential derivatives which possess a similar property, and by prosecuting this inquiry to lay the foundations of the theory of Reciprocation or Reciprocants.
Having regard to the fact that the present theory originated in that of the Schwarzian Derivative, I shall proceed to demonstrate, although this is 8. IV. 20
I
306 Lectures mi the Theory of Recijjrocants [42
not strictly necessary for the theory of Reciprocants, the remarkable identity
fdzy
{y,x)-{z,x)={J^ .{y.z).
This identical relation is the fundamental property of Schwarzians, and from it every other proposition concerning their form is an immediate deduction.
In the following proof*, y and z are regarded as two given functions of any variable t, and x aa & variable function of the same : so that y and z are functions of x for any given function that x is of t.
It will be seen that
fdx\
Ky..)-(...)}(g)'
remains unaltered by any infinitesimal variation 6 of x, that is, when x becomes a; + e^ {x), e being an infinitesimal constant and ^ {x) an arbitrary finite function of a;.
For brevity, let accents denote differential derivation in regard to x, and let any function of x enclosed in a square parenthesis signify the augmented value of that function when x becomes x + 6. In calculating such augmented values, since we suppose that 6 = e<f>(x), it is clear that 6, & , 6" ... are each of them infinitesimals of the first order, and consequently that all products, and all powers higher than the first of these quantities, may be neglected.
We have therefore r^q = ^y = -^ = v' - ^V
^y^ dx+dd i+ff y "^
un - ^^^ dx^'~^^^ y"{\-&)-e"^ ^y^~dx+d0 1 + 6' 1 + 6'
= y"-Wy"-6"y
r,/"i- ^^-^ ^ if - Wy") - 6"y' ^ y,„ ^^ _ ^g,^ _ ^^,y„ _ ^.y, ^y ^~dx + d6 1 + 6' 1 + 6'
= y'" - S6'y"' - Z6"y" - 6"'y'. Hence [yV"] = yy'" - ^6'yy'" - W'y'f - r'y"
W"\ =y'^-(^ffy"'-Z6"y'f [y^] =y'^-26'y\ And since by definition
* As originally given in the Messenger of Mathematics, Vol. xv., this was defaced by so many errata as to render expedient its reproduction in a corrected form.
I
42] Lectures on the Theory of Reciprocants 307
we readily obtain
V(y,x)-\ = ^^,-^6'{y,x)-e"'^{y,x){\-2d-)-e"'.
So also [{z, x)] = (z, x) (1 - 2d') - ff".
Whence by subtraction
\iy, X) - {z, x)-\ = (1 - 2^) [{y, x) - (z, x)].
Dividing the left-hand side of this by [/^], and the right-hand side by /* (1 — 20') which is the equivalent of [z''], our final result is
[{y, x) - (z, x)'\ _ (y, x) - (z, x)
Thus, then, we have seen that the expression
(y, x) - (z, x)
does not vary when x receives an infinitesimal variation €<j)(x), from which it
follows, by the general principle of successive continuous accumulation, that
the same will be true when x undergoes any finite arbitrary variation, and
consequently this expression has a value which is independent of the form of
X regarded as a function o{ t; it will, of course, be remembered that y and z
are supposed to be invariable functions of t. Let x become z, then (y, x)
dz becomes (y, z), while at the same time (z, x) vanishes and -j- becomes unity :
80 that we obtain
\dx) Hence, whatever function x may be of t,
fdz^-'
{y,x)-{z,x) = {^^ .iy.z). (2)
To this fundamental proposition the equation marked (1), itself the import- ant point in regard to the Theory of Reciprocants, is an immediate corollary. For if in (2) we interchange y and z, it becomes
{z,x)-{y,x)=(-£j .{z,y); and now, making x = z, we have
which is the same as (1), except that z occupies the place of x.
I
20—2
308 Lectures on the Theory of Heciproeants [42
But (1) may be obtained more immediately from (2) by substituting in it X for y and y for z, leaving x unaltered ; when it becomes
This is equivalent to saying that
^'f - 3y"' = - y" (2xx,„ - 3x,;), a verification of which has been given already.
Observe that *^-^ — ,^^ or (y, x) contains f^j in its denominator and
(dx\^ . . . 'dy\'
J-) in its denominator, which is the same as [~f-) in the
numerator. Thus it is that the square of -r enters three times.
Let me insist for a moment on the import of the fact brought to light
in the course of this investigation, that ^ — , .^ ' — - is invariable when x, y
\dx) and z being regarded as functions of t, x alters its form, but y and z retain
theirs. Of course we might write i-r-] in the denominator instead of ( j- j ,
and then make the same aflfirmation as before ; as will be evident if we only remember that by hypothesis y and z are both of them constant functions of
(dzX^ -J- j must also be so. This is tantamount to saying
that when the same conditions are fulfilled {{y, x) — (z, x)] (dx)' is invariable, that is, that when x becomes X in virtue of any substitution (including a homographic one) impressed upon it,
1(2/, X) - (z, X)} (dxy = {(y, X) - {z, X)} {dXf,
and thus we see that when x becomes X,
(y, x) - (z, x)
remains unaltered except that it takes to itself the factor (-r- 1 which depends
solely on the particular substitution impressed on x.
If y =f{x), z = ^ (x), and X = to (x),
our formula becomes
{{/x, x) - (<t>x, x)} (dxy = {(/a)-'Z, X) - (</.«-'Z, X)} (dxy,
so that, speaking of Quantics and Covariants with respect to a single variable X, (fx, x)-(<})X, x) is to all intents and purposes a Covariant to the simul- taneous forms /(«) and <^{x), in a sense comprehending but far transcending that in which the term is ordinarily employed ; for it remains a persistent
I
42] Lectures on the Theory of Reciprocants 309
factor of its altered self when for x any arbitrary function of x is substituted, the new factor taken on depending wholly and solely on the particular sub- stitution impressed upon x. In the ordinary theory of invariants, the substitution impressed is limited to be homographic ; in this case it is absolutely general. We might, moreover, add as a corollary that if (y, x), {z, x), (u, x) ... are regarded as roots of any Binary Quantic, every invariant of that Binary Quantic is a covariant in the extended sense in which the word has just been used, in respect to the system of simultaneous forms /(x), <f> (x), -^ (x) For every such invariant will be a function of
(y, x) - (z, x), {y, x) - (u, x), (z, x) - (u, x), ... and will therefore remain a persistent factor of its altered self, taking on a
power of ^— as its extraneous factor. ax
Calling (fx, x) the Schwarzian Derivative of /(x), our theorem may be stated in general terms as follows :
All invariants of a Binary Quantic whose roots are the Schwarzian Deri- vatives of a given set of functions of the same variable are Covariants (in an extended sense) of that set of functions.
The theory of the Schwarzian derivative originates in that of the linear differential equation of the second order,
u"+2Pu+Qu = 0, which becomes, when we write u = ve~!^^,
t;"+/y = 0, where I=Q-F'-P'.
Now, suppose that ttj and u^ are any two particular solutions of the first of these equations, and let z denote their mutual ratio; so that, when v, and v.i are the cori'esponding particular solutions of the second equation, we readily obtain
^ = ^ = -' M, v^'
and therefore, z = z .
A second difiFerentiation gives
,, vX'-%v" 2y,'(«,<-t;X)
IP
V,'
V, . ■ »l" ^«" T
But since — - = — = — i,
the first term of the expression just found vanishes identically, and we have
z = ,
310 Lectures on the Theory of Reciprocanta [42
or. "'=--27-
DiflFerentiating this again, we find
g"' 3 «"« Hence Y-|^ = 2^'
where the left-hand side of the equation is "the Schwarzian Derivative " with z written in the place of y.
S
LECTURE II.
The expression "^y'y"' — 3y"^ which we have called the Schwarzian, may be termed a reciprocant, meaning thereby that on interchanging y, y", y'" with x^, x^^, a;,,, its form remains unaltered, save as to the acquisition of what may be called an extraneous factor, which, in the case before us, is a power of y (with a multiplier — 1). Before we proceed to consider other examples of reciprocants it will be useful to give formulae by means of which the variables may be readily interchanged in any differential expression.
We shall write t for y and t for its reciprocal x^ , using the letters a,b,c,... to denote the second, third, fourth, etc., differential derivatives of y with respect to x, and a, /8, 7, ... to denote those of x with respect to y. The advantage of this notation will be seen in the sequel.
The values of a, 0,y, ... in terms of t, a,b,c,... are given by the formulae o = - a -r i',
7 = - cf + lOabt - I5a' ^ f,
B = -dt' + {loac + 106') t^ - lOoa^bt + 105a* -=- f,
e=-ef+ {21ad + 356c) t^ - (210a'c + 280ab') C + 1260a'bt - 94:5a' h- t",
If, in these equations, we write
a = 1.2.ao, 6=1.2.3.01, c= 1 . 2 . 3 . 4.0,, ... and a=1.2.a„, /3 = 1.2.3.ai, 7 = 1.2.3.4.02,...
42] Lectures on the Theory of Reciprocants 311
they become
Oo = - tto -=- <'.
ai = - ait + 2a^-r-1?,
fltj = — a^t^ + 5a„ait — 50o' -r V,
a, = - a3<^ + (GooO, + 3ai») i^ - 21a„=ffli< + 14a„* ^ <»,
a, = - a^i^ + (TkoOs + 7a,aa) f - (28a„»a3 + 28oh>=) t^ + 84ao»ai< - 42a„» ■¥ i",
Any one of the formulae iu either set may be deduced from the formula immediately preceding it by a simple process of differentiation.
Thus,s,nce ^ = j— '^''^ dy = 1 ' di'
. dB \ d (-htJr So'^N
we have — ' '
dy
_1 d /-bt + 3a'\ ~ f dx\ f / ■
But T^ = 7 and T- = a3{ + 69o + c96+ ...,
ay aa;
80 that y = j(adt + bda+cdb + ...)( j
= -, (- c«' + lOabt - 15a').
By continually operating with - (ddt + bda + cdi, + ...) the table may be
z
extended as far as we please, the expressions on the right-hand side being the successive values of
|J(aae + Wa+ca»+...)}"(-P)
found by giving to n the values 0, 1, 2, 3
Precisely similar reasoning shows that, when the modified letters
Og, O] , a,, ■ ■ ■ are used,
(n + 2) a„ = - (2o,3t + 3a, 3a, + 4a,9a, + • • •) a»-i , z
1 1 " ' OoN
A proof of the formula
r
a„ = -«-»-3(e"«)a„, obtained by Mr Hammond, in which
F = 4 . -^ 3a, + oaoOiSo, + 6 f ajO, + y 1 3a, + 7 (a.aj + aiOj) Sa. + • •• ,
will be given later on, when we treat of this operator, which, in the theory of Reciprocants, is the analogue of the operator 03^ + 263c + 3c3d+ ..., with which we are familiarly acquainted in the theory of Invariants.
and that a„ = ii^, . , „^
" 3.4.5 ... (n + 2)
312 Lectures on the Theory of Reciprocants [42
Consider the expression
ct — 5ab.
If, in 7T — 5a/8, which may be called its transform, we write
1 a a -bt + Sa* -cf + 10aht-15a*
'■=<■ *" — ?' ^ if- ' 'y= e •
this becomes a fraction whose denominator is <*, while its numerator is - c<' + lOabt -lba' + ba (- bt + 3a') = - c^' + haht.
Removing the common factor t from the numerator and denominator of this fraction, we have
. - ct — bob 7T - oa/3 = ^ — .
Here, then, as in the case of the well-known monomial for which < a = -Va,
and the Schwarzian for which
2bt -^a' = -i? (2y3T - Sa"), the expression ct — bah = — V (yr — 5a;S)
changes its sign on reciprocation.
That reciprocation is not always accompanied with a change of sign will be clear if we consider the product of any pair of the three expressions given above. Or we may take, as an example of a reciprocant in which this change of sign does not occur, the form
Zac-hb\
^ o .a, Sa(ct^-10aht + 15a')-5(bt-3a'y
Here 3a7 - .5/3^ = — ^^ — — - — — ^ .
z
In the fraction on the right-hand side the only surviving terms of the numerator are those containing the highest power of t, the rest destroying one another. Thus
Say - 5/3^ = i-(3ac - 56»).
Reciprocants which change their sign when the variables x and y are interchanged, will be said to be of odd character; those, on the contrary, which keep their sign unchanged will be said to be of even character. The distinction is an important one, and will be observed in what follows.
Forms such as the one just considered, whei-e t does not appear in the form itself, but only in the extraneous factor, will be called Pure Reciprocants, in order to distinguish them from those forms (of which the Schwarzian 2tb—3a' is an example) into which t enters, which will be called Mixed Reciprocants. It will be seen hereafter that Pure Reciprocants are the analogues of the invariants of Binary Quantics.
42] Lectures on the Theory of Reciprocants 313
With modified letters (that is, writing a = 2ao, b = Ga^, and c = 24a2)
Sac — 56^ becomes IMaaO^ — ISOai" = 36 (4iaa(ii — Soi").
Operating on this with
F= 2a^da^ + 5a„aj3a2+ •••'
we have F(4aoa2 — S^i") = 0.
We shall prove subsequently that all Pure Reciprocants are, in like manner, subject to annihilation by the operator V.
Hitherto we have only considered homogeneous forms ; let us now take as an example of a non-homogeneous reciprocant the expression
(1 +t')b- 3aH.
Here ^i + .^^ ff - s.^r = [l + -^ [^^—) —^
^ (1 -I- <') (- bt + 3a') - 3a' f
In the numerator of this fraction the terms + 3a' and — 3a' cancel, a factor t divides out, and we have finally
r
In general, a Reciprocant may be defined to be a function F of such a kind that F(t, a, /3, 7, ...) contains F(t, a,b, c, ...) as a factor. An import- ant special case is that in which the other factor is merely numerical ; the function F is then said to be an Absolute Reciprocant.
When we limit ourselves to the case where .F is a rational integral func- tion of the letters, it may be proved that
F(t,a,b,c,...)=±t^F(T,a,0,y, ...).
For, in the first place, since any one of the letters 0, /S, 7, ... is a rational function of t, a,b,c, ... and integral with respect to all of them except t, containing only a power of this letter in the denominator, it is clear that any rational integral function of t, a, /9, 7, ... such a,a F(t, a, >3, 7, ...) is supposed to be, must be a rational integral function of t, a,b,c, ... divided by some power of t. But since ^ is a reciprocant, F(r, a, 0, 7, ...) must contain F(t, a,b, c, ...) as a factor; and if we suppose the other factor to be
<l>{t, a, b, c, ...) we must have
where <f> is rational and integral with respect to all the letters.
I
314 Lectures on the Theory of Reciprocants [42
Moreover, Fit, a, b, c, ...) ^'^(•^' «-^' 7, -) ^(^_ ^ ^_ ^_ ),
Hence we must have identically
<f)(t, a,b, c, ...)^(t, o, )9, 7, ...) = 1. where, on the supposition that the functions ^ contain other letters besides t and T, ^ (t, a,b,c,...) is, and <f) (t, a, ^,y, ...) can be expressed as, a rational function integral as regards the letters a, b, c, .... But this supposition is manifestly inadmissible, for the product of two integral rational functions of a,b, c, ... cannot be identically equal to unity. Hence t is the only letter that can appear in the extraneous factor and we may write
F{T,<x,0,y,...) = ^F{t,a,b,c,...)
where i/r (<) is a rational integral function. ]
The same reasoning as before shows that we must have identically 1
But this cannot be true if t^ (t) has any root diflferent from zero ; for if we give t such a value as will make •</r (t) vanish, this value must also make ■^ (t) infinite ; and since
Vr(T) =4 +BT+Gr^+...+ Mr'"
, B G M
the only value of t for which yjr (t) becomes infinite is a zero value. Hence yfr (t) is of the form Mt"^, and consequently i/r (t) = Mr'". Thus
■^ («) yjr (t) = MH"'t"* = 1, and therefore M^=l.
We have now proved that if i'' is a rational integral reciprocant, ;
F(t, a, b, c, ...)=±t>^F(r, a, A y, ...), ■
or we may say, = (-)««" F(t, a, /3, 7, . . .),
where k=1 or 0 according as the reciprocant is of odd or even character.
It obviously follows that the product or quotient of any two rational integral reciprocants is itself a reciprocant ; but it must be carefully observed that this is not true of their sum or difference unless certain conditions are fulfilled. For if we write
F,(t,a,...) = {-y't^'F(T,a,...) *
and F,(t,a,...)= (-)"tf^F, (t, a, . . .),
we see that pF,{t. a, ...) + qF,{t, a, ...) = {-)"t'^'pF,{T, a, ...) + (_)«. i^-gF,(T, a, ...),
I
42] Lectures on the Theory of Reeiprocants 315
and consequently this expression will be a reciprocant if k, = k^ and /Zi = fi^, but not otherwise. If we call the index of t in the extraneous factor the characteristic, what we have proved is that no linear function of two reeipro- cants can be a reciprocant, unless they have the same characteristic and are of the same character. In dealing with Absolute Reeiprocants, since the characteristic of these is always zero, we need only attend to their character.
I propose for the present to confine myself to homogeneous and isobaric reeiprocants*, that is, to such as are homogeneous and isobaric when the letters t, a, b, c, ... are considered to be each of degree 1, their respective wetghts being —1,0,1,2,.... The letter w will be used to denote the weight of such a reciprocant, i its degree, and j its extent, that is, the weight of the most advanced letter which it contains.
Let any such reciprocant F(t,a,b,c, ...) contain a term .4<"'a'6'"c"..., then
v+l + m + n+... = i,
and —v + m + 2n+ ... =w.
The corresponding term in F(t, a, y8, 7, ...) will be AT^a'^y" ... where
1 a _ 6 c
^=?' " = -?• ^ = -<i + -' 7 = -^,+ -. etc.
Now, if no term of J?* contains a smaller number of the letters a,b,c, ... than are found in the term we are considering, the first terms of /3, 7, etc., may be taken instead of these quantities themselves and AT"a'^"^y" ... may be replaced by
(^^y+m+n+... ^f-t^tt-im-M-... a'b"'c" ... = (-y-^Af^"-^ a'b"'c'*
But since F(t, a, b, c, ...) = (-)'«^Jf(T, a, /3, 7, ...) we must have identically
J.ra'fc^c" . . . = (-)<-»+' ^<''+'^»'-«a'6"'c" ....
Hence the character is even or odd according to the parity of i — v (that is, of the smallest number of letters different from t in any term), and the characteristic /t = .3i + w.
The type of a reciprocant depends on the character, weight, degree and extent. As the extraneous factor is always of the form (— )'<^, where k is 1 or 0, we may define the type of a reciprocant by
l:w:i,j or 0:w:i, j, according as its character is odd or even.
For Pure Reeiprocants the smallest number of letters different from t in any term is (since all the letters are different from t) the same as its degree.
* Here and elsewhere the word reciprocant is used in the sense of rational integral reciprocant : this will always be done when there is no danjier of confusion arising from it.
^
316 Lecttires on the Theory of Reciprocants [42
Hence the character of a Pure Reciprocant is odd or even according to the parity of i, and for this reason the type of a Pure Reciprocant may be defined by w:i,j.
A linear combination of reciprocants of the same type will he a recipro- cant, for when the type is known both the character and characteristic are given.
LECTURE III.
Let F be any function (not necessarily homogeneous or even algebraical) of the differential derivatives which acquires a numerical multiplier il/, but is otherwise unchanged when the reciprocal substitution of x for y and y for x is effected. A second reciprocation multiplies the function again by M, and thus the total effect of both substitutions is to multiply F by AI\ But since the second reciprocation reproduces the original function, we must have M' = 1. Functions of this kind are therefore unaltered by reciprocation (except it may be in sign), and for this reason are called Absolute Reciprocants. These, as we shall presently see, play an important part in the general theory. Like all other reciprocants, they range naturally in two distinct classes, those of odd and those of even character.
It is perhaps worthy of notice that the extraneous factor of a general reciprocant is the exponential of an absolute reciprocant of odd character. For if
F{t,a, b, c, ...)=(f)(t,a, b, c, ...)F(t, a, ^, y, ...),
we must still have, as before,
^(t, a, b, c, ...)</)(t, a, ^, 7, ...)=1;
that is log^(<, a, b, c, ...) = - log (^(r, a, /3, 7, ...);
or log <f> {t, a, b, c, ...) is an absolute reciprocant of odd character.
An absolute reciprocant may be obtained from any pair of rational integral reciprocants in the same waj' that an absolute invariant is found from two ordinary invariants. For let
F,{t, a, b, c, ...) = (-)'.<^.f.(T, a, A 7. •••).
and F, (t, a,h,c, ...) = (-y^tr'F.ir, a, /9, 7. — ).
{FAt.a,b,c,...)}^' _ ,.,,,_,,, (i^,(T,«,A7.-)h .
{F,{t, a, b,c. ...)}'-'- ^ ^ {^,(t,«,/9, 7, ...))"■'
or we may say that jPj*^ -r- F/' is an absolute reciprocant of even or odd character according to the parity of k^ii^ — K^fi^.
I
42] Lectures on the Theory of Reciprocants 317
Thus, for example, from
a = — fa
and Zac -5¥ = f (Say - 5/3=)
we form —, an absolute reciprocant of even character.
a'
From a reciprocant F whose characteristic is /x we obtain an absolute
reciprocant of the same character as F by dividing it by i*.
For if we only remember that t = - , it obviously follows that F(t,a,b.c,...)^±t>^F{T,a,l3,y. ...)
can be written in the form ~
F{t,a,b,c,...)_ ^ F{T,a,^,y,...)
where the original character of the reciprocant F is preserved.
It may be noticed that a reciprocant of odd character cannot be divided
ty ^(_ 1)<2 80 as to give an absolute reciprocant of even character; for, the
i» /^
reciprocal of i" being -«''i^', that of /"-=- V(-l)«* will still be -F' ^'^i-l)T^. The character of a reciprocant is thus seen to be one of its indelible attributes.
As simple examples of absolute reciprocants we may take ,
which becomes on reciprocation — , and -z, which reciprocates into
— -. The character of the former is even, that of the latter odd.
Observing that
1 . 1 , 1 d 1 d
log< = -logTand^^.^ = ^.^.
From this, in like manner, we obtain and so, in general.
318 Lectures on the Theory of Reciprocants [42
Hence l-r . ^ j log t is an absolute reciprocant, and of an odd character,
for all positive integral values oft. We thus obtain a series of fractions with rational integral homogeneous reciprocants in their numerators and powers of t^ in their denominators. It will be sufficient, before proceeding to the more general theory of Eduction, as it may be called, to examine, by way of illustration, the cases in which i = 1, 2 and 3.
Let i = 1 ; then
So that, in the case where i = 2, we have
Wt ' dco) ^^ Wf dx) ti f 2 • <« _ 2bt - 3a'
The numerator of this fraction is the Schwarzian. In like manner, when t = 3, /'I ^Y] f-f^ d\f2bt-Sa''\_2ct-4,ab eabt-da" _2cf-l0aht+9a!'
But here a reduction may be effected, for ( -^ 1 , as well as -; itself, is an
absolute reciprocant of the same character as the whole of the expression just
9 a'. found. Hence we may reject the term ^ . -j without thereby affecting the
reeiprocantive property of the form, and thus obtain
ct — 5ab
an absolute reciprocant of odd character. The corresponding rational integral reciprocant is
ct — 5ab.
We have found that -, and -^ — are each of them reciprocants.
t^ t
2bt Why, then, by parity of reasoning, is not — , and therefore b, a reciprocant ?
It is because — , the square of -j , is of even character, while is of
t t^ "
an odd character, so that no linear combination of the two would be legitimate.
I
42] Lectures on the Theory of Reciprocants 319
If we differentiate any absolute reciprocant with respect to x, we shall obtain another reciprocant of the same character. For let R be any absolute reciprocant and R its transform, then
R=±R';
and since t- = * t- may be written in the equivalent but more symmetrical dec ^y
form
1 i_ = J^ A
i\/t' dx \Jt' dy'
Q
On one side of this identical equation is a function of the differential derivatives of y with respect to a; ; on the other, a precisely similar function
of those of X with respect to y. Hence -^ . -7— is an absolute reciprocant,
and therefore --j- is a reciprocant, the character of each being the same as
that of R.
I will avail myself of the conclusion just obtained, which is the cardinal property of absolute reciprocants, to give a general method of generating from any given Rational Integral Reciprocant an infinity of others — rational integral educts of it, we may say. Let F be such a reciprocant, and /j, its charac-
teristic ; then — is an absolute reciprocant, and consequently t- ( ~ j is a
reciprocant, both of them of the same character as F; that is
^dF fl „r,
*^_-£_^.
or we may say 2t -j fiaF
is a reciprocant of the same character as F.
This is even true for non-homogeneous reciprocants, for the only assump- tion made at present as to the nature of F is that it is a rational integral reciprocant. But if we further assume that it is homogeneous and isobaric*,
we know that
^ = 3t + w.
Now, Euler's equation gives
3i = 3(0« -♦- a3a + Mj + c3„-H ...),
* It will sabseqaently be proved that every rational integral reciprocant which is homo- geneons is also isobaric.
I
320 Lectures on the Theory of Reciprocants [42
and from the similar equation for isobaric functions (remembering that the weights of*, a, 6, c, ... are - 1, 0, 1, 2, ...) we obtain
so that /t = 2<a« + 3aaa + 4636 + 5ca„+....
And since t- =a3( + 69a + c9i, + d9<, + ....
we
may in ( 2< ^ ^ j F replace It -, fia by
- a {2tdt + 3ada + 4-bdb + ^c^c +•••), or by its equivalent
(2bt - 3a») da + (2ct- 4a6) di, + {2dt- 5ac) dc+ ....
The conclusion arrived at is that when ^ is a rational integral homo- geneous reciprocant,
{{2bt - Sa") da + {2ct - iab) dt + {2dt - 5ac) dc + ...}F is another, and that both are of the same character.
It will be convenient to use the letter G to denote the operator just found and to speak of it as the generator for mixed reciprocants. By the repeated operation of this generator on a we may obtain the series Oa, O'a, O^a, ..., whose terms will be mixed reciprocants, since each operation increases the highest power of t by unity. The forms thus obtained will, in general, not be irreducible. It is, in fact, easy to see that a reduction must always take place at every second step. Observing that GF only expresses
the numerator of the absolute reciprocant -j,.-t-( - \ in a convenient form.
and that G^F is equivalent to the numerator of ( -77 . t- ) / - V we have
(1 /7 \ ' Vt • dd
l.aF
^'
42] Lectures on the Theory of Reciprocants 321
The whole of this fraction is an absolute reciprocant of the same character as ^; so also is / the product of the even absolute reciprocant — by — V
We may therefore reject the term ^ . ^ ' . li-F from the numerator, and the
remaining fraction
dF fi „\ /A + 3 dF t -, ^ . aF\ -'-—-.a
dx\ dx 2' j "2, ' dx
?^ will still be an absolute reciprocant of the same character as F. Its numera- tor, which is one degree lower than G-F, may be written in the form
d-'F . .^ dF /j..„ *d^-^^ + ^^^d.--2^^-
This, it may be noticed, is a reciprocant of the same character as F, even when F is non-homogeneous.
Starting with a, we have Ga = 2bt — 3a' (the Schwarzian), G'a = G (2bt - 3«') = - 6a (2bt - 3a») + 2t {2ct - iab) = 4ci' - 20abt + 18a».
But, for the reason previously given, 18a^ may be removed, so that reject- ing this term and dividing out by 4< we obtain the form
ct— oai,
which may be called the Post-Schwarzian.
The next form is obtained by operating on the Post-Schwarzian with G ; thus, we have to calculate the value of G(ct — 5ab), where
G = {2bt - So') da + (2ct - 4a6) d^ + (2dt - hoc) a,.
The working may be arranged as follows :
dfi act m d?h
|
t(2<fe-5ac) = |
2-5 |
. |
from (2(ft-5ac)3j |
|
-5a(2c<-4a6)- |
. -10 |
20 |
„ {ict-Aah)\ |
|
-56(26«-3«2) = |
-10 15 |
„ (2fe-3a2)a, |
|
|
2 -15 |
-10 35 |
||
|
The result should be read thus : |
|||
|
2dt |
* — \hact - |
-\Wt^'. |
ioa'b. |
To obtain the next of this series of reciprocants, we have to operate on jthis with G and at the same time to take account of the reduction that has
8. IV. 21
322
Lectures on the Theory of Reciprocards
[42
to be made at each alternate step. The arrangement of the work is similar to that of the former case.
efi adfi bcfi a'ct aV^l o'6
|
2fi{2et-6ad) = |
4 |
-12 |
. |
. |
. |
from {2et-6ad)di |
|
-I6cu(2dt-bac) = |
. |
-30 |
. 75 |
. |
„ {2dt-&ac)d, |
|
|
(35a»-206<)(2rf-4a6) = |
. |
. |
-40 70 |
80 |
-140 |
„ {2ct-4ab)d,, |
|
(70o6-16c0(26<-3oS)- |
• |
• |
-30 46 |
140 |
-210 |
„ {26<-3a«)a„ |
|
4 |
-42 |
-70 190 |
220 |
-350 |
||
|
-70a2(c<-5a6)= . |
. |
. -70 |
+ 350 |
4 -42 -70 120 220 This divides by 2t, giving the reduced value
2eP -2\adt- 2obct + 60a'c + llOoi*. The next obtained by this process will be seen by the following work
to be 4/i?-5Qaet^
lUbdC" - lOcH' + ^O^a^dt + 995a6c< + 2206»< - 660a'c - ffi aefi hdfl cH^ aHt abet bH a^c a%^
1210a=6».
2<2(2/i!-7a«) =
-i\at{2et-6ad) =
( - Zbht + 60a2) (2ofe - bac) =
( - 35c< + 220a6) (2c< - 4a6) =
(- 21<ft + 120ac+11062) (26<- 3a2) =
|
4 -14 |
, |
|
|
. -42 |
. 126 . . |
. |
|
. |
-70 . 120 175 . |
-300 . |
|
. -70 . 580 . |
-880 |
|
|
-42 . 63 240 220 |
-.360 -330 |
4
from(2/f-7a«)?, „ (2«<-6ad)3j| „ (2rfi— 5ac)r „ (2c<-4aA)r „ (26<-3a«)r„
4-56-112-70 309 995 220 -660 -1210 This cannot be reduced in the same manner as the preceding form, but it must not be supposed that the forms thus obtained are in general irreducible.
Having regard to the circumstance that the forms of the series ,
a, Ga, O^a, ...
(1 d \" -ji- -f-j log t, they
may be called the successive educts, and the reduced forms given above may be called the reduced educts and denoted hy E^, E^, E,.... Thus
E, = a,
Ei = 2bt - Sa-,
Es = ct — Bab,
Et = 2dt^ - load - lObH + Soa^b,
E, = 2ef - 2\adt - Sobct + 60a=c + I10ab\
Et = ifP -56aet'-ll 2bdt' - 70cFf + 309tt'd« + 99oabct
+ 2206»< - 660a'c - 1210(t»6».
42] Lectures on the Theory of Reciprocants 323
LECTURE IV.
We have seen that when F is a. rational integral homogeneous and isobaric reciprocant, GF is another of the same character. It will now appear that the condition of isobarism is implied in that of homogeneity ; for let ^ be a rational integral homogeneous reciprocant, yu. its characteristic and i its degree in the letters t, a, b, c, ..., then, in the identical equation
F{t,a.b,c,...)=±t''F{T,a,^,y, ...)
both members are homogeneous and of the same degree in the letters t, a,b, c, ... ; that is, if 4<*a'6"*c" ... be any term of F{t, a, b, c, ...), its degree must be the same as that of i''.4T*o'/3"'7" ... when t, a, /3, 7, ... are expressed in terms of t, a, b, c, But
1 a r, b c
and so on. The degrees of t, a, /9, 7, ... are therefore — 1, — 2, — 3, — 4, ... respectively. Hence
k-i- l + m + n + ... = fi — k — 21 — Sm — 4in— ...,
or /I = '2k + 31 + 4m + on + ... ,
And by hypothesis i = Ic + 1 + m+ n + ...,
80 that fi — :ii= — k + m+2n+ ... .
Neither ft. nor i is dependent for its value on the selection of a particular term of F, for all terms of jF'(t, o, /3, 7, ...) are multiplied by the same extraneiius factor ±t^, and all terms of F{t, a,b, c, ...) are of the same degree i. Hence — k + vi + 2n + ... must also be the same for each term of F; or, attributing the weights - 1, 0, 1, 2, ... to the letters t, a, b, c, ..., the function F is isobaric.
Next, suppose F to be fractional, and let it be the ratio of the two rational integral homogeneous reciprocants Fi and F,. The operation of G on F will, in this case also, generate another reciprocant of the same character as F. For, since G is linear in the diCFerential operative symbols 9a. ^6. ^e. •••, its operation will be precisely analogous to that of ditferen- tiation, so that, operating with G on
u /> ET F^GFi — FfiF,i we have GF =» — ^rnr-*
21—2
324 Lectures on the Theory of Reciprocants [42
In order to prove that this is a reciprocant, we have to show that the character and characteristic are the same for both terms of the numerator. But OF^ is a reciprocant of the same character as F^, and GF^ is one of the same character as F^; thus the two terms of the numerator are of the same character as F^F^. As regards the characteristic, it should be noticed that G, that is, the operator (26<— 3a')3o + (2c<— 4a6)94+ ..., increases the degree by unity, but does not alter the weight, so that it increases the characteristic of any rational integral homogeneous reciprocant by 3. Thus the characteristic of each term in the numerator exceeds by 3 that of F^F^. Hence GF is a reciprocant, and, taking account of its denominator as well as its numerator, we see that the operation of (? on a rational homogeneous reciprocant, whether fractional or integral, produces another in which the original character is preserved while the characteristic is increased by three units.
More generally, let JP*,, F^, F,, ... be any rational homogeneous recipro- cants whose extraneous factors are (—)"'<''', (—)"'<''', {—Y'V^', ■■■ respectively; and suppose <I> to consist of a series of terms of the form AFj"F2'^F3'' ..., such that the extraneous factor for each term is (-)*t^. Then <I> is a recipro- cant, but not necessarily a rational one; for the indices \,, Xj, \,, ... may be supposed fractional, provided only that they satisfy the conditions
«i\i + «aXa + /CjXj + ... — K= a. positive or negative even integer,
and Jh\ + /^^ + Ms^s + ■ ■ ■ — /J- = 0.
We proceed to show that G^ is also a reciprocant, and that its extraneous factor is (—)"<''+'. Since
we have to prove not only that each term of this expression is a reciprocant, but also that all of them have the same extraneous factor ; otherwise their sum would not be a reciprocant.
Now, in * = lAFi'^'F^'^'F,'"' ...,
the extraneous factor for each term is by hypothesis (—)'<'*, so that the extraneous factor for each term of
^ = 1A\F,''-^F^^FJ^...,
is (—)"""'<''"'", and therefore -rrr is a reciprocant. Also, GFi is a reciprocant
whose extraneous factor is (—)"■<''■+•'. Hence -rw.GFi is a reciprocant having
(_)«^+s for extraneous factor, and in exactly the same way we see that every other term of G<P is also a reciprocant with the same extraneous factor.
«
42] Lectures on the Theory of Reciprocants 325
Thus 0, operating on any homogeneous reciprocant whose extraneous factor is (— )'<^, generates another whose extraneous factor is (—)'*"+''.
If, in the generator for mixed reciprocants,
G = {2bt - 3a=) da + (2c< - 4a6) db + (2rf< - oac) S„ + . . ., we write «=1.2.ao, 6 = 1.2.3.ai, c = 1.2.3.4.a2..., (that is, if we use the system of modified letters previously mentioned), its expression assumes a more elegant form. Substituting for a,b,c,... their values in terms of the modified letters, we have
2bt-Sa'=2.1 .2.3a,t-S(l . 2)»a/ = 1 . 2' . 3 (aj - a^"), and Sa = j-2 • 9ao ;
so that (26( -3a»)a«= 1.2.3 (a,«-ao09ao-
Again, (2ct - 4iab) = 1 . 2' . 3 . 4 (a^t - a^a^)
and 3,= _l__^a„,;
so that (2c< - 4a6) 8^ = 1 . 2 . 4 (a^t — a^a^) 9a,-
Similarly, (2dt - oac) ?« = 1 . 2 . 5 {a^t - a^a^ 9a,-
Thus the modified generator for mixed reciprocants is 1 . 2 . 3(a,«-a,»)aao+ 1 ■ 2 . 4(M - a„ai)9a, + 1 - 2 . 5 (a,« - a,a^)da^ + ---, in which the general term is
1 . 2 (» + 3) (a„+i t - a^Un) 9a„. The factor 1 . 2 may, of course, be rejected, and our modified generator may be written in the simple form
3 (a,< - Co') 9o„ + 4 (cUit — a„ai) 9a, •+• 5 {ajt — aoa^) da^ + .... Operating with this on the homogeneous reciprocant F{t, a„, a,, a^, ...), the result will be another homogeneous reciprocant of the same character as F. When we start with a^ and make the reductions which, as we have seen, occur at every second step, we find a system of reduced educts corresponding iu every particular with those formerly given, but expressed in terms of the
modified letters a„, Or,, a^, ... instead of a, b, c These are as follows:
fflo,
ZOit— Ba^Oj, •2a,<«- «a,a^ - Sa,H + 7a„'a^,
iaft"— la^a^t — 7aia,t + Sa^-a^ + llaoa,', •l4a,(» - .56a„a4«= - oGajO^f - 28aj»«» + 103ao''a3t + 199aoa,aj«
+ 33a,»« - SHao'a, - 121a„'ai^
* It will be observed that in the nnredaced forms, marked with an asterisk, the sam of the Wmerical coefficients is zero. This is a direct consequeuce, as may be easily seen, of the form of the modified generator, in which the sum of the numerical coefficients in each term is also zero.
i
326 Lectures on the Theory of Redprocants [42
It will be found on trial that these modified educts are obtained with greater ease and with less liability to error by a direct application of the generator
3 (a,< - ao') 9ao + 4 {a^t - a^a^) 3o, + 5 (a,< - OoO,) 3a, + • • •. than by making the substitution of 1 .2.0,, 1.2.8. a,, 1 . 2. 3 .4. a,, ... for a,h,c,... in the system of educts already given. For this reason the working by the former method is here performed, instead of being merely indicated.
From ao we obtain immediately
a^t — tto'. Operating on this with the generator, there results
4< {a^t — a(,o,) — Qa„ {a^t — Oo'') = ^a^t^ — lOaofli^ + 6(»(,'- This, when reduced by removing its last term and dividing the others by 2t, gives
la^t — ^a^ai.
The next form is found from this by a simple operation, without subse- quent reduction, and is therefore
\Qt {Oit - ttoOa) — 20ao (Ojf — OoO]) — ISoi {a^t — ao')- J
Or, collecting the terms and rejecting the numerical factor .5,
2a3<^ — Qa^a^t — Sui't + la^Ui.
The operation of the generator on this gives
12<= {aj, — a^aj) - 20aot{ast — a^a^) + 4 (^a^ — 6ai<) (oj^ - aoOi)
+ 3 (14aoai - &a^t) {a^t - ao'). The collection of terms and subsequent reduction is shown below :
a^l? a^ast^ a^a^t^ a^a^t a^a^t a^'Oj 12 -12 .
-30 . 30
-24 28 24 -28
-18 18 42 -42
I
I
12 -42 -42 76 66 -70
- 1400" (2a2< - Saotti) = • • . - 28 .4-70
12* ^^^42 ^^2 48 66 ~T
Removing the factor 6^, the reduced form is
iaj^ — la^a^t — Tuia^t + Sao'Oa +.llaoa]". J
Operating on this with the generator, we have 14t^ {a^t — a^a^ — 42aot {a^t — aoO,) + 5 (8ao' — Ttht) (as* - ajOa)
+ 4 (22aoai - Ta^t) {a^t - a„(h) + 3 (llai=-|- 16aoaj - Tcht) (a,< - a„') = 14a5«'- 56a^a^t' - oGa^a^t'- iSaJ'f + lOSao'Os* + 199aottia2< + 33a,»«- 88aoX- 121a„X', which cannot be reduced in the same manner as the preceding form.
42] Lectures on the Theory of Reciprocants 327
To obtain a generator for passing from pure to pure reciprocants a process is employed similar to that which gave the generator for mixed reciprocants which we have just been using. I state the results before giving the proof, and tiien proceed to speak of generators in the theory of Invariants. The generator for pure reciprocants is
(Sac - 460 Sft + {^ad - obc) dc + (Sae - 6bd) dd+...; or, expressed in terms of the modified, letters,
4 (ttoa-i — a^) dai + o (aotts — (h<h)^at+^ {attti — OiO,) 9o, + —
By oj)erating with this on any pure reciprocant R, we generate another pure reciprocant of opposite character to that of B,.
The connection between the two theories of Reciprocants and Invariants is so close, and these brother-and-sister theories throw so much light upon each other, that I began to inquire whether, in the latter, there did not exist a theory of Generators parallel to that of the former.
Fortunately, Mr Hammond was able to recall a correspondence in which Prof. Cay ley had given such a theory, which he regarded, and justly, as an important invention. Its substance has been subsequently incorporated in the Quarterly Journal (Yo\. XX. p. 212). It offers itself spontaneously in the Reciprocantive Theory ; in the Invariantive one it calls for a distinct act of invention. Prof Cayley has discovered two generators similar in form with those for reciprocants, and one of them strikingly so ; in a letter to me he calls these P and Q. As given by him,
P = abda + acdb + ddde + ... —ib, Q = acdb + 2acldc + . . . — 2wb,
where i is the degree and w the weight, the weights of a, b, c, d, ... being taken to be 0, 1, 2, 3, ... (I supply the a which Cayley turns into unity.) As an example he takes the " Invariant " a'd — 3abc + 26*=/, suppose. We have then
PI = (abda + acdi + addc + a^a- %) I
= ah {lad- 36c) -^^ ac{- 3ac + 66") - 3a'6d + a'e - 36 (a'd - 3a6c + 26')
= a'e - 4a*6d - 3a»c» + \2a¥c - 66*
= a» {ae - ibd + Sc^ - 6 (ac - iPf, and QI = {acdb + 2ada„ + 3ae9d - 66) /
= ac (- 3ac + W) - &a''bd + 3a' e - 66 (a»d - 3a6c + 26»)
= 3a'e - 12a'6d - 3a»c' + 24a6'c - 126*
= 3a^ {ae -^bd + 3c0 - 12 (ac - 6^^. P and Q may be transformed by means of Euler's equation and the similar one for isobaric functions, which enable us to write
i = a9a + ^6 + c9c +ddd+ ..., and w^ hdb + 2(^c + 'iddd + ...;
328 Lecttires on the Theory of Recijirocants [42
P thus becomes
abda + acdb + addc + aedd + ...
— abda — b% — bcde — bdda — ...
= (ac — b') db + (ad - be) dc + (ae— bd)dd + •■.,
the same in fo7~m as either of our generators, except that the arithmetical
coefficients are all made units; a,b,c, ... taking the place of the t, a, b, ...
of the generator for mixed reciprocants.
In like manner, Q becomes
(ac - W) St + 2 (ad - 2bc) 3^+3 (ae - 2bd) da + ....
where the arithmetical series 1, 2, 3, ... takes the place of 3, 4, 6, ... or of 4, 5, 6, ... in the two Reciprocant Generators.
The effect of P and of Q is obviously to raise the degree and the weight
of the operand / each by one unit. But if we take R = - (2wP — iQ), the
terms in Cayley's original formulae containing b cancel, so that 2wP — iQ divides out by a and the weight is raised one unit without the degree being affected. This is mentioned in the Quarterly Journal (loc. dt.) ; but it may also be remarked that when / is a satisfied invariant, it is annihilated by the operation of R ; when the invariant is unsatisfied, each of the three operators P, Q and R increases its extent by an unit, that is, introduces an additional letter. For let j denote the extent, then, writing a^.a^.a^, ... aj for a, b, c, ..., we have
P = a„a,9a„ + aoOjSa, + ... + a^aj+idaj - ia,,
Q = aoOsSa, + 2aoa.ida^ + ... +jataj+ida^ - 2wa^ ;
whence we find
iJ = - (2wP - iQ)
= 2wa,Sfl„ + (2w - i) a^da^ + ...+(2w -ij + i) ajdaj_^ + (2w - ij) aj+^day
But for a satisfied invariant
2w = ij ;
and substituting this value for 2w in the above expression for R, it becomes
i { joi^oo + ( J - 1 ) «29oi + • • • + aj9o^_il> which, as is well known, annihilates any satisfied invariant.
I
42] Lectures on the Theorjf of Reciprocants 329
LECTURE V.
It will be desirable to fill up some of the previous investigations by discussing some points in them that have not yet received our consideration.
There may be some to whom it may appear tedious to watch the com- plete exposition of the algebraical part of the Theory, who are impatient to rush on to its applications. But it is my duty to consider what may be expected to be most useful to the great majority of the class, and for that purpose to make the ground sure under our feet as I proceed. To the greater number it will, I think, be of advantage to have their memories refreshed on the kindred subject of invariants, and probably made acquainted with some important points of that theory which are new to them.
I confess that, to myself, the contemplation of this relationship — the spectacle of a new continent rising from the waters, resembling yet different from the old, familiar one — is a principal source of interest arising out of the new theory. I do not regard Mathematics as a science purely of calculation, but one of ideas, and as the embodiment of a Philosophy. An eminent colleague of mine, in a public lecture in this University, magnifying the importance of classical over mathematical studies, referred to a great mathe- matician as one who might possibly know every foot of distance between the earth and the moon ; and when I was a member, at Woolwich, of the Government Committee of Inventions, one of my colleagues, appealing to me to answer some question as to the number of cubic inches in a pipe, e.xpressed his siirprise that I was not prepared with an immediate answer, and said he had supposed that I had all the tables of weights and measures at my fingers' ends.
I hope that in any class which I may have the pleasure of conducting in this University, other ideas will prevail as to the true scope of mathematical science as a branch of liberal learning; and it will be my endeavour to regulate the pace in a manner which seems to me most conducive to real progress in the order of ideas and philosophical contemplation, thus bringing our noble .science into harmony and in a line with the prevailing tone and studies of this University. Faraday, at the end of his experimental lectures, was accustomed to say — I have myself heard him do so — "We will now leave that to the calculators." So long as we are content to be regarded as mere calculators we shall be the Pariahs of the University, living here on sufferance, instead of being regarded, as is our right and privilege, as the real leaders and pioneers of thought in it.
330 Lectures on the Theory of Reciprocants [42
That Cayley's two operators, which have been called P and Q, are in fact generators, may be proved as follows •{• :
Let a=adf>f 2bdc + Scdd + '^d^e + •••,
and ^ = aQdida+ fjicdb+fdde +...) — Kb,
■where k, \, /j,, v, ... are numbers.
When ic is the degree of the operand, and \ = ij.— v= ... =1, the operator 0 is identical with P ; but 0 is identical with Q when k is twice the weight of the operand and \ = 0, fi=l,v = 2, ....
If now we use * to signify the act of pure differential operation, it is obvious that
n0=(nx 0) + (n*0), 0n=(n X 0)+(0*n),
so that fi0-0il = (fl*0)-(0*n).
But since Ha = 0, Hi = a, He = 26, ...
we have D,¥fB = a {\ada + 2fi,bdb + Svcd^ + ...-«)
and €>* il = a (\hdi+2ficdc + Svddci + ...).
Hence 120 - 0n = a [Xada + {2/jl - X) bdb + (3i/ - 2^) c9, + ...-«) ;
now if the operand I be any invariant (satisfied or unsatisfied), we have 0,1=0,
and therefore 012/ = 0 ; so that we find
il^I = a {\ada + i2/jL-\)bdb + {Sv - 2fi)cde+ ... - ic] I. If in this we write \ = fi = v = ... =1, and k = i, where i is the degree of the operand, 0 becomes P and we have
D.PI==a(ada + bdb + cdc+...-i)I. •
But, by Euler's theorem, the right-hand side of this vanishes, and therefore
npi=o.
Similarly, by means of the corresponding theorem for isobaric functions, we may prove that
nQi=o.
For if, in the general formula, we write \ = 0, /:i = l, v = 2, ... and k = 2w, where w is the weight of the operand, we find
nQI = a{2bdb + 4:cdc + 6ddi + ... - 2w) 1=0.
Thus, when 0 stands either for P or for Q, it is either an annihiiator or a generator (that is, 0/ is either identically zero or else an invariant). But if I be the most advanced, or say the radical letter of /, no term of mdj can cancel with any other term of 0/; and since, for this reason, 0/ cannot vanish identically, it must be an invariant, and the operators P and Q must be generators.
t In the Quarterly Journal (Vol. xx. p. 212) Prof. Cayley only considers a special example, and has not given the proof of the general theorem.
I
42] Lectures on the Theory of Reciprocants 381
The generators previously given for reciprocants also possess this property of introducing a fresh radical letter at each step. The radical letter, on its first introduction, enters in the first degree only, and in the case of the educts of log*, whose values have been calculated, its multiplier is seen to be a power of t. The form of the generator for mixed reciprocants
3 («]< - a^) d„„ + 4 (fflj* - ffloai) 8a, + ..•+(»+ 3) (a„+, t - a„a„) da„
shows this, or it may be seen by considering the successive values of
For let — — — " "^' '"^ denote this expression, and let its radical letter
be a„ ; then, on differentiating again with respect to x, the new letter intro- duced arises solely from a term in the numerator
^^((,a„.a„a„...a„).^^.
But a„ = 5^^2.3...«+2; so that ^ = (n + 3) a„+i .
aar dx
Hence, if when a„ is the radical letter, it occurs in the first degree only '
and multiplied by a power of t, it follows that, since -,— will be a power of
t, the derived expression which contains the radical letter a„+, will contain it in the first degree only and multiplied by a power of t. And since this is
true for the case i = 1, when —r. . -;- log < = -^ , it is true universalh*.
Observe that for t = l, 2, 3, ... the radical letter is ao, Oi, aj, ... respec- tively.
It will be remembered that (71 • j-j log i is an absolute reciprocant. It
may be called the I'th absolute educt, to distinguish it from the rational integral educts E^, E^, E„ ... whose values have already been calculated.
Let R(t, a,, a,, Oj, ... a„) be any homogeneous rational integral recipro- cant, and let the educts be A^, -4,, A^, ... An] then obviously
a„ may be expressed rationally in terms of .4„ and a„_,, a„_2, ... a^, t, (iji-i n n j> » -"n-i 8.nd a„_2, ... ttoi f>
«i „ „ „ „ -4,, a„ and t,
«o „ „ „ „ A^ and t,
where observe that the denominators in these expressions are all powers of t. Hence, by successive substitutions, ii(<, ao, a,, ... a„) may be expressed
332 Lectures on the Tlieory of Reciprocants [42
rationally in terms of A^, ... A^, A,,, and t. Thus any rational integral homogeneous reciprocant is a rational function of educts, and is of the form
E
— , where E \»& rational integral function of the educts.
v
Does not this prove too much, it may be asked, namely, that any function F of the letters is a rational function of the educts, which are themselves reciprocants, and will therefore be a reciprocant? But this is not so; for observe that although F will be expressed as a sum of products of educts, such products will not in general be all of the same character, and their linear combination will be an illicit one, such as is seen in the illicit com- bination of Oo' with the Schwarzian (a,t — a^).
We have seen that by differentiating an absolute reciprocant, or by the use of a generator, we obtain a fresh reciprocant. But there are other methods of finding reciprocants ; as, for example, if the transform of
^{t, a, b, c, ...) is yjr{T, a, /3, 7, ...),
that is, if ^ (t, a, b, c, . . .) = -^Ir (t, a, /9, 7, . . .),
then yjrit, a, b, c, ...)= <f) (t, a, /3, 7, ...).
Whence, by multiplication, ^(t, a, b, c, ...)ir(t, a, b, c, ...) = ^(t, o, /3, 7, ...)y}r{T, a, /3, 7, ...).
Thus <f> .yjr is a reciprocant, and, moreover, an absolute one of even character, although neither <f>, which is a perfectly arbitrary function, nor 1^, its transform, is a reciprocant.
Herein a mixed reciprocant differs from an invariant, which cannot be resolved into non-invariantive factors. It is worth while to give a proof of this proposition; but first I prove its converse, that if p, q, r, ... are all invariants, their product must be so too. This is an immediate consequence of the well-known theorem that
a/=o
is the necessary and sufficient condition that / may be an invariant where, as usual, O is the operator
adi, + 2bde + Scdd+ ■■., and the word invariant has been used in the same extended sense as formerly.
For D,(pqrs...) = (-^ + —3 -^ ^ -I- ...
\ p q r
But since p, q, r, ... are all invariants, we have np = 0, nq = 0, flj- = 0, ., and therefore fi {pqrs . . .) = 0.
Next, suppose that I = P^Q^,
where / is but Qi is not an invariant.
pqrs
I
42] Lectures on the Theory of Reciprocants 333
To meet the case in which P, and Qi are not prime to one another, Q^, if resolved into its factors, must contain one Q' where Q is not an invariant.
Suppose that Pj contains Q\ and let i +j—k; then we may write
/ = PQ*,
where P is prime to Q. But since / is an invariant by hypothesis,
12/ =0, and therefore, Q* HP + kPQ'-' flQ = 0 ;
p=-*ap-
Now P is prime to Q, so that the fraction ^ is in its lowest terms; there- fore flQ contains Q ; but this is impossible, for the weight of fiQ is less than that of Q. Hence / cannot contain any non-invariantive factor Qi.
All this will be equally true for a general function J annihilated by any operator fl which is linear in the differential operators 3a,3t, 9c, •••no matter what its degree in the letters a,h,c, ... themselves ; that is, we shall still have
/=PQ*
and 9-_k^
and p- Ic^^,
where P and Q are piime to each other, and, as before, flQ will contain Q as a factor. But if fl is an operator which diminishes either the degree or the weight, HQ is either of lower degree or of lower weight than Q, and so cannot contain it as a factor. Hence J cannot contain a factor Q not subject to annihilation by fl.
If, however, il does not diminish either the degree or the weight, it may be objected that ilQ might conceivably contain the factor Q ; and were it so, there would be nothing to show the impossibility, in this case, of a function J subject to annihilation by il containing a factor Q, which is not so^ But quaere: Is it possible, when </is a general homogeneous and isobaric function of a, 6, c, ..•, for iU to contain J and at the same time the quotient to be other than a number*? Valde dubitor. But I reserve the point. Setting aside this doubtful case, and considering only such linear partial differential operators as diminish either the degree or the weight of the operand, we see that there cannot exist any universal operator of this kind whose effect in annihilating a form is the necessary and sufficient condition of that form being a reciprocant. But this does not preclude the possibility of the existence of such annihilators for special classes of reciprocants, and in fact
iU . * If il—pad^+ql)d^+redf+..., where p, q, r, ... are in Arithmetical Progression, - is a
number; hot then 0 coald not be an annihilator.
334 Lectures on the Theory of Eeciprocanta [42
(as we have already stated and shall hereafter prove) Pure Reciprocants are definable by means of the Partial Differential Annihilator
V=4-.-^da, + 5a.a,8a, + 6 (0,0, + y j a„, + . . .,
which is linear in the differential operators, and diminishes the weight.
The generator for mixed reciprocants, which we have called G, will not assist us in obtaining pure reciprocants, but generates a mixed reciprocant in every case, even when the one we start with is pure. Thus, starting with the pure reciprocant R, our formula
6R = {3(ait- a,") 800 + 4 (a,< - a^Oi) 9<., + 5 (a,t - a^a,) da,+ ...}R may be written thus
GR = t (3a,9a« + 4as3o, + oa^da, + ...)R - a, (3ao9a<, + 4a,9a, + 5aj9as + •••) ^• Here R heing pure, that is, a function of a^, a,, a^, ... (without t), we see that
(3ao9no + 4ai9aj+ oajSo, + 6a39a3 + ...)R
= 3 (ao9ao + «i9a, + OsSflii + ■■■)R
+ (oi9o, + 2a29a, + 3a89a, + ...)R
= (3i + w)R, where i is the degree and w the weight of R. Hence
GR = t (Sa,da, + ia^da^ + oa,da, + ...)R-(3i + w)aoR, where it should be noticed that a^R is of opposite character bo R (for a„ is of odd character), while GR has been proved to be of the same character as R. Thus we cannot infer that t (3ai3(,„ + ia^da, + SosSo, + ...)iJ is a reciprocant. The mixed reciprocant GR cannot therefore be resolved into the sum of two terms, one of which is a pure reciprocant and the other a pure reciprocant multiplied by t.
LECTURE VI.
Before proceeding to prove that, as was stated in anticipation in Lecture IV, the operator
(3ac - 46^) db + (Sad - 56c) 9, + (Sae - 6bd) dd + ...,
or, when the modified letters are used,
4 (aoOa — tti") 9a, + 5 (aottj — Uia^) da^ + 6 {a^at — aiOj) da, + ... ,
will serve to generate a pure reciprocant from a pure one, it may be useful to briefly recapitulate what has been said concerning the character and
I
42] Lectures on the Theory of Reciprocants 335
characteristic of reciprocants. It will be remembered that the extraneous factor of any rational integral reciprocant is of the form {—Ytf^, that the character is determined by the parity (oddness or evenness) of k, and that /x is what has been called the characteristic.
For homogeneous reciprocants it has been proved that /a = 3i + w, where i is the degree of the reciprocant and w its weight, the weights of the letters t, a,b,c,... being taken to be —1, 0, 1, 2, ... respectively. The character is odd or even according as the number of letters other than t in the principal term or terms is odd or even. By a principal term is to be understood one in which t is contained the greatest number of times. So that, in other words, the character is governed by the parity of the smallest number of non-i letters that can be found in any term. For pure reciprocants, there being no t in any term, the character is determined by the parity of the number of letters in any one term.
Let R be any pure reciprocant, and suppose its characteristic to he fj,; R then — is an absolute reciprocant. If, however, we differentiate this with
respect to x, and thus obtain another reciprocant, the resulting form will not be pure, for its numerator will be identical with the form obtained by the direct operation on R of the generator for mixed reciprocants, and its
denominator will be a power of t. But, remembering that -j , and therefore
a' . . R
— , is an absolute reciprocant, we see that — , which is the quotient of the
t* n, a»
two absolute reciprocants — and — , is so also. Hence t ( — \ is a recipro-
§■ <2
cant, and, since it no longer contains t, a pure one. Now,
dR /i , p
•G)'"
<5)
remains a reciprocant when multiplied by any power of the reciprocant a. Hence the numerator of this expression, or
{sal->.b)R,
is a reciprocant. The general value of -r- has been seen to be
aSt + bda-¥ c3(, + ddc+ ..., but, since R is supposed to be pure, dtR — 0.
336 Lectures on the Theory of Reciprocants [42
We may therefore, in Za -j- — /jh, replace -j- by
Mo + c9» + f^» + c9(i + ••• •
Now, remembering that fj, = 3i + w, and that by Euler's theorem and the similar one for isobaric functions
and w= 69i, + 2c9c + 3d9<i+ ...,
we see that fi is equivalent to
3aa„ + 4636 + ocdc + 6d9d + . . . .
Hence, 3a t — ^6 = 3a (69o + c96 + d9c + e9d + • . .)
- h (3aa„ + 4696 + 5c9c + 6d9d + ...)
= (3ac - 46=) 96 + (3ad - 56c) 9c + (3ae - 66d) 9<i + . . . .
Thus, if R be any pure reciprocant, !
{(3ac - 46=) 96 + (3ad - o6c) 9^ + (3ae - G 6d) 9^ + . . . ) ^
is also a pure reciprocant. If the type of ft be w ; i, j, that of the form derived from it will clearly be w + 1; i + 1, j+1. Its character (which, for pure reciprocants, depends solely on the degree) will therefore be opposite to that of R, and its characteristic will be /x+ 4, that of R being fi.
Beginning with the form 3ac — 56', which was given as an example in Lecture II, a series of pure "educts" may be obtained by the repeated use of the above generator ; and it will be noticed that the successive educts thus formed are alternately of even and odd character, whereas those previously given, namely, a, 2bt— Sa' ..., were all negative. A reduction similar to that which formerly took place when the generator for mixed reciprocants was used, may be effected at each second step in the present case. For, since the
3a T fib) Ria fji + 4i, the next operation will give
Performing the indicated differentiations, this becomes 3a ^ (3a 2 - f^bR^ - 3 (/^ + 4) a6 ^ + ya (/. + 4) b'R
^^""'^ ■*■ ^"^£~ ^'^"^S" ^'^''^~ "^ (fi + 4:)ab^+/x(fi + 4') b'R = 9a'^-S{2fi + l)ab^-SfiacR+ti(fjL+'i)b^R.
42] Lectures on the Theory of Reciprocants 337
Adding /t(/4 + 4) (3ac — olf)R to 5 times tlie above expression, we obtain 45a= -TV - 15 (2u + 1) ab -j- + 3/x {fi — 1) acR, which, when divided by 3a, gives the pure reciprocant
15ag-5(2;. + l)6^+M(M-l)cii.
This form is one degree lower than the second educt from R, the depres- Ision of degree being due to the removal of a factor a by division.
■•
When the modified letters a^, a^, a^, a,, ... are used, the generator
(3ac - 46=) aj + (Sod - 56c) d, + (3ae - Qhd) 9^-1... ( 1 )
is easily transformed by writing in it
a = 2ao, 6 = 2.3.0,, c=2.3.4.a„, rf = 2.3.4.5 a^..., and consequently
06-2.3' '"2.3.4' ''"2.3.4. 5 •■•'
when it becomes
2».3».4, „.. , 2^3^4.5, ,.
2 3 («««2 - «i') 5a. + 2 3 4, ("»"' - «i«s) Oo.
2^.3».4.5.6, ,„
"2 3"4~ 5 ' ^"'''"* ~" "'"'' ''os + • • ••
Dividing each term of this by 2.3, and writing the numerical coefficients in their simplest form, we have
4 (ooa, — a,') 3„, + 5 (aottj - OiOs) 9a, + 6 (aoa^ — a^a,) 9a, + . . ., (2)
which is the modified generator previously mentioned.
The generators formerly used in the theory of mixed reciprocants were
{2tb - 3a«) da + (2fc - 4a6) dt + {2td - bac) Sc + . . . (3)
and 3(<a,-ao')9a„ + 4(<a5-aoai)3a, + 5(ta5-aoa3)9a,+ .... (4)
The memory will be assisted in retaining these formulae if we observe that (1) is obtainable from (3), or (2) from (4), by increasing at the same time each numerical coefficient and the weight of each letter by unity.
It will, I think, be instructive to see how the form 3ac — 56' was found originally by combining mixed reciprocants. The degree alone of a pure reciprocant suffices, as we have seen, to determine its character ; but when we are dealing with mixed reciprocants their character does not depend either on the degree or the weight, so that we require a notation to discri- minate between forms of the same degree-weight, but of opposite character. In what follows, {+) placed before any form signifies that it is a reciprocant of even character, while (— ) signifies that its character is odd.
8. IV. 22
338 Lectures on the Tlieory of Reciprocants [42
I have previously given the three odd reciprocants
(-) «. (A)
(-) 2bt - 3a', (B)
(-) ct - 5ab. (C)
From these we obtain even reciprocants ; thus the product of (A) and
(C)i8
(+) act - 5a% (D)
and the square of (B) is
(+) 4bH'-l2a'bt+9a*.
After subtracting the even reciprocant 9a* from this, we may remove the factor it from the remainder without thereby affecting its character. These reductions give
(+) bH - 3a'b,
which may be combined with the even reciprocant (D) in such a manner that the combination contains a factor t. In (act,
3 (act - 5o'6) -o(bH- 3a»6) = (3ac - 56') t,
so that a legitimate combination of mixed reciprocants can be made to give the pure one
Sac - bb\
Similarly we might find the known form
9a»rf - 45a6c + 406»,
which equated to zero expresses Sextactic Contact at a point x, y. But it is more readily obtained by operating with the generator on 3ac — 56' ; thus,
{(3ac - 46') db + (3ad - 56c) 9,} (3ac - 56') = - 106 (3ac - 46') + 3a (3ad - 56c)
= 9a'd - 45a6c + 406'.
An orthogonal reciprocant may be defined as a mixed reciprocant whose form remains invariable (save as to the acquisition of an extraneous factor when the reciprocant is not absolute) when any orthogonal substitution is impressed on the variables x and y. Concerning such reciprocants, we have
the very beautiful theorem : If R and -7- ai-e both of them reciprocants, then R is an orthogonal reciprocant.
For suppose R to be an absolute reciprocant ; that is, let
R = qR' {q=.± 1),
where iZ is a function of i, a, 6, c, ... and R' the same function of t, a, /9, 7, ...; then, denoting by AiJ the variation of R due to the variation of y by ex, and by DR the variation of R due to the variation of a; by — ey, we have
I
42] Lectures on the Theory of Reciprocants 339
For the variation of 2 is e and the variations of a, 6, c, ... vanish. Similarly
CIT
Now, since R = qR',
DR = qDR'^-eq^,
therefore DR + Ai2 = e {-r- -q-r-j]
that is, the total variation of R (due to the change of x into x — ey and of y
into y + ex) vanishes if
dR_ dR^ dt ~^ dT '
Hence, if R be an absolute orthogonal reciprocant, -7- is also an absolute reciprocant (though it is not orthogonal) of the same character as R.
If R be not absolute, suppose its characteristic to be /i ; then it can be
made absolute by dividing it by a*. The application of the foregoing
dt\
method of variations will now prove that -r.( — \ is an absolute reciprocant
of the same character as — . But-ri\ = r-- Hence — r- is a recipro-
» dt\ ^ ] t dt dt ^
a' \* / a'
cant whose characteristic is ft, and character the same as that of R.
The simplest Orthogonal Reciprocant is the form
(l+<')6-3a%
which occurs on p. 19 of Boole's Differential Equations. When equated to zero it is the general differential equation of a circle. It is noticeable that although Boole obtains this form by equating to zero the differential of the radius of curvature
a '
he does not recognise the fact that it vanishes at points of maximum or minimum curvature of any plane curve, but says that the "geometrical property which this equation expresses is the invariability of the radius of curvature."
Taking this form as an example of our general theorem, let I R={l + P)h- 3aH ;
then ~=^2bt-3a'.
I
22 — 2
340 Lectures on the Theory of Reciprocants [42
which is the familiar Schwarzian. Observe that
(1 + «») * - Sa'f = - <• 1(1 + T») /3 - 3aV} and 2bt -3a' = -f (2/3t - 3a'),
80 that the characteristic and character are the same for both these forms.
The form ct — oab, which we have called the Post-Schwarzian, when multiplied by 2 and integrated with respect to t, gives
ci"- lOabt + 4> (a, b, ...). In order that this may be a reciprocant, we must have
^(o, b, ...) = c + loa\ In this way the Orthogonal Reciprocant
(1 +P)c- lOabt + 15a» was obtained originally.
It will be easy to verify that this is a reciprocant by means of the
identical relations
1
|
t=-, |
|
|
T |
|
|
a |
|
|
« = -;i. |
|
|
^-^- |
-3a> |
7T^-10a/3T+loo^ c = -, .
We shall find that
(1 +f)c- lOabt +l5a' = -(' {(1 + r=) 7 - lOaySr + 15a»),
and comparing this with
ct — oab = — {' (7T — oaS),
it will be noticed that both forms have the same character and the same characteristic.
The complete primitive of the differential equation c{l + tr)-\Oabt+lba^ = 0 has been found by Mr Hammond and Prof Greenhill. The solution may be written in the following forms :
*"~j V{«(l-15<»+15<^-<«) + X(6«-20<»+6t'')}"''^
_ r tdt_
^ ~ J Vff (1 - 15«' + 15<* - <•) + X (6« - IW+Qt')]
cos(e-A)de
+ const.
+ 1'
r coi
_ r_sin^(^^^)d0_ '~J ^J[BcosQ{e-A)]
+ const.
42] Lectures on the Theory of Reciprocants 341
k'Hn\{X, k) = mn'(Y,k'), where i=sinl5°, A' = sin 75°,
and X = lx + my -\-ni,
Y='mx — ly + n^, I, m, «i, Tia being arbitrary constants.
The last two forms of solution are due to Prof. Greenhill.
LECTURE VII.
I have frequently referred to, and occasionally dilated on, the analogy between pure reciprocants and invariants. A new bond of connection between the two theories has been established by Capt. MacMahon, which I will now explain. Let me, by way of preface, so far anticipate what I shall have to say on the Theorem of Aggregation in Invariants (that is, the theorem concerning the number of linearly independent invariants of a given type) as to remark that the proof of this theorem, first given by me in Crelle's Jmimal and subsequently in the Phil. Mag. for March, 1878, depends on the fact that if we take two operators, namely, the Annihilator, say
n = Uodai + 20,80, + 3a,aa, + . . . +jaj^idaj
and its opposite, say
0 = ajdaj_, + 2aj_tdaj_„ + 3a,_,9a^._j + . . . +j(hdao,
then (no - Oil) / is a multiple of I.
Thus, if/ stands for any invariant (that is, if 117=0), it follows imme- diately that nOI is a multiple of I, and consequently n"'0"'/ is also a multiple of /. We may call il and 0, which are exact opposites to each other, reversing operators.
Now, MacMahon has found out the reversor to V, the Annihilator of pure reciprocants. His reversing operator is no longer of a similar, though
opposite, form to F, as 0 is to il, but is simply -=- ; nor is the effect of operating with V -j- on any pure reciprocant R equivalent to multiplication by a merely numerical factor, as was the case with HOI, but {Vj-\R is a
numerical multiple of aR, and as a consequence of this {^'",7";^)^ ^^ ^
numerical multiple of a!^R. Thus the parallelism is like that between the two sexes, the same with a difference, as is usually the case in comparing the two theories.
342 Lectures on, the Theory of Reciprocants [42
This remarkable relation between the operators V and -j- may be seen
a priori if we assume that (as we shall hereafter prove) to each pure recipro- cant R there is an annihilator V of the form
3a»84 + (...)9. + (...)Sd + (...) 3. + -., not containing da and linear in the remaining differential operators 3^ , 3c. 9<j. • • • • For if we call the characteristic /a, by differentiating the absolute pure reci-
procant — with respect to x we obtain, as was shown in the last lecture, the
pure reciprocant
3a -J fihR.
Since this is annihilated by V, we have
3a{V-^R-(iRVh-(ihVR = 0.
But, since ii is a pure reciprocant, VR = 0 ; and from the assumed form of V it follows that
or
Hence Qafv^R- Sfia'R = 0,
Thus the operation of F-y- is equivalent to multiplication by /ta, so that (barring the introduction of a) F restores to -^ the form it had antecedent to the operation of -,- , and may be called a qualified reversor to t- .
For example, suppose that
i2 = 3ac-562.
Since we are using natural letters for the derivatives of y with respect to X, we have
■^ = hda+cdb-\-ddc+ ■■■,
and, as we shall presently see,
F = ^a-'db + lOahdc + (loac + 106'') da+ .... Now, ~ = (ida + cdb + ddc) (3ac - 56=) = 36p - 106c + Sad = Sad - Ihc. Operating on this with F, we find ^dx^^ (3ad - 76c) = - 21a=c - 70a6» + 3a (loac + 106^ =;24a=c - 40a6' ; that is F ~ (3ac - 56=) = 8a (3ac - 56=).
I
42] Lectures on the Theory of Reciprocants 343
Let us now inquire whether it is possible so to determine an operator V that the relation
may be satisfied identically when F is any homogeneous isobaric function of the letters a, b, c, ... of degree i and weight w. If so, we must be able to satisfy each of the equations
|r)a.3..,
\ dx
\ ax da; J
\ dx dx J
which are found by writing a, b, c, d, ... successively in the place of F.
Now -r' = b, -r = c, ^ = d, ... so that the above equations may be dx dx dx
written
F6=3a=+~(Fa),
Vc=^b + ~{Vb),
Vd = 5ac+~(Vc),
Ve = 6ad + ^iVd),
These equations are sufficient to completely determine V on the supposi- tion previously made that it is linear in the dififerential operators and does not contain da', for, since V is linear, it must be of the form
iVa)da + (Vb)d„ + {Vc)d,+ .... and, since it does not contain da, we must have Va = 0, and therefore Vb = 3a\
Vc = 4ab + ^(3a') = 4a6 + 6ab = lOab,
Vd = 5ac + T- (lOab) = 5ac + 106' + lOac = loac + 106',
Ve = Gad + ^ (15ac + 106') = 6ad + 156c + 206c + I5ad = 2lad + 356c,
Hence F= 3a»a6 + 10a6Sc + (15ac-l- 106=)9i + (21ad + 3o6c)8e+ ••••
1
344 Lectures on the Tlieory of Heciprocants [42
When the modified letters a,, o,, a,, ... are used, we shall have, in con- sequence of the change of notation, f V,] R = 2fiaoR (instead of /jmR). If, as before, we seek to satisfy the equation
we shall find, on writing a„ in the place of F,
This condition will be sufficient, as well as necessary, for the satisfaction of (1) when V is linear ; for then
V—-~ V dx dx
will also be linear, its general term being
.dan d
(ydOn d^ „ \ V dx dx^^V'^""
which is equal to 2 (3 + n) UoOnda^ by equation (2). Hence
(J J V
Ft T-Fj^=asumof terms of the form 2 (3 + w) at,a„da^F
= 2a„ (3a„a^ + 3a, 9„, + 3a,a„, + ...)F
+ 2a„ {Oida, + 2aja„, + .■■)F;
that is, equation (1) is satisfied whenever (2) is. Writing in (2)
dcin , „,
^ = (n + 3)a„+„
we obtain (n + 3) Fa„+, = 2 (n + 3) a^a^ + ^^ ( Va^), (3)
from which the values of Va^ may be successively determined. W^hen Fao = 0, the value of Va^, which satisfies (3), is
T7- M + 3 , ,
van = — o— (ttoan-i + (hO-n-i + • ■ • + C-n-Mi + an-^.a„) ;
4 thus Foi = 2 . ao', Faj = 5a„ai , Fas = 6aoa., + 3ai=, . . .
. «o''9a, + ba^aida, + 6 f ajO.^ + o ^ / ^"» "^ '^ ^"'""^ "^ '^'"^^ 9*4 +• ■
and the value of F is therefore 4 2
Now that we are on the subject of parallelism between the old and new worlds of Algebraical Form, I feel tempted to point out yet another very interesting bond of connection between them. There is a theorem concerning Invariants which I am not aware that any one but myself has noticed, or at
42] Lectures on the Theory of Reciprocants 345
all events I do not remember ever seeing it in print*, which is this: If we take any " invariant " and regard its most advanced letter as a variable, or say rather as the ratio of two variables u : v, by multiplying by a proper power of V we obtain a new Quantic in u,v; or, if we take any number of such invariants with the same most advanced letter (or, as we may call it in a double sense, the same radical letter) in common, we shall have a system of binary Quantics in u, v. My theorem is, or was, that an Invariant of any one or more of such Quantics is an Invariant of the original Quantic. I recently found a similar proposition to be true for Reciprocants, namely, forming as before a system of 'pure Reciprocants into Quantics in u, v, any " Invariant " of such system is itself a Reciprocant.
The two theorems may be stated symbolically thus :
//' = /")
IR = R'] ■
On mentioning this to Mr L. J. Rogers, he sent me next day a proof which, although only stated as applicable to Reciprocants, is equally so, mutatis mutandis, to Invariants. Although given for a single invariant, it applies equally to a system.
I give Mr Rogers' proof that any invariant of a pure reciprocant (the proof wll not hold for impure ones) is a pure reciprocant; or rather I use his method to prove the analogous theorem that any invariant of an invariant is itself an invariant. It will be seen hereafter that this same proof applies to ptire reciprocants with only trifling changes ; but the proof as given by Mr Rogers requires some further considerations to be gone into for which we are not yet ripe.
Consider, for the sake of simplicity, the binary Quintic
(a, b, c, d, e, fjx, yf,
and let / be any invariant of it (satisfied or unsatisfied) ; then
/ = a,/" + a,/»-' + a,/"-'' + . . . + a„,
where a,, a,, a,, ... a„ do not contain/, but are functions of a, h, c, d, e alone.
Let the Protomorphs for our Quintic be denoted by A, B, C, D, E, F; then ^ F=a'f- oabe +2acd + 8b'd + 6bcK
Eliminating /from / by means of this equation, we have
Ta^ = AoF» + A,F"-' + A,F'^*+...+An.
where A„, J,, A„, ... An are all of them invariants (not necessarily integral
• The theorem i«, however, given in Vol. xi. p. 98 of the Bulletin de la Sociite Mathematique 4e France, in a paper by M. Perrin, which has only recently come under the lecturer's notice.
\
346 Lectures on the Theory of Reciprocants [42
forms, but this is immaterial to the proof, for SI annihilates fractional and integral invariants alike). For
and, in consequence of /a*" and F being invariants, so that, as regards fl, F may be treated as if it were a constant, this becomes
0 = J?'''fi4„ + J?'"-' fi^i + i^»-»fiii, + . . . + n^„,
in which the coefficients of the several powers of F must be separately equated to zero. In other words, A^, A^, A^, ...An are all of them invariants. Now, any invariant of
il.i?'" + A^F""-' + A^F""-' +...+A„ is a function of ^4,,, .4,, A^, ... An, and therefore an invariant.
(N.B. — We cannot assume that any function of general reciprocants is itself a reciprocant.)
Again, since A^F^ + ... +An, and Oo/" + ... +a„
are connected by the substitution
which is linear in respect to the letters F andy, any invariant of
A,F»+...+An
is (to a factor prh, that factor being a power of a which is itself an invariant) equal to the corresponding invariant of
a„f^+...+an. But every invariant of the former has been shown to be an invariant of the original quantic, and therefore every invariant of the latter is so also.
I add some examples in illustration of this theorem : Ex. 1. Take the invariant of the Quintic a'/"- 10a6e/+ 4<acdf+ 16b'df- 126cy+ I6ace' + 9b'^- 12ad'e-7(ibcde
+ 48c^e + 4i8bd' - S'lc'd". The discriminant of this, considered as a quadratic in /, is a» (16ace= + 96'e^ - Uad'e - 76bcde + 48c'e + 486d' - 32c'd'}
- {oabe - 2acd - 8b^d + dbc'Y = 16a'ce' - 16a»&^e» - 12a'd-e - oGa'bcde + 48a''c'e + SQab'de - QOab-c'e
+ 4>8a'bd' - SGa'c'd' - S2ab-cd^ - 646^d» + 2iabc'd -i- 9%'c-d - 866'^c'. It will be found on trial that this is divisible by the invariant
4 (ae - 4,bd + 3c=), the quotient being
4a^ce - 4,a¥e - Sa^d' + 2abcd + 46»ci - 36V
= 3a {ace - b'e - ad- + 2bcd - c^ + {ac - b') {ae - 46d + 3c^).
|l
I
42] Lectures on the Theory of Heciprocants 347
Thus the discriminant of the quadratic in f, that is, of the invariant
ay - 2/ {babe - lacd + B,¥d - 66c=) + . . .,
is shown to be an invariant. It will further illustrate the proof of the theorem if we remark that precisely the same invariant is obtained by eliminating y between the above form and the protomorph
a"/- babe + 2acd + 86=^ - 66c''.
Ex. 2. If we take the pure reciprocant
45a'd=- 450a^6cd + 400a6'd + 192a^c' + 16oa6=c= - 4006*c,
which, from its similarity to the Discriminant of the Cubic, I have called the Quasi-Discriminant, and form its discriminant, when regarded as a quadratic in d, we find
45a' (192a^c' + l^bab^'c^ - 4006^) - (225a'6c - 200a6»>'.
If, in this expression, we write P = 3ac — 56-, so that 3ac = P + o¥, it becomes
5 . 64a» (P + 56=^)' + 5 . 165a"-6« (P + obj - 15 . 400a='6« (P + 56»)
-625a-b^(SP+7b^y.
On performing the calculation it will be found that all the terms involv- ing b will disappear from this result, and there will remain the single term 320a»P», that is, 320a» (3ac - ob'f, which is a reciprocant.
LECTURE VIII.
In my last lecture the complete expression, both in terms of the modified and unmodified letters, was obtained for V, the annihilator for pure recipro- "l^iants assuming its existence and its form. These assumptions I shall now IHoake good by proving, from first principles, the fundamental theorem that IBbe satisfaction of the equation I' VR = 0
is a necessary and suflBcient condition in order that R may be a pure reciprocant.
It will be advantageous to use the modified system of letters, in which
, . dv 1 d'v 1 d'y 1 d*v
(, flj, a,, a.,, ... stand for /- , zr-^ . -f£, . ^ „ . -~, ^-.r-^—. ■ -A, ■■■ dx 1.2 fte" 1.2.3 da? 1.2.3.4 oar*
J r- 1 d^x 1 d^x 1 d*x
and. .„,«„,„... for —.^„j_^._.j-2-3-^.^,...
respectively. Let the variation due to the change of x into x + ey, where e
348 Lectures on the Theory of Reciprocants [42
is an infinitesimal number, be denoted by A. Obviously this change leaves the value of each of the quantities o,, a,, a^, ... unaltered, and therefore
KR(ao, a,, a,, ...) = 0,
whatever the nature of R may be. But when iJ is a pure reciprocant,
R{ao,a„ Os, ...)=±<^-K(ao, «!, Oj. •••).
whence it immediately follows that
^t-i'Ria,, Ui, a,, ...) = 0*.
Before proceeding to determine the values of
At, Aoo, Aa,, Aoj, ...
it will be useful to remark that since
^-t ^-1 2 a ^-1 2 3 a,
^^^*^^ ^^^''^ :£°=^^'-' {
and generally -r-^ = (« + 3) an+i ■
Now let [t] denote the augmented value of t, and in general let [ ] be used to signify that the augmented value of the quantity enclosed in it is to be taken. Then
so
or -. ro n <^ W d[t] d[t] ,, ..d[t]
also 2 [ao] = [2ao] = jM = . , V \ = ^ /i , >\ = (^~^*)^ L oj L "J ^Tg.-! d(x + €y) da;(l + €t) dx
[x]~ d{x + ey) dx{l + et)
= (1 - eO ^ (< - ei") = (1 - eO (2a, - 4etoo)
= 2ao — 6etoo ; that is [flo] = Oo — ^eta„.
Reasoning precisely similar to that which gave
2K] = (i-60^^W,
leads to the formula
(n + 3)[a„+,] = (l-eO^K]. \
* It has been'suggested by Mr J. Chevallier that the proof might be simpliBed by considering _M _
the variation A0(, S R {a^, a^, a„, ...) instead of At ** i? (Oj, a^, a^, ...)•
42]
Lectures on the Theory of Reciprocants
349
from which the augmented values of a,, a^,a,, ... may be found by giving to n the values 0, 1, 2, ... in succession. Thus, writing w = 0, we have
3 [aj = (1 - et) ^ [aj = (1 - e«) ^ («„ - Seta,)
= il-€t) (3a, - 9eta, - 6ea„0 = Sa,-e (12ta, + 6a„=), or [oi] = a,-e (ita, + ia,").
Similarly, when n = 1,
4 K] = (1 - et) ^ [a,] = (1 - et) ^(ch- 46to, - 2ea„^)
= (1 - €<) (4ai, - 166^2 - 20ea„a,) = 4a, - 206^2 - 20ea„a„ [oj] = aj _ oe (taj + aoO,).
and
k
Again. 5 [a,] = (1 - rf) A [„,] = (i _ ,^) ^ (^^ _ 5^^^ _ -^^^^^
iKdi
= (1 - eO (5a, - 25eta3 - 30ea„a, - loea,-) = 5a, - aOetoj - SOeaoOj - 1 5eai', 80 that [a,] = a, - 6 (6^3 + 6a<,a, + Soi').
In like manner we shall find
[a*] =a,-7e (ta^ + a^a^ + a^a^). lese results may be written in a more symmetrical form ; thus : 2[<] =2t-2et', 2 [a„] = 2a„ - 3e (to„ + a„«), 2 [a,] = 2ai - 4e (to, + a,^ + a^t), 2 [aJ = 20, - 5e (to, + OoOj + Oja, + a^t), 2 [a,] = 2a3 - 6e (to, + 0,0, + a.' + 0,0, + a,0. 2 [a<] = 2a, - 7e (to, + a„a, + 0,0, + OjOi + a,a, + aJ). The general law
2 [a„] = 2a„ - (n + 3) e (to,, + a„ o^, + . . . + a,_, a„ + a„t), or, as it may also be \vritten, A w+3 .
Imits of an easy inductive proof
Assuming the truth of the theorem for [a,J, and writing for brevitv in what follows,
S„ = tan + a^a^-, + a^a^^ + . . . + Un^^a, + a„_iao + a„t,
360
Now,
Lectures on the Theory of Reciprocants
dS„
[42
dx
^ [a„] = (n + 3) a„+, - ^
Hence
But, as we have already seen,
= (n + 3) to„+, + 2a,a„
+ (n + 2) aottn + 3a,a«_,
+ (n + 1) o,a„_i + ia^On-t
+ +
+ 4a,na, + (»+ 1) On-iO,
+ 3a„_,a, + (n + 2) a„a(,
+ 2a„ao + (n + 3) a,n.i« = (n + 4) (te„+i + a^an + aia„_i + . . .
+ an-ia, + Unaa + ttn+it) — 2ten+i
= (n + 4) Sn+i - 2tan+i. n + S
e {(n + 4) Sn+i - 2to„+,}.
consequently,
(« + 3) [a„+,] = (1 - eO ^ [a„] ;
L«n+iJ = (1 - 6<) a„+, g— ebn+i + eton+i = a„+, ~ ebn+i ;
that is, the theorem holds for [a„+i] when it holds for [a„]. But we know that it is true for the cases n=0, 1, 2, 3, 4, and therefore it is true universally.
Resuming the proof of the main theorem, it has been shown that
Ar''i2(a„, a,, a„ ...)=0;
that is - fit-'At + R-'AR = 0,
r>j 1 * J . dR . dR . dR . „
or - fiRt-^/^t + T— Aao + :t- Aa, + J— Aoj + . . . = 0.
otcio dOi dcUi
But At =-e«',
Atto = — 36to„,
Aai = - 6 (4to, + 2a(,'),
Attj = — e (Sfaj + 5aoO,),
Aa, = - e (Gtas + Gaottj + 3a,'),
Aa4 = - e (7<a4 + 7a„a3 + ToiOa),
and consequently
< (n - 3rto9a<, - 4a,9a, - 5a«da^ - Ga^da^ - 7atda, - ...)R - |4 (yj dat + 5 (ttoOi) 8a, + 6 (ooa^ + ^- j 9„,
+ 7 (ttoaj + ttiotj) Sa^ + . . J J? = 0.
42] Lectures on the Theory of Reciprocants 351
This is equivalent to the two conditions
(3ao8ao+ 4ai9a, +00,90^+ ...)R = /j.R and VR = 0,
•where
F = 4 f y j aa, + 5 (a„ai) da^+6 [^a.a^ + y j a^. + 7 (aotts + aittj) d^^
+ ...,
For greater simplicit}' I confine what I have to say to the only essential case, to which every other may be reduced, of a homogeneous pure reciprocant. The equation
(3ao9oo + 4ai9o, + Ba^da^ + ...)R = fiR
shows that for every term w + Si is constant ; that is, tv is constant and therefore the function R is isobaric. This is also immediately deducible from the form of the relations between ao, fli, a,, ... ; ao, ""i, otj, •••, and, what is important to notice, for future purposes,
^Hwhen F is a, homogeneous isobaric function, and /a = w + 3i is itself a homo- ^^Keneous function of (a,,, a,, a,, ...), whose degree is the same as that of i^.
^H The only condition affecting R, a function of ao, a,, a^, ..., supposed
^■homogeneous and isobaric, is
■ VR = 0.
^r I shall now prove the converse, that if jR = ^(a„, a,, a^, ...) (being homo-
' geneous and isobaric) has V for its annihilator, then i? is a pure reciprocant.
Let D be the value of F((io, (h, ch, ■ ■ ■) — V^ F {<x^, a,, a.^, ...) expressed as a
function of a,, a^, a.^, ... alone. Then D will be a function of the same type
as F{ao, a,, a,, ...).
Suppose that Ai) = 0 ;
that is, that the variation of D due to the change of x into x + ey vanishes in virtue of the equation VR — 0.
IH Let D become If when y receives an arbitrary variation y + r)u, where r) "" is an infinitesimal constant and u an arbitrary function of x ; then the varia- tion of ly will vani.sh when x is changed into x+ ey ■\- eiyu, and consequently when X is changed into x+ey the variation of Z)' will also vanish. Hence
AJ)' = 0,
and if we take the difference of the variations of D and D', we shall find
\ aaa aoi aa, /
Now, the arbitrary nature of the function u shows that we must have
A:^i) = 0, A^i) = 0, A:^i) = 0, aoo aoi oo.
352 Lectures on the Theory of Reciprocants [42
and if we reason on -j— D, -j— D, ... in the same way as we have on D, we
see that the variation A of each of the second differential derivatives of D will also vanish ; and, pursuing the same argument further, it will be evident that the A of any derivative of D, of any order whatever, with respect to «o, a,, a,, ... will vanish. Hence
i) = 0;
for if this is not so we may, supposing i) to be a function of degree i in the letters a^, a^, a^, .... take the A of each of the differential derivatives of i) of the order t — 1 ; each of these variations would vanish by what precedes ; that is, the variation due to the change of x into x-\- ey oi each of the letters «o. <*i. «a. ••• contained in D would be identically zero, which is absurd. We see, therefore, that when AZ) = 0 (that is, when R is annihilated by F), D = 0, or
F(o„, a„ o^, ...)=P'F(ao, o,, a,, ...), which proves the converse proposition.
It will not fail to be noticed how much language, and as a consequence algebraical thought (for words are the tools of thought), is facilitated by the use of the concept of annihilation in lieu of that of equality as expressed by a partial differential equation.
It is somewhat to the point that in the recent two grand determinations of the order of precedence among the so-called fixed stars relative to our planet, as approximately represented by the intensities of the light from them which reaches the eye, the one is directed by the principle of annihila- tion, the other by that of equality. Prof. Pritchard's method essentially consists in determining what relative thicknesses of an interposed glass screen, effected by means of a sliding wedge of glass, will serve to extinguish the light of a star ; that employed by Prof. Pickering depends on finding what degree of rotation of an interposed prism of Iceland spar (a Nicol Prism) will serve to bring to an equality the ordinary image of one star with the extraordinary one of another. As these intensities depend on the squared sines and cosines of this angle of rotation measured from the position of non-visibility of one of them, it follows that the tangent squared of the twist measures the relative intensities by this method.
Hereafter it will be shown that if ^ is a homogeneous isobaric function of
y. y'' y"> y'". •••>
whose weights are reckoned as
-2,-1, 0, 1,... then, when x becomes x + hy, where h is any constant quantity, F becomes
{l + hty^e i+^i?*. where t = y', F, = - t^dt + F, and /* = 3i + w,
i being the degree and w the weight of F.
•»»
42] Lectures on the Theor}/ of Reciprocants 353
From this, by an obvious course of reasoning, could be deduced as a particular case the condition of F{ai,, a^, a«, ...) remaining a factor of its altered self when any linear substitutions are impressed on x and y ; namely, the necessary and sufiBcient condition is that F has V for its annihilator.
LECTURE IX.
The prerogative of a Pure Reciprocant is that it continues a factor of its altered self when the variables x and y are subjected to any linear substitu- tion. Its form, like that of any other reciprocant, is of course persistent when the variables are interchanged ; that is, when in the general substitution, in which y is changed into
fy+gx + h
I and X into f y+ 9'^ + '*'>
we give the particular values h—0, h' = 0, f=0, g' =0,/' =1, g = 1, to the constants. Stated geometrically, the theorem is that the evanescence of any pure reciprocant R indicates a property independent of transformation of axes in a plane. We suppose R to be homogeneous and isobaric in a, b, c, .... (If it were not, the theorem could not hold, for either the change of y into Ky or that of x into \x would destroy the form.)
The persistence, under any linear substitution, of the form of pure recipro- cants may be easily established as follows :
By a semi-substitution understand one where one of the variables remains unaltered. There are two such semi-substitutions, namely, where x remains unaltered, and where y does.
(1) Let X remain unaltered and y become fy+gx + h; then a, b, c, ... become fa, fbj/c, ... respectively, and therefore
R(a, b, c, ...) becomes f^R (a, b, c, ...), IBvhere i is the degree of R.
' (2) Let ?/ remain unchanged and a; become y^-f-^r' a; -(- A'. Then, instead
of R, I look to its equal
qt>^R{a,&.y....)(q = ±l);
that is, to qr-^R (a, /3, 7, . . .),
which becomes q (/' + g'r)-''g'^R (o, /3, 7, . . .).
Since iiJ is a reciprocant, this is equal to
or, replacing t by its equivalent - ,
V
U't + 9')-<^g'R{a,b,c, ...).
8. IV. 23
354 Lectures on the Theory of Reciprocants [42
Thus we see that the proposition is true for a semi-substitution of either kind. Consider now the complete substitution made by changing y into
fy-\-gx + h and X into Fy + Gx+H.
d'u d*v , 9 d'x g d'x
If/=0 and (? = 0, then 5J, ^. - become Jk- d^.' J,- df' ■'' ^"^
that ie(o, b,c,...) becomes pfr^ . R(a, 0, y, ...); and since this is equal to
^.qt-'^R{a.b,c,...).
the proposition is true.
But if either of the two letters /, 0 (say/) is not zero, we may combine
two semi-substitutions so as to obtain the complete substitution, in which y
changes into
fy+gx + h,
and X changes into Fy + Gx+ H.
(1) Substitute yi{=fy +gi>: + h) for y, and x,{=x) for x.
(2) Then substitute yi{=yj) fw y^ and Xii=f'yi+ g'x^ + h') for a^. By the first of these semi-substitutions
R(a, b, c, ...) takes up an extraneous factor/'. By the second it acquires the factor
Hence we see that the extraneous factor is a negative power of a linear function of t, which we shall presently particularize, though it is not essential to the present demonstration to do so.
It only remains to show how the combination of these two semi-substi- tutions can be made to give the complete one in question. We have
and x^ =/','/! + g's^i + h' =/' {fy + gx + h) + g'x + h'
==fry+if'9+9')'^ + (f'h + h'). In order that this may be equal to Fy + Gx + H, we must be able to satisfy the equations
f =->, 9 =G-y , h - tl-~j ,
which is always possible, since by hypothesis / is not zero. Similarly it may be shown that when/ vanishes, but G does not, by substituting
(1) a;,(=i^y+ Ga; + //)for «, and y,(=y) for 2/,
(2) X, (= X,) for x^, and y, (=/'Vi + 9"<^i + ^") for yi. ^
we may so determine/", g", h" as to get the complete substitution as before.
I «
I
42] Lectures on the Theory of Reciprocmits 355
In every case, therefore, any linear substitution impressed upon the variables x and y will leave R{a, b, c, ...) unaltered, barring the acquisition of an extraneous factor which is a negative power of a linear function of t.
Now, the first semi-substitution introduces, as we have seen, the constant factor
/•;
the second introduces the factor
where ^'=/* + 5'-
The complete extraneous factor is the product of these two, and is therefore
f'g'Hfft+f'g+g')-''.
To express /' and g' in terms of the constants of the complete substitu- tion we have
Writing these values for/' and g' in the expression just found, we obtain
{fG-gFf{Ft+G)->^,
which is the extraneous factor acquired by R when the complete substitution is made. For example, if x becomes
Fy+Ox+H, and y becomes fy+gie + h,
the altered value of oUhat is, of -j^\ is
{fG-gF){Ft-^G)-'a. Corresponding to the simple interchange of the variables, we have -^=1, (? = 0, H=0;f=0, g=l, h = 0, so that fO — gF= — \,
and the altered value of a is -r- , or
dy^
a
which is right. In this case the general value of the acquired extraneous factor
(/G-gF)*(Ft + G)~'' becomes (-)'r^ thus showing, what we have already proved from other considerations, that the character of a pure reciprocant is odd or even according as its degree is odd or even.
23—2
356 Lectures an tlie Theory of Redprocants [42
We saw in the last lecture that every pure reciprocant necessarily satisfied the two conditions
(where /i is the characteristic), and
VR = 0. We also saw that Ffl = 0 was a sufficient as well as necessary condition that any iMvwgeneom function R of a„ a,, a,, ... should be a pure reciprocant. It will now be shown that every pure reciprocant is either homogeneous and isobaric, or else resoluble into a sum of homogeneous and isobaric recipro- cants. Non-homogeneous mixed ones, it may be observed, are not so resoluble, so that the theorem only holds for pure reciprocants.
(1) Let us suppose that R (a pure reciprocant) is homogeneous in o„. a,, o,... ; then it must be isobaric also. For, if i is the degree of R, Euler's theorem shows that
(3a.9„, + 3a,aa, + 3a,8a, + 3a,9„, + ...)R = 3iR;
and since 72 is a pure reciprocant, the condition
(3ao9a« + ^^<H + Sos^o, + Q(h^<H + ...)R = tiR
is necessarily satisfied. Hence
(a,da, + 2a,da, + 3a,8a, + . . .) ^ =(m - ^i) iJ = a constant multiple of R,
which is the distinctive property of isobaric functions.
And, vice versd, if R is homogeneous and isobaric of weight iv and degree
i, then
(3a„a^ + 4a.a„, + 5a,a„, + ...)R = (w + 3i) R = fiR.
Thus homogeneous pure reciprocants are also isobaric and their character- istic is Si + w. (This property is also true for mixed reciprocants, as we have previously shown.)
(2) Suppose that R is not homogeneous, but made up of the homo- geneous parts
R,, R,,' R,,,
Then, since V{R, + R„ + R,„ -1- . . .) = 0
is satisfied identically, it is obvious that
VR,+ VR„+VR,,, + ... = 0 must also be satisfied identically.
But since all the terms are of different degrees, the only way in which this can happen is by making VR^, VR„, VR^^„ ... separately vanish. Now,
R,, jB„, i?,„, ... are by hypothesis homogeneous functions of Op, Oj, a , and
it has just been shown that each of them is annihilated by V, which has been shown to be a sufficient condition that any homogeneous function of tto, th> <h, ... loay be a pure reciprocant. Thus each part R,, R„, R,„, ... of i2 is a pure reciprocant.
I
c
I
42] Lectures on the Theory of Redprocants 357
Also, the condition
(300800 + 4oj3«, + SoaSo, + ...)R = imR shows that if t\, w,; i,, w,; i,, w,; ... are the deg. weights of R„ R,„ R„,. .-,
^we must have #
Thus non-homogeneous pure reciprocants are severable into parts each of which is a homogeneous and isobaric pure reciprocant, the characteristic of each part being equal to the same quantity fi, which is the characteristic of the whole.
I will now explain what information concerning the number of pure reciprocants of a given type is afforded by the equation VR = 0. Let
Aao'^ai'^m^'^ ... a/^i be a term of a homogeneous isobaric function (with its full number of terms) of Oo, Oi, a» ••• fflj. whose degree is i, extent j, and weight w, and which we will call R.
Then in the entire function there are as many terms as there are solutions in integers of the equations
\o + \i + \i + X3+ ... +>^j = i, X, + 2X^ + 3\, + . . . + jA, = w. In other words, the number of terms in E is equal to the number of ways in which w can be made up of i or fewer parts, none greater than j. This number will be denoted by
(w; if)-
Since the function R is the sum of every possible term of the form
Aoo'^ch''' ... af'i, each multiplied by an arbitrary constant, the number of these arbitrary constants is also
Now, suppose iZ to be a reciprocant ; this imposes the condition
VR = 0. Consider the effect produced by the operation of any term of
F = 4 ("I) da, + 5aoa,9«, + 6 (a„a, + y) a«. + • ■ • , Bay (aoa, + ^)9a, (rejecting the numerical coefficient 6).
Operating on R with da, decreases its weight by 3 and its degree by 1 unit. The subsequent multiplication by a„(h + y , on t^e other hand, in- creases the weight by 2 and the degree by 2 units. Hence the total effect
358 Lectures on the Theory of ReciproA. '-
of \a„a^ + -J- j a^j is to increase the degree by 1 and to diminish the weight
by 1 unit. The same is evidently true for any other term of V. Thus the total effect of V operating on the general homogeneous isobaric function B of weight w, degree i, extent j, is t(f change it into another homogr' isobaric function whose weight, degree and extent are respectively ? i + l,j. Observe that the extent is not altered by the operation of V.
It is easily seen that the coefficients of VR are linear functions Vt coefficients of i? ; for example, if '^ '"
VR = ao'oj {QA + 2 J5) + a,^a^ (34 + 55 + 6(7). "^
Hence the condition VR = 0 gives us {w-\; i+ljyimear'eqm between the (w; i,j} coefficients of i? ; so that, assuming that these equl of condition are all independent, after they have been satisfied the nu of arbitrary constants remaining in R (that is, the number of linearly :^^ pendent reciprocants of the type w ; i,j) is equal to
(w; i,j)-{w-\; i + l,j),
when this difference is positive ; but when it is zero or negative there r reciprocants of the given type.
If, however, any r of the (w-1; i+l,j) equations of condition s not be independent of the rest, these equations would be equivale. (w 1; i + l,j)-r independent conditions, and therefore the numbe linearly independent reciprocants of the type w ; i, j would be
(w; i, j)-(w-l; i+ij)+ r. It is therefore certain that this number cannot he less than
(w; i,j)-{iu~l; i+l,j). We shall assume provisionally that r = 0, or in other words thar the above partition formula is exact, instead of merely giving an inferior limit. Ihough It would be unsafe to rely on its accuracy, no positive grounds for doubting Its exactitude have been revealed by calculation.
Such attempts as I have hitherto made to demonstrate the theorem' have proved infructuous,but it must be remember<?d that more than a quarter of a century elapsed between the promulgation of Cayley's analogous theorem and Its final establishment by myself on a secure basis of demonstration.
j^f,,. Lectures on the Theory of Reciprocants 359
' LECTURE X.
I will commence this lecture with a proof of Capt. MacMahon's theorem that if R is any pure reciprocant and /x its characteristic (that is, its weight ided to three times its degree),
(^"S) -K = 1 • 2 • 3 ... m {m(m+ 2) (m + 4) ... (/i + 2?ft- 2)} {fTR,
'. ere y" may be replaced by either 2ao or a, according as the modified or >nmodifieH system of letters is employed.
. Instead of a pure reciprocant, let us consider any homogeneous isobaric notion F of degree i and weight w; and [for the sake of simplicity writing
^frr T-] -'nstead of the operator F"3^™ let us consider F^a,"- aa»F''». e have identically
( F'-a," - 3,»F») F= F"-' ( FS, - 9xF) a,»-'Jf
° + F-"-* {Vd,-d, V) F'a,»-'F
+
Bi + F ( Fa, - 9, F) F-^a,"-' F
anc + ( Fa, - a, F) F"->a,*-' F
+ a, ( v^d,^' - a.»-' F") F.
Now, the operation of (Fa» — a,F) on any homogeneous isobaric function ^ aose characteristic is ni is equivalent, as we have seen in Lecture VII, to multiplication by /tiy"; so that if the characteristics of
a,"-'i', Fa,"-'f, F»ax»-'/'. ... F'»-'a,»-'^
are /t, , /*> , M» . ■■• Mm>
it follows that
( F^a." - a," F") = (At, + A*, + A*, + • . • + /i») y" F—a^-'F
+ a« ( F"*ax"-' - a,"-' f™) i^.
Observe that
F— > ( FBx - a,F) a,»-'^= F— 'Ai,y"9«"-'^= Miy">^"^'a»""'^.
where the transptosition of the y" is permissible because Fdoes not act on it ; but if y" were preceded by dx it could not be similarly transposed.
The numbers a*i. /*». /^».--- form an arithmetical progression, for each operation of F increases the degree by unity and diminishes the weight by nnity, so that
All = 3t, + tffj becomes a»« = 3 (*, + 1) + (w, — 1) = a^i + 2. Similarly a*i = /^i + *. a*4 = /*i + 6, . . . Atm = AH + 2ot — 2.
360 Lectures on the Theory of Reciprocants [42
The characteristic of F being
/* = 3i + w, that of d./'-^F is /ii = /t + n-l;
for each operatioa of dx leaves the degree unaltered, but adds an unit to the weight ; hence
/*, + /*, + /i, + ... + /i,„ = m (/i + TO + » - 2) ; 80 that
When F= R, a pure reciprocant, so that VR = 0, our formula becomes
V'^dx^R = m (/i + m + n - 2) y" F'»-'a,»-'iJ + d^ F-^a^^-'E. (!)
Suppose that in (2) m>n, then F"'3a:"-R = 0. This is obviously true when n = 0, and when n = 1. When n = 2 we find
V^'d^^R = m{fi + m) y" V^-'d^R + 3* V^d^R
= 0 if TO > 2.
Similarly the case n = 3, to > 3 can be made to depend on n = 2, m > 2, ard in general each case depends on the one immediately preceding it. Next l«t n = wi in (2) ; then, remembering that V^dx^^^R = 0, we have
V'^dx'^R = TO (/x + 2to - 2) y" V^-'d^'"- 'R,
from which MacMahon's theorem that
V"'dx'"R = 1 . 2 . 3 ... TO {/i (/i + 2) (/x + 4) . . . (/t + 2to - 2)} (y'^R
is an immediate consequence.
Another special case of Formula (1) is worthy of notice, namely, that in which we take n = 1, when we obtain the simple formula
( F^a^ -dxV'^)F = m(fi + m-l)y" V'-'F. (3)
If in this we write a„ in the place of F, and (the modified system of letters being used) 2ao for y", fi becomes 3 + n, and we have
( V'^d^ - dx F">) a„ = 2to (to + K + 2) a.V^-^On,
or, as it may also be written,
V^d^an ^ dxV^an 2 (m + n + 2) a,V«>-'an ,,.
1.2. 3... TO 1.2. 3. ..to"*" 1.2.3...(to-1) " ^ ''
Mr Hammond remarks that this last formula may be used to prove the
theorem _Y
«« = -r''-»(e ()a„,
which was given without proof in Lecture II. Assuming that a„ = - t-^-^a^ + t-»-* Va„ - r"-» ^ + . . . ,
42] Lectures on the Theory of Reciprocants 361
we have to prove that the theorem is also true when n is increased by unity. Differentiating both sides of the assumed identity with respect to x, we find
= - t-^-'d^an + r"-* {d^Vun + 2 (« + 3) a„an}
(? Vfi )
- ir'^ ^'~ + 2 (n + 4) a„ Fa„J
+ ;
the general terra being
._.„+, ._„_„_3 [ a»F"-a. 2(m + » + 2)a,F">-aJ
^ ' (1.2.3... wi 1.2.3...(m-l) J
which, by means of (4), reduces to
^ ^ '^ 1.2.3...m'
V'd a Hence 3,a„ = - «-»-'a,o„ + 1r"-*Vd^a„ - f^-' -^^ +■■■>
or, more concisely,
r
axan = -<-'-'(e"0axO». But dxUn = (n + 3) a„+,, and dxO^ = OyO„ = (n + 3) <a„+,,
and therefore
r in + 3) <a„+, = - (n + 3) r»^ (e' ' ) a„+„
r or «„+, = -<-»-* (e"')o»+i-
The theorem is easily seen to be true, for m = 0, 1, 2, and is thus proved to be true universally,
I will now return to the point at which I left off in my previous lecture. We saw that the exactitude of the formula
(w;i,j)-{w-\;i + \,j)
for the number of pure reciprocants of the type w ; i,j could not be inferred with certainty unless we were able to prove that the (w— 1; i + \,j) linear equations between the coefficients of R, found by equating VR to zero, were all of them independent. A similar difficulty presents itself in the proof of the corresponding formula (w; i,j) — (w — 1 ; i.j) in the invariantive theory; but in that case I succeeded in making out a proof of the independence of the equations of condition founded on the fact that Q^O"/ is a numerical multiple of /, where / is any invariant, and ft, 0 are the well-known operators
a,Sa, + 2a,aa, + 30,80, + . . . + jOj-^'^aj
362 Lectures on the Theory of Reciprocants [42
I have since discovered a second proof of the theorem for invariants which, though very interesting, is less simple than my first; but neither of these methods can be extended to the case of reciprocants.
It was suggested by Capt. MacMahon that the fact that V"^dx'^R is a numerical multiple of a"^R ought to lead to a proof of the theorem for reciprocants similar to that obtained for invariants by my first method, alluded to above, but this I find is not the case ; and indeed it is capable of being shown a priori that it cannot lead to a proof. One great distinction between the two theories, which is fatal to the success of the proposed method, is well worthy of notice.
If {w;i,j)-{iu-\;i,j) = >0
(I shall sometimes call this positive), then
(w'; i, j) - (w'- 1; i, j) = > 0 for all values of w' less than w; the condition that this difference, say A(w;t,j) shall be positive being simply that ij—2w is positive (that is, ij—2w = > 0). This is not the case with the difference
{w;i,j)-(w-l;i + l,j), say E{w,i,j); it by no means follows that if this is positive for a given value of w (i, j being kept constant), it will be so for any inferior value of w.
We may illustrate geometrically the condition ij — 2w = > 0, which holds when A (w ; i, j) is non-negative.
Let (r, j) be co-ordinates of a point in a plane and draw the positive branch of the rectangular hyperbola
ij — 2w = 0.
Then, ij — 2w < 0 for all points in the area YOXBA between the curve and its asymptotes; but for points on the curve AB,
ij -2w = 0,
and for all points of the infinite area on the side o{ AB remote from the origin,
ij — 2w>0.
Thus, for all points which lie either on or beyond the curve AB, A (w ; i, j) is non-negative, and for all points between the curve and the asymptotes A(w; i, j) is non-positive.
We have here considered w as constant and i, j as variable, but in the case where all three are variable we should have to consider the hyperbolic paraboloid
ij — 2w = 0,
of which the curve AB is a section, by the plane w= const.; and the condition
I
42] Lectures on the Theory of Reciprocants 363
of A (w ; i, j) being non-negative or non-positive depends on the variable point (i, j, w) lying in the one case on or beyond the surface, and in the other between the surface and the planes of reference.
The function of{,j, w, whose positive or negative sign determines in like manner that of E (w; i, j), cannot be linear in w. What its form is, or whether it is an Algebraical or Transcendental function, no one at present can say. Indeed, except for the light shed on the subject by the Algebraical Theory of Invariants, it would have been exceedingly difficult (as I know from vain efforts made by myself and others in Baltimore) to prove the much simpler theorem that A (w ; i, j) is positive (that is, non-negative) when ij — 2w is so. It amounts to the assertion that the coefficient of
a'af" in the expansion of
1-ar
(1 - a) (1 - or) (1 - ax') ... (1 - (mj>) is always non-negative, provided that ij — 2w is non-negative.
This is a theorem of great importance in the ordinary Theory of Invariants, and may be seen to be a consequence of the fact, which I have proved, that (using [w; i, j] to denote a function of the type w ; i, j having its full number of arbitrary coefficients) there are no linear connections between the coefficients of (1 [w; i,j] when ij —2w = > 0; but no one, as far as I know, has ever found a direct proof of it.
Viewing the connection between the two theories of Invariants and Reciprocants, I think it desirable to recapitulate with some improvements the proof, given in the Phil. Mag. for March, 1878, of the theorem that the number of linearly independent invariants of the type w ; i, j is exactly A (w ; i, j) when this quantity is positive, and exactly zero when it is 0 or negative.
As regards reciprocants, at present we can only say that the number of linearly independent ones of the type w ; i, j is never less than E {w ; i, j), leaving to some gifted member of the cla.ss to prove or disprove that the first is always exactly equal to the second. The eaxict theorem to be proved in the theory of invariants is as follows :
If ij — 2w = > 0, the number of linearly independent invariants of the type w; i,j is A(«;; i,j).
If ij —2w<0, the number of such invariants is zero ; that is, there are none. The proof is made to depend on the properties of
il = a„ao, + 2a,a„, -I- 3a,aa, + ••• +jai-x\ and of 0 = ajda^_^ + 2aj_,daj_ + Suj^da^, + ... + ja,9„„.
If U be any homogeneous isobaric function of degree i and weight w in the letters a^, Oi, aj,...aj, it is easy to prove that
(no - Oil) tr= {ij - 2w) U,
364 Lectures on the Theory of Reciprocants [42
and consequently, if U is an invariant /, so that HI = 0,
nOI^iij-lw)!.
I call ij — Iw the excess and denote it by rj, and shall first show that if rf is negative 7 = 0; that is, there exists no invariant with a negative excess. This will prove that when A(w ; i,j) is negative, that is, when
{w-\;i,j)>iw;i,j),
the number of independent functions of the coefficients of \w ; i, _/] which appear in fl [w; i,j'\ is exactly equal to (w ; i,j), which is the number of the coefficients themselves. Clearly it cannot be greater; for, no matter what the number of linear functions of n quantities may be, only n at the utmost can be independent; there might be fewer, there cannot possibly be more. The complete theorem is that the number of independent coefficients in n[w; i, j] is the subdominant of two numbers: one the number of terms of the type w ; i, j, the other the number of terms of the type w — l;i, j.
N.B. That one of two numbers which is not greater than the other is called the subdominant.
LECTURE XL
We may write for the Annihilator of an Invariant n = ttod, + 2a,d2 + Sa^a-i + ... +jaj_idj and for its opposite
0 =ja^a, + (j - 1) a^di + (j - 2) a^a^ + . . . + ffljO,-,,
where the pointed letters do, d,, dj, ... d^ stand for the partial differential operators
f^aoi Ooi, Ooj) ••• 8o.
Suppose n and 0 to operate on any function U{a„, a^, a,, ... aj); then D,OU={D..O + D.*0)U and On[/'=(0.n + 0*f2)Z7,
where the full stop between 0 and fl signifies multiplication, and the asterisk operation on the unpointed letters only. Thus,
and, consequently, (HO - OH) f/= (fl * 0 - 0 * fi) [7. Now,
ft* 0[r= {1 .ja^a, + 2{j-\) a,d, + S(j - 2)a,a,+ ... +j . la>_id,_,} U, and
0*nU={l .ja^d, + 2 (j - 1) a,rl + . . . + ( j _ i) 2aj_^dj_, + j . lajdj\ U,
42] Lectures on the Theory of Reciprocants 365
whence we readily obtain
(flO — on) f/^ = j (aodo + Oidi + as<>^ + • • • + ffljO;) U
— 2 {a^d-i + ^.a^a^ + Za^d, + ... +jajdj) U. Introducing the conditions of homogeneity and isobarism, namely, (oodo + fliOi + Ost's + ••• + ajdj) U= ill and (a,di + 2aid^ + Sojds + . .. +jajdj) U=wU,
where i and w denote the degree and weight of U, supposed now to be a
rational integral homogeneous and isobaric function (or, to avoid a tedious
periphrasis, say a gradient), we see that if the complete type of the gradient
U\sw;i,j,
(ilO - Oil) U= (ij -^w)U=7,U,
where ij is the excess.
Since the operation of 0 increases the weight of the operand by unity, but does not alter either its degree or its extent, it is clear that the type of O'U \s w + 0; i, j. The excess of O^U is therefore
ij-2{w + 6) = r}-2d,
and the theorem just proved shows that
(no - on) (y>u=:{n- -le) o»u.
From this we pass on to prove that fiO' — O^Vi, acting on any gradient as its objective, is equivalent to 9(1; — 5 + 1)09"'; that is, when q is any positive integer, we shall show that
(nO« - 0«n) U=q(r,-q^\)09-^U.
The subsequent consideration of a special case of this formula, in which U is replaced by any invariant /, will enable us to prove that there can be no invariants for which the excess ij — 2w is negative. Let
(y>-»ilO>V = P,U;
then 09^»-'nO*+' U = P^^ IT.
and therefore (P,+. -Pt)U= 0^-»-'(ilO - On) 0« U.
Substituting in this for
(no - on) 0»U its value (17 - 26) 0»U,
we have (P»^i -Pt)U=>{n- 2^) O?-' U.
Hence
(P, -Po)U= {(p, - p„) + (P, - P.) + (P. - p,) + . . . + (P, - P,-Oj u
= {1, + (,, - 2) + (1, - 4) + ... + (,7 - 2g + 2)} Oi-'U
But since Pg = ilO^ and P, = O'n, this result may be written (no? - (nil) U=q(r,-q + l) O^-'U.
366 Lectures on the Theory of Reciprocants [42
If now U= I, an invariant, we have ClU= 0, and our formula becomes
aOiI=q{ri-q-¥l)0i-'L Writing in succession q = m, m — 1, ... 1, we obtain m(i}-m+l) O^^I = no™/ {m - 1) (t; - m + 2) 0™-^ = nO""-'/ (m - 2) (i? - m + 3) O™"'/ = nO"^^I
1.7?/= no/.
By assigning to m a sufficiently large value we are able to make 0"*/ vanish as well as D.I ; for, the type of / being w; i, j, that of 0"'I is w + m;i,j. But it is evident that no gradient can have a greater weight than ij, the product of its degree and extent, for each term is a product of i letters none of them having a weight greater than j. If, then, we suppose that m = i) — w + l, the weight of O"*/ is
w + m = ij + I. Therefore O'"/=0.
Again, t] —m + l = ij — 2'W — (ij — 2V + 1) + 1 = — w.
If, then, IJ is negative, every term in the series
m(r)-m + l), (m-l)(97-m + 2), ... 2(i7-l), l.t) is negative and can never vanish. Hence we have successively
O'»-i/=0, O'»-''/'=0, ... / = 0; that is, when ij — 2w < 0 no invariant of the type w ; i, j exists.
Observe that the elenchus of the demonstration consists in the fact that the successive numerical factors rj — m + l,r} — m + 2, rj — m + 3, ...t) are all non-zero on account of r/ being negative ; but if ij were positive we should eventually come to a factor 17 — /* which would be zero, and we could not conclude from (fi + 1)(7? - /x)0''/ being zero that 0"/= 0. Since 77 - (m - 1) passes from ij - (ij - w) to rj, that is, from - w to t), it passes through zero when r) is positive.
The second part of Cayley's completed theorem remains to be proved, namely, that when ij - 2w = > 0, tlie number of linearly independent in- variants of the type w; i,j is precisely equal to A (w; i,j); that is, to
(w;i,j)-{w-l;i,j). I show this by proving that if i)(w; i,j) is the number in question, keeping
i and j constant and taking w < = — ,
D(w; i,j) + D{w-l; i, j) + 1) (w - 2; iJ) + ... + D{0; i, j) cannot be greater than
A(w;i,j) + A(w-l;i, /)-(-A(w-2;i, y)+...-hA(0;i, 7),
42] Lectures on the Theory of Reciprocants 367
and consequently, since we know that no single D{iu;i,j) can possibly be less than the corresponding A(w; i,ji), it follows that
D{w; i.j) + D{w-l; i,j) + D{w - 2; i,j) + ... + Z)(0; i,j) = A(w; i,j) + A{w -l;i, j) + A (w - 2; i, j) + ... + A (0; i,j); and, furthermore, that each
D{w;%,j) = £^{iu ;{,]).
For if any D were greater than its corresponding A, some other D would have to be less, which is impossible.
This principle of reasoning may be illustrated by imagining a row of ballot-boxes and supposing it to be ascertained that no single box contains fewer white balls than black ones. If, then, there are not more white than black balls altogether, the total number of whites must be the same as that of the blacks. And since there are just as many whites as blacks distributed among the ballot-boxes, the number of white and black balls must be the same in each box ; for otherwise some box must contain fewer whites than blacks, which is contrary to the hypothesis.
Observe that the sum of these A's is {u};i,j); for («' ■.i,J)-(w-\; I, /) + {w-\; i,j) - (w - 2 ; t, j) -f- ... -h (0 ; x,j) - (- 1 ; i.j)
aud (-l;»,j) = 0,
since there is no way of composing — 1 with parts 0, 1, 2, ...j. Hence what I have to show is that
D{w;i,i) + D{w-\;i,j) + ... + D{\;i,j) + D(Si;i,j) = iw;i.]'). I want preliminarily to express QflCfil as a multiple of /*. This can be done by a formula previously demonstrated, namely,
which gives
r2'0«/ = g (i; - 9 + 1) nO^'/ = 5 (i? - 3 -»- 1 ) (9 - I) (ij - gr + 2) O*-*/; similarly
CVOn =q(v-q + l)(q-l)(,V-q + 2)iq-V(v-q + 3) O^-'I;
and finally, changing the order of the numerical factors,
n->0^I^l.2.S...q{v(v-i)(v-^)-{v-q + l)]I.
This shows that Si^Cfil and d fortiori 0^1 can never vanish unless t) —q + 1 becomes negative.
* The result of operating on / with 0 and 0 each q times, the two operations following each other according to an; law of distribntion whatever, will always be a numerical multiple of / ; but the value of this multiple will differ for different laws of distribution.
368 Lectures on the Theory of Becijtrocants [42
Suppose now that /« means an invariant of the type w — q; i,j; its excess is ij— 2(w~q), and consequently O*/, cannot vanish unless
ij-2(w-q)-q + l
becomes negative, which is impossible. For
ij — 2 {w — q) — q + 1 = ij — 2w + q + 1,
and t; — 2m; = > 0 by hypothesis.
By taking 0'/, as an image, so to say, of Iq we shall be able to obtain a limit to the number of /^'s by obtaining a limit to the number of their images. In fact, taking the image O^Iq of each of the D{w — q; i,j) linearly independent invariants of the type w — q; i,j (this is what is meant by the /,'s) and giving q all possible values from 0 to w inclusive, the total number of these images is obviously
D{w;i j) + D{w-l; i,j) + ... + D(0 ; i,j).
Each of them will be a gradient of the weight w — q + q (that is, of weight w), and will consist of terms of weight w, degree i, and extent j. The total number of such terms will be the number of ways of making up w with i of the numbers 0, 1, 2, 3, ...;', or with the usual notation (w ; i, j). If, then, it can be shown that none of these forms are linearly connected, then, inasmuch as they are all functions of the same (w;i,j) arguments, it will follow that theii' total number cannot exceed {w ; i, j). That is, we shall have shown that
D{w;i,j) + D{w-\; i,j) + D{w -2;i,j) + ... + J){0; i,j)
cannot exceed
A{w;i,j) + A{w-1; i, j) + A(w-2; i,j) + ... + A(0; i, j),
and by the ballot-box principle, as already stated (inasmuch as no D is less than its corresponding A), it will follow that each D is the same as the corresponding A, and the theorem to be proved is established.
The proof of this independence is easy. For (1) suppose that there is any linear relation between the forms
for each of which the value of q is the same. Denoting these forms by
P P ' P " let the relation in question be
Then xn^P^ + Vn'P,' + X"fl?P/' + . . . = 0.
But each argument n«P, is of the form fi'O'/,, and since this is equal
42] Lectures on the Theory of Reciprocants 369
to /, multiplied by a number which does not vanish*, we have a linear relation between /,, /,', /,", ..., namely
\/, + X'/,'+\"V+... = 0;
that is, the IqS, would not be linearly independent, contrary to hypothesis. Thus the images (O^Iq, 0'>lq, O^Iq" ...) belonging to invariants of the same type w — q;i,j cannot be linearly connected.
(2) I say that the images of invariants of different types cannot be linearly connected. For let q, q', q", ... arranged in descending order of magnitude, be the different values of q in the images supposed to be linearly related. The result of operating with f2« on any image of the form 0^1^ is to bring it to the form Cl^-^'Q.^'Oi'I^, which is a multiple of Vit-i'I^, and therefore vanishes. But fl', acting on any of the images O'/,, 0''lq, ..., will, as we have seen, bring back the multiple of /,; thus the operation of H? on the supposed relation will give a linear equation connecting Iq, Iq, Iq, ... , and for the same reason as before this is impossible. Hence there can be no linear relation whatever between the images of the invariants whose types extend from w; i, ; to 0; %, j, and the number of these images will accordingly be not gieater than {w; i,j), as was to be proved.
It is well worthy of notice that D{w\ i,j) may be zero, but obviously cannot be negative, as it denotes a number of things which may have any value from zero upwards. Hence follows a remarkable theorem in the pure theory of partitions which it would be extremely difiScult to prove from first principles, namely, that the difference between the two partition numbers
(M';t,j)-(«;-l;i, ;■) can never be negative when ij — 2w=>0. It may be zero, but cannot be less than zero. This explains what I said about the hyperbolic paraboloid ij — 2w = 0, where i, j, w are treated as co-ordinates of a point in space. We might call the value of (w ; i, j) — {w—l; i, j) the density of any point i, j, w, and the theorem may then be expressed by saying that at points within or upon the hyperbolic paraboloid the density can never be negative ; for points outside this surface it can never be positive.
As regards the analogous formula in the Theory of Reciprocants
(w; i,j) - (w - 1; i + 1, j),
we do not know that any algebraical surface can be constructed which will enable us to discriminate between the cases in which this difference, say E(w ; i, j), is positive or negative. Should such a surface exist, its equation must contain w in a higher degree than the first. Supposing that the above
* In fact, remembering that the excess of the type u -q; i, j ia ij - 2 {a - q) =ji + 2q, we find n«0«/, = 1.2.3...g{(, + 2g)(, + 25-l)...(, + g + l);/„ in which both i; and q are positive integers.
S IV. 24
870 Lectures on the Theory of Reciprocants [42
formula represents the actual number of reciprocants, it will follow (and this is confirmed by experience) that there can be no reciprocants to a type of negative excess. For
{w;i,j)-(w-l;i + l,ji) = {w\i,j)-(w-\; i,j)-[{w-l; t + 1, 7) - (w-1; i, j)] = («» ; i j) - (w - 1 ; i, j)-{w-i-2;i + \,j - 1).
But if ij — 2w is negative, {w ; i, j) — {w — l; i, j) is zero or negative. Hence {w ; i, j) - (w — 1 ; i + 1, j) is non-positive.
For satisfied invariants (those ordinarily so called) it; = ^ , and the formula for their number becomes ( ^; i,j] — (^ — l;i, j) .
As these form a well-defined class apart, it would have seemed very natural to begin with them in endeavouring to establish the theorem, reserving the theory of un.satisfied invariants (sources of covariants) for future consideration. But to all appearance it would have been very diflScult, if not impossible, to have succeeded in dealing with them alone.
This is another example of the law in Heuristic that the whole is easier of deglutition than its part.
LECTURE XII.
Before proceeding further with the development of the pure analytical theory of reciprocants, it may be useful to point out some instances of its relations and applications to geometrical questions.
Using yi, 2/2, 1/3, ... y„ to denote the successive derivatives of y with respect to x*, let the complete primitive of the differential equation
^ {o!, y, yi, Vi, ••• yn) = o
be 4>{x, y, X, /i, v, ...) = 0.
We can in general so determine the n constants X, in., v, ... that the curve (^ may pass through n given points, and if we take these to be consecutive points on the curve
<& («, 2/) = 0, <^ and <I> will have a contact of the (n — l)th order at a given point of <I>. In order that the curves may have a contact of the reth order at a point
* In future y\,yi,yz, ... y» will always have this meaning, the derivatives of x with respect to y will be denoted by x^, Xj, 13, ..., and whenever the letters (, a, 6, c, ... are used they will
stand for j,,. J|?-, ^^ , ^ J\^ ^, ... respectively.
42] Lectures on the Theory of Reciprocants 371
whose abscissa is x, the ordinates of ^ and ip at that point and their 1st, 2nd, ... nth derivatives with respect to x must be the same for both curves. But at every point of ^ its dififerential equation
F{x,y,y^,y.„ ... y„) = 0
has to be satisfied, and therefore the x, y, y,, y^, ... y„ of any point on <5, at which contact of the nth order with ^ is possible, must also satisfy the same equation.
Now, suppose that for x and y we substitute given functions of them, X and F; the curves <^ and 4> become
<f> {X, Y, \,fji,v,...) = Q and ^(X,Y) = 0.
Contact of the nth order with the transformed (f) will therefore be possible at any point of the transformed «1> for which
F{X,7.Y,.Y„...Yn) = 0.
where Y„ Fj, Y,, ... Y„ are the derivatives of Y with respect to X.
But, unless the function F and the substitutions X —f^(x, y), Y=fi (x, y) are so related that the transformed differential equation
F{X,Y,Yr. F„... F„) = 0
is identical with the untransformed one, the property marked by the contact of the transformed curves will not be identical with that marked by the contact of the untransformed ones.
For example, let F=y.j; then the relation between <^ = y + Xa; + /i = 0 (the complete primitive of y, = 0) and an arbitrary curve 4> is that the constants \ and ft may be so chosen that the line y + \x + fi = 0 may have a contact of the second order at any point of <I> for which yi = 0; and the property marked is an inflexion on <1>. But if we make the substitution X = x', Y^y*, so that the differential equation ^^ = 0 is transformed into
||-T— j y* = 0 and its complete primitive into y* + Xa;" + /* = 0, it will still be*
Ipossible so to choose \ and /x that y" + Xa:* + /* = 0 may have a contact of the
second order at any point of an arbitrary curve for which ( j-j) y' = 0, but
the property marked, instead of being an inflexion, will be a contact of the second order with a conic having a pair of conjugate diameters coincident with the co-ordinate axes.
The property remains unaltered when the co-ordinate axes are inter-
(d N' -^ J y» = 0 will be identical
-j—A ar' = 0, in which the variables x and y have changed places. The
24 2
372 Lectures on the Theory of Reciprocants [42
identity of the two differential equations is easily verified, for
\di^) ^^ 2x' dxKx'dxJ 2a!'\x'da^ x \dxj x^ ' dx
so that the differential equation may be written
«yys + icyi" - 2/y, = 0. Interchanging x and y in this, we have
yxXi + yxi — xxi = 0,
in which, if we write Xi = -r- = — , and x^ = j— = — ^ it follows immediately
dy 2/, rfy yi
that
yxx, + yx^^ -xx^ = - — (xyy, + xy^- - yy,),
and the identity in question is established.
Such a form as the above, which merely acquires an exti-aneous factor when the variables are interchanged, might be called a reciprocant, if it were not convenient to restrict the use of the word to forms in which the variables X and y do not appear explicitly. With this limitation, the geometrical property indicated by the evanescence of a reciprocant will be independent of the position of the origin, but not in general independent of the directions of the co-ordinate axes. Thus, we may prove that the equation
22/12/3 -3y/ = 0 indicates the possibility of 4-point contact with a hyperbola whose asymptotes are parallel to the co-ordinate axes. To do this it is sufficient to show that its complete primitive is the equation to such a hyperbola.
Writing the equation in the form
y> •2'yi'
we see that its first integral is
3
log yi=^\ogyi + const.;
or, when prepared for a second integration.
Hence yi~ - = \x + /i,
2/, = (X^ + /x)-», and finally we obtain the complete primitive
\(v-y) = (\x + fi)-\ which proves the proposition.
42] Lectures on the Theory of Reciprocants 373
With the notation previously explained, in which y, = t, y^ = 2a, y^ = 66, the differential equation is ht — a? = 0. We have therefore proved that at all points of a general curve for which the Schwarzian {bt — d^) vanishes, 4-point contact with a hyperbola whose asymptotes are parallel to the co-ordinate axes is possible.
We now consider the important case in which the conditioning differential equation remains unchanged when the axes are orthogonally transformed, and is therefore found by equating to zero an orthogonal reciprocant. The simplest example of this class of equations is that which marks the points of maximum or minimum curvature on a cyrve. Since these points are points of 4-point contact with a circle, the conditioning differential equation will be
that of the circle
{x + \y + {y + tji)-^v = 0.
Differentiating this three times in succession, we have
x + X + iy + 11)1 = 0,
1 -f «' + 2a (y -f- /i) = 0,
at + b(y-¥ii.) = 0.
Eliminating /* from the last two of these equations, y will disappear at the same time, and the condition for points of maximum or minimum curvature is found to be
2aH - 6 (1 + «') = 0.
In Salmon's Higher Plane Curves (2nd edition, p. 357) the "aberrancy of curvature" is given by the formula
tanS = y.-<l±M^ = ,_a4_p^. ^' :iy? 2a»
The above differential equation is therefore equivalent to 8 = 0.
(1 -f- Vi')^ (1 + H^
If we differentiate the radius of curvature p= ^ ^^-^ = ^ — s — - > we find
y. 2a
Hence it follows that
The conditioning equation for points at which -^ or tan S is a maximum
or minimum is -7^ = 0 ; or the same condition may be expressed by
d tan S _
dx Now
d tan S _ rf f, 6(1-1- «')) _ o„ 2c(l -M') 2o6< 36'(I + <')
=l{'-^1=-
dx dx 1 2a' i a' a' a»
374 Lectures on the Theory of Reciprocants [42
is an orthogonal reciprocant, for it can be expressed in terms of legitimate combinations of 1 + f, which is an orthogonal reciprocant of even character, with the three orthogonal reciprocants of odd character,
o, 6 (1 + O - ^0'% c (1 + *') - 5a6« + 5a».
In fact, the above expression for — ^ , when multiplied by a? to clear of
fractions, becomes
2a« - 2a?ht + 36« (1 + <») - 2ac (1 + <»)
= j^, {6 (1 + «») - 2aHY + ^— ^^ - 2a {c (1 + i=) - 5aht + 5a»},
where the right-hand side is a linear function of orthogonal reciprocants of the same (even) character, so that the combination is legitimate.
Quantities such as p, ~, j^ , ..., or p, -^, j£- , . . . , where d<j) is the
angle subtended by the arc ds at the centre of curvature, have values independent of the particular position of the co-ordinate axes (supposed rectangular), and consequently these values, expressed in terms of t, a,b,c,... will be absolute orthogonal reciprocants. A differential equation expressing the condition that any one of these quantities vanishes, or that any one of them has a maximum or minimum value, will also be independent of the position of the rectangular axes, and must therefore be expressible in the form of an orthogonal reciprocant equated to zero.
Mr Hammond remarks that, since the radii of curvature at corresponding
points of a curve and its evolute are p and -J^ , the radius of curvature of its
nth evolute is -^^ . The radius of curvature of the nth evolute of any nth
involute of a circle is constant, and, consequently, the differential equation of an nth involute to a circle is
Writing this in the form
n + t' dy+^ (1 + ff
\ a ' dxj ' a
+1 ^1 4. *2^f
= 0,
to which it is easily reduced, since
d_^ d^^ p d_ _ (1 + f ) d^ d(f} ds (1 -I- i=)^ " da; 2a ' da;'
we see by what precedes that the left-hand member of the differential equation is an orthogonal reciprocant.
As an example of the class of singularities which next presents itself for consideration, let us find the differential condition which holds at points of
42] Lectures on the Theory of Reciprocants 375
contact of the fourth order with a common parabola. This condition is expressible by the differential equation whose complete primitive is
(y + KxJ + 2Xa; + Ifiy + v = 0. Differentiating three times in succession, we obtain (y + KX)(t + K) + \ + fit = 0, 2a {y + KX + fj.) + (t + kY = 0, b(y + icx + fi) + a(t + K) = 0.
The arbitrary constants v and \ do not appear in the last two of these equations, from which, if we eliminate fi, the variables x and y disappear at
the same time, and we find
2a^-b{t + K) = 0.
A final differentiation and elimination give
10a5-4c(<+«) = 0,
4ac - 56= = 0.
Points of 5-point contact with a parabola are therefore indicated by the evanescence of the pure reciprocant 4ac — 56=. And in general the differential equation R = 0, where B is any pure reciprocant, indicates a property of a curve which may be called a descriptive singularity, since it is totally unaffected by the arbitrary choice of any two lines on the plane for the axes of co-ordinates. For it was proved in Lecture IX of the present course that if t be the degree and fi the characteristic of R, the substitution of ly + nuc + n for a; and I'y + m'x + n for y changes R into {I'm - lm'y(lt + m)''^R, 80 that the differential equation R = 0 and the geometrical property corre- sponding to it are left unchanged by the substitution.
Six-point contact with a cubical parabola is another example of a descrip- tive singularity. Its defining differential equation may be written in any of the following forms :
45y,V - VoOy,h/,y,y, + my.'y,' + WOy.y^y, + 165y,y,V " 400y,*y, = 0,
125a'd' - ToOa'bcd + 256aV + 500a6-^d + 165a6»c' - 3006'c = 0,
5 (9y^y, - roy,y,y, + 40y,')' + 64 (Sy,y, - 5y,')> = 0,
125 (a^d - 3a6c -1- Wy + 4 (4ac - 56')> = 0;
or, if we make a?d — Sabc + 2l^ = A and ac — -r¥ = M, the equation may be put in the form
(re)'-©*-.
In the theory of Binary Forms, when the numerical parameter k in {a?d - Sabc + 26")' + K{ac- b'f
376 Lectures on the Theory of Reciprocants [42
is 80 chosen that the highest powers of h cancel each other, the form divides by o* and gives the Discriminant of the Cubic
a'd' - Qabcd + 46»d + 4ac' - St'c'. In the parallel theory of Reciprocants the form
125^' + 2563f' is divisible by a (instead of by a»), giving
125«»d» - 750a»6cd + 500a6»d + 256a»c» + 165a6V - 3006*c, which may be called the Quasi-Discriminant.
A complete discussion of the differential equation
A^ + kM" = 0 is reserved for the next ensuing lecture, in the course of which it will appear that the Quasi-Discriminant equated to zero is the differential equation of the cubical parabola.
LECTURE XIII.
We may integrate the general homogeneous equation in reciprocants extending to d, inclusive, as follows:
Calling ac-^h^=M and a^d - Sabc + 26» = 4,
the equation in question will be of the form
A' + kM' = 0. But if we write /9 = Aa\
where /9, a are general linear functions of the co-ordinates, say
y + ma; + 7i, y + m'x + n, we may eliminate the five constants m, n, m', n', A, and the result will evidently be a pure reciprocant extending to d, inclusive, and, being homo- geneous and isobaric, can only be of the form
A^ + kM' = 0, so that it remains only to determine « in terms of \, or, which is the same
thing, \ in terms of k.
1 1 The solution /3 = Aa* implies a = A ^/3\ Hence the equation between M and A must be of the form
e [(\ + p) (p\ + 1)YM> + {(X + q) (q\ + l)}iA'- = 0, where ^ is a constant, for otherwise there would be more than one general solution to it. It only remains then to determine the values of p, q, 0, i, j, which may be affected by considering the particular solution y = a^.
42]
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377
When \ = 2, J/ and A both vanish, and if \ = 2 + e, where e is an infini- tesimal, M and A will each be of the same order as e (that the first power of e does not vanish in M ov A may be easily verified). Hence 2 + q + e is of the order e, and therefore q= — 2 and / = 1.
When X = — 1 + 6, M remains finite and A is of the order e. Hence p = \ and i = \. Thus, the equation is
0 (X + 1)» if» + (\ - 2) (2\ - 1) ^' = 0. To find 6, let X = 3 and y=sf; then
so that
a = ^x, 6 = 1, c = 0, d = 0, M=-'-, A = 2, -^.^+5.4 = 0, ^=ii,
And finally 16 (X + 1)» Jf» + 25 (2X= - 5X + 2) 4^ = 0
has for its integral /9 = Aat*.
If X = « , we may make
and, consequently, /3 = e**, which contains five independent arbitrary constants, will be the general integral.
For a parallel method of deducing the Integral of J.' + /cA' = 0, where A (our future AG — B^) is the projective reciprocant whose letters go up to /, see Halphen's Thhe sur les Invariants Differentiels, Paris, 1878.
Mr Hammond has succeeded in deducing the equation between A and M from the primitive /8 = Aa* by direct elimination, as shown in what follows. Po8.sibly he, or some other algebraist, may eventually succeed in the more difficult task of obtaining the Differential Equation to y = /S'^a}''^ (that is, the linear relation between A' and A*) by some similar direct process.
Differentiating the equation y9a"* = A three times in succession, and observing that, since a = y + mx + n and 0 = y + m'x + n,
^ da? y*
we have
o/3'-Xa'i9 = 0,
y,(a-X/3) + (l-X)a'/3' = 0,
y, (a - Xy8) + y, {(2 - X) o' + (1 - 2X) /9'} = 0.
From the last two of these three equations we obtain, by eliminating
(« - X/3),
y,(l-X)a'^-y,»{(2-X)a' + (l-2X)^'!=0;
or, writing
Vi = 2a, y, = 6fc, 2 - X = Zq\ 1 - 2X = - 3?^, 1 - X = j' - »•»,
378 Lectures on the Theory of Reciprocants [42
and dividing by a')S', the equation assumes the form
Differentiating again, remembering that
a" = r-2a. and 1 = 36. 1 = 40.
we
The elimination of ^' between this and the equation immediately pre- ceding it gives
Writing in this 4ac - bh'' = 4ilf, we obtain by an easy reduction ^fMa!^ = r" [2a? - ^>a'}^
and, taking the square root of each side,
a:{2q>jM-\-rh)-1a?r=0.
A final differentiation gives
a'{^ + 4cr) + 2a {2q >JM - 5br) = 0.
Finally, eliminating a', we obtain
{2q s/M + rh) {2q s/M- 57-b) + ar {icr + ^) = 0.
Hence 4%» + qr (^ - 86 ^fM) + r' (4ac - 5b') = 0 ; or, 4 iq' + r^)M^ + qr (aM' - 8bM) = 0.
Now ^'=f^=i {'^ - ?) = ^^^ - ^^''
and, consequently,
aM'- 8bM = a (Bad - 76c) - 6 (Sac - 106^ = 5 (a'd - Sabc + 2b') = 5A ;
so that we may write
4 (q' + r') M^ = -qr (aM' - SbM) = - 5qrA ; or. 1 6 (f + r'fM^ - 2bq^r'A^ = 0,
where 3g^ = 2 - \ and - Sr' = 1 - 2\.
Replacing g' and r^ by their expressions in terms of \ the differential equation becomes
16 (\ + lyM' + 25 (2\» - 5\ + 2)^^ = 0. Some special cases may be noticed.
42] Lectures on the Theory of Reciprocants 379
When \ = 2 or ^ , the equation reduces to M=0, which is the differential
equation of the common parabola previously obtained.
When \ = 3 or -, we obtain 256Jf' + 1254''= 0 for the equation of the o
cubical parabola, where the expression on the left-hand side is the Quasi- Discriminant.
When \ = — 1, we find ^4 = 0 for the differential equation of the general conic.
When \ is an imaginary cube root of negative unity, so that X'' — \ + 1 = 0, we have
(\+iy + (2X=-5X + 2) = 0,
and the differential equation becomes
16i/' - 254= = 0.
We shall subsequently avail ourselves of this result in finding the complete primitive of the Haiphenian A.
In the case where \ is infinite, from the complete primitive j8 = e'" we first eliminate the exponential function and afterwards the arbitrary constant I.
Thus we find yS' = ia'/3 and ^ = ^^ § ;
pap
or, y,)8(a'-/8')-a'/3" = 0.
Hence y,/3 (a - /3') - ^2/9' (a' + 2/S') = 0.
The elimination of /9 gives
36 1 2 **'' 2^» = ^' + a'-
Comparing this with the equation previously obtained,
we see that 5* = 1 and r* = — 2. Substituting these values in the differential equation
1 6 (9' + r»)» M" - 255V.4 ' = 0,
it becomes 83P + 25^1' = 0,
which is the differential equation corresponding to the complete primitive
^ = ^.
We shall hereafter consider in detail the theory of that special class of pure reciprocants (M. Halphen's Differential Invariants) which retain their form when any homographic substitution is impressed on the variables ; that is, when, instead of x and y, we write
Ix + my + n , l'x-\- vi'y + n'
l"x + m"y + n" *° l"x + m"y + n" '
380 Lectures on the Theory of Reciprocants [42
Since perspective projection is the geometrical equivalent of homographic substitution, it follows from the definition of Differential Invariants that they are connected with the properties and relations of curves which remain unaffected by perspective projection. For this reason Differential Invariants are sometimes called Projective Reciprocants. Two reciprocants with which we are familiar belong to this important class. One of them, y, or a, vanishes at points of inflexion on the curve y=/(a;); the other,
^y^Vs — ^^yij/ay* + 40ys', or a^d — Sabc + 26*,
which, for reasons given below, we shall call the Mongian, vanishes at sextactic points; that is, at points where a conic can be drawn having 6-point contact with the given curve.
To illustrate the distinction between a projective and a merely descrip- tive singularity, consider for an instant the pure reciprocant 4ac — 56", which, as we have seen, vanishes at all points of a general curve where .5-point contact with a parabola is possible. Now, 5-point contact with a parabola is a descriptive but not a projective singularity ; after projection the parabola becomes a general conic, and 5-point contact with it becomes 5-point contact with a general conic, which is not a singularity at all. But inflexions and sextactic points are indelible by projection, and thus belong to the class of projective singularities.
The differential equation to a conic was originally obtained by Monge in the form
dy^-y, - ^'Oy.y^y, + 40y,' = 0
(see Monge, " Sur les Equations diffi^rentielles des Courbes du Second Degr^," Corresp. sur I'Scole Polytech., Paris, II. 1809-13, pp. 51-54, and Bulletin de la Soc. Philom., Paris, 1810, pp. 87, 88). At the end of the first chapter of his Differential Equations, Boole mentions this form of equation as due to Monge, but without any reference, and adds the remark : " But here our powers of geometrical interpretation fail, and results such as this can scarcely be otherwise useful than as a registry of integrable forms." The theory of Reciprocants, however, furnishes both a simple interpretation of the Mongian equation and an obvious method of integrating it.
To see that the differential equation of a conic is satisfied at the sextactic points of a given curve, we have only to remember that at such points the derivatives of y with respect to x, up to the fifth order, inclusive, are the same for the given curve as for a conic.
We proceed to show how the Mongian may be integrated. "Writing in the above equation
y, = 2a, ys = 2 . 36, y^ = 2 . 3 . 4c, y. = 2 . 3 . 4 . 5d, it becomes d'd - 3abc -I- 26^ = 0,
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Lectures on the Theory of Reciprocants
381
where it can hardly fail to be noticed that the left-hand member of the equation is an ordinary Invariant as well as a Reciprocant. It will be proved hereafter that all Differential Invariants possess this double nature.
Now, if |t = 3» + w, where i is the degree and w the weight of any pure reciprocant R, the ordinary theory of eduction shows that
d / R\__ax____
is another pure reciprocant.
When we consider the letters o, b, c, ... in any invariant / to mean
the parallel theory of generation for Invariants gives
2 ' 2.3' 2.3.4'
the corresponding theorem that if v = 3i + 2w, where i is the degree and w
the weight of /,
dx
©
a -, vol
ax
is also an invariant.
A strict proof of this theorem will subsequently be given. For present
purposes it is suflBcient to notice the easily verified special cases of the two
theorems
d^ /^ac - 56»\ _ 20 (o'd - 3a6c + 26»)
d fac — dx
5(o'd-3a6c + 26») a*
It follows as an immediate consequence that the equation
a^d - 3abc + 26' = 0 admits of the two first integrals
o ~ ' {iac — 5b') = const.
and a~ » (ac — 6") = const.
d , _4.. 1 d'
Now,
a-S(4ac-56.) = £(a-J6) = -25^(a-?);
so that the Mongian equation is equivalent to
|.(a-3) = 0,orto^(yrS) = 0.
We thus obtain an integral of the form
yj~ 9 = i + imx + no?,
382 Lectures on the Theory of Reciprocants [42
from which the complete primitive may be found by two easy integrations.
Thus,
C dx m-Vnx
^' ^ ~ i (i + 2hw; + na?f (In - m») {I + 2rruc + «;b»)* gives y+px + q = ^^^^^,{l + 2mw + ruc')K
which is the equation of a general conic.
By first interchanging the variables x, y in the Mongian equation (whose form remains unaltered by this interchange, since a^d - 3a6c + 26' is a reciprocant) and then integrating three times with respect to x, we should find another integral of the form
iB,-t = i'+2m'y + ny.
The solution may be completed by two integrations, as in the former
method.
2 (ac — 6°) d^ 2 Mr Hammond remarks that ^ ^— ^ = ^^ {a% where < = y,. For, smce
|
d dx d Id |
|
|
dt~ dt' dx 2a' dx' |
|
|
we have |
l,<«')4J--'-^'-^r |
|
and, consequently. |
Hence the integral a~'^(ac - 6=) = const, previously obtained for the Mongian is equivalent to -7- {a^) = constant ; that is, to -j— ^ {y^) = const. Thus we have another integral of the form
y^ = \+2,iy,+ vy,\ from which it is also easy to pass to the complete primitive.
I add a few general remarks relating to the subject-matter of this and the preceding lecture. Instead of the cumbrous terms Projective Kecipro- cants or Differential Invariants, it may be better to use the single word Principiants to denominate that crowning class or order of Reciprocants which remain, to a factor pres, unaltered for any homographic substitutions impressed on the variables. This is the species princeps. If we go back to the species infima, we see the beginning of life in the subject. In general Reciprocants, all that is affirmed is that there exist forms-functions of the derivatives of y in regard to x which (to a factor pres) remain unaltered when the variables x and y are interchanged, so that/(yi, y,, y^, ...) becomes
42] Lectures on the Theory of Reciprocants 383
(^(asj, «,, iTj, ...). The function <^ only dififers from f by the acquisition of an extraneous factor {—Yy^', that is,
fivi, ys, 2/3, •••) = {-Yyffi^'i, a^a, a^s, •••)•
A particular species of these general (mixed) reciprocants arises when f(yi, yt, y,, ■••), differentiated in regard to yx, gives a reciprocant. These are Orthogonal Reciprocants, and in them we see the first dawn of free con- tinuous motion as distinguished from mere displacement (or mere interchange of axes). Orthogonal Reciprocants, when x, y are rectangular co-ordinates, remain unaltered (save as to a factor) when the orthogonal axes are moved continuously. A quarter of a revolution of course will reverse their original positions, so that we see the condition of mutual displacement is fulfilled. Thirdly, Reciprocants into whose form the first derivative 3/1 does not enter are called Pure. Their form is invariable when the axes (now taken generally) undergo separate displacement (instead of turning round together) in a plane. Here there is a further development, so to say, of life in the subject.
Finally, in Principiants, a particular species of Pure Reciprocants, the invariance remains good, not merely for any position of the axes of reference, but for any homographic deformation of the plane in which they lie, so that the evanescence of a Principiant corresponds to some property of a curve not only intrinsic but indelible by projection, as, for example, an inflexion, or a double point, or a sextactic point, and so on.
It is clear from this review that the Theory as we have given it goes to the root of the subject, and that the word Reciprocant is rightly chosen as conveying the notion of a property which is common to the entire continuous series of forms bearing that name. All the links of this connected chain are thus comprehended under the general name of Reciprocants.
LECTURE XIV.
The remaining lectures of the course will be devoted to the theory of Pure and Projective Reciprocants. I shall first treat of the existence and properties of the Protomorphs of Invariants and Reciprocants, using the latter system of protomorphs to obtain all the fundamental forms of Reciprocants in the letters a, b, c, d, e. I shall then pass on to the theory of Projective Reciprocants, or Principiants, with its applications contained in M. Halphen's Thhse pour obtenir le grade de docteur es sciences (Paris, Gauthier-Villars, 1878). It will be seen that M. Halphen's very ingenious methods become greatly simplified when his results are read by the light of an important discovery in the theory of Principiants recently made by myself and Mr Hammond working conjointly, arising out of a theorem put
384 Lectures on the Theonj of Reciprocants [42
forward by one of my hearers. This theorem, on examination, we found was necessarily erroneous and would fail at the very first step of its application. But although the proposition stated was wrong, it contained an Idea which survives and may be incorporated in a valid and extremely important theorem, which I will endeavour to explain.
A Principiant, besides being an Invariant in the original letters a, 6, c, d, ... is also an Invariant in the letters a. A, B, G, D, ... where each capital letter is itself a Reciprocant ; and, conversely, every invariant in the capital letters A, B,G, D, ... ^a & Principiant. The invariants in the capital letters form a system of protomorphs for Principiants, so that every Prin- cipiant is either some such invariant simply, or a rational integral function of such invariants provided by some power of a. Thus, for example, it will be proved that the Cubic Criterium (that is, the Principiant which gives, when equated to zero, the differential equation of a cubic curve) may be expressed as the quotient of
^A' + ^A {A^D -ZABG-ir- 2jB») -{AGE-AD^- B'Ji + 2BGD - G') 64 4
by the fifth power of a.
The proof of this theorem is based upon the fact that we can form a series of terms beginning with the Mongian (namely, aJ'd — Sabc + 26'), say A, B,G,D, ... such that
D.G = 2Bx%,
ni)=3ax|.
where fl = ogft + 269, + Sca^ + . . . ,
coupled with the fact that every Principiant must be a function of the letters in such series and the small a.
Each consequent of the series A, B, G, D, ... is, so to say, an Invariant relative to its antecedent; it becomes an actual Invariant when its ante- cedent vanishes.
In the theorem as originally proposed, each letter of the series was derived by the operation of an eductive generator upon the one which precedes. In the true theorem the scale of relation is between three and not two consecutive terms. Calling the letters u^, u^ ii^, ... u^ we have
(t -I- 7) Mi+, - Gui+i + (t + 1) Mui = 0.
42] Lectures on the Theory of Heciprocants 385
where O is the ordinary eductive generator,
4 {ac — b-) di + 5 (ad -bc)dc+Q{ae — bd)dd+---,
M is the first pure reciprocant after the monomial a, namely, M = ac — -rb-,
u^ = A = a'd- 3a6c + 26», and 6it, = GA.
But although, as I have said, the theorem in the form proposed was absolutely erroneous, its proposer has rendered an invaluable service to the theory by the mere suggestion of what turns out to be true, namely, that every Principiant is an Invariant in regard to a known series of Reciprocauts considered as simple elements.
To this theorem there is a correlative one, for it will be shown that there exists a series of invariants A^, A^, A^, ..., the first term of which, A^, is the same as the Mongian A, each of the other terms of the series being a Reciprocant relative to the one that precedes it. In fact, we have
VA,= 0,
F.4j = -2aM„
VA„ = - na^'An-u
where V= 4 (^j dt + oabdc + 6 foe + ^-j 3rf + ...,
and, as a consequence, every Principiant will be an Invariant in respect to these Invariants and the first small letter a.
Thus, speaking symbolically, we have not only
P = R + I
(a logical equation meaning that P has the same qualities as both R and /, or that a Principiant is both a Reciprocant and an Invariant), but also
P = IR and P = II,
meaning that a Principiant is an Invariant of Reciprocantive elements, and an Invariant whose elements are themselves Invariants.
I may add that the invariantive elements A^, Ai, A^, Aj, ... are defined by the equations
A^ =A,
A=^B-^A.
..= C-2(|)B.(|)'a,
...^-3(|)c.3(|y.-(|)'..
8. IV. 26
386 Lectures on the Theory of Redprocants [4?,
M> that any invariant in the reciprocantive elements A, B, C, D, ... is equa to the corresponding invariant in A^, A^, A„ A Thus,
A = Ao,
AC-B* = AoA,-Ai',
A*D - SABC + IB* = A^A^ - ZA.A^A^ + 2A^,
AE-*BD + ZC^ = A,A^-4iA,A^-{-ZAt\ '
M. Halphen appears not to have noticed the Principiant AE—^BD + 3(7' which presents itself naturally when the theory is viewed from our presen ground of vantf-ge, but A.AG-B" and AW - 2ABG + 2^ occur in his Thes in connection with the curve
in which a, /8, 7 are any linear functions of x,y,\.
When \ = — 1 the dififerential equation of this curve (the conic a/8 = 7*
is 4 = 0, but it is
AG-B^ = 0
when \ is a cube root of negative unity, and
A-'D-^ABG+2B' = 0
when \ has an arbitrary value.
Before ma dng out an exhaustive table of all the irreducible forms of pure reciprocants in the letters a, b, c, d, e similar to, but not identical with, the corresponJing table for invariants, it seems to me desirable to say something of Protomorphs in general ; and this will be better understood if we devote a short space to the protomorphs of Invariants. The simplest forms of these are the following well-known ones of alternately the second and third degrfss:
Ft = ac-b\
P, = a-d - Sabc + 2ft»,
Pt = ae-ibd + 3c»,
P. - a'/- babe + 2acd + 8b^d - 660",
Pt = ag- 6bf+ Ibce - \0d\
P, = a«A - labg + 9oc/- bade + 126^/- 306ce + 206rf
72
The quadratic Protomorphs P^, P4, Pg, ..., are absolutely unique, for the number of invariants of the type j ; 2, j is (j ; 2, j) - (j - 1 ; 2, j) = 1 if j is even, and = 0 if _; is odd. Tiieir form is so well known that there is no need to dilate upon it here.
. 42] Lectures on the Theory of Reciprocanta 387
The cubic ones P„ P^, P.,, ..., may be derived from the quadratic ones by means of Cayley's generators, given early in the course, namely,
P={ac- h") db + (ad - hc)flc + {ae-hd)di + ..., "* Q = (ac - 26») 96 + 2 {ad - 26c) dc + ^(ae-^hd)dd + ....
Let us first use the P generator
P (ac -l^) = a{ad- be) - 26 (ac - b^) = a^d - 3a6c + 26^ ^j P (a« - 46d + 3c^) = a {af- be) - 46 (ae - bd) + 6c {ad - be) - 4d {ac - 6")
g =a]f-5abe+2acd + 8¥d-Qbc\
c< Similarly, we find
P {ag - 66/+ 15c« - 10#) = a'A - labg + 9ac/- oade + 126y - 306ce + 20bd\ e and so on.
Let / be any invariant whatever of the type w; i, j (satisfied or un- satisfied) ; then using the original forms of the generators P and Q as given by Cayley (see Lecture IV), we have
PI=a (bda + cSfc + d3„ + ...) /- ibi, QI=a{cdi + 2ddc + 3e8d + ...)/- 2m;6/, and, consequently,
{jP-Q)I= a [jbda + ij -l)cdb + {j - 2)dd, + ...} I-{{j - 2w)bl. If in this formula we write
0=jbda + {j-l)cSt + {j-2)ddc + .... it becomes {jP-Q)I= aOI - {ij -2w) bl,
which, when / is a satisfied invariant, so that ij — 2w = 0 &nd,,OI=0, reduces to
(jP- Q)7=0,
showing that the forms obtained by operating with eitheif P or Q on any satisfied invariant are the same to a numerical factor pr^. ,._
Now, each quadratic protomorph is a satisfied invariant (for when w = j and i = 2, ij — 2w = 0), and therefore the cubic protomorphs found by operating on the quadratic ones with Q will only differ by a numerical factor from those already obtained by the operation of P. But we must not conclude from this that the cubic protomorphs are unique. Their number is in fact given by the formula
(j; 3, ;)-(i-l; 3,j),
where it is obvious that
O'-l; 3, 7) = (7-l;3, 7-1);
80 that the above formula may be written
(J ; 3. J) - (j - 1 ; 3, 7 - 1), or say A 0' ; 3, ;).
25—2
|
2 |
3 |
4 |
5 |
6 |
7 |
8 |
•9 |
10 |
11 |
12 |
13 |
14 |
15 |
|
2 |
3 |
4 |
6 |
7 |
8 |
10 |
12 |
14 |
16 |
19 |
21 |
24 |
27 |
|
1 |
1 |
1 |
2 |
2 |
2 |
3 |
388 Lectures on the Theory of Reciprocanta [42
Now, there is a simple rule for finding (j; 3, j); it is the nearest integer to , . From the following table, obtained by the use of this rule,
J"
it may be seen that for any odd number j = > \) there are two or more forms of extent j equally entitled to rank as protomorphs. If I be the last letter which occurs in one of these forms, its first term will of course be aH; the difference between any two such forms will not involve the letter I, and will only extend to k, but will still be of the same (potential) extent as L
The property of the protomorphs o, Pj. A. P*, •■•is that every invariant is a rational integral function of them divided by some power of a, as appears from the fact that Q, any given rational integral function whatever of the letters a, b, c, d, e, ..., may obviously be expressed as a rational integral function of a, b, Pj, P„ P4, ... divided by some power of a. Thus, Q^a-'<f>(a,b.P„F„P,....). Suppose Q to be an invariant /; then
Ja» = .^(a, 6,P„P3. P,,...), and, consequently,
where 12 is the annihilator for invariants ; so that
n(/a»') = o, na = o, nP2 = o, np, = o, ....
We have therefore
do do
Hence <f) does not contain b, but is a rational integral function of the protomorphs alone, and
I=a'"^4>{a,P,.P„P ).
I shall show how to obtain a similar scale of forms possessing like properties for pure reciprocants.
i
LECTURE XV.
A Protomorph may be defined as a form whose weight is equal to its actual extent, so that its type is j ; i, j. The first protomorph is a, which corresponds to j = 0. For higher values of j it follows immediately from the definition that every protomorph will contain a term a'~% in which the letter of highest extent appears only in the first degree multiplied by a
42] Lectures on the Theory of Reciprocants 389
power of the first letter. The existence of this term enables us to instantly recognize a protomorph. As in the case of invariants, it will be shown that evei-y pure reciprocant is either a rational integral function of protomorphs or else such a function divided by some power of a. But first it will be better to prove a priori their existence and exhibit examples of them for the earlier values oij.
It was proved, in Lecture IX, that the number of pure reciprocants of the type w ; t, j is at least equal to
(w;i,j)-(w-l;i + l,j).
Now, obviously, the number of partitions of w into i parts not exceeding «; + € is the same as the number of partitions of w into i parts not exceeding
w, so that
(w •,i,w + e) = (w ; i, w) ;
and since, by a well-known theorem, {w\ t,j) = (w,j, i), we see that
{w;w + e,j) = {w;j,w + e) = {w;j,w) = {w; w, j), a result which follows more immediately from the consideration that the partitions of w ; w + e, j differ only from those oi w\ w, j by e columns of zeros, as we see in the annexed example :
3; 5, 3 I 3; 3, 3
30000 300 21000 210 11100 I 111
Hence, if w =j, and i = >j, we have
(w ; i, j) = ( j ; ;, j) and (w - 1 ; I + 1, j) = 0" - 1 ; i - 1. i - 1)-
Thus, the number of pure reciprocants of the iyp&j;j,j is
0'; j. i)-0'-i;i-i.i-i).
in other words, the difference between the indefinite partitions of j and those ofj — l. Expressed by means of generating functions, this difference is the coefficient of a^' in
l-x
(1 - a;) (1 - a;") (1 - a^) . . . (1 - af^) = coefficient of a;-' in the expansion of
1
(l-a?)(l-a:')...(l-a>>)" This coefficient is a positive integer for all values of j (except J = 1, when it is zero), which proves the existence of reciprocants of the type j; j, j when j has any value except unity.
But we wish to prove the existence of one or more reciprocants of the type j; j,j which actually contain a term of the form a^~H, where the letter I
390 Lectures on the Theoi-y of Reciprocants [42
is of extent _;'. The number of such forms is the difference between the number of pure reciprocants of the types j;j,j and j; j, j - 1.
Now, the number of linearly independent pure reciprocants of the type j;j,j has just been shown to be
0'; 7. j)-0"-i; 7-1. ./-I)-
And, in like manner, that of the linearly independent reciprocants of the
0';i.i-i)-0'-i;i + i.i-i) = (;;;. i-i)-(i-i;i-i> i-^)-
The difference between these two numbers is therefore
0';i.i)-(i; j. j-l) = l• For the only partition not common to the two types is j . 0^~', made up of one j and j — 1 zeros, which belongs to the first type, but not to the second. Hence reciprocants of the type ;' ; j, j contain one term which those of the type j ; j, j — 1 do not, and which can only be a^~H. This proves the existence of protomorphs.
In the latter part of the above proof we have assumed the truth of the theorem, which, however probable, is not demonstrated, that the number of reciprocants of the type w; i, j is (w; i, j) — {w — l; i + l,j) and no more [that concerns the subtrahend, namely, {j; j, j — l) — (j — 1, j — 1, j — !)]•
We shall, however, have an independent method of arriving at Proto- morphs by direct generation, just as we saw that all the cubic protomorphs to invariants were derivable by direct operation of generators from the quadratic ones.
The difference between the two cases is that the lowest degree of Invariantive Protomorphs fluctuates alternately between 2 and 3. For Reciprocantive Protomorphs the lowest degree corresponding to a given extent fluctuates, but has a tendency to rise, and goes on progressing until it exceeds any assignable number.
It is interesting to find what the degrees are for successive values of j. The calculations required are greatly facilitated by an extensive table of partitions given by Euler in 1750, and partly reproduced by Cayley in the American Journal of Mathematics, Vol. iv., Part iii. In the table as presented by Cayley, the number in column j and line i means the number of ways of partitioning j into exactly i parts (zeros excluded). Hence, to find the number of ways of partitioning j' into i parts or fewer, that is, to find {jj ». oo) or its equivalent (;; i, j), we must add up the numbers in the Ist, 2nd, 3rd, ... ith lines of column j.
42]
Lectures on the Theory of Reciprocants
391
When these summations are made we obtain the subjoined table :
Extent j = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
I
1 2
.11 3 g 4
|
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 6 |
1 7 |
1 7 |
1 8 |
1 8 |
1 9 |
1 9 |
1 10 |
||
|
2 |
2 |
3 |
3 |
4 |
4 |
5 |
5 |
6 |
||||||||||
|
2 |
3 |
4 |
5 |
7 |
8 |
10 |
12 |
14 |
16 |
19 |
21 |
24 |
27 |
30 |
33 |
37 |
||
|
2 |
3 |
5 |
6 |
9 |
11 |
15 |
18 |
23 |
27 |
34 |
39 |
47 |
64 |
64 |
72 |
84 141 |
||
|
2 |
3 |
6 |
7 |
10 |
13 |
18 |
23 |
30 |
37 |
47 |
57 |
70 |
84 |
101 |
119 |
|||
|
2 2 |
3 3 |
5 5 |
7 7 |
11 |
14 |
20 |
26 |
35 |
44 |
58 |
71 |
90 |
110 |
136 |
163 201 |
199 248 |
||
|
11 |
15 |
21 |
28 |
38 |
49 |
65 |
82 |
105 |
131 |
164 |
||||||||
|
2 |
3 |
5 |
7 11 |
IS |
22 |
29 |
40 |
52 |
70 |
89 |
116 |
146 |
186 |
230 |
288 |
The number of pure reciprocants of the typej; i,j is
(i; ij) - (j - 1; t + 1. j) = (i; ij) - 0" - 1; » + i.i - 1)-
To find the minimum degree for protomorphs of extent j we have there- fore only to see for what value of i any figure in the j column first becomes greater than the figure in the column to the left one place lower down. The fluctuations of the minimum degree are indicated by the dark irregularly waving line which runs through the table.
Accordingly, we find that the types of the protomorphs, omitting w, which is always equal to j, are as follows :
(2, 2), (3, 3), (3, 4), (4, 5), (3, 6), (4, 7), (4. 8), (5, 9), (5, 10), (5, 11), (5, 12), ..., whereas for invariants they are
(2, 2). (3. 3), (2,4), (3, 5), (2, 6), (3,7), (2, 8). (3, 9), (2, 10), (3, 11), (2, 12)
Corresponding to the extents
2,3,4,5,6,7, 8,9,10,11,12, ..., the lowest degrees of the Reciprocantive Protomorphs are 2, 3,3, 4, 3, 4,4, 5,5, 5, 5, .... Contrast this with the regularly fluctuating series 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, ..., which shows the minimum degrees of invariantive protomorphs for successive extents.
It may be proved, from known formulae in the theory of partitions, that as the extent increases the minimum degree of reciprocantive protomorphs increases (on the whole) and ultimately becomes infinite when the extent is so.
392
Lectures on the Theory of Recijrrocants
[42
The apparent number of protomorphs to the several types is
(2, 2), (3, 3), (3, 4), (4, 5), (3, 6). (4,7), (4, 8), (5, 9), (5, 10), (5, 11), (5, 12)
111111234 2 3
The explanation of this multiplicity is the same as that previously given for the case of invariants : the difference between any two protomorphs of a given type j; i, j will be a reciprocant (no longer a protomorph) of the type
For the only term containing the letter I (of extent j) will disappear from the result of subtraction; and, accordingly, the above numbers, each diminished by unity, will give the numbers of a set of reciprocants of the same degree-weight as the protomorphs, but of a smaller (actual) extent.
Assuming that the number of pure reciprocants of the type w ; i, j is correctly given by the formula
(w; i, j)-(w-l; i + l, j),
Euler's great table of partitions, already referred to, enables us to carry on the determination of the minimum degree and multiplicity of protomorphs for all extents as far as 59.
If m is the multiplicity corresponding to the minimum degree i of a reciprocantive protomorph whose extent is j, we form without diflSculty, using only the principles explained above, the following table :
|
J = |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
|
i = |
1 |
- |
2 |
3 |
3 |
4 |
3 |
4 |
4 |
5 |
5 |
5 |
|
m = |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
2 |
3 |
4 |
2 |
|
3 = |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
|
i = |
5 |
6 |
6 |
6 |
6 |
7 |
7 |
7 |
7 |
7 |
8 |
8 |
|
m = |
3 |
6 |
8 |
5 |
5 |
15 |
18 |
12 |
12 |
2 |
40 |
32 |
|
j = |
24 |
25 |
26 |
27 |
28 |
29 |
30 |
31 |
32 |
33 |
34 35 |
|
|
t = |
8 |
8 |
8 |
9 |
9 |
9 |
9 |
10 |
10 |
10 |
10 10 |
|
|
m = |
32 |
14 |
6 |
84 |
82 |
58 |
45 |
207 |
211 |
180 |
161 102 |
|
|
;■ = |
36 |
37 |
38 |
39 |
40 |
41 |
42 |
43 |
44 |
45 |
46 |
47 |
|
1 = |
10 |
11 |
11 |
11 |
11 |
11 |
11 |
12 |
12 |
12 |
12 |
12 |
|
m = |
45 |
482 i 469 |
391 |
320 |
167 |
13 |
1126 |
1064 |
881 |
687 |
337 |
|
|
j = |
48 |
49 |
50 |
51 |
52 |
53 |
54 |
55 |
56 |
57 |
58 |
59 |
|
t = |
13 |
13 |
13 |
13 |
13 |
13 |
13 |
14 |
14 |
14 |
14 |
14 |
|
m<" |
2829 |
2666 |
2492 |
2097 |
1643 |
892 |
26 |
6394 |
6017 |
5227 |
4266 |
2755 |
Notice the repetitions of % indicated by the series
1', 0', 2', 3', 4', 3', 4', 5*, 6^ 7», 8», 9^ 10«, 11«, 12', 13', 14'+'.
42] Lectures on the Theory of Reciprocants 393
It will be observed that there is a general tendency of the number of equal values of i to increase, but that this is subject to occasional fluctua- tions. When j = 5, i = 4; but when j =6, i= 3, so that the miuimum value of i recedes. After this point is reached, i either advances or remains stationary, but never recedes.
In order actually to find the protomorphs, we may use the annihilator V. This was my original method of obtaining them ; a shorter way, analogous to that used by Halphen for differential invariants (principiants), has been previously mentioned, but it will be instructive to begin with the method of indeterminate coefficients. In the first place we have the form a of weight 0, which is annihilated by
V = 2a'db + oabdc + (6ac + 36') da + {lad + 76c) 9, + . . . . For weight 1 there is no pure reciprocant. We could not make R = \a*~'6, for then VR = 2\a*+', which cannot vanish unless X = 0 and consequently R = 0.
To find the Protomorph of extent 2, assume R = Xac + fth' ; then
VR = ^fia'b + oXa^b = (4/t + 5X) a»6. Hence \ and fj, are proportional to 4 and — 5, and we may write
R = 4,ac- 56'. For extent 3, assuming R = \a'd + /labc + 1/6*, we have
VR = 2fia'c + Qva'b' + Bfia'b' + 6\a'c + SXa^b% which vanishes when
2^ + 6X = 0, 6r + 5/i + 3X = 0. We may therefore write X = I, /x = — 3, v = 2, and thus obtain
R = a'd- 3a6c + 26». For extent 4 the table of minimum degrees indicates the existence of a protomorph of degree 3. To find its value we assume R = Ko'e + \abd + ficu^ + vb^c.
Operating with V, we find
a'd a?bc o6* VR = 2X 4i;
10/t bv 6X 3X 7k 7k . In order that VR may vanish, we must have
2\ + 7k = 0, 4j/ + 10/x + 6X + 7/c = 0, and 5i/ + 3X = 0. To avoid fractions, let « = 50 ; then X = — 175, v= 105, and // = 28 ; thus, R = 50a'e - ITBabd + 28ac' + 1056'c ; whereas, the protomorph of extent 4 for Invariants is ae — 4ibd + 3c'. There is no reciprocant of degree 2 weight 4 to correspond to this.
394 Lectures on the Theory of Eeciprocants [42
LECTURE XVI.
By using the generator for pure reciprocants instead of the annihilator V, we readily obtain the protomorph of extent 5 and of the fourth degree whose existence is indicated in the previously given table of minimum degrees. We have only to operate on the protomorph of degree 3 and extent 4 with
G = 4i(ac-b')dt + o(ad-bc)dc + 6(ae-bd)dd + 7{af-be)de+... Thus, 0 (50a»e - lloaAd + 28ac'' + lOob'c)
= 4 (ac - 6») (- I75ad + 2106c) + 5 (ad - be) (56ac + 1056") + 6{ae-bd)(-noab) + 7{af-be){oOa').
Rejecting the numerical factor 35, which is common to all the terms in the result, and at the same time writing the terms themselves in reverse order, we have
lOa" (a/- be) - 30ab (ae - bd) + (ad- be) (Sac + 1 5b-) + 4 (ac - b") (- Bad + (ibc)
= lOay - iOa^be - I2a'cd + 65a¥d + IQabc^ - 396'c, which is the protomorph in question.
The form just found is irreducible, as indeed it ought to be, since the minimum degree for extent 5 is greater than that for extent 4 by unity, which exactly corresponds with the unit increase of degree due to the operation of 0. But if we use G to generate a protomorph of extent 4 from that of extent 3, the resulting form will be reducible. In fact G (a?d - 3a5c + 26») = 4 (ac - 6^) (- 3ac + G6^) + b{ad- be) (- Sab) + 6 (ae - bd) a' = 3 (2a»e - 1a?bd - 4a^c^ + 17ai»c - 86^).
If now we write
5
ac - 7 6'^ = M, 4
aH - 3a6c + 26' = ^,
7 17
a'e - 2 a'-bd - 2aV + -y "^'c - 46* = B,
we have shown that GA = 6B.
But oOB + 128M' = 25 (2a^e - 7a=6d - 4aV + I7a6''c - 86*) + 8 (4ac - 56')-^ = a (oOa'e - 175a6d + 28ac^ + 1056»c) ; so that B is reducible, being expressible as a rational integral function of a, M, and the previously obtained protomorph of degree 3 and extent 4.
42] Lectures on the Theory of Reciproeants 395
The general theory of the generator G is contained in that of the differentiation of absolute reciproeants, in which, ii fi = Zi + w, where w is the weight and i the degree of any pure reciprocant R, we have
and, consequently,
where iJ, and Oj are what R and a become when x and y are interchanged. Hence
a^-ftR^ dx 3 dx
a' and therefore also the numerator of this fraction is a reciprocant. Remembering that
da db ^ dc _ ,
dx = ^^' dx^"^'- -cLo = ^^'-'
the numerator may be written
a^-^R = OR. The ordinary expression for 0 is found by writing
a ^ -/*i = a(36a„ + ^(xit + odS„ + ...)
- 6 (3a3a + 4636 + 5cac + ...). If the actual extent of R is j, that of GR is j + 1 ; for the operation of 0 ' introduces an additional letter. Both the weight and degree are also increased by unity. Thus, the type of R being w ; i, j, that of OR is
w + 1; i + 1, j + 1.
Suppose the weight of R to be equal to its actual extent ; then ^ is a protomorph of the type j ; i, j, and GR, whose type iaj+l; i + l,j + l, is also a protomorph. This proves the existence of protomorphs for every possible extent. Starting with the form 4ac - 56" we obtain, by successive eduction, a series of protomorphs of the tjrpe j; j, j for which the general expression is
G>-»(4ac-56»),
where j has any of the values 2, 3, 4
If /2 is a protomorph of minimum degree, GR (if irreducible) will also be a protomorph of minimum degree. Hence the minimum degree can never increase by more than one unit when the extent is increased by unity.
396 Lectures on the Theory of Reciprocants [42
The second educt 0*R is always reducible ; for
[R.
oa T-;; - 5(2/^ + 1) 6 T- + 4/x(fi-l)c i R
Combining this with M = ac — j^b^, we have
5G'R + V(/x + 4) i/i2 = a I 5a ^ - 5 (2//, + 1) 6 ^ + 4/x (/* - l)c I E,
where the right-hand side is divisible by a, showing that the degree of G'R is always depressible by unity. R being a protoraorph of degree i and extent j,
is one of degree i + 1 and extent j + 2. Hence we may conclude that an increase in the minimum degree for protomorphs cannot be immediately followed by another increase ; for, if this were possible, the minimum degree for extent ^' + 2 would be i + 2, instead of being i+\ at most.
This conclusion is in accordance with the sequence of the values of i in the table of minimum degrees, and as far as it goes confirms the exactitude of the formula (w ; i, j) — {w — \; i + 1, j) for the number of pure reciprocants which was assumed in calculating the table.
The method previously employed to prove that every invariant is a rational integral function of protomorphs, or such function divided by a power of a, may be very easily extended to the case of reciprocants.
In the first place, it is obvious that every rational integral function of the letters a, b, c, d, ... is by successive substitutions reducible to the form
a-o^ia,b,P,.P„P„...Pj),
where Pj means the protomorph of extent j.
Let any reciprocant .K be put under this form ; then
a'R = <^{a,b,P„P,.P„...Pj), and, consequently,
Via'R)^ ^^Va + ^Vb + ^VP, + ... + ^VP,.
Now, F annihilates R, a, P^, Pj, ... P,-, since these are all pure recipro-
/7cf> cants. Hence the above identity reduces to —fy Vb = 0, from which (since
do Vb does not vanish) we conclude that <$> does not contain b explicitly. Thus,
a«ii=4>(a, P„P„P., ... P,.),
and the theorem is established for reciprocants.
42] Lectures on the Theory of Reciprocants 397
The Protomorphs for Reciprocants as far as extent 8 are as follows : Pa = 4ac — 56', Pa = a^d - Sabc + ^¥, P^ = oOa^e - 175a6ci + 28ac» 4- 105fr»c, P5 = 10a'/- 40a=6e - 12a»cd + 6oo6"rf + 16a6c^ - 396^0, P, = 14a»5r - 63a6/- 1350ace + l7826=e + 1470ad» - 41586cci + 2310(r', P. = 7a'A - 35a% - 539a'c/+ 73oa6y^"+ eOSa'de + 306a6ce - 148565e
- 213506^" + lOOlac^d + 346o6=c(i - 19256c», Ps = 420aH' - 2310a»6A - 25648a-c5r + 9240a=c(/ + 21780a-e^ + mQ%Qab-g + 85386a6c/- 191730a6rfe - 59220ac^e + \20bWacd} - 1269456'/+ 2521266^ce + 1692606=rf= - 4190346c'rf + 129360c*. The work necessary for obtaining the first four of these, Pa, P3, P4, P5, has been fully set out. Since P^ is of degree 3, its second educt, G'Pj, is of degree 5 and its reduced second educt of degree 4. A linear combination of this with a form whose leading term is a'^ce becomes divisible by a and gives P,; but as this requires the preliminary calculation of the form {a^ce), it is simpler to find P, directly by the method of indeterminate coefficients, and thence by eduction to get P-, and Pg. Thus (to ia numerical factor prh) P^ is the educt and P, the reduced second educt of P,. Beyond this point the calculation of protomorphs has not at present been carried.
Referring to the table which gives the minimum degree and multiplicity for a Protomorph of any extent, we see that the multiplicity exceeds unity I when the extent j = > 8, and is exactly equal to 2 when j = 8, 11, or 21.
Hence the protomorphs as far as Pj inclusive are unique ; but there are [two forms of extent 8 and degree 4, any linear combination of which I (provided it contains the term a'i) may be regarded as a protomorph. One j of these forms is Pj, whose value is given above ; the other is a linear I combination of P, with a form, whose leading term is a'cg, hereafter to be [set forth.
The irreducible forms for extent 2 are a and P, ; every other form must Ibe simply a power of P, multiplied by a power of a. We proceed to the Icalculation of all the Irreducible Forms for the extents 3 and 4 respectively. I When j = 3, we may combine the protomorphs
4ac — 56^ a'd - Sabc + 26» nth one another.
Adding 125 times the square of the latter to 4 times the cube of the former and dividing by a, there results the form
12ba'd' - loOa'bcd + oOOb'd + 25Ga'c' + 165a6V - 3006*c.
398 Lectures on the Theory of Reciprocants [42
This form is analogous to the discrimiDant of the cubic, but is of a higher degree by one unit. Its type is 6 ; 5, 3, whereas that of the discriminant is 6; 4, 3.
In the case of invariants, we have to combine ac — b" with a'd — Sabc + 26'. The square of the second, added to 4 times the cube of the first, gives a*d* - Qa'bcd + 4o'6»rf + da'b'c' - 12ab*c + 46« + 4aV - Uw'b^c^ + 12ab*c - 46'.
Here the term 12ab*c is nullified by — 12ab*c, so that the result contains o", the other factor being the discriminant
a'd' - Gabcd + Wd + 400" - 36V, which is of the type 6 ; 4, 3.
We may show a priori, assuming the problematical but highly probable formula (w; i, j) — {w — I; i + 1, j), that the type 6 ; 4, 3 does not belong to any reciprocant.
For, as seen in the partitionments set out below,
(6 ; 4, 3) - (5 ; 5. 3) = 5 - 5 = 0 3.3 3.2
3.2.1 3.1.1 3.1.1.1 2.2.1
2.2.2 2.1.1.1 2.2.1.1 1.1.1.1.1
We can by no other means combine the protomorphs with one another or with the Quasi-Discriminant (125a'd*...) so as to obtain additional fundamental forms. Every Rational Integral Pure Reciprocant of extent 3 is therefore necessarily a rational integral function of the four forms
deg. vit. 1.0 a,
2.2 4if = 4ac - 56',
3.3 A =a'd- 3abc + 26»,
6 . 6 (a'd») = 125a^d^ - 750a-bcd + 5006*d + 256a't^ + 165a6'c» - 3006V. These are connected by a syzygy of degree-weight 6 . 6, namely 1254' + 256Jf » = a {a'd"), analogous to the syzygy of the same degree-weight, in the Theory of the Binary Cubic, which connects the Discriminant with a and the Protomorphs of extent 2 and 3.
It will be clearly seen from an inspection of the fundamental forms that! there is no law for the coefficients of Reciprocants akin to that of their] algebraical sum being zero in Invariants.
42] Lectures on the Theory of Reciprocants 399
LECTURE XVII.
The fundamental reciprocants for extent 3, given in the last lecture, agree with the irreducible invariants of a binary cubic both in number and type, with the single exception that the degree of the cubic discriminant is lower by unity than that of the reciprocant corresponding to it. When the extent is raised to 4, both the discriminant and its analogue cease to rank among the irreducible forms, the former being expressible as a rational integral function of invariants of lower degree, and the latter as a similar function of reciprocants. But the increase of extent introduces three addi- tional reciprocants whose leading terms are a'e, a'ce and aV, whereas the additional invariants are only two in number and begin with ae and ace respectively.
The irreducible reciprocants of extent 4 are as follows : deg. wt. 1.0 a,
2 . 2 4i¥ = 4ac - 56',
3.3 A =d}d- 3abc + 2¥,
3 . 4 P4 = hOa^e - llbabd + 28ac» + 1056»c*,
4 . 6 {a^ce) = SOOu'ce - lOOOat'e - 875a»d» + 2450a6c(i - 1344ac» - 356=c',
5 . 8 (a*^) = 625a»e= - 4375a'6rfc - 49700aVe + 128625a6='ce - 787506*e
+ 55125aW- 61250a6''d-^ - \r>Q^00abcFd + 1837506'crf + 84868ac« - 1021656V.
The similar list of invariants for the quartic is deg. wt. 1.0 a,
2.2 oc - 6»,
3 . 3 a'd - 3a6c + 26',
2.4 ae - 46d + 3c^
3 .6 ace- b'e - ad? + 2hcd - &.
To obtain the fundamental forms of extent 4 we have to combine M, A and the Quasi-Discriminant
(a»d») = 125a»d» - 750a='6cd + 500a6'd + 256a»c» + 165a6V - 3006'c with the additional Protomorph
P« = 50a'e - 175a6d + 28ac» + 1056'c
* P4 is the protomorph of minimam degree; the other protomorph, B, which will be used when we treat of Principiants, is, when expressed in terms of the irreducible forms,
B = l(ai',-128Jlf»).
400 Lectures on the Theory of Reciprocants [42
in such a manner that the combination contains a factor a. The removal of this factor gives rise to a form of lower degree, and the process is repeated as often as possible.
Calling that portion of any form which does not contain o its residue, the residue of 43f is - 56", that of {a?d?) being - 3006*c, and that of P^ being 1056'c. Thus
16ilfP4-7(a'd-)
contains the factor a, and leads to {a^ce) of the type 6 ; 4, 4, which is the analogue to the Catalecticant
a h c
bed
c d e
The form (a'd-) now ceases to be a groundform (= irreducible form) and is replaced by the Quasi-Catalecticant {a'ce), for
Similarly, the Cubic Discriminant, a groundform qua the letters a, b, c, d, becomes reducible when a new letter, e, is introduced, and is then replaced by the Catalecticant.
We now come to an extra form which has no analogue in invariants. The residue of the Qiiasi-Catalecticant (a^ce) is — 356V, and consequently
P,^-252M{a'ce)
divides by a numerical multiple of a (as it happens by 4o) and yields the form (aV), whose type is 8 ; 5, 4.
Here the deduction of new fundamental forms comes to an end on account of the appearance of e in the residue of (aV). It would have ended sooner but for the apparently accidental non-appearance of the term b'd(o{ the same type 6 ; 4, 4 as b'c") in the residue of (a'ce). Had this term appeared, no combination could have been made leading to a new groundform after (a'ce). We are able to show from d priori considerations that it cannot exist.
For the arguments in the annihilator V, up to 3, inclusive, are
a^db, ahdc, acdd, b^dd, adde, and 6c3«.
If, now, the term fib^d were to form part of a Pure Reciprocant, b^da operating upon it would give /Jy ; but every other portion of the operator would necessarily give terms containing one or other of the letters a, c. Since such terms cannot destroy fjh\ we must have /i6° = 0. Hence the term in question is necessarily non-existent.
42] Lectures on the Theory of Redprocants 401
The method of combining the protomorphs which we have followed shows that the fundamental reciprocants of extent 4 are connected inter se by the two relations or syzygies
7 (256i¥» + 1 25^0 - IBaifP, + a' {a^ce) = 0,
P^ - 252M {a'ce) - 4>a (a>e') = 0.
The invariants of the binary quartic are connected by only one syzygy, similar to the first of these ; the second has no analogue in the theory of Invariants. It has been shown that the irreducible reciprocants of extent 3 are connected by the syzygy
256.¥' + 125^2 - a (a'd») = 0.
Substituting in this for the Quasi-Discriminant (a*#) its value expressed in terms of the fundamental forms of extent 4, by means of the equation
163/P, - 7 {a'cf) = a (a^'ce), we obtain the first of the above syzygies. By a precisely similar substitution, the syzygy connecting the invariants of the quartic is derived from the one which connects the invariants of the cubic.
Every reciprocant of extent 4 is a rational integral function of the six fundamental forms given in the table ; and, by means of the syzygies, powers, but not products, of A and P^ can be removed from this function. For the first syzygy gives A" and the second gives P^' as a rational integral function of the four remaining forms a, M, (a'ce), and (a'e^). Hence every reciprocant of extent 4 is of one or other of the forms
*. A<P, P,^, AP/i>,
where ^ does not contain either A or P^, but is a rational integral function of the other four fundamental forms.
Let the four forms which appear in ^ occur raised to the powers k, X, fi, v, respectively, in one of its terms. Since the degiee-weights of these four forms are
1,0, 2.2, 4.6 and 5.8,
any such term may be represented by
a'(aV/(aV>'(oV)'. Thus the totality of the terms in 4> will be represented by
2 o'(a»a?)*(aVy (aV)* = .-; ^. — -^ -— — - .
Now, A, Pi and APt have the degree- weights
3.3, 3.4 and 6.7, and consequently the totality of terms in
^, A^, P.* and 4P.<D 8. IV. 26
402 Lectures on the Theory of Redprocants [42
(that is, the totality of the pure reciprocants of extent 4) will be repre- sented by
(1 + a^a? + aV + aW) 2 a''(a»a!»)*(aV>'(aV)''
1 + a?a? + a*af + a'a?
~ (1 - a) (1 - a?x') (1 - aV) (1 - a'a?) ' Hence the number of Pure Reciprocants of the type w; i, 4 is the coefficient of a*x" in the expansion of a fraction whose numerator is
1 + a V + aV + aV, with the denominator
(1 - a) (1 - a'w') (1 - aV) (1 - aW). This fraction is called the Representative Form of the Generating Function, in contradistinction to the Crude Form, which is a fraction with the numerator
1 — a'^ic, having for its denominator
(1 - a) (1 - aa;) (1 - oar") (1 - aa^) (1 - cwr*). The crude form expresses the fact that the number of pure reciprocants of the type
is (w;i,j)-(w-l;i + l,j).
Its numerator is 1 — a"'*' for all extents ; for the general case in which the extent is j, its denominator consists of the j + 1 factors {1 - a)(l - cu;){l - aa^) ... (l-ax^). The removal of the negative terms [corresponding to cases in which (w; i,j) <(w — 1; i+ l,j)] from the crude form would give either the repre- sentative form or one equivalent to it, according as the representative form is or is not in its lowest terms. lu the parallel theory of Invariants the terms to be rejected are those for which ij — 2w< 0; but we do not at present know of any similar criterion for reciprocants, and are thus unable to pass directly from^the crude to the representative form of their generating function.
Knowing both the crude and the representative form for reciprocants of extent 4, we may verify that the difference between these two forms of the generating function is omninegative. It will be found that 1 — a~'jr (i-a)(l-aa!)(l-aa;=')(l-aa;')(l-aar*)
1 -I- a'a? + a'x* + a'af
I
(1 - a) (1 - aW) (1 - aV){l - a'af') 1 /a-'a; + aV a^ + aV\
{l-aa?){\-aa?){\-ax')\ 1 - a V "l-aV/ 1 IX + a'x"> oV + a'x'\
(l-aa^)(l-aV)(l-ttV)Vl-cwr' '^ H^cu^' )
42] Lectures on the Theory of Reciprocants 403
Thus the crude form is seen to consist of an omnipositive part, equal to the representative form, and an omninegative part.
There is no diflSculty in obtaining the representative form of the generating function for pure reciprocants of extents 2 and 3. In the one case every reciprocant is a rational integral function of two forms of degree-weight, 1 . 0 and 2 . 2 respectively. The generating function is therefore
1 (l-a)(l-ttV)" In the other case (that is, for extent 3) every pure reciprocant can be expressed as a rational integral function of four forms, of which the degree-weights are 1.0, 2.2, 3.3 and .5 . 6, no higher power than the first of the form 3 . 3 occurring in the function. Thus the representative form is
l-Ha'a;'
{\-a)(\-a^a?){l-a'-af)'
LECTURE XVIII.
The number of Pure Reciprocants of a given degree is finite ; the number of Invariants of the same degree is infinite. Thus, for example, we have the well-known series of invariants
ac-6», a«-46d + 3c*, .... all of degree 2, but of weights and extents proceeding to infinity. This may be proved from the theory of partitions (see American Journal of Mathematics, Vol. v., No. 1, "On Subinvariants," Excursus on Rational Fractions and Parti- tions). It will be seen in that article that if N{w:i) is the number of ways in which w can be divided into i parts, and if P is the least common multiple
of 2, 3, 4 i, then N(w:i) can be expressed under the form
F(w,i) + F'{w,i,p), where p is the residue of w in respect of P.
„r . . i(i + l) Wntmg w-\ V — = ".
F(w, i) is of the form
2».3'...(i-l)«.i ■■■' all the succeeding indices of the powers of v in F(w, i) decreasing by 2, and their coefficients being transcendental functions of i which involve Bernoulli's Numbers.
In F'{w, I, p) the highest index of v is one unit less than the number of times that i is divisible by 2, that is, is or — ^— , according as i is even
or odd.
26—2
404 Lectures on the Theory of Redprocants [42
Thus, for the partitions of w into 3 parts, we have the formula
JVr(«, : 3) = 1^ - 1} + {I (- D- + \ {p.- + p/)} ,
, 1+2+3 ^„ wliere v = w + 5- — =w + 3.
And, for the partitions of w into 4 parts, I/* 5v
N(wA) =
144 96
+ {m (-)■""" + ^7 ^P^"" + P^"^' - P'~' - P^""^
-^2^'^'^''^''^'''''^'"''''^^'
1+2+3+4 where v= iv-\ ^ = w + 5,
aad pi , p-i are the roots of p* + /> + 1 = 0,
tj , tj „ „ t + 1 =^ 'J i
in other words, p, and p^ are primitive cube roots, and i,, % primitive fourth roots of unity.
The principal term of N^w.Z), regarded as a function of w, is
, that 01 JV{w: 4) bemg
12 2».3' ^ ' ' ^144 2^3^4■
And in general the principal term o( N{w. i) is
W
Hence it follows, from a general algebraical principle, that for all values of w above a certain limit, which depends on the value of i and may be determined by the aid of partition tables, {w; i, oo ) — (w — 1 ; i + 1, oo ) must become negative.
Ultimately, ^^ ; — ^—. ^ = ^-r- — r-, , which must eventually be creater
■" (w; t, oo) t(i+l)' J b
than unity. This shows that beyond a certain value of w there can be no
pure reciprocant, and consequently that the number of pure reciprocants of a
given degree i is finite.
Mr Hammond remarks that the formulae for N{w : 3) and N (wA) may, by the substitution of trigonometrical expressions for the roots of unity, accompanied by some easy reductions, be transformed into
and i\r(«;:4) = j^-- + -sm«^ + 3sm^-g-^3sm^,
where, in the first formula, i- = w + 3, and in the second v = w + o. He also obtains the principal term of iV(w:{) from first principles as follows:
42]
Lectures on the Theory of Redprocants
405
\
The partitions of w into i parts may be separated into two sets, the first containing at least one zero part in each of its partitions, the second consist- ing of partitions in which no zero part occurs.
Suppressing one zero part in each partition of the first set, we see that the number of partitions in which 0 occurs is N {w :% —\). Diminishing each part by unity in those partitions which contain no zeros, their number is seen to be N{w — i:i). The sum of these two numbers is N{w :i), which is the total number of partitions, and consequently
N{w.i) = N'('w:i—l) + N{w-i: i).
Let the principal term of N{w : i — \) be aw'"*, where a is independent of w, and write
w = ix, N(w:{) = Ux, N {w — i:i) — U3^i.
Then m, - u^i = ow'-» + . . . = ai^-^a^-^ + . . . .
Hence, by a simple summation, we find
u^ = ai'-» [x^ + {x - !)•■-' +{x- 2)'-- + ...) + ....
But, since only the principal term of u, is required, this summation may be replaced by an integration. Thus the principal term of u^ is
ai'-» \i
'dx =
Restoring
i-1 ■
w = ix and N(w:i) = Ux,
we see that the principal term of N{w:i) is
aw
Thus the principal
term oi N {w: i) is found from that of i^T (w : i — 1) by multiplying it by
(i-l)i-
When 1 = 3, the principal term is ^r— ;, ; it is therefore -^ j when
i = 4 ; and for the general case it is ,, .,,
w^
2».3«.4>...(t-l)».r
The value of N (w :i) is given in line i and column w of the following table:
|
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
||
|
2 |
2 |
2 |
3 |
8 |
4 |
4 |
S |
5 |
6 |
6 |
7 |
7 |
8 |
|
|
3 |
2 |
3 |
4 |
6 |
7 |
8 |
10 16 |
12 18 |
14 23 |
16 27 |
19 34 |
21 39 |
24 47 |
|
|
4 |
2 |
3 |
6 |
6 |
9 |
11 |
||||||||
|
6 6 |
2 2 |
3 3 |
5 5 |
7 7 |
10 11 |
13 14 |
18 20 |
23 26 |
30 35 |
37 44 |
47 58 |
67 71 |
70 90 |
406
Lectures on the Theory of Reciprocants
[42
From an inspection of the tabulated values oi N{w:i) we see that N {w.2)— N (w — \ :%) is negative or zero when w>2, N{w:2)-N(w-\-A) „ „ w>6,
N(w'A)-N{w -1:5) „ „ w>8,
N(w:5)-N{w-1:6) „ „ w>12.
Hence for pure reciprocants of indefinite extent, whose degrees are
2, 3, 4, 5, the highest possible weights are 2, 6, 8 and 12, respectively.
In like manner, from Euler's table, in his memoir "De Partitione Numero- rum " (published in 1750), it will be found that
for degrees
the highest weights are
Further than this the table, which goes up to w = 59, will not enable us to proceed.
The actual number of pure reciprocants of degree i, weight w, and of indefinite extent, is seen in the following table, which gives the value of ]S'('w:i) — N(w-l:i+l) when positive, blank spaces being left in the table when this difference is zero or negative.
|
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
|
2 |
6 |
8 |
12 |
16 |
21 |
26 |
30 |
36 |
42 |
49 |
55 |
Weight «i = 5 6 7 8 9
10 11 12 13 14
|
1 |
||||||||||||
|
1 1 1 |
1 1 1 |
1 2 2 |
1 |
1 2 3 |
1 2 |
2 4 |
3 |
i |
2 |
3 |
||
|
2 |
Thus, for degree 2, there is only one pure reciprocant, namely
(ac) = 4ac — 5b-. For degree 3 the table shows that, in addition to the compound form
a (ac) = a (4ac — 56'), there are three others whose weights are 3, 4 and 6 respectively. These are the three protomorphs, (a'd) = a»d - Sabc + 2b\ (a^e) = 50a»e - 175a6d+ 28ac= + 1056'c, (a'^r) = IWg - 63abf- 1350ace + I7826»e + 1470ad' - 41586cd + 23100*.
42] Lectures on the Theory of Reciprocants 407
With the above forms and a we are able to form the following compounds of degree 4 :
a''{ac), a^aPd), (acf, a{a?e), a{a'g),
whose weights are 2, 3, 4, 4, 6.
The forms of degree 4 and weights 5, 7, 8, and one of the forms of weight 6, cannot be similarly made up of forms of inferior degree, and are therefore groundforms. Three of them are the protomorphs (a^f), {a^h) and (uH) of weights 5, 7 and 8, whose values were given in Lecture XVI. The ground- form of weight 6 is the Quasi-Catalecticant given in the last lecture. All the forms of degree 4 have thus been accounted for except one of the two forms of weight 8, which will be seen to be of extent 6, and to have a^cg for its leading term.
We know from Euler's table that N{8 : 4) - N(7 : 5) = 2 ; that is,
(8;4,8)-(7;5,8) = 2.
Now, (8 ; 4, 7) = iV(8 : 4) - 1, the omitted partition being 8.0.0.0,
(8 ; 4, 6) = iV^(8 : 4) - 2, the partition 7.1.0.0 being also left out,
iA- i, ti\— TVta i.\ A ^^^^ 6.2.0.0 and 6.1.1.0 are excluded from (8; ,5)-iV( : )- '1(8.4 5)_ but, make their appearance in (8; 4, 6).
Similarly, (7 ; 5, 7) = N(7 : 5),
(7;5,6) = i\'(7:5)-l,
(7;5, 5) = iV(7:5)-2. We have, therefore, (8 ; 4, 8) - (7 ; 5, 8) = 2,
(8;4,7)-(7;5, 7)=1,
(8;4,6)-(7;5,6) = l,
(8; 4, 5) -(7; 5.5) = 0. Hence we may draw the following inferences :
(1) No pure reciprocant exists whose type is 8 ; 4, 5.
(2) The one whose type is 8 ; 4, 6 must contain the letter g.
(3) No fresh form is found by making the extent 7 instead of 6, so that there is no pure reciprocant of weight 8 and degree 4 whose actual extent is 7.
(4) There is a pure reciprocant (the Protomorph whose leading term is aH) whose actual extent is 8.
(5) This, with the one whose actual extent is 6, makes up the two given by(8;4, 8)-(7;5,8)=2.
408 Lectures on the Theory of Reciprocanta [42
LECTURE XIX.
The following is a complete list of the irreducible reciprocants of indefinite extent for the degrees 2, 3 and 4 :
Oeg. wt.
2.2 {ac),
3.3 (a'd),
3.4 (a»e), 3.6 (d'g),
4.5 (a'/),
4.6 {a?ce), 4 . 7 {a^h),
4.8 {aH), {a?cg).
The values of all of them except {a^cg) have been given in previous lectures, and the method of obtaining them sufficiently indicated. Thus (ac), {a?d), (a'e), (a'/), {a^g), (a'h) and (aH) are the Protomorphs of minimum degree P.^, Pj, P4, P„ P,, P7 and Ps, respectively; and (a'ce) is the Quasi-Cata- lecticant whose value has been set forth in the table of irreducible forms of extent 4. It will be remembered that (a^ce) was found by combining the Quasi-Discriminant (a'd^) with P2P4 linearly in such a manner that the combination, which is of the 5th degree, divides by a and gives {a'ce) of the 4th degree. If we try to find (a'cg) by a similar process, it will be necessary to rise as high as the 7th degree, and then to drop down by successive divisions by a to the fourth.
In fact, since to a numerical factor prh the residues of
Pa) Ps) "i, Ps
are b\ b\ h% ¥c,
that of P^P, will be 6«c,
and that of P^^P^ will be 6«c.
Thus a linear combination of P^P^ and P^P^ will be divisible by a, and, taking account of the numerical coefficients, we shall find
26P,»P, + 87oP^P, = 0 (mod. a).
As a result of calculation, it will be seen that the above combination of the protomorphs divided by a,
^(26P,'P, + 875P,P.), has (to a numerical factor pr^) the same residue as P4'.
42] Lectures on the Theory of Reciprocants 409
Making a second combination and division by a, we find
7 / — iZ ?^J _ 25P,» = 0 (mod. a) = a8, suppose.
Then, by actual calculation, the residue of S is found to be - 2625006*e + 612.5006'cd - 3390806^c'. Two reductions have already been made in obtaining this form /S of the 5th degree. A final combination of 8 with P^P^ and the form (aV), whose value was given in a former lecture, enables us to divide out once more by a and thus get the form {a?cg) of the 4th degree.
It is the fact that P^Pf^ and (a'e') have residues which are not the same to a numerical factor pres which necessitates the long calculation above described. No linear combination of P^P^ and (aV) with one another is divisible by a, and it is necessary to find a third form S a linear combination of which with both PyP^ and (a'e") will divide by a.
There is, however, another way of arriving at the form (a^cg) by using the eductive generator
G='i(ac-b')db + o(ad-bc)dc+6{ae-bd)dd+.... Starting with the Quasi-Catalecticant
(a^ce) = SOOa'ce - lOOOoA'e - 875a^d^ + 24i50abcd - 1344oc' - SSfeV, and operating on it with G, we have
6 (a^ce) = 4 (ac - b') (- 2000a6e + 24o0acd - TOftc")
+ 5 (ad- be) (SOOa'e + 2450abd - 4032ac= - 706'c) + 6(ae-bd)(- lloOa'd + 24.50a6c) + 7 (o/- be) (SOOaV - 1000a6=). The terms of this expression contain the common numerical factor 10, which may be rejected ; thus we have
(?(a»ce) = 10 (a'c/X where (d'cf) = o60a'cf- 700a'6'/- 650a'de - 290a'bce + loOOafe'e
+ 227 oa'bO' - lOSGaVd - SllOaH'cd + WHSabc' + 636V.
This form (a'cf) is the first educt of (a'ce), and is irreducible (but, being of the fifth degree, does not appear in our list, which contains no forms of higher degree than the fourth). Operating on it with G, we obtain the educt of (a*cf), which is the second educt of (a^ce). This second educt will be of the 6th degree (its leading term will be a*cc/), but is reducible to the 5th when combined with
(4ac - 56») (a'ce),
as we know from the general theorem concerning the reduction of second educts. We shall thus obtain a form (a'cg), the reduced second educt of (o'ce), of the 5th degree, and a final combination of (a'cg) with one or both of
410 Lectures on the Theory of Reciprocants [42
the forms P^Pt and (aV) will enable us to divide once more by a and thus arrive at {a*cg) of the 4th degree.
By either of these methods we obtain
{a?cg) = 1 lIQa^cg - 8085a'rf/+ 7040aV - iMQah^g + 18963o6c/ - immahde - 27160ac»e + 26460acd» - 9.5556»/ 4 280986»ce + liTWh^cf - .528226c»d + 21560c« ; but the second way, besides being more direct, gives us at the same time the value of the irreducible form (a'cf).
Every Pure Reciprocant is an Invariant of a Binary Quantic whose
coefficients A, B, C, D, ... are functions of the original elements a, b, c, d, ...
such that
VA=0.
VB = A,
rc=2B,
VD = 30,
and conversely, every Invariant of this Binary Quantic, or of a system of such Binary Quantics, is a Pure Reciprocant.
This is a particular case of the more general theoreui, due to Mr Ham- mond, that if ® is the operator,
^, (a) 3t + ^2 (a, h)dc + (t>3{a, b, c)dd + ...,
where </>i, ^, <^s, ... are arbitrary rational integral functions, and if
A, B, G, D, ..., A', B', C, D', ..., A", B", C", ...
be any rational integral functions of the original letters a,b,c, ... which satisfy the conditions
04=0, 04' = 0, @A" = 0,
®B = A, @B' = A', @B" = A",
eC=2B, eC' = 2B', &C" = 2B",
eJ3 = 3C, 0i)' = 3C", @D" = 3G",
then every invariant in respect to the elements
A, B, G, D A', B', G', D', ..., A", B", G", U', ...
is a rational integral solution of the equation
0 = 0.
Obviously, every rational integral solution of 0 = 0 is an invariant in the above elements, so that the converse of the proposition is true. For the only
42] Lectiires on the Theory of Reciprocants 411
conditions imposed upon A, A', A", ... are that they shall be rational integral functions of a, b, c, d, ... annihilated by ©. Let
<i>(A, B, C, D,..., A', B', C, D', ..., A", B", C", D", ...)
be any invariant in the large letters. We have to show that
0<J) = O.
Now, ea> = ^0^+^0£ + ^©C+...
dA dB dU
+
Hence, writing for 0.4, 05, 0C, ..., their values given above, we have e^=(Ads +^Bdc +2Cdj, +...)«>
+ (A'dg + 2B'dc- + 3C'diy+...)^
+
= 0 (.since <I> is an invariant) ; which proves the proposition.
Confining our attention to a single set of letters, the Binary Quantic
(A.B,C,...J,K,Llx.yr, whose coefficients are formed from one another by the successive operation of 0 as above, may be called a Quasi-Covariant ; and it will follow immediately from the Theory of Binary Forms that every Co variant of a Quasi-Covariant is itself a Quasi-Covariant, and that every Invariant of any Quasi-Covariant (or system of Quasi-Covariants) is an Invariant in respect to the letters A, B, C, ..., and therefore, by what precedes, a rational integral solution of 0 = 0.
Writing the terms of
(A.B,C....J.K,Llx,yr in reverse order, we have
Xy» + nKxy"-^ + "y~ ' Jxy""-^ + ...+Ax^,
where %L = nK, %K={n-\)J, ...%A = 0.
Thus the Quasi-Covariant may be written
Ly» + eLxy»-' + j-^ a:»y»-' + ... + ^^ ^ ~w^ = y'*Ke'> ) L,
where 0»+'Z = O.
This is the general symbolic expression for a Quasi-Covariant. An example of a Quasi-Covariant has already been given in Lecture II. [p. 310, above],
412 Lectures on the Theory of Eeciprocants [42
where it was stated, and afterwards proved [p. 360], that the reciprocal of the nth modified derivative could be put under the form
- «-«-» Vc"'^) a„.
The numerator of this reciprocal expression, which may be called the reciprocal function, is
which is identical with the general expression
y" Key) L, if a; = - 1, 2/ = <, i = a„ and 0 = F.
Hence every Invariant of the reciprocal function is a Pure Reciprocant.
This property of the reciprocal function was discovered independently by Mr C. Leudesdorf, who published his results in the Proceedings of the London Mathematical Society (Vol. xvii. p. 208). Mr Hammond's results were given in two letters to me dated January 15th and January 20tb, 1886, and were briefly alluded to by him at a meeting of the London Mathematical Society. They are here published for the first time.
Recalling the form of the operator
0 = ^,(a)34 + <^2(a, 6) a^ + <^3 (a, 6, c)3d + ...,
where <^,, <^j, ^j, ... are rational integral functions, we can form a Quasi- Covariant of extent ^ by a finite number of successive operations on a single letter of that extent.
To fix the ideas, take the letter d of extent 3, and operate on it with 0; then
%d = ^3 (a, 6, c).
Since ^i, (^2, <^3, ... are by definition rational integral functions, we can, by operating a finite number of times with 0, remove fir.st c and then h from ^(a, 6, c), and thus obtain
0"d = funct. a,
where n denotes a finite number of operations. Since 0a = 0, we have
0"+'d = 0.
In this manner we form the Quasi-Covariant of the nth order
y^ie'H') d. If <f)i, (j),, 04, ... do not contain higher powers than the first of the last letter in each, the order of the above Quasi-Covariant will be the same as its extent. This is the case with the reciprocal function, which is a co-recipro- cant (that is, a Quasi-Covariant relative to V).
42] Lectures on the Theory of Reciprocants 413
Ex. y-\ei / c = cy-+ Vcxy + z. — ^ a^ = cy^+ oahxy + oa^a?.
The discriminant of this is the pure reciprocant
5}f\
oa-
( ^n
As an additional example, consider the pair of linear co-reciprocants 4a (4ac — 5¥) x + {oad — Ihc) y, 50a (a'd - 3a6c + 26') x + {25abd - 32ac- + 5¥c) y. The resultant of this pair is
2a (125a'd^ - ToOa'bcd + oOOab^d + 256aV + 165a¥d' - 3006*c), that is, is the Quasi-Discriminant multiplied hy 2a.
LECTURE XX
" Quinteasenced into a finer substance." — Drummond of Hawthomden.
Before proceeding with the proper subject of this day's lecture, I should like to mention a geometrical theorem which has fallen in my way, and which, inter alia, gives an immediate proof of the existence of 27 straight lines on a general cubic surface. It is proved by means of a Lemma (itself of quasi -geometrical origin) which finds its principal application in an ex- tension of Bring's or Tschirnhausen's method, and shows how any number of specified terms, reckoning from either end, can be taken away from any equation of a suflSciently high degree*.
Subjectively speaking, I was led to the Lemma by considering the question, closely connected with Differential Invariants, of the method of depriving a linear differential equation of several terms.
Let ^ be a cubic and u a linear function in x, y, z, t, say
<t>=aa^ + ... ■>rfa?y + ...,
« = ir + my +TUi+pt.
Then, if i^ is a scroll which contains all the straight lines on <^ + \u', fhen the parameter X has any arbitrary numerical value from +00 to — 00 , prove that
yjr=:<l>^A+ <})u'B + u'G,
* I recover all Hamilton's resalts contained in his Report to the British Association, 1836, " On Jerrard'B Method," in a much more clear and concise manner, and make important additions to his theory.
414 Lectures on the Theory of Reciprocants [42
where -^ is of the degree 1 5 in the variables x, y, z, t,
6 in the coefficients (i, m, n, p) of w,
11 (a, )of <^.
Or, more briefly, in x I a
•^ is of degree 15 6 11, and consequently
G 9 0 11.
The intersections of <^ with -^ are its intersections with m« and with C, of
which the intersections with the arbitrary plane w' are clearly foreign to the
question, but the cubic <^ and the ^C intersect in 27 straight lines, which
are the 27 ridges on <^.
C is identical with the covariant found by Clebsch and given in Salmon's
Geometry of Three Dimensions at the end of the chapter on Cubic Surfaces.
It may with propriety be called the Clebschian.
By giving the parameter X (which occurs in ^ + \m') an infinitesimal variation, it is easily proved that
B = -2EC, A=E'C, E?C=^0,
where E is the operator i'9o+ ... + 3i''m3/+ ..., which may be simply and completely defined by its property of changing the general cubic <f> into {Ix + my + nz + pty.
The equation E'C=0 expresses a new property of the Clebschian: it shows that if a, /are the coefficients of a^ and any other term in <j> containing a?, neither a? nor a?f can occur in any one of the terms of C. Defining a principal term in ^ as one which contains the cube of one of the variables, and a term adjacent to it as one which contains the square of the same variable, this is equivalent to saying that neither the cube of the coefficient of a principal term nor its square multiplied by the coefficient of any adjacent term can appear in any of the terras of G.
An interesting special case of the general theorem is when the arbitrary plane u is taken to be one of the planes of reference, say u = x. Then
1=1, m = 0, n = 0, jo = 0, and the operator E becomes simply -=- . Thus we learn that
is a Scroll of the fifteenth order which contains all the Ridges on
^ + Xx' for any arbitrary value of the parameter \.
It also contains 6 times over the curve of intersection of ^ = 0 with a; = 0.
42]
Lectures on the Theory of Reciprocants
415
I now propose to give the substance, with a brief commentary, of some very interesting letters I have recently received from Capt. MacMahon. I abstain from giving a proof of his results, as I am informed that he intends to do this himself at an early meeting of the London Mathematical Society.
Using V to signify the Reciprocant Annihilator and fl the Annihilator of Invariants, we have studied the properties of
|
dx dx |
||||
|
and those of |
dx dx |
|||
|
These may be written in |
the form |
|||
|
-i |
dx |
|||
|
yi |
dx |
and may be called alternants to V, -z- and to fl, -j- respectively. It has been shown in Lecture VII. [p. 341, above] that
The corresponding formula is
dx dx
may be seen by writing k = 0, X = 3, /x=4, j' = 5, ^formula given in Lecture V. [p. 329, above].
in a more general
Observe that operating with the alternant to fl, t- is equivalent to
dx
d
multiplication by a number, and that operating with the alternant to V, -j-
merely introduces a numerical multiple of a as a factor. No such property exists for the Alternant
Fn-fiF,
but one much more extraordinary.
MacMahon has found that this alternant, which he calls J, is a generator to a Reciprocant and a generator to an Invariant ; that is, it converts a Reciprocant into another Reciprocant, and an Invariant into another Invariant. As regards a Differential Invariant, which is at once an Invariant and a Reciprocant, it is an Annihilator. He shows, in fact, that
and VJ-JV=0.
416 Lectures on the Theory of Reciprocants [42
If, then, HR = 0, it follows immediately that fl (JR) = 0 ; that is, if ^ is an invariant, JR is so too. Aod in like manner, if
VR = 0, V{JR) = 0, that is, if ^ is a reciprocant, so is JR.
Of course, if itf is a Differential Invariant,
JM= V(nM) - n ( VAT) = 0. Let me here give a caution which may be necessary : The fact that a form is annihilated by J is not sufficient to show that it is a Differential Invariant, though all Differential Invariants are necessarily annihilated by J. Forms exist which are subject to annihilation by
J=a''dc+^abda+..., but are, notwithstanding, neither invariants nor reciprocants.
Such a form is the monomial b, which is obviously annihilated by J. Another is od — 36c. For, since
a»d - Babe + 26' is a Differential Invariant, we have
J(a»d-3a6c + 260 = 0. But ^6^ = 0 and Ja = 0;
therefore, also, aJ{ad — 36c) = 0.
The general theorem is as follows, and is a most remarkable one : If we write
mP (m, ft, V, n) = fia^d^^ ■\-{ji. + v) ma"^' 63^^^^
m(m— 1) „ „
+ (fi + 3v)\ma"'-^d + m (m - 1) a"^''6c
n+S
where the coefficients of the terms inside the brackets are the same as those of the corresponding terms in the expansion of (a + 6 + c + . . .)"*, and where a„ stands for the nth letter of the series a, 6, c, d, .... then Capt. MacMahon establishes that the alternant of any two P's is another P.
A question here suggests itself naturally : What would be the alternant of three or more P's ? For instance, would the alternant
Pi P. Pa I
P, P, P,\= P,P,P, - P,P,P, + P,P,P, - P,P,P, + P,P,P, - P,P,P,
P, P, P. I be another PI*
' In my Multiple Algebra investigations, which I hope some day to resume, I have made important use of similar Alternants, which, it may be noticed, do not vanish when their elements
42] Lectures on the Theory of Reeiprocants 417
Moreover, he obtains expressions for the parameters m, fi, v, n of the resulting P in terms of the parameters of its two components. He proves that if Pi, Pj are the two components whose alternant is P, supposing ni^, fij, Vi, «! to be the parameters of P,,
WI5, /ij, »2, Jla Pi,
then the parameters m, fi, v, n of their resultant P are given by the equations
m = TMi + Wj — 1,
/t = (»*! + Wj - 1) 1;^ (/I, + n^v,) -^(/i2+ >hvA , , . »w»— 1 mj — 1
V = (Ma-ni)Vif2 M1W2 + — — /ijV,,
n = «! + «2. It will be seen that fl and F are special forms of P. Thus,
n = P(i, 1,1. 1),
F=P(2,4, 1, 1). Now, if the second and third parameters are zero, every term of P vanishes, and MacMahon finds that in the following two cases the second and third parameters of the resultant above given vanish.
(1) Supposing to be an integer, this takes place when the two
component systems of parameters are
(2) When they are
m», »ii»ij, wij — 1, (wg — 1).
Now, P(i, 1, 1, i)=n,
P(2, 4, 1, 1)=F,
and by the law of composition
J=nF- Fn = P(2, 2, 1, 2).
Llso '2212)
i' l' / 1 1 *^^^ ^® found to come under the first case;
ad 2, 2, 1, 21
„' t t' -if the second.
2, 4, 1, 1)
re non-commatatiTe. In thia connection it is well worthy of observation that the P'a (as
Ddeed would be true of any operators linear in the differential inverses) obey the associative law.
It would be interesting to ascertain under what arithmetical conditions, if any, other than
(acMabon's, any two linear operators of the same general form as his P's become commutative.
Perhaps it would also be worthy of inquiry whether the P theory might not admit of extension
in some form to operators non-linear in the differential inverses, and whether to every such
operator of degrees i and j in the letters and their differential inverses there is not correlated
another in which i and j are interchanged.
8 IV. 27
418 Lectures on the Theory of Reciprocants [42
Hence. nj-jn = 0 and VJ-JV = (\
The above theorem is one of extraordinary beauty, and must play an important
part in the future of Algebra.
In another letter Capt Ma«Mahon calls my attention to the fact that the operator called by me Cayley's generator P. in Lecture IV. of this course [p. 323, above], is a particular case of one of a much more general character given by him in the Quarterly Mathematical Journal (Vol. xx., p. 362).
He also states that every pure reciprocant, when multiplied by the needful power of a, is an invariant of the binary quantic {2.(2Ti + l)!}a»+'-n{l!(2n + l)!}a»-'6«
_ n(n-l)(n-2) {3 , (2„ _ 1) .} {a-'d + (n - 3)a'-6c + ^^^"3— «"-'^} «"
+
which I have written in the non-homogeneous form.
But this expression is (to a numerical factor prh) identical with the numerator of ?^5 when t,a,h,... are taken to be the modified differential
derivatives^ IcP^ _l_d^ _ gee my note on Burman's law for the derivatives ^,2^. 2. 3 da,^' ^
Inversion of the Independent Variable [Vol. 11. of this Reprint, p. 44].
The property that its invariants are pure reciprocants has already been proved in the lectures [above, p. 412].
LECTURE XXL
I take blame to myself for not earlier communicating to the class the substance of a note of Mr Hammond's under date of January 20th, 1886, in which he makes an interesting application of the theorem that any invariant of the form
-V
y"(e» )F{a,h,c, ...),
in which the function F is subject to the condition
F»+'jF'=0, or of any combination of such forms, is a pure reciprocant.
42] Lectures on the Theory of Reciprocants 419
Forms such as the above, whose invariants are pure reciprocants, he calls co-reciprocant8. It follows that any covariant of one or more co-reciprocants IS Itself a co-reciprocant, for any invariant of a covariant is an invariant. Taking i?* to be a single letter b, c, d, he forms the functions
bi/ + 2a-x, ^j'j
cy" + 5ahxy + oa'af, /•2)
d!/> + 3 (2ac + 6») xy' + 21a'bai'y + 14a^a^, (3)
in which 2a' = Vb,
5ab= Vc, 5a'=~l.
3{2ac + b^)=rd, 21a=6=p|, 14a«= Z'fL.
On writing y = t,x = -l,it will be observed that these three forms are the numerators of
'Sldf 4:ldy*' h\d^' The Jacobian of (1) and (2) is
(4ac — 56») ay ; the coeflScient of ay is the familiar pure reciprocant 4ac - 56>. The Jacobian of (1) and (3) is the determinant
* 2a' I
df + (4ac - Si*) xy (2ac + 6») y» | ' which is divisible by y, giving the quotient
(2a»rf - 'labc - 6») y + 2a» (4ac - 56») x. (4)
This is ^ y(<^'')(2a'd-2a6c-6'),
the terms involving - , - , ... vanishing identically.
Looking at 2a-d- 2a6c - 6> as the anti-source to a Co-reciprocant • we might at first sight expect that it would give rise to a co-reciprocant of' the third order m x, y, whereas we see it is the anti-source of a linear co- reciprocant.
to O Z^t <';ff"«°ti'''e'' Reciprocants from Invariants is that we have no reverser to K as O is tZ "^ '"""""'• """ "■ "° '""'''' "'""''' •»-« -' -'^"-l"- - additional
wth^O rf?o''°tr °' ."^ "°"'"\°' "' °^'^°''' *'"'" ''<"" '^« ^o"^'"' l"? continnally operating ^nt we aronlvabi? """■■". ' '^""''""""^ °^'''"°« "'"> "" ^"' '° '"« ^^ "^ a co-recipro' trhirirn! "^ , r "l '° "■^ '^"^"°° '"""""y f^"-" '^« -nti-source, or coefficient of
the highest power of y, to the source), as we have only one operator, V, at our disposal.
27—2
420 Lectures on the Theory of Reciprocants [42
We have F(2a»d - 2aic -}f) = 2a? (4ac - 66').
Combining this with
V(a?d - Sabc + 26') = 0 (the well-known Mongian),
and dividing by a, he obtains
V(5ad - 7bc) = 4a (4ac - 56»).
Hence (5ad — 76c) y + 4o (4ac — 56») a; (5)
is a co-reciprocant. It is in fact (4) reduced in degree.
The Jacobian of (5) and of cy^ + babxy + ba^id^, that is, bad — 7 be 4a (4ac — 56") 2cy + 5abx baby + lOa^x will divide by a, and gives the new linear co-reciprocant
(25a6d - 32ac= + 56^) y + 50a {a'd - Sabc + 26») x. (6)
The coeflRcient of y is of weight 4, but instead of giving rise to a co- reciprocant of the 4th order, we see that this again is the anti-source of a linear co-reciprocant.
The resultant of the two linear co-reciprocants (4) and (6) divided by a numerical multiple of a gives the well-known Quasi-Discriminant 12ba'd'+ ..., as was stated at the end of Lecture XIX [above, p. 413].
The noticeable fact is that (including by + 2a:'x) there exist 3 linear independent co-reciprocants of extent 3. Probably there are no more, but this requires proof.
The promised land of Differential Invariants or Projective Reciprocants is now in sight, and the remainder of the course will be devoted to its elucida- tion. Twenty lectures have been given on the underlying matter, and probably ten more, at least, will have to be expended on this higher portion of the theory.
One is surprised to reflect on the change which has come over the face of Algebra in the last quarter of a century. It is now possible to enlarge to an almost unlimited extent on any branch of it. These thirty lectures, embracing only a fragment of the theory of reciprocants, might be compared to an unfinished epic in thirty cantos. Does it not seem as if Algebra had attained to the character of a fine art, in which the workman has a free hand to develop his conceptions as in a musical theme or a subject for painting? Formerly it consisted almost exclusively of detached theorems, but now-a- days it has reached a point in which every properly developed algebraical composition, like a skilful landscape, is expected to suggest the notion of an infinite distance lying beyond the limits of the canvas.
It is quite conceivable that the results we have been investigating may be descended upon from a higher and more general point of view. Many
42] Lectures on the Theory of Reciprocantits 421
circumstances point to such a consummation being probable. But man must creep before he can walk or run, and a house cannot be built downwards from the roof. I think the mere fact that our work enables us to simplify and extend the results obtained by so splendid a genius as M. Halphen, is sufficient to convey to us the assurance that we have not been beating the wind or chasing a phantom, but doing solid work. Let me instance one single point : M. Halphen has succeeded, by a prodigious effort of ingenuity, in obtaining the differential equation to a cubic curve with a given absolute invariant. His method involves the integration of a complicated differential equation. In the method which I employ the same result is obtained by a simple act of substitution in an exceedingly simple special form of Aronhold's /S and T, capable of being executed in the course of a few minutes on half a sheet of paper, without performing any integration whatever. This will be seen to be a simple inference from the theorem invoked under three names, to which allusion has been made in a preceding lecture and the demonstration of which .will shortly occupy our attention.
Before entering upon the theory of Differential Invariants, I think it desirable to bring fonvard the exceedingly valuable and interesting com- munication with which I have been favoured by M. Halphen establishing d priori the existence of invariants in general.
SuK l'Existence des Invariants.
(Extracted from a Letter of M. Halphen to Professor Sylvester.)
Dans des theories diverses on a rencontr^ des Invariants sans qu'on ait pent^tr^ la cause gendrale de leur existence. C'est cette lacune qu'il s'agit ici de faire disparaitre.
1. Soient A, B L des quantity auxquelles on puisse attribuer des
valeurs ad libitum.
Une substitution consiste k remplacer ces quantit^s (A, B, ...,L) par d'autres (a, b, ..., I).
Les substitutions, que Ton doit considf^rer ici, sont d^finies par des rela- tions alg^briques, de forme suppos^e donn^e, mais contenant des paramUres arbitraires p, q, ....
a=f {A,B L\ p.q, ...)]
br=f,(A,B....,L;p.q,...)\. (1)
Soit maintenant une seconds substitution, de mSme espfece, mais avec d'autres paramfetres tt, x< ■■■• et donnant lieu a (a, yS, ..., X), en sorte qu'on ait
a=/(J,5, ...,i; 7r,x, ...)]
0=A(A,B L;7r,x,...)}. (Ibis)
422 Lectures on the Theory of Reciprocants [42
2. DEFINITION. Les substitutions dont il s'agit forment un OROUPE, si,
quels que soient les paramfetres p, q nr, x ai^si que A, B, ..., L, il
existe des quantites P, Q, ... verifiant les egalit^s semblables
a=/(a,b I; P,Q,...)]
^=/,(a,b,...,l;P,Q,...)\. (Iter)
Les invariants sont I'apanage exclusif des substitutions formant groupe. On va le montrer. Mais auparavant, pour ^viter toute confusion, on doit faire une remarque sur la definition.
3. Dans les diverses theories od Ton a rencontr^ des Invariants, les sub- stitutions forment groupe, en effet, suivant cette definition ; mais il s'y rencontre encore une circonstance particuliere de plus, c'est que les paramfetres P, Q, ... de la substitution compos^e (1 ter) dependent uniquement des parametres p, q, ..., tt, ;;^, ... des substitutions composantes (1) et (1 bis). Cette propriety nest pas ndcessaire k I'existence deft Invariants, et nous ne la supposerons pas ici. II sera done entendu que P, Q, ... peuvent dependre, non seulement de j9, q tt, x,, ••■, mais aiissi Ae A, B, ..., L.
EXEMPLES :
I. a = Ap'', b = Apq+Bp, c = Aq^ + 2Bq + G; a = Ait^, ff := A'jrx+ Bit, y = Ax'+2Bx + G; a^aP*, fi^aPQ + bP, y = aQ' + 2bQ + c;
P P
P et Q ne dependent pas de A, B, C.
II. a = Ay, b = A^pq +ABp, c=^ Aq' + 2Bq + C; a = AV, 0 = A"-irx+ ABtt, y = Ax^+2Bx+C; a = a^P^, fi = a'PQ + aiP, y = aQ'+2bQ+c;
P=— ^ 0 = ^^ Ay' ^ Ap '
P et Q dependent de A.
Dans ces deux exemples, il y a un invariant absolu, ^ — .
4. Dans la substitution (1) nous supposerons que le nombre des para- mfetres soit inf^rieur au nombre des quantitds A, B, ..., L.
Soient ain.si m le nombre des parametres p, q, ...,
n le nombre des quantites A,B L,
on suppose m<n.
42] Lectures on the Theory of Eecipi'ocants 423
Cela etant, on peut eliminer les parametres entre les equations (1), et il reste (» — m) Equations
Fia,b, ...,l; A.B,...,L) = 0)
F,{a.b,...,l; A,B,...,L) = 0\. (2)
TH:6oRfeME : Si les substitutions cmisiddrdes forment GROUPE, les (n — m) equations (2) peuvent etre raises sov^ la forme
^{a,b,...,T) = ^{A,B,...,L)\ <P,{a,b,...,l)^^M,B,...,L)\, (3)
I
en d'autres termes, il y a {n — m) invariants absolus.
R^dproquement, s'il y a (n — m) invariants absolus (distincts), les substitu- tions forment groupe.
5. DEMONSTRATION. ProuvoDS d'abord la seconde partie, ou reciproque. Voici I'hypothese : des Equations (1), par Elimination de p, q, ... resultent les equations (3).
Par consequent, A, B, ..., L et a, b, .... I ^tant quelconques, mais satisfaisant aux Equations (3), on peut determiner p, q, au moyen des Equations (1).
Soient A, B, ..., L, p, q, ..., tt, x, ... pris arbitrairement, et a, b, ..., I, a, /8, ..., \ dEterminEs par (1) et (1 bis). Suivant I'hypothese, on a
^(a.b....,l) = <P(A,B,...,L) et <^(a, ^, ...,\) = ^{A, B, ..., L);
done <t> (a, b, ...,l) = ^(a, ^, ..., X), etc.
Done on peut dEterminer P, Q, ... par les Equations (1 ter), ce qu'il fallait dEmontrer.
DEmontrons maintenant la premifere partie, ou thEorfeme direct. Par
hypothese, A, B, ..., L, p, q, ..., tt, x> ■■• Etant pris h. volontE et a, b I,
a, fi, ..., X dEterminEs au moyen de (1) et (1 bis), il en rEsulte les relations (1 ter).
Des Equations (1) rE.sulte le systeme (2); de m^me, de (1 bis) et de (1 ter) resultent
F (a,fi X; A,B,...,L) = 0]
J^,(o,/9, ....X; A,B Z) = oL (2 bis)
F(a,^,...,X;a.b I) =0]
^>(3,y8 \;a.b,...,l) =o\. (2 ter)
Je dis que le systeme (2 ter) rEsulte de (2) et de (2 bis).
424 Lectures on the Theory of Reciprocants [42
En efifet, a, h, ..., I et a, /9, .... X n'^tant definis que par (1) et (1 bis),
le syst^me (2 ter) r6sulte de (1) et de (1 bis) par I'dlimination de p, q
•TT, X. ••• et A, B, ..., L. Mais I'^limination de p, q, ... remplace le syst^me (1) par le systfeme (2), celle de tt, X' ••• remplace le systeme(l bis) par (2 bis); done (2 ter) r^sulte de I'^limination de A, B, ..., L entre (2) et (2 bis).
Le systfeme (2), (2 bis) est form^ par 2 (n - m) (Equations, et cependant r^limination de n lettres A, B, ..., L, au lieu de donner (71 -,2m) Equations, en donne (n — m), les Equations (2 ter). Si done on ^limine seulement (n — m) lettres A, B, ..., G, les mautres H, ..., L disparaitront d'elles-memes. Tirons
A, B 0 des Equations (2), et nous aurons
^=^ (a.b I; H,...,L),
B = Sir,ia,b, ...,l; H, ..., L),
Tirons de m^me A, B, ..., G des Equations (2 bis), et nous aurons A=^ (<x,0,...,\; H,...,L),
Le r^sultat de I'^limination est done represent^ par (n — m) Equations telles
que
^(a.b,...,l; H,...,L) = ^(a,^,...,X; H....,L)]
^,(a.6 i; if, . ..,£) = ^,(a,/8,..., X; Zr,...,X)L (4)
et Ton sait que H, ..., L disparaissent, d'eux-mfimes, de ces Equations.
En assignant done a H, ..., L des valeurs num^riques a volont^, on voit done bien que les equations rdsultantes, equivalentes k (2 ter), ont la forme
* (a, 6, ..., 0 = *(a,/3 \),
*i(a,6 l) = <^,(a,0,...,X),
Cest ce qu'il fallait ddmontrer.
6. Remarques. Si les Equations (4) sent rationnelles, la disparition de IT, ..., L exige que ^ ait la forme suivante
^ = *(a, b, .... 1)®{H, .... L) + 0{H. ..., L),
et de m^me pour '^i, etc. Sous cette forme, on voit que 0 et 5 disparaissent dans les Equations (4), et I'invariant resultant est ^.
Mais, si lea Equations (4) sent irrationnelles, la disparition de H, ..., L peut n'^tre pas immediate. En assignant k H, ..., L des valeurs num^riques k volenti, comme on I'a dit dans la demonstration, c'est-a-diie en consid^rant H, ..., L comme des constantes arbitraires, on voit les invariants se presenter
42] Lectures on the Theory of Reciprocants 426
avec des constantes arbitraires. Ceci ne doit pas etonner, puisqu'il s'agit ici d'invariants absolua, que Ton peut efFectivement modifier en leur ajoutant des constantes arbitraires ou en les multipliant par des constantes arbitraires, sans troubler la propri^te d'invariance.
L'analyse employee dans la demonstration foumit un moyen r^gulier de former les invariants ; ce moyen consiste a dliminer les paramfetres dans les Equations (1), puis a resoudre par rapport a {n — m) quantites A, B, ..., G. Mais, les substitutions formant groupe, on peut aussi resoudre par rapport k a, b, ■.■,g, en ^liminant les parametres.
ExEMPLE: a=Ap', b = Apq + Bp, c = Aq^+2Bq+C. ^
Eb rdsolvant par rapport k c, c'est-i-dire en tirant p, q des deux premieres, on obtient
V Ap 1 Ap Ap^ A a A
B*
Voici Tin variant C — r • A
/' /B' — A0\ En r^olvant par rapport a b, on trouve b = >Ja./ ( ^ — ) +c, ce qui
B'-AC , ,. .
donne 1 invariant ^ — +c, ou c est une constante arbitraire.
LECTURE XXII. £ ptir 81 muove.
The theory still moves on. We have now emerged from the narrows and are entering on the mid-ocean of Differential Invariants, or of Principiants, as I have called them. These, it will now be seen, are perfectly defined by their property of being at one and the same time invariants and pure reciprocants. In other words, if P be a Principiant, it has both fi and V for its annihilators. Thus, for example, the Mongian
A=a^d- 2abc + 2b*
is necessarily a Principiant. For
n^ = (084 + 263. + 3c3d) (a'd - 3aic + 26*) = 0,
and at the same time
VA = {2a'db + oabd, + (6ac + 36') da} {a'd - Sabc + 2b') = 0.
Among Pure Reciprocants, those only are entitled to rank as Principiants whose form is persistent (merely taking up an extraneous factor, but other- wise unchanged) under the most general homographic substitution (see
426 Lectures on the Theory of Reciprocants [42
Lecture XIII. [pp. 379, 382 above]. We have therefore to show that such reciprocants and no others are subject to annihilation by £1.
With this end in view, let us consider the effect of substituting - . for
X and ^ for y in any rational integral function of y and its derivatives
with respect to x. Suppose that, in consequence of this substitution, the function
F(y,yi,y2,y>, ■■■yn)
becomes changed into
^ i'lCa;, y, y„y„y„ ... y„);
then the transformed function will be
^(Y, 3^1, ^2. 5^3, ••• Yn),
where X = , — j-, F= , . , and Fj, Fj, Fj, ... F„ are the successive \-\- hx \-\-hx
derivatives of T with respect to X.
If, for the moment, we agree to consider h as an infinitesimal (we shall afterwards give it a finite value), neglecting squares and higher powers of h, we may write
X=-x— ha?,
Y=y-hxy.
Hence, by n successive differentiations of F with respect to X, neglecting squares of h whenever they occur, we deduce
Yi=y^-\-hxy^-hy,
Y^ = y^+Shxy^,
Yi = y3+ 5hxy3 + Shy^,
Yt^y^+Thxyt + Shys,
Y, = y^ + Qhxyt + l^hyt,
Yn-i = yn-i + (2n - 3) hxyn-^ + (« - 1) (n - 3) %„ I^n = yn + (2n - 1) hxyn + n (w - 2) %„_,. The last of these, for instance, is obtained as follows : dF„_,
We have F„ =
dX
^°^ ~^ " Tx ^^"-' + (2« - 3) hxy^_^ + („ _ 1 ) (n _ 3) Ay„
= yn + (2w - 3) Aa«/„ + n (n - 2) %„_i.
42] Lectures on the Theory of Reciprocants 427
Consequently, }"„ = (1 + 'ihx) - ,"~'
= (1+ Ihx) {y„ + (2n - 3) hxyn + n (n - 2) %„_i} = y„ + (2n - 1) hm/n + n (n - 2) %„-,.
On substituting the above values of F, T^, F„, ... F,. in the transformed function, we find immediately
F{Y, F., F„ ... Y,,) = {l + 1iwv + h%)F{y,y„y„ ... y„),
where v and © are the partial differential operators
v = -y% + yAi + %29s„ + 5y,a„3 + 7^49^, + . . .,
Q = - y9y. + %9»3 + 8y3a»4 + ISy^Si/, + ... + n (n - 2) y„_,ay.. Changing to our usual notation, we writ*
y, = t, y, = 2a, y, = 2.Sb, y,= 2.3 .4c, ..., and then if Fi is what F (a. rational integral function of a,b, c, ...) becomes
when we substitute - — j- , ^, for x, y (regarding h as infinitesimal), we 1 + Aa; 1 + tuc
have
i', = (l + Axi;+A©)Jf,
where r = - y?, + Oj + Soda + oMj + 7c8c + 9ddd + ...,
and e = -ya«+aafc + 26a, + 3cai + 4da<,+ ....
In general v is merely the partial differential operator written above ; but when its subject, F, is homogeneous, of degree i, and isobaric, of weight w, in
the letters y, t, a, b, c, d, ... supposed to be
of degrees 1, 1, 1, 1, 1, 1, ...
and of weights -2, -1, 0, 1, 2, 3,...,
its operation is equivalent to multiplication by the number 3i + 2w. For in this case we have
ydy + tdt + ada + bdi, + cdt + ddi + ...= i, and -2ydy-at + bdb + 2cdc + 3ddi+...=w;
so that we may regard 1/ as a number, simply writing
»' = 3i+2M; when we have occasion to do so.
We are now able to show that if F \a a persistent form, we must neces- sarily have
For 5 = 1+.A^+^;
428 Lectnres on the Theory of Recijyrocants [42
and consequently, if F^ is divisible by F (this is what is meant by saying
%F that jF* is a persistent forn)), unless ^F vanishes, -^r must be a rational
integral function of y, t,a,b,c, .... But since the operation of €> diminishes
%F the weight by unity without altering the degree, —^ must be of degree 0
and weight — 1. The impossibility of the existence of such a function leads to the necessary conclusion that
Let us apply this result to the case of a pure reciprocant. We have
© = - y9t + oS* + 263. + 3cad +... = - 3/3, + n.
Thus when ^ is a pure reciprocant, or indeed any function in which t does not appear, ydtF= 0 and 0 reduces to fi. We have therefore shown, in what precedes, that the condition
£IF=0
is necessary to ensure the peraistence of the form of F under a particular homographic substitution ; d, fortiori, this condition is also necessarily satisfied when the form of F is persistent under the most general homographic sub-
i-t ..• /• I,- u u A ■ ^ Ix + my + n I'x + m'y + n' \
stitution in which x, y are changed into r^, 77 ;; , -it; jf « 1 •
V " ^ rx + m'y + n"l"x + m'y + nj
The satisfaction of ilF = 0 is of itself inadequate to ensure persistence under the general homographic substitution ; the necessary and sufficient condition of pure reciprocants
VF=0
must also be satisfied. This follows from the fact that the general linear substitution, for which all pure reciprocants are persistent, is merely a particular case of the most general homographic substitution.
It only remains to be proved that the two conditions VF=0, ilF=0, taken conjointly, are sufficient as well as necessary.
In what follows I use a method which may be termed that of composition of variations. Its nature and value will be better understood if I first apply it to the rigorous demonstration of the theorem that the substitution of X + hy for x in the Quantic
(a, b, c, ...J_x, yY
changes any function whatever of its coefficients, say
F(a,h,c, ...), into ^^F{a,h,c, ...).
42] Lectures on the Theory of Reciprocants 429
This is not proved, but only verified up to terms of the second order of differentiation, in Salmon's Modern Higher Algebra (3rd ed. 1876, p. 59). Remembering that, whatever the order n of the Quantic may be, the changed values of the coefficients a, h, c,d, ... are
a =a,
b' = b + ah,
c'-c + 2bh + ah\
d' = d + Sch + Sbh^ + ah^,
I
what we have to prove is that, for all values of h,
F(a', b', c', d', ...) = e^F(a. b, c, d, ...)• In other words, if for brevity we write
Fia. b, c, ...) = F, and F{a,b',c',...) = F^,
it is required to show that
F, = F+hnF+^ n'F+ j-|-^ il'F + . ..,
where n = adi + 2bdc + 3cda + ....
When h is infinitesimal, it is obvious that
F, = F+hClF. Hence, when h has a general value, we may assume
F. = F + hnF+^^P^^^Q + ^-^^R+....
Let h be increased by the infinitesimal quantity e ; then, considering this
increase as resulting from a second substitution similar to the first, we see
that Fi becomes
Fi + eilFj.
But it also becomes
F+(h + e)nF+ ^-^ P+ ^-^--^Q+...-F, + e^
h" ^ h'
=>,+e(ni'+AP+^2«+rlr3^ + -)-
Equating this to Fi + eClFi, we obtain
nF,=nF+hP+:^^Q + ^~^R + ....
But iiF, = n(F+hnF+^p+j^Q+...y
430 Lectures on the Theory of Reciproeants [42
The comparison of these two expressions gives
P = O^F, Q = Q,P = il'F,
R = nQ = n*F,
Substituting these values in the assumed expansion for F, , there results F, = F + hQ,F-\ which is the expanded form of
F, = F + hnF+:^il*F+^-^n'F +
Fi = e^°F.
A similar method of procedure will enable us to establish the corresponding but more elaborate formula
fi = (l+/w;)-ei^**jr,
in which F is any hoviogeneous and isoharic function* of degree i and weight w in y and its modified derivatives {t, a,b,c,...) with respect to x; the operator & = — ydt + adb + 269^ + 3cdd + ■■.; the function Fj is what F becomes
in consequence of the substitution of z — j— , - — ^^ for x, y; h is any finite
quantity, and v = 3i+ 2w.
Before giving the proof of this theorem, I will show that, upon the assumption of its truth, two inverse finite substitutions will, as they ought, nullify each other, leaving the function operated upon unaltered in form.
To avoid needless periphrasis, we call the substitution of z — -r- , , , _ *^ 1 +hx 1 +hx
for X, y the substitution h.
Either of the two substitutions, h, — h, reverses the eflfect of the other ; for the substitution — h turns
■•K . ^ X , hx
into r; 7 — H1+:, r- = a'.
\+hx l-hx' 1-hx'
A y • ^ y . -, ^^
\ +]ix 1 — hx' \—hx~^'
The two substitutions h, — h, performed successively on F, ought there- fore to leave its value unaltered. But by hypothesis the substitution h converts F into F^; consequently the substitution -h performed on Fi ought to change it back again into F.
F need not be integral or even rational ; whenever it is homogeneous or isobaric, v will be a number.
42] Lectures on the Theory of Redprocants 431
It must be carefully observed that (since the operation of B decreases the weight by unity, leaving the degree unchanged) the weight of %'F is k units lower than that of F, whilst the degree is the same for both.
Thus for F we have 3i +2w = v,
and for B'F Si + 2{w -k)=^v- 2k.
Hence the substitution — h, which changes
F into {l-hxye ^-'"'F,
also changes ^F „ {I — hx)'^-e'^-'"^&F,
&F „ (1-hxy-^e 1-A:r@2if,
I
and in general ©"l" into (1 - Aa;)'~"e i-'^'&'F.
Moreover, 1 +hx becomes 1 + :; j- = (1 — ha;)~^, so that
1 — fia;
(1 +hxy-''^F becomes (1 - A«)-(-') (1 - hxy-^e i"** S'F
Ae = {l-hx)-'e i-'^&'F
= e i-*« (1 _ fuc)-'&'F (since 6 does not act on x).
Consequently, (l+hxyF becomes e ^-i^F,
Ae {\+}ucy-^%F „ e ^-1" {l-hx)-'BF,
Ae (1 + hxy-^^F „ e~»-** (1 - hx)-^^F,
And since, by the formula to be verified,
F^ = {l+hxyF+h{\ + ha!r-^%F+^{\-\-hxy^B^F+...,
A«
jP, becomes e *"**
1 + A(1-M~'Q + j^(1-M-'0'+---[j!^
A» Ae _g \-kx gl-hx p ^ p^
432 Lectures on the Theory of Reciprocants [42
LECTURE XXIII.
We now proceed to show how the composition of variations can be made to furnish a strict proof of the formula
F, = (1 + hxye^+^F,
which was set forth in the preceding lecture.
As before, calling the change of x, y into ^-^7— . , \ , the substitution
h, it is easy to see that the product of two substitutions, h, e, is the substitution A + e. For
X - X X
H-l+€
\+hx ' 1+hx \+{h + e)x'
\+hx' \-\-ha; l+{h + e)x'
This shows that if
^j is what F becomes on making the substitution h, and Fi „ ^1 » .. » „ e,
then F^ „ F „ „ „ „ h + e.
Thus we can find two expressions for F^, the comparison of which will enable us to assign the coeflScients of all the powers of h in the expanded values of F,.
The first two terms of this expansion were obtained, in the preceding lecture, by treating A as an infinitesimal. We may therefore write
F, = F + h{vx + %)F+^^N,+-^^N, + .... Changing h into A + e, we deduce
F,= F^{h+e){vx + %)F+^^-±^N, + ^^^^^N,+ ....
For greater simplicity, let e be an infinitesimal, and write
^^l^iil=^F,.
Then £i.F, = {vx+%)F+hN, + ^N, +
Now look at each term in the expansion of F^ and find its increment (that is, its A) when x, y undergo the substitution e. We thus obtain
Ai?-, = Ai?- + M (va; + @) ^ + j^ Ai\^, + j-^ AiVr, + . . ..
42] Lectures on the Theory of Reciprocants 433
Comparing these two values of AFi, we find
N, = AN,,
and generally N'r= ANr-i.
These equations are sufficient to determine all the coefficients of F^ ; it only remains to show how the operations A may be performed.
We have in fact
F, = F + hAF+ ~A'F+ — ^ A»^+ ....
where A^ = (vx + 0) F.
But we must not from this rashly infer that
A«F = {ix + SyF.
To do so would be tantamount to regarding i^ as a constant number, whereas its value depends on the degree and weight of the subject of operation.
T This will be clearly seen in the calculation which follows*. We first
generalize the formula
/^F={vx + e)F
by making S'F the operand instead of i^.
Then, since i is the degree and w — k the weight of ©"F, instead of
3t + 2w = V,
we have Si + 2{w — k) = v— 2k.
Thus, AB'i' = [{u -2k)x + 0} <d'F.
Again, since Ax = ( :; x] -i- e = — x^,
II \l + ex J
me find [ax>^%'F=>ji^-'S'F. Ax + x>'A&'F= - \x''+'S^F+ x^ [{p -2k)x+ 0) &'F. Hence we obtain the general formula I Ax^e'F=x>'{(v-2K-X)x + (d]e'F,
I * If oar sole object were to show that QF=0 is a saflicient as well as necessary condition of Bie persistence of F, we might dispense with all further calculation. Thus it is obvious that, since AF^irx + O) F, A"F must be of the form (x, 9)" F ; for the dependence of y on the degree- weight of the operand will not affect the form of A", but only its numerical coefficients. Hence we conclude that Fj is of the form 0 (x, 6) F ; and remembering that (i'^F=0, &>F=0, ... when- ever 6^=0, it is at once seen that not only (as was shown in the last lecture) must QF vanish when F is persistent under the substitution A, but, conversely, that when eF=0, the altered value of F contains the original value as a factor (the other factor being in this case a function of X only) ; that is, F is persistent.
8 IV. 28
434 Lectures on the Theory of Redprocanta [42
by means of which we calculate in succession the values of A'^, A'f , .... Thus,
^vx{{v-\)x + %]F+\{v-2)x + ^^F ■ = [u(v-l)iie'+2(v-l)x& + ^^}F. Hence A»^ =v{v-l) ^x'F+ 2 (i/ - 1) AxSF + A©''^
= v(v-l)x'{{v-2)x + %}F+2(v-l)x{(v-3)x + id]SF
+ {iv-4)x + @]e'F = {v{v-l)(v-2)a^ + S{p-l){v-2)x'S + 'iiiv-2)xe^-i-&'}F. If [i/J^is used to denote j'(«' — l)(i' — 2) ... to Ji factors ([v]' will of course mean v), we have shown that
AF={[vyx + B)F,
A'F = ([vy x^ + 2[v- 1]' xe + 0») F,
A^F = ([vf x' + 3[v-l]'x'@ + S[v-iyx@'' + &>) F, and by induction it may be proved that in general
^»F= |[^]»a;» + m [v - l]"-'a;"-' @ + "'\'~J' [" " 2]»-'a;"-»0*+ ... + B"! 2*'.
That the last term of this expression is &^F is sufficiently obvious ; what we wish to prove is that, when m is any positive integer less than n, the term in A"j^ which involves @"' will be
n(n — 1) ... (n — m + 1) r -u.^^ „ ™,a-.rT
1.2.3 ... m ■■ -"
To find the term involving ©"" in A""*^'^, we need only consider the operation of A on two consecutive terms of A^F; none of the remaining terms will affect the result. Suppose, then, that
A"^ = . . . + pa;"-'" ©""F + ^3^-"*+' ©"'-'i?' + . . . . Operating with A, we find
A"+'i' = . . . + p Aa;"-" ©"■ F + g^ajn-m+i ©m-i 2?" + . . . = . . . + pa;"-™ {(f - 71 - m) a; + ©} ©™F
+ 5'a;»-"'+4(j/ - n - m + l)a; + ©) @'»-^F+ ... = ... + {p(j'-7i-m)+g'}a;»+i-™©"»F+ .... Now, assuming the general term of A^F to be as written above, we have n(n—l)...(n — m + l)r
Ji(n — 1)... (n-m + 2)r n„_™j.,
^ 1.2.3...(n.-l)-^''-"^+^^""""'
., . {m(v~m+ 1)1
so that </ = » s — ^^ -t .
^ ^] n-m + 1 ]
42] Lectures on the Theoi'y of Reciprocants 435
Thus the general term of A^+'-F has for its numerical coefficient
{m(v—m + \)-ir{v — n — 'ni)(n — m + V\] pi.-n-m) + q=p^ ___ 1
■^ ( «-m+l J 1.2.3...m '• -"
which shows that the numerical coefficients in A""'"'jP' obey the same law as those in A"^; and as this law is true for n = 1, 2, 3, it is also true universally.
We have thus shown that the general term in A" J' is
n(n— 1) ... (n — m+ 1) r n„ „ „_™/-»™rr
1 .2.3 ...m ^ -■
and, consequently, the corresponding general term in
is — J^^; -A»-"a;"""
1.2.3...n 1.2.3...(n-m) ■l.2.3...m'
Now, as we have already seen,
F. = (l + AA4.^^A'+^-^A»+...)^,
which, by merely expressing the symbolic factor as a series of powers of 0, may be transformed into
F, = (l + [vjhx + fj| h?:^ + ^J-^ A'^ + . . .) ^ " + (l + [y - l]'/u; + 1^" ~ }^* h'ai' + p^ h'x' + ...)hSF
« +
where, remembering that [v]" stands for v{v —l)(v — 2)... to n factors, it is evident that the functions of x which multiply F, hSF, — . &^F, . . . are all of them binomial expansions. Hence we immediately obtain
F, = {1+ hxYF + (1 + hxY-' h^F + (1 + /ub)-' :f^ e«J?' + . .. = (1 + Aa:)'|l + (1 + A«)-iAe + (H- Aa:)-«^'j^+ ...I .y,
Ae
and finally, .F, = (1 + hxYe^+'^'F.
Mr Hammond has remarked that, with a slight modification, the foregoing . demonstration will serve to establish the analogous theorem, that
Ma. /•,=(! +A<)-''e'»+*'F,
28—2
436 Lectures on the Theory of Reciprocants [42
where, as before, F means any homogeneous and isobaric function of degree i and weight w in the letters y, t, a, b, c, ... ; and Fj is what F becomes when, leaving y unaltered, we change x into x + hy, where h is any finite quantity. Instead of the operator
e = -ydt+c^b+2hSc + ScSd+ ■■■ =-ydt + il we have - F, = ytdy + t^t - 2a»aj - 5a6a„- . . . = yO^ + «»8, - F* ;
and instead of i/ = Zi + 2w, a different number, /x = 3i + w (which I have called the characteristic), taken negatively.
If we suppose that
Fi is what F becomes on changing x into x + hy, and Fi „ F „ „ „ x „ x + ey,
then F^ „ F „ „ „ x „ x + (h + e)y.
Hence, if F, = F+hP + ^^Q + ^^R + ...,
we must have F, = F+(h+ e)P + ^-^'j^Q + ^^'^ R+ ...
Thus, if € be regarded as infinitesimal, and we write
F,-F,
6
= ^F„
it follows that ^Fi = P + hQ+ ^ R+ ....
But, by the direct operation of A, we find
AF = AF+hAP + ^AQ + ...,
and, comparing these two values of A^i,
P = AF, Q = AP = A'F, i? = AQ = A'F,
Hence it follows that
F, = F + hAF+J!-^ A^F+^J^ A»i^+ ....
* This theorem was stated without proof in Lecture VIII, where, through inadvertence, the term ytd, in the expression for Fj was omitted [p. 352, above].
42] Lectures on the Theory of Reciprocants 437
It remains to find the value of A"^. This can be effected by means of formulae given in Lecture VIII. [p. 350, above], where it is shown that
Aa; = y,
Ay = 0,
A< = - 1\
Aa = — 3a<,
A6 = - 46< - 2a=,
Ac = — bet — bah,
AcZ = - Qdt - 6ac - 36',
Ae = - let - lad - 76c,
We now show that
AF=-i^t+V,)F,
where Fi = F - t'dt - yfdy ,
just as in the cognate theorem we had
Since ^ is a function of y, t, a, b, c, ... without x, it is evident that
A r» dF . dF . , '^=Ty^y^di''' + -
= -t(idt + Sada + ibdb + 5cd,+ ...)F - {2a^di + 5abd, + (6ac + W) da + ...}F, where the part of A^ which is independent of < is — VF.
Now, ydy + tdt + ada +63j +c3e +...=i
and -2ydy-tdt +bdb +2c3c+...=w;
so that fdt + 3ada + 4:bdi + 5cdc+ ... = 3i + w-ydy-tdt.
Hence, writing Si + w = fi,
^F=-t{(j.-ydy-i^t)F-VF = -(nt+V,)F. where V,= V- fdt - yfdy.
Observing that V^'F is of degree t + « and weight w — k; since 3 (i + «) + (w - «) = /x + 2/c, we see that A V,'F= - [{/jl + 2/e) « + F,} V^'F.
Again, At* V^'F= \t>^-' V,'F. A< + <* A F.'F
= - \<*+' F.'i'- <*{(/* + 2k) < + F,) V,'F. We thus obtain the formula
A<* F.' /^ = - <* {(/t + \ + 2*:) < + F,} F" iF', (1)
analogous to the one previously employed,
£^^F =a^\{v-2K-\)x + &i e'F. (2)
48B
Lecturer on the Theory of Reciprocants
[42
The remainder of the work will be step for step the same for this as for the previous theorem. In fact, by using (1) just as we used (2), we shall deduce
F, = il + ht)->^e ^+''*F, (3)
just as we deduced the analogous formula
i', = (1 + lixY 6^+** F. (4)
The reason of this is obvious : by interchanging x and (, n and — v, 0 and - Vi, we interchange the formulae (1) and (2), (3) and (4).
It may be well to observe that if we use S^ to denote a substitution of such a nature that
and if (regarding e as an infinitesimal) we write
6
then in general SkF=^'^F.
The proof of this proposition is virtually contained in what precedes.
LECTURE XXIV.
Whenever a rational integral function of x, y, t, a,b,c, ... is pei"sistent in form under the general linear substitution, it cannot contain explicitly either x, y or t, but must be a function of the remaining letters a, h, c, . (the successive modified derivatives, beginning with the second, of y with respect to x) alone.
For if, keeping y unaltered, we change x into x + a, where a is any arbi- trary constant which may be regarded as an infinitesimal, the derivatives t, a,b,c, ... are not affected by this change, and consequently the function
dF F= F{x, y, t, a, h, c, ...) becomes F + a-j- ,
which cannot be divisible by F unless -,- = 0.
•' dx
(The alternative hypothesis of -j- being divisible by F is inadmissible, because i** is a rational integral function.)
Hence .F cannot contain x explicitly; and if we write y + /3 for y, keeping X unchanged, we see, in like manner, that F cannot contain y explicitly.
42] Lectures on the Theory of Reciprocants 439
Again, if in the function
F = F{t,a,h,c,...) we change x, y into x + a, y + ^po + /3, the efifect of this substitution will be to increase t by the arbitrary constant /9,, without altering any of the remain- ing derivatives a,h,c, ....
Hence, in order that the form of F may .still be persistent, we must have T- = 0 ; the reasoning being just the same as that by which -j— was seen to vanish. Thus, F does not contain t explicitly. Moreover, the function
F=F(a,h,c,...) must be both homogeneous and isobaric.
For the substitution of aa;+a, /8„y + /3,a; + ,S for x, y, respectively, will multiply the letters
a , 6 , c , d , ...
by )8„«,-'. A.«r'. A,«^^ /8„ar°. ••••
Each term of F will therefore be multiplied by a positive power of /3„ and a negative power of a^.
Let one of the terms of F be a^b'^'C^d^' It will be multiplied by
Q A,+A, + A,+X,+ ... J, -(2A,+3A,+4\,+fSA,+ ...)_
In order that F may retain its form, this multiplier must be the same for every term of ^, no matter what arbitrary values are assigned to o^ and 0,,. This can only happen when, for all terms of the function F, we have
X, + X, +X3 + Xj + ...= const.
and \, + 2X5 + 3X3 4- . . . = const.,
that is, when F is homogeneous and isobaric.
We have thus proved that among all the rational integral functions of X, y, t, a,b,c, ... the only ones persistent under the substitution of a + a,a;, /8-l-y9,a; + y9,,y for x, y, respectively, are such as simultaneously satisfy the conditions of not explicitly containing x, y or t, and of being homogeneous and isobaric in the remaining letters a,b,c
If F, any function satisfying these conditions, merely acquires an extra- neous factor when, leaving y unaltered, we change x into x + hy, the form of F will be persistent under the general linear substitution. For both a + a,(x + hy) and ^ + ^,{x+hy) + fi,,y are general linear functions of X, y, 1.
Now, the change of a; into x+hy converts (as was shown in the preceding
lecture) F into
Ml F, = {l + ht)-''e'^+>''F,
where F, = F - f 8, - y^y.
440 Lectures on the Theory of Reciprocants [42
But, since neither y nor t occurs in F, we must have
'OyF=() and a«i'=0. Consequently, V,F=VF, V^F=y'F.
and so on. Hence
kV
= (1 +AO-'"^-(l + A<)-''-'AF'F+(H- AO-'^-'^V- ... .
Unless VF, V^F, VF, ... all of them vanish, F^ cannot contain F as a factor. If it could, VF, VF, ... would all have to be divisible by F. But this is impossible ; for VF, a rational integral function of a, b, c, ... whose weight is w — 1, cannot be divisible by F, a rational integral function of weight w.
We must therefore have
VF==0
(which implies V^F<=0, etc.) as the necessary and sufficient condition of the persistence of the form of F under the general linear substitution. In other words, F must be a pure reciprocant.
In order that F may also be pereistent in form under the general homo- graphic substitution, it must (besides being a pure reciprocant) be subject to annihilation by the operator
il = adb + 2bdc + Scda+....
For it was seen, in the preceding lecture, that the special homographic
substitution in which ~ — v-, , are written instead of x, y, respectively,
has the efifect of changing any homogeneous and isobaric function F into F^, where
.F, = (l + Aa;)V+**i!',
® = £l-ydt.
When the letter t does not occur in F, we may write dtF = 0, so that 0 becomes simply fl, and the above formula becomes
no
F.^il+hxYe^-'^'F.
Hence it follows immediately that, when F is a rational integral function of the letters a, b, c, ..., the condition D,F=0 is sufficient as well as necessary to ensure the persistence of the form of .F under the special homographic substitution we have employed.
But when jf is a pure reciprocant it also satisfies the condition VF=0, and it is the simultaneous satisfaction of [IF — 0 and VF = 0 that ensures
42] Lectures on the Theory of Reciprocants 441
the persistence of the form of F under the most general homographic substi-
tution. This may be shown by combining the substitution - — 7- , , , (for
which ^is persistent when, and only when, n^=0) with the general linear substitution (for which VF = 0 is the necessary and suflScient condition of the persistence of the form of F), so as to obtain the most general homo- graphic substitution. Thus the linear substitution
x=lx^ + my^ + « 1
when combined with
gives the substitution
_ lx„ + my„+n{l + hx,) -^
_ I'x,, 4- my, + «.'(! + hx^) y~ l+hx^,
in which both the numerators are general linear functions.
By combining the substitution just obtained with the linear substitution
^/, = ^//, + h-Vn, + ". y,. = «/«/. the denominator of each fraction is changed into a general linear function,
and thus, by combining the special homographic substitution = — 7- , ^,
with two linear substitutions, we arrive at the most general homographic substitution.
This proves that the necessary and sufficient condition of F being a homographically persistent form is the coexistence of the two conditions
VF=0, nF=o.
Thus a Projective Reciprocant, or Principiant, or DiflFerential Invariant, combines the natures of a Pure Reciprocant and Invariant in respect of the elements.
Notice that every Pure Reciprocant is an Invariant of the Reciprocal
d^^x d'u
Function (that is, the numerator of the expression for -j-^ in terms of ~,
■j—, ..., or what is the same in terms of the modified derivatives t, a, b, ...),
but the elements of such invariants are not the original simple elements, but more or less complicated functions of them.
What has just been stated is obvious from the fact that all invariants of the "reciprocal function" have been shown to be pure reciprocants (vide* Lecture XIX.). The ordinary protomorph invariants of this function will
[* above, p. 412.]
442 Lectures on the Theory of Recijrrocants [42
have for their leading term a power of a multiplied by a single letter. Con- sequently, by reasoning previously employed in these lectures, every pure reciprocant will be a rational function of invariants of the Reciprocal Func- tion divided by some power of a. Thus, for example, the Reciprocal Function
Ua* - 21a»6« + 3 (2ac + 6^) <» - d<» = (a, /3, 7, 8^1, - tf
if a = 14a«, j8 = 1a% 7 = 2oc + 6», 8 = rf.
The two protomorph invariants of this reciprocal function are
ay-^ = 7a* (4ac - 56»)
and a'S - 80/87 + 2/3' = 1 96a« (a'd - Sabc + 26»).
All other pure reciprocants of extent 3 may be rationally expressed in terms of a and the two protomorphs 4ac— ofr*, a^d — Sabc + 2b^; that is, all pure reciprocants of extent 3 are invariants of the reciprocal function of extent 3.
The reasoning employed can be applied wilh equal facility to the general case of extent «.
Instead of r- . ■( fi > 1®* "^ consider the special homographic sub-
1 V
stitution - , - employed by M. Halphen.
Z = - and F=^,
X X
Writing
let Fi, Fj, Fj, ... denote the successive derivatives of F with respect to X, and y,, y,, y„ ... those of y with respect to x. Then
F= fl;-'y,
Fj= a^syj,
3
F. = - a;- (^2/, + - y, + — y, + -^ y, j ,
Hence, if a, b, c, d, ... are the successive modified derivatives (beginning with the second) of y with respect to x, and a', V, c', d', ... the corresponding
42] Lectures on the Theory of Reciprocants 443
modified derivatives of 1" with respect to X, it follows immediately that a = a?a,
/ 2 1 \ c'= x'{c + -h + -a], \ X x^ I
\ X a? a? I
Attributing the weights 0, 1, 2, 3, ... to the letters a, b, c, d, ..., it is very easily seen that if F is any homogeneous and isobaric function of degree i and weight w,
F(a',b',c', ...) = (-)«'«"+««/'('a, 6 + ia, c + -6 + ^a, ...V
But we proved (in Lecture XXII.) [above, p. 429] that for all values of h F{a,b + ah, c + 2bh + ah\ ...) = ^°F (a, b,c,...).
Hence, making h = - , we obtain
Q
F{a', b', c', d', . . .) = {-ra^+^^F(a, b, c, . ..). which proves that the satisfaction of
Q.F{a,b,c. ...) = 0
is the nece.ssary and sufficient condition for the persistence of the form of F
1 V under the Halphenian substitution - , - .
'^ XX
Similarly we might prove that F(y, t, a, b, c, ...), which contains y and t,
1 y . but not X, is changed by the substitution - , - into
e (-)^x'e'Fiy,t,a,b,c, ...).
where 0 = — 2/9, + a96 + 2Wc+ ... = fi - y3t;
or we may deduce this result from the formula, demonstrated in the preced- ing lecture of this course,
Fr = (l+hxye^+>^F,
x in which jP, is what F becomes in consequence of the substitution , ^-.'
Y^^T impressed on the variables.
444 Lectures on the Theory of Reeiprocants [42
Let 1 be the degree and a the weight measured by the sum of the orders of differentiation in each term of
F(t/,t,a,b,c, ...)•
If we measure the weight by the sum of the orders of differentiation of every term of F diminished by 2 units for each letter in the term, then
w= (a — 2i and 2a) —i = 3i + 2w = v.
Let F{y. t, a, b, c, ...) become F' (y, t, a, b, c, ...),
when we change
x into qx+p and y into ry ;
then F'Q/, t, a, b, c, ...)=^r^q-''F(y, t, a, b, c, ...).
X y
A further substitution _-t- , ^ , , impressed on the variables in F',
will convert the original variables into
, ^ . + p and _ , , 1+hx ^ 1+hx'
,,.... p(l + hx) + qx , ry
that 18, into ^--^ — ; i ^ and :; — ^v--
1+hx 1+hx
The function F' is at the same time changed into
r*?-" (1 + hxye^+'^Fiy, t, a, b, c, ...). If now, in the above, we write p = h, q = — h^, r = h, we shall have changed
the original variables x, y into , , ^^Y ' ^°^ *^® original function F
into
»«_ _*?L /l+hx\'' **
A* (- A')-*'(l + hxye^+>^F = (-)"/i<-=" (1 + Itx)' e^ *"' F = (-)" (^- ] ei+** F.
h hv fl+hxV **
Let h become infinite; then r-, , ^. and (-)"'( — ; — 6^+*-' F
1+hx 1+hx \ h J
1 V - 1 V
become -, - and (—yx'e'F, showing that the substitution -, ^ changes F
e into (-)" af^F.
42] Lectures on the Theory of Reciprocants 445
LECTURE XXV,
In a letter to me dated June 14th, 1886, M. Halphen calls forms which
1 V are persistent under the substitution - , - , Invariants d'homologie. He uses
the letters
to denote y and its successive modified derivatives with respect to x\ and, supposing them to become
A„, Ai, ^2. ■"3. ••• ■"n>
1 « . . in consequence of the substitution - , - , gives, in the briefest possible man-
ner, two very ingenious proofs of the formula
/ w « , ( «-2 (n-2)(»i-3) )
from which he deduces the theorem that the substitution in question changes any homogeneous and isobaric function y, of degree i and weight « in
flo, Cfj, Oj, Oj, ... On,
e into F =^ {-)" aP^' ^- f,
where © is the partial differential operator
— a<i9o, + a,9aj+ 2as9„, + ... +(»- 2)a„_,9a,. I give the two proofs mentioned above in M. Halphen's own words, adding occasional footnotes, and making slight changes in the literation of his formulae when it seems desirable to do so.
Soient X = -, F=^.
X X
Par une fonnule connue (Schlomilch, Compendium II.)
* An easy indnctiTe proof of this may be obtained as follows : Since ^= _ x» 1 we have f ^ = - ^» ^^ (g) .
Hence, assmning the trath of the formula when n=K, we find
= ( -)««x» j««+>^^i^(x'->j,) + (« + l)x«^(:r«-j,)|
Thns, if the formula is true for n = it, it will be equally so when n = /c + l. But it is obviously true when n = l (when it becomes tj= -** j j • and therefore holds universally.
446
Lectures on the Theory of Redproeants
[42
et puisque il en resulte
F = Zy,
, ,. ^,( n(n-2) n(H-l)(w-2)(n-3)
Si Ton pose il vient
1.2.0!'
d"F
71-2 (w-2)(n-3)
1.x
l.l.af
a„_3+ ...|- ,
(I)
Soit
S/
0/=S(n-2)a^,g^t
on aura 8a„ = (m - 2) a„_i ,
0»a„=(n-2)(«-3)a„_j,
^„ = (- 1)»^"»-' |«« + i^^«'. + Y7^i®'«»+ •••} •
Par consequent, pour une fonction contenant Oo, a^, <h, •■■, de degrd i et de poids w, k chaque terme, on aura
2^=(-l)»';.^"-'|/+j^@/+j— 2-^0'/+...|j. C.Q.F.D.
* For, expanding by Leibnitz's Theorem, ^(«--'y)-n^,'.\(:t»-=y) = x''-»y„ + n(n-l)x«-»j/„_, + ":^^'(n-l)(n-2)a:»-»j(„_,+ ...
- n {x"-" j/„_i + (n - 1) (n - 2) i»-'t/n-s + ■ ■ • }
t The summation extending to all positive integral valaes of n, from 1 to co , so that e = - a^d^^ + 02 3„^ + 2a3S„^ + ^a^d^^ +....
Bemembering that Halphen's a^, a^, a^, 03, ... have the same meaning as our y, t, a, b, ..., this operator is - y 3, + a3(, + 263,. + 3c8j + ... identical with the G used in previous lectures.
X We may show without much diflSoulty that, when 9i, 62, 63, ... are each of them equivalent to e, but 9i acts on u only, 6-2 on v, 63 on w, and so on, Guwic ... = (81 + 62 + 63+ ...jufi" .... From this it can be deduced that G^uvw) ... = (9, + 62 + 63+ ...)"ut)«! ..., when k is any positive integer. Now let the number of the functions u, v, w, ... be i, and suppose that
« = n„, v = ap, w = aj, ...; suppose, also, that the weight n+p + q+ ... = u. Then
A^ApA,... = {-)''x''
■•Ve* a„/ \ex a,,/ \ex <(<,/...=
„ . i(e,+e,+e,+...) (-)"a;-"-'e» an«P«a
(for by what precedes 61 + 62 + 63+ ... may be replaced by 6). Taking a^apaq ... and A„ApAq ... to be corresponding terms of/ and F, we see at once that
F=(-)''x^''-*ez/.
42] Lectures on the Theoi'y of Recipi-ocants 447
Autre Demonstration de la Formule (I)' Si Ton change X et a; en X + fT et a; + /t, on a
X{x + Hy
Maintenant la formule
y = ao + hai + h^a^+ ... +A"a„ + ...
ecrite symholiquement^r
1
y~\-ah
devient y =
X^ + if (JC + ct) ■
D'ailleurs Y = {X+H)y;
done symboliquevient
X(X + HY X' + HiX + a)' ^^
Si Ton d^veloppe le second membre (II) suivant les puissances ascendants de H, le coefficient de H" est An. Or ce d^veloppement est
— <-)-©"Mr(i)'--j
* If z becomes x + h in consequence of the augmentation of X by an arbitrary quantity H, the increment of x will not be a constant, but will depend on X as well as on H. The
value of h may be fonnd at once by eliminating x between X = - and X+H= = , when we
X X + n
Y ZT
obtain X + H=:- — j-^, and consequently h =
1 + hX' -^ "•' " X{X + U)
This increase of X also changes y and Y (functions of x and .Y, whose original values were Og and Ag before the augmentation of X took place) into
and into Y=A(, + HAi + IPAi+ ... + H''A^+ ....
These altered values of y and Y are the ones used in this second proof ; the other letters retain
their orii^inal signification.
+ The word symholiquement indicates, whenever it is used, that powers of a are to be replaced by suffixes of corresponding value. For example, in the final result
Ib to be replaced by /<„= (-)"!'*"' ( a„-i <'n-i+ •■• I •
In our notation the final result is A^.^^= ( - )"x*'+3 la,b,c,d,...\) , 1 j .
448 Lectures on the Theory of Redprocants [42
done symboliqiiement
ee qui est justement la formule (I).
We may regard the coefficients a, 6, c, ... of the ordinary binary Quantic
in u, V,
(a, b, c, ...$«, vY,
as the successive modified derivatives, beginning with the second, of a new variable y with respect to another new variable x.
Any invariant / of this Quantic will then retain its form unaltered, or at most merely acquire an extraneous factor, if
(1) leaving x, y, v unaltered we change u into u + \v,
(2) „ u,v „ „ „ x,y ,. ^,^^,
1 y
(3) „ u,v „ „ „ oc, y „ - , - , where \ and h are arbitrary constants.
For we have seen that these three substitutions will severally convert
anv homogeneous and isobaric function F, of degree i and weight w in the
letters a, h, c, ..., into
ha a
e^^F, {l+hxye^'^'^F, and {-Ya^e'F,
where, in each case, fl = oSj + 269c + 3c9ri + . . ., and i/ = 3i + 2w. From our point of view an invariant is defined as a homogeneous and isobaric solution
of the equation
07 = 0.
Hence the above substitutions convert the invariant / into
/, {l+hx)"!, and {-fx"!, respectively.
An absolute invariant with respect to any substitution is one which, dis- regarding its sign, remains unchanged in absolute value by that substitution. Thus, any invariant for which
1/ = 3i + 2w = 0
is an absolute invariant with respect to each of the three substitutions here considered.
An invariant is of odd or even character with respect to any substitution according as its sign is or is not changed by that substitution. Thus,
1 V invariants are of odd or even character with respect to the substitution - , -
^ . XX
according as their weights are odd or even.
42] Lectures on the Theory of Reciprocants 449
This corresponds to the theorem that the character ( with respect to the interchange of x and y) of a pure reciprocant is odd or even according as its degree is odd or even [p. 316, above].
From any two invariants for which v has the same value we can form an absolute invariant (that is, one for which v=0) by taking their ratio, and then by differentiating the absolute invariant thus formed obtain another invariant.
Suppose Jj to be an invariant of degree I'j and weight w, ,
•'•2 )> » „ ^ ,) Wj,
and let 3ii + 2wi = I'l, 34 + 2wj = v<i ;
then the v for I'^ is the same as that for I^, and consequently 1^*1 ^~'' is an absolute invariant.
We proceed to show that j-(/i'«/a"''') is an invariant, though not an
absolute one.
Using accents to denote differential derivation with respect to x, we have
If, then, we can prove that i/^//, — Vj/i/j' is an invariant, it will follow
that T- (/,'«/,"'•) will be one also, and the proposition will be established.
It may be very easily shown that this is the case by using Cayley's generators P and Q. For [p. 327, above], / being any invariant of degree i and weight w, PI and QI are also invariants where
P = o(6Sa+c36 + c©c + e3d+ ...)-ib, and Q = a(cdb + 2ddc + 3ad + ...)-2wb.
Hence (3P + Q) / is an invariant.
Now, since 369o + 4c36 + 5ctec+ ... = t- ,
and Si + 2w = v,
(3P + Q)/ = o(3W„ + 4ca6 + 5da. + ...)/- (3i + 2w)bl = al' - vbl.
Consequently a// — vjblj and o/j' — v^i/j
are both of them invariants. Hence the combination
vJt {al,' - vMi) - vJi (all - vj)h) = a {vj.'l^ - vJJ,) is also an invariant; that is
is one ; which is the theorem to be demonstrated.
8. IT. 29
450 Lectures on the Theory of Redprocants [42
The invariant al' — vbl, which we generated from /, is of degree i+ I and weight w + 1 ; its i* is therefore the original v increased by 5 units, three for the unit increase in the degree and two for the unit increase in the weight. Hence, on repeating the process of generation, we obtain the invariant
ja^-(. + 5)6
(a/' - vbl) = a"/" -2(v+l)abr- ivaci +1/(^+5) ¥1.
By adding on the invariant v{v-{- 5) {ac — 6') / and dividing the sum by a, the above invariant is reduced to
al" -2(v + l)br + v{v + 1) cl, which is an invariant of lower degree by unity than the unreduced form.
The results obtained above may be compared with the corresponding ones in the theory of reciprocants.
Thus to the invariants / (deg. i, wt. w),
al' - vbl, vJi'Ii — vjil^', ar'-2{v+l)br + v{u + l)cl, where v = Si+ 2w,
correspond the reciprocants
R (deg. i, wt. w), aR'-fj.bR,
fiiRi'Iii — fj^RiRs, 5aR" -o(2fi + l)bR' + 4!fi{fi-l) cR, where /i = 3i + w.
Defining a plenarily absolute form to be one whose degree and weight are both zero {i = 0, w = 0), the theorem I shall now prove may be stated as follows :
By differentiating a plenarily absolute principiant we obtain another jnnncipiant.
Let P be any principiant of degree i and weight w. Then, by what pre- cedes, since P is both an invariant and a reciprocant,
a -j vbP is an invariant,
ax
and a -^ fibP is a reciprocant.
Hence, when v = 0 (that is, when 3i + 2w = 0),
dP. . . ^ -J- is an invariant,
and when /t = 0 (that is, when 3i + w = 0),
When both /t = 0 and v = Q (which happens when i =0, tv = 0),
dP . ^ ^ , . .
-J— is both a reciprocant and an invariant ;
, . dP . ...
that IS, -T- IS a principiant.
I
I
42] Lectures on the Theory of Reeiprocants 461
LECTURE XXVI.
In the theory of Invariants the annihilator fl has two independent reversors any linear combination of which will also be a reversor. To each of these reversors there corresponds a generator for invariants. Thus Cayley's two generators
a{hda-\-cdi, +ddc + edd+ ...)—{b,
a(cdb+2ddc + S^a +...)-2wb,
correspond to the two reversors
bda + cdb +ddc + edci + ...,
(0i + 2ddc + 3edd +....
The only linear combination of these which does not increase the extent j as well as the weight of the operand is
0=jbda + (j-l)cdi + {j-2)dd, + ....
It is convenient to take this for one of our reversors, and for the other
^ = 3bda + icdt + 5ddc + ...,
which is a reversor to V, the annihilator for reeiprocants, as well aa to il, the annihilator for invariants.
We saw in Lecture XL [p. 364, above] that when F is any homogeneous and isobaric function of degree i and weight w in the_; + 1 letters a, b, c, ...
{£10 - OD.) F= {ij - 2w) F.
The method employed in proving this can also be applied to show that
where v=Si + 2w.
Corresponding to the reversors 0 and -7- we have the two generators for invariants
a T — vb and aO — {ij — 2w) b,
which are linear combinations of Cayley's generators. Thus, if / be any invariant,
V^dx~ "V ^ *°^ ^"^ ~ ^^^ ~ ^^^ ^' ^ are also invariants.
29—2
462
Lectures on the Theory of Reciprocants
[42
The operator -=- has, but 0 has not, analogous properties in the theory of
Reciprocants ; namely, -v- is a reversor to V and a-^ — /lA is a generator for
reciprocants. Thus, we have shown in previous lectures that
d d
(
^i-^'^)^-''^''-
where F is any homogeneous and isobaric function, and fi = Si + w, and that if R is any pure reciprocant ( a j /ib) Ria one also.
Now, Mr Hammond has found that if
^ b^ 2ac-b>. Sa'd-:iabc+b\ , ^=a^''+ a' ^*+ a' ^'+--
TT is a reversor to V, and a^W — ib is a generator for pure reciprocants. In fact we have
vw-wv=vi-]da
Va,
2ac-6=\
+ +
But, since fb
-0
■f
Sa'd - Sabc + b'
f „ fSa'd - Babe + lfi\ ,,, , . , , Y[ ^i )-W{oab)
= 2a,
= 106-46=66,
= (iSc + 9 -) - (lo^V 6cj + 6 ^ = 12c,
and
W (2a«) = 46,
W(oab) = o - + 5 = 10c,
a \ a J
it follows that
VW- FF= 2ada + 2636 +2ca„ + ... = 2i. Thus If is a reversor to V. Moreover, a^W — ib acting on any pure reciprocant generates another.
Let i2 be a pure reciprocant of degree i ; then, by what precedes, (VW- WV)R = 2iR.
I tl
42]
Lectures on the Theory of Reciprocants
453
But, since iJ is a pure reciprocant, VR = 0, and consequently VWR = 2iR.
Now, V{a^W- ib) R = a^ VWR -iRVb = a?. '2iR - iR . 2a^ = 0.
Hence {a^W-ib)R
is a pure reciprocant ; that is a'W— ib is a generator for pure reciprocants.
Mr Hammond shows that TT is a reversor to V in the following manner : Let w=ao +0,6* +a2e^ +a3e^ +...,
yjt (u) = Ao' + A,'e^ + 4,V» + ^a'e^ + . . ., and consider the operators
P = XA,da„ + (X + /i) A,da„^^ + (X + 2/i) 4 A„+, + • • ■.
Q = x'4o'a„„, + (X' + /) ^,'aa„.+, + (X' + 2/) 4;a„„.+, + • . •■
Regarding e* as an operative symbol defined by the equation
we may write
P = {\Aoe^ + (\ + fi) il,e'»+"« + (X + 2^) il,e"»+«« + . ..) [aj
= e»»X(^„ + ^,e» + 4^''+ ...)[3ao] + e"»^ (4ie» + 24 je"' + . . . ) [3 J
Similarly, Q = e»'» (\' + /*' ^) ^ («) PJ-
Now, PQ-QP= |Pe»-» (x' + / ^) ^/r («) - Qe»» (x + /. |^) <^ («)} [9a,]
= je--" (x' + ^' ^) PV' («) - e»» (x + M J^) Q.^ (m)| [da,]. Q4,(u)= QA, + e^QA, + e">QA,+ ... ;
For
so that
and so that
e"* J^ Q(f> (u) = e'^{<^QA, + 2e''QA,+ ...)
«"• J^ <^ (m) = e"» (e«il, + 2e»»^ + . . .) ; Qe"» 4 <^ (m) = e»* (e*Q4, + 2e'»Q4, + . . . )
de
Q<^(«).
Similarly,
Pe-■o~f{u)=e'^■'^yP^|r(u).
454 Lectures on the Theory of Eeeiprocants [42
Moreover,
= yfr' («) {e"»X^, + «<"+"• (X + m) ^1 + e'"^'* ('^ + 2/*) ^2 + • • ■}
Similarly, Q<t> («) = «»'* f («) (x' + m' rf^) f («)•
Hence / d \
PQ _ QP = |e"-« (x' + / rf^) e- 1' W (^ + '^ ^z^j ^ W
- (x + /.n' + M ^) </>' («) (>^' + m' J'g) >/^ («)| Kl-
If in this we write
<f> = ^, X = 4, /t=l, n = l.
i{r = logM, X' = 0, /=!, n'=-l,
;«_«p-|(i+I).-.(.^|)I-(s4,)4.'«^«1p-i
= 2mK].
2m [S J = 2 (a„ + Oie" + o^e^' + . . .) Poo] = 2 (ao9ao + «i9<«i + °^<«s +••■)• P = 4^„aa, + 54ia„, + 6^ As + • • • . Q = ^/a„„ + 2^,'aa. + 3^s'9a, + • • •. i(a„ + a,e» + 026^" +...)' = ^0 + 416" + ^26^" + ...
log (tto + aie« + a^« + . . .) = log Oo + ^i'e» + ^^'e^* + • • ••
we have
Now, Also
where and
Equating coefficients, we have
. , _ Oi • , _ 2a(ia3 - Oi" ^'-"a ' "*"" 2a„" ' ■•■■
1 . A ^
42] Lectures on the Theory of Reciprocants 455
It is easily seen by expanding the logarithm that the general value of J.„' is (—)"+' — where ;Si„ denotes the sum of the rath powers of the roots of
aox" + <h!>^~^ + Oi^""^ + . . . + a,;.
Thus we have shown that if
P - 2a„^da, + oa^da, + (Gttoa^ + Soi") So,
J n aj. 2a^ - «!% , Soo'a, - 3a^(h + «i% , and Q=-da„+ —^ ao, + ~, ea,+ ---,
then PQ-QP = 2 {afiao + <hda, + a^a,+ ---)= 2i.
The general formula obtained for PQ— QP is an extension of a result of Capt. MacMahon's, who considers the case in which
^ m m
When (f) (u) and i/r (u) have these values, the general formula becomes
PQ-QP = «<»-■'« {(V + A + / ^) (^^ + M"-'"'- d^)
[9J-
But (V + A + / ^,) g «— + M«— ^-1)
„m+m'— 1
V dp/ \m TO + m — 1 ctr/
Consequently
- I w"^"'-' [5 J.
In Capt. MacMahon's notation
P = (m, X, /*, ra), Q = (to', X', /, n') ;
I in our notation
If now we write
which is equivalent to
PQ-QP = {m + m' - 1, X,, /Xi, n + n'),
456 Lectures on the Theory of Redprocants [42
we have (5^' + A + /*' J^) {^('» + "^'-l)+/*d5}
-(x + ^n' + /.J^)||,(m + m'-l) + /*'^}=X. + ;t, Hence we obtain
\, = (m + m' - 1 ) 1^ (\' + /n) - ^, (\ + /tn')| .
iii = u.u'(n-n') + -^(m'-l) ^ (?» - 1).
r- r-r- \ ' VI W,
This agrees with Capt. MacMahon's result, a statement of which was given in Lecture XX. [above, p. 417].
Let Q be a reversor to the operator P= \a'"9(, + (...)3c + (...) Si + ..., and suppose that
(PQ-QP)F = «a'»-'jP,
where F is any homogeneous and isobaric function and k some number depending on its degree and weight. Then \aQ — xb will be the generator corresponding to Q. In other words, we have to prove that
P{\aQ-Kb)F=0 whenever PF=0.
Now, by hypothesis, Pa = 0, Pb = Xa™, and when PF = 0,
PQF=Ka'"-'F.
Thus, P{-KaQ-Kb)F=\aPQF-KF.Pb
= XkO^F- \icd^F= 0.
As an example, consider the case of the reversor -r- in the theory of reciprocants. Here
and since
P = F, \ = 2, m = 2 ;
we have « = 2/t. Hence the corresponding generator is 2 ( a j /t6 j ; or, dis- regarding the numerical factor 2, we may take a -j /ii for the generator in
question, which is usually denoted by the letter O. We may also write 0 in the equivalent form
0 = 4,(ac-b') di+o (ad -bc)dc + 6 (ae-bd)dd+..., which it is sometimes more convenient to use. I shall now show that
nG-Gn = aw- bil, where w is the weight of the operand.
42] Lectures on the Theory of Reciprocants 457
It is very easily seen that
fl {ac - b^) = 0, fl {ad -be) = 2 (ac - ¥), fl (ae -bd)=S (ad - be), n (af— be) = 4 (ae — bd),
Hence it follows, by a direct and very simple calculation, that
no - Gn = 2 (ae-¥) do + 3 (ad - be) 9^ + 4(ae - bd)de + .... But, since 696 + 2c9c + Sddd + 4e9« + . . . = w,
and adi+2bdc + Sedd+'^dde + ...=0,,
aw-bil = 2 (ac - ¥) de+S(ad - be) 9^ + 4 (ae - bd) 9.+ .... Consequently D,G — G^ = aw — bil.
The use of this formula will be seen in a subsequent lecture.
We may also prove an analogous theorem relating to the invariant generator a t vb, which we shall call 0'.
Let the operand be F, a homogeneous and isobaric function of degree i
and weight w. Then VF is of degree i+1 and weight w — 1; its v is
therefore
5(i + l) + 2(w-l)= v+1.
Thus. (VO'-G'V)F=\^v{a^-vb)-[a^-vb-b)v]F
= 0(7^-^ V^ F- v(Vb-bV) F + bVF.
id VbF=brF+2a^F.
Consequently VG'- G' F= 2 (3i + w) a'F - 2m-'F+ bVF
= 2 (3i + w - v)a^F + bVF
= -2wa^F+bVF.
It is perhaps worthy of notice that if / is an invariant of weight w and R pure reciprocant, also of weight w, then
aOI = awT and VG'R = - 2a^R ;
rhereas £167 = 0 and VGR = 0.
468 Lectures on the Theory of Reciprocants [42
LECTURE XXVII.
I should like to make a momentary pause in the development of the theory which now engages our attention and to revert to the proof of Cayley's theorem for the enumemtion of linearly independent invariants contained in Lecture XI. and expressed by the formula (w; i,j) — (w— 1 ; i,j).
Since that proof was written out I have endeavoured to obtain one that might be capable of being extended to the supposed analogous theorem, regarding pure reciprocants, expressed by the formula (w; i, j) — {w — \;i-\-\,j), but all my efforts and those of another and most skilful algebraist in this direction have hitherto proved ineffectual.
In aiming at this object, however, I obtained a second proof of Cayley's theorem, less compendious than the previous one, and subject to the drawback that it assumes the law of Reciprocity, but which possesses the advantage over it of being more direct and of looking the question, so to say, more squarely in the face. The forms of thought employed in it seem to me too peculiar and precious to be consigned to oblivion. I am not one of those who look upon Analysis as only valuable for the positive results to which it leads, and who regard proofs as almost a superfluity, thinking it sufficient that mathe- matical formulae should be obtained, no matter how, and duly entered on a register.
I look upon Mathematics not merely as a language, an art, and a science, but also as a branch of Philosophy, and regard the forms of reasoning which it embodies and enshrines as among the most valuable possessions of the human mind. Add to this that it is scarcely possible that a well-reasoned mathematical proof shall not contain within itself subordinate theorems — germs of thought of intrinsic value and capable of extended application.
That such was the opinion of our High Pontiff is shown by the publica- tion of his seven proofs of the Theorem of Reciprocity, a number to which subsequent researches have made almost annual additions (like so many continually augmenting asteroids in the Arithmetical Firmament) to such an extent that it would seem to be an interesting task for some one to undertake to form a corolla of these various proofs and to construct a reasoned bibliography, a catalogue raisonnee, of this one single theorem. For these reasons, I shall venture to put on record (valeat quantum) the following Second Proof of Cayley's Theorem.
The notation which I proceed to explain will be found very convenient. A rational integral homogeneous isobaric function will be called a gradient; its weight, degree, extent (extent meaning the number of letters after the first) will be denoted by w; i, j and spoken of as the type of the gradient. Either a single letter, such as <f), will be employed to denote a gradient, or
42] Lectures on the Theory of Reciprocants 459
else its type enclosed in a parenthesis thus [w; i,j]. The abbreviation T(^ signifies the type of </>; thus, T6 = iu; i, j.
The number of terms in the most general gradient whose type is the same as that of <f> will be spoken of as the denumerant of if). The letter N will be used to denote such a denumerant; thus, ^^ signifies the denumerant of^.
In like manner, the letter A will be used to denote the number of linear
relations between the coefficients of any gradient, whenever such relations
exist. Hence N(p — i\(f> expresses the number of terms in <j) whose coefficients
are left arbitrary. Obviously, when <f> is the most general gradient of its
type, we have
A<^ = 0.
We also use E to denote the ij — 2w, which may be called the excess, of the gradient of type w; i, j. Thus, if T^ = w,i, j, we write E<f) = ij — 2w.
The operators which we shall employ, namely, fl and D.', are defined by the equations
il = ao9o, + (hda^ + a^dat + •■■,
fi'= a,9a^ + a^3o3 + ....
The first of these is of course an equivalent, but for present purposes more convenient, form of adi, + 2bde + Scda + ■■■, the ordinary invariant
annihilator f2 ( as will be evident on writing af, = a, Oj = ^ , 02 = ;; — 9 1 •••)', the second of them, il', is merely il deprived of its first terra.
We may now give the following enunciation of the theorem to be proved: If (f) is the most general gradient of its type, il(f> is also tlie most general gradient of its type whenever E<j) is not negative. In other words, we shall prove that, subject to the condition stated above, Afl^ = 0 whenever A^ = 0. This is equivalent to Cayley's Theorem on the number of linearly independent invariants. For the number of forms of the same type as ^, and subject to annihilation by ft, is " N^- Nn^ + AO<^ ;
and Cayley's Theorem states that the number of such forms ia N^ — Nflfj), which will be the case when
An^ = 0.
The theorem of Reciprocity enables us to dispense with the discussion of those cases in which the extent j is greater than the degree i. For since [Vol. m. of this Reprint, p. 151] the number of linearly independent invariants for the type w ; j, i is the same as for the type w ; i, j, we can substitute the first of these types for the second, using 1^, whose type is w-fj, i, instead of <^, whose type is w; i, j. Thus we have
N'f - Nn^jr + Afl-v^ = N<]>- N[l<f> + An</).
460 Lectttres on the Theory of Eeciprocants [42
But by Ferrers' proof of Euler's Theorem (vide "A Constructive Theory of Partitions" [p. 1, above]),
Ny}r==N(f> and iVfli^ = i\rn<^. It obviously follows that
Cases for which the extent is greater than the degree may therefore be made to depend on those for which the degree is greater than the extent. Hence Cayley's Theorem depends on the proof that ^il^ = 0 when i = >j and t; = > 2w.
In the course of the demonstration, the following Lemma will be used :
If T<f> = w; i, j and T-^ = ij — w; i, j, then iV<^ = Nyfr.
The types of the two gradients we are now considering may be said to be complementary, and then the Lemma may be enunciated in words as follows :
The denumerants of two gradients are equal when the types of the gradients are complementary.
The proof consists in showing that to each term of the type w, i, j there corresponds a term of the typet; — w; i,j. Let a^^ai^ia^ ... a/y be any term of the type w,i,y, then
w= X, + 2X3+3\, + ... +JX5 and i = X, + Xj+ 7^_+ \s+ ... + \j.
Writing the suffixes of the letters ao, Ui, a^, ... a, in reverse order, every- thing else being kept unchanged, we obtain the term a;\aj_,\a,_2S... Oo^-, I whose weight we will call w'. Then
w' =3K + (i - l)>-i + ( j - 2) X, + ... + Vi
= j(\o+ \, + X, + ... +\,) - (\i + 2\, + 3X3 + ... +ix,) = ij — w.
The degree of the transformed term is still i, and its extent is still j, while its weight has become ij—w; its type is therefore complementary to that of I the original term. Hence to each term of any given type there corresponds j a term of the complementary type, and consequently the total number of j possible terms (that is, the Denumerant) for each type is the same.
By means of this Lemma it can be shown that AD,^ = 0 when E(}) = — l- Let
T(f> = w; i, j where ij—2w = — l;
then, since T[l<f>=^v=l•, i,j, the types T<p and Tfl^ are complementary (thej sum of the weights being w + w — 1 = ij).
It follows from the Lemma that the Denumerants of <}> and fl<^ are equal] Hence
An</) = 0.
I
42] Lectures on the Theory of Reciprocants 461
For if not, the number of independent terms in fl^ being less than the dennraerant of fl<^, will also be less than its equal, the denumerant of <^, and therefore there will be one or more invariants of the type w,i,j for which the excess is negative. Since this is known to be impossible, we must have
An^ = 0. We next prove that, in all cases for which i=>w, the number of linearly independent invariants of the type w; i,j is correctly given by the formula
(w;i,j)-(iv-l; i, j), which is equivalent (as we showed at the beginning of Lecture XV.) to
(w;w,j)-(w-l;w,j), or, what is the same thing, to the coefficient of a"x^ in the expansion of
p^ 1-a;
(1 -a) (1 - aa;){l - ax'yil - aa?) ...(1 - axiy Let the expansion of
Q^ 1 -X
(1 -ax)(l- aar')(l - aa^) ... (1 - axJ)
be l + (a-l)x + A^+ ...+Ay,x'^ + ....
The expansion of F is obtained by multiplying that of 0 by the infinite geometrical series
l+a + a' + a'+ ....
But we only require the coefficient of a"*" in the expansion of F, so that we need only retain the portion
.^^^"(l + a + a' + .-. + a")
of the above product instead of its complete expression.
It is of importance to notice here that ^„, which is independent of x, cannot contain any higher power of a than a". (That this is so will be evident from the constitution of the fraction G, for clearly no power of a in the expansion of G can be associated with a lower power of x.) Thus we see _fchat
Ay,= aa'' + 0a^'+ya"-^+ ... +«a+X,
and consequently
A^x^il+a+a'-i- ... + a«')=...+a^x»'{a+^ + y+ ... + K + X)+ .... Hence the coefficient of a^x" in the expansion of F is a + y3 + 7+ ... + K+\, yhich is the value assumed by ^„ when in it we write a= 1. Call this value \Av,'y and let the value of G when a= 1 be denoted by G'. Then A^' is the oefficient of x'" in
1
(?' =
(l-aj'Xl -a')...(l-ar>)'
482 Lectures on the Theory of Redprocants [42
Hence we see that, when i = >w, the value of (w ; i, j) -(w-l; i, j) is the total number of ways in which w can be made up of the parts 2, 3, ..._;".
We have yet to show that this number is the same as that of the linearly independent invariants of the type w; i,j when i=>w.
This follows from the known theorem that every invariant is either a rational integral function of the Protomorphs a, P^, P„ ... Pj (meaning the invariant a and those of the second and third degrees alternately whose first terms are ac, a'd, ae, a^f, ...), or can be made so by multiplying it by a suit- able power of a. Thus, if / be any invariant of degree i and weight w,
Ia^'=^{a,P„P„...Pj),
where "Jj, which is of degree- weight w.wwhen expressed in terms of a,6, c, ..., is rational and integral as regards the protomorphs.
When i = > w, writing
/=a-"<t>(a,P„P„...P^),
4> consists of a series of terms of the form Aa^P^'^P/- ... Pf, each with an arbitrary coefficient, where, since
2X + 2fi, + 4:v+ ... +jp = w,
the number of arbitrary constants in <I> is the total number of partitions of w into parts 2, 3, ... j. Hence the number of linearly independent invariants of the type w; i,j is also this number of partitions, that is, by what precedes is (w; i, j) — {w — 1 ; i, j). This proves Cayley's theorem for cases in which i = >w.
But when i < w, the equation
/a»-' = <I>(a.P„P„... P,) shows that the coefficients of <I> are not all arbitrary, but must be so chosen that ^ may be divisible by a"*"*, and the reasoning employed in the case of t = > w no longer holds.
It will be convenient at this point of the investigation to review the results we have hitherto obtained and to see what remains to be proved.
Cayley's Theorem has been demonstrated for cases in which the degree is not less than the weight. This will be expressed by saying that
Afl \w; i, j] = 0 when i=> w. We have also proved that
Ail [w; i,j] = 0 when ij — 2w = — 1. The law of reciprocity has been expressed in the form An,[w;i,j] = Ail[w;j,il where [w; i, j] denotes the most general gradient of the type w; i,j.
42] Lectures on the Theory of Reciprocants 463
The theorem to be proved is that
AH [w; i, j] = 0 when ij — 2w=>0; but we may at once dismiss those cases in which i = > w, and (assuming the theorem to have been proved for Quantics of order inferior to j) those in which i<j, for these depend on the truth of the theorem for a Quantic of order i.
It remains, then, to prove that, when ij— 2w = > 0, Afi [w; i, j] = 0 for values of i inferior to w, but not inferior to j. This may be effected as follows :
Let <j> be the most general gradient of the type w;i + l,j, and suppose ^ = P + Qa+Ra'' + Sa\
where P, Q and R do not contain the letter a, though S may do so. Then, writing
^ is the most general gradient of the type w; i,j.
Now, it n = adb + bdc+cdd + ■■; and D,' = bdc + cdd+ ■•■, we have
n^=n'P-H(n'Q + f).-H(n'ie4.f)a»4.(r2^-,f)a3, (i)
and n<^ = n'Q + (n'i2 + ^) a + (0^ + ^)0'.
Confining our attention for the present to iltf>i, it is clear that if no linear relations exist among the coeflBcients of il'R (that is, if Ail'R — 0) the coeffi- cients of il'Q are not connected with those of il'R + ;^ by any linear relation.
For the coefficient of each term of il'R + -^ is the sum of a single coefficient of Q and an independent linear function of the coefficients of R. Moreover, obviously the coefficients of il'Q are unconnected with those of flS + -rj- .
If, then, the coefficients of il'Q are not related inter ae (that is, if AH'Q = 0), we have
Afi^ = A((n'E + f)a+(n^+f)a=}. (2)
Looking now to the expression (1) for il<f>, we see immediately from (2)
Ithat any linear relation subsisting between the coefficients of O^i will also
subsist between those of il<p, and therefore that Af2^ is not greater than
If, then, An<^ = 0, it follows that Afl<^, = 0, provided that both the supplementary conditions AH'Q = 0 and Afl'R = 0 are also satisfied.
484 Lectures on the Theory of Reciprocants [42
Now, since ^i = Q + Ra-\- Sa* is the most general gradient of the type
Q will be the most general gradient of the type w — i;i,j—\, andi2 „ „ „ „ „ w-iJr\;i-\,j -\,
when in Q and R we change b, c, d, ... into a, b, c, .... This change converts Cl' = bdc+cda+ ... into il = (idb+bdc+ .... Hence the conditions A.[1'Q = 0 and An'7i = 0 are respectively equivalent to
An[w-i;i,j-l] = i) and Ail[w-i+ 1; i-l,j -l]= 0.
Supposing these supplementary conditions to be satisfied, what we have proved is that when
An [w; i 4- 1, j] = 0 (that is, An<^ = 0), then also Afl [w; i,j] = 0 (that is, An<^i = 0).
Now, T^ = w; i + l,j, so that E(f>=(i+l)j-2w = {ij - 2w) +j, TQ=w-i; i,j- 1, so that EQ = i{j -l)-2(w-i) = (ij -2w) + {, TR = w-i + l;i-l,j-l,so th&t ER =(i - 1) (j - 1)- 2(w -i + 1)
= (ij - 2w) + i -j - 1. Thus, when ij — 2w=>0 and i = > ;,
E^ and EQ are both positive.
ER is in general = > 0, but in the special case where ij—2w = 0 and i =j, we have ER = — 1. Except in this case (which gives us no trouble, since we have seen that AD,R = 0 in consequence of ER = — 1), we have never to deal with a type of which the excess is negative.
Hence, if we assume Cayley's Theorem to have been proved for all extents up to j—1 inclusive, we have
An [w-i; i,j- 1]=0,
and An[w-i + l;i-l,j-l]=0,
(that is, the two supplementary conditions are satisfied).
We wish to extend the theorem to the extent j.
Subject to the conditions i= >j and ij— 2w=>0, we have
An [w; i, j] = 0 if An [w ; i + 1, j] = 0.
But we need consider no value of i greater than w, as we have proved that
An [w; w, j] = 0 = An [iv\ w + K, j] ;
therefore An [w;^u-\, j] = 0,
An [«;;«;- 2, j] = 0.
^"[M';j,i] =0.
42] Lectures on the Theory of Reciprocanta 465
As previously shown, the theorem is true for all values of t inferior to / if it is true for all Quantics of inferior order. Thus the theorem is true for a Quantic of order j and for evenr value of t if it is true for all Quantics of order inferior to j. But it is true for the Quadric (where j = 2)»; therefore also for the Cubic (j = 3); therefore also for the Quartic (/=4), and so universally. Hence the theorem to be proved is demonstrated.
LECTURE XXVIII.
We now resume the theory of Principiants and proceed to prove the important theorem that every Principiant U either amply an invariant in respect to a known series of pure reciprocants, which we call A, B, C, D, or else becomes such an invariant when multiplied by o-^'. where to is the weight and i the degree of the Principiant in question. Using the letter M to denote the pure reciprocant ac- ^6». and 6 the ordinary eductive generator,
4(ac-6')a» + 5(o<i-6c)a. + 6(a«-W)3-+7(a/-6e)a. + ... (which, it will be remembered, is only another form of o-£-/iA, with the advantage of the /* being suppressed, that is, only implicitly contained), we obUm m succession the values of A, B, C, D. ... from the following equations:
oA = GM,
fiB=GA,
7C=GB-MA,
8D = GC-2MB.
9E^GI)-3MC,
On performing the calculations indicated by these equations we shall 6nd il=a»d-3aAc + 26'.
B = a'e-2aV-'^aH>d+~alfc-4b^,
C = «•/- ooVxi - 4a»6e + 13a»6c» + ^ a-W - 1?? ai'c + ** f
■» 4 2 '
25
^ = oV - 8 a*d' - 6a*ce+7af(f + terms involving b,
15 E = a*h-^a*de- 7tfyf+ 29aV<i + terms involving 6.
«J .S?J7h*,,!^ «»>iMo. »=>» b««M. id«tk.l with i= ,,: bot we h.Te ^^,
30
466 Lectures on the Theory of Reciprocants [42
The fact that Z) is a pure reciprocant enables us to calculate the terms in E which are independent of b without a previous knowledge of the values of those terms in D which involve b. For, since
Q = 4i{ac-h')db+ ... and V= 2ci:'di + ••■,
a*G — 2 (ac — 6") F does not contain 3j.
Hence the operation of a}0 — 2 {ac — 6*) F" on terms involving 6 cannot give rise to terms independent of 6. But,
D being a pure reciprocant, VD = 0 ;
80 that [a'O - 2 (ac - 6») r} 2) = a'GD,
and the terms of a}GD which do not involve b are found by operating with
[a»(?-2(ac-6')F]j_,
on the terms of D which do not involve b.
If, now, we use Jf„ A^.B^.C^,... to denote those portions oi M,A,B,C, ... which are independent of b, and write
[a»G-2(ac-6^)F]j.„ = a'G„,
we shall still have QE, = GJ)^ - SM.Co;
and in general the law of successive derivation for A,,, £„, (7„, i)„, ... is the same as that for A,B,G,D,... except that G^ takes the place of G.
We have
a»(?„ = [a»G'-2(ac-6»)F]4.„
= a? {badZo. + ^aedd + 7a/a« + 8agdf + 9ahdg + ...) - 2ac {GacSa + Todd, + (8«e + 4c») df + (9a/+ 9cd) 3^ + . . . j ; so that
Go = oodS. + 6 (ae - 2^) da + 1 {of- 2cd) de
8 9
+ - {a'g - 2ace -(?)df+'- (a'h - 2acf- 2c»d) 9^ + . . . ;
and consequently (since M„ = ac),
5A, = GJifo gives A^^a'd,
6B, = GoAo „ jB, = a»e-2a>c», J
1C, = GA-M,A, „ C„=ay-5a'cd,
8Do = G,C7. - 2MoB, „ D, = a'^r _ ^ a'd' - Qa*ce + 7aV, " 9E,^GJ)o-3M,C, „ Eo = aVi-~a'de-7a'cf+29a*c%
42] Lectures on the Theory of Reciprocants 467
Thus, for example,
8Z)„ = G, (a*f- 5a^cd) - 2ac (a'e - 2a'c*)
= - 25a«d« - 30a'c {ae - 20") + 8a' {a?g - 2ace -r?)- 2ac (a'e - 2aV) ;
25 whence D„ = a'a - -5- a*d'' - Qa*ce + laV.
o
Again, 9£„ = (?„ (a'^f - ^ 0*^' - 6a^ce + 7a'<A - Sac (a*f- aa'cd)
= oad (- 6a*e + 21a'c') - ^ (ac - 2c0 a*(^ - 42 (a/- 2cci) a*c
it
+ 9 (a^A - 2ac/- 2c'(i) a* - 3ac (a*/- oa'cd),
15 gives jE'o = a«/i. - -^ a'de - 7a'c/ + 29a''cV.
Similarly, from the known values of D^ and E^ we may deduce that of the next letter, ^0, and so on to any extent.
It may be noticed that each of the pure reciprocants A, B, G, D, ... can be determined without ambiguity, by means of the annihilator V, when the portions of them, A„, Bo, C„, Dj, ... independent of b are known.
For suppose R and R' to be two reciprocants, of weight w, for each of which the terms independent of b are the same. Then their difference is divisible by b. Let
R-R'= b(f>; then V{b(f)) = 0 ; that is, 2a'<f> + bV^ = 0.
Hence <j) is divisible by b, and jB — i?' is divisible by 6"; say R — R' = b'^. Then
F(6^/r) = ia'byjr + b^Vyfr = 0,
showing that y]r is divisible by b, and R- R' hy 6'.
By continually reasoning in this manner, we prove that R — R' must be divisible by 6"; and then the remaining factor (being of weight 0) is neces- sarily of the form Xa*, where \ and 0 are numerical constants. Thus
R - J?' = \a*6«', and consequently F(\a*6«') = 0.
This is impossible unless \ = 0, when the two reciprocants R, R' become equal, showing that there cannot be two different reciprocants for which the terms independent of b are the same. When, therefore, the terms which do not involve b of any pure reciprocant are known, the complete expres.sion of that reciprocant can be determined without ambiguity.
Each reciprocant of the series A, B, C, D, ... possesses the property of being, so to say, an Invariant relative to the one which precedes it, meaning that the operation of Q, = adi, + 2bde + 3cdd + ■ ■ ■ on any letter gives (to a
30—2
468 Lectures on the Theory of Reciprocants [42
factor pris) the one immediately preceding it. The first letter, A, is an Invariant in the oi-dinary sense. We can in fact show that
£IA = 0,
nB=A x|, fl(?=2£x|, ni)=3(7x|,
The proof depends on a formula established in Lecture XXVI. of this course [p. 457, above], namely
ilG-Gn = wa~ bn,
where G is the generator 4(ac — 6^)86 + 5 (ad — 6c)3e + •••> and w is the weight of the operand.
Thus, observing that the weights of A, £, C, D, ... are 3, 4, 5, 6, ... respectively, we have
(SlO-Gn)A={Za-hD.)A,
(fiG - Gil) B = (4>a- bil) B,
(nG-Gn)C=(5a-bil)G,
Now, since A is the well-known invariant a'd— Sabc + 26', we may write ilA =0 in the first of these equations, which then reduces to
ilGA = 3aA.
But, since 6B = GA,
we have GilB = ilGA = 3aA.
Thus nB = Ax^.
Again, substituting for [IB in the formula
{nG-Gn)B = {4>a-bn)B,
we find nGB-G(^^'^ = 4aB-jA,
where, since G (which is linear in 3;,, 9^, ... and does not contain da) does not operate on a,
G(^).|M=.,„i>.
and consequently n,GB+-^ A = 7aB. ■
I
42] Lectures on the Theory of Reciprocants 469
Now, IC^GB-MA;
so that inC = nOB - AHM - MnA.
But, since nM = D. ( ac — -r-j = — -^ a.nd ilA=0,
7nG=nGB+jA = 7aB.
Thus nC=2Bx^.
We may, in exactly the same way, prove that
nZ)=30x|,
n^=4i)x|,
and so on to any extent.
In the following inductive proof it will be convenient to denote the letters
A, B, 0, D, E, ...
by M„, w,, M„ Ws, Ut, ...,
and then the theorem to be proved is that
nu„=min-iX g.
When this notation is used, the law of successive derivation which defines the capital letters is expressed by the equation
(n + 7)Un+,-GUn+i + {n + l)MUn = 0, (1)
where 0 is the generator
4c{ac—b^)di,+ 5(ad — bc)dc+ ..., and M = ac — -r- .
Operating with il on the above equation, we obtain
(n + 7) n?<„+, - Q.Gun+1 + {n + l)(Mnu„ + u„nM) = 0. (2)
Now, the weights of u,,, m,, Mj, ... are 3, 4, 5, ... respectively, and conse- quently the operation of
nG-Gn = im- bn
on t(„+i (whose weight is n + 4) gives
(ilG — Gil) u„+i = (n + 4) a'i„+i — 6I1m„+,.
Or, assuming that ilu^ — icu,^i x - for all values of « as far as n + 1 inclu- sive (it has previously been shown that Q,B = A x x and nC=2B x^, so that the theorem is true for *c = 1 and k = 2),
Iil(?M„+l = GilUn+t + (n + 4) aM„+i - bilUn+i = (n + 1)G(^ M„j +{n + 4,) aun+i - (m -|- 1) ^ u„.
470
Lectures on the Theory of Redprocants
[42
But (remembering that 0 does not operate on a, so that G . ^ "n = s^'Mn) we have, in virtue of equation (1),
0 f| «nj = I {(« + 6) Un+i + nMun-^].
Hence it follows that Xl(?u„+, = '^- a {{n + 6) it„+, + nil/M„_,} + (n + 4) att„+, - (n + 1)
2
a6
ah
(n + 2)(n + 7) 'w(»i+l) „ , ..<
= ^ '-^ aun+^ + -^^"2 — - a3/«„_, - (»i + 1 ) "2 M„.
On substituting this in (2) we obtain
(n + 7) |nu„+, - (n + 2)| «„+, •
• +(w + l)ilf jnM„-n|M^,l ah
+ {n+\)uAnM + "^\=0.
This reduces to
f^«n+2 = («+2)5M,.+i.
a
2
For, according to the assumption previously made in the course of the demonstration,
so that the second term vanishes ; and the third term vanishes because
56a ah
ilM
( b¥\ ah
We have therefore proved that if the theorem is true for Hm,, when k has any value up to m + 1 inclusive, it is also true for Hun+i- But the theorem holds for /c = 1, and for /c = 2. It therefore holds universally for any positive integer value of k.
Recalling the known values of the reciprocants M, A, B, C, B, ... we observe that their principal terms are ac, a^d, a'e, a*f, a'g, ..., where it is to be noticed that the most advanced of the small letters in the expression for any capital letter occurs only in the first degree multiplied by a power of a. In other words, M, A, B,C, B, ... form a series of Protomorphs, and consequently every Pure Reciprocant can, as we have already seen (vide [p. 384, above]), be expressed as a function of a, M, A, B, C, B, ... rational in all of them and integral in all except a.
42] Lechires on the Theory of Reciprocants 471
But it is further to be noticed that whereas
a is of degree 1 and weight 0,
|
M |
» |
2 |
)J |
2. |
|
A |
i> |
3 |
>* |
3. |
|
B |
» |
4 |
1} |
4, |
and in fact that every capital letter is of equal weight and degree.
From this it will follow that every Pure Reciprocant will be the product of a power of a into a function of the capital letters alone.
For let i be the degree and w the weight of any pure reciprocant ex- pressed in terms of a, M, A, B, C, ..., and suppose one of its terms to be
a^M^A'B'^C'^...; then Ti + 2d + 3K + 4!K + Ofi + ... =i,
and 20 + SK + iX + ofj.+ ...=w.
Hence r] = i — w,
which is the same for every term of the pure reciprocant in question. Thus each term contains a'~" as a factor, and the reciprocant is of the form
a*-^^{M,A,B, C',D, ...).
Let us now consider any Principiant P; since P is a pure reciprocant, we must have
P = a'-^<i>(M,A,B,C,D, ...).
But Principiants are subject to annihilation by fl, and consequently nP = 0, which gives
On writing for ilM, ilA, ilB, nC, ...
their values -bx^, 0, -^ ^^> 25 x-,...
we obtain | (- bdn + Adg + 2Bdc + SCdj, + ...) * = 0.
From this it would follow that 4> is an invariant in the two sets of letters
-b,M&vdA,B,C,I), ...;
but it is easy to see that it is an invariant in the latter set exclusively. For M and A, B, C, D, ... being all of them pure reciprocants,
O and 8«*, 9s*. 9c*, ^d^, ••■> which are functions of M, A, B, C, ... exclusively, must also be pure reciprocants.
472 Lectures on the Tlieory of ReciprocarUs [42
If, then, we operate with V on
we shall find V{—hd„)^ = 0 (every other term being annihilated by V). Thus
V{bdM) * = (?M^) yb = 2a=ajtf* = 0,
and consequently 9jj4> = 0. Hence
(Ads + 253c + 3CSi> + ...) ^ = 0. The equation 3a,4> = 0 shows that M does not appear in the expression for any principiant in terms of the capital letters, while
{Ads + 'i-Bdc + 3CSo +...)* = 0 shows that 4> is an invariant in A, B, G, D, .... We have thus shown that every invariant of
{A,B,G,...){x,yy is a principiant, and conversely that every principiant is an invariant of
{A,B,C,...){x,yy, or such an invariant multiplied by a power of a.
LECTURE XXIX.
From the theorem that every Principiant is (to a power of a prh) an Invariant in the reciprocantive elements A, B, G, ... we readily deduce its correlative in which, everything else remaining unchanged, the reciprocantive elements A,B, G, ... are replaced by a set of invariantive elements which we
call A^, Ai, A The equations connecting the new elements with the
old ones are as follows :
Ao = A,
..=o-2(|)..(|y., ..=z,-s(|)c.3(|)-i,-g^,
..-^-4(|)...ego-.(|)-..(|)'.,
42]
Lectures on the Theory of Reciprocants
473
We have, in the first place, to prove that A^, A^, A^, ... are all of them invariants in the small letters a, h, c, .... This is an immediate consequence of the identities
ilA = 0,
Q.B = Ax nC=2Bx
2'
a 2'
established in the preceding Lecture, coupled with the fact that D,b = a. Thus
nA, = nA = o,
nA, = -^^ilA+(nB-A x|)=0,
G
and in general, writing the equation which gives i4» in the form
^-=(-2) ^+H-2) ^+-i:2^(-2
n(n -!)(«- 2) /_ ftNj"-' T72T3 I 2, and operating on it with fi, we find
i) + ...,
n (n - 1) (n - 2) / 6y-»
= 0 (each term vanishing separately). We next observe that Mo, ^,, 4„ ...)(a;, yK being equal to (^, B, C, ...)\a;- ^y, y\ ,
is a linear transformation of (^, B, G, ...)(x, yV,
b and that the determinant of the transformation
1-2 0 1
is equal to unity.
Hence every invariant in A^, A^, A^, ... is equal to the corresponding invariant in A, B, C, ..., which proves the theorem in question.
474 Lectures on the Theory of Reciprocants [42
Each of the invariantive elements A^, A^, A,, ... is, so to say, a recipro- cant relative to the one which immediately precedes it, just as in the cognate theorem each of the capital letters A, B, C, ... was an invariant relative to its antecedent. It is in fact easily seen that
VAo = 0.
VAt = -2A,a'', VAt = - 3A^a\
and in general VA„ = — n.4„_ia
Thus, for example, if we operate with V on
A=i>-a(|)c.s(|)'^-g.,
remembering that A, B, C, D are pure reciprocants, we shall find
VA..-l]C-2QB^I^^A\VB.
But C-1 (I) B + (I)' A = A^ and Vb = 2a» ;
so that VA^ = -^A^a\
In like manner, operating with V on
A„ = {A,B,G,...)(-\,\j ,
we obtain VAn = -\{A, B, G, ...){-\, 1 )""' Vb
= — nAn-^a''.
This property enables us to give a proof (exactly similar to the proof of the cognate theorem in the preceding Lecture) of the theorem that every principiant is expressible as the product of an invariant in A^, A^, A^, ... by a suitable power of a. We first observe that, using N to denote axi — b-,
N, Ac Ai, Ai, ...
form a series of invariantive protomorphs of equal degree and weight.
Hence it follows that any invariant of degree i and weight w can be expressed in the form
ai-^^(N,Ao,A„A„ ...),
and consequently that every Principiant can be expressed in this form, pro- vided only that
t1
42] Lectures on the Theory of Reciprocants 475
Substituting for VA^, VA^, VA^, ... their values given above, and at the same time observing that
VN=V {ac - ¥) = ba% - 4.a^b = a%
we find F* = a' (bdy - A,d^^ - 2A,dj, - 3A,d^, -...)* = 0.
Finall}', we prove that <I> does not contain N, but is an invariant in Ao, Aj, A2, ... alone, by operating with fl on
(bdj,-A,d^^-2A,d^,-SA,d^,-...)^ = 0, when it is easily seen that every term vanishes except the first, which gives
where, nb = a being difi"erent from zero, we must have 9^^^'!' = 0-
The invariants N, A„, Aj, A^, ... obey a law of successive derivation similar to that which holds for the reciprocants M, A, B, C, ....
Starting with N= ac — b" and operating continually with
(?' = a T^ - (3i + 2w)b = (4ac - ob^) dt + (bad - Ibc) 9^ + • • •,
we shall find G'N= bA„,
G'A, = QA„ G'A, = 7A,-NA,. G'A, = 8A,-2NA„ G'A, = 9A,-SNA„
and generally G'An = (n + 6) An+i — nNAn-i.
These equations are exactly analogous to
GM=5A, GA = 65, GB = 7G + MA, GC = 8D-\-2MB, GD = 9E + SMG,
5
in which #=00 — 76', and GM, GA, GB, ... are the educts of if. A, B, ... 4
obtained by operating with
G = aj--(3i + w)b = 4!{ac-b')di, + 5(ad-bc)dc+....
It should be noticed that the two generators G and G' are connected by the relation
G' = G- wb,
where w is the weight of the operand.
476 Lectures on the Theory of Reciprocanta
Also, that
Oh = i{ac-h') = 4N, and O'b = 4ac - o6»= 4il/.
We may easily verify that
0'N= 5 A, = 5 (a'd - Sabc + 2b') by operating with G' = (4ac — 56') 9(, + (Sad - The) dc on N = ac- 6".
To prove that G'Ao = 6Ai,
we operate on Ao = A,
for which the weight is 3, with
0'=0-3b. Thus 0'Ao = (G-Sb)A = 6B-SbA=6A,.
fb^
[42
For by definition
A,-B-[^.
In general, to find G'A„, we have by definition
b
4„=(A£,C,...)(-|,l)",
and, since the weight of An is w + 3,
G'An = GAn-(n + 3)bAn. Now
b \»
G4„ = (?(4,B,C, ...)(- 1 , l)"
= (G^,(?£,(?0,...)(-|, l)"-|(^,5,C', ...)(-|, l)" '
Gb.
Substituting for GA, GB, GO, ... their known values, and remembering that Gb = iN and that {A, B, G, ...)(-^, iT ' = A„., , we have
(?4„ = (6£,7C',8i), ...)(- 1 , l)"
+ M{0,A, 25, 3(7, ...)(- 1 , l)" - 2niV^4„_.
= 6(£,C,A...)(-|. l)" + (0,C,2A3^, ...)(-|, l)"
+ M{0, 4, 25, 3C, ...)(- 1 , l)" - 2HiVril„_,.
But (0, C,22),3£?, ...)(-|, l)"
/^/ *V'"' / i\n/ ^\"~' n(«-l)(w-2) „/ 6\"-»
= n((7,A^,...)(-|.i)"~';
42] Lectures on the Theory of Reciprocants 477
and similarly
(0, A, IB, 3C, ...) (-|, \f = n{A, B, C, ...) (-|, l)"""' =„4„_,.
Hence
GA, = 6(B. C, D, ...) (-|, \y + n{C, D, E, ...)[-\, l)""
Now let U = {A,B,G,...) (u, i;)» ;
then ^-^=n{A.B,G,...){u,vr-\
du
dU dv
and ^=n(B,C,D,...)iu,v)-^;
' ?7= (il, 5, C, ...)(u, v)" = u(A, B, C, ...)(«, f)"-'
whence it follows that
{u,vY = uiA,B,C
+ v{B,G,D,...){u,vT-\ (1)
Similarly, we see that (fi, 0, A . . .) (u, vY =u(B,C,D,...) {u, v)"-'
+ v(C,D,E,...)(u,v)-^-\ (2) Writing ii = — - and u = 1 in the above equations, and remembering that
(4,5, C, ...)(-|,l)" = 4„, we obtain immediately from (1)
b ,\"-' . b
(fi,C,A...)(-|.i)" ' = ^« +
■"»— 1.
and then (2) gives
(c, i>, iS;, . . .) (- 1 , i)"~' = (4^, + 1 4„) + 1 (4„ + 1 4^,)
= 4„+, + 64„ + -4„_,.
But it has been shown that
G4„ = 6(£,C, D,...)(-|,l)" + n(C,i),i:,...)(-|.l)""
+ n(ilf-2i\0^»-i. Hence, by substitution,
0A„ = 6 (a^, + I 4„) + n ^4 „+, + 6 J„ + ^' 4^,) + n (Jlf - 2N) 4„_. = (n + 6) 4„., + in + .3) 6il„ + „ (^Jlf + 1' - 2iV^) 4„_,.
478 Lectures on the Theory of Reciprocants [42
Now, O'A^ = Gil„ - (n + 3) 6il„ = (JJ + 6) 4„+i + n (if + ^ - 2A') 4„_, ,
where Jf + j = ac-^6' + |- = ac - 6'=iV^.
Thus 0'An = (n + %)An+i-nNAn-i,
which proves the law of successive derivation for the invariantive elements A„, Ai, At, ... .
We now proceed to explain the method of transforming a Principiant, given in terms of the small letters a, b, c, ..., into one expressed in terms of a, A, B, C, ....
Remembering that the expressions for
A,B, G,D,E, ...
have for their most advanced small letters
d, e,f, g, h, ...,
and that, in each capital letter, the most advanced letter occurs only in the first degree, multiplied by a power of a, it follows, as an immediate conse- quence, that we may, by continually substituting for the most advanced letter, eliminate d, e,f, g, h, ... from any rational integral function
<f}(a, b, c, d, e,f, g, h, ...)
and thus transform it into another function whose arguments are
a,b.c,A,B,C,D,E, ...
and which is rational in all its arguments, and integral in all of them, with the possible exception of the first argument, a.
But (see Lecture XXVIII.) [above, p. 471] the result of this elimination is known to be
ai-«'^{A,B,C,B,E,...)
in the case where ^ is a Principiant of known degree i and weight w. Hence b and c must disappear spontaneously during the process of elimination.
This being so, we can give b and c any arbitrary values, without thereby affecting the result, and it will greatly simplify the work to take 6 = 0 and c=0.
It is also permissible to take a = 1 ; for, although the factor a*""" is thereby lost, it can always be restored in the final result because both i and
The establishment of the scale of relation between the terms of the A^, Ax, A^, ... series, and the above proof of it, is due exclasively to Mr Hammond.
II
42]
Lectures on the Theory of Reciprocants
479
w are known numbers. Now, if we write a = 1, 6 = 0, c = 0 in the known expressions for A, B, C, D, ..., we shall find
A = d,
B = e,
G=f.
25
E = h-^de,
Hence we have to eliminate d, e, /, g, h, ... between the above equations and
P = ,f>{l,O,0,d,e,f,g,h, ...),
where P stands for the given Principiant. In other words, we have to
|
substitute |
for |
||||||||||
|
a, |
b, |
c, |
d. |
e, |
/ |
9. |
h. |
||||
|
1. |
0, |
0, |
A, |
B, |
c, |
D + |
¥-■ |
E^ |
15 2 |
AB, |
|
|
in |
P = |
<f>{a |
b, c |
, d, e |
/, 9, h. |
...). |
The result of this substitution will be
P = ^iA,B, C,D,E, ...), where, to compensate for the factor lost by taking a = 1, we must multiply 4> by a'~". As an easy example, consider the Principiant which Halphen calls A, and for which he obtains the expression
|
h |
c |
d |
e |
f |
|
a |
b |
c |
d |
e |
|
-a» |
0 |
b' |
2bc |
2bd + c' |
|
0 |
a" |
2ab |
2ac + b' |
2ad + 26c |
|
0 |
0 |
a' |
Sab |
36= + 3ac |
i Here the degree i = 8and the weight w=8; so that i — w = 0, and no factor has to be restored. On making the substitutions spoken of, the determinant becomes
|
0 |
0 |
A |
B |
G |
|
1 |
0 |
0 |
A |
B |
|
-1 |
0 |
0 |
0 |
0 |
|
0 |
1 |
0 |
0 |
2A |
|
0 |
0 |
1 |
0 |
0 |
which immediately reduces to AG—& by striking out the first three columns and the last three rows.
Of this Principiant we shall have more to say hereafter.
480
Lectures on the Theory of Reciprocants
[42
LECTURE XXX.
The method of substituting large letters for small ones will be better understood if we employ it to obtain an expression of the form a'-^<S>{M,A,B,C,D,E,...)
for any pure reciprocant
^(a, 6, c, d, e,f,g,h, ...)
of known degree i and weight w in the small letters.
The transformation is effected by substituting in </> for c, d, e,f, g,h, ...
their values (which are perfectly definite) in terms of a, b,M,A, B, C,D, E, ...
But since h does not appear in the final result, we are at liberty to give it
any arbitrary value, and it will be convenient to take 6 = 0, for then (see
Lecture XXVI IL) [above, p. 463] we have
M= ac,
A = a^d,
B = a'e- 2a?(?,
C=a*f-5a'cd,
25 D = a'>g- — a*d^ - 6a*ce + 7a^c\ o
E=a''h-Y a'de - 7a»c/+ 29a*c-d,
There is an additional advantage in taking b = 0, namely, that then the values of the invai-iants N, A„, A^, A^, ■■■ (see their definition at the begin- ning of* Lecture XXIX.) exactly coincide with those of the reciprocants M, A, B,G, ... set forth above. Hence, merely interchanging the capital letters, the same substitutions enable us to express any invariant in terms of a, N, A„, Ai, ..., as well as any reciprocant in terms of a, M, A, B, ....
c d 6 The solution of the above equations will give - , - , - ,
a 0/ cb
in terms of
M A
a?' a? a shall find
B
4'
.; but we can, without loss of generality, put a=l, when we
a = l,
6 = 0,
c = M,
d = A,
e = B+2M\
f=G+ 5MA,
g = D+~^A' + 6MB + 5M\
O
h = E + ^1 AB + IMC + QMA\
[* p. 472, above.]
i
42]
Lectures on the Theory of Reciprocants
481
The substitution of these values in the pure reciprocant
</)(«, 6, c, d, e,f,g, h, ...) will convert it into
<^{M,A,B,C,D,E, ...).
We havewritten a= 1 for the sake of simplicity ; but without doing this we have, since ^ is homogeneous of degree i,
(c d e ^ 1, 0, -, -, -, ...
Hence, substituting for
a a a
,M A B
in terms of — , -j, -,, a? a" a*
4>{a, 0, c, d, e, ...) = a'4) [-,,-,, -,,
or, since M, A, B, ... are of weights 2, 3, 4, ... and <i> is of weight w, <p (a, 0, c, d,e,...) = a'-«'^ (M, A, B, ...). Thus, in consequence of writing a=\, the factor a'~" has been lost ; but this factor can always be restored, both i and w being known numbers.
When ^ is a Principiant, M will not appear in the final result, which will be identical with that obtained by the simpler substitutions of the preceding Lecture. If, for example, we substitute for
a, h, c, d, e, f,
1, 0, if, A, B + 2M\ C + 5MA, instead of 1, 0, 0, ^, B, G,
in the determinant expression for Halphen's A, previously given, it becomes
0 M A B + 2M' C+5MA
1 0 i¥ A B + 2M' -10 0 0 M^
0 10 2M 2A
0 0 1 0 3if
Subtracting the 4th row multiplied by M from the first, the determinant reduces to
^ A B C + 31/.4
1 MA B + 2M'' -10 0 M^
0 10 SM
Again, subtracting the 2nd column multiplied by .3if from the last, and reducing, the determinant becomes
0, B, C
1, A, B-M^ =AG~&, - 1, 0, M''
where M disappears, as it ought to do, because A is a Principiant.
8. IV 31
482 Lectures on the Theory of Reciprocants [42
In what follows we shall have frequent occasion to make use of the fact that if iia is an absolute pure reciprocant, -jj- , which we know is a pure reciprocant, is also an absolute one.
This is very easily proved. For let R be any pure reciprocant, of degree t and weight w, which becomes ii» when made absolute by division by a power of a, then
R„ = — , where /i. = 3i + w, a« and, using 0 as usual to denote the generator for pure reciprocants,
dRa OR
Hence
da; ^+1
dRa _ GR
a ^
which is an absolute pure reciprocant because GR, which is of degree i + l
and weight w + 1, must be divided by a^~ in order to make it absolute. Thus, if Ma, Aa, Ba, Ga, ... are what M, A, B, G, ... become when each of them is made absolute by division by a power of a, we have
a~^^Ma = 5Aa,
a'^-j-Aa-QBa,
dx
a-i~f-Ba = 1Ga + MaAa,
t I
We shall use these results in deducing the complete primitive of the
differential equation
AG-B' = 0
from that of the equation in pure reciprocants,
25A^-16M' = 0.
This equation may be written in the form
25Aa' = l6Ma'; whence, by difiTerentiation, we obtain
^OAa{a-^lAa)=^SMa^{a-^lM,
which gives oOAa . QBa = 48Jlf„' . oAa]
that is, 5Ba = 4>Ma'.
a I >
42] Lectures on the Theory of Reciprocants 483
Difierentiating this result, we find
which gives Ca = MaAa-
We now restore the non-absolute reciprocants M, A, B, C; that is, we
55 = 4i/^ and G = MA.
Hence 25 (AG - 3^) = M {2b A- -16M') = 0 (hecsiuse 25A^='16M^).
Now, the equation AC — B'=0 remains unaltered by any homographic substitution, so that it will be satisfied not only by any solution of the equation in pure reciprocants 25^1^ — 16M' = 0, but also by any homographic transformation of such solution. But it has been shown (in Lecture XIIL, [p. 379, above]) that the complete primitive of 25A^ — 16if' = 0 is a linear transformation of y = a^, where \' — \ + 1 = 0 (that is, where \ is a cube root of negative unity).
Consequently any homographic transformation of y = a;* is a solution of
AG-:^ = 0.
Moreover, this is its complete primitive; for the highest letter,/, which occurs in AG — B^, corresponds to the seventh order of differentiation, and if we write
F X
y=z- ^=:?'
where X, Y, Z are general linear functions of x, y, 1 (that is, if we make the most general homographic substitution), y = a^ becomes Y=X*'Z^~*', which will be found to contain exactly 7 independent arbitrary constants. Thus the complete primitive of AG—B^ = 0 is F = Z*Z'-\ where X, Y, Z are general linear functions of a;, y, 1, and X is a cube root of negative unity.
Observe that although any solution of Jl/ = 0 also makes A, B,G, ... aX\ vanish, and so satisfies AG — B' = 0, we cannot from this infer that a homo- graphic transformation of the parabola y = 3? will be the complete primitive of.4C-jB» = 0. For, though YZ=X^ is a solution of AC-B' = 0, it oWy contains 5 independent arbitrary constants, and therefore cannot be its complete primitive. Neither can YZ= X' be obtained from the complete primitive by giving special values to the arbitrary constants. Hence YZ=X' is a singular solution of AC — B' = 0.
We may also deduce the differential equation of the curve F = X''Z~'', where \ has a general value, from the corresponding equation in pure reciprocants,
25 {2X' - 5\ + 2) -4' -f 16 (X + 1)* if' = 0,
which has (see [p. 377, above]) for its complete primitive any linear trans- formation of the general parabola y = a^.
31—2
484 Lectures on the Theory of Eedprocants [42
Writing for shortness
2\'-5\+2=p and (\ + iy=q, and at the same time making both ^ and ^ absolute, the above equation
becomes ,, „
2opAa'+16qMa' = 0.
Hence, by differentiation, we obtain
50pAa . 6fi„ + 4>8qMa' ■ oAa = 0, which gives ^pSa + ^S-^a' = 0.
After a second differentiation we find
5p {ICa + MaAa) + iOqMaAa = 0 :
that is, ^pCa + {P + 89) ^^aAa = 0.
We now replace the absolute reciprocants Ma, Aa, Ba, Ca hy M, A, B, C, and thus write the original equation and its two differentials in the form
2opA''=-l6qM', opB = - ^qM\ 7pC=-{p + Sq)MA. Hence we find
5^ 7 . p» (AG- B') = - 2op ip + 8q) MA' - 16 . Iq'M* = I6q (p + q) M\ 5'.7\p'{AC- B'f = 1 6»9» ip + qf M'\ oYA' = l(iYM", and, eliminating J/ from the two last equations,
2* . 7» . p-'q (AC- ^Y =5Hp + qY A\ Now restoring p = 2V - 5X + 2 = (\ - 2) (2X - 1) and q = {\ + lY,
we have p + g = 3 (\^ — \ + l) ;
so that the final equation becomes
2V7»(\+1)»(>^-2)M2X-1)H^C'-£')^ = 3».5«(X''-X + 1)'^'- The same reasoning as before will show that, for a general value of X, the complete primitive of this equation is the general homographic transforma- tion y = Z*Z'~^ of the curve y = a;^.
There is, however, a special exceptional case in which the differential
equation becomes
2'.V{AG-B'Y = ^'-o^^', the corresponding value of the parameter \ being either 0, 1 or co , as may be seen by solving the equation
(X + 1)" (X - 2)' (2X - 1)== = 4 (X'^ - X + 1)'.
42] Lectures on the Theory of Reciprocants 485
In the case where A. = 0 or oo we can, in the same manner as before, show that the complete primitive is a homographic transformation of the curve y=e' hj deducing the differential equation from the corresponding equation in pure reciprocants,
25A' + 8M' = 0,
whose complete primitive is (see Lecture XIII.) [p. 379 above] a linear transformation of y = 6^.
When \ = 1 the corresponding equation in pure reciprocants is
25 A^ - 64Jif = 0, whose complete primitive may be shown to be a linear transformation of y = x]ogx. The reason why these two distinct equations in pure recipro- cants lead to the same equation in principiants is that the two curves y = e^ and i/ = x log x are homographically equivalent but not linearly trans- formable into one another. For we may write the equation y—x log x in the
V
form x = e*, which is a homographic transformation of y = e*.
Besides the special case just considered, in which the complete primitive
Y - of the equation in Principiants is y = e^, we may notice that in which the
parameter \ is either - 1, 2, or 2. the differential equation reducing to
^ =0 simply, and its complete primitive Y= X'^Z^-'' being the equation to a conic, as it should be. The case where X^-X + 1 = 0 and the differential equation reduces to AC-IP=0 has been considered already. There remains the case in which \ = 3, when the complete primitive becomes YZ^ = X^ (the equation of the general cuspidal cubic) and the differential equation assumes the simple form
fAC-B'^' fAy
1—3— y^UJ-
which is therefore the differential equation of cuspidal cubics.
We shall hereafter show that in this case the Principiant
2»(4C-5»)>-3M»,
which is apparently of the 24th degree, loses a factor a* and so sinks to the 20th degree. It is, however, generally difficult to determine the power of a contained as a factor in a Principiant given in terms of the large letters.
The results obtained in the present Lecture agree with those of M. Halphen contained in his Thhe sur les Invariants diffirentiels (Paris, Gauthier-Villars, 1878), which contains a complete investigation of the properties of the Principiant AC-B', which he calls A. But our point of
486 Lectures on the TJieory of Reciprocants [42
view 18 diflFerent from his. He obtains A in the form of a determinant from geometrical considerations. With him A=0 is the differential equation which expresses the condition that, at a point x, y on any curve, a nodal cubic shall exist, having its node at x, y, and such that one of its branches shall have 8-point contact with the curve at that point. With us AC — B' is the simplest example, after the Mongian A, of an invariant in the capital letters il, B, C, ....
LECTURE XXXI.
We may include X. among the arbitrary constants in the primitive equation Y = X^Z'~\ which can also be written in the form
X log Z - log F + (1 - X) log Z = 0, or (X, T, Z being general linear functions of a;, y, 1) in the equivalent form X log (y + ox + ;8) - log (y + o'a; + /S') + ( 1 - X) log (y + d'x + /S") = const., which evidently contains 8 independent arbitrary constants.
One of these will be made to disappear by differentiation, and thus we shall obtain a differential equation of the first order, containing 7 arbitrary constants, identical (when the constants are rearranged) with
(y - xt) (Ix + my) + t {I'x + my + n') + l"x + m"y + n" = 0, which is known as Jacobi's Equation.
For, by differentiating the primitive equation, we obtain X (« + a) (y + oa; + /3)-' - (« + a') (y + a'a; + yS')-'
+ (1 - X) (« + o") (y + a."x + ;9")- = 0, which, when cleared of negative indices by multiplication, becomes X (y + a'x + ^') {(y + al'x + /S") (< + a) - (y + cu; + /3) (< + a"))
+ {y + ax + ^)[{y + a'x + /3') {t + a") - (y + o^'x + /3") {t + a')} = 0. Writing this equation in the equivalent form X (y + a'x + 0) ((a - a") (y - xt) + (^" - /8) < + (a/3" - a"/3)!
+ (y + cu; + ,9) {(a" - a') {y - xt) + (/3' - ,3") t + (a"y8' - a'/3")l = 0, it is easily seen to be identical with Jacobi's equation given above.
The seven arbitrary constants which occur in Jacobi's equation are the mutual ratios of the eight coefficients I, m, V, m\ n', I", to", n", any one of which may have an arbitrarily chosen value assigned to it.
Taking m = — 1, the equation may be written in the form Pt + lxy-y'' + l"x + m"y + n" = 0, where P=l'x + my + n' - Ix' + xy.
42]
Lectures on the Theory of Reciprocants
487
In order to eliminate n" and I", we differentiate the above equation twice. The first differentiation gives
■2aP + t(r + la:-2>/+m') + ly + l" = 0,
where P" =^!- = 1' + m't-2la; + y + xt, and the second differentiation gives ax
6bP + 2a {2P' + lx-2y + m") + 1 (P" + 21- 2t) = 0. l^ow, P" = ~-=2a{m' + x)+2(t-l); so that, on substituting this
value, the above equation becomes
SbP + aQ = 0, where Q=2F + lx-2y+m" ■\-m't + xt
= 21' + 3m't - Zlx + ^xt + m". Differentiating (1) we have
tl2cP + SbP + SbQ + aQ: = 0, where Q' = 3 (i - 0 + 6a (x + m) =2R + 6a8, suppose
Thus we have 4cP + bF + bQ + aR + 2a'S = 0.
(1)
tfot
(2) Differentiating this 4 times in succession, and at each step substituting
P*. Of, R', S',
their values 2R + 2aS, 3R + QaS, 2a, 1,
■we obtain 4 more equations, from which, combined with the 2 previously
obtained, we can eliminate
P, P', Q, R, S.
Thus, differentiating (2), we find
20dP + 8cF + b(2R + 2aS) +4>cQ+b {SR + 6aS)
+ SbR + 2a* +I2ab8 + 2a'' = 0; that is, odP + 2cP' + cQ + 2bR + habS + a" = 0,
and continuing the same process,
6eP + 3dP' + dQ + ScR + (6ac + 36') S + 3a6 = 0, 7/P + 4eP' + eQ + MR + (7ad + 7bc)S + (4ac + 26») = 0, 8gP + S/P +fQ + 5eR + (8ae + 8bd + 4c») 8 + (5ad + 56c) = 0. The result of elimination is
0 0
2a' 0
5ab a'
6ac + 36=" 3a6
lad + 76c *ac + 26'
8ae + 86d + 4c' Sac? + 56c where the determinant equated to zero is a Principiant.
(3)
(4) (5) (6)
|
36 |
0 |
a |
0 |
|
4c |
6 |
b |
a |
|
od |
2c |
c |
26 |
|
6e |
M |
d |
3c |
|
7/ |
4e |
e |
U |
|
% |
5/ |
f |
5e |
= 0,
488
Lectures on the Theory of Reciprocanta
[42
In his Thiae swr les Invariants diffdrentiels, p. 42, M. Halpben states that this equation can be found by eliminating the constants from Jacobi's equa- tion, but he does not set out the work. When in the above determinant twice the 3rd column is added to the second, it becomes exactly identical with the one given by Halphen, which he calls T.
We proceed to express the above result in terms of the capital letters, using the method explained in Lecture XXIX., and observing that the deter- minant is of degree 8 and of weight 12; so that in this case i— w=8— 12=— 4, showing that the final result has to be multiplied by a~*.
Substituting in the determinant for
it becomes
a b 1 0
0
0
5A
6B
1C
8D + 25.4''
c d
0 A
0
0
0
2A
4,B
oG
B
1
0 0 A B G
f C
..^-
0
1
0 0
44 oB
0
2
0
0
7A
8B
0 0 1 0 0 5A
Subtracting the last column multiplied by 5.4 from the first, and the 4th column multiplied by 2 from the 5th, and then striking out rows and columns, we obtain
0
0
1
0
0
5A
|
0 |
0 |
1 |
0 |
0 |
|
|
0 |
0 |
0 |
1 |
0 |
|
|
0 |
0 |
0 |
0 |
0 |
|
|
6B |
:iA |
A |
0 |
0 |
|
|
70 |
45 |
B |
4J. |
-A |
|
|
8D |
50 |
G |
5B |
-2B |
|
|
0 |
0 |
1 |
0 |
0 |
|
|
0 |
0 |
0 |
0 |
1 |
|
|
6B |
3A |
0 |
0 |
0 |
|
|
7G |
45 |
iA |
— |
A |
0 |
8D 50 5B -IB
0 0 0 1
65 34 0 0
7C 45 -4 0
8D hG -25 54
24(4»i)- 3450-1- 25»).
65 34 0 70 45 4 8D bG 25
42] Lectures on the Theon-y of Reciprocants 489
If, using Halphen's notation, we call the principiant now under considera- tion T, what we have proved is that
T = 24a-* {AW -SABC + 2B'),
and consequently that AW-SABC+2B^ is divisible by a*.
The differential equation T^O corresponds, as we have seen, to the com- plete primitive Y = X^Z'~^, in which X is counted as one of the arbitrary constants.
This result may be otherwise obtained. For we have shown in the pre- ceding Lecture that the differential equation of the seventh order, from which all the arbitrary constants except X have disappeared, has the form
(AG-B'y = KA\ ■where k depends solely on \.
Writing this equation in the fonn
{AC - :^) A- i = const.,
I and differentiating with respect to x, we remove the remaining arbitrary con- 1 Btant, and thus obtain the differential equation of the 8th order free from all farbiti-ary constants, a result which, to a factor ^res, must coincide with
T=0.
We proceed to show how this differentiation may be performed without introducing any of the small letters. In the first place, it is clear that since
(r = 4 (ac - b') dt+5{ad- be) dc + 6{ae-bd)dd+ ... does not contain da and is linear in the other differential reciprocals db, 9c, ..., Ga»^{A, B, C, ...) = a'G^{A, B, C, ...)
fd^ ^ . d^ „^ d<^
-S«^-S«''-S«<'--)'
And since we have
GA = 65,
GB = 1G-^MA,
GC = 8D + 2MB,
it follows immediately that
Ga'^ {A, B, G, ...) = a» {QBd^ + TGde + ^Ddc + ...) ^ + am{AdB+ 2Bdc + SGdD + ...) ^. This is true for any function of the capital letters, whatever its nature may be ; but when 4> is a principiant, it is also an invariant in the large letters ; so that in this case we have
{AdB + 2Bdc + ^Cdj,+ ...)^ = 0 and Ga*^ = a»{6Bd^+7CdB + SDdc+.-.)^.
490 Lectures on the Theory of Reciprocants [42
Now, the operation of 6 on a function of degree i and weight w is equi- valent to that of a -j (3i + w) b, or to that of a -=- , when both i = 0 and
cm: dx
w = 0 (which happens in the case of a plenarily absolute form). Hence, if we suppose ^ to be a plenarily absolute principiant, 0^ is also a principiant, though not a plenarily absolute one.
For a is a principiant, and -r- is a principiant ; therefore o -r- or G^ ia
one also*. Thus
6Bd^ + 7CdB + SDdc+..., acting on any plenarily absolute principiant, generates another principiant, but not a plenarily absolute one.
We now resume the consideration of the equation
{AC -B')A-^ = const. Differentiating and multiplying by a, we have
a^{(AG-B')A-n = 0.
Hence, by what precedes,
(653^ + 7Cde + 8Ddc) {{AC-&') A'i} =0; or, using 0 to denote the operator,
6Bdj, + '7GdB + 8Ddc+..., A-iB(AG-B^)-^A-'^(AG-B')®A = 0; or, observing that &A = QB,
A%(AC-&)-lQB{AC-B') = 0. This gives A (6BC - UBC + HAD) - 16B {AC - B') = 0 ; or finally AW-3ABG + 2B' = 0.
We may find a generator for principiants expressed in terms of the large letters similar to the expression for the reciprocant generator G in terms of
* See the concluding paragraph of Lecture XXV. [p. 450 above], where it was shown that P,
(IP being a principiant (of degree i and weight w), a- (3i + v>)bP is a reciprocant, and
dP a -^-(Si + 2w)bP an invariant. This proves, what we omitted to mention there, that P
being a zero-weight principiant,
QP= la- — 3i6 j P is a principiant.
It may here be remarked that a principiant of degree i and of zero loeUjht is equal to the corresponding plenarily absolute principiant (which is a function of the large letters only) multiplied by the factor o*, on which the operator Q does not act.
I
42] Lectures on the Theory of Reciprocants 491
the small letters. For let P be any principiant, of weight w, which, when reduced to zero weight by division by A *, becomes PA * ; then
&(PA~»)
is a principiant. But
ir-l
e(PA ») = 4 3 (A@-2wB)P,
-1
I
where, remembering that A '* is a principiant, (A® — 2wB) P is one also.
Now, the weights of A,B,G,D,...
being 3, 4, 5, 6,...,
we may write w = ZAd^ + 459^ + bCdc + ^Ddn + ... ,
and consequently
= (lAG-B,B')dB + iSAD-\0BG)dc-\-{^AE-\1BD)ds + ..., which is the generator in question.
»As an easy example of its use, suppose it to operate ouAG — B^; then [{1AG-%B'')dB + {^AD-\0BG)dc]{AC-B^) = - 25 (7^ a - 8fi») + ^ (8 J i) - 1050) = S{A-'D-ZABG+2B'). The generator just obtained,
OAG-SB'')dB + {^AD-\0BG)dc + {9AE-\2BD)dD + -... is a linear combination of Cayley's two generators (given in Lecture IV., [p. 327, above]), which, when we write A, B, C, ... instead of the correspond- ing small letters, become
{AG-B')dB + {AD-BC)dc + {AE-BD)di> + ...
and (AG-2B')dB + (2AD-4,BG)dc + (SAE-6BD)di, + ....
Thus we shall obtain the principiant generator by adding the second of Cayley's generators to six times the first. Either of Cayley's generators acting on a principiant would of course give an invariant in the large letters (that is, a principiant), but the combination we have used has special relation to the theory of the generation of principiants by differentiation.
492 Lectures on the Theory of Beciprocants [42
LECTURE XXXII.
I will now pass on to the consideration of the Principiant which, when equated to zero, gives the Differential Equation to the most general Algebraic Curve of any order.
The Differential Equation to a Conic (see the reference given [p. 380, above]) was obtained by Monge in the first decade of this century. This was followed by the determination, in 1868, by Mr Samuel Roberts, of the Differential Equation to the general Cubic (see Vol. X. p. 47 of Mathematical Questions and Solutions from the Educational Times). I do not consider that any substantial advance was made upon this by Mr Muir, in the Philosophical Magazine for February, 1886, except that he sets out explicitly the quantities to be eliminated in obtaining the final result. These may, of course, be collected from the processes indicated by Mr Roberts, but are not set forth by him. In speaking of the history of this part of the subject, I pass over M. Halphen's process for obtaining the Differential Equation to a Conic. It is very ingenious, like everything that proceeds from his pen, but, being founded on the solution of a quadratic equation, does not admit of being extended to forms of a higher degree, and consequently, viewed in the light of subsequent experience, must be regarded as faulty in point of method.
Let the Differential Equation to a curve of any order, when written in its simplest form, containing no extraneous factor, be ;^; = 0. It is convenient to give ;)^ a single name ; I call it the Criterion. The integral of the Criterion to a curve of order n must contain as many arbitrary constants as there are ratios between the coefficients of a curve of the «th order. The number of
these ratios being ^ 1, the order of the Criterion ought to be
«' + 3w 2
It must be independent of Perspective Projection, because projection does not affect the order of a curve. Hence it is a Principiant, and as such ought not (when y is regarded as the dependent and x as the independent variable)
to contain either x,y or ^ (see Lecture XXIV. [p. 438, above]).
Let Z7 = 0 be an algebraical equation of the 7ith order between x, y. I write symbolically
U=(p + qx-\ryY = m", ]
where the different powers and products of p, q, 1 which occur in the expan-
42] Lectures on the Theory of Reciprocants 493
sion of m" are considered as representing the different coefficients in U\ so that, for example, if n = 3 the coefficients of
y», ^^x, 'Sy", Syaf, 6yx, Sy, a?, 3a;=, Zx, 1
are represented by
1. 9. P' <f' p(i' p"' y*. Pf' P% P^-
The number of terms in U is
(n + !)(« + 2)
1 + 2 + 3 + .. .+(«+!) =
2
The number of these containing y is
l+2 + 3 + ...+n = "i^\
To obtain the Differential Equation we equate to zero the Differential Derivatives of U of all orders from n + 1 to J(n^ + 3B) inclusive, and from the ^ {n^ + n) equations thus formed eliminate the ^ (n^ + n) coefficients of the terms in U containing y.
1 All the coefficients of pure powers of x will obviously disappear under dififerentiation ; for no power of x higher than x" occurs in U, and no differential derivative of U of lower order than n. + 1 is taken.
We thus find a differential equation of the order J^ (n^ + 3n), free from all
e ^(w'+3n+2) coefficients of U. This equation might conceivably contain
', y and all the successive differential derivatives of y with respect to x.
But we know d priori that it ought not to contain either x, y or -— ; and in
fact we shall be able so to conduct the elimination that x, y and ~ appear only in the quantities to be eliminated and not in the final result.
Treating u = p + qx + y as an ordinary algebraical quantity, we have, by Taylor's theorem,
1 d'u" ,.. / , h^ h?
I
1.2.3...r' daf
I ft" A' \"
= CO. A' in ( M + Mi/t + itj P2 + "» 1 2~3'^""j '
where u,, Ma, Uj, ... are the successive differential derivatives of ?< with respect to X. And this result will remain true when for m" we write V, meaning
thereby that - . -r—^ will be the quantitative interpretation of the
function of m, Mj, u^, ... which multiplies A*" in the expansion of
/ , A' \»
I M + M,/i + Mi, ^— 2 + ...1 ,
subject to the condition that this function shall be linear in the coefficients
494 Lectures on the Theory of Reciprocanta [42
of U. This condition can be fulfilled in only one way, so that there is no ambiguity in such interpretation. Hence the equations obtained by equating to zero the successive diflferential derivatives of U of all orders from n + 1 to ^ (n' + 3n) inclusive may be written under the form
u + ttiA + «, J— 2 + «^ p-y-g + ...j =0,
where r = n + 1, n + 2, n + 3, ... ^ (n« + 3n).
Now, using y,, y^, y,, ... to denote the successive differential derivatives of y with respect to x, we have
ih = q + yu Ui = y2, «8 = y3, •••,
and, in general, Ui = y,- when i is any positive integer greater than 1. Thus
/ A' h* \"
CO. h" in f M + MaA + y, j-^ + y, ^ ^ S "*" ' " j "" ^ '
or, employing the usual modified derivatives a,h,c, ... ,
CO. hr in (u + «iA + ah'' + hh^ + ch* + ...)" = 0.
Writing now Q= ah^ + h¥ + ch* -{■... ,
and expanding (m + u^h + Q)" in ascending powers of Q, we have
CO. hr in {(« + «, A)» + « (m + M,A)»-' Q + "^"~^^ (m + m,A)"-» Q^ + . . .1 = 0,
where, remembering that r > n, the value of co. /i*" in (u + Uih)" is zero; so that, omitting this term, we may write
CO. h'- in L {u + «,A)"-> Q + '"^^y^ (" + "^''^"~'' Q' + ... + Q"} = 0.
The quantities to be eliminated will now be combinations of the various powers of u, u^ and 1. Their number will be the same as that of the terms in («, w,, 1)""', which is ^ (n* + n), the same number as that of the equations between which the elimination is to be performed.
We now use {m . fi) to denote the coefficient of A™ in Q** (which, since
Q = ah^ + bli' + ch*+...,
will be independent of the combinations of u and «i to be eliminated), and in writing out the ^ (n' + n) equations which result from making the coefficients
of A"+',A»+«, ...A « in
n (u + «. A)»- Q + 1(^^ („ + u,hy-^ Q» + . . . + Q«
42]
Lectures on the Theory of Reciprocants
495
vanish, we arrange their terms according to ascending values of m and ft,. Thus, making the coeflScient of /i"+' vanish, we find
nwi»-' (2 . 1) + n (n - 1) V"'^ (3.1) +
nin— V\
' (3. 2)+. ..+(« + !. n) = 0,
and similarly, making the coefficient of A""*"' vanish,
n(n— 1)
mt,»-'(3.1) + n(n-l)iii»-»M(4.1) +
1.2
iii"-= ( -t . 2 ) + . . . + (n + 2 . « ) = 0.
So in general the equation obtained by making the coefficient of A"+'' vanish consists of a series of numerical multiples (which are independent of the value of k) of u^~'u'~'^ (0 + k, ij) where t; has all values from 1 to ^ inclusive, and 8 all values from 1 to n inclusive. Hence, by elimination, we find
|
(2.1) |
(3. |
1) |
(3.2) |
(4.1) |
(4.2) |
(4.3) (5 |
1) (5 |
2) (5.3) |
(5.4)... |
|
(3.1) |
(4.1) |
(4.2) |
(5.1) |
(5.2) |
(5.3) (6 |
1) (6 |
2) (6.3) |
(6.4)... |
|
|
(4.1) |
(5. |
(5.2) |
(6.1) |
(6.2) |
(6.3) (7 |
1) (7 |
2) (7.3) |
(7.4)... |
|
|
(5.1) |
(6 |
(6.2) |
(7.1) |
(7.2) |
(7.3) (8 |
1) (8 |
.2) (8.3) |
(8.4)... |
|
|
(6.1) |
(7. |
(7.2) |
(8.1) |
(8.2) |
(8.3) (9 |
1) (9 |
2) (9.3) |
(9.4)... |
|
|
(7.1) |
(8 |
(8.2) |
(9.1) |
(9.2) |
(9 . 3) (10 |
1) (10.2) (10.3) (10.4)... |
|||
|
(8.1) |
(9 |
(9.2)(10.1)(10.2)(10.3)(11 |
1)(11 |
.2) (11.3) (11.4)... |
|||||
|
(9 . 1) (10 . 1) (10 . 2) (11 . 1) (11 . 2) (11 . 3) (12 . 1) (12 . 2) (12 . 3) (12 . 4) . . . |
|||||||||
|
(10.1) (11 |
. 1) (11 . 2) (12 . 1) (12 . 2) (12 . 3) (13 |
1) (13.2) (13.3) (13.4)... |
|||||||
|
(11 . 1) (12 . 1) (12 . 2) (13 . 1) (13 . 2) (13 . 3) (14 . 1) (14 |
.2) (14.3) (14.4)... |
= 0,
where the determinant on the left-hand side, consisting of ^(n' + w) rows and columns, is the Criterion of the curve of the wth order.
Thus in the case of the Cubic Criterion, which we shall specially consider, we have w = 3, and the elimination of 3ui', 6u,m, 3mi, 3u', 3m and 1 between the six equations
3V(2.1) + 6ji,u(3.1) + 3w,(3.2) + 3«»(4.1) + 3M(4.2)-|-(4.3) = 0, 3V(3.1) + 6«,m(4.1) + .3!*,(4.2) + .3m»(5.1) + 3m(5.2) + (5.3) = 0, 3tt,»(4.1)+6u,«(5.1) + 3«,(5.2)4-3Mn6.1) + 3M(6.2) + (6.3) = 0, 3m,>(5.1) + 6m,m(6.1) + 3i*,(6.2) + 3m'(7.1) + 3m(7.2) + (7.3) = 0, 3w,» (6 . 1) + 6m, M. (7 . 1 ) + 3m, (7.2) + 3u' (8 . 1) + 3m (8 . 2) + (8 . 3) = 0, 3u,'(7.1) + 6«im(8.1) + .3m,(8.2) + 3«''(9.1) + 3m(9.2) + (9.3) = 0,
gives the Cubic Criterion in the form of the determinant
|
(2.1) |
(3.1) |
(3.2) |
(4.1) |
(4.2) |
(4.3) |
|
(3.1) |
(4.1) |
(4.2) |
(5.1) |
(5.2) |
(5.3) |
|
(4.1) |
(5.1) |
(5.2) |
(6.1) |
(6.2) |
(6.3) |
|
(5.1) |
(6.1) |
(6.2) |
(7.1) |
(7.2) |
(7.3) |
|
(6.1) |
(7.1) |
(7.2) |
(8.1) |
(8.2) |
(8.3) |
|
(7.1) |
(8.1) |
(8.2) |
(9.1) |
(9.2) |
(9.3) |
496 Lectures on the Theory of Reciprocants [42
Remembering that
{m.ii) = CO. A" in (aA' + bh' + ch* + ...y, it is easy to express the Criterion explicitly in terms of o, 6, c, ....
Thus, since {ah? + bh? + ch* + ...)> = a?h* + 2a6A» + (2ac + 6») h* + (2ad + 26c) A"
+ (2ae + 26d + c») A» + (2a/+ 2be + 2cd) A» + . . . and
(aA» + 6A» + cA^ + ...)' = a» A« + 3a»6A' + (3a»c + 3o6») A»
+ (3a»d + 6a6c + t^) A» + ...,
the Cubic Criterion may be written in the form
o i!> 0 CO? 0
6 c o» d lab 0
c d 2a6 e 2ac + 6' a'
a
d e 2ac+b-' f 2ad + 2bc 3a>6
e f 2ad + 26c ^r 2ae + 26d + c= 3a'c + Sat"
/ jr 2ae + 26d + c= A 2a/+26e + 2cd 3a=d + 6a6c + 6'
in which it was originally obtained by Mr Roberts.
M. Halphen has remarked that the minor of A in the Cubic Criterion is the Principiant which he calls A (our AC — B-) multiplied by a (see p. 50 of his These).
We proceed to determine the degree and weight of the Criterion of the curve of the nth order. These are the same as the degree and weight of its diagonal
(2.1)(4.1)(5.2)(7.1)(8.2)(9.3)(11.1)(12.2)(13.3)(14.4).... which consists of ^ (?;' + n) factors, separable into n groups,
(2.1), (4.1)(5.2), (7.1)(8.2)(9.3), (11 .1)(12.2)(13.3)(14.4), ... containing 1, 2, 3, 4, ... n factors respectively. Now,
(m . /n) = CO. A" in (aA» + bh' + ch* + . . .)>•■ = CO. A"^'" in (a + bh + cA^ + . . .)^
and consequently (m . /*) is of degree /^ and weight m — 2/j,. Hence the degree of the Criterion (found by adding together the second numbers of the duads which occur in the diagonal) is
1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + ... + (1 + 2 + 3 + ... + »/)
= 1 + 3 + 6 + 10 + ...+*^
_n(w + l)(w + 2) 6
42]
Lectures on the Theory of Reciprocants
497
To find the weight of the Criterion, we begin by arranging the factors of its diagonal according to their weight. This is done by writing each group of factors in reverse order, so that the diagonal is written thus :
(2.1)(5.2)(4.1)(9.3)(8.2)(7.1)(14.4)(13.3)(12.2)(11.1)....
The weights of the factors are now seen to be 0, 1, 2, 3, ... — ^ 1 ;
there being i(n''+n) factors in the diagonal, one of them of zero weight. Hence the weight of the Criterion is
1 + 2 + 3 + ...+I
C
ri' + n
-1
n' + n
{n-\)n (7i.+ l)(ft+2)
I
If, in the above formulae, we make w = 2, we shall find that the degree is 4 and the weight 3, whereas the Mongian a^d — 3aic + 26' (which is the Criterion of the second order) is of degree 3 and weight 3.
To account for this discrepancy, observe that in this case
(2.1) (3.1) (3.2)
(3.1) (4.1) (4.2)
(4.1) (5.1) (5.2) which is divisible by a, the other factor being the Mongian, as may easily be verified. This is the only case in which the determinant expression for the Criterion contains an irrelevant factor.
|
a |
b |
0 |
|
b |
c |
a^ |
|
c |
d |
lab |
To express the Cubic Criterion in terms of a. A, B, C, D, E, we first
15. Thus
3 4 5 2 3 4.5
remark that its degree is ' '- = 10, and its weight
8
deg. wt.
the Cubic Criterion is expres-sible as the product of a~°(10 — 15 = — 5) into a function of the capital letters, which we determine by the usual method of substituting for
a, b, c, d, e, f, g ^ > ^ . >
1, 0, 0, A, B, C, I> + ^A\ E + ^AB.
When these substitutions are made, the Cubic Criterion becomes
|
1 |
0 |
0 |
0 |
1 |
0 |
||||
|
0 |
0 |
1 |
A |
0 |
0 |
||||
|
0 |
A |
0 |
B |
0 |
1 |
||||
|
A |
B |
0 |
C |
2A |
0 |
||||
|
B |
C |
2A |
D + |
25 8 |
A^ |
IB' |
0 |
||
|
C |
D + |
25 8 |
A^ |
2B |
E + |
15 2 |
AB |
2C |
3 |
8. IV.
32
498
Lectures on the Theory of Redprocants
[42
Subtracting the first column of this determinant from the fifth and reducing, we obtain
0 \ A 0 0
A OB 0 1
B 0 C ^0
2A D + ^A'
O
B 0
D + ~A^ 2B E+^AB 0 34
Again, subtracting the second column multiplied by A from the third and reducing, there results
A B 0 1
B C A 0
d + Ia^
B 0
B + ^A' E + ^AB C SA
which, after subtracting the first row multiplied by 3A from the last and reducing, becomes
BOA
C
i) + I 4» B
B + ^A' E + ^AB C
= b(cb + ^a'C-be-^ab''\ + c(bd + ^A'B-cA
+ a{ce+Iabg-B'-Iaw-Ia*)
= {AGE-B'E- AD^ + 2BGD -C')-^A (AW - 8ABC + 2B') - ^ A'.
This expression, which is of degree-weight 15 . 15, instead of 10 . 15, must be divided by a' to give the correct value of the Cubic Criterion.
42] Lectures on the Theory of Reciprocants 499
LECTURE XXXIII.
In this Lecture it is proposed to investigate the differential equation of a cubic curve having a given absolute invariant ™^ .
Since the value of „-, is the same for any homographic transformation
of the cubic as for the original curve, the dififerential equation in question
must be of the form
... S"
Plenarily absolute priacipiant = m^.
This equation is (as we see at once by dififerentiating it) the integral of another of the form
Principiant = 0,
which is satisfied, independently of the value of the absolute invariant, at all points on a perfectly general cubic.
Now, the differential equation of the general cubic is of the 9th order, and when expressed in terms of A, B, C, ... contains no letter beyond E. Hence the integral of this equation, which we are in search of, will be of the 8th order and will contain no capital letter beyond D.
When no letters beyond D are involved, all plenarily absolute principiants are functions of the two fundamental, or protomorphic, ones,
AG-B* A*D-SABC+2B'
Ai ' A*
Thus the differential equation of a cubic with a given absolute invariant is of the form
„fAC-B' A^D-SABC+2B^\_S' \ Ai ' A* IT''
M. Halphen actually integrates the differential equation of the general cubic, which he shows (on p. 52 of hia These sur les Invariants Diffirentiels) may be put und^r the form
^rdf+|?-|(?+3)(f+27)Jdr=o.
where, in our notation,
^ A' • ^~ A» •
32—2
500 Lectures on the Theory of Eeciprocants [42
The integral of this equation, which M. Halphen obtains partly from geo- metrical considerations, involves an arbitrary parameter depending on ^. His result is as follows :
where
2«Q = 2" f » + 2* (^ + 3») (? - 3» . 5) ? + (f + 3')*, and T' - eihS" = 0.
(Two misprints, which are here corrected, occur in the expression for R as given on p. 54 of the Thhe.)
In this result the invariant S differs in sign from the invariant usually denoted by that letter. Thus the discriminant is T' — 64<S' instead of T' + 64S'.
When h = l the discriminant vanishes and the differential equation
becomes
R^-Q>= 0.
This is divisible by a numerical multiple of f ; in fact,
R'=Q''+2\3'^'P,
where 2»P = (2»§- + ^» - 2 . 3'| - 3")^ + 2» . 3f ' = 0
is the differential equation of a nodal cubic, previously obtained by Halphen.
It is from a knowledge of the fact that F = 0 and another algebraic relation between f and f, which he finds by trial to be Q = 0, constitute two particular integrals of the differential equation to the general cubic, that he arrives, not by any regular method but by repeated strokes of penetrative genius, at the general integral
R'' = hQ'.
In establishing the relation T" — 64:hS-' = 0 he supposes that, by means of. the equation to the cubic and its differentials as far as the 8th order inclusive, 1 the coefficients of the cubic have been expressed in terms of the variables x, y I and the derivatives of y with respect to x up to the 8th order, and that the j values thus obtained for the coefficients have been substituted in Aronhold's^ S and T.
The abbreviations introduced by the use of our notation enable us tos actually perform this calculation, which would otherwise be impracticable in' consequence of the enormous amount of labour required ; and we shall use this method to obtain the plenarily absolute principiant which, equated
to ™^, gives the differential equation to a cubic with a known absolute
invariant.
1
42] Lectures on the Theory of Reciprocants 501
Using the symbolic notation explained in Lecture XXXII. [above, p. 492], the equation of the cubic and its first eight differentials are m' = 0, M'zt, = 0, 2mV + w'Mj = 0, 2m,' + 6mMjM2 + u'Mj = 0,
3V(2 . 1) + 6mjw(3 . 1) + 3m, (3 . 2) + 3m'(4 . 1) + 3m(4 . 2) + (4. 3) = 0, 3m,»(3 . 1) + 6m,m (4.1) + 3?^ (4.2) + 3m''(5 . 1) + 3m (5 . 2) + (5 . 3) = 0, 3m,» (4 . 1) + 6m,m (5 . 1 ) + 3w, (.5 . 2) + 3m= (6 . 1) + 3m (6 . 2) + (6 . 3) = 0, 3«.»(5 . 1) + 6m,m(6 . 1) + 3m. (6 . 2) + 3m^(7 . 1) + 3m(7 . 2) + (7 . 3) = 0, 3u,H6 . 1) + 6m,m (7 . 1) + 3m, (7 . 2) + 3m» (8 . 1) + 3m (8 . 2) + (8 . 3) = 0, where u=p + qx + y, m, = g + t, Mj = 2a, itj = 66 ;
, ^ dy 1 d^y , 1 d'y
tasusual. t = £, a = ^.^, 6 = ^.^,...;
(m./i) denotes the coeflBcient of A"* in (oA' + 6A* + cA* + ...Y; and if, as in Salmon's Higher Plane Curves (2nd edit., p. 187), the equation of the cubic is taken to be r + ^a^x + 3<Xiy + 360** + ^hioyy + 36jy^ + c^a? + Scjar'y + Sc^xy' + Csy' = 0, then, in the above equations, the symbols .
I p', ffq, p', pq\ pq, p, q', ^, q, 1
stand for r, a„, a^, bo, 6,, 62. Co, c,, Cj, c,.
These nine equations are sufiBcient to determine the values of the
S' coefficients of the cubic which have to be substituted in ^ in order to
obtain our differential equation, which will be, as we have seen, of the form
\ A* ' A* ) T^'
Since this equation contains nothing which involves x, y, or t, these letters must have disappeared spontaneously in the process of forming it, and con- sequently we may, at any stage of the work, give x, y, and t any arbitrary values without thereby affecting the result. Let, then,
x = 0, y = 0, t — 0, so that u=p, Ui = q, u^=2a, m, = 66, and the first four equations become
u' = p' =r —0, u'ui = p'q = Oj = 0, ^ (2uui' + u'Ui) = pq' + p'a = 60 + a, a = 0,
^ (2mi' + 6mm, M, + m'Mj) = g* + 6pqa + 3;/6 = c„ + 661a + 3a,6 == 0.
502 Lectures on the Theory of Reeiprocants [42
Writing in the last five equations
v? = f =<h, u=p =6„
1 =c„ we have
3c(2.1) + 66,(3.1) + 3c(3.2) + 3ai(4.1) + 36,(4.2) + c,(4.3) = 0, 3c(3.1) + 66.(4.1) + 3c,(4.2) + 3a,(5.1) + 36,(5.2) + c,(5.3) = 0, 3c(4.1) + 66,(5.1) + 3c,(5.2) + 3a,(6.1) + 36,(6.2) + c,(6.3) = 0, 3c (5 . 1) + 66, (6 . 1) + 3c, (6 . 2) + 3a, (7 . 1) + 36, (7 . 2) + c (7 . 3) = 0, 3c, (6. 1) + 66, (7.1) + 3c (7. 2) + 3a, (8.1) + 36,(8. 2) + Ca(8. 3) = 0*.
. S' Substituting in ^ for r, a„, 6o, c^ their values given by the equations
r = 0, ao = 0, 60 + 0,0 = 0, Co + 66,a + 3a,6 = 0, and for the mutual ratios of a,, 6,, 62, c,, c,, C3 their values found by solving the last five equations, we obtain the differential equation required.
Referring to Salmon's Higher Plane Curves, p. 188, we see that, when r = 0,
8 = (c^a^) + (ch^a) - {¥f,
r = 4 {&a?) - 3 (c»6«a») - 12 (6^ (c6^a) + 8 (6=)', where (c^a?'), (chi'a),... are functions of Oo, a,, 60, 6,, 6,, Co, c,, c,, C3, which, when Co = 0, become
{(^a^) = (coCa - c,») a,», (c6'a) = (6o''c3 - 36o6iCa + 6062C1 + 26i^Ci - 6,6200) Oj,
(6>) = 6.6,-6,=, {(^a*) = (co'c, - 3coC,c, + 2c«) o,», {(^h^a?) = (00*6," - 4coC,6,6, - 2000,606, - 4coC,6,' + 8coC36„6,
+ 8c,''6,'' + 4c,«6„6, - 12c,c,6o6, - 80,0360^ + 90,^60") a,«. We have now reached a point at which the work will be greatly facilitated by the introduction of the capital letters A, B, C, D. This is usually done by writing for
a, 6, c, d, e, f, g, ^
1, 0, 0, A£. G, i> + ^^'.
• These equations are only set out for the sake of distinotnesa ; when our abbreviations are introduced, only two terms survive in the first three, and only three terms in the last two of these five equations.
I
42]
Lectures on the Theory of Reciprocants
503
But in the present instance we may make a further simplification by writing
^ = 1, 5 = 0, C=G„ D = D„ for the only effect of this will be to make the final result take the form
^ , „(AC-B^ A^D-SABC+2B'\ -S' instead of F 5 — , = T^r •
The form of the function will not be affected by writing in it ^ = 1,5 = 0, and the letters A, B can be restored at pleasure by making
C,=
AG-B^
A^
i>> =
A^D-^ABC-¥2B^
Hence we may write for
a, h, c, d, e, /, g, 1, 0, 0. 1, 0, C,, A +
25 8 •
I
Instead of the coefficient of
A™ in (ah^ + bh" + ch* + ...y, (wi . fi) will now signify
Thus we have
CO. A™ in Ia' + h' + G,hJ + (a + ^) h'
(2.1)=1 (3.1) = 0 (4.1) = 0 (o.l)=l (6.1) = 0 (7.1) = C,
(8.1) = A +
25
(3 . 2) = 0
(4.2)= 1 (5.2) = 0 (6 . 2) = 0 (7. 2)= 2 (8.2) = 0
(4.3) = 0, (5.3) = 0, (6.3) = 1, (7.3) = 0, (8.3) = 0.
Hence the equations which give Oj, 61, 62. c,, Cj, Cj become
c, + 6a = 0,
Cj + a, = 0,
66, + Cj = 0,
C + a,C. + 26, = 0,
2.5^
A + ^1 = 0.
26,C, + 2c, + a,fA
From the first four of these, coupled with the equations
60 + «i = 0, c„ + 66, = 0,
504 Lecttires on the Theory of Reciprocants [42
obtained by making a = 1 and 6 = 0 in the original equations which give 6,, Co, we find
Co = Cj = — 66,,
C = _6. = -C,'.
Ca= 6, = — a, = (7,, by asauming ai = — Ci (which we are at liberty to do since any one of the coefficients may be chosen arbitrarily).
The last equation then gives
Substituting these values in the previously given expressions for (c^a'), (cb^a), ... we have
(d'a*) (66, + C.')(7.»,
(cb'a) = - (W - 9b, - C.») C,
(6») = C,»-V, (c»a») = (2166,» + 186, C7,» + 20,') (?,», (c»6»a») = (3126,» + 206,'' (7,' - 246, C,' + 9C,' + 40,') C,'. Hence S = (c^a") + (d>''a) - (6'')=
= - C," + 36, C' - 26,'^C,' - 6,^ and r = 4 (c'a') - 3 (c^'b^a') - 12 (6^ (c6''a) + 8 (6»)»
= - 8C,» - 3 (86,^ - 126, + 9) C," - 126,' (26, - 3) 0,» - 86i«. To express S and T in terms of A, B, C, D, we write
fy _ 4^5» A 9 _ JAD-3ABC + 2S' 9
'" Ai ' '~2+16~ 23^ +16'
or, if we use Halphen's notation in which
^ _288 (AG -B^y ^_2'i(AW-3ABC + 2B»)
we have 2» . 3« C,» = f , 2« . 36, = | + 3',
and consequently,
2^3(26,-3) = f-3^5,
2» . 3» (86,» - 126, + 9) = (2* . 36, - 2» . 3*)» + 2* . 3* = (f - S')" + 2* . 3«. Hence
- 2" . 3*-S = 2" . 3*(7,« + 2" . 3*6, (26, - 3) (7,» + 2'" . 3*6,*
= 2-?^ + 2^^ + 3»)(f - 3^ 5) ?+ (f + 3»)*,
- 2" . 3«r = 2" . 3« C,» + 2« . 3' (86i« - 1 26, + 9) C,»
+ 2«» . 3'6,» (26, - 3) 0,> + 2« . 3«6,« = 2»f > + 2« . 3 [(^ - 3»)» + 2< . 3*] ?»
+ 2».3(f + 3»)'(f - 3'. 5) ? + (^+ 3»)«,
I
42] Lectures on the Theory of Redprocants 505
where the expressions on the right-hand side are 2'Q and 2^2? in Halphen's notation. Thus
-2i°.3*<S=Q, -2>=.3«r=iJ;
so that ^2--2«.3ijy»-~"T^-
This result agrees exactly with Halphen's, if we remember that his 8 is taken with a dififerent sign from ours.
Since «'^ = T + i6= 2Z^ - + 2^.
we may write
* = -I'A'b, = 2' (^^i) - ^ABG + 25») + 3U^ and in like manner
^ = ^'C,' = (4G-B»)». Now 2M « (6,2 + Ci') = * ' + 28 ^,
I which is divisible by A-. Hence if $2 + 2'^ = A^®, we have 0 = 2M« (6i» + C,») = 2«(.4''i)» - QABCD + 4^C" + 45»i) - 35^(7") + 2* . ^-"A^A^D - 3 ABC + 25») + 3*A'. The equations which give S and T in terms of b, and C, may be written -S = (b,' + G,y-3b,C,\
- r = 2» (6,'' + G.'f -2KS' (6,» + (7,») 6, C,» + 3'C',«, and consequently,
- 2'M"S = e^ - 2" . 3<[>^,
- 2"A"T= e» - 2" . 3'©4>S[' + 2'" . 3M=^^ where ©, O, ^ are the rational integral principiants
e = 2« (A'D' - 6ABGD + 4,AC' + 4>B'D - SB'-G'')
+ 2* . 3U' (A"-D -3ABC+ 2B») + S*A', ^=2' (AW- SABC + 2B') + S'A\ 'ir={AG-B'y, which, as we have seen, are connected by the relation
^^ + 2'"^ = A'B. The diflFerential equation of cubics with a given absolute invariant is
(0» - 2" ■ 3<&^)'' 2'/S'
(©' - 2" . 3^0*N[^ + 2*' . S'^'^')" ~ ~ iT^ ' or, as it may also be written,
(©a _ 2" . 3*^)» 2" + 2'/S' (0» - 2" . 3''0*'^ + 2^' . 3'4=^)= = 0.
506 Lectures on the Theory of Reciprocants [42
For a nodal cubic, the discriminant T*-\-2'S* vanishes. Hence the differential equation of a nodal cubic is
(e» - 2" . 3"©*^ + 2" . 3M»^0» - (0' - 2" . 3*^)' = 0. When expanded, and divided by 2''. 3'S^', this reduces to
^>e» - 0=4>» - 2" . 3M'0<I>^ + 2"4>»^ + 2* . 3M«^» = 0, which (since ^'0 — ^=: 2'*') divides out by 2'^, giving
e" - 2» . S'uA'e* + 2'^ + 2" . 2,'A*^ = 0, or, what is the same thing,
©» - 2' . 3U''@4> + 2«*' + 1* . S'A' {A'B - ^2) = 0. This may also be written in the form
(0 - 2= . 2,^A'^ + 2» . Z^Af + 2« (* - 3M«)» = 0, or, replacing 0 and <I> by their values in terms of A, B, C, D, {2«(-4'i)»- 6^£C2) + 4iiC" + 4fi'i) + SB^C")
- 2« . 3M» {A^D - SABC + 2B') - S'A'Y + 2" {AW - MBG + 2fi»)' = 0. For a cubic whose invariant S vanishes, the differential equation is
02-2>».34>^ = O, and for a cubic whose invariant T vanishes,
0» - 2" . S'©*^ + 2=" . 3M=^' = 0. For the cuspidal cubic, both S and T vanish, so that the algebraic equation of the cuspidal cubic is a particular solution of each of these equations. We can, however, replace the system
02 - 2'^ 3<I>^ = 0, (1)
0» - 2" . 3*0*^ + 2^ . 3»4«^» = 0, (2)
by another pair of equations, for one of which the cuspidal cubic is a particular solution, and for the other the complete primitive.
Multiplying the first equation by 0 and subtracting the second from it, we have, after dividing by 2" . 3^,
0«1>-2".3M^^ = O. (3)
From (1) and (3) we obtain
@2(j)2 ^ 2" . 34>»'SP^ = 2^ . 3*A*'9\ Hence *»= 2'.3»Jl*^. (4)
But ^"0 = 4>2 + 2»^,
80 that ^"©O = tf + 2<'ci>'¥.
Substituting in this the values of €><t> and <!>' found from (3) and (4) and dividing by ■^, we have
2".3'.4' = 2«.3'^*+2»<t>,. which gives 4> = 3'A*. (5)
42] Lectures on the Theory of Reciprocants 507
Substituting this value of <I> in (4) and rejecting the factor 3M*, we obtain
3^4'= 28^;
In the course of the work we have only rejected powers of ^ (that is of AC — B') and of ^, of which neither corresponds to the cuspidal cubic.
Since ^=S^A*, it follows that AW - SABC + 2B' = 0. The equation to the cuspidal cubic above obtained is a particular solution of this, its complete primitive being (see Lecture XXXI. [above, p. 486]), F= X''Z^~'', where \ is an arbitrary constant.
LECTURE XXXIV.
I
The preceding 3.3 lectures contain the substance of the lectures on Reciprocants actually delivered, entire or in abstract, in the course of three terms, to a class at the University of Oxford.
A good deal of material remains over which the lecturer has lacked leisure or energy to throw into form, which he hopes to be able to recover and annex to what has gone before as supplemental matter in the convenient form of lectures numbered on from those which have already appeared.
The one that follows is entirely due to Mr Hammond, who has rendered invalxiable aid in compiling, and in many cases bettering, the lectures previously published.
It constitutes probably the most difficult problem in elimination which has been effected up to the present time. J. J. S.
The problem in question is to obtain the differential equation correspond- ing to the complete primitive
(I'ai + m'y + »') = (lx + my + «)* (l"x + m"y + n")'-* (say F=X*Z'~*) by the process of eliminating all the arbitrary constants except \.
The eliminations to be performed become greatly simplified by aid of the following Lemma. If X be any linear function of x and y, and M^ the absolute pure reciprocant corresponding to M\ then
Z,-4i/„Z. = 0,
where S = "*^- S-'="*^- S' = "*^'-
For if we suppose X =lx + my + n,
two SQCcessive differentiations give
a^Xi = I + mt and a^X, + a~ ^bXj = 2ma.
608 Lectures on the Theory of Reciprocants [42
Writing the second of these equations in the form
a~^Xt + a~^bXt = 2m, and differentiating again, we find
Z, - a - ♦6Zi, + o - Hz, + (4.ac - 5&») a - *Z, = 0, or, since 4iMa = (4ac — 56') a - 1^
Zs + iMaX, = 0.
N.B. — Throughout the following work all letters with numerical suffixes are to be considered as derived from the corresponding unsuffixed letters in the same way as, in what precedes, Xj, X^, and Z, are derived from Z; namely by successive differentiations, each of which is accompanied by a division by a*.
Writing the equation
F=Z^Z'-^
(in which Z, T, Z denote any three linear functions of x, y) in the form
logF=XlogZ + (l-\)log^, we obtain by differentiation and division by u^,
^ = X§ + (1-X)§. (1)
Let now Zj = uX,
Z^ = wZ, so that (1) takes the form
v = \u + (1 — \) w, and consequently v^ = \Uj + ( 1 — X) Wj ,
fa = Xttj + (1 — \) W2. By means of the Lemma it can be shown that
M' + 3mMi + Mj + 4ifaM = 0, (2)
x^ + Zvv^ + ?;2 + 4Jlfa?) = 0, (3)
Vfi + BWW, + Wa + ^MaW = 0. (4)
For, since Z, = Xu,
we have Z, = ZjU + Zm, = Z (w= + «,)
and Z, = Zj« + 2Z,m, + Zm^ = Z («» + 3mUi + w,).
Substituting these values for Z, and Z, in
Z, + 4Jlf„Z, = 0, we obtain «' + 3umi + «, + 4ilfaM = 0,
1
I
42] Lectures on the Theory of Reciprocants 509
which proves equation (2). The equations (3) and (4) connecting v, v^, v^ and w, w,, w, are similarly established. We now write
M+ D + w= So)] u — w = ^z) These, combined with v = \u + {\ — \)w,
give u = <ii — {\ — 2)z
t) = o) — (1 — 2X)z ,
w = O) — (\ + 1)0
which, when operated on by a"* -j- twice in succession, peld
Itl = W, — (\ — 2) ^1 ^ Mj = OJg — (\ — 2) Zj
Vi = u>i — {\ — 2,\)zA , i;2=a>j — (1 — 2X)z„_ .
Wi = a)i-(\ + l)5', J W3=fl)j — (X,+ l)z2
When expressed in terms of w, Wi, w^ and ^^, z^, z^, equations (2), (3), and (4) become transformed into
P-(\-2)Q +{\-iyR -(\-2)'0^ =0, (5)
P - (1 - 2\) Q + (1 - 2\fR- (1 - 2X)'0» = 0, (6)
P-(\ + l)Q +(\+l)=ii -(\ + l)'2' =0, (7)
where, for the sake of brevity, we write
a)» + 3a)ft), + a)2 + 4Jl/a«i) = P, Soj'z + 3ft)0i + 3e<),2 + 0^2 + 4ifa2 = Q, 3a)«= + 32:^, = iJ. In order to simplify (5), (6), and (7), we multiply the first of them by \, the second by — 1, and the third by 1 —X, and take their sum, which is obviously independent of P, and from which it is easily seen that the terms containing Q and s? will also disappear. For
\(X-2) -(1-2X) +(1_X)(X + 1) =0, and X(X-2)'-(l-2X)'+(l-X)(X + l)' = 0.
We are thus left with
{X (X - 2)' - (1 - 2X)^ + (1 - X) (X + 1)'') ii = 0, which, on restoring the value of R and reducing, becomes
X(X— \)z{(oz + Zi) = 0.
X Y Z
Now the values of u, v, w, which are equal to -y? , ^ , -^ respectively, being
distinct from each other, z cannot vanish ; for z — 0 would imply u = v = w. Hence, considering X to have any finite numerical value except 1 or 0, we may write
(oz + Zi = 0
610 Lectures on the Theory of Reciprocants [42
in equations (5), (6), (7), which will then become
P-(X-2) (3w,^+z, + 4ilf„z)-(\-2)»«» =0, (8)
P - (1 - 2X) (3«,2: + ^, + ^Maz) - (1 - 2\)' ^' = 0. (9)
P-(\ + l) (3w,z + Zj+4Jf„^)-(\ + l)«a? =0. (10)
Adding these together, we find
3P = ((X - 2)' + (1 - 2X)> + (X + 1 )«} ^> = 8(X-2)(1-2X)(X + 1)0'. Restoring the value of P, and writing for shortness
(X-2)(X + 1)(2X-1) = ^, there results at* + 3wa)j + «*, + ^sMato + pis' = 0.
From any pair of the equations (8), (9), (10) we obtain by subtraction
3ft),2r + z, + 4J/„^ + 3 (X> - X + 1) 2> = 0. Thus, for example, subtracting (10) from (8), we have
3 {Zm,z + z^ + UIaZ) = {(X - 2)» - (X + 1)"} 2» = - 9 (X» - X + l)z'.
Collecting our results, we see that equations (5), (6), (7) may be replaced by
(o' + 8(oa)x + a>, + 4>Ma(»+pz' = 0, (11)
3tOiZ + z^ + 4MaZ + Sqz' = 0, (12)
a>z + Zr = 0, (13)
where i) = (X-2)(X + 1)(2X- 1),
and g = X^ - X + 1,
Differentiating (13), we obtain
<OiZ+ a)Zi + Z3 = 0.
Subtracting this from (12) and adding (13) multiplied by to, the result divides by z, and we find
o)' + 2(o, + 'iMa + Sqz' = 0, (14)
which, when multiplied by to and subtracted from (11), reduces it to
cBWi + co^+pz' — Sqz^to = 0. (15)
Now it has been shown in Lecture XXX. [above, p. 482] that
whence it follows that (14) gives on differentiation WW, + Wj + lOAa + SqzZi = 0.
I
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42] Lectures on the Theory of Reciprocants 511
Combining this with (15) we have
lOAa =ps^ — Sqz (qsz + Zi), or, finally, since wz + 2^ = 0,
10Aa=p^.
Dififerentiating this, we have
20Ba=pz''Zi = -pz'a>; that is 2Ba + Aa<o = 0, (16)
whence, by differentiation,
140„ + 2M^A^ + QBaCo + A^m, = 0. Subtracting (14) multiplied by Aa from the double of this, we have
28Ca - Aaco'' + 12Baa> - Sqz'Aa = 0.
2B Substituting in this for to its value — j-^ , found from (16), there results
Aa
28 {AaCa-Ba') = Sqz'Aa'. But it has been shown that
10Aa=pz'. Hence the elimination of z gives
2Sy (AaCa - Bif = 3'9'pV^„« = lO'SY-^o'. Or restoring for p and q their values in terms of X, and replacing the absolute reciprocants A a, Ba, Ca by the non-absolute ones A, B, C (which is effected by merely multiplying throughout by a power of a), we have
2*.T(\-2y(\ + iy{2\-iy(AG-BJ = S'.5*(\'-\+iyA\ (17) For other methods of obtaining this differential equation see Halphen's These sur les Invariants Diffh-entiels, p. 30, and Lecture XXX. of the present course. It corresponds in general (that is unless \ = 0, 1, oo ) to the complete primitive
When \= 0, 1, oo , the differential equation (17) becomes
28'(i4C-5'>' = 3».5»il», (18)
which corresponds to the complete primitive
Y=X^. (19)
This case has been discussed in the Thhe and in Lecture XXX. [above, p. 480].
We may obtain (18) from (19) by a method of elimination similar to that employed in deducing (17) from its complete primitive. Thus the first differential of (19) may be written
1 1 -4 J ^1 Ji. — ^JL J
which becomes v = u + Sz
when we assume X, = Xu, Yj = Yv, Zi = Zu + SXz.
612 Lectures on the Theory of Reciprocanls [42
By means of the Lemma we obtain
M» + 3mm, + m, + 4i/oM = 0, (20)
V* + 3Wi + I'j + A:MaV = 0, (21)
3m'« + 3m,^^ + 3m«, + z, + ^MaZ = 0. (22)
The first two of these are identical with (2) and (3) previously given ; the third is found as follows. Since Z^ = Zu + ZXz, Z, = Z,m + Zm, + 3Z,« + ZXz, = Z (m» + tt,) + 3Z (2m^ + ^i). Hence
Zt = Z, (tt« + M,) + .^(2mm, + m,) + 3Z, (2m^: + z^) + 3Z (2m,0 + 2mj, + ^a) = Z (m" + 3mmi + Wj) + 3Z {^u^z + 3uiZ + Suz^ + z^. Thus we have
Z, + 4i»f„Zi = Z(m» + ^XlXh + Mj + 4JlfaM) + 3Z(3m»z + 3mi^ + 3m«, + ^a + Mdaz). But ^3 + 4tMaZi = 0, and m' + 3mm, + v^ + iiMati = 0, which shows that Siv'z + Su^z + 3m2i +z^ + 4MaZ = 0. Equations (20), (21), and (22), of which we have just proved the last, are merely convenient expressions of the fact that X, Y, Z are linear functions of X, y. We combine them with the first, second, and third differentials of the primitive equation (19) by writing
V = U + 3^: ' t), = M, + ^Zi
Vi = 'u^ + 3^2 When this is done (21) becomes
(m' + 3mm, + Mj + 4ifoM) + 3 (3M'^^r + 3m0, + ZUiZ + Zi + iMaZ)
+ 27z(uz + i!' + Zi) = 0, which, in consequence of the identities (20) and (22), reduces to
(m + ^) ^^ + ^i = 0. Let now u = o)-z (so that (oz + Zj = 0). Substituting in (20) and (22) we find
oj' + 3<»a), + Q)s + 4MaCo — S{to — z) {wz + z^ — z^ — 30),^ — z^ — 4'MaZ = 0, and (Sw - 62) (a>z + z,) + 3z' + 3a),z + z^ + ^M^z = 0
respectively. Adding both equations together, and remembering that
a>z + ^, = 0, we obtain a>» + 3ft>6), + 0)2 + 4 J/,,© + 22» = 0, (23)
3«,ir + ^2 + ^^az + 3«» = 0, (24)
which, combined with w^ + ^, = 0, (25)
replace the system (20), (21), (22).
I
42] Lectures on the Theory of Reciprocants 513
Comparing these equations with (11), (12), (13), we see that the two sets are identical if we make \ = 0, when p becomes 2 and 9 = 1. Hence, by performing exactly the same work as in the previous case, we shall find
bAa = z' (instead of 10^„ = p:^)
and 28 {A^Ca - B^") = ^z^A^^ (instead of ^z-A^").
And, finally, eliminating z between this pair of equations, at the same time replacing the absolute reciprocants Ag,, Ba, C^ by the corresponding non-absolute ones A, B, C, we have
2S>(AC-B^y = 3K5^A\ which is what (17) becomes when \ has any of the values 0, 1, or oo .
s. IV. 33
43.
SUR LES RECIPROCANTS PURS IRR^DUCTIBLES DU QUATRIl^ME ORDRE.
[Comptes Rendus, cil. (1886), pp. 152, 153.]
Dans une Note pr^cedente* nous avons voulu donner le systeme de rdciprocants iir^ductibles par rapport aux lettres a, b, c, d, e.
Malheureusement une erreur de calcul s'est gliss^e dans la determination de la forme num^rot^e (5) [p. 248, above], et consdquemment la forme (6) qui, d'aprfes notre m^thode de calcul, depend en partie de la forme (5) est aussi erron^. L'erreur est grave, car, en consequence, un terme contenant 6'd se trouve dans cette dernifere forme qui ne doit pas y paraitre ; cela empecherait une combinaison ulterieure lin^aire de cette forme avec le carr6 de la forme (4), qui donne naissance a une nouvelle forme irr^ductible.
Dans la forme (5) dounee, au lieu de loSoaWc' on doit lire 1485a6'c^, et, au lieu de — 180006*c, on doit lire — 36006*c. Ainsi corrigee, la forme, en divisant par 9, devient
ioa'd' - ^bOa'bcd + 192aV + Uoab^c^ + 400a6»d - 4006*c, et, en combinant celle-ci lin^airement avec le produit de (2) et (4), on obtient, en divisant par a, pour la forme (6),
240a'ce - iOOab^e - 315a'd^ + U70abcd - 1008ac» - S5bV.
Sans aucun calcul arithmdtique, on aurait du prevoir que I'argument 6'd ne doit pas paraitre la-dedans ; car le terme qui contient 6^8(j dans V, operant sur 6'd, donne b', et ^videmment aucune autre partie de V, operant sur un terme quelconque de la forme commen^ant par a'ce, ne peut donner ce mdme argument.
En combinant lineairement le produit de cette forme par la forme ac — 6' avec le carre de (4) [p. 248, above], on obtient, en divisant par a, une nouvelle forme irreductible (7). C'est M. Hammond qui m'a averti de nion erreur de calcul et qui a calcule lui-meme cette nouvelle forme dont il a verifie I'exacti- tude par le moyen de I'^quation diff^rentielle partielle. On peut done accepter avec pleine confiance pour (7) la forme
25a»e= - 350a^bde - 4970aVe + 17150a6=ce + eGloa'cd" - 9800ab^d' - 31360o6c=d + 21217ac* - 140006*e + 49O006»cd - 340556V.
Avec ces conventions le systeme complet de Griindformen, pour le systfeme de lettres a, b, c, d, e, sera constitu^ par les formes (1), (2), (3), (4), (6). (7).
[* Above, p. 242.]
44.
SUR UNE EXTENSION DU THEOREME RELATIF AU NOMBRE D'INVARIANTS ASYZYGETIQUES D'UN TYPE DONNE A UNE CLASSE DE FORMES ANALOGUES.
[Comptes Rendus, cii. (1886), pp. 1430—1435.]
[Cf. p. 459, above.]
Nous employons toujours aujourd'hui le mot invariant pour designer les sous-invariants et les invariants (ainsi ordinairement nomm^s) sans dis- tinction.
Le type d'un invariant est I'ensemble de trois elements, le poids, le degr^ et r^tendue, que nous d^signeron.<« ordinairement par les lettres w,i,j, et nous nous servons de cat ensemble entre parenthfeses (w : i, j) pour signifier le nombre de manieres de composer w avec i des chiffres 0, 1, 2, ... , _^' ou bien, ce qui revient au meme, avecj des chiffres 0, 1, 2, ..., i.
II est quelquefois utile d'ajouter a ces trois ^l^ments un autre dont il est fonction, a savoir X'excea qu'on prend dgal k ij—2w.
Quand on considfere un invariant comma source d'un covariant, I'exces coincide avec I'ordre dans les variables de ce dernier.
Le th^oreme connu, dont nous parlous dans le titre de cette Note, se divise en deux parties:
(1) II n'existe aucun invariant dont I'excfes du type soit negatif ;
(2) Quand I'excfes est positif, le nombre des invariants asyzygetiques du type w : i, j est (w : t, j) — {w — \ : i, j) qu'on peut repr^senter par A (w : i, j).
^videmment, ces r^sultats peuvent etre etendus au cas des formes ration- nellea et entieres qui sont aneanties par I'opdrateur
pourvu qu'aucun des \ ne soit nul ; car alors, en rempla^ant les a par des multiples num^riques convenables, I'an^antisseur peut etre changd dans la forme tto8a -(- 2a,Sa, + . . . -l-i«,-i 8o^.
Quand tous les \ dans I'op^rateur sont pris ^gaux k I'unit^, on peut doDoer aux formes qu'il an^antit le nom de binariants.
33—2
616 Sur une extension du theoreme rdatif au [44
De mSme, on peut consid^rer un an^antisseur
et donner aux formes qu'il an^antit le nom de binariants de raison k* ; en particulier, quaud i = 2, on peut les nommer transbinariants. C'est sur les trausbinariants pour lesquels I'^tendue j est un nombre pair que nous aliens ddmontrer un th^orfeme analogue k celni que nous avons enonce sur les binariants ordinaires.
Si nous consid^rons les binariants de raison k, voici comment on pourrait proc^der pour trouver toutes les formes du type {w:i,j):
On prendra la forme la plus generale de ce type qui contiendra {w:i,j} constantes disponibles. On operera sur elle avec Taneantisseur OoSat + ...> ce qui dounera une forme du type (w — k: i,j) dont les (w — k: i,j) coeflBcients seront des fonctions lineaires de ceux de la forme primitive, et Ton ^galera a z6ro tous ces coefficients. Ainsi Ton pourrait etre porte a croire que, pourvu que le nombre des coefficients de la forme primitive excede le nombre de coefficients de la d^riv^e, la difference de ces deux nombres doit etre le nombre de binariants de raison k asyzygetiques. Mais tout ce qu'on peut legitimement conclure dans ce cas, c'est que ce dernier nombre ne peut pas 6tre moindre que cette difference ; car les Equations dont on a parld ne sont pas necessairement ind^pendantes. Cette precaution n'est nullement sur^- rogatoire ; un seul exemple suffira a le d^montrer. Prenons A = 2 et cherchons le nombre des transbinariants du type (6 : 2, 5).
On a
(6 : 2, 5) = 3, car 6 peut etre compose avec 5+1, 4+2, 3+3,
(4 : 2, 5) = 3, car 4 „ 4 + 0, 3 + 1, 2 + 2.
Done (6:2, 5)-(4:2, 5) = 0.
Cependant le nombre des transbinariants du type donn^ n'est pas zero, mais 1 ; car, dvidemment, 2hf— d? est an^anti par I'operateur
ahc + hhd + che + dhf.
On voit done que c'est un theoreme bien r^el et nullement nugatoire, qui enonce que, pour le cas onj est un nombre pair, le nombre des transbinariants du type (w.i,j) est ^gal exactement a {w:i,j) — {w—2:i,j) quand cette difference n'est pas negative. On peut ajouter que cette difference est negative seulement dans le cas ou Vexces du type est ndgatif et qu'alors (comme on va le demontrer) il n'y a pas de binariants de ce type.
Sil'ona ® = a<,8a, + (h^a^+--.+a^,S^^,
on peut ecrire 0 = ^j + 0^^
* Le thi^or^me de Brioschi montre qu'un binariant de raison k est une fonction de »1. »2> ■■•> »i-l> »*+!' •••. "j-'o *tant la somme des poissanoes e'*"** des racines de I'^qnation a^x' + aix'-^ + ... + ay = 0.
I
44] nombre ctinvariants asyzygetiques cPun type donnd 517
en posant ^, = a^K^ + flsS,,* + O't^a^ + ■ • • + a^ri-iK^,
^j = 01^03 + as S„j.+ +a2^,Sa^_j.
En faisant <=^ + ij,
avec <, = 1 . TjajSa, + 2 (77 — 1) 0480, + 3 (77 - 2) a^hat + ... + t? . 1 . a2,8o2,_3.
<,= 1 (,, -l)a,S„,+ 2(77 -2)a,8a,+ ... +(77 - 1)1 .a^iS<^_3, on trouvera
^i<i -tidi= ■vaoSa, + (77 - 2) OjSa, + ... - (77 - 2) a.j,_2S„j^ - i?aj,8a^, ^jt, - tA = (17 - 1) OiSo, + (77 - 3) OjSo, + ... - (77 - 1) aj^iSoj^. Done, si 7 est una fonction homogene et isobarique dans les lettres a du I type w ; i, j, on aura
(er- r©) /= [77a„8<^+ (77 - l)a,S„, + ... - (77- l)a^-,8a^, -77«^S„J/
. . - 2771 - 2w , = (177-^)7 = -^-2 7;
fear on remarquera que ni I'un ni I'autre 0 n'agit sur I'un ou I'autre t, et que ni [I'un ni I'autre t n'agit sur I'un ou I'autre 0.
Le coefficient de 7, on le remarquera, est la moiti^ de I'exc^ au type w : i, 2r).
U est bon d'observer qu'il n'est pas possible d'obtenir un resultat semblable dans le cas ou j est impair, c'est-a-dire qu'on ne peut pas trouver, dans ce cas, une forme T telle que le resultat de I'op^ration (©T — T&) sur une forme homogene et isobarique soit Equivalent au produit de cette forme par une fonction quelconque de w ; i, j.
Avec I'aide de la formule ci-dessus, suivant la raeme marche que nous avons prise pour les invariants dans le Philosophical Magazine * (mars 1878), on parvient a des r^suitats tout a fait semblabies.
En appelant e la moitiE de I'exces et en supposant que I est un trans- binariant, on trouve
€7 = 077
et, plus gdn^ralement, (iT^-^I =^T^I,
oii fi = q{e — q + l).
Or il est Evident que, puisque I'effet de Test d'augmenter (par deux unites) le poids de la forme sur laquelle il agit sans en changer le degrd ni I'Etendue, et que le poids d'une forme homogene et isobarique ne peut pas excEder le produit du degr6 par I'^tendue, en prenant q suffisamment grand, on aura
TI = 0, et, k plus forte raison, 02'7 = 0.
On trouvera done successivement 7^'7 = 0, T9-"I = 0, ..., 77 = 0,7 = 0, pourvu que le /j, ne devienne pas nul dans le cours de cette deduction: ceci [• Vol. III. of this Beprint, p. 117.]
518 Sur une extension du th^reme relatif au [44
ne peut pas arriver quand e est n^gatif, car on trouvera que les valeurs de fi, dans ce cas, resteront toujours negatives.
Cela d^montre qu'un transbinariant, dont le type a un excfes ndgatif, ne peut pas etre autre que z6ro, c'est-k-dire n'a pas d'existence actuelle quand I'excfes est non n^gatif; en d&ignant par E{w : i,j) le nombre
(w:i,j)-iw-2:i,j), et par D (w : i, j) le nombre de transbinariants du type (w : i, j), on prouve que D {w : i, j) = E(w: i, j) de la maniere suivante.
En remarquant que, pour w negatif, E (w.i, j) = 0, on trouve imm^- diatement
9=0
2 E{w -2q:t, j) = (w : i,j),
q = ao
et, puisque cbaque D est au moins 6gal au E correspondant, on a
9=0
2 D(w-2q: i,j) ? (w : i,j).
9=00
Or on peut demontrer facilement que, si ij — 2w est non ndgatif, en appelant /«>:ij un transbinariant du type {w : i,j), ©«2'»/u,_2g.^jSera ^gal a un multiple num^rique de Iv-tq-.i,] different de zero pour toutes les valeurs de q qu'on a besoin de considdrer.
Or, dans I'ensemble des transbinariants asyzyg^tiques, dont le type est w—2q: i,j, on peut substituer h, chacun, pour ainsi dire, son image T^I„f.^.,ij. Le nombre de ces images sera
X'n(w-2q:i,j).
9=00
De plus, chaque image sera du meme type (w : i, j).
On d^montre facilement qu'il ne peut pas exister entre ces images une relation lin^aire ; car, dans le cas contraire, en opdrant sur I'^quation qui les lie ensemble avec une puissance convenable de 0, on tomberait sur une Equation lindaire entre les transbinariants asyzyg^tiques eux-memes. Done, 6videmment, le nombre des images ne peut pas exc^er la valeur de (ia:i,j). Done
TD(w-2q;t,j)
9=00
9 = 0
nest ni plus grand ni plus petit que S E{w — 2q;i,j); il lui est done dgal,
9=00
et consequemment, puisque aucun D ne peut 6tre moins que le E qui lui correspond pour chaque valeur de q,
D(w-2q;{,j) = E{w-2q;iJ);
car si un D quelconque dtait plus grand que le E qui lui correspond, un autre D serait n^cessairement plus petit, ce qui est inadmissible.
44] nombre cTmvariants aspzyg^tiques d'un type donne 519
On aura done D (w : i, j) = E{w: i, j),
ponrvu que ij — 2w ne soit pas ndgatif. CQ.f.d.
On demontre facilement les memes thdorfemes pour des formes an^antiss- ables par une somme d'opdrateurs
f^o^ot + ••• + aj-2^a,-> Oo'Sa,' + . . . + a'y^^ha'j.
Eu supposant que chaque _;' soit pair et en regardant w : i, j : i', j', . . . comme leur type, on paT^ient a cette conclusion qu'aucun transbinariant d'un tel type n'existe dans le cas ou ij + i'j + ... — 2m; est negatif et que, quand cette quantity n'est pas negative, le nombre des transbinariants asyzyg^tiques est egal a {w : i, j : %', /:...)- (w - 2 : i, j : i', /;...), oil {w : i, j : i', j':.. .) d^signe le nombre de maniferes de composer w avec i des chiffres 0, 1, 2, ....combines avec i' des chiffres 0, 1, 2, ...,j', etc.
P II est utile de remarquer que les formes et les syzygies fondamentales des int^grales de I'equation U (ao5a, + a,Sa,+ ...+a,^Sa^)/=0
sont des m^mes types que les invariants et les syzygies fondamentales d'un systfeme formd avec deux qualities d'ordres 17 et ?/ — 1 respectivement ; ce qui donne un moyen facile de verifier la formule que nous avons demontr^e pour le nombre de transbinariants asyzygetiques d'un type donne. II va sans dire que nous n'avons pas neglig6 de nous servir de cette methode pour verifier la justesse de nos conclusions.
45.
NOTE SUR LES INVARIANTS DIFFERENTIELS.
[Comptes Rendus, cii. (1886), pp. 31 — 34.]
En aflSrmant, dans notre Lettre k M. Hermite (dont un Extrait a paru dans les Comptes rendus), que les invariants diffdrentiels de M. Halphen sont identiques avec nos reciprocants purs, nous sommes alle trop loin ; nous aurions d(i dire qu'ils sont identiques avec la classe sp^ciale de ces derniers que nous avons nommds reciprocants projectifs; en effet, en prenant pour dSments
1 #y _1_ ^ 1 d*y
1.2^' l727Sd^' 1.2.3.4ek*' "■'
regardes comme quantites algdbriques, lesquelles on peut nommer (selon I'usage quand on parle de formes binaires) a, b, c, d, ..., un invariant dif- f^rentiel possfede la propri^te vraiment etonnante d'etre en m^me temps un r^ciprocant et un sous-invariant ordinaire.
En accommodant la valeur de F a cette notation nouvelle, il devient
4aaSj + 5 (ab + ba) Sc + 6 (ac + bb + ca) Sd, . . . ;
et, en posant aS^ + 2bSc + 3cS<j +...=11,
un invariant differentiel / satisfait en meme temps aux deux equations partielles diff^rentielles
v.i=o, n.i = o.
Voici comment on peut ^tabiir le fait que n . / = 0.
En commen9ant avec les trois premiers invariants difFi6rentiels, c'est-a-dire a, a^d — Sabc + 2b', et le A de M. Halphen (dans sa thhse immortelle), on sait que les deux premiers, et Ton v^rifie sans trop de peine que le troisi^me sont tons les trois des sous-invariants.
45] Note sur les invariants diff4rentiels 521
De plus, on sait que, en commen9ant avec ces trois invariants que nous nommerons /„, /i, I^, on peut former une suite ind^finie de formes proto- morphiques
■'(l> -'l; J-2> Jsi •••> ■'p> •••>
dont tous les autres seront des fonctions rationnelles.
Pour obtenir cette suite, on n'a qu'a former une fonction J de I„, 1^, ..., Ip, ..., dont le degre et le poids soient tous deux z^ro; en opdrant alors sur J" (considere comme fonction des derivees de y par rapport a x) avec hx, on obtient I^i.
Si done on peut d^montrer que ClhxJ = B^iU, il s'ensuivra que Ip+i sera un sous-invariant, pourvu que Ip en soit un, et le thdoreme en question sera demontre.
Or remarquons en premier lieu que, a cause de la valeur z6ro du degre et du poids de J, la quantite
(\aBa + /J^bSi, + vcSc + ...)J
sera nulle si X, fj., v, ... forment une progression arithm^tique quelconque ; et, en second lieu, que (par rapport k une fonction de ddriv^es de J par rapport k x), hx = 36Sa + 4cSj + bdhe + ... identiquement.
Cons^uemment
(nS« - S^) J = [(3aa, + 86S(, + locS, + ...)- {^hh + ScS. + . . .)] J
= (3o3a + 5686 + lch, + ...)J=Q,
ce qu'il fallait d^montrer.
M. Halphen, h. qui j'avais communique ce r^sultat, en a trouv^ une tout autre demonstration qu'il m'autorise k communiquer k l' Academic. EUe possede sur la mienne I'avantage d'aller plus au fond de la question, en faisant voir que lequation H .1=0 ^quivaut a dire que, en se servant de x, y, z au lieu de x, y, 1, un invariant diff^rentiel peut subir le changement entre eux de a; et z. Or, puisque F./=0 signifie qu'on peut imposer des substitutions lindaires quelconques sur x et y, il s'ensuit, en combinant les deux Equations, que la mSme chose aura lieu quand x, y, z subissent tous les trois des substitutions lin^aires quelconques. Voici la demonstration tr^s dl^gante de M. Halphen :
" Si Ton fait le changement de variables
X X
et qu'on 6crive
dx y' da? y ' •■•' dx^ y • ■■•'
522 Note m/r les invaricmts diff^entids
on a Y= + arhf,
[45
dX
g = (- 1)"^-' [y<»' +"<"^ ^)y(n-»+«y.n-.,+|y(n^., + ...j ,
"Posant
d»F ,, rf"y , 1
dZ"
» = "'^»' 5^ = "'""' x=^'
on a il„ = (- l)»a;»^' K + (n - 2) eOn-i + aVon-j +...]•
" Soit une fonction/(4o. -^i, ... , ^n) dont tous les termes soient de poids et de degr^ constants p,i; en supposant e infiniment petit, on aura
/•(A, J„ ...,^„) = (-l)Pa;^-«|/(a„,a, a„)
" Done, pour que f soit invariant pour la substitution consid^r^e, il faut qu'on ait
«=a^+2a,g + 3a,|: + ...+(n-2)a„_,g = ao|.
" En particulier, si / ne contient pas a,, ce qui est le cas des rddprocants purs, on aura
Ainsi, Ton voit qu'un invariant diffdrentiel est en mSme temps reciprocant et sous-invariant ; ce n'est nullement un melange ou une combinaison de deux choses difKrentes, mais plutot, pour ainsi dire, une personnalitd seule et indivisible doude de deux natures tout k fait distinctes.
Afin de completer la th^orie, il faut d^montrer la r^ciproque, c'est-i-dire que toute forme dou^e de ces deux natures est un reciprocant projectif. M. Halphen efifectue cela en trouvant le d^veloppement complet de sa serie et en faiaant voir que, quand le coeflBcient de la premiere puissance de e disparait,
45] Note 8ur les invariants differentiels 523
la meme chose aura lieu pour tous les coefficients suivants. Voici notre methode, a nous de I'effectuer.
Soit H una forme rationnelle et entifere dent le terme principal (c'est-a- dire celui qui contient la plus haute puissance du terme le plus avanc^) est Gh>. On suppose que le theoreme a demontrer est vrai jusqu'a la lettre g incluse, et que VA =0, ilH=0 sans que H soit projectif.
Alors evidemment VG = 0, ilG = 0 et G, par hypothese, sera projectif. Soit H' une puissance d'un protomorphe pour laquelle le terme principal est G'h\ alors, si Hj = G'H — GH', G, G', H' sont projectifs, mais H non projectif ; done, Zfi (qui, comme H, est aneanti par V et par H) sera non projectif : de plus, dans H^ le degr6 du terme principal en h est abaiss^. De la meme manifere on pent construire H.^, H,, ... jusqu'a ce qu'on parvienne k une forme* qui ne contient pas h, laquelle possedera les m^mes caracteres que H, ce qui est impossible par hypothfese. Done, si le'' theoreme a demontrer est vrai pour un nombre quelconque donn^ de lettres, il sera vrai universelle- ment : mais il est evidemment vrai pour la fonction a qui est le seul recipro- cant k une lettre. Done, si VI = 0 et RI = 0, 1 est un reciprocant projectif, c'est-i-dire un invariant diflPSrentiel. Ce qui ^tait k demontrer.
* Cette forme sera, en effet, le resultant de H et de la premiere puissance da protomorphe. Nons avons jag£ inntile de dire dans le texte que G', comme G, sera aneanti par V et par Ct et oons^qnemment, par hypothise, sera loi aussi projectif.
46.
SUR L'^QUATION DIFFfeRENTIELLE D'UNE COURBE D'ORDRE QUELCONQUE.
[Comptes Rendus, cm. (1886), pp. 408—411.]
[Also, above, p. 492.]
On peut obtenir une solution directe et universelle de ce problfeme : Trouver I'iquation diffdrentielle d'une courbe de I'ordre n, en repr^sentant la fonction de I'^quation (avec I'unit^ pour terme constant), soit U ou (x, y, 1)",
/ d \" sous la forme symbolique u^,ohu = a-\-hx + y. Alors, en mettant ( -r- j y=yr>
du , d*+ht .
^galons k z6to les d^rivees de m" des degr^s n + l,n + 2, ..., ~ ;
il en r^sultera — ^ — Equations entre lesquelles on peut ^liminer le meme
nombre de coefficients, c'est-a-dire tous les coefficients en U, sauf ceux qui ne contiennent nulle puissance de y, lesquels ne paraitraient pas dans les equations dont nous parlons.
Pour obtenir le determinant qui correspond k ce syst^me d'^quations, remarquons que le th^orfeme de Taylor doune imm^diatement-f-
jj^9/M»= CO, [u + u'h+u" J— 2 + u'" j-^-g + ...j
= cOr Uu + u'hy + n.{u + u'hy-^ F + n . ^^^ (m + m'A)"-» F' + . . . j , ou Ton peut prendre
ce qui suffit k r^soudre le probl^me.
* On remarquera qu'avec cette notation toute fonction entiSre de u et 3j.« representera sans ambiguity une quantity alg^brique ordinaire, pourvu que I'on sache a priori qu'elle doit fitre lin^ire dans les coefficients de u". C'est pourquoi dans le texte on est libre d'exprimer tonte diriv^ diff^rentielle de U comme fonction de u et u'.
t Par cOf on sous-entend les mots "le coefficient de h'' dans."
46] Siir Vitiation differentielle (Vune corn-he 525
Pour cela, on considere toutes les ddrivees de JJ comme fonctions lineaires des termes qui paraissent dans le d^veloppement de («, w', 1)"~^*.
Alors, en representant par m . y. le coeflScient de A™ dans
on trouvera, sans calcul algebrique aucun, que la ^'^^^e ligne du determinant cherche pent etre prise sous la forme
(1+9).! (2+g).l (2+9).2 (3+g).l (3+5).2 (3+9).3 ...
(n+g).! (n+g).2 ... {n-\-(().n. Par exemple, prenons le cas de »i = 4 ; le determinant
|
2 |
3 |
3.2 |
4.1 |
4.2 |
4.3 |
5.1 |
5.2 |
5.3 |
5.4 |
||
|
3 |
4 |
4.2 |
5.1 |
5.2 |
5.3 |
6.1 |
6.2 |
6.3 |
6.4 |
||
|
4 |
•5 |
5.2 |
6.1 |
6.2 |
6.3 |
7.1 |
7.2 |
7.3 |
7.4 |
||
|
•5 |
6 |
6.2 |
7.1 |
7.2 |
7.3 |
8.1 |
8.2 |
8.3 |
8.4 |
||
|
6 |
7 |
7.2 |
8.1 |
8.2 |
8.3 |
9.1 |
9.2 |
9.3 |
9.4 |
||
|
7 |
8 |
8.2 |
9.1 |
9.2 |
9.3 |
10.1 |
10.2 |
10.3 |
10.4 |
||
|
8 |
9 |
9.2 |
10.1 |
10.2 |
10.3 |
11.1 |
11.2 |
11.3 |
11.4 |
||
|
9 |
10 |
10.2 |
11.1 |
11.2 |
11.3 |
12.1 |
12.2 |
12.3 |
12.4 |
||
|
10 |
11 |
11.2 |
12.1 |
12.2 |
12.3 |
13.1 |
13.2 |
13.3 |
13.4 |
||
|
11 |
12 |
12.2 |
13.1 |
13.2 |
13.3 |
14.1 |
14.2 |
14.3 |
14.4 |
sera le premier membre de I'^quation diflferentielle (disons le criterium differentiel) d'une courbe du quatrieme degre.
Si I'on se borne aux termes contenus dans les six premieres lignes et colonnes, on aura le crit^rium pour la cubique, et, en se bornant aux termes contenus dans les trois premieres lignes et colonnes, celui pour la conique, ou plutdt ce criterium multipli^ par 2. 1, ce qui constitue un cas exceptionnel.
2 . 1 lui-mSme, c'est-a-dire -^ , est naturellement le criterium pour la ligne
droite. On remarquera que 3 . 2, 4 . 3, 5 . 3, 5 . 4, 6 . 4, 7 . 4 sont des com- binaisons pour ainsi dire fictives, qui ont pour valeur zdro-f-. De mime, en general, il y aura toujours des termes nuls dans les (n — 1) premieres lignes du criterium de la courbe de degrd n\ au-dessous de la (n — 1)'**"" ligne, toutes les places seront remplies par des combinaisons qui correspondent hk des non-z6ros.
v" y'" v"
Quand m = 3, en-substituant pour :r^ , ^ , :j — ^ . , ... les lettres
a,b,c, ... , on retombe sur la formule trouv^e pour la cubique par M. Samuel
* On platdt les termes avec lenrs coefficients num^riques de (u, u', 1)", en omettant les (n + 1) teimee da degi^ n.
t Evidenunent m . /i est z^ro qaand m < 2/i.
526 Sur rSqiuztion differentielle dune courhe [46
Roberts (voir MathematiccU Questions from the Educational Times, t. x. p. 47)*, c'est-i-dire la meme matrice que celle donn^e par M. Roberts, mais avec ses colonnes autrement presentees.
On voit imm^diatemenl que le degr^ du crit^rium pour une courbe du
n**°^ ordre sera — ^ ^ et, par un calcul facile, que son poids sera
(n-l)n(« + l)(n+2) n(n+l)(n + 2). p, , • ,
^ i — i i-^ + -^ ^ -J. Ce dernier nombre suppose que
le poids de dg^y est compt6 comme i. Dans le calcul des r^ciprocants, on le
compte toujours comme ^tant i — 2 et, en faisant cette reduction, le poids
. , (n-l)n(n + l)(n + 2) devient tout simplement g .
M. Halphen nous a appris que les formules qu'il a donn^es dans son M^moire intitule : Recherches des points d'une courhe algdbrique plane, etc. (Journal de Mathematiques, 3* serie, t. ii. pp. 373, 374 et 400 ; 1876) four- nissent un moyen pour calculer le degr6 et le poids du crit^rium n'*"® et conduisent aux memes r^sultats que ceux donn^ ci-dessus. Dans le cas de la conique, le determinant, comme nous I'avons dit, se divise par y", de sorte que son poids-degr^ s'abaisse et, au lieu d'etre 3 . 4, devient 3.3; en efifet, c'est la forme bien connue a^d — Sabc + 26», trouv^e par Monge.
* Ce travail a 6tii oit£ et reproduit dans le Philosophical Magazine de f^vrier 1886, par M. Mnir, qai y constrait pour ainsi dire le tableau dn oalcul dont M. Roberts avait ddjk fait le proc^s- verbal.
t Car le degr^ sera la somme de n termes de la s^rie 1 +3 + 6+ ..., c'est-i-dire — ^ ^ ,
et le poids, moins deux fois le degr^, la somme de n termes de la s^rie 0+ (2 + 1) + (5 + 4 + 3) + (9 + 8 + 7 + 6)+...
ou bien de ^ termes de la progression natureUe 1 + 2 + 3 + 4 + 5 + ..., c'est-sl-dire
n''+n-2 n^ + n
47.
SUR UNE EXTENSION D'UN TH^OREME DE CLEBSCH RELATIF AUX COURBES DU QUATRIEME DEGRE.
[Comptes Rendus, Cll. (1886), pp. 1532—1534.] En appliquant un terme quelconque du d^veloppement de
(K, S». Sz, •••)
ii
au quantic {x, y, z, .. .)*', on obtient autant de fonctions de degr^ t} qu'il y a de termes dans chaque fonction. L'ensemble de leurs coefficients pent done etre regarde comme la matrice d'un determinant auquel nous donnerons le mfime nom de catalecticant, dent on fait usage dans le cas des formes binaires.
On voit trfes ais^ment que la matrice catalectique, pour une puissance d'une fonction lineaire de variables, possede cette propri^td que chaque determinant mineur du second ordre qu'elle contient s'^vanouit. Conse- quemment, deux colounes quelconques d'une telle matrice, associees k d'autres colonnes arbitraires, en nombre suffisant pour former une matrice carr^e nouvelle, feront s'^vanouir le determinant de cette derniere.
Or la matrice catalectique d'une somme de puissances de fonctions lin^aires des mdmes variables est la somme des matrices qui appartiennent k chacune prise s^par^ment; done, comme consequence immediate de cette propriety dont nous avons parl^, si le nombre de ces matrices est moindre que I'ordre de chacune, le determinant de leur somme s'^vanouira, car il pourra 6tre resolu dans une somme de determinants dont chacun aura la valeur z^ro *.
' S'il y a n matrices, chacane de I'ordre N (de sorte que N eat le nombre des colonnes dans chaque matrice), on associera k volont^ la premiere colonne d'une quelconque des n matrices aveo la seconde, avec la troisi^me, etc. colonne, prises ou dans la mime on dans aucune autre matrice, en sorte que le nombre des nouvelles matrices partielles sera n". II est Evident que, N ^tant par hypoth^se plus grand que n, deux colonnes au moim de chaque matrice ainsi form^e appartien- dront k une mime matrice fondameutale, c'est-&-dire a la matrice catalectique d'une puissance d'une fonction lineaire des variables. Voili la raison pour laquelle chacun des n" determinants partiels est ^gal & ziro.
528 Sur une extension d'un ttUorenie de Clebsch [47
(1 ) Prenons deux variables. Le catalecticant sera de I'ordre »? + 1 ; on retrouve ainsi cette rfegle bien connue, et qui ue contieut rien d'exceptionnel ni de paradoxal : pour qu'une forme binaire d'ordre 21; soit dquivaleate k la sonime de i; puissauces de functions lin^ires, 11 faut que le catalecticant de la forme soit nul.
(2) Prenons trois variables et faisons 17 = 2 : I'ordre du determinant catalectique de {ax + by + cz)* ^tant 6, le catalecticant de
2 (o,a; + bgy + CgzY = 0.
e=5
Cela donne le th^orfeme de Clebsch, k savoir que le premier membre de r^uation d'une courbe du quatrieme degr^ n'est pas, en g^ndral, exprimable en une somme de cinq puissances de fonctions lin^aires des variables.
(3) Prenons cinq variables, en faisant encore 77 = 2. L'ordre du deter- minant catalectique (ax + by + cz + dt + eu)* ^tant 15, le catalecticant de
2 (agx + bgy + CgZ + dgt + egu)* s'^vanouit.
K /^ IT u
Or 5 X 14 = 70, ce qui est justement le nombre 5— s— o~7 ^^^ coefficients
de {x, y, z, t, «)«.
On arrive ainsi k cette conclusion nouvelle, et un peu paradoxale, que r^quation d'une hypersurface du quatrieme degrd, bien que contenant le meme nombre de constantes que la somme de 14 puissances biquadratiques de fonctions lin^aires des variables, ne pout pas en g^ndral §tre exprim^e comme une telle somme ; car, pour que cela fftt possible, il faudrait que le catalecticant de I'hypersurface s'^vanouit.
(4) Prenons encore 1? = 2, et consid^rons la somme de 9 puissances quatrifemes de fonctions lindaires de x, y, z, t. Le catalecticant de cette somme sera de I'ordre 10 et, cons^quemment, z^ro.
Done le premier membre de I'^quation d'une surface du quatrieme degr^ qui ne contient que 35 constantes ne peut pas en general etre mis sous la forme d'une somme de 9 puissances de fonctions lin^aires des variables, quoique cette somme contienne 36 constantes disponibles.
Ce r^sultat pour les surfaces est, on le voit, un peu plus paradoxal, en apparence, que le th^orfeme de Clebsch, sur les courbes du quatrieme degr^, quoiqu'en effet il n'y ait aucun paradoxe, ni dans I'un ni dans I'autre de ces th^orfemes, pour ceux qui sont convaincus qu'on ne doit jamais se fiei, sans controle, aux conclusions apparentes, fournies par la comparaison num^rique de constantes.
48.
ON THE DIFFERENTIAL EQUATION TO A CURVE OF
ANY ORDER.
[NcUure, xxxiv. (1886), pp. 365, 366.]
To Mr Samuel Roberts (see Reprint of Educational Times, X. p, 47) is due the credit of having been the first to show that a direct method of elimination properly conducted leads to the differential equation for a cubic curve ; but he has not attempted to obtain the general formula for a curve of any order. By aid of a very simple idea explained in a paper intended to appear in the Comptes Rendiis of the Institute, I find* without calculation the general form of this equation. The left-hand member of it may be conveniently termed the differential criterion to the curve. One single matrix will then serve to express the criteria for all curves whose order does not exceed any prescribed number. For instance, suppose we wish to have the criteria for the orders 1,2,3.4:-
Let m'/i be used in general to denote the coefficient of A" in
(oy"^'+ri3^"^'-^ 1-72^^'"^' + •••)"•
Write down the matrix —
2-1 31 3-2 41 4-2 43 51 52 6-3 5-4
31 41 4-2 51 5-2 5-3 61 62 6-3 64
41 51 5-2 6-1 6-2 63 71 72 73 74
6-1 6-1 6-2 7-1 7-2 73 81 8-2 8-3 84
6-1 7-1 7-2 8-1 8-2 83 91 92 93 94
7-1 8-1 8-2 9-1 9-2 93 101 102 103 104
8-1 91 9-2 101 10-2 10-3 11-1 112 11-3 114
91 101 10-2 11-1 11-2 11-3 121 12-2 123 124
101 111 11-2 121 12-2 12-3 13-1 132 ISS 13-4
11-1 121 12-2 131 13-2 13-3 14-1 142 143 14-4
[* Cf. pp. 492, 524 above.] 8. IV. 34
530 On the Differential Equation to a Curve [48
The determinant of the entire matrix, which is of the tenth order, is the criterion for a quartic curve. The determinant of the minor of the sixth order, comprised within the first six lines and columns, is the criterion for a cubic. The determinant of the third order, comprised within the first three lines and columns (subject to a remark about to be made) will furnish the criterion for a conic, and the apex of the matrix is the criterion for the straight line. By adding on five more lines and columns, according to an obvious law, the matrix may be extended so as to give the criterion for a quintic ; then six more lines and columns a sextic, and so on as far as may be required.
The remark to be made concerning the determinant of the third order
y" referred to is that it contains the irrelevant factor 21, that is, ^ , so that the
criterion for a conic (Monge's) is this determinant divested of such factor. It is certain that the next determinant is indecomposable, and is therefore the criterion for a cubic. There is no reason that I know of to suppose that any other determinant except that one which corresponds to the conic, is decom- posable into factors. If this is made out, then, observing that the single term which is the criterion for the right line is indecomposable, we have another example of what may be called, in Babbage's words, a miraculous exception to a general law.
A well-known similar case of such miraculous exception I had occasion many years ago to notice in connection with the criteria for determining the number of real and imaginary roots in an algebraical equation. Such criteria may, with one single exception, be expressed by means of invariants. The case of exception is the biquadratic equation, for which it is impossible to assign an invariantive criterion that shall serve to distinguish between the cases of all the roots being real and all imaginary.
It is proper to notice that it follows, from the definition of the symbol m-fi, that its value is zero whenever m is less than 2(1. Thus, in the matrix written out above, the symbols ;32, 4-3, oS, 5-4, 64, 74 may be replaced by zeros.
The above general result for a curve of any order is actually obtained by a far less expenditure of thought and labour than was employed by Monge, Halphen, and others to obtain it for the trifling case of a conic. I touch a secret spring, and the doors of the cabinet fly wide open*.
• Adopting the convention for degree and weight of a differential coefficient nsaal in the theory of reciprocants the deg : weight of the differential criterion of the nth order will be easily found to be
n.n+1 .n+2 n-1. n.n+l.n+2 6 • 8
except that for n = 2 it ia 3 : 3 instead of 4 : 3.
I
49.
ON THE SO-CALLED TSCHIRNHAUSEN TRANSFORMATION.
[Grelle's Journal, c. (1887), pp. 465 — 486.]
Exactly one hundred years ago, E. S. Bring (Dissertation, tTniversity of Lund, 1786, Meletemata quaedam mathematica circa transformationem aequationum algebraicarum) gave the method to which the name of Tschirn- hausen by a common consent in error is now usually attached*. Sometimes but more rarely the method is attributed to Jerrard who came much later into the field. This is especially the case in England ; Hamilton for instance in his " Report on Jerrard's method " published exactly 50 years ago in the
* The expression P, - i„_, g„_,_g + J/.B,_2_,
where L, M are given entire functions in x of degrees n - 1, n,
P,Q,R „ disposable ,, „ „ ,, 0, n-1-0, n-2-e,
may be made identically zero by solving 2n - 1 - 9 homogeneous linear equations between the 3n - 0 disposable constants contained collectively in P, Q, R, and when this is done we have
Pj 7, = i«-i [mod. Af,].
Hence it follows that the Tschimhansen substitution has a one-to-one correspondence with any fractional substitution containing the requisite number of disposable constants : so for instance in the case of a quintic the Bring substitution
Ix* + mx^ + rufi +px + q
is only another name for the general quadratic tubititutian -r-^ ■ .
ux + ex -^-J
This change of form in the substitution, supposed to be generalised, is interesting for the teason that it completes the analogy between the Tschirnhausen method of simplifying an algebraical equation and Combescure's method of simplifying a linear differential equation. Bir James Cockle appears to have arrived at the same result as M. Combescure in a paper on liiuear Differential Equations. (Quarterly Journal of Mathematics, Aug. 1864.)
This method involves two quadratures, the integration of a differential equation of the second order, and substitutions impressed simultaneously upon the two variables.
The quadratures and solution of an equation of the second order are, of course, analogous to the solution of two simple and one quadratic algebraical equation ; the substitutions impressed on the two variables run parallel to the two integral substitutions to be performed upon the two Tariables of the algebraical equation put under the form of a quantic which are equivalent to a fractional substitution performed upon the single variable of a non-homogeneous form.
34—2
532 On the so-called Tschirnhansen Transformation [49
Reports of the British Association makes hardly any mention of any other author but Jerrard in connexion with the subject.
In the following memoir I propose to present Hamilton's process under what appears to me to be a clearer and more ea-sily intelligible form, to extend his numerical results and to establish the principles of a more general method than that to which he has confined himself.
But previously to entering upon this part of my work I think it may be well to call attention to a circumstance connected with the so-called Tschirn- hausen transformation, as bearing upon the character of the transformed equation to which it leads, which hitherto appears to have escaped observation, arid which is of particular interest as regards the application of the method to the equation of the 5th degree when it is reduced to the form
f + By + C=0, for I shall be able to show in that case that in general the coefficients which remain (notwithstanding the large element of indeterminateness of which the method admits) cannot be made real when more than one of the roots of the original equation is real ; this remark will be found to apply whether the method be used under its original form or under the modified form employed so advantageously by Hermite.
In order to make out this proposition it will be useful to give a somewhat more extended statement of the Law of Inertia (Tragheitsgesetz) for quadratic forms than that originally presented by me in the memoir : " On a theory of the syzygetic relations of two rational integral functions comprising an application to the theory of Sturm's functions and that of the greatest algebraical common measure" (Phil. Trans, for 18.53)*.
Let us suppose a quadratic function of m + n letters, either independent or connected by linear relations which in the latter case reduce the number of independent quantities to fj. + v.
Let the function be supposed to be expressed
(1) by the sum of m positive and n negative squares,
(2) by the sum of fi positive and v negative squares of real linear functions of the variables.
Then I affirm the impossibility of either of the two inequalities
fi > m ; V > n. (1) I say that the conjunction of the inequalities m>fi, v>n is impossible.
For suppose the two expressions of the same quadratic function to be tti" + aj» + ... + a„^ - 6,' - 6,^ - ... - 6„» and «,"+ 0^=+ ... +«/- A'- /3,»- ... -/3/'.
[* Vol. I. of this Reprint, p. 611.]
49] On the so-called Tschirnhmisen Transformation 533
Then Oi' + a^^ + ... + a™^ + /3r + /32= + ... + /3,'
= 6,»+6j^ + ... + V + ai- + «2'+ ... + V- By hypothesis ft,+ n< jjl+v,
li + n<m + v.
By virtue of the first inequality it must be possible to establish fi + n relations between the fi + v independent variables.
jHL Consequently we may equate each square on the right-hand side of the equation to some distinct square on the other side, and then by virtue of the second inequality some squares will remain over on the left-hand side of the equation whose sum will be identically zero. Which is impossible. Hence the inequalities 7n > fi, v > n cannot exist simultaneously. In like manner it follows that n> V, /i >m cannot exist simultaneously.
Now the only suppositions of combined relations of greater and less that can connect m, n ; fi, v are the following :
Im < fi, n < v; m< fi, n = v; tn < fi, n > v ; m. = fi, n < V ; m = fi, n =v; m = fx, n > v ; m > fi, n < V ; m> fi, n =v; m > fi, n > v. f these 9 suppo.sitions the 1st, 2nd, and 4th are excluded by the condition vi + n= or > fjL + v, and the 3rd and 7th by virtue of what has just been proved. Hence the only hypotheses admissible are the four contained in the negative statements :
fi not >m and v not >n. Q.E.D
Although the only application which I shall have to make of this Lemma is to the case where m + n = fi + v + 1, I have thought that it is of sufficient interest in itself and collaterally in the logical process of its proof to deserve setting out in full.
Suppose now that we have the equation f{x) = {x, 1)" = 0 where all the coefficients in /are supposed to be real, and that we write in conformity with the ordinary so-called Tschirnhausen process :
y = «,« -f WjX" + ... -I- «n-i *""*.— iSf, where nS=u^l.x + u^l,a?+ ...+Un-i1af^^
80 that the transformed equation will be of the form :
jr + 5,2/»-» -f- 5,y»-» -J- . . . 4- J5„ = 0, where £< is a quantic of degree i in the letters v^, u^, ... ««-!• Let us consider the projective character of the quadratic function B^. This character is determined by the nature of the succession of algebraical signs in the sum of positive and negative squares to which B, regarded as a function of the n — 1 letters u may be reduced by real linear transformations.
634 On the so-called Tschirnhavsen Transformation [49 Since yi + yaH- ..-+^11 = 0,
80 that it is the character of 2y' which determines the projective character of 5,. The number of real vahies of y is the same as of x. Hence if / has t pairs of imaginary roots, 2y' will be the sum of n — t positive and » negative squares of real linear functions of Wj, it,, ... ii,i_i.
Consequently, by virtue of the lemma above proved, there is only one element of uncertainty as to the character of 2y', that is, it must we know d priori, when reduced to a sum of n — 1 positive and negative squares of linear functions of u,, ttj, ... «„_,, contain either i or i—1 negative squares. This uncertainty may be removed by means of a second lemma, namely, that the discriminant of JB^ is a numerical multiplier of the discriminant of/.
When two of the roots of/are equal, two of the values of y become equal so that 2y' becomes reducible to a sum of n — 2 instead of a sum of n — 1 squares.
Hence the former contains the latter as a factor : moreover it is obvious from the form of each value of y that its discriminant regarded as a function of the n roots of / will be of the degree 2 {1 + 2 + ... + (n - 1)), that is, n(ra — 1) which is the same as that of the squared product of the differences of the roots of/. Hence B^ is a.numerical multiplier of such squared product. To find the value of the multiplier, I observe that in general it follows from known algebraical principles that if ^ is a sum of the squares of n linear functions of w — 1 variables the discriminant of F may be found as follows. Form an oblong matrix with the coefficients of the several linear functions. The determinant represented by what Cauchy would have called the square of this matrix, but which is more correctly to be called the product of this matrix by its transverse, will be the discriminant in question, or which is the same thing this discriminant is the sum of the squares of all the complete minors that are contained in the oblong matrix.
In the case before us if we make /= «" — 1 * it will easily be seen that
When /=x"- 1 the value of S (the mean of the values of y) is obviously zero. Suppose now by way of illustration that n = 5, then calling the imaginary 5th roots of unity pi, Pi, pi, Pt, one of the complete minors referred to in the text will be the determinant of the matrix
Pi P2 P3 P4
|
h" |
pi' |
P^ |
P^ |
|
p.' |
P2' |
ft' |
Pt' |
|
PI* |
P2* |
ft* |
P.* |
and when the columns of this matrix are divided respectively by p,*, pj*, ft', P4*, [9 = 1, 2, 3, 4], which will leave the value of the determinant unaltered, the determinant of the matrix so modified will represent in succession each of the other 4 minors.
The value of the one above written, paying no attention to the algebraical sign, is by a well known theorem the product of the differences ot p^, p^, p^, pt, that is, inasmuch as
a-/'i)(l-P=)(l-ft)(l-/)J = 6
49] Oil the so-called Tschirnhausen Transformation 535
the n minors in question, paying no regard to algebraical sign, become all equal, and each will be the product of the differences of the roots of a;" — 1 when the root 1 is excluded, or which is the same thing will be the product of the differences of all the roots (not excluding 1) divided by n.
Hence the sum of the n squared minors will be the nth part of the square of the products of the differences of the roots of a;" — 1. Consequently in general the discriminant of l,y^ is the nth part of the product of the squares of the differences of the roots of the function /, and therefore by the process of reduction of — "Ey^ to a sum of n — 1 squares it is the positive sign always which will undergo the diminution of a unit, the number of negative signs remaining unaltered.
Hence when there are no imaginary roots in/, — B^ will have all its signs positive ; but when there are i pairs of imaginary roots in /, i of the signs in
I— B^ will be negative, and thus the character of B^, or of the quadratic contour (that is, curve, surface, hypersurface, etc.) represented by 5^ = 0 is completely determined when the number of real and imaginary roots in / is given. If we suppose n = 5 we see that according as the number of real roots in / is 5, 3, or 1, the signs of - B^ regarded as a sum of positive and negative squares of real linear functions of 4 letters will be :
+ + + + + + + -
+ + - -
In the first case the contour B^ is completely imaginary, and it is not only not possible to apply the Bring-Tschimhausen method so as to make simultaneously £3 = 0, 5, = 0 by real quantities Ui.u^, u,, M4, but it is also the case that such values of w,, 11^, u,, «< do not exist. This indeed is evident a priori, from the fact that the equation
y' + B,y + B, = 0 must have at least two imaginary roots and therefore the equation in x would have at least two imaginary roots if the quantities u^, u^, M3, M4 were all real and unequal ; whereas all the roots of that equation are supposed to be real.
In the second case the intersection of the contours B^, B, may be real or imaginary : but even if it be real the method will not serve to determine any
it is the oth part of the product of the differences of 1, pi, pj, ps, p^, and consequently the sum of the squares of the 5 minors is 5 times the 25th part of the squared product of the differences of
the 5 roots. Here — represents the general numerical multiplier -^ , that is, - .
536 On the so-called Tschhiihausen Transformation [49
single point in such section, because no real right line can be drawn to 5, at any point which shall lie on the surface.
In the 3rd case at each point of B, two real right lines can be drawn each of which will intersect 5, in one real point at least, and accordingly there will be a duplex-infinity of systems of real values of the m's which will make 5, = 0, 5i= 0 capable of being found by solving only a quadratic and a cubic equation in succession, and any one of such systems will lead to an equation of the form
where 5«, £5 (which it is hardly necessary to notice become respectively ^Sy*, — ^2y°) will each be real.
The B, found by Hermite's method may be obtained from the B3 above given by a real linear substitution impressed on the letters m,, tt,, «,, U4, and consequently the same conclusions continue to apply, that is, the coefficient of y and the constant will not in general be real unless four of the roots of the equation in x are imaginary *.
I will now proceed to the principal object of this paper, namely, the elucidation and extension of the method, contained in Hamilton's report, for determining the least number of letters which must be contained in one or more equations in order that they may admit of being solved by means of equations whose degrees are subject to satisfy certain prescribed conditions.
Before proceeding to the Lemma upon which all that follows is based, it will be useful to give one or two definitions.
1. Let (Si be a system of homogeneous equations in an indefinite number of variables x, y, ..., and let x = a,y = h, ... satisfy all the equations. I call a,b, ... a solution of <S.
2. If a, 6, ... is a given solution of 8, 1 call the equation obtained by operating upon any of those in 8 with {adx-¥hdy+ ...)'' where q has any integer value whatever not excluding zero, an emanant of such equation in respect to the solution a, b, ..., and the new system 81 which contains all the emanants of all the equations in >S an emanant to 8 in respect to the given solution.
* Hamilton remarks (Report of British Association, 1836, p. 307) that " the coefiicients of the new or transformed equation will often be imaginary even when the coefficients of the original equation are real." Apparently he was not aware that the criterion for determining when this is BO, depends solely on the intrinsic character of the equation to be transformed.
It should have been noticed before that when two of the roots in the given quintic are equal the quadratic surface represented by the coefficient of y^ in the transformed equation becomes a cone and the reasoning employed in the text falls to the ground. But inasmuch as in this case two of the values of y become equal, we know a priori that the equation in y must be reducible to a form with real coefficients, namely,
yi-5y + i = 0.
49] On the so-called Tschirnhausen Transformation 537
3. Let o, h, ..., Oi, bi, ..., be any two solutions of a systena of equations. I call
a + \ai, b+'XLi, c + XCi, ...
an alliance of the two solutions.
We may now state the following Sub-lemma. The alliance of any solution of a system S with a solution of Si, its emanant in respect to the first named solution, is a solution of S^ .
For let a,b, ... be the solution of S which gives rise to S^. Then, calling
adx + bd„+... = E, the general form of the equations in iS, is Ei^{x, y, ...), and, supposing ^ to be of n dimensions,
Eii^^x+Xa, y + \b, ...)
Hence the effect of substituting x + \a, y + \h, ... for x, y, ... in Si is merely to effect upon it a linear transformation, and consequently the alliance of the solution a,b,c, ... with any solution of Si will be a solution of that system. If now we find a solution Oj, 62, ... of Si and form an emanant S^ of it in respect to that solution, it will follow from the sub-lemma that an alliance of the solutions a,b, ... ; a,, 61, ... ; with any solution a^, 63, ... of S^, that is, the solution a + \a, + fj,a^, b + X6, + /i6,, . . . will be a solution of »Si2, and so in general. This I call the Lemma.
An ordinary solution of a system of equations may be called a point solution, an alliance in which 1, 2, 3, ... parameters enter a line, a plane, a hyperplane, ... solution.
It will of course be observed that any solution of an emanant to a system is d fortiori a solution of the primitive which as observed forms a portion of its emanant.
If <Si be a system of g,-, g',_,, ... 5, equations of degrees t, i—1, ... 1 respectively in the variables, it is obvious the 1st emanant will consist of
qi, qi + qi~u qi + qi-i + qi-2, ••• qi + qi-i+ •■■ + qi
equations of the degrees i, i—l,i—2, ... 1 respectively, and more generally in the »-th emanant the number of equations of the degree
i will be qi
t - 1 ,, ,. rqi + g.-.i
i-2 „ „ ^r(r+l)qi + (r-l)qi-i + qi^
1 » ., [r + i-l]iqi + [r + i-2]i^iqi-i + ... + qi,
0(6-1). ..(d-i+1)
where in general [d]i is used to denote
l.2...i
538 On the so-called Tschirnhausen Transformation [49
The question now arises as to what must be the number of variables in a system S in order that its rth emanant Sr may admit of a general solution. If the total number of equations in Sr be called N, it might at first sight be supposed that the number of variables, or letters as I prefer to call them, in S must have N + \ as an inferior limit : but the case is not so — the least number of variables required will be r greater than this, that is, i\r + r+ 1.
Thus, for example, suppose we consider a first emanant <S, ; then if Oj, 6,, Cj , ... is a solution we know that o, + Xa, 6, + \6, c^ + Xc, ... is also a solution
whatever \ may be. Hence making \ = ^ and remembering that the equa- tions are homogeneous we see that zero associated with any system of independent minors of the matrix
a b c ...,
a, 6, Ci
will constitute a solution, as for instance 0 ; abi—boi; aci — coj ; . . . *. Hence the number of independent quantities in /S, will be 1 less than the number of letters in S.
* As an iUastration suppose 4> is a quantic of degree n in (n + 2) letters representing what may be termed a contour, the analogue in general space of a curve in 2-dimen8ional or a surface in S-dimensional space. If we take all the successive emanants of 4> in respect to a point upon it a, 6, c, ... the n resulting functions [4> included] being functions of the n+ 1 minors to the matrix [(n+ 2) places in length]
la b c ... I \ X y z ... I the contours which they represent will intersect in a faisceau of right lines — showing that on a contour of the nth degree in (n + l)-dimensional space 1 . 2 . 3 ... n right lines lying in the contour will pass through every point thereof, a fact we are familiar with in the case of a quadric surface where n=2. We might with equal propriety and more convenience say that IIii straight lines may be drawn upon and at every point of an n-fold contour of the nth order.
As I have already referred in this footnote to right lines drawn on contours I venture upon a slight digression connected with this conception. If we have a cubic twofold contour (an ordinary cubic surface) expressed as a quantic in x, y, z, t, we see that on writing x, y as linear functions of z, t and substituting their values in 4> in order to make the result, a cubic function of z, t vanish, we have to satisfy 4 equations between the 4 coefiBcients of substitution, which at once shows that a finite number of right lines may be drawn upon such contour of which the number we see at once cannot exceed S* and which we know aliunde is 3'.
It would seem then that for a contour in n letters of the degree 2n - 5 (unless there is some lurking fallacy in the counting of the constants) we ought in like manner to be able, by expressing n - 2 of the letters as linear functions of the two remaining ones, to make the result vanish by solving 2n - 4 non-homogeneous equations of the degree 2n - 5 between the like number of co- efScients of substitution, and as if upon such a contour we must be able to draw a definite number of straight lines of which the number, supposing that there is no latent fallacy of constant- counting, would be not greater and in all probability less than (2n - 5)^-*, in fact (2n - S)'*'-^
Also it may be shown that, as by Bedetti's theorem we know that every twofold contour (an ordinary surface) is cut by its linear polar (its tangent plane) in respect to a point upon it, in a curve having a double point thereat, so a contour of the 3rd order will be out by its linear and quadratic polars in respect to any point upon it in a curve having a sextuple point thereat, and 80 in general an n-fold contour will be cut by n - 1 consecutive polars (starting from the tangential
49] On the so-called Tschirnhamen Transformation 539
Similarly for the system Sr) r zeros associated with any independent system of complete minors of the matrix
a h c ...
Oj 61 Ci
Wo On Cn • • •
a, br Cr ...
may be taken as the variables, and consequently it is iV - r - 1 and not iV - 1 which has for its inferior limit the number of equations in S,.. We may restore to the variables their independence by associating with the equations in Sr r additional perfectly arbitrary linear functions and there is sometimes a convenience in substituting in place of the rth emanant as it stands such emanant augmented by r arbitrary linear functions, which may be called the completed emanant.
For the purpose of greater clearness of exposition there will be an advantage in ignoring in the first instance all considerations based upon any other alliance except of the 1st order, that is, involving only one arbitrary parameter.
Suppose a system of equations S^ consisting of a system S and one equation more Q. If we are in possession of a linear solution of S, that is, a solution
x = ai+\a, y = 6, + \6, ... by substituting these values in Q, X may be found by solving an equation whose degree is that of Q, and thus a point (or ordinary) solution of S^ will have been found.
Let us now consider the question of a linear solution of S containing qt, qu-j, ... q^ equations of degree i,i-l, ... 1 respectively. This we shall call of the type {qi, qi_^, ... q,]. Let
a, h, ... be any point solution of 8, *Dd o„ 61, ... any point solution of E8,
homaloid as the first of them) in respect to any point upon it in a curve haying thereat a point of mnltiplicity 1 . 2. 3 ... n.
It may be well here to notice that a nni-parametrical solution of *=0 corresponds to drawing a straight line upon the contour represented by *, and in like manner a bi-parametric solution corresponds to drawing a plane upon the contour, a tri-parametric solution to drawing a hyper- plane upon the contour, and so in general. This is why I call such solutions linear, planar, hyperplanar, etc.
So again in this connexion it may be remarked that upon a quadratic contour in trans-hyper- space 6 planes lying on the contour pass through every point and in like manner upon a quadratic oontour in 2n letters, 1.2.3...n n-fold homaloids may be drawn upon the contour through every point thereon.
540 On the so-called Tschirnhatiseti Transformation [49
the completed emanant of S in regard to a, 6 This will be of the type
g,-; qi-k-qi-i; ...; 5'< + gi-i+ ... +3»; ?<+ gr,_i+ ... +9, + 1. The alliance of these two point solutions will be a linear solution of S. Again, the number of variables required for the point solution of ES need not exceed the number required for the linear solution of the system to which S is reduced by the abstraction of one equation of the degree i. Hence if we use [p, q, r, ... rj, 0] to denote the number of variables sufficient for the solution of a system of ^ equations of degree i, q of degree i — 1, etc. (i being the number of indices p, q, ... 0) we obtain the formula of reduction
[p, q,r, ... 0] = [p-l,p + q,p + q + r, ... p + q+r + ... + d] + l*. Continuing this process of reduction until the first index is reduced to zero a very easy calculation leads to the formula of obliteration
[p,q, r,s, ... ^]=p + [?„r„s,, ... ^i]t. where
p(p + l)^ ?'= 2 "'■^'
n = ^^^^ —^ +pq + r,
_p(p + l)ip + 2)(3p + l) p(p+l)
ff _p{p + l)...(p + i-l)(ip + l) p(p + l)...(p + i-2) .
"' - 17273 T:7(VT1) ^ 1.2.3...(t-l) ? + - + ''•
In applying the process of reduction in the way indicated, the system S will have been replaced by two systems which we may call a diminished S and a diminished emanant of S, that is, S and ES each deprived of an equation (not necessarily the same in both) of degree i.
In like manner each of these will give rise to two systems, namely, a diminished self-system and a diminished emanant-system ; but as the object is to obtain a formula of reduction for the number of letters required to obtain a linear solution of S, and as this number is greater for an emanant of any system than for the system itself, it was sufficient to follow the main- stream of deduction, in which the first alone is taken account of, in order to arrive at the required formula. In doing so, 2^ independent equations of the degree i will have been set apart each of which will have to be solved in its proper turn.
In the formula of obliteration the index in the first place has disappeared. Repeating the process we shall come to
p + qi + b;,s^, ... 0.,]
* In thie and all subsequent formulae of reduction or obliteration the sign " = " is to be understood to mean " not greater than."
+ Compare Hamilton, Report of British Association, 1836, p. 335, formula 244, and p. 345, formula 320.
49] On the so-called Tschirnhauaen Transformation 541
where r,, Sj, ... 6^ are derived from ^i, r,, «,, ... ^i in the same way as q^,ri, s, , ... di from p, q,r, s, ... 6 except that i will be i-eplaced by t — 1 ; and thus pursuing the same process we shall arrive at
or say [cr]. The number of variables required for a solution involving one arbitrary parameter of a homogeneous linear equations being o" + 2, this latter will be the number sufficient for S to admit of a linear solution without giving occasion to solve any equation of a degree exceeding i, and also with- out having occasion to solve any simultaneous system of equations other than linear ones.
Suppose a system of equations of the respective degrees 1, 2, 3, ... i and a single equation of the degree i + 1.
The type of the former will be 1, 1, 1, ... 1 to i places, and of the latter 1, 0, 0, 0, ... 0 „ i + 1 „ . By the rule which has been established the number of letters required for the linear solution of the latter will be one more than for the former.
Hence the determination of the Tschirnhausen question of finding what the degree of an equation must be in order that i consecutive terms following immediately after the first term in the transformed equation, conjoined with any more advanced term, may admit of a solution of minimum weight, contains a determination of the number of variables required to ensure the possibility of obtaining a linear solution by a system of equations of minimum weight of a single equation of degree i+ 1 ; for the latter number will be the former increased by a unit*. The first form of the question is the more simple in itself; but as the other is more immediately connected with the object in which the theory originated, I prefer to put it in the latter form.
We may apply the obliteration formula to the indefinite type and obtain the annexed Table.
Triangle of Obliteration.
11111 1 1
2 3 4 5 6 7
6 15 29 49 76
36 210 804 2449
876 24570 401134
408696 246382080
83762796636
* For example, to take away the 2nd, 3rd, and another term the degree required is 5 : and to obtain a linear solution of a cubic the number of variables required is 6.
To take away the 2nd, 3rd, 4th, and another term, employing a solution of the lowest weight, 11 Tariablcs are required; in order to obtain a solution, of lowest weight, of a single function of the fourth degree, 12 variables are required, and so on.
542 On the so-called Tschirnhausen Transformation [49
The degree of the equation sufficient to allow
2, 3, 4, 5, 6, 7, ...
consecutive terms following the first to be removed by a solution of minimum weight of the auxiliary equations, will be the continued sum of
1, 2, 6, 36, 876, 408 696, 83 762796 636, ... each increased by 2, that is,
3, 5,11, 47, 923, 409 619, 83 763 206 255, ...
These numbers up to 923 agree with those found by Hamilton {Report, p. 346), the two last have been calculated here probably for the first time.
It would be too arduous a task to seek to give a much further extension to the table inasmuch as each successive term in the series 1, 2, 6, 36, ... is a fraction converging to \ of the square of the preceding term. This becomes obvious from inspection of the series formed by dividing each number in the above series by the square of the one before it ; we thus obtain the fractions: 4 6 36 _876^ 408696 83762796636 T' 4' 36' 1296' 767376' 167032420416'
which are continually diminishing.
But if we call two successive and infinitely distant rows of the Triangle of Obliteration
a 6 ...
B ....
B b
Hence-; converges to i + — , which is always greater than J. Moreover
— , calculated for the successive values as far as the table extends, will be seen to be a continually decreasing fraction and assuming (what awaits exact proof) that it eventually vanishes, — must converge to ^ .
The successive values of — for the different rows are
w
3 15 210^ 24570 ^46382080 4' 36' 1296' 767376' 167^32420416"
Inverting these fractions the values, to the nearest integer, become 1, 2, 6, 31, 678, so that there can be no doubt of the truth of the law that the asymptotic value of the square of each term divided by the square of its antecedent is \.
49] On the so-called Tschirnhausen Transformation 543
Moreover the numbers last found themselves obviously obey a parallel law to that of the original series which raises a presumption that it may be possible to obtain an exact expression for the general term in the original series or even in the Obliteration Table in its entirety. But be that as it may, as evidently the asymptotic law is equally true for the sums of the terms in the first diagonal as for the terms themselves, we arrive at the interesting fact that if <E> (i) is the minimum degree of an equation from which i consecutive terms immediately following the first can be removed, 2<l>(i + l) converges to a ratio of equality with <I>(i)- when i increases indefinitely.
The minimum number of letters thus found is we see a minimum, at all events in this sense that the method employed to obtain a solution is in- applicable if that number of letters be reduced. In the words of Jerrard as quoted by Hamilton (Report, pp. 326, 327) " to discover m — 1 ratios of m disposable quantities,
fl], (Z^, ... dni
which shall satisfy a given system of h^ rational and integral and homogeneous equations of the first degree
^' = 0, A"=0, ... ^<*>i = 0, hi such equations of the second degree
B' = 0, B" = 0, ... £!*«•= 0, A, of the third degree
C'^0, C" = 0, ... C""'=0, and 80 on, as far as ht equations of the tth degree
r=o, r" = o, ... r<v = o
without being obliged, in any part of the process, to introduce any elevation of degree by elimination."
But this definition may be superseded by another in which only the intrinsic character of the result arrived at is in question, and not the particular method pursued to reach it.
Let us agree to consider all equations of the same degree to have the aame weight and that this weight is infinitely greater than that of an equation of any lower degree. The weight of a system of equations to be regarded as the sum of the weights of the equations which it contains.
We may, extending but not altering the meaning previously attached to the word " solution," call the ensemble of the equations to be solved in order to obtain any solution of the given system a solution thereof If now a system of equations is given in number and in the degree of each, and each equation is supposed to be the most general of its kind, but the number of variables in the system is left disposable, it is easy to see that the above
544 On the so-called TschirnJiansen Traiisfoiination [49
process, when it is practicable, leads to a solution of the lowest weight, so that no increase in the number of letters will have any effect in diminishing the weight of the solution, whatever may be the process employed to obtain it. Thus the numbers given by the linear method are minima in regard to solutions of the lowest weight.
We may however suppose another and more natural condition attached to the solution to be obtained ; let n be the highest degree of any equation in a given general system proposed for solution ; we know that it is impossible to avoid the solution of one or more equations of the nth degree. We may therefore propose to ourselves the problem of determining what is the least number of letters necessary in order that no equation in the solution shall be of a degree exceeding n. The minimum thus obtained will in general be inferior to the minimum required for obtaining a solution of the lowest weight, and to arrive at it in any particular case it becomes necessary to make use of the Lemma in its general form which introduces the notion of alliances above the first order. Hamilton has not touched upon this part of the subject except in a single case which it was impossible to overlook : namely, where he considers the problem of taking away four consecutive terms from the general equation of the tenth or any higher degree.
The process we have seen leads to the conclusion that as many letters are required as are needed for the solution of two quadratics and seven linear equations. The solution of one biquadratic equation in the application of the process being indispensable, he felt the absurdity (if I may use the word) of stickling at the introduction of one biquadratic more, the use of which has the effect of lowering the minimum from 11 to 10. See Report of British Association, 1836, p. 326.
The linear method however or theory of solutions of lowest weight enjoys this prerogative that the reduction formulae are of a purely algebraical kind, whereas when the other condition above referred to is introduced, questions of numerical equality and inequality have to be considered and the theory ceases to be strictly algebraical. In what follows therefore I shall confine myself to the only case of any particular interest, namely, that which arises from the original problem of removing any given number of consecutive terms (immediately following the first) from an algebraical equation.
We may accept as the general condition to be observed that the degree of no equation appearing in the solution of a system of equations shall exceed the highest degree which must perforce figure in such solution, that is, the highest degree in the system of equations to be solved. In the case then of n equations of the successive degrees 1, 2, 3, ... i the condition will be that no equation in the solution shall be of a higher degree than i.
49] On the so-called Tscldrnhausen Transformation 545
Thus, for example, if we look back to the easy case of a quaternary succession of such terms to be removed, we find that the problem reduces itself to finding the number of letters required to obtain a line-solutiou of the system whose type is 1, 1, 1, and that again to finding the number of letters required to obtain a line-solution to its augmented emanant 2, 4, that is, a system of 2 quadratic and 4 linear solutions, that is, a point solution of the completed emanant to this system which will be of the type 2, 7. The condition imposed here is that no equation shall appear of a higher degree than a biquadratic. Consequently subject to this condition the number of letters required to solve a system of one linear, one quadratic, and one cubic equation, is that sufficient for the plane-solution of a system of 7 linear equations, that is, 10, which is less by 1 than the number required in order to obtain a solution of the same system which shall be of the lowest weight.
It might at first sight be supposed that in general the introduction of sulutions involving 2 or more parameters would lead to a very considerable reduction of the numbers found in the obliteration table ; this however is not the case, the reduction in the values obtained by this extended method bears in general a very small ratio to the number reduced. This is a consequence of the following rule : —
In passing from the point solution of a system to a solution of any kind with a reduced type, the reduction is effected by segregating a certain number i of the given equations and obtaining a solution of the remainder which shall contain i arbitrary parameters.
Now it will be found that the literant (by which I mean the number of letters sufficient for the solution) will never be diminished by any other kind of segregation than what may be termed an external segregation*.
* Imagine the type of a set of equations to be represented by a broad ribbon, in which each gronp of equations of the same degree is represented by a band of a distinct colour occupying as many units of space as there are units in the group. The legitimate process of segregation will then consist in dividing the band into two, obeying the same conditions as the original one, and the rule of " external segregation " amounts to saying that this separation must be effected by a single straight cut so that no middle portion is to be cut out.
According to this (which is a perfectly natural) representation the rule of external segregation may in the language of logic be described as the rule of the excluded middle. Thus, for example, suppose we wish to find the smallest number of variables required for the solution of a system of equations of which the type is 1, 1, 1, 0 without solving an equation beyond the 8th degree. The number required may be made equal to (cf. p. 547)
. H, 1, 0] or to :[1, 0, 0]. But .[1,1,0] = [1,2, 3]=: [3],
•nd :[1, 0, 0] = [1, 2, 5]=:[5].
Thus the simultaneons segregation of the equations of the 4th and 2nd degrees contrary to the rale not only raises the weight of the solution but also increases the number of variables required in the given system in order that the solution may be possible.
Aa a consequence of this rule it may easily be seen (in (he problem of determining the
8. IV. 35
546 On the so-called Tscfnrnhmisen Transformation [49
Let f, g, ... k, I be the type of the system of equations segregated, this will have no effect in diminishing the literant unless f, g, ... k are the initial numbers of the type of the given system, in such case I call the segregation ea;temal.
Thus in starting with a system of the type 1, 1, ... 1, 1 the first act of segregation must consist in setting apart the equation of the highest degree and finding a line-solution of the system thus reduced. Suppose, to fix the ideas, that the highest degree is 6 and that we have arrived in the course of the deduction at a system of linear, quadratic, and cubic equations denoted by the type m, n, p.
So far as regards observance of the limit 6 for the highest degree in any substituted system, it would be permissible to segregate one cubic and one quadratic, but according to the rule of external segregation this will not be profitable (it will in general be quite the reverse unless m=l) and so in general.
Let us now proceed to obtain the literant required for the point-solution of a sequence of i equations of all degrees from 1 to i subject to the condition that no auxiliary system shall contain an equation of degree higher than i for the values i = 5, 6, 7, 8 which is as far as the table of obliteration extends. The rule teaches that this is the same as the literant of a line-solution of a system of i — 1 equations whose degrees extend from 1 to i — 1.
It will be useful in what follows to obtain a general formula for the plane- literant of a system of i quadratics denoted by the type i, 0.
Let us signify by a symbol consisting of a type preceded by q points the literant to the form of solution containing q parameters of the system to which the type refers.
Then calling the plane-literant for [i, 0] t;,-, we have by virtue of the Lemma
Vi = : [i, 0] = [i - 2, 2i -f 2] = Vi^ -1- % -|- 2, t;, = :[l,0] = .[l,2] =.[4]=6,
z;, = :[2,0] = .[2,3] = .[l, 6] = [8] = 9. Hence by integrating t;,- — Vi^ = 2i -I- 2 we shall easily obtain :
v^ =25^-1-45 + 3,
v^^ = 25» -I- 25- -I- 2.
In treating of the literant to[l, 1, 1, 1, 1, 1, 1, l]it will be convenient to find
minimum degree of the equation required for taking away t cousecutive terms without any eqnation in the solution exceeding the I'th degree) that the occasion can never arise in the act of segregation to take account of any other numerical equalities and inequalities than one or the other of the two following
g«= or <n, 5'(g-l}''= or <«.
49] On the so-called Tschirnhausen Transformation 547
the value of • [t, 0, 0] the general expression of which rid of exponentials will give rise to 3 cases.
Not being desirous of encumbering this memoir with formulae, and as we shall only have occasion to consider a single case of these formulae, I adjourn the calculation until we know what the form is of i in regard to 3 in the case to be calculated, and shall obtain the value of \ [t, 0, 0] for that case alone.
[ will now consider in succession the literants denoted by .[1,1,1,1] .[1,1,1,1,1] .[1,1,1,1,1,1] .[1,1,1,1,1,1,1] subject to the conditions of the solution containing no equation of a degree higher than the 5th, 6th, 7th, 8th respectively
. [1, 1, 1, 1] = . [2, 3, 5] = . [1, 5, 11] = . [6, 18]
= :[4, 25] = 25 + 2.2-+4. 2 + 3 = 44.
This is the literant for the solution of minimum highest degree and is 3 units less than 47, the literant for the solution of lowest weight.
It will be observed that . [6, 18] has been expressed in the course of the deduction by : [4, 25] instead of . [5, 25]. In fact . [6, 18] = [6, 25] and this latter according as we segregate 1 or 2 of the quadratics is expressible by .[5, 25] or by: [4, 25].
The expression .[6, 18] might have been obtained immediately from the triangle of obliteration
1111
2 3 4 6 15
by simply substituting 1 + 2 + 15 for 18. (It is worth noticing that in the table of obliteration after the 2nd line every initial number in any line ends with 6 and after the 3rd line every second number in each line ends with 0.)
So in like manner observing that 1 + 2 + 6 + 210 = 219, we have .[1,1,1, 1,1] = . [36, 219] which must have been led up to from
[1, 36, 219]. Hence . [1, 1, 1, 1,1] = . [1, 35, 182] = [1, 36, 219] = : [35, 219]
= 219 + 2 . IS'' + 2 . 18 + 2 = 905 which is 18 units less than the corresponding literant of lowest weight 923. Similarly observing that
1 + 2 + 6 + 36 + 24 570 = 24 615,
[1,1,1,1,1,1] = : [875, 24 615] = 24 615 + 2 (438)» + 2 (438) + 2 = 409 181
35—2
548 On the so-called Tschimhausen Transformation [49
which is 438 less than the corresponding literant of lowest weight 409 619.
In like manner calling
246 382 080 + 876 + 36 + 6 + 2 + 1 = 246 383 17 5 = s,
. [1, 1, 1, 1, 1. 1, 1] = « + : [408 695, 0] = [408 695, 0] + < = I [408 692, 0] + t
where « = « + 2 x 408 695 + 2 = 247 200 567.
Here 408 695 = 2 [mod. 3].
But in general j [3^ + 2, 0] = ; [3gr - i, o] + 9^ + 9
= :[2,0] + 9{(q + l) + q + (q-l)+ ... + 2} = -[2,0] I 9(g' + 39)^9g' + 27g + 24^
m + 2y + 5(Sq + 2) , , = — 2 + ^-
Therefore . [1, 1, 1, 1, 1, 1, 1] = < + 5 + (204 346) (408 697)
= 247 200 572 + 83 515 597 162
= 83 762 797 734.
This number is the minimum degree of equation which admits of 8 of its terms being removed without solving any equation above the 8th degree in the same sense as 5 is the minimum degree of equation from which 3 terms can be removed without solving an equation above the 3rd degree.
The Hamiltonian numbers corresponding to the solutions of lowest weight, have been found to be
3, 5, 11, 47, 923, 409 619, 83 763 206 255
the reduced numbers due to the introduction of planar and hyperplanar
solutions
3, 5, 10, 44, 905, 409181, 83 762 797 734,
the differences are 1, 3, 18, 438, 408 521.
The ratio of these last numbers to the numbers above them constituting a rapidly decreasing series, it is obvious that the " asymptotic law " will remain good for the second as well as for the first line of numbers : so that if <f) (i) expresses the minimum degree of an equation from which i terms can be
abstracted without solving an equation above the ith degree, ,. ..^ — will
continually decrease towards and finally (when i is infinite) coincide with unity.
I have already defined the weight of a solution. According to analogy (as, for example, in the case of a given symmetric function Sa' .b^ .c^ ...) the degree of the equation of highest degree in a solution may be termed its order.
* For I [2, 0] = [2, 9]=:[9] = 12.
49] On the so-called Tschirnhausen Transformation 549
Thus then the two first series of numbers which have been given express the first of them the literant of the solution of lowest weight, the second the hterant of the solution of lowest order. The numbers in the first series up to 923 and m the second series up to 10 appear in Hamilton's Report, all the others are here presented (it is believed) for the first time.
A solution is of course to be understood to mean a non-simultaneous but not independent system of equations from which a solution of a given system of equations may be derived. The equations in the solution-system form an arborescence or a ramification of consecutive systems, meaning thereby that the solution of any one of them depends upon a successive process of substitution of values of variables deduced from equations which precede it in such ramification. Some of the simpler of these arborescences I propose to delineate graphically in a subsequent communication.
Invited to participate in the centenary number of the 'leading Mathe- matical Journal in the world, it occurred to me that compatibly with my feeble means no more suitable contribution could be made than one which at the same time celebrates the centenary of the discovery due to the long and persistently ignored author of the method which it is the object of this memoir to elucidate and extend. I offer it (an aloe-flower of 100 years- growth) as a tardy Bessarabian "satisfaction to the Manes of" Bring.
50.
SUR UNE D^COUVERTE DE M. JAMES HAMMOND RELATIVE A UNE CERTAINE S^RIE DE NOMBRES QUI FIGURENT DANS LA TH^ORIE DE LA TRANSFORMATION TSCHIRN- HAUSEN.
[Comptes Rendus, civ. (1887), pp. 1228—1231.]
On peut se proposer le probl^me suivant :
JStant donni un quantic, le /aire disparaitre en exprimant chaque variable comme une fonction lin6aire et homoghne de deux variables.
Si le nombre des variables dans le quantic est suflSsamment grand, quel que soit son degre n, ce probl^me peut s'efiFectuer au moyeu d'un systeme auxiliaire d'^quations, tel que pour r^soudre le systeme on n'aura jamais occasion de rdsoudre une Equation d'un degr^ sup^rieur a n.
En nommant N le nombre minimum des variables necessaire pour que cela soit possible, cette question se prdsente : trouver la valeur de N pour une valeur donnde de n.
Par exemple, pour ra = 2, on voit bien que N est 4.
Pour n = 3, on peut d^montrer que iV est 6 ; pour n = 4, JV= 11, etc.
Mais on peut imposer une condition plus rigoureuse sur le caractere du systfeme auxiliaire d'^quations qui aura I'effet d'augmenter la valeur minimum i\r. On peut exiger que le type du systeme auxiliaire d'equations sera le plus simple possible ou, comme je pr^fere le dire, sera d'un poids minimum. Le poids d'une Equation depend seulement de son degr^ i et peut dtre pris ^gal k /3*', oil p est une constante indefiniment grande. De plus, le poids d'un systfeme d'equations peut etre d^fini comme ^tant la somme des poids des Equations individuelles qu'il contient.
On a ainsi un criterium exact pour determiner lequel des deux syst^mes a son poids infdrieur k celui d'un autre; le terme poids minimum devient exempt de toute ambiguite, et Ton comprend ce que veut dire le systeme d'dquations le plus simple d'un nombre quelconque de tels syst^mes.
50] Sur line decouverte de M. James Hammond 551
Avec la premiere ddfinitioa de N, ses valeurs successives seront 3, 4, 6, 11, 45, 906, 409182, 83762797735, ....
En imposant la condition la plus rigoureuse, on obtient la se'rie moins transcendante
3, 4, 6, 12, 48, 924, 409620, 83763206256, ...
que je nommerai E^, E^, E^, E,
En diminuant ees derniers chifFres de I'unite, on trouve la serie de nombres
2, 3, 5, 11, 47, 923, 409619, 83763206255, ...,
dont les six premiers ont ^te calculus par Hamilton (voir Report of 6th Meeting of British Association, pp. 346 — 7, 1837).
Hamilton a, en effet, montr4 que le degre d'une Equation algebrique, ^tant pris suceessivement egal a 2, 3, 5, 11, 47, ..., on peut, par la methode dite de Tschimhausen, la transformer dans une autre ou 1, 2, 3, 4, 5, . . . termes consecutifs, aprfes le premier, manquent, sans avoir occasion de resoudre aucune Equation au-dessus des degr^s 1, 2, 3, 4, 5, ... respectivement.
J'ajoute que le systeme d'equations auxiliaires, auquel on parvient par la mdthode qu'il emploie, sera du type le plus simple possible. Si, pour 6ter i termes consecutifs, on voulait se borner a la seule condition de n'avoir pas a resoudre une Equation au-dessus du degr^ i, alors, au lieu des nombres 2, 3, 5, 11, 47, ..., on aurait les nombres plus transcendants 2, 3, 5, 10, 44, .... C'est la s4ne 2, 3, 5, 11, 47, ... que je nomme les nombres de Hamilton, et
que je designe par Ho, Hj, H^, H^, H^ Pour les obtenir (ou plut6t
leurs differences) par la mdthode de Hamilton, on a besoin de construire un triangle de chiffres (voir mon M^moire dans le Journal de Kronecker, t. c. p. 477 [above, p. 541]).
Mon collaborateur, M. James Hammond, a trouvd un tres beau th^oreme pour d6duire les N immediatement et suceessivement les uns des autres, sans introduire de nombres Strangers.
En se servant de /9r (?) pour repr^senter ^^ — ^ '"' ^^ ' , il a trouve
J. ■ ^ • • • 7*
la formule vraiment remarquable
Hi^2 + 0, (Zr,_,) - ,83 (Hi.,) + A (F,_3) - . . . .
A ce th^oreme, j'ajoute comme corollaire une formule qui se rapportekla s^rie de nombres E (qui ne sont autre chose que les nombres H, augmentes chacun de I'unit^), qui est bonne pour toutes les valeurs de r sup^rieures a I'unite,
^, (Er) - yS, (Er-,) + A (Er^) -...+ (-)'/3, (E,) = 0,
c'est-i-dire .£,_. = 1-1-/9, (Er-,) - /3, (Er-,) +...+ {-Y^r (^o).
552 Stir une decouverte de M. James Hammond [50
4 3.2 Par exemple, 1 - r + y" s ~ ^>
* 1^1.2 1.2.3 '
12 6^5 _ 4.3.2 _ . ^~ 1 "*"1.2 1.2.3 '
1 "*" 1.2 1 . 2 . 3 "*" 1717374 '
924 48.47 12.11.10 6.5.4.3 1 "*" 1.2 1.2.3 ■'"1.2.3.4"
C'est par la methode de fonctions generatrices que M. Hammond a r^ussi a etablir eette ^chelle de relation entre les nombres de Hamilton, lequel evidemment n'avait pas le moindre soup9on de I'existence d'une ^chelle pareille.
Si Ton prend les differences des nombres de Hamilton, on obtient la s^rie
1, 2, 6, 36, 876, ..., qu'on peut nommer Aj, h^, h,, ht, h^, On savait deja
par demonstration que Aj+i -r- hi^ est plus grand que ^ pour toute valeur finie de t et avec certitude morale que ce rapport devient \ quand i est infini. Avec la formule de M. Hammond, on peut donner une demonstration rigour- euse de ce dernier fait et en meme temps etablir ce nouveau theoreme : Hi^i -r- jff," est plus petit que ^ pour toute valeur de i finie et plus grande que I'unitd, et dgal a ^ quand i est infini.
51.
ON HAMILTON'S NUMBERS.
Bv J. J. Sylvester and James Hammond.
[Philosophical Transactions of the Royal Society of London, CLXXViii. (1887), pp. 285—312; CLXXix. (1888), pp. 65—71.]
Inteoduction.
In the year 1786 Eriand Samuel Bring, Professor at the University of Lund in Sweden, showed how by an extension of the method of Tschirn- hausen it was possible to deprive the general algebraical equation of the 5th degree of three of its terms without solving an equation higher than the 3rd degree. By a well-understood, however singular, academical fiction, this discovery was ascribed by him to one of his own pupils, a certain Sven Gustaf Sommelius, and embodied in a thesis humbly submitted to himself for approval by that pupil, as a preliminary to his obtaining his degree of Doctor of Philosophy in the University*. The process for efifecting this reduction seems to have been overlooked or forgotten, and was subsequently rediscovered many years later by Mr Jerrard. In a memoir contained in the Report of the British Association, for 1836, Sir William Hamilton showed that Mr Jerrard was mistaken in supposing that the method was adequate to taking away more than three terms of the equation of the 5th degree, but supplemented this somewhat unnecessary refutation of a result known d priori to be impossible, by an extremely valuable discussion of a question raised by Mr Jerrard as to the number of variables required in order that any system of equations of given degrees in those variables shall
* Bring's "Reduction of the Quintic Equation " was republished by the Rev. Robert Harley, P.R.8., in the Quarterly Journal of Pure and Applied Mathematio, vol. vi. 1864, p. 45. The full title of the Lund Thesis, as given by Mr Harley (see Quart. Journ. of Math., pp. 44, 4.5) is as follows : "B. cum D. Meletemata quaedam mathematica circa transformationem aequationum algebraicarum, quae consent. Ampliss. Facult. Philos. in Regia Academia Carolina Praeside D. Eriand Sam. Bring, Hist. Profess. Reg. & Ord. publico Eruditorum Examini modeste ■abjicit Sven Gustaf Sommelias, Stipeodiarius Regius & Palmcrentzianus Lundensis. Die xiv Decemb., mdcclxxxti, L.H.Q.S. — Lundae, typis Berlingianis."
564 On Hamilton's Numbers [51
admit of being satisfied without solving any equation of a degree higher than the highest of the given degrees.
In the year 1886 the senior author of this memoir showed in a paper* in Kronecker's (better known as Crelle's) Journal that the trinomial equation of the 5th degree, upon which by Bring's method the general equation of that degree can be meide to depend, has necessarily imaginary coefficients except in the case where four of the roots of the original equation are imaginary, and also pointed out a method of obtaining the absolute minimum degree M of an equation from which any given number of specified terms can be taken away subject to the condition of not having to solve any equation of a degree higher than Mf. The numbers furnished by Hamilton's method, it is to be observed, are not minima unless a more stringent condition than this is substituted, namely, that the system of equations which have to be resolved in order to take away the proposed terms shall be the simplest possible, that is, of the lowest possible weight and not merely of the lowest order ; in the memoir in Grelle, above referred to, the author has explained in what sense the words weight and order are here employed. He has given the name of Hamilton's Numbers to these relative minima (minima, that is, in regard to weight) for the case where the terms to be taken away from the equation occupy consecutive places in it, beginning with the second.
Mr James Hammond has quite recently discovered by the method of generating functions a very simple formula of reduction, or scale of relation, whereby any one of these numbers may be expressed in terms of those that precede it : his investigation will be found in the second section of this paper, and constitutes its most valuable portion. The principal results obtained by its senior author, consequential in great measure to Mr Hammond's remarkable and unexpected discovery, refer to the proof of a theorem left undemonstrated in the memoir in Grelle above referred to, and the establishment of certain other asymptotic laws to which Hamilton's Numbers and their differences are subject, by a mixed kind of reasoning, in the main apodictic, but in part also founded on observation J. It thus
[' Above, p. 531.]
t For instance, an equation of not lower than the 90oth degree may be transformed into another o( that degree, in which the 2nd, 3rd, 4th, 5th, 6th, 7th, terms are all wanting, by means of the successive solution of a ramificatory system of equations, of no one of which the degree exceeds 6, whereas by the Jerrard-Hamiltonian method this transformation could not be effected for the general equation of degree lower than the 6th Hamiltonian Number, namely, 923. So for the analogous removal of 5 consecutive terms the inferior limit of degree of the equation to be transformed would be 47 by the one method, but 44 (the lowest possible) by the other. In the case of 4 consecutive terms Hamilton could not avoid being aware that 11, the 4th number which I have named after him, might be replaced by 10, as the lowest possible inferior limit of the equation to be transformed.
X In the 3rd section, communicated to the Society after the 1st and 2nd had gone to press, the empirical element is entirely eliminated, and the results reduced to apodictio certainty.
» I
51] On Hamilton's Numbers 555
became necessary to calculate out the 10th Hamiltonian Number, which contains 43 places of figures. The highest number calculated by Hamilton (the 6th) was the number 923, which comes third in order after 5 (the Bring Number), 11 and 47 being the two intervening numbers. It is to be hoped that some one will be found willing to undertake the labour (consider- able, but not overwhelming) of calculating some further numbers in the scale.
The theory has been " a plant of slow growth." The Lund Thesis, of December 1786 (a matter of a couple of pages), Hamilton's Report of 1836, with the tract of Mr Jerrard therein referred to, and the memoir in Crelle, of December 1886, constitute, as far as we are aware, the complete biblio- graphy of the subject up to the present date.
§ 1. On the Asymptotic Laws of the Numbers of Hamilton and
their Differences.
Consider the following Table : —
100000 0 0
11111 1 1
2 3 4 5 6 7
6 15 29 49 76
36 210 804 2449
876 24570 401134
408696 246382080
83762796636
Any line of figures, say p, q, r, s,t, ... 0, in the Table being given, to form the subsequent line qi, r,, s,, ti, ... 6^, we write
„-P(P+l) , „
^■° 1.2.3 +^^ + ^'
_p(/) + l)(2j> + l) 1.2.3
_p(p + l)(p + 2)(3p + l) p{p + \) *' 1.2.3.4 + 1.2 9+^^^ + *'
, _y(p+l)(p + 2)(j> + 3)(4p-H) p(p + l)(p + 2) p{p + \)
'- 1.2.3.4.5 + T72T3 ?+ 1.2 ^+^* + «'
a _io(p + l)-(p + t-l)(»>+l) , p(;? + l)...(p + t-2) g
"' 1.2.3...(t + l) + 1.2.3...(t-l) ^ + - + ''-
556 On Hamilton's Numbers [51
If we call the nth term of the j»th line [m, n], the general law of deduc- tion may be expressed by the formula
[m + 1, n] = - Bn+i ([m, 1] - 1) + T[m. n + 1 - i] 5< [m. 1],
where Bik means the coeflBcient of z* in (1 — ^)~*.
The negative term — 5„+, ([to, 1] — I), it may be noticed, arises from decomposing the first term of [to + 1, «], as given by the original formulae, into two parts, of which it is one. Thus, for example,
p(p + l)(p+2)(p + S){*p + l) 1.2.3.4.5 is changed into
_ {p-l)p(p+l)(p + 2)(p + S) pip + l){p + 2)(p + S)
1.2.3.4.5 1.2.3.4 ^'
The numbers in the hypothenuse of this infinite triangle, namely,
1, 1, 2, 6, 36, 876, 408696, 83762796636, 3508125906207095591916,
6153473687096578758445014683368786661634996
are what I call the Hamiltonian Differences, or Hypothenusal Numbers*; and their continued sums augmented by unity, namely,
2, 3, 5, 11, 47, 923, 409619, 83763206255, 35081259062.90858798171,
61534736870965787.58448522809275077520433167
are what I call the Hamiltonian Numbers. The two latter of these have been calculated by means of Mr Hammond's formula, presently to be mentioned, and the corresponding Hypothenusal Numbers deduced from them by .simple subtraction. Their connection with the theory of the Tschimhausen Transformation will be found fully explained in my memoir on the subject in Vol. c. of Grelle. My present object is to speak of the numbers as they stand, without reference to their origin or application f.
* The other numbers of the "triangle," whose properties it maybe some day desirable to investigate, may be termed oo-hypothenusal numbers of order measured by their horizontal distance from the hypothenuse — their vertical distance below the top line denoting their rank. In the sequel the development is given of the half of a hypothenusal number (of the first order) in a descending series of powers (with fractional indices) of the half of its antecedent, the coeffi- cients in the principal part of such series being (not, as might have been the case, functions of the rank, but) absolute constants. These may be termed the hypothenusal constants. The' values of the first four of them are shown to be 1, ^, {^, Jf .
t The reader will be disappointed who seeks in Hamilton's Report any systematic deduction of the numbers which I have called after his name. He treats therein the more general question of finding the number of letters sufficient for satisfying any system of equations of given degrees by means of a certain prescribed uniform process whereby the necessity is obviated of solving any equation of a higher degree than the highest one of the given equations, and among, and mixed up with, other examples considers systems of equations of degrees 1, 2, 3; 1, 2, 3, 4;
51]
On Hamilton's Numbers 557
The question arises as to whether it is possible to deduce the Hamiltonian Differences, or to deduce the Hamiltonian Numbers, directly in a continued chain from one another without the use of any intermediate numbers. Mr James Hammond has shown that it is possible, and has made the remarkable discovery that it is the Numbers of Hamilton, and not the Hypothenusal Numbers, which are subject to a very simple scale of relation. These being found, of course the Differences become known. This is contrary to what one would have expected. A priori, one would have anticipated that the determination of the Hypothenusal Numbers would have preceded that of their sums.
I leave Mr Hammond to give his own account of his mode of obtaining the wonderful formula of reduction, which, by a slight modification, I find, may be expressed as follows: — Using Ei to denote the {i + l)th Hamiltonian Number augmented by unity, .so that £"0=8, ii\=4<, ^2=6, £^3=12, £"4=48,...;
1, 2, 3, 4, 5; 1, 2, 3, 4, 5, 6; for which the minimum nnmbers of letters required to make such process possible (when the equations are homogeneous) are 5, 11, 47, 923, respectively. Accord- ingly he has no occasion to employ the infinitely developable Triangle which gives unity and cohesion to the problem which deals with an indefinite number of equations of all consecutive degrees from 1 upwards. This triangle, which plays an important part in the systematic treat- ment of the problem, first appears in my memoir on the subject in the 100th volume of Crelle.
It is proper also again to notice that what I call the Numbers of Hamilton (at all events those subsequent to the number 5) are not the smallest numbers requisite for fulfilling the condition above specified. Smaller numbers will serve to satisfy that condition taken alone ; but when such smaller numbers are substituted for Hamilton's the resolving equations will be less simple, inasmuch as they will contain a greater number of equations of the higher degrees than when the larger Hamiltonian numbers are employed. This distinction will be found fully explained in the memoir cited, and the smallest numbers substitutable for Hamilton's are there actually determined for r equations of degrees extending from 1 to r for all values of r up to 8 inclusive.
I have added nothing (for there is nothing to be added) to the fundamental formula of Hamilton expressed by the equation
[X, /i, o, ... ir]=i-(-[\-i, ^-^/i, X+M+ 1-, •■•, ^ +/*+>' + ■•■ + «■].
where, sapposing the letters X, /t, j/, . . . ir, to be i in number, [X, /x, », ... ir] means the number of letters required in order that it may be possible to satisfy, according to the process employed by Hamilton (in conformity with a certain stipulation of Jerrard), a system of X equations of degree t, H equations of degree i-l, v equations of degree t - 2, ..., ir equations of the degree 1, without solving any single equation of a degree higher than t. This formula, applied X times successively, will have the effect of abolishing X and causing [X, m> "i •■• "■] 'o depend on [/i', v', ... ir'], where ;*', >•', ... t' are connected with X, /i, v, ... x by means of the formulee given at the commence- ment of the present paper, but where instead of the letters X, ju, c, ... I have used the letters
V, 1, r
It is presumable that the reduced Hamiltonian numbers would be found much less amenable to algebraical treatment than the Hamiltonian numbers proper ; for numerical equalities and inequalities have to be taken account of, in determining them, which have no place in the deter- mination of the latter numbers. Hamilton, as already stated, expressly alludes to the reduction of U to 10, bnt with that exception has avoided the general question of finding the absolutely lowest number of letters required in order that a system of equations (expressed in terms of those letters) of given degrees may admit of being satisfied without the necessity arising to solve any equation of a higher degree than the highest of the given ones.
558
On Hamilton' 8 Numbers
[51
and ^itn to signify the coefl&cient of If in (1 + <)"•; then, for any value of i greater than unity,
^,Ei - A Ei^, + ^,Ei^ -/3,Ei., + ... + i-y ^iE, = 0.
Or in other words, writing ^„Ei=\, i8,^,_i=^i_,, and replacing i— 1 by i, Ei=l + ^,Ei_, - /3,^i_ + ... + (-)'+'/9.+i a; for all values of i greater than zero.
This is eminently a practical formula, as all the numerical calculations made use of to obtain any E are available for finding the E which follows. Dispensing with the symbol /9, we may deduce all the values of E succes- sively from those that go before by means of the equivalence
S = (l- «)*• + 1 (1 -0^' + t'{l- 1)^' + ... = 1 - 2t,
which, by equating the powers of ( on the two sides of the equivalence, gives
^0 = 3,
E, = l +
|
4.3 1.2 6.5 |
3 ~1 4. |
.2. .2 3. |
2 3, |
,2.1.0 |
|
1.2 |
1. |
2. |
3 ' I, |
,2.3.4 |
= 12,
and so on.
I use the terra equivalence and its symbol in order to convey the neces- sary caution that the relation indicated is not one of quantitative equality ; for, although the series on the left-hand side of the symbol converges for all positive values of t less than 2, it is never equal to the expression on the right-hand side except when t = 0. Thus, for example, when t is unity the two terms of the equivalence are 0 and — 1, and when t = ^ they are
2-^0 + 2--E1-1 + 2-E.-2 + ... and 0, respectively ; and for all values of t within the limits of convergence the value of the left- hand side is in excess of the value of the right-hand side of the equivalence by a finite quantity which decreases continuously as t decreases from 2 to 0, and which vanishes when t= 0*.
In a word, the generating equation is not an equation in the usual sense of the term. Conceiving each term of the series S to be expanded in ascending powers of t, and like powers of t to be placed in columns under and above
* Of the truth of the statement that the excess never changes sign, and continually decreases, I have scarcely a doubt, bat it requires proof. Mr Hammond remarks that
where F^ (() is positive for all positive values of t. Probably a proof of the point in question might be deduced from this expression, but I have not thought it necessary to investigate the matter.
51] On Hamilton's Ntimbers 559
each other, the double sum may be taken as a vertical sum of line-sums or as a horizontal sum of column-sums, and, although for licit values of t each sum has a finite value, the two iinite values are not identical, just as a double definite integral may undergo a change of value when the order of its inte- grations is reversed *.
I have noticed [see above, p. 542] that the value of any Hamiltonian Difference divided by the square of the preceding one was always greater than ^, and stated as morally certain, but "awaiting exact proof," that this ratio ultimately becomes ^. By aid of Mr Hammond's formula for the numbers, I shall now be able to supply this proof, and at the same time to show that the ratio of a Hamiltonian Number to the square of its antecedent (which, of course, converges to the same asymptotic value |) is always less than that limitf.
We must in the first place prove that in the series
A^i-i - ^,Ei^ + ^,Ei_, - A^.-4 + • ■ . the absolute value of each term is greater than that of the one which follows it.
In proving this, I shall avail myself of the property of the Hypothenusal Numbers disclosed in the process of forming the triangle given at the outset of the memoir, namely, that Ei — £",_, is greater than (^,-_i — Ei^^^j2.
Let us suppose that the law to be established holds good for a certain value of i. For the sake of brevity, I denote Ei, Ei^i, Ei^^, Ei_„ ... by N,P,Q,R
We have then
p 1 Q(Q-l) R(R-l)iR-2) S{S-l)(S-2){S-S)
2 2.3 2.3.4
^ ^^P(P-l) Q(Q-l)iQ-i) ^R(R-1){R-2)(R-^)
2.3 2.3.4
S(S-l)(S-2){S-S){8-4>) 2 . 3 . 4 . .5
+
• Professor Cayley has brought under my notice a not altogether dissimilar, but perhaps less striking, phenomenon, pointed out by Cauchy, that, although the series
is convergent, its square
«o' + {2«o«i) + (2"o«2 + V) + •■•. thatis, i-^2 + (^ + i)--...
is divergent.
f The fortunate circumstance of the two ratios in question being always respectively less and greater than the common asymptotic value of each of them enables us to find the value of the constant in the expression c^, which is asymptotically equivalent to the half of the xth Hamil- tonian or Hypothenusal Number by a method exactly analogous to that of exhaustions for find- ing the Archimedian constant correct to any required number of decimal places. See end of this section [p. 566, below].
560 On Hamilton's Number 8 [51
If, then, the law to be proved is true for all the consecutive terms of the upper series it will obviously' be true for the second series, abstraction being made of its first term, provided that no antecedent is less than its consequent in the series
Q-2 R-Z 5-4 3 ' 4 ' 5 ' •■■'
which is true a fortiori if
Q R S 3' 4' 5""
continually decrease, as is obviously the case, inasmuch as
Q, R. S,...
form a descending series.
In order, then, to establish the necessary chain of induction, it only remains to show that
3P(P-l)-Q(Q-l)(Q-2) is positive.
Now (P-Q)- ^^-^' , and d foHiori P - ^^~^^' .
is positive for a reason previously given.
And, if in the series 3, 4, 6, 12, 48, 924, ... we make exclusion of the first three terms, we have always
R= or <^, 4
and consequently P > ^ *.
And, since under the same condition (P — 1)/{Q — 1) > 4, 3P (P - 1) - QHQ - 1), and A foHioriSP(P-l)-Q(Q-l)(Q- 2), is positive if 12P — Q* is positive, which is the case, since P > 9Q»/32,
Hence, since the theorem to be proved is true for the several series
4_J 3.2.1 ^' 1.2
(2) ^ ^^ 1.2
._. 12.11 6.5.4 4.3.2
(4)
|
1 . |
.2. |
3* |
|
4. |
3. |
2 |
|
1. |
2. |
3' |
|
6 |
.5 |
.4 |
|
1 |
.2 |
.3 |
|
12. |
11 |
.10 |
1.2 1.2.3 1.2.3.4'
48.47 _ 12.11.10 6.5.4.3 1.2 1.2.3 '*"l.2.3.4'
* The proof that the ratio of each term of the series 4, 6, 12, 48, 924, ... to its antecedent continual!; increases is too easy and too tedious to be worth setting forth in the text.
51]
On Hamilton's Numbers
561
I
it will be true universally ; for in all the succeeding series the term we have called R will be higher than the term 6 in the scale 3, 4, 6, 12, 48
Hence P - 1 = or < i (Q" - Q).
For the initial values of Q, P, (namely, 3, 4)
(When P represents any term beyond the first it is very easy to prove, but too tedious to set out the proof, that the sum of all the terms after the first in the series equated to P — 1 will be less than — 2 ; so that, except in the case stated, P <h(jQ^- Q).)
For the series 12, 48, 924, ... we have seen that P > 9Q-/32.
Hence, for the series 48, 924,
■32Q
/
But
Hence
Q'-Q R(R-l)(R-2) 2 6
2 6 ■
P>^^-'^Qi,.ndP<'^-^
2 SI" "^ ' 2
Hence, when P, Q, are at an infinite distance from the origin,
P
Hence, also,
Q . P
ultimately = tv = i.
which proves the theorem left over for "exact proof" in the memoir referred to.
It is convenient to deal with the halves of the sharpened* Numbers of Hamilton, which may be called the reduced Hamiltonian Numbers, and denoted by h with a subscript, or, when required, by p, q, r, ... (the halves of P, Q, R, ... respectively).
We have then
- 45» - 2q
or
p<q'-l:
p>q'-l-Wq^
* NamberB increased by unity may conveniently be denominated sharpened numbers, and nnmbers diminished by unity flattened numbers.
S IV. 36
562
On Hamilton's Numbers
[51
We may find a closer superior limit to p in terms of 9 as follows —
P 1 nr.-<^-^ R(R-1){R-2) S(S-l){S-2)(S-fi) ^ ^ 2 6 "^ 24
in which inequality it may be shown by inspection up to a certain point, and after that by demonstration, the tedium of writing out or reading which I spare my readers and myself, that P may be substituted for its flattened value P —1.
We have then
2 6 ■^24'
Let us suppose that 8, R, are not lower in the scale of the ^s than 12, 48, respectively; so that P is not lower than Eg, which is 409620.
Then, as we have previously shown,
Q»<^P, R'<^Q, S-'KS^R.
Moreover, we have
P < i (Q" - Q), whence it follows that Q'>2P + Q,
|
and, d fortiori, |
Q'>2P. |
|
Similarly |
R^>2Q, |
|
and |
^>2R. |
|
Now |
p^Q'-Q R'R^^s* 2 6'^2"^24 |
that is,
This result, expressed in terms of the reduced numbers p, q, takes theJ form
and we have previously shown that
2
■ 9'
p>q'-Wr
at all events when P is not lower in the scale than E..
The fraction J^ arises from our having substituted for R* the inferiorJ value (^ Q)* ; but, the higher we advance P in the scale, the nearer i?j approaches to 2Q, and is ultimately in a ratio of equality with it. But, if ' had written (2Q)' for i?', the coefficient, which now stands at — ^, would
51] On Hamilton's Numbers 563
have been — |. In like manner, as P and Q are travelled on in the scale, If and 5* become indefinitely near to 2Q and {2Rf, that is, 8Q, so that the coefficient of Q in the superior limit approximates indefinitely near to
- i + 1 + J, that is, f , and the two limits of p which have been obtained become
Inhere ultimately e and tj are infinitesimals*. Hence it follows that the ultimate value of
{p-q')^q^ is _|,
that is, — ^ — ~ = — */§ when i = oo .
Let \, fi, V, ... represent the halves of the Hypothenusal Numbers in the triangle given at the commencement of the paper, that is, the diflferences of the numbers which we have called p, q, r, .. ,.
Since P = 'f~ §9* ^'i*^ q = r' — |r*
p — qz^q' — ^qi—q, and q — r = r' — ^r^—r.
Obviously, therefore, as a first approximation when \, fi, are very ad- vanced terms in the hypothenuse,
Let us write \ = fi' + Kfi'
for a second approximation.
Then ^-§?* - 9 = (^ - §r*-r)»-l- « (r' - f r*- r)«,
or, neglecting terms of lower dimensions than r',
(r.-|r*)'-|,-(l-^ + ^-...)=(r«-Jr»-r)» +
KT"
Therefore - §r» = - 2/^ + kt^.
Consequently * = ^ ^°<i * = J-
Thus, then, for the consecutive Hypothenusal Numbers \, /j,,
X = fi^ + ^fJ + .... Let \ = ^' + J^* + ^/x,
or say Vx+i = '/x' + ^Vx^ + Pxnx, where t/^ is the artih term in the series ^, 1, 3, 18
* As a matter of fact, it will be foand that, as soon as q and p attain the values 6, 24, J* - 1 5* may be taken as a superior limit. It may be noticed also, to prevent a wrong inference lieing drawn from the above expressions, that, as will hereafter appear, tj is an infinitesimal of the Older Ijqi, when q is infinite.
36—2
564
On Hamilton's Numbers
[51
The successive values of px and their diflFerences are given in the annexed Table.
|
s |
1. |
P« |
^. |
|
1 |
•5 |
•55719096 |
|
|
S |
1 |
•66666666 |
+ •10947670 |
|
3 |
3 |
•69059893 |
+ 02393227 |
|
4 |
18 |
•67647909 |
-•01411984 |
|
f, |
438 |
•64334761 |
- -03313148 |
|
6 |
204348 |
•61769722 |
- •02565039 |
|
7 |
41881398318 |
•61139243 |
-•00630479 |
|
8 |
1754062963103547795958 |
•61111171 |
- -00028072 |
The decimal figures following those given in p^, required for ulterior purposes, being 5795.
An examination of the column of differences for x= 5, 6, 7, 8, shows that the ratios of each to the rest go on decreasing somewhat faster than their squares: this makes it almost certain that pa — p» will be between the 400th and 500th part of 000280, and that accordingly the value of p, will be •6111111, &c. I believe it is beyond all moral doubt that the ultimate value of p is exactly fj; and, indeed, it was the conviction I entertained of this being its true value, when I had calculated pr, that led me to undertake the very considerable labour of ascertaining the 10th Hamiltonian Number in order to deduce from it the value of pg. This being taken for granted*, we may proceed to ascertain a further term in the asymptotic value of rj^+i expressed as a function of r)^.
For, calling Px — H^^x and ^rjx = Qx,
we have 8,= -00658611,
8,= 00028132r Bg = -0000006047,
9. = 21, I
q, = 452, \ neglecting decimals.
g, = 204649,)
{Bq\=-m2, (Bq\= -12375.
(Bq\-{Bq), being -0111, {Bq-h-iSq\ „ 0035,
Thus
The value of and of
* It is reduced to certainty in the supplemental 3rd section.
51]
On Hamilton's Numbers
565
I
we may feel tolerably certain, from the Law of Squares, that {hq\ - {8q\ will be somewhere in the neighbourhood of the tenth part of -0035, and accordingly that {Sq\ is about -1234, so that the probable value of (Sg)^ is •1234 ....
Thus we have found
Vx+l = Vx^ + iVx^ + HVx + [ ]Vx^ + --;
tte only moral doubt being as to the degree of closeness of propinquity of the coefficient of t;^;^ to the decimal '1234 ...*.
For the benefit of those who may wish to carry on the work, I give the following numerical results which have been employed in the preceding arithmetical determinations : —
= 615347368719452970289576400111588468.5871706
1.2
E,{Ej-l)(E,-2) ^ 97950944448414216137607200637520 ^.(^,-l)(£'.-2)(^,-3)
1.2.3.4
= 1173024302352295838445
^.(^.-l)(^-2)(^-3)(^.-4) ^ 5552272910184 1.2.3.4.0
E,(E,-l)(E,-2)(E,-3){E,-^HE,-o) ^ 12271512 1.2.3.4,5.6
E, {E,-l) (E, - 2) (E, - 3) (E, - i)(E,-5)(E,-6)
= 792
1.2.3.4.5.6.7 17, -77, = 24-33333333 ... ^, H- 975 = 466-54794520 ... 177 ^17, = 204951-34925714 ... 178 -=-»7, = 41881671184-54776412 ... 17, -r 17, = 17540629531 59389842293-346657805 . . . V»74 = 4-24264068 ... ^17,= 20-92844819 ... ^17, = 452-04866994 ... ^^77 = 204649-45227877 ... ^179 = 41881534751051659567667 ... Finally, it is interesting to find the asymptotic values of hx and 77, (the halves of the sharpened Hamiltonian and of the Hypothenusal Numbers), which are ultimately in a ratio of equality to each other, in terms of x.
* The exact value of the coefficient of ij^ji, left blank in the text, is proved in section 3 to be J{, that is, the recurring decimal -123456790.
566 On Hamilton's Numbers [51
Obviously each of these is ultimately in a ratio of equality with M^, where Jlf is a constant to be determined.
Let M = l(y^ and «, = 10''+'.
Then, for finite values of x, remembering that (in the preceding notation)
p < j' and \ > /*',
Uj, must be intermediate between the corresponding terms of the two series
17 =i 1, 3, 18, 438, 204348, 41881398318, ....
A = 2, 3, 6, 24, 462, 204810, 41881603128, ....
By means of this formula, writing for Ux corresponding values of ij and h, and retaining so much of the two corresponding determinations of a as is common to both, we can find o precisely to any desired number of places of decimals, as shown in the following Table, in which 18 and 24 are taken as the terms of place zero in the respective series :
Ux= 18, 438, 204348, 41881398318,
a = -32, -401, -4088, -4089863...
M,= 24, 463, 204810, 41881603128,
a = -46, -41.3, -4090, -4089866....
Hence, if we now change the origin, taking ^ and 2 as the zero terms, we have approximately
and 81ogilf=2-«'««,
which gives M= 14654433 ...*.
As a verification, since 2' = 8, (1-46544)* should lie between 18 and 24; and, as a matter of fact, a rough calculation gives
(1-46544)== 2-1473 .... (2-1473)== 4-608 ..., (4-608)» = 21-234 ..., which is about midway between the two limits. — J. J. S.
§ 2. Proof of the Formula for the Successive Determination of each in turn of Hamilton's Numbers from its Antecedents.
Let 1+ x+ x^+ x^+ x*+ x'+ af+... = F„(x),
2a; + 3«=+ ix'+ ox'+ 6a^+ 1a*+ ...=F^{x\
%(i?+lha?+ 29a^+ 49«»+ lQa*+ ...=F^{x),
Z&a? + 210a:' + 804a^ + 2449a:* + . . . = J^, {x),
• See Note 1, p. 578 [below].
I
51] On Hamilton's Numhers 567
where the coefficients of the various powers of x are the numbers set out in the triangular Table at the commencement of this paper.
If, in general, we write
Fn (a;) = OnX'' + 6„a;"+' + CnX^+^ + d„a;"+^ + . . ., the coefficients of ^„+, («), expressed in terms of those of F„{x), are as follows : —
_, o„(a„+l)
A -„ ^.. h ^ «n ((In + 1) (2a« + 1) "n+i — Cn + "nOn + ""i — o o
n -^ j.r, . _l«»(«»+1)a , an (ttn +!)(«» + 2) (3a„ + l)
Cn+i — ^n + "nCn + " , g 0„ + 1 2 S 4
Now (l-.)-n=l+a„. + ^^i^^).^ + «-^^^;§^
when multiplied by
Fn {x) = a„a;» + 6„a;"+> + c„a;"+^ + d„a;''+» + . . ., gives (1 - x)-<^ Fn(x)-a„ai"+ &„«"+' + CnX"+' + d^af^' + ...
+ a,i'x''+' + a„ b„x"+'' + a^c„x"+' + ... ^ a.''(an + l)^..^«n (an + 1) ^^^„,3 ^ ...
, an'(an+l)(an + 2)^.^, , ■*■ 17273 "*"•••
+ .... Comparing this with
^„+, (j;) = 6„a:»+' + Cnx'^^+ d„a;»+» + . . .
+ °"^°"^^^ 0?'+' 4- a„6„a^+' + a„c„a;»-t' + . . .
an(an+l)(2a„+l) «« («« + 1) , ,+,
+ 1.2.3 "^ ^ 1.2 ^"^ +•••
a„ (a„ + 1) (ffl„ + 2) (3a„ + 1) + IT2T374 "^ ■^•••
+ ..., we see that the difiference of the two expressions is
^ 1.2 1.2.3
.1 —■ ':
+ 1.2.3.4 '"^ ■^•■••
which is equal to «""' (1 — a;)"*"^-" — a;"~' (1 — a;).
668 On Hamilton's Numbers [51
Thus /"„+, {x) = (1 - x)-^, Fn (.x) - a;"-' (1 - ar)-«n+' + a?""' (1 - a;) *.
Multiplying this equation by (1 — j;)'ii+i, where
«»+, = tto + a, + o, + . .. + On-i + c, we obtain
(1 - xy^i Fn+i (x) = (1 - «)•« Jf„ («) + x'^' (1 - «)'-+i+'- af^' (1 -«)•»+•, which gives, when we write successively n— 1, n — 2, n— 3, ... 0 in the place of n,
(!-«)•« F, («) = (1 - a;)*.-! JVi («) + a^-' (1 - «;)»»+■ - «»-* (1 - a;)»»-i+' ; (1 - a;)',^, F^t (a;) = (1 - a;)»«-2 ^,h-j1«) + a^"' (1 - a;)'»-i+' - a;"-' (1 - a!)'»-8+' ;
(1 - xY' Fr(x)=(l- xY' F„ (x) + a;-' (1 - aj/.+i - a;-' (1 - xf'+K
Hence, by addition of these n equations, we find (l-xy, F„ (x) = (1 - xy<, F, {x) + a;"-^ (1 - «)««+• - a;-> (1 - a;)«o+'
+ a;"-' (1 - a;)'»-i+= + «»-♦ (1 - a;)«»-2+» + . . . + a;-' ( 1 - a;)»i-^, where it has been assumed that it is possible to assign to «„ (previously un- defined) such a value as will make the last of the above n equations, namely,
(1 - a;)'- F^{x) = (l- xY' F, {x) + X-' (1 - «)»■+• - a;"' (1 - xY'+\
identically true. That this can be done is obvious ; for, if in that equation we write for J'i(a;), Fn{x), and Sj their values, namely,
Fi {x) = (1 - a;)-= - 1, F, {x) = (1 - a;)-', and Si = «„ = 1. then, on making s^ = 0, the equation becomes
(1 - a;)-» - (1 - a;) = (1 - a;)-» + a;-' (1 - a;) (1 - a; - 1). Thus the general value of Fn {x) is given by the equation (1 - a;)',. Fn {x) = (1 - a;)-> + a;"-» (1 - a;)»»+i - a;-' (1 - a;)
+ a;»-» (1 - a;)«»-i+2 + a;"-* (1 - «)»„-a+'' + . . . + a;-' (1 - a;/!*', which is equivalent to (1 - a;)»n Jf„ (a;) - ( 1 - a;)-' + a:"' ( 1 - a;) - a;"-' (1 - a;)».+i
= a;"-' (1 - a;)«»+> + a;"-« (1 - a;)«„-i+^ + a;""^ (1 - a:)»»-2+= + . . . + a;"' (1 - a;)'i+',
where, ao, Oj, Oj, a,, ... being the Hypothenusal Numbers 1, 2, 6, 36, ... we have
fii = fflo = 1.
«2 = ao + «! = 3,
Ss = Cto + «1 + ^2 = 9,
that is, the successive values of s„ + 2 are the Hamiltonian Numbers 3, .5, 11,47 ....
• See Note 2, p. 578 [below].
I
V
51] On HamiUon's Numbers 569
Now F^ (x) = a^ai^ + . . ., so that the coeflBcient of x" in (1 - xfn Fn (x) is the same as the coefficient of «" in F„ (x), namely, a„. Consequently, equat- ing coeflScients of a;" on each side of the equation just obtained, we find
a»-l+(«» + l)- ^"2 17273
+ .
, / wi(^. + 2)(^i + l)...(gi + 2-n) ^ ' 1.2.3...(n + l)
Remembering that a„ + s„ = s„+i,
if we call the Hamiltonian Number Sn + 2, Hn, the above relation may be written thus :
n.^, - 1 ,„^ ___
^„_, (^„_, - 1) (^„_, - 2) (/f„_, - 3)
"+' ~ 1.2 1.2.3
+ (-)'
,n+l
1.2.3.4 H,{H,-\){H,-2) ..AH,-n)
1.2.3...(n+l)
To obtain Professor Sylvester's modification of this formula given in the preceding portion of this memoir, we multiply the equation from which it was obtained by 1 — a; before proceeding to equate coefiicients. Thus we have to equate coefiScients of «" on both sides of
(1 - xfn-^^ F„ (x)-l+x-m-xy- x"-' (1 - xyn+'
= «»-«(l -ar)'«+» + a;»-'(l -ar/n-i+'-f- a;"-*(l -«)»«-»+»+.., + «-' (1 -a;)'i+'. Or, writing Sn + 3 = En,
we equate coefficients on both sides of (1 - x)^n-^ Fn (a:) - 1 + a;-' (1 - xf - x^-' (I - ic)^»-'
= x"-^(l-x)^n + a:»-> (1 - a;)«»-i + x«-* (1 - x)K-2 + ... +x-'(l-x)^i. This equation is easily transformed into {l-x)^o + x(l -X)^l + X'(l-X)^i + ...+x''(l -X)^n
= 1 - 2a; + a;» (1 - a;)»»-« F„ (x) - a;"+' (1 - x)^n-', from which, as Professor Sylvester has pointed out in this memoir, by equating coefficients of all powers of x from 0 to n, we can obtain the succes- sive values of En-
The general formula
E„.,(E„^-1) ^^E,(K-l)...(E,-n + l)_^
i-E^,+ r:2 ••• + (-) 17277^1 "
arises from equating the coefficients of x". — J. H.*
• See Note 3, p. 578 [below].
670 On Hamilton's Numbers [51
§ 3. Sequel to the Asymptotic Theory contained in § 1.
The relation p = q''- ^q^, etc.
previously obtained supplies only the two first terms of the remarkable asymptotic development
S^^ZP = ^{qi + qi+qi + ... + qW) + aq,
where i is any assigned integer and 3 is of a lower order of magnitude than the lowest power of q in the series which precedes it. This may be easily established as follows : —
By the scale of relation proved in the preceding section we have p = f-^r' + ^ + ... = (f — fr" + terms whose maximum order is that of r*. Let. now, P = f-lq^- IH -^kq" -^Iqy ...;
therefore . g = r'-§r» -§Ar«-§Ar^-f irr ...
and p = q* — ^r' (1 - r~^ - hv-^ - kr^-' - Irr-' . ..) + ...
-^hr^-^kr^-^lr^ ... = j2 _ ^r" + |(ri + hr^+'- + kr^+^ + Zr''+' + ...) + ...
-^hr^-^h^-^lr^-
Therefore h — 1, k=l, 1 = 1, m=l, ...
2«=§, 2/3 = l+a. 27=1 +y3, 2S=l+7, ... that is. a = f. /3 = |. 7 = -^. 8 = lf. -
and thus p = q' - ^q{q^ + q^ + q^ + q^^ + ... + 5**'') + Eq,
as was to be shown*.
* This theorem may be rigorously demonstrated, and reduced to a more precise analytical form, as follows : —
For the sake of brevity, we may call -pjq + q the relative deficiency of p, and denote it by A. First it may be noticed that, if in the equation
we wntelog q=k,
(fc' jf Jt7 \
*'''l. 2. 3. T"*"!. 2. 3. 4. 5. 31 "'"1.2. 3. 4. 0.6. 7. 127 "'■•■7' which is always convergent.
Moreover, the value o{ F{q) may be calculated for any given value of q within close limits. For, if we call U the right-hand branch of the series in q, beginning with z - z~', the terms of V will easily be seen to lie between those of two geometrical series of which z - z~^ is the first term, and of one of which J, and of the other {z'-i-z~ =)'', is the common ratio. Hence U is intermediate between 2 {z^ - 1)/* and («' - 1) (z + l)/i {z-z^ + 1).
51] On Hamilton's Numbers 571
It is interesting to notice that the formula apparently remains arith- metically true for finite values of p and q, provided that q is not less than
The difierence between these limits, it may be parenthetically observed, is
-which, when z is nearly unity (the limit to which q(i)' converges), is nearly equal lo -^(z- z~'^Y ; that is, if 2 = 1 + T, the difference between the limits (for t small) is very near to t^/2.
Now on p. 575 (post) it is shown that sfq-r + s-%^r=e, and that, when the rank of q is taken indefinitely great, e converges to J. Hence e always lies between finite limits.
For, in general, x being any one of a series of increasing numbers, and i/- (x) a function of x which is always finite for finite values of x, but ultimately converges to c, by taking for x a value of L sufficiently great, we make the series of terms for x>L intermediate between c + d and c-S, where 5 is any assigned positive quantity ; and consequently, if /i, y, are the greatest and least values of ^ {x) when x does not exceed L, the greater of the two values, c + 6, /i, and the lesser of the two, c - S, y, will be superior and inferior limits to the value of -^ (x) for all values of x.
Henoe, writing v'(p) -? + ''- W(9) = fi.
^(q)-r + »-lJ(r) = (2, V(r)-»+«-|v/(»)=c3,
>/(6)-3 + 2-W(3) = 6,_i, we obtain, by summation,
>/(jp)-« + l{v'(9)+N/('-) + V(«) + -+v/{6)} = 2f-2 + W(3), and, consequently, VCp)-9 + i {%/(?) + n/W + v'(*) + ■•• + V(6)1 =P^.
where p is always a finite quantity lying between determinable limits. But again (p. 573)-
p={q-ej(q)]\
where $ (whose ultimate value is \) is always a proper fraction. Hence
Hence, from what has been shown above,
^=1 W(9) + V('') + VW + - + n/(6)}-V^-
In this equation we may write
v/(«)=9* +*:
^ f (x — 3 equations),
where ij, k^, ^3, ... are all of them finite (and, as a matter of fact, of no consequence for our immediate object, positive proper fractions). For, ultimately,
*i=V('-)-9*=4^ =-r^=4*'=* (««* P- 574).
r' + q* r4 + 9* and consequently the finiteness of each k is a direct inference from the general principle previously applied in the case of the e's.
Applying this result to the equation previously given, it follows that qi^qt + ,,, + q\h) =|4-i;x (where xi is finite)
=F(?) + (g-4 + g-i + ...+g-(i)'"")-{(2-z-i) + (j4-2-i) + (ii-2-i) + ...}, where z lies between 1 and 2.
The series of negative powers of q is obviously less than x, and the z-series, which follows it, is less than the finite quantity 2 (z-ljz), that is, <2(2-i). Henoe 3A=i*'(4) + ex, where 9 is
572 On Hamilton's Numbers [51
24, when we replace each term in the formula by its integer portion, and in the series on the right stop at the term immediately preceding the first term for which
Thus, when p = 462 and q = 24,
T^((f-P\ E./'o76-462\ „/114\ . we have E (?--^) = E [—^^) = ^ ( 24 ) = *'
and E [I (Eqi + Eqi)} = ^ {§ (4 + 2)} = 4.
So also, when p = 41881603128, q = 204810,
and E {§ (Eqi + Eqi + Eqi + ^5^)} = i5; {|(4o2 + 21 + 4 + 2)} = ^ (^f a) = 319. But, if we had included the term Eq^''^, the result would have been E [I (452 + 21 + 4 + 2 + l)j = 320.
a namber lying between fixed limits, and x, the rank of q, is of the same order of magnitude as log log q. This equation contains as a consequence the asymptotic theorem to be proved ; for, using t to denote any positive integer,
iA - 2.*9<4)'=f (?) - 2/ 3<4>* - ex=«(4)*'^' + 'T (5(J)' - r(i)') - 2.* l/s'^^''^' - 6^. Hence, remembering that x is of the same order of magnitude as log log q, and that
»=»
» = «+2 which is of a lower order of magnitude than g'^)'"*"^, it follows that fA-Sj'g"'' for all values of
i is ultimately in a ratio of equality with ?(4)'''"', which is the theorem to be proved.
We have thought it desirable to obtain the formula f A = F'/ +ex for its own sake, but, so far as regards the proof in question, that might be obtained more expeditiously from the expression given for 3 A/2 - vx without introducing the series Fq.
It is easy to ascertain the ultimate value to which 6 converges. In the first place, the series
of fractions \lq^ + \jq* + \jq'' + ... tox-2 terms (where x is the rank of q) maybe shown to be always finite, and consequently, when divided by x, converges to zero.
For we know that (|) - 5) > (5 - r)^> (r - «)* ... > (6 - Z)'^"^. Hence the last term of the series 94,ji, ji... (namely, q(^^^~^)>i. Hence the finite series l/g4 + l/gi + l/j^ + ... for a double a fortiori reason is less than the infinite geometrical series 1 + ^ + 5^ + . •< 4-
In fact, from § 1 (p. 566) it may easily be shown that the last term of the series
g4, gi, g* ... >Jtf*>(l-465)*>4-608,
BO that the sum is really less than _ „-. .
o*dUo
Hence, retracing the steps by which 0 has been obtained, and observing that p' differs from
p by a finite multiple of l/x, we have ultimately 9 = u = J; - 3p' = ft - 3p = fc - 3e = J - f = - j'j. If,
then (using u, to denote the half of the sharpened xth Hamiltonian number), we write
«j - 1/U;j=»j(, and understand by G (t - 1/t) the infinite series
(ti - ri) + («i - ri) + (t* - r *) + ...,
it is easily seen that the principal part of v/(»i+i), regarded as a function of », and x, is
b
^1] On Hamilton's Numbers 573
Again, when
p = 3076736843548289379224261404637538760216.584, g = 1754062953145429399086,
^(^^) = 27921159919,
and E{l{Eqi + Eqi + Eqi + Eq^^ + Eq^"^ + Eq^ )}
= E{I (41881534751 + 204649 + 452 + 21 + 4 + 2)} = 27921159919*.
We will now proceed to consider afresh the asymptotic development of any Hypothenusal Number p-qm terms of its antecedent q-r, and to reduce to apodictic certainty results which in the first section were partly obtained by observation. It has already been shown in that section that
p>q^-l^qi-l
jwhen p is not lower than 204810 in the scale 2, 3, 6, 24, 462, 204810, ..., |that is, when q is not less than 462.
Hence J" > 9' - 2^ + ? + (f^g* - f ?),
|or, since ffj^-fg is a positive quantity,
p>{q-^qf, ■t all events when ^ = or > 462,
It will be found also on trial that this formula remains true for all the palues of q inferior to 462.
Thus 462>(24- V24)»,
24>(6-V6)^ 6 > (3 - v/3)', 3 > (2 - V2)l Hence, universally, P>i9— V?)'+.
But we know that p < q".
We may therefore write p = (q — d ^qf^ where 0 is some quantity between 0 and 1. Similarly, q = {r-d, s/rf.
where 0,, ^,, ... are also positive fractions.
* The authors must be understood merely to affirm the potsibility of the theorem being true •od to offer no opinion on the strength of the presumption raised that it is so.
t Had this inequality been true only for values of q sufficiently great, it would have been enough for the purposes of the text.
574 On Hamilton's Numbers [61
When p and q become infinite,
T
Hence the ultimate value of 0 is ^. Similarly, ^,, 6^, ... all of them converge to the value ^.
This agrees with the result previously demonstrated (p. 563), and is the starting point of all that follows.
We know that letters p, q, r, s, .... being used to denote the halves of the augmented Hamiltonian Numbers, they are connected by the scale of relation
2.3.4 and T stands for the remaining terms, involving
t, u, V, .... Considering q, r, s, t, ...
to be of the order 1, ^, J, ^, ..,,
we may reject the term ^, which is of zero order, and write
P = q'-^r^; -l + r'-l + S-T.
Hence, rejecting terms of order less than | (which have, however, to be retained in obtaining the subsequent approximations),
(p-9)L| q'-^^; -^q + i^-l+s-T
-{q-ryj [-q^+2qr; -r'
= (25r-§r»);
that is, (i'"?) — (? — »"?= I (?*
when q is infinite.
Again, writing for (S its expanded value, namely.
s* , . .. . s we have
1 <>3 J. II.
3 ^ + i-^*-4'
(p-q) \ ( 2qr-lr^-iq^ Order ,,
3 •
- (9-'-y l = J + 2?*r-f3+is* „ 1;
-Hq-r)^} [-^-^r'-^^-s'+^s'-ls-T „ < 1,
rejecting the terms q'^r', q~^r*, ... in the expansion of (q - r)^ because the order of none of them is superior to zero.
51] On Hamilton's Numbers 575
We now write 5 = (' — ^i V^)^
so that 2.qr - f r^ - f 9^ = (2r» - 4^,r^ + 2^i''?-=') - |r» _ (|r> - 40, r^ + 40i»r' - f ^jV*) = - 2(9i»r= + |6'/rl Hence
{p-q)\ /■- 20iV» + 25'ir - I? + ^s* Order 1; - {q-r)A U^efri-s' „ f;
Since q=r^ = s* (ultimately),
the terms of Order 1 (which are the only ones with which we have to do at present) are ultimately equal to
(-20,= +2-f + i)5;
or, giving 0, its ultimate value ^, to ^q, or to the same order of approxima- tion to II (g- — r).
Hence, ultimately,
(p-q) = (q-ry + i{q-r)^+mq-r)\
We use this result to obtain a closer approximation to V? than r—O^ kJt, and to find the relation between the general values of 0, and 6^.
Thus, assuming V(? — r) = r—8-\-\ >J{r — s) + k,
we have, ultimately,
^ - r = (r - s)' + 1 (r - «)* + (I + 2k) (r - s)
= (r - s)» + f (r - «)* + ||(r - s).
Consequently, as r becomes indefinitely great, k converges to the value
Now ^{q - r) = >J(q) -^-r ■■■ = \/{q) - J ultimately ;
yq
and similarly V(^ — «) = V(>') — i ultimately.
Hence, ultimately,
V(9) = r--s-|-JVW-|-A+i-| = r-s + |V(r) + i.
We may therefore write
>J{q) = r — » + § \/{r) + e (where ultimately e = \).
But V(9) = r-di >Jr,
and therefore 5, </(r) = s - 1 V(^) - «•
* Ag previously obtained by observation in § 1 (p. 563). It will, of course, be understood that in the above and similar passages the sign = is to be interpreted to mean "is in a ratio of equality with."
676
On Hamilton's Numbers
>J{r) = 8-63 -Js,
[51
Moreover whence it follows that
0j V(r) = ^+^, V(») - e (where e = J ultimately).
Resuming the development of (p - q) in terms of (q — r), we have
8*
(p-q) - (q-ry
-H(q-r)
-2^,r' + 2g4r--yig+ o Order 1,
i-i9-M + ^r+|j5'-|-r „ <i
The terms of order inferior to | are of no value for present purposes, and are only retained for the benefit of those who may wish to carry on the work.
To reduce the terms of Order 1, we write, in succession,
q = {r-e,^/ry,
^.\/(r) = is + f^.V(s)-f, Thus 5-2^iV' + 2g4r-J^^
O
= J - 26',V + 2r» - J^r^ ; - 28,ri + ^e,r^ ; -^Oi'r o
= J - J - 2r Q + f ^, Vs)' ; + 4er g + 1^, Vs) + ^'O^r^ ; - 2£'r - J#^,«r
+ ^^js^ (ir' + 4,0,8^ + ^e^^s) - f d,=s if + 4^25* + 4^j»5) ; 4 f ers + ^ ^,r* ; + f e6'3r- V(s) - 2eV - -^^,V = Jer« + ^^,r* Order f,
+ ^ ^/s' + f e<?,r V(s) - e^'s" - 2e'r - J^ d.'r „ < J.
Hence ip-q)\ ( i 0,'r^ - s' + ^ ers + 3^ e^ri Order J,
- (q-ry -i(q- »•)'
--B(9-'-)
51] On Hamilton's Numbers 577
Here the terms of Order \ are ultimately equal to
which, when ^i and e receive their ultimate values, \ and \, becomes
From this it follows immediately that (rejecting terms of an order of magnitude inferior to that of (g' — r)*)
The law of the indices in the complete development is easily deduced from the relation
\ ^ g(27-l) r(2r-l)(2r-2) s (2g- 1) (2s - 2)(2g-3)
^~^^ 2 2.3 ■*■ 2.3.4
The terms carrying the arguments
q', q, r^, r\ r, s*, sr>, ^, s, 1^, ...
furnish the indices 2, 1, |, 1, ^, 1, |, ^, {, |, ...,
which, arranged in order of magnitude, become
2, f, 1, f, f, J, J, -fg, i, —
Thus, calling p— q and q—ry and x respectively, the expansion for y in terms of x will be of the form
2m+1
y = XAx ^ ,
where n has all values from 0 to oo , and 2m + 1 does not exceed n + 2, that is, m has all positive values from 0 to n/2 or ^ (n + 1), according as n is even or odd.
But, besides this expressed portion of the development of a Hypothenusal Number, say ■j;x+ii as a function of its antecedent, rjx, there will be another portion, consisting of terms with zero and negative indices of tj^ having functions of x for their coeflficients, which observation is incompetent to reveal, and with the nature of which we are at present unacquainted. The study of Hamilton's Numbers, far from being exhausted, has, in leaving our bands, little more than reached its first stage, and it is believed will furnish a plentiful aftermath to those who may feel hereafter inclined to pursue to the end the thorny path we have here contented ourselves with indicating, which lies so remote from the beaten track of research, and offers an example and suggestion of infinite series (as far as we are aware) wholly unlike any 1 which have previously engaged the attention of mathematicians.
J. J. S. and J. H. * Agreeing closely with what had been previonsly fonnd by observation in § 1 (p. 563). 8. IV. 37
578 Oil Hamilton's Numbers [61
Note 1, p. 566.
It is easy to see that, if iM and 8a are corresponding errors in the values of M and a respectively,
ZM={M log, M log, 2) 8a = (-38822 ...)ha (since Jlf= 1-46544 .... log.ilf = -38220 ..., and loge2 = -69314 ...).
Hence, 8o being intermediate between -0000003 and -0000006, m lies between -000000116 and 000000233.
The value of M (the base of the Hamiltonian Numbers) is thus found to be 1-465443 ..., correct to the last figure inclusive. — J. J. S.
Note 2, p. 568.
This equation may be obtained more simply from the fundamental formula of Hamilton (middle of above note). It follows from the law of derivation there given that, if we write ^Fn = (1 — a;)"' -^n — ^". and, in general, J+'.f„ = (1 -«)"' •''i'„ - a;", then F^+i = ""Fn ; and, consequently,
-f»+i - (1 - a;)-«"Fn = - «» {1 + (1 - a;)-' + (1 - x)-' + . . . + (1 - a;)-««+'} = af^'{{l-x)-(l- x)-<h>+'}.—J. J. S.
Note 3, p. 569.
It is curious to notice the sort of affinity which exists between a form of writing the scale of relation for Bernoulli's Numbers and that given at p. 569 for Hamilton's.
If we write
(?„=1, (?, = -l, G, = (-4)A, ^3 = 0, (?, = (- 4)»£a, G, = 0, Ge = (-4)»£„ ...
then, using /3, in the same sense as at p. 558, we shall find the scale of relation between the B's (Bernoulli's Numbers) is given by the equation
2 (— )* /3^ i . Gi-g = 0, provided i is odd. «=o
On striking out the i which intervenes between /3, and G,_,, so as to make the former operate on the latter, the equation becomes that given at p. 569 for the El's,, the sharpened numbers of Hamilton. — J. J. S.
I
51] On Hamilton's Numbers 579
§ 4. Continuation, to an infinite number of terms, of the Asymptotic Development for Hypothenusal Numbers.
" This was sometime a paradox, but now the time gives it proof."
Hamlet, Act in., Scene 1.
In the third section of this paper [above, p. 575] it was stated, on what is now seen to be insufficient evidence, that the asymptotic development of p — q, the half of any Hypothenusal Number, could be expressed as a series of powers of q — r, the half of its antecedent, in which the indices followed the sequence
■"> 5) 1; t> S' 2> •••
It was there shown that, when quantities of an order of magnitude inferior to that of {q — r)^ are neglected,
p-^=(^-r)» + J(3-r)*4-ii(5-r) + if(g-r)i; bnt, on attempting to carry this development further, it was found that, though the next term came out )§^. {q — ry, there was an infinite series of terms interposed between this one and (g — r)^, namely, as proved in the present section, between {q — r)* and (q — r)' there lies an infinite series of terms whose indices are
a 9 17 33 flS
S» T7» 3T« S?> TJH' •••'
and whose coefficients form a geometrical series of which the first term is jff^ and the common ratio §.
We shall assume the law of the indices (which, it may be remarked, is identical with that given in the introduction to this paper as originally printed in the Proceedings*, but subsequently altered in the Transactions) and write
p-q = {q-ry + ^{q-r)^ + \^{q-r) + ^{q-r)^
+ $A{q-rf+tB{q-r)^+$G{q-r)^ Jt$D{q- r)** + %E{q- r)^^^ + &c., ad inf. + «t- (1)
The law of the coefficients will then be established by proving that A = B==C = D = E=...=:^.
If there were any terms, of an order superior to that of {q — r)^, whose indices did not obey the assumed law, any such term would make its presence felt in the course of the work; for, in the process we shall employ, the coefficient of each term has to be determined before that of any subsequent
[* See footnote, p. 584, below.]
f In the text above 6 represents some unknown fanotion, the asymptotic value of whose ntio to (3 - r)i is not infinite.
37—2
680 On Hamilton' 8 Numbers [51
terra can be found. lb was in this way that the existence of terms between (q — r)* and {q — r)^ was made manifest in the unsuccessful attempt to calculate the coefficient of (q — r)'. It thus appeai-s that the assumed law of the indices is the true one.
It will be remembered that p, q, r, ..., are the halves of the sharpened Hamiltonian Numbers En+i, En,En-\, ..., and that consequently the relation
^^,= 1 + — j-2 j-273 + •••
may be written in the form
= 14. g(2?-l) _ r(2r-l)(2r-2) s(2a - l)(2s- 2)(2s-3) ^ * 2 2.3 "^ 2.3.4
t {It - 1)(2< - 2){2t - 3)(2< - 4) M (2m - 1 )(2m - 2)(2h-3)(2m - 4) (2m-5) 2.3.4.5 "^ 2.3.4.5.6
(2)
The comparison of this value of p with that given by (1) furnishes an equation which, after several reductions have been made, in which special attention must be paid to the order of the quantities under consideration, ultimately leads to the determination of the values A, B, C in succession.
Taking unity to represent the order of q, the orders of
p, q, r, s, t, u, V, w, ...
will be 2, 1, i, \, i, ^, ^, ^, ...
Hence, after expanding each of the binomials on the right-hand side of (1) and arranging the terms in descending order, retaining only terms for which the order is superior to ^, we shall find
Order 2 p = q''
|
f |
-2qr + iqi |
|
1 |
+ r" - 2qir + ^q |
|
f |
+ i^q^ |
|
1 |
+ $Aqi |
|
A |
+ $Bq^ |
|
U |
+ %Cqii |
|
M |
■^$Dq^i |
|
M |
+ $Eq^'^ + |
(3)
Again, retaining only those terms of (2) whose order is superior to J, we have
P = q'; -^r'; -^q + r^ + ^s*; -s';-^e (4)
Order 2; f ; 1 ; f ; f .
51] On Hamilton's Numbers 581
From (3) and (4) we obtain by subtraction
Order f 0=^7^ -2qr + ^q^
1 -^g^-2q^r + i^q
I +^ + ^qi
„ I +^i^ + f,Aqi
., -nr +P-B?"
.. ^ +$Cq'^
„ T^ +f;£g^V.+ .... (5)
Changing p, q, r, ... into q, r, s, ... respectively, equation (4) becomes
so that, if we assume q=r'(l — a), the order of a will be the same as that of r-" s', namely, — f + f = — i-
Hence, if we substitute t-* (1 — o) for q in (5), neglecting in the result quantities of the order ^, we shall find
^r^-2qr + iqi-j,s*-2qir + i^q ■ =|r'_2r»(l-a) + fr'(l-fa + |a» + ^a»)
while at the same time, since the order of r^a does not exceed J, we have
q^=ri(l-af = r\ and in like manner q^ = r*, q^^ = r*, and so on.
Thus equation (5) becomes
Order 1 0 = | r'o'-^s^ + ^r^
„ I +^i^ + f,Ari
„ ^ +|!£:r** + ... (6)
where a^^r-'^; +ir->-r-'s= -^r-»«*; +r-»<^ +^r-^u\ Older -i; -J ;-f;-ii'
Let a = f r-'fiJCl + a')
then «' = !«-' (^r-s^-^e^ + f' + Am")
where terms as far as, but not beyond, - -^ (which is the order of s~'m') have been retained. .
582 On Hamilton's Numbers [51
Now 2> consists of terms whose orders are 2, f, 1, f, |, ^, ... ? » .» .. 1. f. i f. Vk. i. •••
« »» » .. -i. -i. -|. --B. -f' •••
" " » ~ i' ~ I' ~ T7' ~ 7
Thus the order of o' is - \, and in the above expression all terms of a! superior to - ^ have been retained, and consequently (rejecting the square of o' whose order is - J) in the first line of (6) we may write
^ r»a» = I r-V(l + 2a')
= I r-"s« + § r-V (i r -«»- ^ (< + <»+ Y^ M»)
In the second line of (6) we may reject the whole of a, since its order is
— i, and write
After substituting their values for the terms in (6) which contain a, and at the same time dividing throughout by §, we shall obtain
Order 1 o = ^r-'s«-| s< + Jr»
.. ^ +fl^rH + ... (7)
We now write
r- = s=(l-/3) and y9 = | s"* <» (1 + y3') where, observing that the values of /3 and /S' can be immediately deduced from those of a and a' by changing r,s,t,... into s, «, m, .... it is evident that /3 and 0' are both of the order - i ; for a and a' are both of the order
- i- Thus (neglecting quantities whose order is equal to, or less than, i) we have
= K + K(i«-<'-i «* + «' + A «0 + As-=«'
= K; +hst'-^t'-^t'u* + ^s-H'>: +§«»«»; H-^jV*''''-
51] On Hamilton's Numbers 583
Order f ; I •
_2_ r-» J 16
23
r- y* w» + |-; Sr» = ^ SM" + 1.
and so on.
Hence (7) becomes
Order f Q = li^-\sf + ^^
B ■' ^ +i-:^«M + .... (8)
Dividing this throughout by §s, and then writing
8 = P{l-y) and 7 = f f-''it'(l+7'), we obtain in exactly the same manner as before, merely altering the letters in the previous work,
is-H'-iP + ls' = |w«; +^«u>-fM»-|wV + ^r'M«; +§mV; +:^u'w'. Order | ; -^ ; ^ ; H
where quantities of the order ^, or less, are now neglected.
Similarly ^ «-»f _ ^s"' «»- ^ s-'t'u* + ^t?-is-^f + ^ As^
Order f ; ^
and so on.
Thus (8) becomes Order | o = f m«- J<u' + (§^ -^i^)*' „ T!|r +^tr'u>-^u'-^u'v* + (^-IA)tu'-lir'u' + $Bti
„ ^ +!-:£«« + ....
584 On Hamilton's Numbers [51
Now the terms of the highest order in this equation must vanish when we write t = u\ and therefore f-i + ^^-iiif^O, which gives A=\\,. Substituting this value for A, we find Order | Q = \u* -^tu^-¥^f
which is a mere repetition of equation (8), with all the letters moved forward one place. Hence it is evident that, if we treat this equation as we treated (8), we shall find B = \^, arriving, at the same time, at another equation which will be merely a repetition of (8), with all its letters moved forward two places ; and this process can be continued as long as we please.
Thus we arrive at the result — and the asymptotic development for Hypothenusal Numbers
is established.
Comparing this with the corresponding formula for Hamiltonian Numbers,
given at the beginning of the third section [p. 570], it will be noticed that each of the two developments begins with an irregular portion consisting respectively of four and one term.s, followed by a regular series. In the one case the regular portion is ^(q — r)^, multiplied by a series whose general term is l^iq—r)^^^'; in the other it consists of a series of terms of the form j'i)" multiplied by — §5.
[To p. 679, footnote*. The reference is to Proceedings of the Royal Society, Vol. 42 (1887), pp. 470, 471, where is printed an Abstract identical with the Introduction to this paper (pp. 553- 666 above) save for the insertion after the word "scale" (p. 555 above) of the words "in order to establish or disprove conclusively the presumptive law of the asymptotic branch of the series connecting any two consecutive semi-differences tj^, ij^^., of the Hamiltonian Numbers, viz. : —
r=0 There is also a paper, Proceedingt of the Royal Society, Vol. 44 (1888), pp. 99 — 101, containing what is here given on p. 679 and the first half of p. 580.]
52.
SUR LES NOMBRES DITS DE HAMILTON.
[Compte Rendu de I'Assoc. Frangaise (Toulouse), 1887, pp. 164 — 168.]
CoNSiD^RONS ce tableau form^ en bas par un proc^d^ qui h peu prfes s'explique de soi-mSrae :
10 0 0 0 0 0
3 5
4 9
5
14
6 20
6 5 4 3 2 1
15 29 49
21 50 89
26 76 175
30 106 231
33 139 420
35 174 594
36 210 804
Ce tableau peut Stre etendu ind^finiment.
On voit qu'il se divise en Stages et que les nombres initiaux des premieres lignes de ces Stages sont :
1, 1, 2, 6, 36.
En les additionnant et en ajoutant I'unit^ aux sommes, on obtient les nombres 2, 3, 5, 11, 47
Ces nombres sont ce que j'appelle les nombres de Hamilton qui a trouve les nombres 11, 47, et encore le nombre qui vient apres 47, c'est-a-dire 923, dans un rapport qu'il a publid dans les Reports of the British Association 1836, sur la m^thode de Jerrard pour reduire les Equations du cinquieme degre, mdthode qui remonte, en effet, k Bring, professeur k Lund, qui I'a public dans un opuscule en 1786 qui restait inconnu ou oubli^ De meme qu'on peut 6ter 3 termes d'une equation dont le degr^ est au moins 5 sans r^soudre aucune Equation d'un degr^ superieur a 3, de m^me aussi on peut
586 Sur les Nombres dits de Hamilton [52
dter 4 termes d'une ^uation dont le degrd est au moins II sans r^soudre des equations d'un degr^ sup^rieur k 4 ; 5 termes d'une equation dont le degre est au moins 47 sans r6soudre des Equations d'un degr6 sup^rieur k 5 et ainsi de suite.
Mais il est n^cessaire d'avertir ici que la mSme chose aura lieu pour des Equations de degr^s moindres, en g^n^ral, que ceux fournis par les nombres de Hamilton. En effet, au lieu de 11, 47, 923 ... on pent substituer 10, 44, 905 ...: mais le systeme d'^q nations r^solvantes deviendra plus compliqu6 quand on fait cette diminution du degre minimum. Ainsi, par exemple, il est bien vrai que pour oter 4 termes k une Equation du degr^ 10, le systeme d'^quations k r^soudre ne contiendra nuUe Equation d'un degr^ supdrieur a 4 : mais il y aura 3 Equations de ce degre k rdsoudre tandis que quand I'dquation donnde est du degre 11 ou plus haut que 11, on n'aura a r^soudre (en combinaison bien entendu avec des Equations cubiques quadratiques et lindaires) qu'une seule Equation biquadratique au lieu de trois: et ainsi en general.
Pour trouver les nombres de Hamilton, mon coadjuteur, M. Hammond a trouv6 une ^chelle de relation d'une simplicity merveilleuse.
On peut former avec les lignes successives du tableau les fonctions 1 + Oa; + Gar" + Oar" + Oa;*... disons F^ (qui en efifet est I'unite).
|
x+ a?+ a?+ a^... |
i> |
F, |
|
2ar' + 3a^+ 4a;*... |
J) |
F, |
|
ar' + oa^+ 9a^... |
}> |
'F, |
|
6a;>+15a^... |
>l |
'F, = F, |
|
5a^ + 21a:*... |
yj |
'F, |
|
4ar'+26a^... |
» |
-F, |
|
3a;» + 30a;'... |
» |
'F, |
|
2a^ + 35a;*... |
)) |
'F, |
|
«' + 35a^... |
i> |
'F, |
|
36ar>... |
i» |
'F, = F, |
|
et ainsi de suite. |
Donnons k 1, 1, 2, 6, 36 ... les noms «», tii, a^, a,, at ... alors il est facile k voir qu'en g^ndral "F^ = F^+i ; mais aussi on voit que
«+ii'„=(l_a;)-i'i^„-a;». Done
F,^,-(l-x)-<^.Fn «»{H-(l-a;)-' + (l-a;)-'+...+(l-a;)-«n+>J
= a;"-' {(1 - a;) - (1 - a;)-«»+'}.
Faisons a„ + ai + a2+...+a„ = Sn+i alors en multipliant I'c^quation par (1 — a;)*"*!, on obtient :
(1 -a;)«»+i.^„+i-(l -a;)«». Jf„ = a;»(l -a;)«»+i+»-a!»(l -a;)««+i.
b
52] Sur les Nomhres dits de Hamilton 587
Cette Equation qui existe pour toutes les valeurs iS,, jusqu'a Si exclusif reste vraie comme identity m^me pour S„ si on met So = 0. Alors en donnant a n toutes les valeurs depuis n — 1 jusqu'a 0 inclusivement et en faisant la sommation des Equations ainsi form^es, on obtient facilement :
{l-x)^F„-l + X-' (l-x)- a;»-i (1 - a;)S»+i
= af^ (1 - x)^+- + «"-' (1 - xyn-i+"- + a;"-" (1 - a;)'S'n-«+2 + . . . . Si dans cette equation on compare les coefficients de a;" en se rappelant que le coefficient de x" en Fn est a„, c'est-a-dire Sn+i — Sn, et que )S„+ 1 est le nombre n"" de M. Hamilton, de sorte que /S„+ 2 que je nommerai E^ est ce nombre augment^ de I'unite, on trouve :
c 1 1 c -^" ~ 1 ^"-1 i^nr-i — 1) (^n-i ~ 2) Ai+i = 1 + -c-n • ,^ 2~3 '"■■■
formule de relation entre les nombres de Hamilton qu'on peut ecrire sous la
forme syra^trique K 1 - {En\ + (£„-,). - {E^\ . . . = 0.
^^ En augmentant les nombres de Hamilton de I'unit^, on obtient pour E ^Hes valeurs successives ■ 3, 4, 6, 12, 48, 924
^ qu'on trouve tres facilement par la formule de la relation donn^e. Ainsi par exemple :
3.2.1
|
1 |
.2. |
3 |
|
4 |
.3 |
2 |
|
1 |
.2 |
3 |
|
6 |
.5 |
4 |
|
1 |
.2 |
3 |
|
12. |
11 |
.10 |
|
1 |
2. |
3 |
|
48. |
67 |
.44 |
4 - 1 = 3
6-1= 5
12-1= 11
+ ^ll] = 48-1= 47
1.2.3.4 6.5.4.3 1 . 2.3 ^4
924 - 1 = 923
, + ^-yj-JV^ - "4444 = 409620 - 1 = 409619.
2 %.:i 1.2.3.4 1.2.3.4.5
Les nombres de Hamilton ainsi calcules sont :
2, 3, 5, 11, 47, 923, 409619, 83763206255 ... oil comme premiere approximation asymptotique on peut remarquer que si ft, est le nombre de rang x, n^+i h- v^' devient de plus en plus prfes de B mais toujours moindre que I'unit^ quand x croit ind^finiment.
Telle est la formule bien remarquable trouvde par M. Hammond, dont j'ai nn peu simplifi«^ et abrdg^ la demonstration.
Un travail sur les nombres de Hamilton, fait par M. Hammond et moi-m^me va prochainement paraltre dans les Philosophical Transactions [above, p. 553].
53.
NOTE ON A PROPOSED ADDITION TO THE VOCABULARY OF ORDINARY ARITHMETIC*.
[Nature, xxxvii. (1888), pp. 152, 153.]
The total number of distinct primes which divide a given number I call its Manifoldness or Multiplicity.
A number whose Manifoldness is n I call an n-fold number. It may also be called an w-ary number, and for n= 1, 2, 3, 4, ... a unitary (or primary), a binary, a ternary, a quaternary,... number. Its prime divisors I call the elements of a number ; the highest powers of these elements which divide a number its components ; the degrees of these powers its indices ; so that the indices of a number are the totality of the indices of its several components. Thus, we may say, a prime is a one-fold number whose index is unity.
So, too, we may say that all the components but one of an odd perfect number must have even indices, and that the excepted one must have its base and index each of them congruous to 1 to modulus 4.
Again, a remarkable theorem of Euler, contained in a memoir relating to the Divisors of Numbers {Opriscula Minora, li. p. 514), may be expressed by saying that every even perfect number is a two-fold number, one of whose components is a prim,e, and such that when augmented by unity it becomes a power of 2, and double the other componentf.
* Perhaps I may without immodesty lay claim to the appellation of the Mathematical Adam, as I believe that I have given more names (passed into general circulation) to the creatures of the mathematical reason than all the other mathematicians of the age combined.
t It may be well to recall that a perfect number is one which is the half of the sum of its divisors. The converse of the theorem in the text, namely that 2" (2"+' - 1), when 2"+' - 1 is a prime, is a perfect number, is enunciated and proved by Euclid in the 36th (the last) proposition of the 9th Book of the "Elements," the second factor being expressed by him as the sum of a geometric series whose first term is unity and the common ratio 2. In Isaac Barrow's English translation, published in 1660, the enunciation is as follows : "If from a unite be taken how many numbers soever 1, A, B, C, D, in double proportion continually, until the whole
53] Note on a Proposed Addition, etc. 589
Euler's function <^(n), which means the number of numbers not exceeding n and prime to it, I call the totient of n ; and in the new nomenclature we may enunciate that the totient of a number is equal to the product of that number multiplied by the several excesses of vnity above the reciprocals of its elements. The numbers prime to a number and less than it, I call its totitives.
Thus we may express Wilson's generalized theorem by saying that any number is contained as a factor in the product of its totitives increased by unity if it is the number 4, or a prime, or the double of a prime, and diminished by unity in every other case.
I am in the habit of representing the totient of n by the symbol tk, t (taken from the initial of the word it denotes) being a less hackneyed letter than Euler's <^, which has no claim to preference over any other letter of the Greek alphabet, but rather the reverse.
It is easy to prove that the half of any perfect number must exceed in magnitude its totient.
Hence, since f . J is less than 2, it follows that no odd two-fold perfect number exists.
added together E be a prime number; and if this whole E multiplying the last produce a number F, that which is produced F shall be a perfect number."
The direct theorem that every even perfect number is of the above form could probably only have been proved with extreme difiSculty, if at all, by the resources of Greek Arithmetic. Euler's proof is not very easy to follow in his own words, but is substantially as follows :
Suppose P (an even perfect number) = 2"^. Then, using in general \X to denote the sum of
the divisors of X,
/P_j2«.p^2»+'-l [A
P V^A ~ 2» 'A' JA 2*+' Q + 1
Hence J = 2«Tn' ^7= "g" •
Hence A=nQ, and jA = l + ii+Q + nQ+ ...(if n be supposed >1). Hence unless ^ = 1 and at
the same time Q is a prime
jA>pi(Q + l),
that is i— is greater than itself.
Hence an even nnmber P cannot be a perfect number if it is not of the form 2"(2"+'-l), where 2*+' - 1 is a prime, which of course implies that n + 1 must itself be a prime.
It is remarkable that Euler makes no reference to Euclid in proving his own theorem. It must always stand to the credit of the Greek geometeis that they succeeded in discovering a class of perfect numbers which in all probability are the only numbers which are perfect. Beference is made to so-called perfect numbers in Plato's "Republic," H, 546 B, and also by Aristotle, Probl. I E 3 and "Metaph." A 5. Mr Margoliouth has pointed out to me that Mnhamad Al-Sharastani, in his Book of Religious and Philosopkical Sects, Careton, 1856, p. 267 of the Arabic text, assigns reasons for regarding all the numbers up to 10 inclusive as perfect numbers, which he attributes to Pythagoras, but which are purely fanciful and entitled to no more serious consideration than the late Dr Cummings's ingenious speculations on the Dumber of the Beast. My particular attention was called to perfect numbers by a letter from Mr Christie, dated from "Carlton, Selby," containing some inquiries relative to the subject.
590 Note on a Proposed Addition to the [53
Similarly, the fact of | . J . |J being less than 2 is sufficient to show that 3, 5 must be the two least elements of any three-fold perfect number; furthermore, ^.f -ii being less than 2, shows that 11 or 13 must be the third element of any such number if it exists* — each of which hypotheses admits of an easy disproof But to disprove the existence of a four-fold perfect number by my actual method makes a somewhat long and intricate, but still highly interesting, investigation of a multitude of special cases. I hope, numine favente, sooner or later to discover a general principle which may serve as a key to a universal proof of the non-existence of any other than the Euclidean perfect numbers, for a prolonged meditation on the subject has satisfied me that the existence of any one such — its escape, so to say, from the complex web of conditions which hem it in on all sides — would be little short of a miracle. Thus then there seems every reason to believe that Euclid's perfect numbers are the only perfect numbers which exist !
In the higher theory of congruences (see Serret's Cours d'Algibre Supdrieure) there is frequent occasion to speak of " a number n which does not contain any prime factor other than those which are contained in another number M."
In the new nomenclature n would be defined as a number whose elements are all of them elements of M.
As tJV is used to denote the totient of N, so we may use i^N to denote its multiplicity, and then a well-known theorem in congruences may be expressed as follows.
TJie number of solutions of the congruence
• fl^-l = 0(modP) is 2"^ if Pis odd,
2/«p-i if p ig tjjg double of an odd number, 2"^ if P is the quadruple of an odd number, and 2''^+i in every other case.
In the memoir above referred to, Euler says that no one has demon- strated whether or not any odd perfect numbers exist. I have found a method for determining what (if any) odd perfect numbers exist of any specified order of manifoldness. Thus, for example, I have proved that there exist no perfect odd numbers of the 1st, 2nd, 3rd, or 4th orders of manifold-
* 3, S, 7 can never co-exist as elements in any perfect number as shown by the fact that
1 + 3 + 3' 1 + 5 1 + 7 + 49 ^, ^. 26/, 1 1\ . .^ „ m,.
g — . — g- . — Tg — , that IS Tc(i + 7 + 4q)>'8 greater than 2. Thus we see that no perfect
number can be a multiple of 105. So again the fact that f .J. H-if •il-il is less than 2 is sufficient to prove that any odd perfect number of multiplicity less than 7 must be divisible by 3.
63] Vocabulary of Ordinary Arithmetic 591
ness, or in other words, no odd primary, binary, ternary, or quaternary number can be a perfect number. Had any such existed, my method must infallibly have dragged each of them to light*.
In connection with the theory of perfect numbers I have found it useful to denote /»' — 1 when p and i are left general as the Fermatian function, and when p and i have specific values as the t'th Fermatian of p. In such case p may be called the base, and i the index of the Fermatian.
Then we may express Fermat's theorem by saying [cf. p. 625 below] that either the Fermatian itself whose index is one unit below a given prime or else its base must be divisible by that prinwf.
It is also convenient to speak of a Fermatian divided by the excess of its base above unity as a Reduced Fermatian and of that excess itself as the Reducing Factor.
I The spirit of my actual method of disproving the existence of odd perfect numbers consists in showing that an w-fold perfect number must have more (ihan n elements, which is absurd. The chief instruments of the investigation are the two inequalities to which the elements of any perfect number must be subject and the properties of the prime divisors of a Reduced Fermatian with an odd prime index.
* I have, since the above was in print, extended the proof to quinary numbers, and anticipate no difficulty in doing bo for numbers of higher degrees of multiplicity, so that it is to be hoped that the way is now paved towards obtaining a general proof of this palmary theorem.
t So too we may state the important theorem that if an element of a Fermatian is its index the component which has that index for its base must be its square.
54.
ON CERTAIN INEQUALITIES RELATING TO PRIME NUMBERS.
[Nature, xxxviii. (1888), pp. 259—262.]
I SHALL begin with a method of proving that the number of prime numbers is infinite, which is not new, but which it is worth while to recall as an introduction to a similar method, by series, which will subsequently be employed in order to prove that the number of primes of the form 4n + 3, as also of the form 6n + 5, is infinite.
It is obvious that the reciprocal of the product
fl_i)(l_i)(l_i)...(l-^)
\ pj \ PJ \ P3I \ PN.pJ
(where pi means the ith in the natural succession of primes, and p^.p means the highest prime number not exceeding N)* will be equal to
and therefore greater than log N (R consisting exclusively of positive terms).
where ^- (' "^O (' "S") - (' ".lb)'
2 and is therefore greater than - .
Hence the number of terms in the product must increase indefinitely with N.
By taking the logarithms of both sides we obtain the inequality
S, - Is, + Is, _ Is, + ... > log log N + log M, * N .p itself of conrse denotes in the above notation the number of primes (p) not exceeding N.
54] On Certain Inequalities relating to Prime Numbers 593
where in general Si means the sum of inverse ith powers of all the primes not exceeding N ; and accordingly is finite, except when i= \, for any value of N. We have therefore
Si > log log N + Const.
The actual value of S^ is observed to differ only by a limited quantity from the second logarithm of N, but I am not aware whether this has ever been strictly proved.
Legeiidre has found that for large values of N
^^1
3 A 5J-V PN.pl \o^N- Consequently
V pj \ pj \ Pn.p' log ^ This would show that the value of our R bears a finite ratio to log N i Uing it 6 log N we obtain, according to Legendre's formula,
— ^ = •552, which gives ^ = '811,
so that the nebulous matter, so to say, in the expansion of the reciprocal of the product of the differences between unity and the reciprocals of all the primes not exceeding a given number, stands in the relation of about 4 to 5 to the condensed portion consisting of the reciprocals of the natural numbers.
I will now proceed to establish similar inequalities relating to prime numbers of the respective forms 4n + 3 and 6ft + 5.
Beginning with the case 4n + 3, I shall use qj to signify the jth in the natural succession of primes of the form 4n + 3, and ^y.g to signify the highest q not exceeding N, N.q itself signifying the number of q's not exceeding N.
Let us first, without any reference to convergence, consider the product obtained by the usual mode of multiplication of the infinite series
by the product
ad inf.
ad inf.
It is clear that the effect of the multiplication of S by the numerator of the above product will be to deprive the series S of all its negative terms. Then the effect of dividing by the denominator of the product, with the 8. IV. 38
504 On Certain Inequalities [54
exception of the factor 1 - i^, will be to restore all the obliterated terms, but with the sign + instead of -. Lastly, the effect of multiplying by the reciprocal of (1 — i) will be to supply the even numbers that were wanting in the denominators of the terms of S, and we shall thus get the indefinite
series
, 111 , w
1 + 2 + 3 + 4 + •••«'^*"/-
Call now
Qjf, which is finite when N is finite, may be expanded into an infinite aggregate of positive terms, found by multiplying together the series
111
1+2+4 + 8 +•••
2 2 2
1+ - + - + -,+••■
2 2 2
1+ + -, + — s +••
2 2 2
1 + + -, + -, + ... .
9N.q iN.q' HN.q
Let ^^' = l-3 + 5~7 + 9"n+- -F-
then from what has been said it is obvious that we may write
g^'S^=l+J + | + 5+.- + jr+^-^-
where V and R may be constructed according to the following rule : Let the denominator of any term in the aggregate Q^ be called t, and let 6 be the smallest odd number which, multiplied by t, makes td greater than N \ then if 6 is of the form 4n + 1 it will contribute to F a portion represented by the product of the term by some portion of the series S^ of the form
111
e e + 2 ' (9 + 4 •■■
and if 6 is of the form 4»i + 3 it will contribute to - R a, portion equal to the term multiplied by a series of the form
_1 J 1_
e'^0 + 2 0 + 4i
54] relating to Prime Numbers
595
Hence R is made up of the sum of products of portions of the aggregate Qy multiplied respectively by the series
3 5"^7 9'^11"13"^--
1_1 1 _2.
7 9"^11 IS"*""-
J__ J^ 11 13"^"' of which the greatest is obviously the first, whose value is \-Si,.
Consequently R must be less than the total aggregate Q^ multiplied
I
Therefore
11.1 1
e^-Sf^ + Q.v(l - -S^) > 1 + i + ^ + ^ + ... + ^ > log iV^, that is, QN>\ogN,
from which it follows that when N increases indefinitely the number of factors in Q^, also increases indefinitely, and there must therefore be an infinite number of primes of the form 4« + 3.
Denoting by ify the quantity
('-^^)(-,7)-('-,i;.)
we obtain the inequality
and taking the logarithms of both sides
^' ~ 2^' ■'' 3^'" - ^^^''EHN + ^logMy-^^ log 2,
where in general 2^ denotes the sum of the ith powers of the reciprocals of all prime numbers of the form 4n + 3 not surpassing N.
Hence it follows that S, > ^ log log N.
I If we could determine the ultimate ratio of the sura of those terms of Q^ whose denominators are greater than N to the total aggregate, and should find that ft., the limiting value of this ratio, is not unity, then the method '•mployed to find an inferior limit would enable us also to find a superior lin.it to fc; for we should have V < fj^Q^ added to the sum of portions
38—2
596 On Certain Ineqtcalities [64
of what remains of the aggregate when /xQjy is taken from it multiplied respectively by the several series
11.1 1 ... 1 1 ^ J w 1111 , . .
9-n+i3-i5 + -"^^"^-
the total value of the sum of which products would evidently be less than
Hence the total value of V would be less than
2\
f^Q^-S + (l-^)Q^(s-l),
2 that is, less than Q^S — ^{1 — fi) Qjf,
and consequently we should have
|(l-/.)Qj,<logi\^
that is Qif < ^ '_ log N.
From which we may draw the important conclusion that if fi is less than 1, that is, if when N is infinite the portion of the aggregate SyQy comprising the terms whose denominators exceed N does not become infinitely greater than the remaining portion, the sum of the reciprocals of all the prime numbers of the form 4n + 3 not exceeding N w.ould diflfer by a limited quantity from half the second logarithm of N.
A precisely similar treatment may be applied to prime numbers of the form 6n + 5. We begin with making
We write
„ _ 1 1 1 1 1 1
T 1,1, 1
, 1 1+- 1+- 1+
Q 1 1 ^ rn ry,r
1-9 1-^ 1-- 1-- 1- —
2 3 r, r» r
N.r
Wemake Q^S^^l +^ + ^ + ^ + ,„ + }^+ V - R.
We prove as before that R < {1 — S) Q^y,
and thus obtain Qj^ > log N,
i^Ksci
54] relating to Prime Numbers 597
and then putting M^={l-^^{l- 1) ... ^1 - -i-,) ,
and finally noticing that - — - . = 3,
we obtain (l + ^J^(l + 1)^ ... (l + ±-J > \m,Xo^N.
Taking the logarithms of both sides of the equation, we find
0.-2®2 + |@3- ••■ >|loglogi\r + ilogif^-2log3,
where ©,■ means the sum of I'th powers of the reciprocals of all the prime numbers, not exceeding N, of the form Qn + 5.
Either from this equation or from the one from which it is derived it at once follows that the number of primes of the form 6ri+ 5 is greater than auy assignable limit.
Parallel to what has been shown in the preceding case, if it could be lertained that the sum of the terms of the aggregate Q^ whose denomina- tors do not exceed N bears a ratio which becomes indefinitely small to the total aggregate, it would follow by strict demonstration that the sum of the reciprocals of the primes of the form 6n + o inferior to N would always differ by a limited quantity from the half of the second logarithm of N.
It is perhaps worthy of remark that the infinitude of primes of the forms 4n + 3 and 6n + 5 may be regarded as a simple rider to Euclid's pnx)f (Book IX., Prop. 20) of the infinitude of the number of primes in general.
The point of this is somewhat blunted in the way in which it is presented in our ordinary text-books on arithmetic and algebra.
^1 What Euclid gives is something more than this * : his statement is, ^P There are more prime numbers than any proposed multitude (ttX^^o?) of prime numbers" ; which he establishes by giving a formula for finding at least one more than any proposed number. He does not say, as our text- book writers do, "if possible let A, B, ... G be all the prime numbers," &c., but simply that i{ A, B, ... C are any proposed prime numbers, one or more additional ones may be found by adding unity to their product which will either itself be a prime number, or contain at least one additional prime; which is all that can correctly be said, inasmuch as the augmented product may be the power of a prime.
* Whereas the English elementary book writers content themselves with showing that to suppose the number of primes finite involves an absurdity, Euclid shows how from any given prime or primes to generate an infinite succession of primes.
598 On Certain Inequalities [54
Thus from one prime number arbitrarily chosen, a progression may be instituted in which one new prime number at least is gained at each step, and so an indefinite number may be found by Euclid's formula : for example, 17 gives birth to 2 and 3; 2, 3, 17 to 103; 2, 3, 17, 103 to 7, 19, 79; and so on.
We may vary Euclid's mode of generation and avoid the transcendental process of decomposing a number into its prime factors by using the more general formula, a,b, ... c + 1, where a,h, ... c, are any numbers relatively prime to each other; for this formula will obviously be a prime number or contain one or more distinct factors relatively prime to a,b, ... c.
The effect of this process will be to generate a continued series of numbers all of which remain prime to each other: if we form the progression
a,a + l, a^ + a + 1, a(a + l)(a' + a+ 1) + 1, ...
and call these successive numbers
M„ «2, «3, Ut, ...
we shall obviously have u^+i = Ux" — Ux + 1-
It follows at once from Euclid's point of view that no primes contained in any term up to Mx can appear in Wj,^,, so that all the terms must be relatively prime to each other. The same consequence follows a posteriori from the scale of relation above given ; for, as I had occasion to observe in the Comptes Rendus for April 1888 [see p. 620, below], if dealing only with rational integer polynomials,
^{x) = {x - a) f {x) + a,
then, whatever value, c, we give to x, no two forms <^' (c), <^ (c) can have any common measure not contained in a: in this case ^{x) = {x—\)x + \; so that </)'(c) and <^-'(c) must be relative primes for all values of i and j*.
It is worthy of remark that all the primes, other than 3, implicitly obtained by this process will be of the form 6i+l.
Euclid's own process, or the modified and less transcendental one, may be applied in like manner to obtain a continual succession of primes of the form 4n + 3 and 6« + 5.
As regards the former, we may use the formula
2.a.6...c + l (where a,h, ... c are any "proposed" primes of the form 4n + 3), which will necessarily be of the form 4ji + 3, and must therefore contain one factor at least of that form.
* Another theorem of a similar kind is that, whatever integer polynomial ip{x) may be, if i,j have for their greatest common measure k, then ^'^[^(0)] will be the greatest common measure of
54] relating to Prime Numbers 599
As regards the latter, we may employ the formula
3.a.6 ...c + 2
(where a,b, ... c are each of the form 6n + 5), which will necessarily itself be, and therefore contain one factor at least, of that form.
The scale of relation in the first of these cases will be, as before,
Mx+i = ^<■J' -Ux + l;
so that each term in the progression, abstracting 3, will be of the form 4i + 3 and 6j + 1 conjointly, and consequently of the form 12n + 7 ; as for example,
3, 7, 43, 1807
In the latter case the scale of relation is
it,+i = uj' — 2ux + 2,
hich is of the form (ux -2)ux + 2. It is obvious that in each progression ,t each step one new prime will be generated, and thus the number of ertained primes of the given form go on indefinitely increasing, as also ight be deduced a posteriori by aid of the general formula above referred from the scale of relation applicable to each. Each term in the second case (the term 3, if it appears, excepted) will be simultaneously of the form 6t — 1 and 4-j + l, and consequently of the form 12ra + 5, as in the example 5, 17,257, 65537,....
The same simple considerations cease to apply to the genesis of primes of the forms 4m + 1, Qn + 1. We may indeed apply to them the formulae
(2.a.6...c)^+l and 3 (a.6 ... c)» + 1 respectively, but then we have to draw upon the theory of quadratic forms in order to learn that their divisors are of the form 4n + 1 and 6« + 1 respectively.
Of course the difference in their favour is that in their case all the divisors locked up in the successive terms of the two progressions respectively are of the prescribed form ; whereas in the other two progressions, whose , theory admits of so much simpler treatment, we can only be assured of the esence of one such factor in each of the several terms.
w
Euler has given the values of two infinite products, without any evidence of their truth except such as according to the lax method of dealing with series without regard to the laws of convergence prevalent in his day, and still held in honour in Cambridge down to the times of Peacock, De Morgan, and Herschel inclusive (and this long after Abel had justly denounced the use of divergent series as a crime against reason), was erroneously supposed to amount to a proof, from which the same consequeuces may be derived
600 On Certain Inequalities [54
as shown in the foregoing pages, and something more besides*. These two theorems are
_3 5 7_ 11 13 ^TT
^^' 3 + l"5-l"7 + lll + 113-l' ■ 4
(where, corresponding to the primes 3, 7, 11, ... of the form 4?j+3, the factors of the product on the left are
3 7 11
3 + 1' 7 + 1' 11 + I'""
all of them with the sign + in the denominator; while the fractions corresponding to primes of the form 4ft + 1 have the — sign in their denominators).
,„. _5 7_ 11 13 17 ^TT
^^' 5 + 1*7- 1 "11 + r 13-1 -17 + I"" 2^
where, as in the previous product, the sign in the denominator of each fraction depends on the form of the prime to which it corresponds (being + for primes of the form 6n — 1, and — for primes of the form 671 + 1).
Dr J. P. Gram {Memoires de I'Acaddmie Royale de Copenhague, 6me sdrie. Vol. II. p. 191) refers to a paper by Mertens (" Ein Beitrag zur analytischen Zahlentheorie," Borchardt's Journal, Bd 78), as one in which the truth of the first of the two theorems is demonstrated — " fuldstcendigt Bevis af Mertens " are Gram's words f.
• It follows from the first of these theorems that with the understanding that no denominator is to exceed n (an indefinitely great number),
(-i)('4)(;4)('4)--
bears a finite ratio to ( ^"'"^) ( ^"'"is ) (^"'"17) '•
so that as their proditct is known to be infinite, each of these two partial products must be separately infinite ; in like manner from Euler's second theorem a similar conclusion may be inferred in regard to each of the two products
('4) ('4) ('4) K) ('4) ('-A)- (-,-)('4.)('4,)('4)-
+ It always seems to me absurd to speak of a complete proof, or of a theorem being rigorously demonstrated. An incomplete proof is no proof, and a mathematical truth not riKorously demonstrated is not demonstrated at all. I do not mean to deny that there are mathematical truths, morally certain, which defy and will probably to the end of time continue to defy proof, as, for example, that every indecomposable integer polynomial function must represent an infinitude of primes. I have sometimes thought that the profound mystery which envelops our conceptions relative to prime numbers depends upon the limitation of our faculties in regard to time, which like space may be in its essence poly-dimensional, and that this and such sort of truths would become self-evident to a being whose mode of perception is according to luperficially as distinguished from our own limitation to linearly extended time.
54] relating to Prime Numbers 601
Assuming this to be the case, we shall easily find when N is indefinitely 4'
TT
great, so that Sy^ becomes — ,
QnSs- ^
which, according to Legendre's empirical law (Legendre, TMorie des Nombres,
2 \os N 3rd edition. Vol. ii. p. 67, Art. 397), is equal to — ^ — , where K = M04 ;
and as we have written Qi^Sy^ = log N + {V — R), we may deduce, upon I the above assumptions,
V-R=(~-l\ logi\^ = 0-811 ... logiV^.
R, we know, is demonstrably less than (l — t) 'og -^^> consequently V [must be less than (0-812 + 0-21 5) log JV, that is, less than 1-027 log iV, and la fortion the portion of the omnipositive aggregate Q^, which consists of [terms whose denominators exceed N, when N is indefinitely great, cannot be
I less than -U-"^ log N, that is, 0273 log iV.
Before concluding, let me add a word on Legendre's empirical formula for the value of
('-5)(-b)-(-S-.)'
referred to in the early part of this article.
If N is any odd number, the condition of its being a prime number is that when divided by any odd prime less than its own square root, it shall not leave a remainder zero. Now if N (an unknown odd number) is divided by p, its remainder is equally likely to be 0, 1, 2, 3, ... or (p— 1). Hence the
chance that it is not divisible by ja is [ 1 j , and, if we were at liberty to
regard the like thing happening or not for any two values of p within the stated limit as independent events, the expectation of N being a prime number would be represented by
which, according to the formula referred to, for infinitely large values of N is equal to r. It is rather more convenient to regard N as entirely
unknown instead of being given as odd, on which supposition the chance of
•* u • ij u 1104 1-104
Its bemg a prime would be ; or , ^, .
« *^ 21ogiVi ^ogN
602 On Certain Inequalities [54
Hence for very large values of N the sum of the logarithms of all the primes inferior to N might be expected to be something like (1104)i\r. This does not contravene Tchebycheff's formula (Serret, Cours d'Algibre Sup^rieure, 4me ed., Vol. Ii. p. 233), which gives for the limits of this sum
6A AN and BN, where A = 0-921292, and B = -^ = llOooo ; but does contra- vene the narrower limits given by my advance upon Tchebycheff's method [see Vol. III. of this Reprint, p. 530], according to which for A, B,-we may write Ai, Bi, where
^, = 0-921423, 5, = 1076577*.
That the method of probabilities may sometimes be successfully applied to questions concerning prime numbers I have shown reason for believing in the two tables published by me [above, p. 101] in the Philosophical Magazine for 1883t.
* Namely Ai= _„„„^ A, and Bi= ,„„>>» '^, the values of which are incorrectly stated in the ooyyy ouyyy
memoir. Strange to say, Dr Oram, in his prize essay, previously quoted, on the number of prime numbers under a given limit, has omitted all reference to this paper in his bibliographical summary of the subject, which is only to be accounted for by its having escaped his notice; a narrowing of the asymptotic limits assigned to the sum of the logarithms of the prime numbers series being the most notable fact in the history of the subject since the publication of Tchebycheff's memoir. Subjectively, this paper has a peculiar claim upon the regard of its author, for it was his meditation upon the two simultaneous di&erence-equations which occur in it that formed the starting-point, or incunabulum, of that new and boundless world of thought to which he has given the name of Universal Algebra. But, apart from this, that the superior limit given by Tchebycheff as I'lOSo should be brought down by a more stringent solution of his own inequalities to only 1-076577 — in other words, that the excess above the probable mean value (unity) should be reduced to little more than §rds of its original amount — is in itself a surprising fact. Perhaps the numerous (or innumerable) misprints and arithmetical mis- calculations which disfigure the paper may help to account for the singular neglect which it has experienced. It will be noticed that the mean of the limits of Tchebycheff is 1-01342, the mean of the new limits being 0-99900. The excess in the one case above and the defect in the other below the probable true mean are respectively 0-01342 and 0-00100.
t A principle precisely similar to that employed above if applied to determining the number of reduced proper fractions whose denominators do not exceed a given number n, leads to a correct result. The expectation of two numbers being prime to each other will be the product of the expectations of their not being each divisible by any the same prime number. But the
probability of one of them being divisible by t is - , and therefore of two of them being not each divisible by t is -ij . Hence the probability of their having no common factor is
('-D ('-9 (^-i) 0-m) •■•'"* '•"•^■' '^''' '«. 'V-
If, then, we take two sets of numbers, each limited to n, the probable number of relatively prime combinations of each of one set with each of the other should be —^ , and the number of reduced
proper fractions whose denominators do not exceed n should be the half of this or — r . I believe
r-'
M. C^saro has claimed the prior publication of this mode of reasoning, to which he is heartily welcome. The number of these fractions is the same thing as the sum of the totients of all
>4]
relating to Prime Numbers
603
numbers not exceeding n. In the Philosophical Magazine for 1883 (Vol. xv. p. 251), a table of these sums of totients has been published by me for all values of jt not exceeding 500, and [above, p. 101] in the same year (Vol. xvi. p. 231) the table was extended to values of n not exceeding 1000. In every case without any exception the estimated valne of this totient sum is found to be intermediate between
3n8 , 3(n + l)2
—~ and .^ ' .
[Calling the totient sum to n, T(n), I stated the exact equation
r(«).r(«).r(|)..(»)
n'+n
[ from which it is capable of proof, without making any assumption as to the form of Tn, that its
asymptotic value is -^ . The functional equation itself is merely an integration (so to say) of
[ the well-known theorem that any number is equal to the sum of the totients of its several
I divisors. The introduction to these tables will be found very suggestive, and besides contains an
interesting bibliography of the subject of Farey series (suites de Farey), comprising, among other
writers upon it, the names of Cauchy, Glaisher, and Sir G. Airy, the last-named as author
of a paper on toothed wheels, published, I believe, in the "Selected Papers" of the Institute
I of Mechanical Engineers. The last word on the subject, as far as I am aware, forms one of the
tinterludes, or rather the postscript, to my "Constructive Theory of Partitions," published in the
\ American Journal of Mathematics [above, p. 55].
55.
SUE LES NOMBRES PARFAITS. [Comptes Rendus, cvi. (1888), pp. 403—405.]*
ExiSTK-T-lL des nombres parfaits impairs ? C'est une question qui reste ind^ise.
Dans un article interessant de M. Servais, paru dans le journal Mathesis en octobre 1887, on trouve cette proposition qu'un nombre par/ait (s'il y en a) qui ne contient que trois facteurs premiers distincts est necessairement divisible par 3 et 5. Je vais d^montrer ici qu'un tel nombre n'existe pas, au moyen d'un genre de raisonnement qui m'a fourni aussi une demonstration de ce th^oreme qii'il n'eadste pas de nomhre parfait qui contienne moins de six facteurs premiers distincts.
On voit facilement que la somme de la s6rie g^ometrique
l+c + c»+... + c',
ou c est impair, sera elle-m§me paire quand i est impair; de plus, quand t est pair, cette somme sera toujours paire, mais impairement paire seulement dans le cas oil c = i s 1 (mod 4).
Done, si un nombre parfait impair est de la forme p^qir^ ... , {p, q,r, ... etant des nombres premiers distincts), tous les indices i,j, k, ... doivent etre pairs a I'exception d'un seul, soit i, lequel, de meme que sa base p, sera congru k 1 par rapport au module 4 ; car on doit avoir
Jp'jq^J')-^ ...=2p*q^r^ ...,
«'■*■' — 1
/a;* representant l+x+ ... + x\ c'est-i-dire =- .
X —1
Ainsi, on voit qu'un nombre parfait impair (si un tel nombre existe) sera de la forme
M'i4>q + iy+\
4g + 1 ^tant un nombre premier qui ne divise pas M.
[* See also below, p. 615.]
55] Sur les Nomhres parfaits 605
Corame corollaire, on peut deduire qu'aucun nombre parfait impair ne peut 6tre divisible par 105; en eflfet, soit un tel nombre
/3"75*/7^ /, 1 IN/, 1\/, 1 1\
c'est-k-dire g ', ' , c'est-k-dire ^^ ; qui est plus grand que 2.
Remarquons qu'en gdn^ral, si p'-qh-'' ... est un nombre parfait, il fait
I p^^ o-''*"^ p Q r
Fflue -rr- — ^ </ ,\ ••• . c'est-i-dire -^^ , t- ... , soit plus grand
[^ p^{p-l)q^{q-l) p-lq-lr-l '^ °
[que 2.
Ainsi, a raoins que le plus petit des ^l^ments p, q, r, ... ne soit plus ind que 3, on doit avoir
5711131719
4 6 10 12 16 18 ■■■^'
Imais en ne d^passant pas 19, ce produit est moindre que 1,94963. Conse- I quemment le nombre des ^Idments, dans ce cas, doit ^tre 7, au moins.
Puisque 1,95 x (1 + ^Ti) < 2, on voit imm^diatement que, si un nombre
[par&it k 7 ^l^ments parmi lesquels 3 ne figurent pas existe, le septieme {Element ne pourrait pas d^passer 37.
Passons au cas de 3 61dments 3, q, r d'un nombre parfait impair.
3 7 11 231 Puisque h 5 ^0 ~ T^n "^ ^' ^^ ^^^^ ^"^ 3*7^11*, et k plus forte raison S^p^q'',
oil p, q sont des nombres quelconques autres que 3 ou 5, ne peut Stre un nombre parfait.
Supposons done que 3, 5, q sont les dl^ments d'un nombre parfait :
I . 3 5 17 255 „ .^ ^ -^ • T-T ■
puisque K T TTj = Too < ". o^ ^''1* 9"^ 9 D6 psu* ^tre m 17, ni un nombre
quelconque plus grand que 17. Done 5= 11 ou g' = 13 ; car nous avons vu que 3, 5, 7 ne peuvent jamais se trouver r^uois comme ^l^ments d'un nombre parfait quelconque.
(1) Soient 3, 5, 13 les ^l^ments. L'indice de 13 ne peut pas etre
impair, car alors le nombre 113**+'= fo' ^ contiendrait le facteur 7, et 7
devrait ^tre uu des ^Idments. II s'ensuit que (3"+' — 1) (13'-'''"' — 1) devrait coiitenir 5 ; mais, par rapport au module 5, une puissance impaire quel- conque de 3 ou 13 est congrue a 3 ou k 2. Done la combinaison 3, 5, 13 est inadmissible.
606 Sur les Nonibres parfaits [55
(2) Soient 3, 5, 11 les ^l^ments.
L'indice de 5 doit 6tre de la forme 4; + 1 ; mais, si j > 0,
/'
5*>" = •
5-1
contiendra les trois nombres itnpaii's premiers entre eux*
5''-'+' - 1 5^J+' + 1 5 + 1 6-1 * 5+1 ' 2 ■
Cons^quemment, il y aura au moins trois autres ^Idments en plus de 5, ce qui est inadmissible: done le nombre sera de la forme 3"5 ll**.
Done (1 +5) (11'*+' — 1) doit eontenir 9, ce qui est impossible; car ll'*+' = 2(mod3).
Ainsi, on voit qu'un nombre impair avec 3 ^l^ments seulement ne pent exister.
Quant aux nombres parfaits pairs, Euclide a demontrd que 2"/2", c'est-^-dire 2"(2"+' — 1), est un nombre parfait pourvu que 2"+' — 1 soit un nombre premier. Mais on doit k Euler la seule preuve que je connaisse de la proposition r6ciproque qu'il n'existe pas de nombres pairs parfaits autres que ceiuc d' Euclide.
[• See below, p. 615.]
56.
SUR UNE CLASSE SPfeCIALE DES DIVISEURS DE LA SOMME D'UNE SERIE GE0M:^TRIQUE.
>
[Cmnptes Rendus, cvi. (1888), pp. 446—450.]
En I'honneur du grand et surprenant Fermat, dont j'ai vu avec une
fmotion indicible graves sur le buste au musde de Toulouse les mots qui lui
Staient adresses par Blaise Pascal: "Au plus grand homme de I'Europe," je
ae propose de nommer la fonction fondamentale de la haute Arithm^tique
' — 1 \efermatien a la base © et a Vindice M.
0^-1 De plus, je nommerai la fonction -^ — , qui n'est autre chose que la
smme d'une serie g^om^trique dont la raison est un entier, le fermatien iduit. M (bien entendu) est un entier positif quelconque, mais 0 un entier positif ou negatif.
Les nombrea premiers qui divisent un nombre quelconque, je les nomme ses eliments.
On sait, d'apres Euler, que tout diviseur d'un fermatien sera de la forme xfji + l, oil /x est M ou bien un diviseur quelconque de M. Parmi ces diviseurs, il y a une classe toute sp&iale qui correspond aux cas de ^= 1 et de /A = — 1. Le caractfere special de ces diviseurs du fermatien, c'est qu'ils doivent necessairement etre (comme on verra imm^diatement) en meme temps diviseurs de son indice. Je remarque prealablement que, ^v —\ (oil p est un nombre premier) etant, par rapport au module p, congru a B — 1, afin que ce fermatien contienne p, il faut que © — 1 le contienne.
(1) Soit M = p un nombre premier impair: je dis que le fermatien . 0p _ 1 r^uit — -~ contiendra p, mais non pas ^. Car, en mettant 0 = ^p + 1, on
. ©P— 1 .
Toit que le fermatien r^duit -^ — =- , envisage comme la somme d'une s^rie
¥5—1
g^om^trique, sera congru par rapport au module p' k p + k . p, c'est- ' i-dire a p.
608 Sur uiie classe »p4ciale des diviseura [56
(2) Soit M la puissance d'un nombre premier impair p'. En supposaut toujours que 8 — 1 contient /), 0* — 1 le contiendra.
^ , «P"_1 0P"_i 0P— '-1 Qp-l ., .
Consequemment, puisque „ — ^ = — —:, —-; ... „ - , , il suit
^ ^ W — 1 0p" ' _ I %p*~^ _ 1 M — 1
comme consequence de ce qui pr^cfede que -^r — ^ sera divisible par /)•, mais non pas parp*''''.
(3) Soit M = iVp", oil N est premier a p ; on a
le premier facteur peut etre envisage comme f'onction de 0-"^ et par le cas
precedent sera divisible par j9", mais non pas par p''+'. Le second facteur,
envisage comme la somme d'une s^rie g^ometrique, sera congru k N par
rapport k p (quel que soit N pair ou impair) et consequemment ne contiendra
0iVp" _ 1 pas p. Done —^ — - sera divisible par p', mais non par p"*'. " — 1
Ainsi, si p est un element quelconque impair de 0 — 1 et p" la plus haute
0itf _ 1
puissance de p contenu dans M, le fermatien r^duit -^ — r- contiendra p",
mais ne contiendra pas p''*'^ et, comme consequence particulifere, ne contiendra mil element de 0 — 1 qui n'est pas un diviseur de M.
On peut aussi supposer que 0 — 1 contient chaque element de M, et Ton obtient le th6oreme suivant :
Un fermatien riduit d indice impair, dont le denominateur est divisible par chaque dUment de son indice, sera lui-meme divisible par cet indice, et de pliis le quotient qui resulte de la division de I'une de ces quantites par I'autre sera premier relatif a I'indice.
C'est dans les recherches sur la possibilite de I'existence de nombres parfaits autres que ceux d'Euclide que se rencontre cette theorie des feimatiens r^duits qui y joue un role indispensable. Comme exemple de son utility, je vais faire voir qii'un nombre de la forme '3N ± 1 a 7 elements ne peut pas ^tre un nombre parfait.
Remarquons que, si g est un des nombres gaussiens 3, 5, 17, 2.57,..., c'est-a-dire un nombre premier de la forme 2" + 1, g ne peut pas diviser un fermatien riduit k indice impair s'il ne divise pas le denominateur; car, afin que cela eAt lieu, ^ — 1 par le theoreme d6}k cite d'Euler devrait contenir un facteur impair.
Done un tel fermatien riduit sera de la forme —. ^ — ; .
(gra;+l)-l
56] de la somme cFune serie giometrique 609
Or nous avons vu, dans la Note precMente [p. 604, above], qu'un nombre oN + 1 a 6 dl^ments ne peut pas etre un nombre parfait, et que, si un tel uombre a 7 Elements est un nombre parfait, le plus grand d'entre eux ne peut pas excdder 37.
II est facile de voir que ce nombre doit contenir 5, parce que
7111317192329
6 ■ 10 ■ 12 ■ 16 ■ 18" 22 ■ 28^ ' en effet, ce produit est raoindre que 1,69.
Soit done, s'il est possible, 3iV + 1 un nombre parfait a 7 elements.
Les nombres premiers de la forme 4a; + 1 pas plus grands que 37 sont 13, 17, 29, 37. Mais 17 ne peut pas ^tre l'^16ment exceptionnel de 3iV + 1 parce que la somme des diviseurs du component* qui repond a 17 sera la somme d'un nombre pair de termes de la serie 1 + 17 + 17^ + 17' + . .. , laquelle necessairement contient 3. La meme chose est dvidemment vraie pour un nombre quelconque, com me 2q, qui est de la forme 12a;+ 5.
Done le component exceptionnel aura pour ^l^ment ou 13 ou 37 ; mais ni 13' — 1 ni 37' — 1 ne contient 5. II faut done que la somme des diviseurs du component ou k I'dlement 11 ou sinon a I'element 31 soit respectivement
11''' — 1 31*" — 1 de la forme ^ ou ^ :,- , car 11 et 31 sont les seuls nombres pas plus
11 — 1 ol — 1
grands que 37 de la forme 5a; + 1. Consequemment tous les diviseurs d'une
11° — 1 31° — 1 au moins des deux quantit^s -pj — r- ou -^r= — — seront compris parmi les
11 — 1 ol — 1
elements de ZN + 1.
I*
Selon notre theoreme, les diviseurs ni de I'un ni de I'autre de ces deux fonctions ne peuvent contenir -5 et consequemment par le thdoreme d'Euler seront de la forme 10a; + 1.
Or, puisque 11 n'est pas un r^sidu quadratique de 31, 11°— 1 ne peut
11» — 1
pas contenir 31 ; done les diviseurs de r- — sont compris parmi les
nombres 41, 61, 71, 101, ....
31° — 1 .
^- contiendra 11, mais ne peut pas etre une puissance de 11, car
ol — 1
au module 11' s
4°(31» - 1) = 3° - 4° = 1 - 4"»'H - 1023,
c'est-a-dire - 11 . 93,
de sorte que 31'— 1 n'est pas divisible m^me par 11'.
31° — 1
Done les diviseurs de .^^j r- sont aussi compris parmi les nombres
41, 61, 71, 101
* La pins hante puissance d'nn ^l^ment d'nn nombie qu'il contient se nomme nn component de ce nombre.
8. IV. 39
610 8ur une classe sp^ciale des diviseura [56
Cons^quemment il y aura au moins un ^l^ment du nombre parfait 3JV + 1 qui n'est pas moindre que 41 ; cette conclusion est contradictoire k I'existence de la limite sup6rieure 37 k la grandeur des dl^inents. Done on peut affirmer en toute s<iret6 qu'un nombre non divisible par 3 qui contient moins que 8 facteurs premiers distincts ne peut pas Stre un nombre parfait.
II y a une m^thode un peu plus expdditive pour parvenir au rdsultat demiferement acquis; mais, tout de mfime, supprimer la premiere m^thode serait un proc^d6 mal avis^, puisque son principe est applicable k d'autres cas oil celui dont je vais faire usage se trouverait en d^faut ; par exemple en combinant les deux m^thodes, c'est-^-dire en tenant compte en mdme temps des consj^quences de la presence de 17 quand il figure comme ^l^ment, et de la presence de I'^lement 5 dans le cas oh. 17 manque, je crois avoir demontr^ qu'un eutier 3A'' ± 1 a 8 ^l^ments ne peut pas §tre un nombre parfait.
Remarquons que, puisque le produit suivant, a 7 termes, oil 17 manque
5 7 11 13 19 23 29 ^ . , , „„„ i,
dans le num6rateur, 4 g fo 12 18 22 28' ™oiiidre que 1,988, un nombre
parfait h 7 616ments non divisible par 3 ne peut pas exister sans I'^lement 17, Supposons qu'un tel nombre exists. Soit r} un de ses 616ments (autre que 17). La somme des diviseurs du component qui y correspond
sera de la forme ~ si iy est un element ordinaire, et de la forme
•>? -1
i5Z — ^^- (i; + l) si 1} est I'el^ment exceptionnel. ij'— 1
Dans I'un et dans I'autre cas, cette somme ne peut contenir 17 que sous la condition que if-\ soit divisible par 17.
Done, puisque le produit des sommes des diviseurs des components d'un nombre parfait doit contenir tous ses ^16ments, il existe au moins un dl^ment 77 tel que 17= - 1 contient 17, c'est-a-dire il y a un ^l^ment qui est un nombre premier compris dans I'une ou I'autre des forinules 17ir+l, 17a! - 1 ; mais le plus petit nombre premier contenu dans ces formules est 67*. Ainsi, puisque
; (1,95) (l + ^)< 1,98,
I'existence d'un nombre parfait 3iV + 1 ^ 7 ^l^ments est impossible.
• On pourrait facilement prouver (s'il ^tait n^cessaire pour les besoins de la demonstration du thior^me) que »j doit 6tre un nombre premier de la forme Xlx + 1 ou un nombre premier en mSme temps de la forme 17x - 1 et 12i/ + 1, c'est-i-dire de la forme 204x + 169, et ainsi il y aurait au moins an element plus grand que 103. t
571113171967 4 6 10 12 16 18 66 "^
b7.
SUR L'IMPOSSIBILITE DE L'EXISTENCE D'UN NOMBRE PAR- FAIT IMPAIR QUI NE CONTIENT PAS AU MOINS 5 DIVISEURS PREMIERS DISTINCTS.
[Gomptes Rendus, cvi. (1888), pp. 522—526.]
Nous avons vu, dans une Note pr^c^dente, qu'un nombre parfait impair avec moins de 7 facteurs doit ^tre divisible par 3, et aussi que nul nombre parfait ue peut etre divisible par 105. Ajoutons que, puisque
3 11 13 17_15Hf^2 2 ■ 10 ■ 12 • 16 80 ^
et que, en changeant 11, 13, 17 pour d'autres dMments, on ne peut diminuer ce produit qu'en empi^tant sur les chiffres 5 ou 7, il s'ensuit que I'el^meut 3 doit etre associe ou avec 7 ou avec 5 dans un nombre parfait k quatre ^^ments, s'il y en a.
Supposons done qu'un tel nombre N existe.
(1) Soient 3 et 7 deux de ses elements. Le troisifeme 61^ment en ordre de grandeur ne peut pas exceder 13 ; car
3 7 17 19_119/. 1\ 126 „ 2 • 6 • 16 • 18 ~ 64 V^ + I8J ^ "64 < ^•
(a) Soit 11 le troisieme ^l^ment; puisque
3 7 11 29 _ 77 / l\
2 • 6 • 10 • 28 ~ 40 I "^ 28; *^ •
on voit que le quatrifeme ^l^ment ne peut gtre qu'un des nombres 13 17 19, 23. ^ . ,
Mais, parmi les Elements, un au moins doit 6tre de la forme 4a! + 1.
De plus, nous avons vu dans une Note prdcedente que nul nombre parfait ne peut contenir I'dl^ment 17 sans contenir en meme temps un ^Idment pas plus petit que 67. Done les quatre ^Idments seront 3, 7, 11, 13.
39—2
612. Sur rimpossibiliU de Fexiatence [57
Le diviseur-somme* i 7 ne peut pas contenir le fecteur alg^brique 7»— 1,
1 7' — 1 1 7' — 1 car alore ^ . -= — r- , ^ . ^. — =^ seront diviseurs de cette somme premiers entre 37-13 7' -1
eux, ^ 3 et ^ 7, et en plus ne contenant pas 13 parce que 13 n'est ni une
fonctiou unilin^airef de q ni diviseur de 7'— 1. Ainsi sur cette supposition
il y aurait au moins cinq Elements distincts. Done le diviseur-somme a 7 ne
peut pas contenir 9, mais le component k 3 contient n^cessairement 3^ ;
cons^quemment, puisque le diviseur-somme all (Element ordinaire et non
pas de la forme 3a; + 1) ne peut pas contenir 3, le diviseur-somme k 13
13'— 1 contiendra un facteur algebrique de la forme r^ — - quiest^gal k 169-1-13-1-1.
Done 61 sera un Element en plus de 3, 7, 11, 13 qui est contraire k I'hypothese.
1. (/S) Soit 13 le troisieme 616ment.
„ . 3 7 13 23 91 /, 1 > „ , ^ ., ...
ruisque ^.-.y^.^^ = jk|1+9h)<2, le quatneme element sera neces-
sairement moins que 23, et le systeme des ^l^ments sera 3, 7, 13, 19, car 17
est exclus.
Les diviseurs-sommes, ni k 13 ni k 19, ne peuvent pas contenir 3; parce
13' — 1 19'— 1 qu'ils contiendraient n^cessairement les facteurs ^s — ? ^* i q — > ^^ ainsi
1 -H3 -I- 13' , ,. ,. «, 1-I-19-H9' . , . ,. ,„_ 5 , cest-a-dire 61, et ;5 -, cest-a-dire 127.
Done le diviseur-somme k 7 doit contenir alg^briquement les facteurs
1 7° — 1 1 7' — 1
K . =2 — r , n ■ n — T' "^ dernier est ^gal k 19; le premier sera n^cessaireraent
o 7 — 1 o 7 — 1
premier k 3, 7, 19 et, pour la raison ddjk dounde, k 13.
II est done d^montr^ que 7 ne peut pas ^tre un element de N. (2) Supposons que 3 et 5 sont deux de ses elements.
2. A. Soit 5 r^lement exceptionnel. fl|
2. A (a). Si I'indice a I'^l^ment 3 est 2, alors, puisque 1 + 3 + 3' = 13, on aura les dl^ments 3, 5, 13; done le diviseur-somme k IS doit contenir 3,
13' -f 13 -H 1 et, cons^quemment, contiendra alg^riquement le facteur x , c'est-
k-dire 61.
Ainsi on aura les ^Idments 3, 5, 13, 61.
Mais -^ . — - — . Y5 • 7^ < 2, ce qui est inadmissible.
* Si p eat un ^l^ment et p' un component d'un nombre N, on nomme p* le component k p, et . ;— le dtvueur-somme a p.
t n est trAs commode, dans ce genre de recherches, de se servir de la phrase " fonction uni- lin€aire de x " ponr signifier kx + 1.
-
57] <fw?i nomhre parfait 613
k2. A (/9). On peut done supposer I'indice du component a 3 au moins 4. Soient 3, 5,p les trois elements; I'indice du diviseur-somme h, p ne peut IS etre 9, car alors on aurait en plus de 3, 5,^ deux autres Elements au oins premiers entre eux et a 3, 5, p.
Soit q le quatri^me Element ; la meme chose sera vraie du diviseur- somme a q.
Done le produit des diviseurs-sommes a 3, b,p, q ne peut pas contenir une plus haute puissance de 3 que 3^ ; mais elle doit contenir au moins 3*.
Ainsi I'hypothese que 5 est I'^l^ment exceptionnel est inadmissible.
2. B. Passons k I'hypothese que 5 est un ^Idment ordinaire.
Remarquons 9"^ ^ . ^ . ^ . ^^ < 1,992 < 2.
^_ Consequemment, il y aura au moins un ^l^ment, disons p, qui n'excede ^feas 29 : je dis que j) ne peut pas etre contenu dans le diviseur-somme de 5 ; ^■Bar, si cela avait lieu, I'indice de cette somme serait necessairement un ^■diviseur impair de I'excfes au-dessus de I'unit^ de quelque nombre premier inferieur a 31, c'est-a-dire 3, 5, 7, 9 ou 11, dont les quatre demiers correspon- dent respectivement aux nombres premiers 11, 29, 19 et 23.
5' — 1 5' — 1
II ne peut pas Stre 3, car ^ — ^ = 31 ; ni 5, car ^ — j- = 11 . 71 (et Ton
aurait une combinaison d'^l^ments 3, 5, 11, 71 ; laquelle est inadmissible, parce que 5 est, par hypothfee, non exceptionnel, et les autres ^l^ments sont de la forme ix + 3).
II ne peut pas etre 7, car on tronve facilement que 5' — 1 ne contient pas 29 ni 9 ; car, quoiqu'il soit vrai que (a ^tant rdsidu quadratique de 19) 5"— 1 contient 19, il contient en m^me temps 5' — 1, et Ton aurait la combinaison 3, 5, 19, 31, qui est defendue par la meme raison que Test 3, 5, 11, 71.
Reste seulement 11, mais 5" — 1 ne peut pas contenir 23, parce que 5 n'est pas r^sidu quadratique de 23.
Ainsi r^lement 5 ne peut pas engendrer (au moyen du diviseur-somme qui lui r^pond) un ^l^ment qui n'est pas en dehors de la limite 29.
Le diviseur-somme k un tel Element (s'il est 11 et seulement dans ce cas-lk) peut contenir 5, mais non pas 5' ; car, s'il contenait 5', on aurait au moins deux diviseurs de cette somme premiers entre eux et k 3, 5, 11.
Remarquons que le component a I'^l^ment exceptionnel ne peut pas §tre une puissance {k exponent 4; -|- 1) d'un nombre; car, si _;' > 0, 5^^+' — 1 con- tiendrait necessairement deux facteurs premiers distincts en addition k 3, 5 etp; doncj'=0; ainsi Ton voit que q+\ doit contenir au moins les puissances de 3 et .3 contenues en 3' . 5', qui ne sont pas contenues dans le diviseur- somme de I'autre ^l^ment indetermind, lequel on montre facilement ne
614 Sur TimpoasihiliU de F existence d'un nombreparfait [57
pouvoir contenir que 3 ou 6 et non pas 3', 3 . 5, ou 5' ; car, sur la premiere ou la demifere de ces trois hypotheses, le nombre des ^l^ments serait plus grand que 4, et sur I'hypothese qui reste plus grand meme que 5. Done r^ldment exceptionnel augment^ par I'unit^ sera de la forme ou 2A . 3' . 5 - 1 ou 2ifc . 3 . 5' — 1 : cons^quemment sa valeur doit exc^er 89 ; cela prouve que le p dont nous avons parl6 n'est pas I'^l^raent exceptionnel.
Soit q cet ^l^ment, on aura
q = 30\ - 1.
Or le diviseur-somme a 5 ne contient ni 3 ni p.
On aura done forc6ment
^—j^q = 30\-l,
c' est-^-dire 5* - 120X + 3=0,
ce qui est impossible.
Cela d6montre que I'hypothese 2. B est inadmissible, et finalement le r^sultat est acquis qu'il n'existe pas de nombres parfaits impairs qui soient divisibles par moins de 5 facteurs premiers; car ce th^oreme, pour les cas d'une multiplicity 3, 2, 1, a d^a ete d^montre.
Ajoutons quelques mots sur les nombres parfaits k cinq ^Idments.
Ici, puisque
3 n 13 17 23
2 • 10 ■ 12 • 16 • 22 ^ ' ■
3 11 13 17 19 „„_,
On voit qu'un nombre parfait k cinq ^l^ments, ou 5 et 7 manquent, ne pent avoir pour ces ^l^ments que les chiffres 3, 11, 13, 17, 19.
Mais 17 (un nombre cyclotomique de Gauss) ne peut pas exister sans un 614ment satellite de la forme ITk ± 1. Done un nombre parfait k cinq ^i^ments, s'il existe, aura n^cessairement ou les ^Idments 3, 5 ou les ^Idments 3,7.
J'ai rdussi a d^montrer Timpossibilitd de I'une et de I'autre de ces hypotheses; mais la preuve est trop longue pour 6tre ins^rde ici.
58.
SUR LES NOMBRES PARFAITS.
^1 [Giymptes Rendus, cvi. (1888), pp. 641, 642; Mathesis, viii. (1888), ■ pp. 57-61.]
^T Dans la demonstration de rimpossibilit^ qu'un nombre a .3 Elements soit un nombre parfait, qui a paru dans les Comptes reiidios du 6 fevrier dernier, il y a une petite omission que M. Mansion a eu la bonte de me signaler. II
est dit Lp. 606, above J, que les nombres -— — — , — ;; — — , — ^ — sont
premiers entre eux.
Cela n'est pas vrai si, 2; + 1 contient 3, mais, dans ce cas-lk, 5^^+' + 1 con- tiendra 5' + 1 qui contient 7 : cons^quemment, on aura les quatre ^^ments 3, 5, 7, 11. Done la demonstration reste bonne.
M. Sylvester vient de publier [p. 604, above], dans les Comptes Rendus de I'Acadimie des Sciences de Paris (stance du 6 fevrier 1888, t. cvi. pp. 403 — 405), une importante contribution a I'etude des nombres parfaits, k I'occasion de remarques de notre collaborateur M. Servais {Mathesis, t. vii. pp. 228 — 230).
Nous sommes heureux de reproduire ici les considerations ddveloppdes par I'illustre georaetre anglais, comme complement des articles publics a ce sujet dans Mathesis (t. vi. pp. 100—101, 145—148, 178, 248—250, et t. vii. pp. 228—230, 245—246).
La notation c h i = 1 (mod 4) est ^quivalente h. la notation plus explicite:
c = i + itl4=l+iW4
et se prononce : c est congru d i et d 1, suivant le module 4.
Nous ajoutons quelques notes a I'article un peu bref de M. Sylvester pour en faciliter I'intelligence*. P. Mansion.
Existe-t-il des nombres parfaits impairs ? C'est une question qui reste ind^cise.
* Dans les nos. des C. R. dn 13 et du 20 Kvrier, M. Sylvester a pnbli^ de nonvelles recherohes ■or les nombres parfaits dont nous ne pouvons, f aute d'espaee, que signaler plus has, les conclusions en note. II s'est aussi occupy des nombres parfaits dans les nos. de Nature, dn 15 et du 22 d^oembre 1887, ct dans VEducational Timet du 1" mars 1888. .
616 Sur les nombres par/aits [58
Dans ua article int^ressant de M. Servais, paru dans le journal Mathesis, en octobre 1887, on trouve cette proposition qu'un nombre parfait impair (s'il y en a) qui ne contient que trois facteurs premiers distincts est ndcessaire- ment divisible par S et 5. Je vais d^raontrer ici qu'un tel nombre n'existe pas, au moyen d'un genre de raisonnement qui m'a foumi aussi une demon- stration de ce th^orfeme qu'il neodste pas de nombre parfait impair qui contienne moins de six facteurs premiers distincts.
On voit facilement que la somme de la s^rie gdom^trique
1 + c + d'+ ... + c' oil c est impair, sera elle-meme paire quand i est impair; de plus, quand i est pair, cette somme sera toujours impaire, mais impairement paire seulement dans le cas ou c = i = 1 (mod 4).
Done, si un nombre parfait impair est de la forme p*qir^... (p,q,r, ... ^tant des nombres premiers distincts), tous les indices i,j, k, ... doivent etre pairs k I'exception d'un seul, soit i, lequel, de mSme que sa base p, sera congru a 1 par rapport au module 4 ; car on doit avoir
Jp*jqifr^ ... = 2p'qJr^ ...,
a;'"''' — 1
/a^ repr^sentant 1 + x + ... +a;\ c'est-a-dire ^— .
X — 1
Ainsi, on voit qu'un nombre parfait impair (si un tel nombre existe) sera
de la forme ilf'(45 + 1)^+', 45^ + 1 etant un nombre premier qui ne divise
pas M*.
Comme corollaire, on pent d^duire qu'aucun nombre parfait impair ne pent etre divisible par 105. En effet, soit un tel nombre S^o^l^...; on aura
y3^;5^j7^^^^_^l_^l
3"'5«7«* "* V 3 3»
D(i + ^+y«);
, . , ,. 2.13.19 ,,.,. 494 . ^ , ,
c est-a-dire § — — ; c est-a-dire ^j^ , qui est plus grand que 2.
Remarquons qu'en general, si p*gi-'r* ... est un nombre parfait, il faut que
TT — ~T\—rr — TT--- c'est-a-dire — ^ — ^-^ — r--- p*{p-l)q}(q-l) p — lq-lr-l
soit plus grand que 2-f.
* Th^or^me d^montr^ anssi, en 1886, par M. Stern, dans Mathesit, t. vi. pp. 248—250, mais que Ton trouve ^galement au no. 109, du chapitre UI, de I'opuscule d'Euler: Tractatm de numerorum doctrina, public dans les Commentationes arithmeticae coUectae (voir t. ii. pp. 514—515).
II en r^sulte que, si 3, 7, ou 11, etc. entrent comme facteur dans un nombre parfait impair, lis y entrent avec un ezposant pair, car ils sont de la forme {ip + 'S).
t Voir, par exemple, I'artiole de M. Servais, p. 230. D'apr^s la definition des nombres parfaits, on a
I
58] 8w les nomhres parfaits 617
Ainsi, k moins que le plus petit des Elements ^, g, r ... ne soit pas plus grand que 3, on doit avoir
5711131719 4610121618"""^ '
mais en ne depassant pas 19, ce produit est moindre que 1,94963. Con- s^quemment le nombre des ^I^ments, dans ce eas, doit etre 7 au moins. Piiisque
1.9.5x(l + ^)<2,
on voit immediatement que, si un nombre parfait a 7 Elements parmi lesquels 3 ne figure pas, existe, le septieme ^Idment ne pourrait pas d^passer 37*.
Passons au cas de 3 61^ments 3, q, r d'un nombre parfait impair. Puisque
3711^231
2 610 120*^ '
on voit que 3* . 7^ . 1 1* et a plus forte raison S'p^q'', oil p, q sont des nombres quelconques autres que 3 ou 5, ne peut etre un nombre parfait.
Supposons done que 3, 5, q sont les ^l^ments d'un nombre parfait ; ipuisque
3 517^2.55 24 16 ~ 128 '^'
[on voit que q ne peut Stre ni 17, ni un nombre quelconque plus grand que 117. Done g = 11 ou ^ = 13 ; car nous avons vu que 3, 5, 7 ne peuvent jamais [se trouver r^unis comme elements d'un nombre parfait quelconque.
(1) Soient 3, 5, 13 les elements. L'indice de 13 ne peut pas ^tre impair, alors le nombre
' 13-1
j^l_l ^+1-1 T*<-»-l -
on encore Vr ^,, . -4- ^^ • -r tv=2.
P^ip-l) 9'(?-l) ri(r-l)
On d^duit ais^ment de U (1) que {qi*^ - 1) (H^' - 1) doit Stre diviBible par p. (2) En supprimant
( - 1) dans les nam^ratenrs,
p-l g-l r-l * Dan» les C. R. du 1.3 f^vrier, M. Sylvester a pronvfi qn'il ne peut y avoir de nombre parfait premier aveo 3, ayant mSme 7 ou 8 il^ments. II se sert pour arriver 4 ce resultat de propri^t^s (d^doites du thiorfeme de Fermat) des expressions ff"-l, [(^- 1) : (9- 1)]; il norame oes expressions /erTnatien de base 6 et d'indice n, etfermatien riduit en I'honneur du grand g^om^tre de Toulouse. II rappelle, a ce propos, les rants adresses k. celui-ce par Pascal : " Au plus grand homme de I'Europe," mots graves snr le buste de Fermat au mus^e de Toulouse. La citation exacte de Pascal est : "Quoique vous soyez celai de toute I'Europe que je tiens pour le plus grand gionAtre, etc." (Lettre da 10 aodt 1660).
618 Sur les noinbres parjaits [58
contiendrait le facteur 7, et 7 devrait etre un des ^l^ments*. II s'ensuit q»e (3»+i _ 1^(13»;+' — 1) devrait contenir 5 ; mais, par rapport au module 5, une puissaDce impaire quelconque de 3 ou 13 est congrue & 3 ou k 2. Douc la combinaison 3, 5, 13 est inadmissible.
(2) Soient 3, 5, 11 f les dldments. L'indice de 5 doit etre de la forme 47 + 1 ; mais, si j > 0,
contiendra les trois nombres impairs premiers entre euxj
50+' - 1 5^+' + 1 5 + 1 5-1 ' 6 + 1 ' 2" [pourvu que 2j+l ne soit pas divisible par 3 ; dans ce cas, 5^' + 1 contien- drait 5'+ 1 = 18 . 7, de sorte que 7 serait un element].
Cons^uemment, il y aura au moins trois autres 616ments en plus de 5, ce qui est inadmissible ; done le nombre sera de la forme 3^^ . 5 . 11^.
Done (1 + 5)(11'*+'— 1) doit coutenir 9, ce qui est impossible; car ll«*+> = 2 (mod 3).
Ainsi, on voit qu'un nombre parfait impair avec 3 ^l^ments seulement ne peut exister§.
Quant aux nombres parfaits pairs, Euclide a demontrd que 2"/2'', c'est-4- dire 2"(2""'"' — 1) est un nombre parfait pourvu que 2"+'— 1 soit un nombre premier. Mais on doit a Euler la seule preuve|| que je connaisse de la pro- position r^ciproque qu'i7 n'existe pas de nombres pairs parfaits autres que
ceux d' Euclide.
5>H-i _ 1 Note. On peut encore ^tablir le (2) comme il suit. Le nombre
introduit dans le premier membre de I'dgalite hypothetique 3W+1 _ 1 54;+2 _ 1 ii2t+i _ 1
o
3-1 ■ 5-1 ■ 11-1
•=2.3»*..5<'+'. 11^
♦ isa+a _ 1 est divisible par 132 - 1 = 168 = 7 x 24. t 3 et 11 out des exposants pairs (voir la premiere note, p. (616]).
X Les nombres 5^+'-l, 5''J+' + l n'ont d'autre diviseur commun que leur difference 2; enanite on a
5«»+»=iIK3 + 2,
done S*'*' - 1 et J (5%'+' - 1) ne sont pas divisibles par —5— = 3. Mais — j — ^ n'est pas toajonrs
premier aveo 3 ; en effet, 5^'+' + 1 est un multiple de 9 plus 6, 0 ou 3 suivant que 2j + 1 est de la forme 3p + 1, ip, ou 3p + 2. Les lignes entre crochets manquent dans les C. R. ; elles nous ont M obligeamment communiqu^es par I'auteur, pour completer la demonstration, dans le cas oil S?*'*' est divisible par 9. (Voir aussi la note a la suite de I'article.)
§ Dans les C. R. dn 20 f^vrier, M. Sylvester d^montre qu'il n'y a pas nombre parfait impair aveo quatre Aliments et annonne qu'il a prouve qu'il n'en existe pas mdme avec cinq elements.
II Commentationes arithm. coll., p. 514, no. 107 (cite par M. Sylvester, Nature, 15 dec. 1887, p. 152). Voir une autre demonstration due k M. Lucas, dans Mathesis, t. vi. pp. 146 — 147.
58] Sur les nombres par f aits 619
527+1 _ 1 au moins un facteur different de 3, 5, 11. En effet, ce nonibre — ^ r~ ^'il
0 — 1
n'est pas divisible par 11, introduit im autre facteur que 3, 5, 11, puisqu'il est premier avec 3, 5, 11. D'autre part, s'il est divisible par 11, il est aussi divisible par 71 ; car on a
5 -1=4, 55(i>P+i)-4 - 1 = Jtl 11 + 4,
5'-l = i(ttll+3, 5»t'*+"-^-l=iJtlH-3,
5»-l = 4. 11.71, 50W+" -l = iW(5»-l) =im71,
5'-l=imi+2, 5»'«^'i+=-l=i«lH-2,
5»-l=i(Wll + 8, 5'«*+"+^-l=ittll + 8.
P. Mansion.
59.
PREUVE ]6l6mentaire du theoreme de dirichlet sur
LES PROGRESSIONS ARITHM^TIQUES DANS LES CAS Ot LA RAJSON EST 8 OU 12.
[Gomptes Rendm, cvi. (1888), pp. 1278—1281, 1385—1386.]
Le principe (ou pour ainsi dire le moment intellectuel) dont nous nous servons est le suivant :
Pour d&mojitrer que le nomhre de nombres premiers d'une forme donnie est infini, cherchons a construire une progression infinie d'entiers relativement premiers entre eux, et dont chacun contiendra un nomhre premier {au moins) de la forme donnSe.
Dans ce qui suit, / signifie une forme fonctionnelle rationnelle entiere et ne contenant que des coeflScients rationnels.
Lehhe I. — Si Ux+i=fux et si ffO—fO, alors, r et s etant deux entiers quelconques, le plus grand diviseur commun d, Ur et Ug sera un diviseur defO.
Car 6videmment Ur+t=ff ...fO (c'est-k-dire f'O) [modw^]. Mais/'O, par hypothese, =/0.
Cons^quemment, tout di\dseur de m^ et m, sera un diviseur defO.
Lemme II. — Si Ux+i=fux et si, de plus, Mi=/0, le plus grand diviseur commun de Uf et «, sera Ut, oil t est le plus grand diviseur commun de r et s.
(1) On aura ^videmment
Us+^ = u, (mod u,). Cons^quemment Mj sera un diviseur de u^, rjht,..., u^t quel que soit m.
(2) Ecrivons un schdma pareil k celui qui s'applique k la recherche du plus grand diviseur de r et s, c'est-k-dire
r — hs = v, s—kv=w, ..., z — ly = t, y — mt = Q; alors, en vertu de ce qui pr^cfede, Ut sera un diviseur de Ur et u,, et tout diviseur de m, et de w, sera un diviseur de Ut.
Done, si t est le plus grand diviseur commun ^ r et s, Ut sera le plus grand diviseur commun k Ur et u„ ce qui 6tait k d^montrer. II s'ensuit que, si r est premier relativement a s, w, et «, auront Mj pour leur plus grand diviseur commun.
59] Preuve Mementaire du theoreme de Dirichlet 621
Je vais faire I'application de ce principe : (A) aux progressions arithm^- tiques a la raison 8, (B) a la raison 12.
A. 1. Cos de 8a; + 3. — Ecrivons
M, = 1, Wa = 2zti^ + 1 = 3, «3 = 2itj- + 1 = 19, ....
On demontre facilement que tout u est de la forme 8m + 3, et Ton salt que les facteurs premiers de tout u sont de la forme 8n + 1 ou 8n + 3.
Cons^quemment, tout u contiendra au moins un facteur de la forme 8m+3, et tout terme de la progression infinie
«3, Ms, Uj, Mil. Mi3. •••
contiendra un facteur premier de la forme voulue.
De plus, en vertu du second lemme, tons ces facteurs seront distincts I'uu de I'autre; car sinon m, et u,, oh r est premier k s, auraient un facteur commun autre que if,.
On pourrait prendre une s6ne plus generale en ecrivant m, egal a un produit d'un nombre quelconque de nombres premiers dont aucun n'est de la forme 8m + 3, tellement combinfe que «j = 1 [mod 8] ; le rdsultat restera acquis que chaque terme de la progression des u contiendra un facteur premier de la forme 8x + 3, et que tous ces facteurs seront distincts entre eux.
A. 2. Cos de 8x+ 7. — Ecrivons
M, = l, u^=2{u, + iy-l = 7, M3 = 2(m,+ 1)»- 1 = 127
Tout M = 7 [mod 8] : chaque diviseur premier de tout u sera de la forme 8m + 1 ou 8ni + 7. Done il entrera dans chaque terme de la progression
M», «s, «», M?. ••• un facteur de la forme 8a: + 7, et de plus, en vertu du second lemme (puisque /O = 1), tous ces facteurs seront distincts.
A. 3. Cos de 8x + 1. — Ecrivons
M,= l, Uj = M]*+l = 2, M3=«,*+l = l7, ....
Tous les facteurs de chaque u, a 1' exception de 2, seront de la forme Scc+l, et, en vertu du second lemme «,, m„ u,, itu, u^, Uu, seront premiers entre eux.
A. 4. Cos de 8x+ 5. — Ecrivons
itj=l, itj=V+l = 2, Ms = «•/+! =5, M4 = u,»+1 = 26, M5=u«»+ 1 = 677, .... Chaque u^+i sera de la forme 8to + 5, et chaque diviseur premier sera ou de la forme 8n + 1 ou 8m + 5, de sorte qu'il s'en trouvera un au moins de la forme 8x + 5. Done par le second lemme la progression
M.. Mi, W7, Mil, Mi3, ...
contiendra un nombre infini de nombres premiers distincts de cette forme.
622 Preuve Mimenlaire [59
B. 1. Gas de 12a;+5. — On d^montre facilement par induction que chaque terme de rang pair de la progression pr^c^dente au delk du second sera de la forme 2 (24/i + 13), et chaque terme de rang impair au dela du premier de la forme 24m + 6.
Les diviseurs premiers de chaque u seront de Tune ou I'autre des six formes 24a; + 1, 5, 19, 17, 13, 21.
Supposons qu'il n'existe aucun facteur premier de la forme 24a; + 17 ni de la forme 24a;+5. Alors les r^sidus des facteurs (par rapport k 12) appartien- dront au groupe 1, 9, 13, 21. Mais on voit facilement que ce groupe est un groupe fermd : car toutes ces combinaisons binaires ne font que reproduire ces memes nombres.
Cons^quemment, tout terme de rang impair contiendra n^cessairement un facteur ou de la forme 24ia; + 5 ou de la forme 24a; + 17, et ainsi, en vertu du second lemme, on voit que la progression dejk ecrite contiendra ua nombre infini de nombres premiers de la forme 127i + 5.
B. 2. Gas de 12a; + 7. — Ecrivons
Ml = 7, t^ = It," — Wi + 1 = 43, Ms = U2^ — Uj + 1 = 1807, ....
Les diviseurs premiers de chaque u seront de la forme 12n + 1 ou 12n+ 7 et u lui-meme de la forme 12m + 7. Done, en vertu du premier lemme, la suite Ml, M,, u,, Ut, ... contiendra un nombre infini de nombres premiers de la forme 12a; + 7*.
B. 3. Gas de 12a; + 11. — Ecrivons
Ml = — 1, Ui = 3Mi^ —1 = 2, Ms = Su^ —1 = 11, u, = 3u,'-l = S62
Tous les M de rang impair seront de la forme 12wi + ll, de sorte que leurs diviseurs premiers dtant, ou de la forme 12a; + 1 ou 12a; + 11, il y aura un nombre infini de nombres premiers distincts contenus dans les termes de la progression
Ms, Mj, Uy, Ml]
B. 4. Gas de 12a; + 1. — Ecrivons
U^ = 6*-d^+l, Ut=llT*-Ui^+l, Ms = Mj« - Mj' + 1, ....
Chaque m, selon la loi cyclotomique, ne contiendra que des facteurs de la forme 12a; + 1 et, en vertu du premier lemme, Mi, Mj, U3, «<, m,, ... seront tous
* Par un proc6d6 analogue a celui que nous avons appliqu^ & la progression dont nous nous sommea servia dans lea oas A. 4 et B. 1 ; on pent d^montrer avec I'aide de la progression 7, 43, 1807,..., donn^e plus haut, que le nombre de nombres premiers dans la doable progression arithm^tique k raison 30,
7, 13, 37, 43, 67, 73
contient on nombre infini de nombres premiers : a plus forte raison cette conclusion s'applique k la doable progression k raison 5
2, 3, 7, 8, 12, 13
f
59] du theoreme de Dirichlet 623
premiers entre eux: done cette progression contiendra un nombre infini de facteurs de la forme 12a; + 1.
L'application du principe general ^noned au commencement n'est nulle-
ment astreinte aux progressions de la forme <^6, (f><f)0, (j}<f><f>d C'est ce
que j'ai montre au Congres scientifique d'Oran.
Au Congres scientifique d'Oran nous avons indique :
(1) Une demonstration instantanee du theorfeme de Diiichlet pour le cas Ax + \, quel que soit A, en nous servant des fonctions cyclotomiques de I'espece ordinaire en u, en prenant pour les indices successifs A, 2A, SA, ... et en donnant a « une valeur quelconque. Ces fonctions cyclotomiques sont les facteurs irreductibles des fermatiens. Par exemple, en prenant 3 pour la base des fonctions cyclotomiques, et en otant de chaque cyclotome dont I'indice est une puissance de 2 le facteur singulier 2, on obtient la progression 2, 2, 13, 5, 121, 7, 1093, ..., dont tons les termes, en omettant le second, sont premiers entre eux, et ou le terme h, I'indice i (le second excepte) ne contient d'autres facteurs premiers que ceux de la forme ix + 1. Cons^quemment, en se bornant aux t*™^, (2i)*°"', (Si)*™®, (4i)*'"'', ... termes, et en decomposant chacun de ces termes dans un produit de facteurs premiers distincts, la totalite de ces facteurs fournira un nombre infini de nombres premiers de la forme ix-\-\;
(2) Une demonstration beaucoup plus cachee pour le cas Ax — 1, quand A est une puissance d'un nombre premier, au moyen des fonctions cyclo- tomiques qui se deduisent des fonctions dont nous avons parle en les divisant par une puissance convenable de u, en exprimant le quotient comme fonction
de w + - , disons v, et en attribuant k v une valeur constante dont la forme par u
rapport au module A ou bien a un multiple de A (capable de grandir ind^- finiment) depend de la forme du nombre premier dont A est une puissance, par rapport au module 8.
Plus r^cemment, nous avons dtendu la mfeme demonstration aux cas ou A est une combinaison de puissances de 2, 3, 5, 7, de sorte qu'il nous parait peu douteux que les propri^tes cyclotomiques donnent le moyen de prouver le theoreme de Dirichlet aussi bien pour le cas de 4a;— 1, comme pour le cas de Ax+1, quelle que soit la forme de A. II nous semble done qu'il y a quelqiie lieu d'esp^rer que le principe gdn^ral (qu'on pent nommer constructif ou cosmoth^tique) pent servir k donner une demonstration pour le cas le plus general du theoreme de Dirichlet. En addition k la m^thode ici donn^e et celle fournie par la theorie cyclotomique pour obtenir des progressions infinies de nombres relativement premiers entre eux, on pent se servir comme troisieme m^thode des cumulanla (les numerateurs et ddnominateurs de fractions
A
624 Preuve eUmentaire du tMoreme de Dirichlet [69
continues) et sans doute d'une infinite d'autres espfeces de fonctions. Toute la difficult^ consiste k trouver la forme de progression convenable a chaque cas donn4
En ce qui regarde la theorie g^nerale des diviseurs des fonctions cyclo- tomiques de toute espece, nous renvoyons a notre article, intitule : Excursus A : On the divisws of cyclotomic functions [Vol. in. of this Reprint, p. 317] ; et en ce qui regarde la propri^t^ des nombres cyclotomiques de la premiere et seconde espfece, privds de leur facteur singulier, d'etre relativement premiers entre eux, k un article paru dans le journal Nature [see pp. 591, 625 of this Volume] du mois de mars de cette annee*.
* Le cas de 12z + 5 (page [622] de la Note pr^c^ente) est mal expliqu^. Afin de d^montrer le th^or^me de Dirichlet pour ce cas il sufilt de remarquer que chaque terme de rang impair (apris le premier) dans la progression 1, 2, 5, 26, 677, ... est de la forme 12m 4- 5, et chacun de ses facteurs pr<mt>r« de la forme 4a: + 1, c'est-a-dire de la forme 12x + l ou 12x + 5 ; consequemment il contiendra aa moins nn facteur premier de la forme 12x + 5.
I
60.
ON THE DIVISORS OF THE SUM OF A GEOMETRICAL SERIES WHOSE FIRST TERM IS UNITY AND COMMON RATIO ANY POSITIVE OR NEGATIVE INTEGER.
[Nature, xxxvii. (1888), pp. 417, 418.]
" Nein ! Wir sind Dichter*."
—Kronecker in Berlin.
rP — l. A REDUCED Fermatianf, r- , is obviously only another name for the
sum of a geometrical series whose first term is unity and common ratio an integer, r.
If /> is a prime number, it is easily seen that the above reduced Fermatian will not be divisible by p, unless r— 1 is so, in which case (unless p is 2) it will be divisible by p, but not by p^.
This is the theorem which I meant to express [p. 591, above] in the footnote to the second column of this journal for December 15, 1887, p. 153, but by an oversight, committed in the act of committing the idea to paper, the expression there given to it is en-oneous.
Following up this simple and almost self-evident theorem, I have been led to a theory of the divisors of a reduced Fermatian, and consequently of the Fermatian itself, which very far transcends in completeness the condition
* Such were the pregnant words recently uttered by the youngest of the splendid triumvirate of Berlin, when challenged to declare if he still held the opinion advanced in his early inaugural thesis (to the effect that mathematics consists exclusively in the setting out of self-evident truths, — in fact, amounts to no more than showing that two and two make four), and maintained unflinchingly by him in the face of the elegant raillery of the late M. Duhamel at a dinner in Paris, where his interrogator — the writer of these lines^was present. This doctoral thesis ought to be capable of being found in the archives of the University (I believe) of Breslau.
t The word Fermatian, formed in analogy with the words Hessian, Jacobian, Pfaffian, Bezoutiant, Cayleyan, is derived from the name of Fermat, to whom it owes its existence among recognized algebraical forms.
8. IV. 40
626 On the Divisors of the Sum of [60
in which the subject was left by Euler (see Legendre's Theory of Numbers, 3rd edition, vol. i. chap. 2, § 5, pp. 223 — 27, of Maser's literal translation, Leipzig, 1886)*, and must, I think, in many particulars be here stated for the first time. This theory was called for to overcome certain difficulties which beset my phantom-chase in the chimerical region haunted by those doubtful or supposititious entities called odd perfect numbers. Whoever shall succeed in demonstrating their absolute non- existence will have solved a problem of the ages comparable in difficulty to that which previously to the labours of Hermite and Lindemann (whom I am wont to call the Vanquisher of PI, a prouder title in my eyes than if he had been the conqueror at Solferino or Sadowa) environed the subject of the quadrature of the circle. Lambert had proved that the Ludolphianf number could not be a fraction nor the square root of a fraction. Lindemann within the last few years, standing on the shoulders of Hermite, has succeeded in showing that it cannot be the root of any algebraical equation with rational coefficients (see Weierstrass' abridgment of Lindemann's method, Sitzungs- berichte der A. D. W. Berlin, Dec. 3. 1885).
It had already been shown by M. Servais (Mathesis, Liege, October 1887), that no one-fold integer or two-fold odd integer could be a perfect number, of which the proof is extremely simple. The proof for three-fold and four- fold numbers will be seen in articles of mine in the course of publication in the Comptes Rendus [above, pp. 604 — 619], and I have been able also to extend the proof to five-fold numbers. I have also proved that no odd number not divisible by 3 containing less than eight elements can be a perfect number, and see my way to extending the proof to the case of nine elements.
How little had previously been done in this direction is obvious from the fact that, in the paper by M. Servais referred to, the non-existence of three- fold perfect numbers is still considered as problematical ; for it contains a " Theorem " that if such form of perfect number exists it must be divisible by fifteen : the ascertained fact, as we must know, being that this hypothetical
* I find, not without surprise, that some of the theorems here produced, including the one contained in the corrected footnote, have been previously stated by myself in a portion of a paper "On certain Ternary Cubic Form Equations," entitled "Excursus A — On the divisors of Cyclotomic Functions " [Vol. iii. of this Reprint, p. 317] the contents and almost the existence of which I had forgotten : but the mode of presentation of the theory is different, and I think clearer and more compact here than in the preceding paper ; the concluding theorem (which is the important one for the theory of perfect numbers) and the propositions immediately leading up to it in this, are undoubtedly not contained in the previous paper.
I need hardly add that the term cyclotomic function is employed to designate the core or primitive factor of a Ferraatian, because the resolution into factors of such function, whose index is a given number, is virtually the same problem as to divide a circle into that number of equal parts.
+ So the Germans wisely name jr, after Ludolph van Ceulen, best known to us by his second name, as the calculator of r up to thirty-six places of decimals.
60] a Geometrical Series, etc. 627
theorem is the first step in the reductio ad absurdum proof of the non- existence of perfect numbers of this sort (see Nature, December 15, 1887, p. 153, written before I knew of M. Servais' paper, and recent numbers of the Comptes Rendus).
But after this digression it is time to return to the subject of the numerical divisors of a reduced Fermatian.
We know that it can be separated algebraically into as many irreducible functions as there are divisors in the index (unity not counting as a divisor, but a number being counted as a divisor of itself), so that if the components of the index be a", hP, c'', ...the number of such functions augmented by unity is
(a+l)(;S+l)(7 + l)....
All but one of these algebraical divisors, with the exception of a single one, will also be a divisor of some other reduced Fermatian with a lower index : that one, the core so to say (or, as it is more commonly called, the irreducible primitive factor), I call a cyclotomic function of the base, or, taken absolutely, a cyclotome whose index is the index of the Fermatian in which it is contained.
It is obvious that the whole infinite number of such cyclotomes form a single infinite complex. Now it is of high importance in the inquiry into the existability of perfect numbers to ascertain under what circumstances the divisors of the same reduced Fermatian, that is, cyclotomes of different indices to the same base, can have any, and what, numerical factor in common. For this purpose I distinguish such divisors into superior or external and inferior or internal divisors, the former being greater, and the latter less, than the index.
As regards the superior divisors, the rule is that any one such cannot be other than a unilinear function of the index (I call kx+\ a. unilinear function of X, and k the unilinear coefficient) and that a prime number which is a uni- linear function of the index will be a divisor of the cyclotome when the base in regard to the index as modulus is congruous to a power of an integer whose exponent is equal to the unilinear coefficient.
As regards the inferior divisors, the case stands thus. If the index is a prime, or the power of a prime, such index will be itself a divisor. If the index is not a prime, or power of a prime, then the only possible internal divisor is the largest element contained in the index, and such element will not be a divisor unless it is a unilinear function of the product of the highest powers of all the other elements contained in the index.
It must be understood that such internal divisor in either case only appears in the first power ; its square cannot be a divisor of the cyclotome.
40—2
628 On the Divisors of the Sum of [60
It is easy to prove the important theorem that no two cyclotomes to the same base can have any the same external divisor*.
We thus arrive at a result of great importance for the investigation into the existence or otherwise of perfect odd numbers, which (it being borne in mind that in this theorem the divisors of a number include the number itself, but not unity) may be expressed as follows :
The sum of a geometrical series whose first term is unity and common ratio anil positive or negative integer other than 4-1 or — 1 m,ust contain at least as many distinct prime divisors as the number of its terms contains divisors of all kinds; except when the common ratio is —2 or 2, and the number of terms is
• The proof of this valuable theorem is extremely simple. It rests on the following principles :
(1) That any number which is a common measure to two cyclotomes to the same base must divide the Fermatian to that base whose index is their greatest common measure. This theorem needs only to be stated for the proof to become apparent.
(2) That any cyclotome is contained in the quotient of a Fermatian of the same index by another Fermatian whose index is an aliquot part of the former one. The truth of this will become apparent on considering the form of the linear factors of a cyclotome.
Suppose now that any prime number, k, is a common measure to two cyclotomes whose indices are PQ, PR respectively, where Q is prime to B, and whose common base is 6. Then k
must measure 0^-1 and also —^ — - ; it will therefore measure Q, and similarly it will measure
B; therefore i = l [unless (3 = 1 or JJ = 1 ; for suppose (? = 1, then ,,— ,- is unity, and no longer
contains the core of G'*- 1]. Hence k being contained in R can only be an internal factor to one of the cyclotomes (namely, the one whose index is the greater of the two). (See footnote at end.) The other theorem preceding this one in the text, and already given in the " Excursus," may be proved as follows :
Let k, any non-unilinear function of P, the index of a cyclotome Xi be a divisor thereto.
p Then, by Euler's law, there exists some number, ii, such that k divides a^ - 1, but the cyclotome
is contained algebraically in —^ ; hence k must be contained in n, and therefore in P. Also,
- x'-l -
k will be a divisor of a* - 1 and of -^ , which contain xi^ -1 and x respectively ; consequently,
i*-l
I'-l if k is odd, fc' will not be a divisor of -f — , and a fortiori not of x- (A proof may easily be
a*-l given applicable to the case of fc = 2.)
Again, let P=Qk*, where Q does not contain k. Then, by Fermat's theorem, x**=.r [mod. k] and therefore k divides xO-1; but it is prime to Q. Hence, by what has been shown, k must be an external divisor of this function, and consequently a unilinear function of Q. Thus, it is seen that a cyclotome can have only one internal divisor, for this divisor, as has been shown, must be an element of the index, and a unilinear function of the product of the highest powers of all the other elements which are contained in the index.
For an extension of this law to "cyclotomes of the second order and conjugate species," see the "Excursus," where I find the words extrinsic and intrimic are used instead of external and internal.
60] a Geometrical Series, etc. 629
even in the first case, and 6 or a multiple of 6 in the other, in which cases the number ofprim£ divisors may be one less than in the general case*.
In the theory of odd perfect numbers, the fact that, in every geometrical series which has to be considered, the common ratio (which is an element of the supposed perfect number) is necessarily odd prevents the exceptional case from ever arising.
The establishment of these laws concerning the divisors and mutual relations of cyclotomes, so far as they are new, has taken its origin in the felt necessity of proving a purely negative and seemingly barren theorem, namely the non-existence of certain classes of those probably altogether imaginary entities called odd perfect numbers : the moral is obvious, that every genuine effort to arrive at a secure basis even of a negative proposition, whether the object of the pursuit is attained or not, and however unimportant such truth, if it were established, may appear in itself, is not to be regarded as a mere gymnastic effort of the intellect, but is almost certain to bring about the discovery of solid and positive knowledge that might otherwise have remained hidden t-
* A reduced Fermatian obviously may be resolved into as many cyclotomes, less one, as its index contains divisors (unity and the number itself as usual counting among the divisors). But, barring the internal divisors, all these cyclotomes to a given base have been proved to be prime to one another, and, consequently, there must be at least as many distinct prime divisors as there are cyclotomes, except in the very special case where the base and index are such that one at least of the cyclotomes becomes equal to its internal divisor or to unity. It may easily be shown that this case only happens when the base is - 2 and the index any even number, or when the base is +2 and the index divisible by 6; and that in either of these cases there is only a single unit lost in the inferior limit to the number of the elements in the reduced Fermatian.
t Since receiving the revise, I have noticed that it is easy to prove that the algebraical resultant of two cyclotomes to the same base is unity, except when their indices are respectively of the forms Q(/cQ + l)* and Q{kQ + lY, where {kQ+1) is a prime'number, and Q any number (unity no( excluded), in which case the resultant is kQ + 1. This theorem supplies the raison TaUonnie of the proposition proved otherwise in the first part of the long footnote.
61.
NOTE ON CERTAIN DIFFERENCE EQUATIONS WHICH POSSESS AN UNIQUE INTEGRAL.
[Messenger of Mathematics, xviii. (1888-9), pp. 113 — 122,]
For greater simplicity suppose in what follows that a difference equation is expressed in terms of the arguments
I shall call Ux+i the highest and Ux the lowest argument respectively, or collectively the extreme or principal arguments, and the degrees in which they enter into the equation the upper and lower or extreme or principal degrees. It is these partial degrees rather than the total degree of the entire equation which determine the essential character of the solution.
If TO is the upper degree and Wq, Mi, ... «,-_i be given it is obvious that for any value of x higher than (i — 1), Ux will have m*~'+' values, and conse- quently in general there will be an infinite number of integrals whether complete or of a given order of deficiency (the deficiency being estimated by the number of relations connecting the initial values «„, Mj, ... m,_i); but it may be, and is in some cases, possible to assign an integral which shall' have m*~'+' values, and in such case there can exist no other ; such an integral may be called an unique or exhaustive one, and the equations which possess such integrals may be termed uni-solutional.
As the simplest example of such, suppose
where m and n are integers.
(-Y If we write Ux = a\"'^
( (-^T
we have u^+j = \a )
or w"»«+, = M«».
(-Y Here m^ = o^"'' is the one and sole complete integral of the equation;
for it possesses to* values so that there can be no other integrals whatever.
61] Note on Certain Difference Equations 631
Let us now seek to form diflference uni-sohitional equations of the 2nd order.
To this end let u^=G{of- 0^), where ayS = 1.
Then calling of = P a.nd ^ = Q, PQ = 1,
M, = C(P-Q),
M,+,= C'(P»-(2'),
«.+« = 0(2^-0*).
Hence '^ = P + Q, '^ = P»+ Q^= (P + Q)^- 2,
and ^^=(^Y-2.
Hence the equation
Ux'Ux+i - M'aH-I + 2Ma;'M,+i = 0
has for its complete integral Ma, = (7 (a*^ — a~^), and there can be no other because when u„, u, are given Ux is absolutely determined.
But furthermore we may invert the above equation by interchanging m, and Ux+i, which gives the equation
(Ux + 2Ux+i) u'x+, - Wx+i = 0, of which the solution will obviously he Ux=C(P — p\ , where P = a''^ . Suppose Mo) J*i to be given ; then
c(«-i)=«. 0(a»-A) = «»
andcalling ^=2r, a^+\ = 2r, a*--i= 2 V(r'-l).
ih a* a*
C =
Hence
"' = w(^ f ^^ -^ ^^^ - '>5"^'"' - {'•- ^^^° - '^^^*''"'^'
has exactly 2*~' values, for the change of VC'"'— 1) into - VC^ - 1) changes simultaneously the signs of the numerator and denominator of this fraction. But by the general principle Ux ought to have 2*-' values in terms of Mo, Wi- Hence the above integral is exhaustive.
Suppose now we were to write
Ux=C{af + ^) with a/3=l;
632 Note on Certain Difference Equations [61
for brevity sake call u^ =/, m,+, = g, ««+» = K then
C{P* + <^) = h, PQ=\. Hence /'=Cg + 2C,
g'=Ch + 2C,
J - g-h • g-k • or fcf- 2/'gh +f*h' = 2/* - 3/y + g* -fgh + g>h.
or /%'-(f+fg)h-g* + if'g'-2/* = 0.
of which the correlative equation is
- 2u*^+i + (4u'^, - «ir+,U, + Mx') U'«+, - U'^iMj, - M*x+, = 0.
A complete solution of the former of these will therefore be
and of the latter u^= C (o'*>' + fi^^^"),
but neither of these will be an exhaustive solution, for in the one the most general value of u^ ought to be a 2*~'-valued functfon and in the latter a 4^'-valued function, whereas the actual value is only one-valued in the one case and 2*~'- valued in the other. Suppose again we write
Ug = C(pf — yS^), where ay3 = 1, as before, say Ut = C(P - Q), where PQ=1.
Then with the same notation as before
C(P-Q)=f, C(P'-Q') = g, G(P»-Qr)'=h.
fh-Sfg=g*-Sg'f. h-^g _g'
61] which possess an Unique Integral 633
Whence it follows that the intesrals of
= 0,
t*n., - 3«, M,
and of ^^- 3"x^.-"«+i ^ Q
"•x+i 3m^, - Ux
are respectively «,= C(a^— ar**),
and • M,= C(a<*>'- a- <*>').
with the understanding that o~ * . a* = 1.
These integrals are evidently exhaustive.
By writing VC— !)«, —'J{— l)or' for a, or' respectively, /, g, h become increased in the ratio of V(- 1), - V(— 1), V(— IX respectively. Hence the equations
H J. SlJ . ««
"x+< + 3it,H.i uVi .
and !^_?'^±!±i^' = 0
have for their solutions
ti,= C(a^ + a-^ and u, = (7(o<*>' + a" <*>*).
Hitherto we have been dealing with homogeneotia uni-solutional equations. It is easy, however, to form non-homogeneous ones by an obvious process. For, if we write
«»= a,"** + a,"* + ... -I-Oj"^ (m being an integer), by eliminating between
/. = Sa, /. = 2a- /, = 2a"', . . ./ = 2a"\ we shall obtain a relation between the/'s of the first degree vafi and of the degree m' in/,, correspondiug to which there will be a difference equation of the tth order in which the upper extreme degree is unity and the lower one m', of which the integral will be the value of u, above written, and by inter- changing u„ Ua+i, ... «,+,- respectively with u,+,-, u^+t-K ••■ "«. another in which the lower degree is unity and the upper one m', of which the integral will be
(1)* /!)' (If
each of which equations will evidently be uni-solutional.
Or, again, if instead of the as being independent we make their product equal to unity we shall obtain uni-solutional equations of the (t— l)th instead of the tth order.
Thus, for example, let
Ug — a^ + b^ + tf' with the condition ahc = 1.
634 Note on Certain Difference Equations [61
Then writing «,=/, M,+, = 5f, ««+, = A,
/=A+B + C, g='A' + B' + G', h :>' A* + B* + C*,
f'-g = 2(AB+AG + BC),
2(f-h) = 4> (A'B' + A'C + B'C)
=(r-ffy-s/.
Hence we obtain the uni-solutional equations
2w,+s - w'!c+, - 2u,+, M/ + u^* - 8u, = 0,
mVs - 2w,+,m\+j - SMj+j - M»a+, + 2ua = 0,
of which the integrals are known and are exhaustive.
We may in a similar manner obtain uni-solutional simultaneous difference equations.
Thus let u^ = G {of - ^), v^ =C'{(i^ + ^),
and call Ux, Ux+i, ««+3 as before /, g, h,
and Vx, Vx+i, Vx+i I, m, n.
Then ^ = P^ + P(2 + Q». ^ = p2_pQ+Q.^
g ^ m
„ h n . (g my
Hence = i l?- T >
g m * V/ I J
--f- = 2(P« + Q«) g m
= 2 (P» + Q') {{P' + Qr-y - 3P^Q»}
=*f7-i){(^T)'-K'7-?n
Hence ^ = i (-/s + 3-?, . y + 9| . ^ -3 -^^) .
Obviously, when «„, Mj; t)o, Vi are given, each Ux and Vx deduced from the above system of equations has only one value, so that their exhaustive integrals will be
w, = C {of - ^\ Vx = C (0^ + ^).
\
i
61] which possess an Unique Integral 636
The related system found by interchanging /with h and I with n will be
g~^{~ h^'^'^h'- n^^h- n' "^ n'> J '
When /, g; I, m are given the system t , - may be found by solving an
equation of the 9th degree. Hence, when w„, u^; Vo, Vi are given, u^, y, will have 9; it,, v,, 81, and in general u^, Vx will have 3^'^" values which will correspond to the 3*~' . 3*~' values of Ux, Vx-
The apparent number of values of each of these is (3*)^, which, however, must be reducible to 3*~' . 3*~' when expressed in terms of the two initial values of u and of v, similarly to what was noticed at the outset on the reduction of the apparent multiplicity 2* to a multiplicity 2*~^.
In fact, we write
«. = C(a-y8), M, = (7(a*-/3*); v, = C'{a + ^), 7^, = C (a' + <9'),
|
a^^* |
1 /M„ V„\ |
|
|
a^ + ^'' |
=*£;-»• |
|
|
a*-^^ = |
VKt- |
-3}. |
|
«i+)8* = |
VK^:- |
foV |
|
Ml |
r" |
V, |
636 Note on Certain Difference Equations
and thus for the final values of Uz and Vx, we find
[61
-7i;(|ii)j''lM'("'-r;)lVKt-3t]"""'
each of which is una6Fected by a change in the signs of the square roots, so that Mj, and Vx are seen to be 3*~'-valued functions, and («,, Vx) a 9*~'-valued system, as should be the case for an exhaustive solution of the last written difference equations.
Let us tentatively go a step further in the same direction and suppose that we are given
ux =C{of- ^), Vx = G'(af + ^), and use/, g, h; I, m, n in the same way as before, and furthermore, write
we shall find
L = A* + A'B' + B\ N = A« + A"'B"' + B^, M=AB(A'' + B'), F = A'B'(A"> + B^).
(where A = of and B = fi'").
Let A"- + B' = \, AB = /i.
Then L = X'-fi\ M=\fj,,
and it will be seen that
iV= (X" - 5X>» + 5X^i*y - fi^", P = \'fj.'-5Xfji'{X>-fju').
For A' + B' = X' - SXfi',
and consequently
X^ = A"> + B'o + oA'B'{A' + B')+ 10 A*B* {A" + B') = A"> + B"> + 5/4,» {X' - SXfi?) + lOXfi*, that is A'o + Ro = X'- 5XV' + 5X/i*.
61] which possess an Unique Integral 637
The above values of N and P (remembering that AB = fi) are found by substituting the expression just obtained for A^" + B^" in
P = A'B^ (A"> + B">). From P = X>= - oXfj.' (X= - /j.%
(remembering that Xfi = M, X- — /i- = L), we obtain
P=M''-r,LMfl\
„ M'-P ,j?^
Hence , , ... = ^
L \ n
M fji \ ■
From these equations we obtain by elimination
[mI
+ {2M* + UDM^ + oL*) ^ + M* (M* - lOL'M' + 5L*) = 0. (1)
Similarly by an elimination into the details of which it is unnecessary to enter we obtain
SLMP + (L' + M')N=L (D - 2M') {D - M^f, (2)
which gives a linear relation between N and P.
Equations (1) and (2) form a non-uni-solutional system of which (as also of its inverse) we are in possession of one complete integral, and I have some grounds for suspecting that it may be possible to obtain from this a second (so-called indirect) integral, but am unable for the present to pursue the subject further.
The preceding investigation originated in my attention happening to be called to Vieta's well known theorem for approximating to the Archimedean constant (tt) by means of an indefinite product of cosines of continually bisected angles. The implied connection of ideas will become apparent when one considers that any one of such cosines may be expressed as a sura of two binary exponentials with \ for the first index, and that thus Vieta's theorem (although presumably obtained by him as a very simple consequence of the method of exhaustions) in its essence depends on the integrability of a uni- solutional difference equation of the 2nd order of the form treated of at the outset of this paper.
62.
SUR LA REDUCTION BIORTHOGONALE D'UNE FORME LINEO-LINEAIRE A SA FORME CANONIQUE.
[Gomptes Rendus, CVill. (1889), pp. 651—653.]
SoiT F une fonction lin^o-lin^aire des deux series de lettres
alors F contiendra w' termes. En assujettissant les x et les ^ respective- ment h. deux substitutions orthogonales ind^pendantes, on introduit dans la transformee n^ — n quantit^s arbitraires, de sorte que, en leur donnant des valeurs convenables, on doit pouvoir faire disparaitre ce nombre de termes en ne conservant que les n paires dont les arguments seront (par exemple)
On peut nommer les multiples de ces arguments les inultiplicateurs canoniques ; je vais donner la rfegle pour les determiner, et en meme temps pour trouver les deux substitutions orthogonales simultanees qui amenent la forme canonique. La marche k suivre sera parfaitement analogue k celle qui s'applique a la rMuction d'une forme quadrique a n lettres a sa forme canonique au moyen d'une seule substitution orthogonale ; mais on remarquera, a priori, une distinction essentielle entre les deux questions. Pour le cas d'une seule quadrique, les multiplicateurs canoniques sont absolument d^ terminus; mais, pour le cas actuel, il est Evident que chacun de ces multiplicateurs peut changer son signe, de sorte que ce sont les carr^s de ces multiplicateurs qui doivent se presenter dans le r^sultat.
II sera utile de rappeler quelques faits elementaires sur les matrices. Le carre d'une matrice est la matrice qui se produit par la multiplication des lignes par les colonnes ; il sera une matrice non sym^trique dont les racines latentes seront les carr^s des racines latentes d'une matrice donn^e. Au contraire, le produit d'une matrice par son transverse donnera (selon I'ordre de la multiplication) lieu k deux matrices sym^triques qu'on obtient par la multiplication des lignes par des lignes ou bien par celle des colonnes par
62J Sur la reduction hiorthogotiale (Tune forme lineo-lineaire 639
les colonnes; ces matrices seront distinctes, mais poss^deront les memes racines latentes, c'est-a-dire en aflfectant tous les termes dans la diagonale de sym^trie de I'un ou de I'autre avec la meme addition, soit — \, le determinant d'une matrice ainsi affect^e sera le mSme pour I'un comme pour I'autre*.
En diflferentiant F par rapport aux x et aux ^, on obtient deux matrices, dont I'une sera la transverse de I'autre, que je nommerai les matrices deter- minatives. Avec I'aide de ces matrices on obtient une solution complete du probleme voulu.
(1) Pour determiner les multiplicateurs canoniques :
Je dis que les racines latentes de leur produit seront les carrfe des multiplicateurs canoniques.
II peut arriver qu'un de ces multiplicateurs soit z^ro; alors le dernier terme de I'^quation aux racines latentes, qui n'est autre chose que le carre du determinant d'une matrice determinative, s'evanouit ; et Ton voit que le cas de la disparition d'un des n termes dans la r^duite canonique est indiqu6 par revanouissement du determinant de la matrice determinative.
(2) Pour trouver les deux substitutions orthogonales canoniques:
Prenons une des deux matrices sym^triques affectees de — X dans chaque terme de sa diagonale; en supprimant une quelconque de ses lignes, les n premiers raineurs de la matrice diminuee qui restent divises chacun par la racine carree de la somme de leurs carres (fonctions de \), en donnant k \ successivement les valeurs des n racines latentes, fourniront les n? termes d'une des substitutions orthogonales, et de meme on obtient I'autre sub- stitution orthogonale en agissant semblablement pas cL pas sur I'autre matrice affectee : ainsi le problfeme de la reduction voulue est completement resolu.
Prenons, par exemple,
F — 8a;f — xtj — ^y^ -f- Tyr).
* Tontes ces racines latentes seront non seulement r^elles (comme elles doivent I'fltre k cause de la forme s;m£trique de la matrice), mais aussi positives ; car, en substituant \ ^ - \, les eoefficients de I'e'qnation latente (en commengant avec le dernier) sont, respectivement, le carr^ du determinant complet, la somme des carres des premiers mineurs, des seconds mineurs, etc., de la matrice determinative (le premier coefficient etant I'anite et le second la somme des carres des coefficients de la forme bilin^aire). Chacune de ce.s sommes sera un invariant biorthogonal, et le determinant de la matrice determinative lui-m£me sera un invariant gauche de la forme bilineaire.
Ajoutons que leg deax matrices qui sont les carres cancbiens de cette matrice, envisagees comme discriminants, fourniront deux quadriqucs (dont chacune contiendra un seul des deux qrst^mes donnas de lettres) qui seront des covariants orthogonaux simultanes de la fonction bilin^re donnee.
|
8; - |
4 |
|
|
-1; |
7' |
|
|
80-\: |
1 |
36 |
|
-36 ; |
; 50 |
-X |
640 Sur la reduction biorthogonale (Tune forme lin^o-lin^aire [62
(1) Pour trouver lea multiplicateurs canoniques :
On prend la matrice determinative dans ses deux formes
8; -1
-4; 7'
dont les produits afifect^s seront
65 -X; -39 -39 ; 65-X'
Ainsi, en se servant de I'un ou de I'autre, on obtient
X= - 130X + 2704 = 0,
dont les racines sont 26 et 104, de sorte que \/26 et 2 \/26 seront les multi- plicateurs canoniques.
(2) Pour trouver les substitutions, on assigne ses deux valeurs a
39 : 65 - X, c'est-^-dire 39 : 39 et 39 : - 39
36 : 80 - X, c'est-k-dire 36 : 54 et 36 : - 24. Ainsi Ton aura, pour les deux matrices de substitution,
J_. J_ _^. _L
V2'' */2 V13' \/13
et
V2' V2 V13' V13
et, en eflfet, on v^rifie facilement que
= 8x^ — XT] — iy^ + 7yr).
Si Ton donne les deux matrices symetriques ayant les memes racines latentes qui doivent representor respectivenient les deux produits cauchiens d'une matrice de I'ordre n par elle-meme, on verra facilement que le probleme de trouver cette dernifere matrice a 6t^ virtuellement r&olu plus haut, et que, comme le probleme de trouver la veritable racine carree d'une seule matrice g^ni^rale donnee, il admet 2" solutions.
63.
SUR LA CORRESPONDANCE COMPLETE ENTRE LES FRAC- TIONS CONTINUES QUI EXPRIMENT LES DEUX RACINES D'UNE EQUATION QUADRATIQUE DONT LES COEFFI- CIENTS SONT DES NOMBRES RATIONNELS.
[Comptes Rendus, cviii. (1889), pp. 1037—1041.]
Si «,■ = X,«i_i -f Mf_3 et M_i = 0, Mo=l, on peut appeler «< un cumulant dont la succession \, \^, X3, ..., X^ est le type; d^signons-le par t.
Alors on peut representer
Par '< la succession X,, X,, ..., X,-
"'^T I „ X, , X,, Xj, ... , Xi_i
'8J t „ X2, Xj, ..., X<_i.
De plus, on peut representer par 6t la reunion du t}'pe 6 suivi par le type t ; par dOt ce que deviant dt quand on intercale un z^ro entre la succes- sion 6 et la succession t; par d(Oty la fcuccession B suivie par la succession 0* rdp^tee i fois ; et par t (06)' t ce que devient tr quand on intercale 0^ t fois entre le t et le t.
T ^tant un type quelconque, on peut designer par [T] le cumulant dont T est le type.
Ainsi, si les elements en T sont regard^s comme les quotients partiels d'une fraction continue, et que, suivant la notation de rimmortel Lejeune- Dirichlet, on repr^sente par (7") la derni^re convergente k cette fraction, on aura
D^signons par 0 ce que devient 0 quand on renverse I'ordre, et par 0 ce qu'il devient quand on change le signe de chacun de ses dl^ments. Posons
Ti=0t{Oty0;
j'ai trouv^ et de'montrd le lemme suivant* :
* Poor ^tablir cette proposition, on n'a besoin que de se servir des deux identit^s Buivantes. Bi T=te,
m=[t][e]+[t']m.
8. IV. 41
642 Sur la corre^ondance complhte entre [63
Lee rapports des trois quantitis [TJ : [TJ - [T/] : \^T{] sont indSpendants de i ; c'est-d-dire sont les mSmes que les rapports de
[et0]:r0td]-[.^t^]-£^^l
Avec I'aide de ce th^orfeme et de I'dquation qui exprime une proprit^td bien connue des convergentes success! ves de fractions continues, savoir
[7"] [T'] - [T] [7"] = ± 1, on ^tablit facilement le th6or^me suivant :
On peat dcrire et d'une seule manure les deux radnes d'une Equation quadratique simultandment sous les formes "•
(et{otr). -idt{otj),
oii tous les iUments de 0, sauf le dernier {qui pewt itre zh-o), et tous les dements de t sont positifs.
Comme un simple corollaire de ce th^orfeme de correspondance, en appli- quant k la seconde forme la mdthode donn^e par Dirichlet pour rigvlariser une succession de quotients partiels dont quelques-uns au commencement sont n^gatifs, on voit que les p6riodes des deux fractions convergentes contiendront les memes 61dments, mais en ordre inverse.
Ud exemple fera mieux comprendre la port^e du th^orfeme.
Prenons I'^quation
23ar' - 68a; + 50 = 0, dont les racines sont
34 + V6 34-^/6
23 ' 23 ■
On trouve, pour le d^veloppement de ces deux quantites, les fractions p^riodiques en fractions continues
(1,2. 1,2; 4, 2; 4, 2; 4, 2;...) et (1,1, 1,2; 2, 4; 2, 4; 2, 4;...)
respectivement.
Si T=ter, [21 =[«] [9] [r]+[t']['e]H+mOT[V] +['']['«'][''•]•
On peut cependant ajouter que, de mSme, si T = tOru,
[T] = [(] [9] [t] [H + [«'] [■»] [t] [CO] + [(] m [V] [u] + [t] m [t'] [•«]
+ [«'] ['9'] [V] [«] + [«'] ['9] [t'] [-co] + [t] [»'] [V] ['o,] + [f] ['»'] [V] ['«], oi I'on remarqaera que les trois premiers produits de la deuxifeme ligne sont composes de deux (le premier et le dernier) de formes analognes, et d'un troisi^me d'une forme diff^rente, ct ainsi, en gtoiral, si le uombre des types partiels t, 6, t, ... est j, on aura 2«-' produits de cumnlants partiels et de leurs d^riv^s simples et doubles ; ear il y aura (t - 1) intervalles entre les ; types sur lesquels on doit faire tomber dans cbaque mani^re possible 1, 2, 3, ... (i-1) paires d'accents. Quand les types partiels devienneat monomiaux, les termes avec les accents doubles dans la somme des produits deviennent z^ros, et Ton retrouve la regie connue pour exprimer un cumulant comme somme des produits des agr^gats de ses Elements, en ^lisant on en traitant comme unites des paires et oombinaisous de paires d'^l^ments cons^eutifs.
63] certaines fractions continues 643
Or, en ^crivant
0=1,2, <=1, 2, 3, on aura
(^(00") = (1, 2, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, ...) = (1,2,1,2; 4,2; 4,2; 4,2;...),
ce qui r^pond a la premiere racine.
On aura aussi
(0tiOtr) = {-l,-2,S, 2,1,0,3, 2,1,0,3,...) = (-1,-2, 3; 2, 4; 2, 4;...),
laquelle convergente, rigularisee selon les regies de Dirichlet*, peut ^tre
remplacee par
(-2, 1,0, 1,2; 2, 4; 2, 4;...), c'est-£l-dire
(-2, 2, 2; 2, 4; 2, 4;...),
\ee qui, selon les m^mes rfegles, ^quivaut k
-(1,1,1, 2; 2, 4; 2, 4;...),
iquelle est la valeur prise n^gativement de la seconde racine.
Terminons par I'exemple tr^s simple
a?!- 10a;- 1 = 0,
iont les deux racines sont 5 + \/26, 5 — V26, qui Univalent aux fractions "continues
(10,10,10,...), -(0,10,10,10,...).
Faisons 0=9,0, «=1, 9.
Alor8(0«(O«)'')devient
(9,0;1, 9;0, 1, 9;0, 1, 9;...),
c'est-^ire
(10; 10; 10;...),
la premiere racine ; et (0<(O<)") devient
(-9,0;9, 1 ;0, 9,1; 0,9, 1 ;...),
ce qui 6quivaut &
(0,10,10,...),
laquelle est la valeur prise n^gativement de la seconde racine.
On comprendra que dans les formules pour une racine et la negative de I'autre, rien n'emp^che que le 0 disparaisse et qu'ainsi les formules deviennent
respectivement.
* VorUtungen liber ZahUntheorie, § 80 ; 1871.
41—2
644 Correspondarice complete entre certaines fractions [63
Dans le cas oil les deux racines sont ^gales, mais de signes contraires, Don seulement le ^ disparait, mais aussi le t devient symetrique : ainsi Ton retrouve la forme applicable ^ I'^quation Aa^ — y9 = 0, pour lequel cas la racine positive peut ^tre mise sous la forme
(phc, ..., cba, 0, abc, .... cba, 0, oic), c'est-Ji-dire
{ajbc,..., cb, 2ajbc, cb, 2aj).
On peut encore simplifier un peu les expressions pour x et x' (oh x et a/ sont les racines de la m^me Equation quadratique) en dcrivant
x = (e(t,or), x' = -(e(t,or),
formule vraiment surprenante par sa simplicity et sa sym^trie.
64.
SUR LA REPRESENTATION DES FRACTIONS CONTINUES QUI EXPRIMENT LES DEUX RACINES D'UNE EQUATION QUADRATIQUE.
[Comptes Rendus, cviii. (1889), pp. 1084—1086.]
Nous avons donn^ dans une Note pr^c^dente [p. 644, above], pour les deux cines x et x' d'une Equation quadratique k coefficients entiers, les formulas imelles
x = (t{rOT), -x' = (i(Tpr).
Mais ces formules admettent encore une simplification importante au loyen des considerations suivantes.
Un type peut etre nomm^ onmi-positi/ ou omni-nigatif quand tous ses Elements sont pnsitifs pour uu des cas et tous n^gatifs pour I'autre : il sera nomm^ homonyme quand il est ou omni-positif ou omni-n^gatif sans specifier lequel des deux il est.
Le z^ro sera regard^ comme un nombre (non pas neutre, mais) amphi- bolique, c'est-k-dire qui est en meme temps positif et ndgatif, de sorte qu'un type omni-positif ou omni-n^gatif ne cesse pas d'etre homonyme en y ajoutant ou y entremSlant un ou plusieurs zeros.
De plus, on remarquera que {T) = — (T).
Alors la th^orie, atteignant son dernier terme de simplicity et de g^n^- ralit^, donne lieu k I'^nonc^ suivant :
En supposant que t est un type homonyme quelconque et t un autre, et que
w, x' sont les deux racines d'une Equation quadratique a, coefficients entiers, on
aura toujours
x = {tT'), a:' = (<0(?r)
avec la facidti a t de disparaitre.
Ainsi, par exemple, en supposant que t disparaisse et que t devienne
monomial et ^gal k a, si
x = {a, a, a, ...,ad infinitum),
on aura a/ = (0, —a, —a, ..., ad infinitum).
646 Sur la repr^entation des certaines fractions contimies [64
c'est-k-dire de sorte que
c'est-k-dire x' = — (0, a, a, a, ..., ad infinitum) ;
1
X
On remarquera que les types It", tOr" sont mutuellement inverses I'un de I'autre, car (<OOT") = (tT'°).
Nous nous sommes ddjk servif dans nos conferences, tenues k King's College London en 1859, sur la determination du nombre de solutions en nomhres entiers d'un systfeme d'equations num^riquesj, avec grand avantage de cette id^e d'une sdrie de quantit^s omni-positive, omni-n^gative ou homo- nyme et de la conception du caract^re du z^ro comma appartenant aux deux categories des quantit^s positives et negatives k la fois.
Dans une Note k suivre, nous nous proposons de faire connaitre la con- nexion § remarquable qui subsiste entre les racines de I'^quation
aa? + Ibx + c = 0
et les developpements en fractions continues des fractions ordinaires ^ ~ ^ ,
oil p, q sont les nombres de Pell qui appartiennent au determinant b* — ac, et, si nous ne nous sommes pas tromp^, nous esp^rons fonder la-dessus une rhgle pour lextraction simultanee des deux racines de I'equation au moyen de ces deux developpements.
• Et, en general, qaand x= — -, , on aura
x={(er),
oA 0 est on type sym^triqne, oe qni est le th^or^me de Gallois {Journal de LiouvilU, t. ii. p. 385). De mdme, si x = ((eO)") {$ 6tant sym^trique) et ainsi 9 = S, on aura
-x'={(ooeo)°")={(eo)")=x,
de sorte que ((00)°°) est la forme g^n^rale de la fraction continue qui exprime la racine carr^e d'une quantity rationnelle quelcouque.
[+ See Vol. II. of this Reprint, p. 122.]
J In^dites juBqn'i ce jour, mais qui doivent paraitre proohainement dans VAmerican Journal of Mathematics. Cast dans nos recherehes sur ce sujet que nous avona rencontr^ et discut^ la tb^orie g^om^trique de dispositions de points dans un plan et dans I'espace que notre Eminent Confrere M. Halphen a retrouv^e ind^pendamment depuis et k laquelle il a donn£ le nom de tMorie d'atpectt. C'est en r^duisant la determination du nombre de solutions en nombres entiers d'un syst^me de 3 Equations k dependre d'un agr^gat de pareilles determinations pour des syst^mes de 2 Equations que cette theorie s'est forc^ment mise en Evidence pour les points dans un plan. Oe meme, en faisant dependre le probl^me pour un syst^me de 4 de celui de syst^mes de 3 equations, on est amen^ a une theorie semblable pour I'espace; bien entendu, I'ceil regarde comme un seul point dans la theorie pour le plan devient lineaire, oa, oe qui revient i la m6me chose, un systeme de deux points, pour I'espace.
§ Pour Vetablir, nous nous servons encore de notre theoreme de I'immutabilite des rapports de [T'] : [r']-['r] : [T] quand T=tT (Ot)' | pour toute valeur positive et enti6re de i.
65.
SUR LA VALEUR D'UNE FRACTION CONTINUE FINIE ET PUREMENT P^RIODIQUE.
[Gomptes Rendus, cviii. (1889), pp. 1195—1198.]
On sait que ]a valear de la fii^ction purement p^riodique infinie (<"), ou t est un type (c'est-k-dire une succession) d'^l^ments quelconques, est la racine positive de I'dquation
M^-(w-m)*-w=o. (1)
Cela conduit naturellement k la question de trouver la valeur de la fraction continue analogue p^riodique mais finie (<").
Avec I'aide de notre formule donn^e dans une Note pr^c^dente, qui sert k exprimer un cumulant k un type compose de i types partiels comme une somme de 2'~' produits des i cumulants partiels et leurs deriv^es simples et doubles, on peut resoudre cette question sans aucune difficult^.
Ona (tn)^m^JlL
Soient [«»] = m„, ['«"-■] = 'tVn,
on trouve que v„ sera une fonction enti^re et Ton 6tablit, au moyen de la
formule cit^e, entre m„ et i;„ les equations aux differences
oh a = [f\, 5=[^<][f'], c = [Y].
Done BVn-i = Un — fflMn-1 ,
av„ + (B-ac) v„-i = u„ = t;„+, - cv„,
Vn+l -ia + c)Vn + {-y~^ Vn-, = 0
[car B — ac = (—Y~^, /* etant le nombre d'^l^ments en f].
Cons^quemment, par un principe bien connu, «„ et m„ seront les coefficients de A" dans le ddveloppement d'une fraction de la forme
A+Bk l-{a + c)k-€l(^'
648 Sur la valeur dune fraction continue [66
oil e = {—Y, A et B 6tant convenablement d^termin^s pour I'un et pour I'autre cas.
Or «o = 1, M, = a,
% = 0, Vi = l.
1—ek Done Vn est le coeflBcient de k^ en ^ — 7 , \u U ®* "» ^® coefficient de i"
JL " yCti ^ C) fC ~' 6Ar
/fc
en :; — J r-i r; . de sorte que, si Ton ^crit
l-(a + c)k-ek'' ^
4>„ (a;) = a;» + (n - 1) ea;--" + (" - ^H^ " ^) ^s^n-* + . , . jusqu'au premier terme qui devient zdro, on aura
Vn = ^n-i (a + C)
et ?*„=*„(a + c)-c*„_i(a + c).
Ainsi Ton voit que
On peut aussi exprimer «„ et t)„ au moyen des racines de I'equation
vi' - ([«] 4- r«']) m - 6 = 0,
dont on remarquera que le determinant J ([t] + [H'Jf + e est le mSme que celui de I'equation (1), puisque
car, en supposant que p et o- sent les deux racines, on aura
M„ _Ap"- Ba'*
ou A, B sont des quantites connues; et, en supposant que p''= > o^, on aura
^ = ^et(0 = 4.1« positive de I'equation
M^-(W-m)^-w=o,
Si Ton suppose que les Elements de t sont m en nombre et tons identiques avec runit6, on aura
r«] = [1™-'], [H^] = [i™"-'],
et Ton obtient la formule peut-etre nouvelle
I
-* = ^ et (t") = y- , laquelle valeur on identifiera facilement avec la racine
^mn-i(l) ^ /^ X
I
oii^„ = <l>„(l) + <I>^(l).
65] finie et purement p4riodique 649
Si I'oii suppose que m est impair, e sera positif et "9^ prendra la forme
m — 3 (m — 4) (m — 5) l+m + m-^— + m!^ ^ ' ^■...,
en s'arretant au premier terme qui deviant z^ro.
Cette formule donne naissance k un corollaire int^ressant. Supposons que la somme de deux termes separes par un seul dans la s^rie phyllotactique 1, 2, 3, 5, 8, 13, 21, ... est un nombre premier p. Soit m, m — 2 Tordre de ces deux termes ; alors je dis que le quotient du nombre de I'ordre mi — 1 par celui de I'ordre m — 1 (nombre toujours entier) par rapport au module p sera congru k I'unit^ si i est impair et k z6to si i est pair ; de plus, dans ce dernier cas ou i = 2j, le quotient de ee quotient divis6 par p sera congru a (~y{j + 1) P^ rapport au meme module p.
On pourrait tirer sans doute d'autres th^oremes analogues, mais appa- remment moins simples, au moyen de I'equation
C'est une chose qu'on n'avait nul droit (a priori) d'attendre que le quotient [*<"]-;- f^], au lieu d'etre une fonction rationnelle et entiere de quatre quantitds [t], f<], [<'], ['t'] ou (ce qui est Equivalent) rationnelle et fractionnelle de [<], [t], [f], est en effet une fonction rationnelle et entifere d'une seule quantity, savoir de [il + T^], c'est-a-dire est un nombre phyllo- tactique affect^ ou parametrique, nom qu'on peut convenablement donner a la valeur de [«"], oil x est monomial et entier, [1"] prenant alors le nom de nombre phyllotactique simple ou unitaire.
Q6.
A NEW PROOF THAT A GENERAL QUADRIC MAY BE RE- DUCED TO ITS CANONICAL FORM (THAT IS, A LINEAR FUNCTION OF SQUARES) BY MEANS OF A REAL ORTHO- GONAL SUBSTITUTION.
[Messenger of Mathematics, xix. (1890), pp. 1 — 5.]
All the proofs that I am acquainted with (and their name is legion) of the possibility of depriving a quadric, in three or more variables, of its mixed terms by a real orthogonal transformation are made to depend on the theorem that the " latent roots" of any symmetrical matrix are all real.
By the latent roots is understood the roots of the determinant expressed by tacking on a variable — \ to each term in the diagonal of symmetry to such matrix.
I shall show that the same conclusion may be established d priori by purely algebraical ratiocination and without constructing any equation, by the method of cumulative variation. The proof I employ is inductive : that is, if the theorem is true for two or any number of variables I prove that it will be true for one more.
To illusti^te the method let us begin with two variables. Consider the form ax* + Ihxy -f by*. J
If in any such form h = a, then by an obvious orthogonal transformation,
namely, writing ^ and — ^ for x and y, the form becomes
a {ac' + y') + h{x'- y% or (^a + h)x' + {a- h) y\
Now in general on imposing on x, y any orthogonal infinitesimal substi- tution, so that
X becomes x + ey,
y ., y- ex,
I
66] General Quadric map be redticed to its Canonical Form 651
h in the new form becomes h+(a — b)e, or say Sh = (a — b) e, and
^S(h') = (a-b)he; the variations of a and b need not be set forth.
Let an infinite succession of such transformations be instituted ; then either a and 6 become equal and the orthogonal substitution above referred to reduces the quadric to its canonical form, in which case this one combined with the preceding infinite series of such substitutions may be compounded into a single substitution, or else by giving e the sign of (b — a) the variation of h^ may at each step be made negative so that k' continually decreases, unless h vanishes. If h does not vanish it must have a minimum value, and this minimum value may be diminished, which involves a contradiction: hence, in the infinite series of substitutions supposed, either a and b become equal or h vanishes, and in either case the quadric is reduced or reducible to its canonical form.
Let us now take the case of three variables x, y, z.
I^b Obviously, by the preceding case, we may make the term involving ooy "isappear and commence with the initial form
cur' + 6/ + %fxz + 2gyz + cz"^.
If / or g become zero the quadric may be canonified by virtue of the preceding case.
Again, if 6 = a, by imposing on x, y the orthogonal substitution
g f
/ 9
i^+irTTT-^y'
the term involving xz will disappear and the final result is the same as if / were zero.
Let us now introduce the infinitesimal orthogonal substitution which changes
X into a; + ey + rjz,
y „ -ex+ y + dz,
z „ -Tix-0y+ z, where e, ij, 0 are supposed to be of the same order of magnitude so that only first powers of them have to be considered.
Then S/= (a-c)v- ge,
8g = {b-c)e+/e,
also the coefficient of 2xy becomes (a — 6) e —ft) — grj.
Now whatever t), 6 may be, we may determine e in terms of rj, 0 so that this may be made to vanish, and the initial form of the quadric will be maintained, provided that 6 is not equal to a.
652 A New Proof that a General Quadrie [66
Hence instituting an infinite series of these infinitesimal substitutions, provided we do not reach a stage where a and b become equal, we may maintain the original form keeping j;, 6 arbitrary, and shall have
iS ( P +f) = (« - o)fv + (6 - c)g0. Suppose a and 6 to be unequal ; therefore (a — c), (6 — c) do not vanish simultaneously, and consequently we may make ^(f' + g') negative unless at least one of the two quantities/, g vanishes.
If neither of them vanishes /' + g' may be made continually to decrease and will have a minimum other than zero, which involves a contradiction.
Hence the infinite series of infinitesimal orthog(mal substitutions may be so conducted that either a — b or one at least of the letters/, g shall become zero ; and then two additional orthogonal substitutions at most will serve to reduce the Quadric immediately to its canonical form.
I shall go one step further to the case of four variables x, y, z, t and then the course of the induction will become manifest. We may, by virtue of what has been shown, take as our quadric
aa? + by'' + cz^ + yxt + ^yi + IKzt + d«». Here, if any one of the mixed terms disappears, the quadric is im- mediately reducible by the preceding case, and if any two of the grouped pure coefficients a, b, c become equal (as for instance a, b), then by an orthogonal transformation one of the mixed terms (/ or g in the case supposed) may be got rid of; so that this supposition merges in the preced- ing one.
Impose on x, y, z, t an infinitesimal orthogonal substitution, writing
X + ey + 9z + \t for x, -ex+ y + r]Z + /j.t „ y, — 0x — Tjy+ z + vt „ z, —\x — ixy—vz+ t „ t.
Then Sf={a-d)X-ge-he,
Sg = {b-d)fji. +fe - hr), Bh = {c-d)v +f0+gv- Also the coefficients of 2xy, 2xz, 2yz respectively become
(a - b) e -fn - g\, (a — c)6 —fv — h\, (b — c) 7) — gv — h/M.
Suppose that no two of the grouped pure coefficients a, b, c are equal; then €, 0, T] can be, and are to be, expressed in terms of \, fi, v so as to make these three expressions vanish ; that being done the initial form of the Quadric is maintained throughout the series of substitutions and we may write fB/+ gSg + hBh = (a- d)fX + {b-d) g^i + (c- d) hv.
66] may he reduced to its Canonical Form 653
Of the three quantities \, /t, v it is sufficient for the purpose of the argument to retain any two as \, fi and to suppose v=0.
Then, since we suppose that a and h are not equal,
(a — d)f\ -\-{h — d) cffi
(where X, fi are arbitrary) can always be made negative unless /, g are none of them zero ; so that if a and h never become equal nor f or g vanish /' + g' + h'' cannot have any minimum value other than zero, which involves a contradiction ; hence in the course of the series of infinitesimal trans- formations either a and b must become equal, or / or ^ or both of them vanish. If/ and g vanish simultaneously or even if one only of them vanish, then one succeeding substitution, and if a and b become equal two succeeding substitutions, will effect the reduction to the canonical form. This proves the theorem for four variables.
The method is obviously extendible to any number of variables; in the case just considered it is seen that in the infinitesimal orthogonal matrix of substitution for the exceptional line or column (that which relates to the excepted variable the t) it is not necessary to employ more than two arbitrary infinitesimals and a like remark applies to the general case, so that if there are n variables, whilst ^ («* — n) is the number of infinitesimals that would appear in the complete matrix, ^(w^— 3n + 6), that is ^ {(n — l)(n — 2)} + 2, are sufficient for the purpose of the demonstration.
Thus then without recourse to any theorem of Equations it is proved that any Quadric may be reduced by a real orthogonal substitution to its canonical form*.
* I hare applied the same method to prove that by two real independent orthogonal Bubsti- tntions operated on
X,, Xj, ... x„; y,, !/j, ... j/„
the general lineo-linear Qnantic in the z's and y'a (with real coefficients) may be reduced to the canonical form Sx^t/j, and have sent for insertion in the Comptes Rendus of the Institute a Mote in which I give the rule for effecting this reduction [above, p. 638].
It may be aafficient here to mention that if U is the given Uneo-linear Quantic, its n canonical
mnltipliers are the square roots of the n canonical multipliers of the Quadric 2 I -3- I , or if we please of 2 I -j- ) , which it may easily be shown h posteriori are necessarily omni-positive ;
and I need hardly add that although these two Quadrics.are different, their canonical multipliers are the same.
67.
ON THE REDUCTION OF A BILINEAR QUANTIC OF THE TjTH ORDER TO THE FORM OF A SUM OF n PRODUCTS BY A DOUBLE ORTHOGONAL SUBSTITUTION*.
[Messenger of Mathematics, xix. (1890), pp. 42 — 46.]
A HOMOGENEOUS lineo-linear function in two sets of variables
X, y,...z; u, v,...w will contain n" terms : two independent orthogonal substitutions performed on the two sets will introduce twice ^n{n — 1) disposable constants, and by a suitable choice of these, n' — n terms of the transformed function may be made to vanish so as to leave a sum of products of the new x, y, ... z paired with the new u,v,,..w. it will of course be found in general impossible to obliterate any arbitrarily chosen (w' — n) terms in the transformed function ; since if in the n remaining products one letter of one set were combined with more than one of the other set, this would (by means of a further super- imposed orthogonal substitution) be equivalent to taking away more than («' — n) terms by means of only (n^ — n) disposable constants. It is very easy to effect the transformation indicated by a method very analogous to that of reducing a quadric in n variables by an orthogonal substitution to its canonical form, and to show a posteriori that the substitutions are always real in this case as in the other, when the original coefficients are real ; but it will, I think (although not necessary), be found interesting and instructive to prove d priori the latter assertion by a similar method to that applied to Quadrics in the last number of the Messenger. I will begin then with this proof, reserving the complete solution of the problem to the end of the article. The leading idea in this as in the preceding article is to regard a finite orthogonal substitution as the product of an infinite number of infinitesimal ones. ^
For ajcu + aaw + fiyu + byv. ^
Let x, y; u, v become x + ey, — ex + y; u + Xv, —Xu + v respectively,
then
Sa=aX — be, S^ = a€ — bX,
o8a + ySSyS = (aa - 6/3) X + (a^ - ba) e.
[* Cf. p. 638 above.]
67] On the Reduction of Bilinear Quantics 655
Heuce o' + /S" may be made to decrease unless a = 0, 6 = 0, or o = 0, yS = 0,
or 5- = -j; = + 1, in which case since b P
(a + o) (a; + y) (u + v) +(a — a){x — y) (m -v) = 2a {xu + yv) + 2a {xv + yu),
(a — a) (x + y) {u — v)-{-{a + o) (x — y) (m + 1;) = 2a {xu — yv) + 2a (xv — ?/m),
the form is immediately canonizable.
Hence in the infinite succession of infinitesimal orthogonal substitutions (equivalent to a single one) either a and 6 or o and ^ must vanish simul- taneously, on which supposition the form is canonical or else it is reducible to the canonical form by a second finite orthogonal substitution.
Let us now proceed to the case of a ternary bilinear form in x, y, z\ u, V, w.
I suppose by the previous case the form to be deprived of two terms, and that we have to deal with the form
axu + hyv + fxw + guz + hyw + kvz + czw.
Lemma. K f= 0, ^ = 0, or h = 0, k = 0 the above form is reducible by the previous case. Also if a- = b^ and /= 0, A = 0, or ^r = 0, k = 0, or a' = b^
and (^j = (t) the form is reducible to the previous case by a single additional finite orthogonal transformation.
For the sake of brevity I leave the proof to my readers. Introducing now two infinitesimal orthogonal substitutions with para- meters e,r], 6; \, ft, v*, we obtain the variations
Bf = afi — h€ — erf, Bh=bv+/e — cd, Bg=ar) — k\— cfi, BIc = bd + gX — cv, also in order to keep the coefficients of ocv, yu at null, we must have
a\ — be — fv — kr) == 0, — bX + a€ — gd — hfi = 0. From the previous equations we obtain /¥+ 9^9 + ^SA + kBk = (of- eg) fi+(bh- ck) v + {ag- cf) rj + (bk - ch) 6.
(1) Suppose a'-6» not zero; then /t, v, r), 0 will be independent and their coefficients cannot all become zero unless /' = g^ and h? = A*, or else /= 0 and ^ = 0, or A = 0 and A = 0, on either of which suppositions the form becomes canonizable by virtue of the Lemma.
(2) Let a' = fr". Then we must have fi> + krj ± (gO + hfi.) = 0,
l| which I shall satisfy by making /«/ ±g0 = O, kij ± hfi = 0.
* The positive valnes of the parameters in each system are supposed to belong to the npper, and the negative values to the lower half of each orthogonal matrix.
856 On the Reduction of Bilinear Quantics [67
Hence
2/y = {(of -cg)kT (ag -cf)h]p + [{aJi -ck)g^ (ak - ch)/] r, p, T being two arbitrary infinitesimals.
Therefore X/Bf may be made negative unless the multipliers of p and t are both zero, in which case by addition or subtraction we obtain fk = gh;
consequently two out of the four variables /, g, h, k are zero, or else j = t,
and on either of these suppositions the transformed function may be canon- ized by virtue of what has been proved in the case of two biliteral sets, or may by a finite orthogonal substitution be brought to a form so canonizable.
Hence it is clear that either /, g, h, k may all be made to vanish, or else we must pass through a form known to be canonizable. This is the proof for a bilinear function of triliteral sets, which may be easily extended to a bilinear function of w-literal sets.
I will now give the method for effecting the reduction which is thus proved to be always capable of being effected by real substitutions.
Let Sar,«ir,3/8 be the given bilinear function B.
Then 2(-t-J , which is an orthogonal invariant of B quiL the y'a, is a
Quadratic function of the x's, which will have an orthogonal substitute quH the a;'s of the form X [Xra;r']-
If then B is reducible by a double orthogonal substitution to the form 2[5ra;,i/,], we must have 2[^r^r? orthogonally equivalent to 2[\ra;r*]. and this can only be the case when the ^'s are respectively (in any order) the squares of the Vs.
The ^'s I call the Canonical Multipliers to B.
This gives rise to the following rule :
Form the Matrix [m].
'^,11 ^,t> ••• ^, I >
From this derive a Matrix [M], a false square of [to], obtained by multi- plying each line in it by all the lines (according to Cauchy's rule, in fact, fori the multiplication of Determinants). Then the latent roots of [M] are the squares of the Canonical Multipliers to B.
But if instead of 2(-r-| we take Sf-;— ) and deal with it in like \dyj _ \dxrJ
manner, we shall obtain a matrix [n], such that [to] and [n] are transverse
i
67] to their Canonical Form 667
to each other, the lines and columns of the one being the columns and lines of the other: the Cauchian Square of [n] will give rise to a matrix [i\r] different from [itf] but having the same latent roots : in fact the coefficients of the equation to the latent roots alike of [m] and of [n] virith the signs in the alternate places changed will be unity, the sum of the squares of all the terms in [m] or [n], the sum of the squares of the minors of the 2nd, 3rd, ... orders in [m] or [n] ; and finally the last coefficient will be the square of the determinant to [m] or [n] : so that we shall obtain as we ought the same set of canonical multipliers whichever matrix [3/] or [N^ we employ ; but in order to obtain the substitutions which must be impressed on the x set and the y set to arrive at the Canonical form in which only n products appear we shall want both [J/] and [A"]. Let me, however, pause for a moment to call attention to the interesting fact that the sum of the squares of the coefficients in B by virtue of being a coefficient of the latent function to \^M'\ or [iV^] is necessarily a bi-orthogonal invariant to 5 ; so, too, all the other coefficients in this function are such invariants : and among them the last, which is the square of the determinant to [rw] or [n]. Thus then this determinant (which may be termed the discriminant) is an invariant alike for the two theories ; namely the better known one in which the x set and the y set are subjected to the same general substitution, and the one here considered where these sets are subjected to two independent orthogonal substitutions.
In either theory the vanishing of the discriminant is the signal of the Canonical form becoming short of one term.
It is also proper to notice that the latent roots of [ilf] or [JV], which by virtue of [Jf ] and [i\r] being symmetrical matrices are neces.sarily real, are for these particular forms of [J/] and [jV^] positive as well as real since the coefficients with the alternate signs changed are all positive, being the sums of squares of real numbers.
To complete the solution it remains to find the two canonizing orthogonal matrices, but these are known by the ordinary theory for quadrics : thus the m substitution will be that which canonizes [J/] and the y substitution that which canonizes [iV^].
Conversely, if [3f] and [iV^] are supposed given, we shall know the linear functions of the x's which substituted for a;,, x^, ... x„ and the linear function of the ^s which substituted for yi, yi,...y„, such that XXj^x^yi shall be identical with B, the \'s being the latent roots common to [M] and [iV]. There will be 2" systems of values represented by Xj*, \^, ... \„^ : thus then 2" matrices transverse to one another can be found such that their false squares shall be respectively identical with any two given symmetrical matrices having the same latent roots, and we are thus enabled indirectly, through the theory of bi-orthogonal canonization, to obtain the solution of 8. IV. 42
658 On the Reduction of Bilinear Quantics [67
a problem which intrinsically has or seems to have nothing to do with orthogonal or other transformation.
It is worthy of observation that this problem of finding the so-to-say false square root common to two given symmetrical matrices having the same latent equation, admits of precisely the same number (2") solutions as the problem of finding the true square root of one general matrix. For if [M] be any given matrix of order n and [1] represents the unit matrix of that order, namely the matrix all of whose terms are zeros except those in the principal diagonal which are units, we know by virtue of a general theorem that calling X,, X,,...X„ its n latent roots, each true square root of [M] is represented by
.([M]-\,[l])([M]-\,[l])...([M]-K[l]) {\-\)(K-\)...(\-\,,)
68.
ON AN ARITHMETICAL THEOREM IN PERIODIC CONTINUED FRACTIONS.
[Messenger of Mathematics, xix. (1890), pp. 63 — 67.]
The well-known form of continued fraction for the square root of N, an integer, is
(a; b, c, d, ..., d, c, b, 2a; b, c, d, ..., d, c, b, 2a; indefinitely continued)
which, if we denote the type a, b, c, d, ..., d, c, b, a by t, may be written under the more convenient form
(t,0,t, 0, t,0,...adinf.).
If now we use [t] to signify the cumulant of which t is the type, and ['<], V]' r^l respectively, the cumulants of the types got by cutting off a from either end and from both ends of t, it is easily shown that whatever numbers
«, 6, c, ... represent, the value of the continued fraction {{t, 0)"} is ^/ hJ=i>
so that if {{t, 0)"] represents the square root of an integer, [t] must be divisible by \^t'].
At first sight one would imagine that it would be a difficult matter to give a rule for determining whether such condition is fulfilled or not by any assigned value of the symmetrical type t, but Mr C. E. Bickmore, of New College, Oxford, has noticed that the case is quite otherwise, for that if we put t under the form a, t, a, then, in order that {(a, t, a, 0)"} may satisfy the requirement of being the square root of an integer, the sufficient and necessary condition is the equivalence
2a = (->'[T'][V'](mod.[T]),
where fi is the number of elements in t.
Consequently t may be taken quite arbitrarily, and then an infinite number of values be assigned to a, except in the case where [t] is even, and at the same time [t'] and [t] are each of them odd.
i2— 2
660 On an Arithmetical Theorem in [68
The proof in my notation is as follows :
Since < = a, t, a, we have 't' = t, and consequently fM, will be an
L * J integer if
[a, T, a] = 0 (mod. [t]).
Expanding and remembering that rT] = [t'] (the type t being symmetrical), we obtain
a» [t] + 2a [t'] + [V] = 0 (mod. [t]).
Hence 2a [t'J + fr'] = 0 (mod. [t]), (1)
and 2a [t']» + [t'] [t] = 0 (mod. [t]). (2)
But [t?-[t][V] = (-1)^+>,
so that [t']" = (- !>*+' (mod. [t]),
and therefore (2) becomes
2a = (-y[T'][V](mod. [t]), (3)
which is thus shown to be a necessary condition.
It is also a suflScient condition, for multiplying (8) by [t'] we have 2a[T'] = (->'[T'P[V](mod.[T]), or, since [t']' = (-)"+' (mod. [t]),
2a[T'] = -|:T'](mod.[T]), which is the same as (1).
Suppose now that V is given and that we wish to ascertain if a can be found of such a value that the congruence (3) sliall be soluble. This will obviously be the case if [t] is odd. It will also be the case if [t] is even, provided [V] is also even, and only in that case ; for, when [t] is even, then by virtue of the equation
MH-[r?=±l. [t'] must be odd.
We have, therefore, to find under what circumstances [V] will be odd and [t] even ; in all other cases but these the congruence (3) will be soluble, and then the most general value of a will be any term in an arithmetical series of which the common difference is [t], unless [t] and [V] are both of them even, in which case the common difference will be ^ [t].
I proceed now to give a rule for determining the possible and impossible cases of the solution of (3), to explain the grounds of which the following statement will suffice.
(1) The value of a cumulant is not affected by striking out any even number of consecutive zeros from its type.
(2) The parity (that is the character qud the modulus 2) of any cumu- lant will not be affected if we strike out three consecutive odd terms, whether
68] Periodic Continued Fractions 661
they occur in the middle or at either extremity. For if t, t be any two types, the cumulant
[t, 1, 1, 1, t] = 3 [t] [t] + 2 \t'] [r] + 2 [t\ [V] + [«'] [V] = MH + [«'][V](mod.2), that is = [t, t] (mod. 2).
Also
[1, 1, 1, t\ = [t, 1, L 1] = 3 [<] + 2 pi] = [<] (mod. 2).
(3) The value of any cumulant in the type of which 1, 0, 1 occurs any- where is the same as if 2 is substituted for 1, 0, 1 ; and therefore its parity is not affected if the units on each side of the 0 are omitted.
In what precedes in Nos. (1), (2), (3) the result, to modulus 2, is obviously unaffected if for 0 we write any even and for 1 any odd number.
In order then to determine the parity of [V] and of [t] we may proceed as follows :
Let T be any assigned symmetrical type, V will then represent the type divested of its two equal terminals.
Rules — (1) for each even number in V write 0, and for each odd number, 1 ;
(2) elide any even number of consecutive zeros, and any number
divisible by 3 of consecutive units ;
(3) elide any pair of units lying on each side of a zero ;
(4) repeat these processes as often as possible ;
then, I say, eventually we must arrive at one or other of the six following irreducible types, namely
( ); 0; 1; 1, 1; 0,1,0; 0,1,1,0*,
where ( ) means absolute vacuity ; accordingly V may be said to be affected with one or the other of these six characters.
If now the reduced form of V is 0; 1, 1 ; 0, 1, 0, [V] is even, and the congruence (3) will be soluble. In the other three cases [V] is odd, but [t] will also be odd unless its terminal elements are odd in the case where the reduced form of V is ( ), and even for the reduced forms 1, and 0.1,1.0.
In the following exhaustive table the second column indicates the even- ness or oddness of the terminals of t denoted by e and u respectively.
The third and fourth columns indicate the evenness or oddness (denoted
as above) of [V] and [t], along with the character of V in the third column.
In the fifth column the answer is given as to the determining congruence
* Except for the symmetrical form of r there wonld be two additional (virtually undistin- gaishable) reduced forms 0, 1 and 1, 0.
662 Arithmetical Theorem in Periodic Continued Fractions [68
being soluble or insoluble, denoted by s and i respectively; and the last column shows whether the common difference of the arithmetical series of the values of either terminal, in the case of solubility, is equal to the modulus [t] or its moiety.
|
Gases |
TerminalB |
>' |
[t] |
Sol. or Insol. |
CD. |
|
|
1 |
e |
( ) |
u |
u |
H |
|
|
2 |
u |
( ) |
u |
e |
||
|
3 4 |
e u |
1 1 |
u u |
e u |
W |
|
|
6 |
e |
0, 1, 1, 0 |
u |
e |
||
|
6 |
u |
0, 1, 1, 0 |
u |
u |
W |
|
|
7 |
e |
0 |
e |
e |
iW |
|
|
8 |
u |
0 |
e |
e |
iH |
|
|
9 |
e |
1, 1 |
e |
u |
H |
|
|
10 |
u |
1. 1 |
e |
u |
H |
|
|
11 |
e |
0, 1,0 |
e |
u |
H |
|
|
12 |
u |
0, 1, 0 |
e |
u |
H |
The following examples are given to prevent the possibility of mis- apprehension in the application of the Algorithm.
(a) Let
T = 1, 9, 1, 1, 1, 2, 1, 7, 4, 2, 2, 2, 4, 7, 1, 2, 1, 1, 1, 9, 1. Then V = 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1
= 0, 1, 0,1, 0
= 0, 0, 0
H 0.
This corresponds to case (8), which is a soluble one, and accordingly we* have from Degen's Table
{(15, T, 15, 0)-} = V(251), 15 being the first term of an arithmetical series whose common difference isHr].
(/8) Let T = 2, 3, 1, 2, 4, 1, 6, 6, 1, 4, 2, 1, 3. 2.
Then V = 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1
= 1,1. 1, 1, 1,1
( )• This corresponds to the soluble case (1), and accordingly we find from Degen's Table {(10, t, 10, Of}=>J(109); 10 being the first term of an arithmetical series whose common difference is [t].
69.
ON A FUNICULAR SOLUTION OF BUFFON'S "PROBLEM OF THE NEEDLE " IN ITS MOST GENERAL FORM.
[Acta Mathematica, xiv. (1890-1), pp. 185 — 205.]
"...quaintly made of cords."
{Two OerUlemen of Verona, Act iii. Sc. 1.)
The founder of the theory of Local Probability appears to have been Buffon (better known as a Naturalist, but who began his career as a Mathe- matician). Among a few other questions of a similar kind, which he proposed in his Essai d' Arithm^ique Morale, the one which has obtained the greatest notoriety is the celebrated one which goes by the name of the ProbUme de V Aiguille, the purport of which is as follows.
On an area of indefinite extent (say a planked floor) a number of parallel straight lines are ruled at equal distances, upon which a needle, not long enough to cross more than one of the parallels at the same time, is thrown down : the probability is required of its falling in such a position as to be intersected by one of the parallels.
An easier question of the same kind, which Buffon treats before the other, is when a circle is used instead of the needle. This latter question he solves by simple geometrical considerations too obvious to need recapitulation ; to obtain a solution of the former he, and after him Laplace, had recourse to a process of integration.
In a question given in the late Mr Todhunter's Integral Calculus (1st edition, 1857, p. 268) the solution of the problem is correctly stated for an ellipse, whose major axis is less than the distance between two consecutive parallels, instead of for a circle or straight line : this important step in the development of the theory is, I am informed, currently attributed to the late Mr Leslie Ellis, of the University of Cambridge.
In the year 1860, Lam^ propo.sed to give a course of lectures on the subject at the Sorbonne, and, apparently without knowledge of the result contained in Todhunter's treatise, reproduced the solution for the ellipse and for any equilateral polygon. In the same year M. Emile Barbier, whose lamented decease occurred in the course of the present year and who had
664 On Buffon's Problem of the Needle [69
attended Lamp's lectures, discovered and published in Liouvilles Jouiiud for that year a universal solution for an undivided plane contour of any form whatever.
The subsequent history I am not able to trace further than to state that in Czuber's Geometrische Wahrsckeinlichkeiten (Leipzig, 1884) Barbier's solu- tion is extended to the case of any two rigidly connected convex figures (in a plane)*. I propose to give here the finishing stroke to the theory as regards plane figures by extending it to any number of them, rigidly connected and of any forms, in the same plane. It is always to be understood, in what precedes as in what follows, that the greatest diameter of the figure, or system of figures, is less than the distance between two consecutive parallels.
Barbier's principle (see Czuber, pp. 117, 125) leads at once to the conclusion that the probability of any figure (subject to the restriction above stated) intersecting the system of parallels is to certainty as the length of a cord stretched round the figure is to the circumference of a circle touched by two adjoining parallelsf. This circumference (with a view to simplicity of expression) we shall adopt as the unit of length in all subsequent formulae.
By the disjunctive probability of a set of figures I shall understand the probability of one or more of them intersecting one of the parallels : by the conjunctive probability of the same, the probability of all of them intersecting one of the parallels.
I start from Barbier's theorem that for a single figure the probability of intersection is measured by the length of a stretched string passing round it: this, it should be observed, is universally true whether the contour be curvi- linear or rectilinear or mixtilinear, composed of a single line straight or curved or of any number of such — a theorem almost unexampled for its generality. The disjunctive probability for any number of figures A,B,C,...,H] I shall for the present denote by A:B:G:...:H, the conjunctive byj A.B.G...H.
Let there he n+1 figures given, let pi be the sum of the conjunctive and ■BTj of the disjunctive probabilities for these figures taken i and t together ; 80 that OTi and pi are identical, and ■sr,,^,, pn+i are monomial quantities. Then by a universal theorem of logic we have the reciprocal formulae
^„+. ='T\-)'+'i)i, (1)
1 = 1
i— »+l
i>n+>= 2 (-y+^^i. (2)
t=l
• See Post$eriptuni, p. [679, below].
t The case of a straight line (the original question of the needle) may be made to fall under this rule : for the line, as Barbier has observed, may be regarded as an indefinitely narrow ellipse or other oval.
^
69] On Buffon's Problem of the Needle 665
Let us now suppose that we have obtained expressions for the disjunctive and conjunctive probabilities of any number not exceeding n figures of any kind : we may extend these to the case of n + 1 figures as follows.
(1) When the n + 1 figures are so situated that it is impossible for all of them to be cut by the same straight line, we have pn+i = 0 so that ct„4.i can be found immediately in terms of p^, p^, ...,pn by using formula (1), or in terms of Ui, ct,, ..., «r„ by using (2) ; that is w„+, can be found in terms of knovvn quantities ; for by hypothesis all the terms of pi or of •sTf are known when i is any number not exceeding n.
(2) When all the re+1 figures are capable of being cut by the same straight line, let XF be some straight line which cuts them all and call the figures taken in the order in which they are cut by XF
•"li -^2) -4si •••, -"n+i •
Let a stretched string be made to wind round these n + l contours passing alternately from one side of XF to the other, as in Fig. 1, and crossing itself
I
Fig. 1. in the n points i„ i^, ..., i„ lying between A^, A,; A^, A^; ... An, An+i respectively. Let us call the figures enclosed by the successive n + l loops of the winding string
/},, XJj, B^, ..., Jjn+i-
It is obvious that any straight line which cuts all these loops will cut all the given figures, and vice versd.
Hence A^.A^.A,... .(4„+, = 5, .£j.5, ... 5^,.
Let Pi, Hi represent what pi, or,- become when for the figures A we sub- stitute the loops B, so that
i=n + l
n^,= s (-)•+■ Pi,
«=n+l Pn+r= 2 (-)•+' Hi, 1=1
and Pn+i = pn+i-
* It may be well to draw at once attention to the fact that different systems of straight lines do not necessarily cat the figures J,, A.,, A,, ,,, in the same order; as, for example, if three eireleg touch, or bo nearly touch one another that each blocks the channel between the other two, ■traigbt lines may be drawn whose intersections with any one of the three shall be intermediate to their intersections with the other two.
666 On Bufon's Problem of the Needle [69
nn+, is known by Barbier's rule, because the loops taken together form a single figure, in fact
where L is the length of the uncrossed string stretched round the system of figures B, which is no other than that stretched round the given figures A. Also, by hypothesis, I!,- is known for all values of i not exceeding n. We therefore know pn+i which is the same as Pn+i- Hence Wn+i is known from (1) : thus then p„+, and «•„+, are both known, so that when the conjunctive and disjunctive probabilities are known in general for n figures they become known for n + 1 figures ; but when n=\,pi and ct, are equal to one another and to the length of a given stretched string. Hence, by the usual process of induction, we may conclude that the conjunctive and disjunctive probabilities for any number of figures can always be expressed as a linear function with positive and negative integer coeflScients, or in a word as a Diophantine linear function, of a finite number of lengths of certain stretched strings.
When there are only two figures A^, A^ we pass a stretched string between them crossing itself in i (see Fig. 2): then using {Ai){A^ to
Fig. 2.
denote the length of this string, and {A^A,^ to denote the length of the uncrossed string (indicated by dots in the figure) stretched round A^, A^ we have
n,=(A,x^.)-P.
and ^, = (A,) + (A,)-p,
(where (.4,), {A^) denote the lengths of the separate bands round A^, 4,
respectively).
But n,=(^,^),
and consequently
p,=^P, = (A,XA,)-(A,A,),
vT, = (AO + (A,) + (A,A,)-(A,XA,).
We will now proceed to consider in detail the application of the inductive
method to the case of three figures for which, since each of these may be
replaced by a convex band passing round it, we may if we please for greater
I
69] On Buffons Problem of the Needle 667
graphical simplicity substitute three convexes (that is contours which any secant must intersect in exactly two points). Many cases requiring separate discussion will arise, but one important consequence, rising to the dignity of a principle, which holds good whatever may be the number of figures, governs them all ; namely that the final result for either probability is a linear homogeneous function of lengths of stretched bands drawn in various ways round the given figures and depending for their course on the forms and disposition of these figures exclusively, wholly uninfluenced by the presence of any points external to them. Lines drawn from the pointed ends, or apices, of the loops enclosing them do it is true make their appearance in the com- putations but, either coalesce into portions of the bands referred to, or else, entering in pairs with opposite algebraical signs, di.sappear from the final result. As a consequence, if for the sake of illustration we suppose the figures to be any closed curves without singular points, the probability, disjunctive or conjunctive, to be ascertained is a function exclusively of the complete system of lengths of double tangents that can be drawn between the curves and of the arcs into which they are severally divided by their points of contact with those tangents.
We have for all the cases of three figures
where p, = (4 1) + (ils) + {A,)
and p,^{A,y,A,)-{A,A,) + {A,){A,)-{A,A,)^{A,){A,)-{A,A,).
Thus ws - P3 = (^i) + (^2) + {A,) + {A^A^) + (^3^1)
+ (^,4,)-(.4,X^)-(^X^i)-(^.XA)- (3) Similarly H, - P, = (5,) + ( fi,) + {B,) + (B^B,) + {B,B,)
+ (5,5,) - (B, X 5,) - (5, X A) - (5, X B,), where 5,, B,, B, are the loops of the string which passes round the figures Ai, A,, Ai and crosses itself at i and j, as shown in Fig. 3. But P,=p3,
Fig. 3.
and n, is the length of an uncrossed band stretched round the entire system of figures Ai, A^, A, (which will be expressed in symbols by writing
n, = {A,A,A,).)
668
On Biiffon's Problem of the Needle
[69
Hence p, = {A,A,A,) + (B, X 5.) + (5, X A)
+ (5, X B,) - (5.) - (A) - (£,) - (5,5,) - (£,£,) - (5,5,). Moreover (ii, X 5») = W + W
and {B,XB^) = {B,) + (B,),
because Bi, B^ and B^, B, are pairs of consecutive loops. And whenever the three given figures are capable of being cut by a straight line in the order A^, Aj, At (that is except in the case^ = 0, which is separately considered)
(B,B,) = {A,A,),
because both the crossing points, i and _;', of the looped string necessarily fall inside the uncrossed band round -4,, A3. Thus the value of p, is given by the equation
p, = {A,A,A,)-(A,A,)+{B,XB^) + (B,) - (B,B,) -(B,B,) (4)
which, for immediate purposes, we shall find convenient to write under the form
p, = (A, A, A,) - (A, A,) + (B, X B,) - (B.B,) + (B, X A) - (B,B,) - (£,). (5)
We shall apply the formula to the two classes which between them comprise all the cases of three figures, namely
Cliiss A. One of the figures, which we call A2, lies either wholly or partially inside the crossed band round the other two.
Class B. Each figure lies entirely outside the crossed band round the other two.
In Class A we recognize three species, namely
Aa. The figure A^ does not cut either of the crossed strings ab, cd of the band looped round ^,,^3 (Fig. 4), but lies wholly in the same loop as one of them, which we call Ai.
y
— -d
'0./K"
Fig. 4.
69]
On Buffon's Problem of the Needle
669
Ab. The figure A^ cuts one, but not both, of the crossed strings ah, cd (Fig. 5), and part of it lies in the same loop as Ai.
)
Fig. 5,
Ac. The figure ^jcuts both the crossed strings ab, cd (Figs. 6 and 7) and part of it lies in the same loop as .4,.
"••-. J
Fig. 6.
To avoid complicating these figures (4, 5, 6, 7) the band (looped round A,, Aj, At as shown in Fig. 3) which crosses itself at i,j is not given, but the position of each crossing point is marked by a small cross. It should be
Fig. 7.
observed that in Fig. 5 (species Ab) j lies outside the crossed band round Ai, Af ; in Fig. 4 (.species Aa) i and j lie in the .same loop, and in Figs. 6, 7 (species Ac) i and j lie in opposite loops of the crossed band round 4i. A,.
670
On Buffon'8 Problem of the Needle
[69
The discussion of species Aa is very simple ; for it is clear that the con- junctive probability is
since it is obviously impossible for a straight line to cut A^ and J, withouj cutting A^. Substituting this value for p, in formula (3) we obtain the disjunctive probability
nT, = {A,) + {A^)+{A,)+{A,A^) + {A,A,)-{A,){A,)-{A,y,A,).
The remaining two species belonging to class A may be discussed simul- taneously ; for we have in all the cases (see Fig. 8), using e, / to denote the
Fig. 8.
points of contact with the figure A^ of the strings which cross at the point i (between A^ and A^),
iB,XB,) = (A,XA,)+fi + ie-e/, {B,B,) = (A.A,) +fi + ie - ef, so that (B, X B,) - {B,B,) = (A,XA,)- (A,A,).
Hence, for all the species of class A, formula (5) becomes
p, = {A,A,A,) - iA,A,) + (A,XA,) - (A,A,} + {B, X B,) - {B,B,) - (B,). In reducing the last three terms of this expression to a form which involves the lengths of bands round the A's, a slight difference arises between species Ab (in which, see Fig. 5, the point j and the figure J., are on the .same side of the string a6) and species Ac (in which ; and Ai are on opposite sides of the string ab, see Figs. 6 and 7).
Thus, for species Ac, the crossed band round B^, B, will not encounter either of the points i, j, but will be identical with the crossed band (abcda, Figs. 6 and 7) round Ai, A,; that is
(b,xb;)=(a,xao.
Moreover, a moment's reflexion will show that the uncrossed band round Bi, Bi will combine with the loop B^ so as to form a single band: in fact we have
{B,B,) + {B,) = D,
where D is the crossed band round A^, A^ with the loop which contains Ai distended until it also contains A^.
69]
On Bufforis Problem of the Needle
671
But in species Ab (see Fig. 9), let the points of contact with A^ of the strings which cross at j (between A^, A^ be g,h; and let a string jk, in contact with A^ at k, be stretched from j to the figure A-^ : then
(B, XB,) = iA,XA)+ 9J +jk +ka-ab- hg, and (Bi jBj) + (^s) = D+gj+ jk + ka-ab - hg,
where D is the band (abgchjlmna), derived from the crossed band (abgcdna) round Ai, A^ by distending the loop which contains A^ until it also con- tains A^.
Fig. 9.
Hence (B, X B,) - {B,B,) - (B,) = (A, X A,) - D,
and the general formula for the conjunctive probability (for class A) becomes
p,^(A,A,A,) + (A,XA,) + (A,XA,)-(A,A,)-(A,A,)-D. (6)
Combining this with formula (3), which belongs to all cases of three figures, we obtain
^, = (4.) + (A.) + (A,) + (.1,^,) + {A,A,A,) - (A, X A,) - D.
The species Aa, Ab, Ac are distinguishable from one another by the difference in shape of the band D belonging to each. Thus in Aa the band
9
Fig. 10.
D is not distended at all, but is simply (j4,X A^\ in Ab the loop containing Ai is distended on one aide only ; and in Ac is distended on both sides (see Figs. 10 and 11). This difference in shape will be denoted by writing D^
672
On Buffon's Problem of the Needle
[69
for D in the general formula when the species is Ab, and Dj for D when the species is Ac.
The dotted bands (pqjghjlmnp) of Fig. 10, and (ahhlmna) of Fig. 11 are what the dotted bands of Fig. 7 (species Ac) and Fig. 5 (species Ab) become, when the former is doubly and the latter singly distended.
Fig. 11.
Varieties of the species in class A (namely one variety for Aa, two for Ab, and three for Ac, making 6 cases in all) occur when we consider the situation of the figure A^ with respect to the uncrossed band round A^, A^. In all cases where A^ lies wholly inside this band we have (4i42.4,)= (i4,.4j), so that in all such cases the general formula (6), which gives the conjunctive probability, becomes
p, = (A,XA,) + (A,XA,)- (A,A,) - D.
Aa. We have i) = (^iX^)
so that p, = (A^X^i) - {^2 A,)
(the same as the result previously obtained from a priori consider-] ations).
Ab. 1. The figure A^ lies wholly within the uncrossed band round .4,,^!,] p, = (A, X A,) + {A, X A,) - (A,A,) - D,
Ab. 2. The figure A^ cuts the uncrossed band round 4,, J.,
p, = (A,A,A,) + (A,XA,) + {A,XAs)-{A,A,)-(A,A,)-D,.
Ac. 1. The figure A^ lies wholly witiiin the uncrossed band round Ai, .4,«
Ac. 2. The figure A^ cuts only one string of the uncrossed band round Ai, Af In these two cases the formulae which give p, are the same as in the corresponding varieties of Ab, except that D2 takes the place of A. .J
69]
On Buffon's Problem of the Needle
678
Ac. 3. The figure A^ cuts both strings of the uncrossed band round A^, A3. In this case the formula for the conjunctive probability
p, = (A.A.A,) + (A, X A,) + (A,X ^3) - {A A,) - (A,A,) - A becomes greatly simplified ; for (see Fig. 12)
Da - (AjA^A,) = rsjgu + vljht -rt — vu= (J2X -^s) - (A,iA^ 80 that /)3 = (^1X^3) -(^1^3),
which is evidently true, since every straight line which cuts both A^ and A3 must also (in this ca.se) cut A^.
Fig. 12. We have now enumerated all the six cases of Class A, and given in each case the formula for the conjunctive probability (from which, by means of formula (3), the disjunctive probability may be determined immediately). We proceed to the discussion of Class B.
In Class B (that is in the cla.ss where each figure lies entirely outside the crossed band round the other two) we recognize four species, and in one of them two varieties, making five cases in all. The enumeration is as follows.
Ba. There is one definite order of succession in which the three figures can be cut by a system of straight lines. There are two varieties of this .species, namely
Ba. 1. The middle figure (J.^, see Fig. 13) lies wholly inside the uncrossed band round the other two. The small cresses in this figure, as in others, indicate the positions of the points i, j where the string looped round .4,, A,, A, (see Fig. 3) crosses itself.
Fig. 13.
8. IV.
43
674
On Buffon's Problem of the Needle
[69
Ba. 2. The middle figure cute the uncrossed band round the other two as shown in Fig. 14. In this, as in the preceding case, both laiaAj lie outside the crossed, but inside the uncrossed, band round
Fig. 14.
Bb. The figures may be cut in two diflFerent orders by two distinct systems of straight lines (see Fig. 15). One system of straight lines cuts the figures in the order A^, A^, A^, the other system cuts them in the order .43, A^, A^.
2U==^
Fig. 15.
Be. The figures may be cut by three distinct systems of straight lines (Fig. 16).
Bd. The three figures cannot all be cut by any straight line (Fig. 17). In all cases with the exception of Bd, which will be treated separately, we have (see formula (4) ante [p. 668])
p, = {A,A,A,) - {A,A,) + (fi, X ^3) + (J5,) - {B,B,) - (B,B,).
* This circnmstance enables us to discuss Ba. 1 and Ba. 2 simultaneously.
69]
On Buffo7i's Problem of the Needle
675
In Ba (see Fig. 18) we have
{B^B,) = (^a 4 3) + Ai + ik -kc-cd- dh , (BjBi) = (AjA,) + mj+jn — nf—fe — em, (B, X -B,) = (fii) + (^3) + ik - kc -or- rj +jn - nf-/p -pi.
Fig. 16.
Fig. 17.
Fig. 18.
Substituting these values in the general expression for ps, we obtain
p^^(A,A,A,) - {A,A,)-(A,A,)-(A,A,)
+ (Bi) + (5,) + (B,) -mr-rc+cd+dh- hp -pf+fe + em
where the term — vir comes from — mj — rj, and the term — hp comes from — hi — pi ; the other terms involving the points i, j or the points of contact k, n of tangents drawn from them to the original figures disappear in pairs. The terms
(5.) + (B,) + (B,) - mr -re + cd + dh - hp -pf+fe + em will be seen to coalesce into a single band (whose course is marked in Fig. 18 by the continuous line aqigljsbkcdhmefna, all other lines in the figure being dotted). This band we shall call A,.
43—2
676
On Buffona Problem of the Needle
[69
Fig. 18 is drawn for the case Ba. 2, but the investigation of case Ba. 1 is precisely the same as that of Ba. 2. In both cases we find
p, = {A,A,A,)-(A,A,)-(A3A,)-(A,A,) + £i, for the conjunctive probability, and consequently
^,^(A,) + (A,) + (A,) + (A,A,A,)-{A,XA,)-iA,XA,)-(A,XA,)-i-A, gives the disjunctive probability in both cases.
The band A, for the case Ba. 1 is shown by the continuous line of Fig. 19, that is A, is the band atqglsvhxcdwuefya : its course is precisely the same as that of the Aj for the case Ba. 2.
Fig. 19.
The difference between the iwo cases is this : in Ba. 1 we have
{A,A,A,) = {A,A,) so that ps = Ai - {A^A^ - {A^A^)*,
whereas in Ba. 2 (and in all the cases to be subsequently considered) the terms {A^A^ + {A^Ai)-{-{AiA.i) — {AiA3Ai)coa.\esce\nio a, single band which we shall call A, so that
J93 = Ai - A. ^
The course of the band A is marked by the letters ahkcdhmefna in Fig. 18. The band Aj may be derived from A by supposing its rectilinear portion ah to be pressed inwards by the figiire A2 so as to occupy the position aqglsb.
The investigation of the case Bb proceeds on exactly the same lines as
that of Ba. 2; we start from the same general formula and, by performing
precisely similar work, obtain the result
^3= Aj-A,
where (see Fig. 15) A is the band ahxcdzefya whose course is indicated by
dots, and Aj is the band derived from A by supposing two of its rectilinear
portions ah, cd to be pressed inwards by the figures .4, and A^.
* By an easy rearrangement of the bands the value of p^ for this case may be eipre-saed as the difference of the two bands, atuelgdwvbxya and atqgleuwdgUvbxya (see Fig. 19), derived from the uncrossed band abxya round A^, A^ by twiatiiuj its rectilinear portion ab right round A, in opposite directions.
69] On Buf oil's Problem of the Needle 677
In the case Be (Fig. 16) the work is simplified by observing that each of the figures Ai, A2, A, blocks the channel between the other two (that is, no straight line can pass between any two of them without cutting the third). Hence every straight line which cuts the uncrossed band round all the figures must cut one or more of them ; that is
and consequently formula (3) gives
p. = (A,A,A,) - (A,A,) - iA,A,) - {A,A,)
+ (A,XA,) + {A,XAr) + iA,XA,)-(AO-{A,)-(A,). Now it is easily seen that
(A,A,) + {A,A,) + {A,A,) - (A,A,A,) = A and (^, X A,) + (A, X A,) + {A, X A,) - (A,) - (4,) - {A,) = A,
where A is the band abxcdzefya (shown by the dotted line in Fig. 16) and A, is what A becomes when its rectilinear portions ah, cd, e/are pressed inwards by the figures A,, A^, A3.
Thus Pa = A3 — A.
The sole remaining case of three figures is Bd (Fig. 17), the case in which no straigiit line can possibly cut all three figures. In it we have obviou.sly
p.= 0, and therefore
«r, = {A,) + (A,) + (A,) + iA,A,) + (A.A,) + {A,A,)
-{A,XA,)-(A,XA,)-{AXA,). This case forms no exception to the general rule for finding the conjunctive probability in cases belonging to class B.
We have A = abxcdzefya
(that is, A is the dotted band of Fig. 17), and since this band is not pressed inwards by any of the figures the conjunctive probability according to the rule would be A — A = 0, which is right.
Having thus pointed out the general method of procedure, and illustrated it by treating in detail the case of three figures, it does not seem desirable to pursue the subject further in this direction for the present ; but, before concluding, it may be worth while to notice that, in the general case of n limited right lines, the probabilities with which we have to do become Diophantine linear functions of the sides of the complete 2m-gonal figure of which the n pairs of extremities of the lines are the angles. There will be a group of such linear functions depending on the mutual disposition of the n lines, but the number of formulae in any such group will be much greater than in the case of n general figures : for, when we pass from these to indefinitely narrow ovals, the portion of a definite band (appearing in any
678
On Buffon's Problem of the Needle
[69
formula), partially surrounding any one of such ovals, may, according to the mutual disposition uf their major axes, have in common with it an infinitesimal arc in some cases, in others an arc (to an infinitesimal pris) equal to a cir- cumference, and again in others to a semicircumference of the oval ; which latter is ultimately the same as the length of the line whose double the complete circumference represents.
By way of illustration let us consider the question of two needles or limited straight lines rigidly connected. Neglecting the limiting cases, where one of the lines terminates in the other, there will remain three hypotheses :
A. The lines intersect.
B. The lines tend to intersect in a point external to each of them.
C. One of the lines tends towards a point lying within the other.
Let Pi denote the chance of both the needles AB, CD being cut by one of the parallels, tr^ the chance of one or other of them being cut : then we have the general formulae applicable to all cases
vr^=2AB + 2GD-p^, Pa = difference between the crossed and tmcrossed bands round AB, CD.
A. When the lines intersect
iST,=^AD + DB + BC+CA,
p^ = 2AB + 2GD-AD-BB-BG-CA.
o
C«ni
■iD
Fig. 20.
Fig. 21.
B.
When the lines tend to intersect in a point external to each of them
p^ = (AB + BG+CD + DA)-(AB + BD + DC+CA)
= BG-GA + AD-DB*, rs^=2AB + 2GD-BG-\-GA-AD-\-DB.
* Imagine a string passing from B to <7, from C to ^, from A to D, and from D to B. Tliis string oannot be kept tight unless fastened by pins at A, B, C, D. Inserting the necessary pins and tightening the string, we agree to consider the oonsecatire portions of the string as alternately positive and negative.
On these suppositions p.^ is the algebraical length of the band BCADB stretched round the pins. The method of representation by means of pinned bands may be extended to the case of two (or any number of) general figures.
69] On Bufon's Problem of the Needle 679
C. When one of the lines tends towards a point lying within the other
p, = {1AB + BC+CD + DB) - (AG + GB + DA)
= 2AB + BG-GA-AI) + DB, ^, = 2CD -BG+GA+AD- DB.
I
ipD
Fig. 22.
The complexity of cases for three right lines is such as would require a separate study even to obtain a perfect enumeration of them ; consequently W shall leave it to others to pursue the subject further whether as regards principles or details. I will only add that the ascertainment of the general law that the formulae contain no other arguments than lengths of tight endless bands variously drawn round the given contours appears to me a distinct step achieved in the prosecution of this extensive theory, and one that is far from being obvious d priori. Buffon's problem of the needle, it will be seen, has now expanded into a problem of n needles rigidly connected, which may be treated as a corollary to that of n entirely separate general contours, the mode of solution of which, it is believed, has been sufficiently indicated in the investigations which form the subject of this memoir.
PoSTSCBiPTUM. Since the above was set up in print my attention has been called to the fact that the extension of Barbier's theorem referred to on p. [664] is due to Prof Crofton and is given by him in his celebrated paper on the Theory of Local Probability contained in the Philosophical Transactions for 1868. Strange to say, no reference to this, so far as I can find, is made in Czuber's treatise. It is the more singular that I should have overlooked the feet inasmuch as it was an outcome of conversations with myself, when Prof. Crofton was serving under me in the Royal Military Academy at Woolwich, that he was put upon the track of investigations in local probability in which he has since earned for himself so great and well merited celebrity. It may be added that Prof. Crofton seems to have written in entire ignorance of Barbier's discovery as he makes no allusion to it in his paper.
It is indeed a romantic incident in mathematical history that Buffon's problem of the needle should have led up (as is undoubtedly the case) to Crofton's new and striking theorems in the integral calculus reproduced in Bertrand's Calcvi integral.
70.
SUR LE RAPPORT DE LA CIRCONF^RENCE AU DTAM^TRE.
[Comptes Rendus, cxi. (1890), pp. 778—780.]
[See p. 682, below ; footnote.]
En ^tudiant la preuve de Lambert, du theorfeme que ir ne peut pas 6tre
la racine carr^e d'un nombre entier, je crois avoir trouvd le moyen d'en faire
rextension au th^orfeme de Lindemann, c'est-ti-dire que v ne peut pas 6tre la
racine d'une equation rationnelle. Par exemple, supposons que ir soit une
racine de I'^quation
Ax' + Bx+C^O,
ou en mettant Ax = p, que Air soit une racine de
p'' + Bp + AG=0;
prenons un nombre entier K, tel que K (B — Air) soit de la forme
2m7r+(l-0)|,
0 ^tant < 1 ; en mettant Kp = R, nous aurons I'dquation
R' + BR + E^O, (1)
dont KAtt sera une racine et I'autre une quantity dont la tangente sera J positive, rj.
Considt^rons la fraction continue
en mettant R = KAir, on aura
,S = 0; en mettant R = 'r), on aura
S' = v.
Or, prenons un nombre v tel que 2v > R^ et consid^rons les deux fractions
continues
^ R:' R' R^
"» = .
S'.=
2v+l-2v + 'S- 2«/ + 5"
■R'" fi' i?"
2j; + 1 - 2i' + 3- 2i/ + 5 ■■
70] Sur le rapport de la circonference au diametre 681
R, R' ^tant les deux racines de I'^quation quadratique (1)
A, B,G, D, ... 6tant des fonctions lineaires avec des coefficients en tiers de R, et Ton aura
„, _ B' - B\rj _ C - C\t) ' A'-A'iq' '^'~R-B\f,
A', R, C ^tant les memes fonctions de R' que le sent A, B, G de R.
Or, on pent ddmontrer que A', B", C, ... seront des nombres positifs, et
A' R C
-T7- . w , pT ■■■ cnacune > t}.
B' — B" n
De plus, toutes les fractions -p -— seront des quantites positives et
A — A it}
moindres que I'unit^.
Mais -r, — T-, rr^ = ^-77 — i dont le d^uominateur seia n^ces-
A A-A^r, ^.(i_^.,)
sairement positif.
jy
Done la quantity positive -j-, ^gale une fraction positive diminu^e d'une
A
autre fraction positive.
jy fi' T\'
Done -T, et les quantites serablables, „, , ^ seront toutes des fractions
positives et moindres que I'unit^.
„ BB CC Diy .AC.- iA .
Done -r-Tj , ^5-5, , -^j^, , ... seront des fractions possedant ce meme carac- AA BB CC
tere.
Mais tons ces produits AA', BB", CC seront des nombres entiers, ce qui est impossible.
Je crois pouvoir faire une demonstration tout k fait semblable pour ^tablir que TT ne peut pas ^tre la racine d'une Equation d'un degre quelconque dont toutes les racines sont r^elles. Pour le cas d'^quations avec des racines imaginaires, il y aura quelque chose de plus k faire pour achever la demon- stration ; mais j'ai lieu de croire qu'avec I'aide de la theorie des modules de quantites imaginaires il n'y aura paa de grosses difficult^s a vaincre. Enfin j'ajoute que deux quantites reelles ou imaginaires, dont I'une est la tangente ou le logarithme nepdrien de I'autre, ne peuvent etre toutes les deux fonctions alg^briques des racines de la m^me Equation irr^ductible, h coefficients entiers.
71.
PREUVE QUE TT NE PEUT PAS £tRE RACINE D'UNE EQUATION ALG^BRIQUE A COEFFICIENTS ENTIERS*.
[Gomptes Rmdus, cxi. (1890), pp. 866—871,]
Lehme. Sort
J = - ^
6 m n + -
// //
n +■
n"+. • »
ott 6° = e* = e"^= ... = 1 ; n,n', n" , ... sont des nombres reels positifs et plus
grands que I'uniU; m, m, m", ..., des nombres rdels ou complexes, et oil chaque
quotient partiel est assujetti d, la condition que n—\ est plus grand que le
module de m.
Alors je dis que le module de J sera moindre que I'unit^.
Supposons que ces conditions soient satisfaites par — , — ^ . Soit m = a + t/9.
Par hypothfese n-l>^{a'+ /3*).
Servons-nous de M (x) pour signifier le module de x, alors
\n/ n n
de sorte que, si — = a, + i/8„ «-' + yS," < 1 et, a plus forte raison, «,'<!,
n
I in I
V'^nl
M(m,) M(m,) M(m,)
* Cette Note doit «tre substitute k la Note de I'anteur qui a M ins^r^e, par suite d'un malentendn, dans lea Comptes rendus du 24 novembre dernier. La Note pr^o^dente, qui ne traitait que le cas le plus restreint du th^orfime du texte, est affect^e d'inexactitudes qui la rendent de nuUe valeur.
71] Racine dune equation algebrique a coefficients entiers 683
car {n, + ay, quand a, est compris entre les limites 1, — 1, est plus grand que Done, par hypothese,
et, evidemraent, par le meme raisonnetnent, on trouve suceessivement
^0-
If™, Jf/-S_\
"*' ' m/ "*"
m n„ +
m n,
ou, ce qui revient a la meme chose, toutes les quantites
n„ +
evi
seront moindres que I'unit^*.
Nous allons d^montrer, a I'aide de ce lemme, que, si 6 est une racine d'une Equation irreductible k coefBcients entiers, tang 6 ne peut pas etre rationnel ou meme une fonction rationnelle k coefficients rationnels de 0.
Supposons que A6^+ Bd"~^ + ... + L=0 et que tang 6 soit une fonction rationnelle de 6. On peut supposer que A = \, car, si nous ^crivons 6' = A6'\, alors I'equation pour 6' peut s'exprimer semblablement a celle pour 6, mais avec le premier coefficient ^gal a I'unitd De plus, si Ton peut d^montrer que tang ff ne peut pas Stre une fonction rationnelle de 9', alors, puisque ff = A6, et cons^quemment tang^', est une fonction rationnelle de tang^, il a'ensuivra que, si tang^ est une fonction rationnelle de 6, tang ^' sera uue fonction rationnelle de d", ce qui est contraire a la supposition faitej.
* Ce lemme peat Stre envisage comme une application de la proposition 8, ni d'Euclide. En prenaut 0 le centre d'an cercle k rayon nnit^ et A' un point ext^rieur k ce cerele, Euclide y eiueigne que le segment de ON, compris entre N et le contour convexe, sera moindre que toute •ntre ligne droite menfee de N au cercle : a plus forte raison il sera moindre que la distance de N i nn point quelconque d'un cercle int^rieur au premier. Voir la Note au bas de la pai^e [685, below] poor one addition qn'on doit faire k ce lemme.
t Voir le scolie pour le cas plus g^niral ofi les coefiicients de I'equation en 6 sont des nombres complexes [p. 686, below].
X Ij'illustre Legendre aarait, il me semble, d(i faire une transformation analogue dans sa presentation calibre de la preuve de Lambert de son th^orfeme (Note IV, EUmenta de Geometric). Pour avoir n^gligi cette precaution, la succession infinie de quantites toujours decroissantes qu'il tronve par le moyen du lemme de Lambert ne forme pas nicessairement une succession de nombres entiers, mais de tels nombres divis^s par des puissances toujours croissantes de A, le denominateur de 6, suppose rationnel, exprime comme fraction vulgaire reduite, ce qui n'est nnllement impossible.
684 Preuve qtie w ne pent pas etre rachie d'uiie [71
Done, nous pouvons supposer que I'^quation en 0 soil de la forme
d^ + B0"-' + ...+L = O. fividemraent on peut aussi supposer que I'^uation en 6 soit irr^ductible.
ferivons 0 tang 0 = r (0), de sorte que
0'
r(0) =
1— ^
3- "
5-.
on trouvera
^
^ 5 - . t(0)-0'
T{0)
0*
7-. _r{0)(S-0')-S0'
t(6>)-^
(9»
et, en noinmant
"9-. _T(0)(15-6ff')-U0'+0* T(e)(S-e')-:i0'
6"-
2rH-l-
2r + 3-.
C) (ft\ _Ar+,{e)r{8)-Br^,{0) ^"-^"^ Ar{0)T{e)-Br{d) '
^'■+'^"^-Ar^AO)r{0)-Br+,(0y
Soit 0^_i(0) ce que devient &r(0) quand on substitue 0i pour 0 dans la valeur de t(^). Si, pour une certaine racine 0i de I'^quation supposee en 0, 'rr,i{0) = Tr(0i), alors T^<(5)en vertu du lemme aura un module moindre que I'unite ; sinon, ce module deviendra eventuellement et restera, pour une certaine valeur r, et pour toute valeur sup^rieure, au-dessous d'une certaine limite, parce que dans ce cas ^r,i(0) diflP^rera et continuera k difKrer par
une quantity aussi petite qu'on veut de -J*'A ! (dont le module a une limite
supdrieure dependant de la grandeur de 6i) quand r est pris suffisamment grand. Cela sera d^veloppe au long dans une Communication ult^rieure. Supposons que N soit le plus grand des modules carr^s des n racines,
71] equation algebrique a coefficients entiers 685
6\, 02, ^j. •••> ^n les n racines de I'equation propos^e en 0. Prenons 2r >N; alors, en vertii du lemme* et a cause du principe enonc^ plus haut, on aura eventuellement (en prenant 2r — N suffisamment grand) le produit des modules de H^C^i), ©r (^2)1 •••. ®r{0n) moindre que I'unit^ pour une ceitaine valeur de r et toute valeur de r sup^rieure a celle-ci.
Or, remarquons que, a cause de la valeur I'unM du coefficient de 0^ dans I'equation en 0, tous les A (0) et les £(6) seront des fonctions lineaires et entieres de 6, ff', ..., 0"~\ car si /i >n— 1, ^'' devient une fonction lineaire et entiere de (9, 0^, ..., 6"-'.
Ainsi, en supposantque k soit un nombre tel qui rende krid) une fonction lineaire entiere de 6, &', ..., d"~^, pour toute valeur de r,
k[Ar{e)T(e)-Br(0)]
sera une fonction rationnelle et entiere de 6 ; or, en vertu de ce qui a 6te dit, le produit des modules de
sera moindre que I'unit^ quand fi est plus grand que le nombre que nous avons nommd r. Mais le produit des modules de n quantites est le module de leur produit ; done
k'^U[Ar(0)Tid)-Br(0)], k-U[Ar+,ie)Tid)-Br+,id)l
k-n[Ar^,i0)T(d)-Br+,ie)i
formeront une succession iniinie de nombres entiers decroissants, ce qui est impossiblef.
Ainsi T [0) et consequemment tang 6 ne peut pas etre une fonction ration- nelle de 0 quand 0 est meiue d'une Equation a coefficients entiers.
Si nous supposons que tang^ soit une quantite rationnelle pure et simple, cela ne fait nul changement dans notre raisonnement ; ainsi, puisque
tang IT (ou bien si I'on veut tang^j est rationnel, tt ne peut pas etre laracine
d'une Equation algebrique k coefficients entiers.
Je d^montre par un proc^d^ k peu pres pareil k ce qui precede, la pro- position invers6, c'est-a-dire que, si tang 6^ est racine d'une equation algebrique, alors 0 ne peut pas etre une fonction rationnelle a coefficients rationnels de tang 0. Or, dans cette theorie, il n'y a nuUe distinction entre les quantites r^eiles et complexes, de sorte que V(~ 1) compte comme quantity eiitifere. Done tang V(— 1), et consequemment e, base des logarithmes nep^riens (qui
* On doit souB-entendre par le lemme la proposition ainsi nomm^e aa commenoement de cette Note, mais avec I'addition essentielle, facilement prouvde, que quand les n croissent continuelle- ment el les m restent constants, alors, en commeni;ant avec un r sufBsammeiit grand, le module de J deviendra une quantity aussi petite que Ton veut.
t Voir le scolie [p. 686, below] pour le cas plus g^n^ral oil I'equation en 0 a des coefficients complexes.
686 Racine dune Equation algebrique a coefficients erUiers [71
en est une fonction algebrique) ne peut pas ^tre racine d'une Equation algebrique k coefiRcients entiers. En r^unissant les deux precedes applicables k ces deux cas, on parvient k d^montrer un theor^me plus g^n^ral, k savoir :
Si une fonction trigonomitrique qaelconque el son amplitude sont lides ensemble par une Equation algihrique a coefficients entiers, ni I'une ni Vautre ne peut satis/aire d une equation algebrique a coeffixiients entiers, et comme cas particulier compris dans ce thdoreme, une fonction trigonometrique et son amplitude ne peuvent pas etre I'une une racine d'une Equation algibrique A coefficients entiers et Vautre aussi une racine d'une telle dquMtion*.
II y a un th^oreme un peu plus general, au moins en apparence, qu'on peut ddmontrer par un raisonnement tout a fait semblable.
Nommons une quantity qui est racine d'une Equation algebrique irr^- ductible k coefficients entiers, simples ou complexes, quantiti equationnelle, et les racines de la raeme Equation algebrique irreductible a coefficients entiers, quantit4s equationnelles associees ; de plus, nommons une quantity qui est racine d'une Equation dont les coefficients sout fonctions rationnelles d'un nombre quelconque d'autres quantites donn^es fonction equationnelle de ces quantites ; alors on peut affirmer qu'une fonction trigonometrique et son amplitude ne peuvent pas etre, toutes les deux, fonctions Equationnelles d'un meme systeme de quantites equationnelles associees. Cette proposition donne lieu de soup^onner qu'au moyen de formules propres aux fonctions elliptiques on pourrait demontrer qu'une fonction elliptique, son amplitude et son paramfetre ne peuvent pas etre, tous les trois, fonctions equationnelles d'un meme systeme de quantites equationnelles associees.
Scolie. On ne doit nullement exclure le cas od 6 serait propose comme racine d'une Equation k coefficients entiers, mais complexes.
Dans ce cas, si le coefficient dn premier terme en cette equation est a + 1/3, alors afin de pouvoir reduire I'equation a sa forme canonique ou ce coefficient est I'uuite, sans que le tangent du nouveau 6 cesse d'etre fonction rationneile de tang 6, il faut ecrire 6' = (a* + ^) Q.
On remarquera aussi que les produits [p. (j85, above]
au lieu d'etre entiers et rEels, deviendrout quantites complexes, mais entieres, dont les modules vont a I'infini en decroissant; de sorte que la demonstration donnee, pour le cas oil les coefficients de i'equation en d sont des nombres ordinaires, reste bonne pour le cas general.
• Ainsi on peut affirmer qu'une fonction trigonometrique et son amplitude, ou bien un nombre et son logarithme, ne peuvent pas etre tous les deux racines de deux Equations algiSbriques quelconques 4 coefficients entiers. Par exemple, cosfcosXTr) ne peut pas 6tre un nombre algebrique de Kroneoker, quand X est rationnel, car son amplitude cosXx est un tel nombre.
De m6me eN/^ + \/M + v'>' + ... ne peut pas etre racine d'une Equation algebrique k coefficients entiers.
72.
ON ARITHMETICAL SERIES.
[Messenger of Mathematics, xxi. (1892), pp. 1 — 19, 87 — 120.]
The first part of this article relates to the prime numbers (or so to say latent primes) contained as factors of the terms of given arithmetical series; the second part will deal with the actual or, say, visible primes included among such terms. Both investigations repose alike upon certain elementary theorems concerning the "index-sums" (relative to any given prime) of arithmetical series, whether simple and continuous as in the case of series ordinarily so called or compound and interstitial as such before named series become when subjected to certain periodic and uniform interruptions.
m PART I.
§ 1. Preliminary Notions.
Consider any given sequence
m+1, TO + 2, TO +3, ..., m + n, in relation to any given prime number q.
Let r be the sum of the indices of the highest powers of q which are con- tained in the several terms of the natural sequence
1, 2, 3. ..., n, a the sum of the indices of the highest powers of q contained in the given sequence.
Then it is almost immediately obvious that s = or > r, that is s > r — 1.
For the index-sum of the natural sequence will be represented by
and the index-sum of the given sequence by
-^(?) - -© - K5) -■
and this is at least equal to
that is « = or > r.
-© - ^(?) - ^© -
688 On Arithmetical Series [72
But there is another and more important theorem, less immediately obvious, and more germane to the subject-matter of the following section, which I proceed to explain.
Suppose <r,, (Tj, <r„ ..., <rn to be the several exponents of the highest powers of q which are contained in
x+1, x+2, x + S, ..., x+n, and let o- be one of these n exponents which is not less than any other of them. Call any term in the sequence
x + 1, x+2, X +3, ..., X + n
which contains q', say P, a principal ^'-term.
On one side of P the terms are less, on the other greater than P ; in lieu of any term substitute the difference between it and P, then I say that the g- index of such altered term will be the same as when it was unaltered.
For let the principal term, or the chosen principal term if there are more than one, be Xq", and let ftqi' be any other term.
If p < (7, 'Kg''*' fiq'' will obviously have p for its g'-index ; also if p = a- the same will be true, that is supposing fiqi' — \qi' to be positive, p will be its g-index : for if we write \ = aq + b and fi = cq + d, where b< q and d<q, a and c must be equal, since otherwise between X^*" and fjtqi' there would be a term (a + l)q.q'' containing a higher power of q than the principal term : hence fi — \ = d — b and does not contain q. In like manner if Xg' — /tg'' is positive, p is its g-index for the same reason as before.
Hence the index-sum, qud any prime q, of the two sequences
m + 1, m + 2 P-1; P+l, P + 2, ..., m + n-1, m + n
is the same as the sum of the index-sums of
1, 2, 3, ..., P-m-1, 1, 2, ..., 7n+ n -P.
Call the sum of these two index-sums s', then
and this is
= or < r.
72] On Arithmetical Series 689
Hence s'= or < r. But the original index-sum of the sequence is diminished by <7 on account of P being omitted.
Hence s — o- or s' = or < r.
Thus we have s>r — l,s~cr<r+l.
But this is not all : we may for certain relative values of m, «, and q (without regard to the situation of the principal term) establish the inequality s — a <r.
I premise the obviously true statement that if f + g < h, then
Let now h be the number of terms in the natural sequence from 1 to « which contain q.
Then in the given sequence the number will be
and the sum of the number of terms divisible by q in the partial sequences on each side of P will be h+e— 1, where e = 1 or 0 ; let the respective numbers be/, g. Then f + g = h — \ +e, where e = 0 or 1, and, using the same notation as before,
and
...«(|).^(«). "'-^©-^W)--
\q/ \q-
Hence if e = 0, s — a<r,
if e=l, 8 — <T<r+l,
the former inequality subsisting whenever
^(T)-^(?)-ni)-«-
If for example m = n, then s— a-<r when
/2n
^(|)-.^Q).o.
which it is easily seen happens whenever El — j is an even number.
8. IV. 44
690 On Arithmetical Series [72
§ 2. Proof that (m+ l)(w + 2) ... (m + n) when m>n—l contains a prime not contained in 1.2.S ...n*.
The universal condition independent of the relation between m, n, q, above found, namely, « — <r = or < r will be found sufficient to establish the theorem which constitutes the object of this section and which is as follows: —
"If the first term of a sequence is greater than the number of terms in j it, then one term at least must be a prime or a multiple of a prime greater J than that number.*'
When the first term exceeds by unity the number of terms, the sequence
takes the form m+1, m+2 2m— 1, and since no term in this sequence
can be a multiple of in, the theorem for such case is tantamoimt to affirming that one term at least is a prime number which is in accord with and an easy inference from the well-known " postulate of Bertrand," that between m and 2m — 2 there must always be included some prime number when m>|.
Suppose if possible that m + l,m-|-2, ..., m + n contains no other primes than such as are not greater than n, and which therefore divide some of the numbers from 1 to n.
Let q be any such prime, and P, a principal term of the sequence
m + 1, m+ 2, ..., m + «., qua gr. Then, by virtue of the proposition above established, (m-f-l)(m + 2)...(w + w) ^,
will contain no higher power of q than does 1 . 2 . 3 . . . w, and consequently if P be the least common multiple of the principal terms in respect to the several primes, say v in number (unity not being reckoned one of them), none greater than n, we may infer that
(m + l)(m + 2)...(m-l-w) P will be wholly contained in, and therefore not greater than 1.2.3... n, if the sequence m +1, m+ 2, ..., m + n contains no prime or multiple of a prime greater than n. To fix the ideas let us agree to consider that term in the sequence which contains the highest power of q, and is the greatest of all that do the same (if there be more than one), the principal q-term. The least common multiple cannot be greater than the product of the principal terms which are distinct from each other, and since even if they are all distinct, their number cannot exceed v (the number of primes other than
• It will readily be seen that, if this theorem is true, for n any prime, it will be so & fortiori when n is a composite number.
72] On Arithmetical Series 691
unity less than n + 1), it follows that P cannot be greater than the product of the highest v terms in the given sequence. Hence we may infer that unless
(m + l)(m+ 2)... {m + n — v)
is less than 1 . 2 . 3 . . . n, some prime greater than n must divide one term at least of the sequence
m + 1, m + 2, ..., m + n.
We might go further and say that unless 1 . 2 . 3 . . . n is greater than
{m+\){m + i) ...{m + n-v)D,
^ ^ i+^C?) +£(?). B(t!H^) where D=llq ^<i> V ^ i ' ,
{q being made successively each of the v primes between 2 and n inclusive and n being used in the ordinary sense of indicating products), this same conclusion must obtain.
Conversely the theorem is true when either of these inequalities is denied. The denial of the first of them, which is sufficient for the object in view, is implied in the inequality
(n» + l)(m + 2)... (m + «-!')> 1.2.3.. . n,
which, since v depends only on n, may be written under the form
J?'(w, n)> 1.2.3. ..n.
This will be referred to hereafter, in this section, as the fundamental inequality*.
Since F(rn,n) increases with m, the theorem if true for m must be true for any greater value of m, when n remains constant.
From this it will be seen at once that the theorem must be true when m has any value exceediug n" and n > 7.
For when n = 8 the number of primes in the range from 1 to 8 is 4 and is equal to ^n : but as n increases the number of new primes being less than the number of odd numbers must be less than ^n.
Hence if n > 7 and m > n',
F{m, n) > m"-' > (n')*" > n» > 1 . 2 . 3 . . . n.
This result enables us to prove that the theorem is true when
13 < n< 3000.
The theorem it will be borne in mind is true if some prime number
occurs in the sequence m + 1, m+2 m + n, or in other words if the
above sequence does not consist exclu-sively of composite numbers. But
* The sabeiBtAnce of the fandamental inequality for any given value of n implies for that value of n the truth of the theorem to be established: but the converse does not necessarily hold. The theorem may be true when the fundamental inequality is not satisfied.
44—2
692 On Arithmetical Series [72
Dr Glaisher has found* that the highest sequence of composite numbers within the first 9000000 contains only 153 terms, namely, the sequence 4652354 to 4652506 (both inclusive). Hence if the theorem is not true when n < 3000, in which case n' + to< 9000000, we must have n => or < 153, and there ought to be a sequence of n composite numbers in which the first term is less than (153)' which is 23409. But the longest sequence of com- posite numbers under 23409 is that which extends from 19610 to 196G0 containing 51 terms, the square of 51 is 2601 and the longest sequence under this number is that which extends from 1328 to 1360 comprising .S3 terms. The square of 33 is 1089, the longest sequence below which is from 888 to 906 comprising 19 terms: the square of 19 is 361, the longest sequence below which stretches from 114 to 126 comprising 1.'} terms. Hence the theorem is true for all values of n not greater than 3000 and not less than 13.
It is easy to show that the theorem is true for all values of n not greater than 13.
(1) Suppose M= 13, which gives v = 6.
The theorem must be true when m is taken so great that (m + 1) (m + 2) (m + 3) (m + 4) (m + 5) (m + 6) (m + 7)
> 1.2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13, which is easily seen to be satisfied when m= or > 100.
But there is no sequence of 13 composite numbers till we come to the sequence 114 to 126, so that when m < 100 the theorem must be true as well as when »t = or > 100.
(2) Suppose n= 11, for which value oi n,v = 5. The theorem is true if
(TO + l)(m + 2)(m + 3)(m + 4)(m+5)(m + 6)
>1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11, which is obviously satisfied as before when m= 100, but there is no sequence of 11 which precedes the sequence before named from 114 to 126. Hence the theorem is true generally for n = 11.
When n=7, 1^=4 and the theorem is true for all values of m which make (rn + 1) (m + 2) (m + 3) > 1 . 2 . 3 . 4 . 5 . 6 . 7, that is, > 5040, which is obviously the case if m = or > 20, but there is no sequence of 7 composite numbers till we come to 89 to 97. Hence the theorem is proved for 71 = 7.
When w = 5, i* = 3 and the condition of the theorem is satisfied if (m + 1) (m + 2) > 2 . 3 . 4 . 5, that is, > 120, * See table at the end of this section.
I
72] On Arithmetical Series 698
as is the case if ?»= or > 10, but the first composite sequence of 5 terms is 24 to 28. In lilie manner when w = 3, i; = 2 and the theorem is true when TO + 1= or >1.2.3, that is, tw = or > 5, but 8, 9, 10 is the first composite sequence of 3 terms. Similarly, when w = 2, ^=1 and the condition TO + 1 = or > 2 is necessarily satisfied since m = or > n by hypothesis.
Finally, the theorem is obviously true when n = 1, because m + 1, what- ever TO may be, contains a factor greater than 1.
Being true for the prime numbers not exceeding 13, the slightest con- sideration will serve to prove that, as previously remarked in a footnote, it must be true a fortiori for all the composite numbers between them. Hence the theorem is verified for all values of n not greater than 3000, and it only remains to establish it for values of n exceeding that limit.
To prove it for this case we must begin with finding a superior limit to
V, when n > 3000, under the convenient form of a multiple of ; .
logn
If we multiply together the first 9 prime numbers from 2 to 23 and divide their product by that of the natural numbers up to 9 increased in the ratio of 1 to 2', the quotient will be found to exceed unity; and since the following primes are all more than twice the corresponding natural numbers, if we denote by Pi,p,,pt, ..., the prime numbers 2, 3, 5, ..., we must have
p^.p,.p,...p,>2'{1.2.3...v), (provided that v>22, sb is the case if ra = or > 89), or log(l . 2. 3 ... v) + (log 2) K < log {p^.p^.p,...p,).
But by Stirling's theorem (Serret, Cours d'Alg. Sup., ed. 4, vol. II. p. 226), i/log 1/ — 1/ — ^ log K + 1 log 27r < log (1 .2. 3 ...J/), and by Tchebycheff's theorem (Serret, vol. ii. p. 236)*,
^og(p^.p,.p2...py)<n',
where n' = f ^n + .-j-^ (log n)» + 1 log n + 2, and .4 =921292 ....
Hence (log ,>) („ - ^) - (1 - log 2) (i- - i) + (^ log 27r - ^ log ^e) < n', and a fortiori log {i> — \)(y — \) — (log \e)(v—\)< n',
(^-i)log|?(,.-i)|<.
2 , Hence, if we write /x. log /x. = - w = n,
we shall have i; — ^ < \efi..
* For greater simplicity I have left out the term - Aifi, and thereby increased the superior limit.
or ?(^_^)Iog|?(^-i)l<?,,'.
694 On Arithmetical Series [72
But /* = ; "■
log /i ' and therefore log /i = log m, - log log /* = log »! - log (log n, - log log fi) > log n, - log log re, .
Hence
'^ log n, - log log n,
< -
2 w^
e 2
log «' — log log n' + log-
log re' — log log w' — (1 — log 2)
Hence, observing that -, — =— , , -^ — ^— all decrease as the
° u u u log w
denominators increase (provided as regards the second of these fractions that w > e, as regards the third that w > e", and as regards the fourth that u > e*), we may find a superior limit to v in the case before us, where n > 3000, by writing in the numerator of v — ^,
(log 3000)' log 3000 2
3000 "' ^lOOCT"' 3000 "'
for (logn)', logn, 2,
and in its denominator, first, log n — log log n for log n' — log log re', and then log log 3000, J l-log2,
-ki^looo-^"^'^ ^°^I3p&^°s"'
for log log n and 1 — log 2 respectively.
Making the calculations it will be found that we shall get
v-^< 1-606 p^.
With the aid of this limit it will now be easy to prove the truth of the theorem when n = or > 3000.
Let us suppose n = or > 3000.
(1) Suppose m< 2n, then m + n> fm and the theorem will be proved for this case, if it can be shown that in the range of numbers from m to |m, there is at least one prime number when m = or > 3000.
• From this it will be seen that the asymptotic ratio of v to . — — is less than the asymptotic ratio which any saperior limit to the sum of the logarithms of the primes not exceeding n bears to n : this perhaps is a new result, at all events it is not to be found in Serret nor indeed is it wanted for Tohebycheff's proof of the famous postulate which Serret has so lucidly expounded.
The correlative theorem that the asymptotic ratio of v to . is always greater than the
asymptotic ratio which any inferior limit to the sum aforesaid bears to n is of course an obvious and familiar fact.
72] Oil Arithmetical Series 695
This will be the case (Serret, vol. ii. p. 239), if (on that supposition) f .|» — n, that is, if
« o ,/« > 25(log|?i)^ 125,, „ , 25
where .1 = -921 29202....
But when n — 3000, it will be found that the terms on the second side of the inequality are respectively less than
134164.1, 66-9773, 475546, 4-5227, whose sura is less than 750.
Hence, the inequality is satisfied, and accordingly the theorem is true when m < 2n and n is equal to or greater than 3000 ; for when n satisfies that condition the derivative in respect to n of the right-hand side of the above inequality will be always less than J.
(2) Suppose wi = or > 2w, then it is only necessary to prove that log(2n+ l)(27i + 2)...(37i-i/)>log(1.2.3... n), or, what is the same thing, that
log {1 . 2 . 3 . 4 ... (3n - i;)) > log (1 . 2 . 3 ... n) + log (1 . 2 . 3 ... 2m), V being the number of primes not greater than n, and n being at least 3000.
Call the two sides of the inequality P and Q. Then (Serret, vol. ii. p. 226) P > log V(27r) + (3n - v) log (3n -v)-{Zn-v)-^ log (3n - v)
> log V(27r) + (3n - v) log 3n + (3n - v) log (l " ^) - 3" + i* - ^ l<ig
3n
> log v'(27r) + 3 (log n) w + (3 log 3 - 3) ji - (log n) v
+ (1 - log 3) V - i log 3 - I log n - v,
W_<3,.-.„^(,-3i)..{l-i(£)-i(3^)--A(J-...i<».
On the other hand,
Q < log ij(2ir) + ft log n- n + ^ log n +^ + log \/(27r) + 2n log 2n - 2n + ^ log 2n + ri^ < \2 log V(27r) + J^ log 2 + ^} + 3 (log n) n + (2 log 2 - 3) n + log n. Hence P - Q > (3 log 3 - 2 log 2) 71 - (log «) 1/ - f log « - (log 3) I' - (i log (127r) + it} > (3 log 3 - 2 log 2) ?i - log n (i/ - i)
- 2 log n - log 3 (V - i) - (i log (367r) + ^J
where k - A < 1-606,-^. logn
696
But Heuce*
On Arithmetical Series 3 log 3 - 2 log 2 = 1-909.5415 > 1-909.
[72
P-Q> (-303) n - (1-606 log 3) j-— - 2 log n - {i log (367r) + ^j.
log n
say P-Q>f(n)>0 when n = 3000.
Also the derivative with respect to n of (log n)f{n) being
(-303) (1 + log «) - 1-606 log 3 - ii^ - ii^Hi^^^'^^±i ,
P — Q will increase as n increases and will remain positive for all values of n superior to 8000.
Hence the theorem is true, whatever m may be, when n = or > 3000, and since it has been proved previously for the case of n< 3000, it is true universally.
I subjoin the valuable table, kindly communicated to me by Dr Glaisher, referred to in the text above.
Table of Increasing Sequences of Composite Numbers interposed between Consecutive Primes included in the first nine million numbers.
|
Limits to sequence |
Number of terms |
|
|
7 to |
11 |
3 |
|
23 „ |
29 |
5 |
|
89 „ |
97 |
7 |
|
113 „ |
127 |
13 |
|
523 „ |
541 |
■ 17 |
|
887 „ |
907 |
19 |
|
1129 „ |
1151 |
21 |
|
1327 „ |
1361 |
33 |
|
9551 „ |
9587 |
35 |
|
15683 „ |
15727 |
43 |
|
19609 „ |
19661 |
51 |
|
31397 „ |
31469 |
71 |
|
155921 „ |
156007 |
85 |
|
373261 „ |
373373 |
HI |
|
492113 „ |
492227 |
113 |
|
1349533 „ |
1349651 |
117 |
|
1357201 „ |
1357333 |
131 |
|
2010733 „ |
2010881 |
147 |
|
4652353 „ |
4652507 |
153 |
* It will now be seen why I take separately the two cases of m greater and m less than in. K we were to take limpliciter m= or > n and were to attempt to prove log {1 . 2 . 3 ... (2n - 1-)} > 2 log (1 . 2 . 3 ... n) the inferior limit to the difference between these two quantities would then have for its principal term, not (3 log 3 - 2 log 2 - 1-606) n but (2 log 2 - 1-606) n, which would be negative.
Of course there is no special reason except of convenience (in dealing with an integer instead of a fraction) for making 'in the dividing point between the two suppositions separately con- sidered in the text ; xn where k as far as regards the second inequality does not fall short of some
I
72 J Oil Arithmetical Series 697
The table is to be understood as follows. The lowest sequence of as many as 3 consecutive composite numbers is tliat included between 7 and 1 1 : the lowest of as many as 5 is that included between 23 and 29, of as many as 7 that included between 89 and 97 ; between 13 and 17 there is a break — this indicates that the lowest sequence of as many as 15, or as many as 17 first occurs in the sequence of 17 interposed between 523, •541. Similarly the break between 21 and 33 indicates that the lowest sequence containing 23 or 25 or 27 or 29 or 31 or 33 terms first occurs in the sequence of 33 composite numbers interposed between the primes 1327, 1361.
It is also necessary to add that in the first nine million numbers tbere is no succession oi more than 153 consecutive composite numbers.
§ 3. Relating to irreducible arithmetical series in general*.
Let P be a principal term qua q in any irreducible arithmetical series whose common difference is i, N any other term greater or less than P, and 1) their difference. If q is not prime to i, no term in the series will be divisible by q.
Just as in the case of a natural sequence when there is only one principal terra in the series it may be shown that the index of D quS, q will be the same as that of N\ when there is more than one principal term it appears by the same reasoning as before that the index of N CHunot be greater than that of D: (it will not now necessarily be equal unless q is greater than the common difference i).
The index-sum qua q is zero when q has a common measure with i, and we may therefore consider only the case where q is relatively prime to i :
certain limit, wonid have served as well : this inferior limit to k wonld be some quantity a little greater (how macb exactly would have to be foand by trial) than the quantity 6 which makes
B log 9 -{$-!) log {6-1) equal to the coefiBcient of j in the superior limit to y. As regards
the first ineqoality «t wonld have to be a quantity somewhat less (how much less to be found by
trial) than the quantity 17 which makes ^ = |, that is, j) = 5. This is on the supposition made
throughout of nsing Tchebycheff's own limits, but if we use the more general, but less compact, limits iadicated in my paper in vol. iv. of the American Journal of Mathematicitt, any fraction not less than { and not so great as ^HH would take the place of f, and the extreme value of i; wonld be ViW. which is a trifle under 6. By a judicious choice of the value given to k, a value of n could be found considerably less than 3000, which would satisfy both inequalities, and this in the absence of Dr Ulaisher's table would have been a matter of some practical importance, but is of next to none when we have that table to draw upon. How low down in the scale of number, n may be taken, without interruption of the existence of the fundamental inequality for the minimum value of n in the case treated of in this section, it has not been necessary for the purpose in hand to ascertain. That it holds good for all values of n above a certain limit follows from the fact that 2 log 2 is greater than the coefficient of the leading term in the superior functional limit to the sum of the logarithms of the primes not greater than n.
* An irreducible arithmetical series is one whose terms are prime to their common difference. [X Vol. m. of this Reprint, p. 530.]
698
On Arithmetical Series
[72
on this supposition, by virtue of what has been stated above, the index-sum qu& q of the series whose firet term is m + i, and number of terms n, will be equal to or less than
E
P-m-i\
rr'h^n^-^)^^rr')
xq
m + ni — P Iq .
iq- + E
fm + ni — P\
and therefore d, fortiori
< or:
< or:
E
i)--(5
+ E
\qj \qV V^y
that is, not greater than the index-sum of 2, 3, .... n quS, q.
Consequently, by the same reasoning as that employed in the last section, the theorem now to be proved, namely, that if m (prime to i) = or > n, then (m + t){in + 2i)...(m + ni) must contain some one or more prime numbers greater than n, must be true whenever
(m + i) (wi + 20 (m + 3i) ... {m + (n-v,)i\ > 1 .2.3 ... m (0)* where vi is the number of prime numbers not exceeding n, and not contained in i, and d fortiori when for v^, we substitute, as for the present we shall do, V the entire number of primes not greater than n. This I term the fundamental inequality for the general case now under consideration.
Suppose n = or > 3000. The logarithm of the first side of the funda- mental inequality when we write v for v, is obviously greater than the I'th part of the logarithm of
(m + 1) (m +2) ... {m + i) (m + i+l) ...{m + (n - v) i] ; and the inequality (subject to certain suppositions) to be established will be satisfied, if on the same suppositions,
- log [1 . 2 . 3 ... {7ft -I- (n - v)i}] > log (1 . 2 . 3 ... >i) + i log(l . 2 .3 ... m).
V J,
Suppose m = », and make
log[1.2.3...1(i-f l)n-tVj]=r, (i + l)log{1.2.S...n)=U,
F{n,i)=^T- U.
* If it had been necessary the condition in the text might have been stated in the more stringent form that some aliquot part of the factorial of n (namely, this factorial divested of all powers of prime numbers contained in f) would have to be greater than
{m + i) (m + 2t) ...{m+ (n-yj) i} if the theorem were not true for any specified values of m, n, i.
It will be noticed that when i is relatively prime to n, »i is less than v so that n - vj > « - » : some use will be made of the formula in the text when dealing with certain small values of n and m - n towards the end of the section.
72] On Arithmetical Series 699
Then T > log (27r) + {{i +l)n- iv} log {{i+l)n- iv}
- {(i + 1) n - iv} - I log [(i + l)n- iv}, U-cii+l) log V(27r) + (i + 1) w log n - (i + 1 ) n + ^ (i + 1 ) log n 4- tV C'^' + !)• Hence F{n, i) >-i log v'(27r) + {(i + 1) w - rVj log {(i + 1) n}
+ {(i+l)n-iv}\og[l-^^\yJ^
•i- iv - (i + 1) K log n - i log {(i + 1) n - iv} - H» + 1) log " - tj (* + 1) > j(i + 1) log (i + 1)} n - i log {(i + 1) «} V - J log {(i + 1) n} - J (i + 1) log w
-i^log(27^)-J^(^■+l). that is > {{i + 1) log {i + 1)} ti-i log {(i +l)n}v-^(i + 2) log ra - ^ log (i + 1)
-^ilog{27r)-i^ii+l) (H),
so that when n >3000 and consequently v <^ + (1606) -, , the inequality
to be established will be true d fortiori if
F(n,i)>
(i+l)\ogii + \)-(l-60Q)i
\ , iog(^+iy
log n
I n — {i + 1) logn
-[i{i+l) log (i + 1) + i {i log (27r)) + ^{i+ 1)].
When I = 1 or 2 or 3 the coefficient of n is negative ; consequently the limit to V before found is no longer applicable to bring out the desired result.
The case of t = l has been already disposed of; that of i = 2 may be disposed of, as I shall show, in a similar manner; when i = S, I shall raise the limit n from 3000 to 8100 of which the logarithm is so near to 9 that it may, for the purpose of the proof in hand, be regarded as equal to 9 without introducing any error in the inequality to be established, as the error involved will only affect the result in a figure beyond the 4th or 5th place of decimals, whereas the inequality in question depends on figures in the first decimal place. When this is done the theorem will be in effect demonstrated for the case of i = 3 ami n > 8100. For all values of n not greater than 8100 I shall be able to show that the fundamental inequality (0) is satisfied by employing the actual value of Vi or v instead of a limiting value of the latter.
Thus the fundamental inequality will be shown to subsist for all values of n when t = 3 and m = n, and d fortiori therefore for all values of m and i not leas than n and 3 respectively.
Case of i-=1.
Suppose n = or > 3000, and take separately the cases m < or = 2n, to > In.
(1) Let m be not greater than 2n so that m + 2n is greater than 2wj — 1.
700
On Arithmetical Series
[72
By hypothesis m must be odd, and by Bertrand's Postulate
m + 2, m + 3, m + 4, ..., 2m,
and therefore ni+2, m+4, wi + 6, .... (2m— 1)
(seeing that the interpolated terms are all even) must contain a prime, and thus the first case is disposed of.
(2) Since the fundamental inequality has been shown to be satisfied when n > 3000, vi > 2n, t = 1 , it will d fortiori be so when n > 3000, m > 2n, i = 2.
Hence the theorem is established for i = 2 when n > 3000. Finally as regards values of n inferior to 3000, the reasoning employed for the case of t = 1 applies a fortiori to the case of i = 2.
To see this let us recall the first step of the reasoning applicable to the supposition of i = 1.
Because in the first nine million numbers there is no sequence of 3000 composite numbers, from Dr Glaisher's Table of Sequences (taken in con- junction with the fact that when 'm,>n^, the theorem has been proved to be true whatever n may be), we were able to infer that it must be true when n does not exceed 1-53: in the present case, if the theorem were not true when 3000 >M> 153, there would be a sequence of 1.53 composite odd numbers and therefore of over 305 composite consecutive numbers in the first 9000000 numbers, whereas there are not more than 153, and so we may proceed step by step till we arrive at the conclusion that the theorem must be true when n > 13; and when «= 13, 11, 7, 5, 3, 2, 1 a like method of disproof (but briefer) will apply as for the case of i = 1.
Case ofi^ or > 3.
Let n = or >8100. Then we may without ultimate error write
5 81 . , 9 2_
n
1-1056 +
+
v-k<.
4 log 6 8100 ^8100 8100
1-
log9 l-log2 ~9~ 9
log»
< 1-546
logn'
and accordingly i'(n, 3)>■
41og4-(3 X 1-546) (1 +
log 4
"gT
- 4 log « - (2 log 4 + f log 27r + J)
and ^^(8100, 3) > (5-545 - 5-352) (8100) - 36 - 5863 > 0.
Hence the Fundamental Inequality is satisfied when n = or >8100.
To prove that it is satisfied for values inferior to 8100, observe that by virtue of the formula (H) it will be so, ex abundantid, for all values of n not
72]
On Arithmetical Series
701
less than 'n and not greater than n', provided that, calling n\ the number of primes not exceeding n,
I (5-545) 'w — 3 log (4n') n'„ — | log w' — 0 > 0,
where G = ^ + log 2 + f log (27r) = 3783.
On trial it will be found that the above inequality is satisfied when we successively substitute for 'n, n, and for n\ (found from any Table for the enumeration of primes) the values given in the annexed table :
|
n' |
ll'r |
'n |
|
8100 |
1018 |
5725 |
|
5724 |
753 |
4096 |
|
4095 |
564 |
2967 |
|
2966 |
427 |
2172 |
|
2171 |
326 |
1604 |
|
1603 |
252 |
1200 |
|
1199 |
196 |
903 |
|
902 |
154 |
687 |
|
686 |
124 |
535 |
|
534 |
99 |
415 |
|
414 |
80 |
325 |
|
324 |
66 |
260 |
|
259 |
55 |
210 |
|
209 |
46 |
171 |
|
170 |
39 |
141 |
|
140 |
34 |
111 |
|
110 |
29 |
99 |
|
98 |
25 |
84 |
|
83 |
23 |
76 |
|
75 |
21 |
68 |
|
67 |
19 |
62 |
|
61 |
18 |
57 |
|
56 |
16 |
50 |
|
49 |
15 |
46 |
|
45 |
14 |
42 |
|
41 |
13 |
39 |
|
38 |
12 |
36 |
|
35 |
11 |
32 |
|
31 |
11 |
31 |
|
30 |
10 |
30 |
|
29 |
10 |
29 1 |
The fundamental theorem is therefore established when i > 2 for all values of n down to 29 inclusive.
It remains to consider the case where n is any prime number less than 29.
Calling fi the difference between n and the number of primes (exclusive of 1) not greater than n, to
n = 2, 3, 11, 17, 23 will correspond
At = 1,1, 6,10,14
702 On Arithmetical Series [72
and for each combination of these corresponding numbers it will be found
that
1.2.3...»i = or<(n + 3)(n + 6)...(n + 3/i).
Hence the theorem is proved for these values of n, whatever n may be,
when t = or > 3. To
n = 13, n = 19
corresponds
fx,= 7, /i=ll,
and fur these combinations of n and /* it will be found that
1.2.3...n.<(»i+4)(n + 7)...(n + l+ 3/ti),
80 that the theorem is true for
ft = 13, 19, except in the case where
m= 13, 19.
That it is true in these excepted cases follows from inspection of the
series,
16, 19, 22, 25, &c.,
22, 25, 28, 31, &c.,
where 19 > 13, 31 > 19 : or it might be proved, but more cumbrousl)-, by
the same method as that applied below to the only two values of n remaining
to be considered, namely
n = 5, n = 7,
for which we have respectively
ytl = 2, /ti = 3.
If n = 5 and i has no common measure with 2.3.4.5, i must be not less than7,but 1.2. 3.4.5 < 12.19.
On the other hand, if i has a common measure with 2.3.4.5, then what we have called Vi, in formula (0), is less than v, so that « — i^, > 2, but
1.2.3.4.-5<8.11.14.
These two inequalities combined serve to prove that, whatever i may be, the inequality (©) is satisfied, and the theorem is consequently proved for ■ft = 5.
So again, when n = 7, if i has no common measure with 2.3.4.5.6.7 it must be 11 at least. In that case the inequality 2. 3. 4. 5. 6. 7< 18. 29. 40, and in the contrary case the inequality 2 . 3 . 4 . 5 . 6 . 7 < 10 . 13 . 16 . 19 serves to prove the theorem.
When ?i = 1 the truth of the theorem is obvious : hence combining the results obtained in this and the preceding section, it will be seen we have proved that whatever n and whatever i may be, provided that m is relatively prime to i and not less than n, the product
(m + i) (m + 2t) ...(?« + ni)
72] On Arithmetical Series 703
must contain some prime number by which 2 . 3 . . . n is not divisible, and the wearisome proof is thus brought to a close. It will not surprise the author of it, if his work should sooner or later be superseded by one of a less piece-meal character — but he has sought in vain for any more compendious proof. He has not thought it necessary to produce the figures or refer in detail to the calculations giving the numerical results inserted in various places in the text : had he done so the number of pages, already exceeding what he had any previous idea of, would probably have been more than doubled *.
PART II f. Explicit Primes.
In this part I shall consider the asymptotic limits to the number of primes of certain irreducible linear forms mz + r comprised between a number x and a given fractional multiple thereof kx, the method of investigation being such that the asymptotic limits determined will be unaffected by the value of r, and will be the same for all values of m which
* The author was wandering in an endless maze in his attempts at a general proof of his theorem, nntil in an auspicious hour when taking a walk on the Banbury road (which leads out of Oxford) the Law of Ademption flashed upon his brain : meaning thereby the law (the nerve, 80 to say, of the preceding investigation) that if all the terms of a natural arithmetical series be incmaed by the same quantity so as to form a second such series, no prime number can enter in a higher power as a factor of the product of the terms of this latter series, when a suitable term has been taken away from it, than the highest power in which it enters as a factor into the product of the terms of the original series.
In Part II. I shall be able to apply the same method to demonstrate a theorem showing that it is always possible to split up an infinite arithmetical series, if not in the general case, at least for certain values of the common difference, into an infinite number of successive finite and determinable segments such that one or more primes shall be found in each such segment : a theorem which is, so to say, Dirichlet's theorem on arithmetical progressions cut up into slices.
The whole matter is thus made to rest on an elementary fundamental equality (Tchebycheff's) which, with the aid of an application of Stirling's theorem, leads (as the former has so admirably shown) inter alia to a superior limit to the sum of the logarithms of the primes not exceeding a given number, from which as has been seen in § 2, a superior limit may be deduced to the number of such primes. With the aid of this last limit together with an elementary funda- mental inequality and a renewed application of Stirling's theorem, all my results are made to flow. Thus a theorem of pure form is brought to depend on considerations of greater and less, or as we may express it, Quality is made to stoop its neck to the levelling yoke of Quantity.
Long and vain were my previous efforts to make the desired results hinge upon the properties of transposed Eratosthenes' scales : now we may hope to reverse the process and compel these scales to reveal the secret of their laws ander the new light shed upon them by the successful application of the Quantitative method.
t I ought to have stated that the theorem contained in section 2 of Part I. originally appeared in the form of a question (No. 10951) in the Educational Times for April of this year.
704
On Arithmetical Series
[72
have the same totient. The simplest case, and the foundation of all that follows, is that in which k = 0 and m = 2 : this will form the subject of the ensuing chapter which may be regarded as a supplement to Tchebycheff's celebrated memoir of 1850*, and as superseding my article thereon in vol. iv. of the Amer. Math. Journ. [Vol. in. of this Reprint, p. 530].
CHAPTER I.
ON THE ASYMPTOTIC LIMITS TO THE NUMBER OF PRIMES INFERIOR TO A GIVEN NUMBER.
§ 1. Crude determination of the asymptotic limits.
Call the .sum of the logarithms of primes not exceeding x (any real positive quantity) the prime-nuraber-logarithmic sum, or more briefly the prime-log-sum to x, and the sum of such sums to x and all its positive integer roots the prime-log-sum-sum, which in Serret is called i|r {x).
Then it follows from elementary arithmetical principles that the sum of this sum-sum to x and all its aliquot parts, that is
^(^) + V^(|) + V^(|)
which we may call the natural series of sum-sums and denote by T{x\ is identical with the logarithm of the factorial of the highest integer not exceeding x, and accordingly from Stirling's theorem may be shown to have for its asymptotic limit x\ogx — x. the superior and interior limits being this quantity with a residue which, as well for the one as for the other, is a known linear function of loga:. Serret, vol. ii. p. 226.
If now we take two sets of positive integers,
p,p',p", ...; q,q',q"
together forming what may be termed a harmonic scheme, meaning thereby that the sum of the reciprocals of the numbers in the two sets is the same, and extend the T series over x divided by the respective numbers in each set and take the difference between the two sums thus obtained, there will result a new series of the form
« = » //r\
. ".■^<"'*©:
of which the asymptotic limit will be x multiplied by
and the value o{/(n) will be
^logp ^\ogq
p q '
•9 '"' —S — ""p: q:'
* Published in the St Petersburg Transaction* for 1854.
72] On Arithmetical Series 705
where, in general, - means 1 or 0 according as n does or does not contain t, or in other words the " denumerant " of the equation ty = n.
»I shall call the ^'s and q's the stigmata of the scheme :
P 1
the stigmatic multiplier, and the new series in ■\jr{x) a stigmatic series of sum-sums (obtained, it will be noticed, by a four-fold process of summa- tion— namely, two infinite and two finite summations).
It is possible, in general (as will hereafter appear), to deduce from the asymptotic value of a stigmatic series of sum-sums, superior and inferior asymptotic limits to the sum-sum itself. The asymptotic limits to the simple sum will then be the same as those last named (Serret, vol. II. p. 236, formulae (8) and (9)*) and will be multiples of a;: dividing these respectively by log a;, we obtain superior and inferior asymptotic limits to the number of primes not exceeding x {Messenger, May 1891, p. 9, footnote [above, p. 694]).
It is obviously simplest always to take unity as one of the stigmata; those employed by Tchebycheff are 1, 30; 2, 3, 5; this scheme as I term it leads to the relation
-Hf3)-^(s)-nn)-^(f8)-t(s)-t(
-1
20)
+
+
= aiog2-Hilog3-l-^log5-^log30)a; + t,
the series extending to infinity but consisting of repetitions (with a differ- ence) of the above period, obtained by adding for the second period 30, for the third period 60, for the fourth period 90, and so on, to each denominator in the period set out. We may call this a period of 30 terms in which the coeflBcients are + 1, 0, or — 1. So, in general, whatever the stigmata may be, the stigmatic series will consist of periods of terms in each of which the total number of terms will be the least common multiple of the stigmata.
* The fonrtb edition, 1879, of Serret's Court d'Algibre Supirieure is referred to here and tfaioaghoat the paper.
t The + is used to denote that a quantity is omitted of inferior order of magnitude to x. The strict interpretation of the "relation" is that the sum of the stigmatic series less the stigmatic mnltiplier into x is intermediate to two known linear functions of logx.
8. IV. 45
706 On Arithmetical Series [72
Thus, for example, the schemes 1; 2, 2 and 1, 6; 2, 3, 3 would give rise to the relations
= (Jlog24-ilog2)x + = (log2)ar+...,
t«-tS)-t(f)-+(i).^(f)-+©+t(^)-t(s)-...
= (ilog2 + flog3-^log6)aj + = aiog2+.Hog3)^+...
of which the periods are 2 and 6 respectively.
The three schemes above given, whose keys, so to say, are 2, 3, 5 respec- tively (these being the highest prime numbers contained in the stigmata), possess the property that their effective coefficients are alternately plus and minus 1, and, in consequence thereof, we may immediately deduce from them asymptotic limits superior and inferior to the logarithmic sum-sum -«/r(a;).
Thus, calling the stigmatic multipliers in the three cases
8t^, Stj, Stt, we obtain as limits to the coefficient of a; in i^ (x), Sti and 2/Stj from the first, St^ ,, ^St^ „ „ second, and Sts „ ^Stf „ „ third scheme.
(Compare Serret, pp. 233, 234, where the A is the present Stt.) The three pairs of limits will thus be
•6931472 : 1-3862944,
•78035.52 : 1170.5328,
•9212920 : 11055504,
which are in regular order of closer and closer propinquity to unity on eacl
side of it*.
I
The question then arises can no further schemes be discovered which will enable us to bring the asymptotic coefficients still nearer to this empirical limit -f?
* Mr Hammond has noticed that the harmonic scheme 1, 12; 2, 3, 4 will also give rise to a stigmatic series in which the effective terms are alternately positive and negative units, namelv,
the stigmatic multiplier corresponding to which, say .S'(i2, is -8522758..., and therefore will
furnish the asymptotic coefficients Sti^ and ^Sti2, that is, •8522758 ... and 1-1363687
•^ The true asymptotic limit to the number of primes below x being according to Legendre'i
empirical rule , , the asymptotic value of iji (x) should presumably be x.
72] On Arithmetical Series 707
It would, I believe, be perfectly futile to seek foi" stigmatic schemes, involving higher prime numbers than 5, that should give rise to stigmatic series of sum-sums in which the successive coefficients should be alternately positive and negative unity, as in the above instances, but this although a sufficient is not a necessary condition in order that limits to a sum-sum may be capable of being extracted from the known limits to the sum of a series of such sum-sums.
This will be most easily explained by actually exhibiting a new scheme which is effective to the end in view, and showing why it is so.
Such a scheme is 1, 6, 70; 2, 3, 5, 7, 210, which, it will be observed, satisfies the necessary harmonic condition : for we have
^ + ff + 77 ~ 210 ~ ""'
The stigmatic multiplier is here i log 2 -f i log 3 -I- ^ log 5 -t- 1 log 7 + 5-hj log 210 - ^ log 6 - tV log 70 = -9787955, which I shall call D.
The stigmatic series arranged in sets in two different ways then becomes as a first arrangement
■f-W-f(^i
i«)^
*HBM&y-MS)-HB'MB-H^y-
Mm)Mm)-Hws)-H^y-*Hm)
45—2
708
On Arithmetical Series
+*(l55)- + (lTo)- + (lT2)' +'''(Tf3)-*(llo)'
[72
-*(if-,)
- ^ (ill) = + ^ (if?) + ^ (1I9/ • M*U/ ■ M*.
-t(lf7)-^(Tl9)-^(llo)-^(lfi)-^(if^) Kllo)+^(r63)-^(lf5)+^(ll7)
168/ ■ ■'^ (1I9) - ^ (ifo) + ^ (ifa) - ^ (ifs)
■ MT9)-^(iIo) + ^(iIt)-^(iI2)+^(iI7)
-^(i|-9)-^(ilo); ^'^{m)'-^{m)-'^{m)
-^(lle); +^(lf7)+^(ll9)-^(2^) + ^(2r9) -^(2fo)-^(2fo)' +^(2n)-^(2lo);
the correlative arrangement being
*w-*(ra)+*(n)++(ni)'
-*(n)-+(f5)-^(S)-^(n)- + (fo)-^(-S) -t(6)-t(5)+t©-t(S)++(i)-t(S)
— V
109/'
72] On Arithmetical Series 709
-^(lIo)-^(lf2) + ^(ll3)-^(ilo) + ^(ll[)
-^(lf6)-^(lfr)-^(ifo)-^(lfi)-^(lfe)
^^(ll7)+^(ll9)^-^(llo)-^(ll3)^-^(ll7) -^(ilo)-'-^(lfo)-^(lfi)=-^(lfe)-^(lf7)^
-^(ilo) + ^(ilr3); -^{w^^^{wi)-^ -^(ils)
-^(iIo) + ^(iIt); -^(ll2) + ^(lf7)= -^(im)
- ^ (ilo) + ^ (ifi) + ^ (A) ' - ^ (lis) - ^ (lie)
+ ^(ll7)+^(ll9); -^(2^)+^(m9)' -^(210) -^(2fo) + ^(2fl)-^(2-|o)+^(2fi)+^(2f3) =
The terms in each arrangement, it will be seen, are separated by marks of punctuation into groups : omitting the first group in either of them, which may be called the outstanding group, in each of the others the sum of the coefBcients is zero.
Moreover, the sum of the coefficients from the beginning of each group is always homonymous in sign, that is, will be non-negative in the first and non-positive in the second arrangement : the consequence of this is that all the terms of such groups may be resolved into pairs, whose sum will be necessarily positive in the one and negative in the other.
Thus, for example, in the first arrangement the last but one of the groups may be resolved into the pairs
^ (1S7) - ^ (2^0) ' ^ (199) - ^ (210) ■' ^ (4) - ^ (210) •
* Each of these arrangements is to be regarded as made up of the outstanding group and an infinite saccession of periodic groups. In the text we have set out the outstanding group and the first period, the other periods will be formed from this one by adding to each denominator in it taccessive multiples of 210.
710 On Arithmetical Series [72
etwjh of which is equal to zero or a positive quantity. So the eighth group of the second arrangement is resoluble into the pairs
-^(gD + ^dS)' -^(i?o) + ^(ro3)'
each of which is zero or a negative quantit)'.
It may be as well to notice in this place that the sum of the coefficients, reckoning from the first term of the outstanding group to the term whose denominator is n, is
which by virtue of the obvious identity,
:?;(r>^©.
hequalto x{£(p-£(p-.
This formula supplies an easy and valuable test for ascertaining the correctness of the determination of the coefficients up to any given term in the series.
These observations may be extended to any harmonic scheme whatever : for it will be observed that
2^£
^(p-^(i)}
is a periodic quantity, and therefore possesses both a maximum and a J minimum ; whence it is easy to see that, by taking the outstanding group ofl terms sufficiently extensive, all the remaining terms in either kind of j arrangement may be separated into groups similar to those above set out;j namely, such that the complete sum of the coefficients in each group from its first to its end term is zero and up to any intermediate term is homonymousA that is, always positive in one and always negative in the other arrangement*.
* For example, from the harmonic scheme 1, 15; 2, 3, 5, 30, we may derive a stigmatio series under the two forms of arrangement
.(x)-.g):..(f)-.(f„); ..(^)-.(^). -Kn)-Krv)-Kl"8)
-K3-6)^Ka-Kro)-Ho)-Ka^Kf3)-Hf7)^*«- In the above arrangements the groups are separated by semicolons and the period is marked oQt by the colons. In this instance it will be observed that minimum and maximum values of
72] On Arithmetical Series 711
The consequence of this is that the outstanding group in the first arrange- ment will always be less, and in the second arrangement always greater, than a function of which the principal, or, as we may call it, the asymptotic term, is the product of x by the stigmatic multiplier, say {St), the complete function being in each case of the form {St)x associated with a known linear function of log x. (Compare Serret, vol. ii. p. 232.)
The importance of this observation will become apparent in a subsequent section.
In the case before us (that is, for the scheme in the key of 7) confining our attention to the principal term of either limit, the first arrangement leads immediately (Serret, p. 234) to the superior asymptotic limit ^Dx.
As regards the inferior limit, we have
■>\r{x)>Dx-^. i$-Dx > \^Dx*.
Substituting for D its value 'dTSldoo, we obtain the asymptotic limits 1 0873505 and -8951370.
The corresponding values got from the TchebycheflSan scheme (1, 30 ; 2, 3, 5) being 1-1055504 and -9212920, which are the f^ and A of Serret.
We know aliunde that the true asymptotic values are each of them presumably unity. The superior value above obtained by the new scheme is thus seen to be better, and the inferior value worse than those given by Tchebycheff's scheme. But these values correspond to what may be termed the crude determination of the limits which the schemes are capable of affording. The contraction of these asymptotic limits by a method of continual successive approximation will form the subject of the following section f.
^i'') + '^(l^-^(^-^('s)-^('i)-^(^) are 0 and 2, and accordingly in the first
arrangement the outstanding group has to be continued until the snm of the coefficients of the terms which it contains is 0, and in the second until such sum is 2.
Writing g = ilog2 + Jlog3 + ilog5 + ,VIog30-yVlogl5='96750...,
we may deduce from the above, the asymptotic coefficients f Q and Q - iV ■ f Q ! ''^^t is, 1-1610 ... and -8992...
* Compare the determination of the limits for the harmonic scheme 1 ; 2, 3, 6 {American Journal of Mathematics, vol. iv. pp. 243, 244 [Vol. iii. of this Reprint, p. 642]).
t By the method about to be explained, it should be noticed, we may not merely improve upon the results obtained by the crude method from certain harmonic schemes (which form a very restricted class) but may also obtain limits to ^(x)-7-x from harmonic schemes which with- out its aid would be absolutely sterile (see p. [715]).
712
On Arithmetical Series
[72
^2. On a method of obtaining continually contracting asymptotic limits to
X
To fix the ideas let us consider the scheme (1, 30; 2, 3, 5) which leads to the stigmatic series
(1) - (6) +(7) - (10) + (11) - (12) + (13)- (15) + (17)- (18) + (19)
- (20) + (23) - (24) + (29) - (30) + (31) . . . ,
in which for brevity (n) is used to denote V^ (-) •
The sum of this series is, we know, intermediate between Dx + R (log x) and DiX + iJ, (log x), where i) = -9212920 .... A = 11055504 ... = f A
and R, Ri signify two known quantities which for uniformity may both be regarded as quadratic functions of log a; (in the first of which the coefficient of (loga;)^ is zero). (Serret, pp. 233, 235.)
Omitting every pair of consecutive terms — (m) + (/x) in which — < ^, and using [yfr (x)] to signify the asymptotic value of yfr (x), we find
[V.(.)]>i>.+ [^(£)]-[^(5)]>i>. + i>^-Ag.
say > B'x.
Similarly, omitting every consecutive pair of terms (rni)- (/*) in which
— <i, we find m "
[Vr(^)]<i)^ + A|-i)f+A~,
say < Di'x.
If instead of [y}t (x)] we had deduced limits to -^jr (x) in the manner indi- cated above, we should have found
1^ (x) >D'x + R' (log x), yjr (x) < Di'x + R^ (log x) ; the added terms being each of them quadratic functions of log x. Repeating this process we shall obtain
\^\r{x)\>D"x, {>^{x)]<D^'x, where D"= D + ^D'- ^^D,', A"= -D + iA'-|^'+ AA'-
Similarly we may write
[f{x)]>D"'x, [itix)]<D,"x. . where iy"^D + ^D"- ^\;D,", Dr=D + iD,"-}iy'+^D^". and so on.
72] On Arithmetical Series 713
If then we write for D, B', D", ..., v,,, «,, v^, ..., and for Di, i)/, D", ... , u„, u^, u.^, ... ,
we shall find in general
I i^ Vi Ui
where „.,^^ = Z) + ^^ - ^g ,
the complete statement of the inequalities being
^fr (x) > ViX + R * (log x), yfr (x) < UiX + Ri'" (log x),
where it is to be noticed that the supplemental terms always remain quadratic functions of log x.
(The result thus obtained differs in this particular from that stated by me in the Amer. Math. Jour. (vol. IV. p. 241)*; the process therein employed giving as supplemental terms rational integral functions of continually rising degrees of log a;. I am indebted to Mr Hammond for drawing my attention to this simple but important circumstance which had strangely escaped my attention previously.) To integrate the equations in u, v we have only to writ6
Vi=Vi + F, Ui=Ui + E,
F{l-^) + ^E = D, F.- = C,p,i + C,p,\ and to take for p,, p^ the two roots of the equation
\p-^' A |=p'-a+A+A)/'+A(i+TV)-ifk=o,
I 7 . P-i-lV I
that is />' - ^p + jUf^ = 0.
The roots of this equation being each less than 1, on making i= oo
we obtain Wa, = F, u^ = E, where E, F are deduced from the two algebraic
equations
^F+i^E = D,
This gives
^-m_l^_m_ , >, ^ 137 X 145 ^ 19865 _ p - (If f ) : (14 ti) - 304 X 56 17024 ^
(compare Amer. Math. Jour., vol. iv. p. 242),
^=Hm^= 10765779...,
-P'=imi^= -9226107...;
whence we may infer that •^ (x) may be made intermediate between two
[* See Vol. HI. of this Reprint, p. 539.]
714 On Arithmetical Series [72
known functions UiX + r (log tc), vtx + sQogx), where «,-, w,- may be brought indefinitely near to the numbers
10765779..., -9226107...;
and the supplemental terms are quadratic functions of logx depending upon the value of i that may be employed. We may, therefore (subject to au
obvious interpretation), treat E and F as asymptotic limits to > .*
If we examine the ratio of the denominators m, fj, of any pair of con- secutive terms throughout the entire infinite series, whether of the form
(m) — (/t) or — (m) + (/a), we shall find that - is always less than q (namely
1'16688...), except in the case of the pairs that have been retained in forming the equations between E and F, from which we may infer that if any of the discarded pairs had been retained we should have obtained values of E and F respectively greater and less than those above set forth.
If, on the other hand, q had turned out to be so much less than | as to cause
— in any rejected pair to be greater than q, in such case in order to obtain a
value of E the least, and of F the greatest, capable of being extracted fi-om the given scheme, it would have been necessary to take account of every such pair and perform the calculations afresh, thereby obtaining a new value of q (say q) less than the former one ; we should then have bad to continue the process of examining the rejected pairs and reinstating those (if any) whose
denominators furnished a ratio — greater than q', thereby obtaining a still
smaller value g". Repeating these operations toties quoties we should at
last arrive at a value of q superior to every ratio ~ throughout the entire
stigmatic series ; the corresponding values of the asymptotic limits would then be the best capable of being deduced from the given scheme.
Per contra had we retained at the start any of the discarded pairs of
terms, we should have found for q a value greater than the value of —
in some of the terms retained, which would be a sure indication that the retention of those terms had led to a greater value of q than was necessary ; those pairs would then have to be omitted ; the q calculated from the reformed equations would be diminished by so doing and the resulting values of E, F
* For the complete analytical determination of the limits to ^ (x) see § 3 of this chapter.
By making i sufficiently great u^, t'j may be brought indefinitely near to E, F: furthermore, ' when the superior and inferior limits of ^ (x)-^x are expressed as functions of .r and t of the form mentioned in the text, these limits may, by taking x sufficiently great, be brought indefinitely ' near \aui,v^, and therefore to E, F, which I therefore speak of throughout as asymptotic limits to \li (x)-i-x. But more strictly the optimistic limits actually arrived at are £ ' as little as we please greater than £, and F' as little as we please less than F.
I
72] On Arithmetical Series 715
would be the best attainable, provided that care was taken at the outset that no rejected pair gave a larger value to — than any pair that had been retained.
In the case we have considered initial asymptotic limits (namely D and 2)i) to ^—^ — were obtained from the scheme itself, but it will not always be possible to do this when we are dealing with any harmonic scheme.
Thus, for example, from the fact that the minor arrangement of the stigmatic series corresponding to the scheme [1, 6, 10, 210, 231, 1155 ; 2, 3, 5, 7, 11, 105] has (1) + (13) for its outstanding group [see p. 718], we may
deduce that i^ (a;) + i/r ( — J has Nx for its inferior asymptotic limit, but are
unable from this arrangement to obtain an initial inferior asymptotic limit to •<^{x) itself, and still less shall we be able to obtain an initial superior asymptotic limit to i/r (x) from the major arrangement of the same stigmatic series. It is therefore important to notice that the final asymptotic limits arrived at by the method explained in this section, depend only on the stigmatic multiplier and the coefficients of the stigmatic series, being quite independent of the initial values employed, so that in the general case we
may start from any given asymptotic limits to ^-^ — , however obtained, with- out thereby producing any eflfect in the final result. The limits i<„ = 21og2 and v^ = log 2 obtained from the scheme [1 ; 2, 2] will do as well as any others
... ^ (x)
for our initial asymptotic limits to "^-- , and we may, by substituting these
limits in the retained portion of the stigmatic series, arrive at new limits M,, Vi which in their turn will give rise to fresh limits u^, Vj and so on. We shall in this way obtain a pair of difference equations (connecting Mi+,, Vi+i with rii, Vi) which will be of the same form as those previously given [p. 713], and it is to be noticed that in the solutions of these equations, namely
Ui = Cp* + C,p,* +E, Vi = Kp' + K, pi* + F, only the values of G, C, , K, K^ will depend on the initial values of w, v; so that, provided the roots of the quadratic in p (which are always real) are each less than unity, we may, by taking i sufficiently great, make Ui and Vi approach as near as we please to E and F respectively ; that is as near as we please to two quantities whose values depend solely on the stigmatic series employed.
The positive and negative divergences from unity of the E and F pre- viously found are respectively
•0765779..., -0773893...; these divergences as found by Tchebycheff being
•1055504..., •0787080...,
716 On Arithmetical Series [72
which is already an important gain; but by varying the scheme we shall obtain still better results.
Let us apply the method of indefinite successive approximation to the scheme in the key of 7 treated of in the preceding section, namely [1, 6, 70 ; 2, 3, 5, 7, 210], for which the stigmatic multiplier (the D of p. [707]), namely
ilog2+^log3 + ^log5+|log7+^log210-^log6-7Vlog70 is -9787955....
Preliminary calculations having served to satisfy me that the asymptotic ratio -=, (the q) for this system was not likely to differ much from 110, which may be called the regulator, I form the corresponding equations for E and F by retaining only those pairs (m) — (/*) in the stigmatic series for which — is greater than 1-10.
As previously explained no error can result whatever regulator we employ ; the worst that can happen will be that the result will not be the best attain- able from the scheme, and such imperfection can be ascertained by means of the method previously explained; the result, if the best possible, will prove itself to be so, and, if not the best, will indicate whether the regulator (or criterion of retention) has been taken too small or too great.
Let us examine separately the two arrangements set out in the previous section, the first being employed to obtain by successive approximations the superior, and the second the inferior, limit.
Consider 1° the periodic part of the first arrangement : in the group (11) + (13) -(14) -(15), the pair (13) -(14) being rejected, (11) -(15) remains. Similarly, in the following group (19) — (20) being rejected, (17) -(21) remains; in the third and fourth groups (23) - (28) and (31) — (85) are to be retained. In the following group, all the consecutive pairs from (73) to (98) both inclusive are to be rejected, leaving (71) — (100) available. (The corresponding pair to this in the next period, namely (281) —(310), gives |^, which is less than the assumed regulator.) All the groups in the first period, following — (100), will have to be rejected until we come to the group beginning with (137), which leads to the available pair (137) — (190): in the next period all the ratios will be too small with the exception of (347) — (400) which must be retained, but the term corre- sponding to this in the third period, namely (557) — (610), will have to be neglected.
Hence, in approximating to the superior limit, we may write
«i+, = if + (tV + iV + ^V + 5^ + 3^ + Tk + T b + ?k) «i
- (tV + Vt + ]!V + bV + W + lir + sir) «»< •
72]
On Arithmetical Series
717
2°. In the second arrangement, the first group in the periodic part being
- (14) - (15) + (17) + (19), and {| (and d, fortiori f|) exceeding the regulator, all these terms are to be preserved.
In addition to these, we shall find in the first period the available couples
- (20) + (73) and -(110) + (139), and in the second period - (230) + (283) ; no other couples will be available, and accordingly, we shall have
- (iV + tV + iV + 1^ + tV + rk + Tfi^) w<- If then we write a, b for the coefficients of it,-, —vi in the first, and c, d for the coefficients of «i, — «< in the second of the above equations, and make «,- = Ui + E, Vi= r< + F, we shall obtain
Vi = Kpi + K,p: + F, where p, />, are the roots of the equation
p-a, b [ = 0, d , p-c I that is p^ — (a + c) p + (ac — bd) — 0,
and E, F are subject to the equations
(l-a)E + bF=^M, dE + {l -c)F=:M,
which give
E =
1-6-
,M, F =
\ — a — d
(l-a)(l -c)-6rf'" * (\-a)(l-c)-bd On performing the calculations, we shall find
M.
6= -24973..., d = -30371..., l-a-rf = -39995..., fed =07584..., (l-a)(l-c)-6fZ = -41563,
a = -29633..
c = -30153..
l_6-c = -44873..
ac = -08935..
a + c = -59786..
p, pi will therefore be the roots of
p=- -59786/3 + -01350 = 0, which are each less than unity.
Also j5:= 10.567 265..., if= -9418543 ...,
1-6-c
? =
= 1-12196,
\—a—d This last number being greater than the assumed regulator I'lO, and less
than any of the retained ratios
it follows that no better limits
718 On Arithmetical Series [72
than E, F can be extracted from the scheme [1, 6, 70; 2, 3, 5, 7, 210] ; or (as we may phrase it) E, F are the optimistic asymptotic limits to that scheme.
Obviously, there is no reason to suppose that these are the closest asymptotic limits that can be obtained from the infinite choice of schemes at our disposal : on the contrary, there is every reason to suppose that these limits may by schemes in higher and higher keys be brought to coincide as nearly as may be desired to each other and to unity.
We shall presently obtain by aid of a new scheme a better result than the E, F of the preceding investigation. But first it should be observed that instead of forming the difference equations mu,v from the two arrange- ments, say' the major and minor, of one and the same stigmatic series (the former meaning the one used to find the superior and the latter the inferior asymptotic limit), we may take these two arrangements, if we please, from two distinct series corresponding to two different schemes.
I have had calculated, from beginning to end, the value of the coeflBcient of each term in the stigmatic series of sum-sums corresponding to the first natural period, containing 2310 terms of the .scheme (1, 6, 10, 210, 231, 1155; 2, 3, 5, 7, 11, 105), the stigmatic multiplier to which, namely
i log 2 + i log 3 -f- i log 5 + 1 log 7 -t- tV log 11 + T*? log 105
- i log 6 - tV log 10 - jI^ log 210 - ^jir log 231 - ttVu log 1155,
is -9909532... (say N).
This stigmatic series, though too long for printing at full in the restricted space of this Journal, is given later on in a condensed tabular form (see Table A, p. 721). I will proceed to describe its essential features and the use made of it to bring the asymptotic limits closer together. The maximum and minimum sums of its coefficients are 2 and — 2 : the first terms being (1) -I- (13) — (14) —(15), the maximum is first reached at the second terra ; so that the outstanding group in the minor arrangement will be (1) + (13). But the minimum sum, — 2, is not reached before the term whose argument is (616). The outstanding group in the major arrangement will therefore contain a very great number of tenns, and there might be some trouble in handling the groups, so as to secure the greatest possible advantage. For this reason, I have thought it sufficient for the present to combine the major arrangement of the scheme [1, 6, 70; 2, 3, 5, 7, 210] with the minor one of the scheme [1, 6, 10, 210, 231, 1155 ; 2. 3, 5, 7, 11, 10.5].
Maintaining the regulator still at the same value as before, namely I'lO, the major arrangement will remain unaltered from what it was in the preceding case. In the minor arrangement there will be found to exist the
72]
On Arithmetical Series
719
following 17 available pairs, all of which, except the last, belong to the first period (the last one belonging to the second period), namely
(14) -(19). (15) -(17), (21) -(31), (33) -(41), (44) -(53), (63) -(73),
(84)-(97), (105)-(241), ^110)-(131), (195)-(223), (315) - (481),
(525) -(703), (735) -(943), (945) - (1231), (1484) - (1693),
(1694) -(2323), (4004) - (4633).
We may accordingly write
Ui+i = M + aui - bvi,
where
= N + yVi — Sui,
1
" 10 "•" 15 "•" 21 "*" 28 "^ 35 "^ 100 ■•" 190 "^ 400 ' 11 "^ 17 "^ 23 "^ 31 "^ 7l "^ 137 "•" 347 '
1
1
1
'^ 14 ■*■ 15 "^ 21 "^ 33 "^ 44 "*" 63 "*" 84 "^ 105 "*" 1 10 1
- J_ J_ J_ J_ _J_
195 ■*■ 315 "^ 525 "^ 735 "^ 945 "^ 1484
+
1694 4004'
k
13 "^ 17 "^ 19 "^ 31 "^ 41 "•" 53 "^ 73 ■•" 97 "^ 131 "^ 223
JL _L J_ J_ _J_
"^ 241 "•" 481 "'' 703 "^ 943 "*■ 1231
+
+
1
+
1
1693 2323 4633'
from which, writing we shall find
(l-a)E+bF=M, BE + {l-y)F=N,
where p, p, are the roots of
P - «. b ^ Q S , p-y
that is p^ — {a + y)p + arf-bB = 0.
The values of a, 6 ; y, S are respectively
•2963346..., -2497346...; -2992774..., -3107808...,
from which we see that p, p, being each less than unity the values of Mx, f« will be E, F, where
E = F=
(1 -y)M-hN (l-a)(l -7)-6S'
{\-a)N-hM (l-a)(l-7)-6S'
720
On Arithmetical Series
[72
and on performing the calculation it will be found that ^=10551851..., ^=-9461974.
Also
E
j = ^ = lllol8...,
which being greater than the assumed regulator, but less than any of the retained ratios — , the results thus obtained are optimistic, that is no better can be found without having recourse to some other harmonic scheme.
The advance made upon the determination of the asymptotic limits beyond what was known previously is already remarkable. TchebycheflF's asymptotic numbers stood at
1-1055504..., •9212920..., coiTesponding to a divergence from unity
1055504... in excess, and 0787080... in defect;
by the combined effect of scheme variation and successive substitution we have succeeded iu reducing these divergences to
■0551851 ... in excess,
and -0538026... in defect;
in which it will be noticed that the divergence for the superior limit is only a little more than half the original one.
The mean of the two limits, it will be seen, is now less than
10007.
The annexed table, in which for brevity c is written for — c, gives in a condensed form the stigmatic series to the scheme [1, 6, 10, 210, 231, 1155; 2, 3, 5, 7, 11, 105].
The coefficients, for all the terms "^ [ — ] from m = \ to m=llo5 (the
half modulus), are written down in regular batches of 10. The coefficients for the ensuing terms up to 2309 can be got from these by the formula Cnis5+e = Cn55_j, the term following will have the coefficient zero; the rest of the infinite series is then known from the formula Ce+.aioi = c«-
72]
On Arithmetical Series
721
Table A.
The coefficients of the first 1155 terms of the stigmatic series to [1, 6, 10, 210, 231, 1155; 2, 3, 5, 7, 11, 105]*.
1000000000 OOlTTOlOlO TTlOOOOTlO lOTOTOl^OOO
iTlTIOlOOO OOIOTTOOIO lOlOOTlOOT lOloTOTOlO
OOlTOOOTlO OOOOOOlTIO 101030101T OTlOOOOOOO
OOOOOllOOO lIOOTOlOlT OOOOOOTOIO 1002001000
0010201110 OOlOTTOOlO ITOOOOOOTO lOlOTTlTlO
OOOOOOOOOT 1000000001 lOlTTOlOlO TOlOOOOTlO
ITOOTOIOOO ITOOTOIOOO OOlTOlOOlO lOToTOlOOr
lOlOTTOOlO OOlTOOTOlO 0000001200 1010301000
OTlOOOOOOT lOOOOTlOOO ooooToioil OTlOOOTOlO
lOlTOOlOOO OOlTTOlTlO 0010200010 iTOOOllOlO
lOlOTTOOlO OOOOOOOTlI lOOOOOOOTO lOlTTOlOlT
loiooooTio ooooToiooo iTioToiooo oooooTooio
lOlTOOlOOT 1010200010 ooiToTooio OOOOOOOTOO
10ia30lTlO OTlOOOOOlO lOOOOTlOOT lOOOTOlOlT
TOIOOOTOIO ITOIOOIOOO OOOOTOlIlO OOlTTOOOlO
iTooToioIo 1010T21010 OOOOOOlOlT lOOOOOOTOO
loiTToiooo ToiooooTiT 1000101000 orioToiooo
oliooiooio lOTOOOlOOT lOlTTOOOlO ooiTToooio
OOOOOTlTOO 1010300010 oTiooooToo lOOOOllOTO
IOOOTOIOI2 0010001010 OOOTOOIOOO OllOlOlTlO
OOOOTOOOIO ITOIOOIOTO 1010211010 OOOOOTOOlT
1000001000 lOlITOlTlO TOlOOOOlOO lOOOlOlOOl
iTioioiooo loiooTooio illoooiool lOOOTOOOlO
0011000010 OOOOlOlTOO 10103T1010 OllOOOTOOO
lOOOOTllOO lOOOTOlOOl OOlOOOlOlI lOOTOOlOOO
loioToiiio OllOlOOOlO ITTOOOIOIO lOlTITlOlO
0000100011 1000010000 loiTioooio T010000210
1000101010 IIIOIOIOOT 0010010010 ooToooiool
lllOTOOOlO OOOTOOOOIO OOOlOOlTOO 10102+.
* This table is to be read off to lines. The first three lines set out in full (omitting the null terms) will mean
t By actual summation it will be found as stated above [p. 718] that the sum reckoned from the beginning of the positive and negative integers in this table always lies between 3 and -2 (both inclusive).
8. IV. 46
722 On Arithmetical Series [72
If we confine our attention exclusively to the outstanding group of the Major Arrangement, which extends to the 616th term mc\\xs\ve, without taking advantage of any of the other groups, we shall find, on making E= 10551851, F= -9461974, and N (the stigmatic multiplier) = -9909532,
X ^ ^ U5 22 ^ 28 35 45 ^ 56 ^ 66 ^ 77 ^ 88 ^ 99
, J_ , J_ . J_ . J_U. ■*" 105 ■*■ 126 "^ 525 ^ 616/
~ ll7 ■*" 23 "^ 29 ■*" 37 "^ 47 ■'' 59 "'' 71 "^ 79 "^ 89 ■'■ 1 13 ■*■ 227 j < 10542390 ... which is inferior in value to E. This is enough to assure us that a better result than the one last found would be obtained by using the above scheme to furnish the major as well as the minor arrangement, instead of combining it, as we have done, with the scheme [1, 6, 70 ; 2, 3. 5, 7, 210].
Mr Hammond has been good enough to work out for me in the annexed scholium the complete approximation to the limits to -v^ {x) given by the original scheme of TchebychefF [1, 30; 2, 3, 5]: this approximation pre- serves precisely the same form as that obtained by the crude method, and, although it lies a little out of the track which I had marked out for myself in this paper, will, I think, besides being possibly valuable for future purposes in a more or less remote future, serve as an example to clear up any obscurity that may have pervaded the previous exposition of the purely asymptotic portion of these limits*.
§ 3. Scholium. Containing an example of the complete ith approximation to the limits to tlie prime-log-sum-sum, to x.
Using 8 to denote the stigmatic series we have the inequalities
which, as explained in the preceding section, may be replaced by
' x\ . I X
^{x)>Ax-\\ogx-\^^[^^—<Sf\^'^ (1),
^{x)<Ax^\\ogx^<fi^-<f{^+^(^ (2).
* In the paragraph [last but one] of p. [709] in the preceding number, a theorem (too simple to require a formal proof) is tacitly assumed which virtually amounts to saying :
If an equal number of black and white beads be strung upon a wire, in such a way that on tellinij them all, from left to right, more white than black mies are never told off, then the whole number of beads, as they stand, may be sorted into pairs, in each of which a bUick bead lies to the left of a white one.
72] On Arithmetical Series 723
If now we assume
i/r {x) > fiAx + qi (log xj + Ti (log X) + s,- (3),
ti/r {x) < <,.4a; + Wj (log x)- + ?;< (log x) + w; (4),
e obtain, by combining these inequalities with (1), ■^(x)>Ax —flog a; -1
— ^ + iiPiAx + qi (log X - log 24)2 4. ^. (j^g ^ _ i^g 24) + s,.
^" — ^tiAx — Ui (log X — log 29)^ — Vi (log a; — log 29) — Wj.
Say -\/r (x) > pi+iAx + qi+^ (log x)' + n+i (log x) + Si+i,
where
i'i+i = Ai'i-^'i + l. qi+i = qi-Ui,
I^i+i = r-i - 1?< + 2Mi log 29 - 2qi log 24 - 1, Si+i = Si - w,- + qi (log 24)2 _ j^. (log 29)2 _ n log 24 + Vi log 29 - 1. Similarly, combining (3) and (4) with (2), we find ■^ (x) < Ax + f log X
+ ^tiAx + Ui{\ogx-\og 6)= + z;; (log a: - log 6) + Wi - ^piAx-qi(\ogx -log 7)2 -n (log a; -log 7)-Si + ^tiAx+ Ui (log X — log 10)2 + Vi (log a; — log 10) + Wj. Say yfr(x)<ti+iAx + M,+, (log a;)2 + z;,+, (log a;) + Wi+„
where
M,+, = 2m,- - 5i,
Vi+i = 2vi - n + 2(7.- log 7 - 2m< log 60 + 1,
Wf+i = 2Wi - Si - qt (log 7)2 + iti {(log 6)' + (log 10)'} + n log 7-Vi log 60. These, together with the four given above, constitute a set of eight differ- ence equations for the determination of pi, qi, r,-, s,-, tt, Ui, Vi, Wf. Their initial values are furnished by the inequalities
■<fr (x) >Ax — ^\ogx —1 1
n-)<iAx + ^ (log .)2 + f log . + 1 1 (Serret, p. 236),
which give Po=l, 9o = 0 , ro = -f, s„ = -l,
The values of pi, ti will be found to be
50999 [^^'"^'^ + 801^ (p' + P,0 + 190IMJ (7=7:)} '
4Q— 2
724 On Arithmetical Series [72
where p, p, are the roots of the equation
(/'-T^)(P-1!V)=203'
and it is easy to verify that these values (which agree with the general ones, involving arbitrary constants, obtained in the preceding section) satisfy the initial conditions
The values of qi and Wj, obtained from the equations
qi+i = qi- Wi, Mt+i = 2Mi - qi ,
witii the initial conditions
^° = ^' ""^fills'
"'■ = -4146'
5 / . , o'-a-'N
Mi= n-T — w a* + a-' + ,
8 log 6 V o — a ' /
where a, a~' are the roots of the equation
a" - 3a + 1 = 0.
The values of r,-, Sj, Vi, wi are linear functions of q^, Ui whose coeflScients are linear functions of i in the case of ?•,•, Vi and quadratic functions of i in the case of s,-, Wj.
Thus we find, when the constants are properly determined,
n- = - (2 log 6 + Xi) Ui+\K-X-2 log 29 + log 6 - (« + \) i} qt, Vi = (3 log 6 - Ki) Mf + (2 log 10 + X - 2« - \i) qi - \,
where ''='^^^^\r.^)' ^^^^AvT^^)-
The substitution of these values of ?-j and Vi in the equations for deter- mining Si and Wi, will give a pair of equations of the form
«t+i = «t - Wf + (a + 6*) g'i + (c + di) «,• - (1 + f log 29), Wf+i = Iwi - Si + (e + /t) g'i + (5r + /a) m^ - 1 log 60,
where a, 6, c, d, e,f, g, h are known constants, and qi, Mj are known linear functions of a*, a~*.
For example, the value of a is
(log 24)'' - (/c - \ - 2 log 29 + log 6) log 24 + (2 log 10 + X - 2k) log 29.
From these equations we should obtain a result of the form
Wi = Q.pt> + JR20-' + C2,
72] On Arithmetical Series 725
in which (7,, C^ are constants and Qi, Q^, R^, R, quadratic functions of i, but the complete determination of these would occupy too much space to be given here.
Sequel to Fart II., Chapter I. § 2.
Since § 2 of this chapter was sent to press I have had asymptotic limits to •^(x)-i-x computed by means of a scheme whose stigmata contain simply and in combination all the prime numbers up to 13 inclusive. The numerical results obtained on the one hand and on the other the process employed to <letermine d priori (so as to save the labour of working out the 30030 terms of a complete period) the minimum and maximum values (— 1 and 4) of the sum of the coefficients of any number of consecutive terms (the first included) in the stigmatic series proper to the scheme, appear to me too noteworthy to be consigned to oblivion.
This calculation differs from those that precede it in the circumstance that it does not attempt to give the optimistic limits which the scheme will aflFord, notwithstanding which the limits actually obtained will be found to be each of them materially closer to unity than the optimistic limits furnished by any of the preceding schemes.
The scheme I adopt is [1, 6, 10, 14, 105; 2, 3, 5, 7, 11, 13, 385, 1001], which satisfies the necessary condition that the sums of the reciprocals of the numbers on the two sides of the .semicolon are equal to one another.
The first thing to be done is to discover the maximum and minimum values of
-^(fs)-^(A)-^(u^)-
On taking n equal to 66, it will be found that the value of /Si„ is — 1 : I shall proceed to show that this is the minimum, in other words that — Sn cannot be so great as 2.
Denote the fractional part of any quantity x by F(x) : if — Sn is not less than 2, then it may be shown that d fortiori
'©-^(n)-^(H)-^©-n3)-^©-^©*^(ife)'
8*y ^(") ■'■■^ (tT^c) roust not be less than 2, and therefore Q(n) must be
726 On Arithmetical Series [72
greater than 1 : now it is not difficult to show that Q (n) is only greater than
unity when
71=106 + 210/1: or n=136 + 210«
(« being a positive integer). But corresponding to these two values it will
be found that
Q(106) + J^@ = i + | + f + ^.
Q(136) + ^@=i + f + Hi%. so that on either supposition Q{n)-^F (rxr) is less than 2.
Hence the minimum value of S,, is — 1, and consequently, since the stigmatic excess is here 8 — 5, the maximum value, as appears from the foot- note below, will be 8 — 5 + 1, that is 4*. (By the stigmatic excess for any scheme I mean the number of stigmata in the right-hand less the number of those in the left-hand set. This excess is obviously equal to the coefficient,
with its sign changed, of i^ (-] in the stigmatic series, where /x is any
common multiple of the stigmata.)
It will be found, on summing up the numbers in Table B, that (Si„ first attains the value 4 when n = 1891, and the value — 1 when n = 66.
For the inferior limit the outstanding group consists of all the terms up to 1891 inclusive, and for the superior limit all the terms up to 66 inclusive. But in obtaining this limit advantage has been taken of the next three groups, which end with 78, 418, and 20G8 respectively. Thus the extreme limit of the following table is 2068, instead of being 30030 (that is 2.3.5.7.11.13) which is the number of terms in a complete period. It contains the coefficients of the first 2068 terms of the stigmatic series for the .scheme [1, 6, 10, 14, 105; 2, 3,5, 7, 11, 13, 385, 1001] written down in horizontal order in regular batches of ten, as was done in Table A for the
• If we call c„ the coefficient of ^ ( - J and S„ the sum of such coefficients up to c„ inclusive
(regarding Cg and S„ as zero), and take ix the least common multiple of the stigmata, we have, obviously,
*'m = 0. Cn = CM-„, and (S„ + S>i_,_„)-(S„_] + S;x_„) = c„-c^-„=0.
Consequently, S„ + S^_,_„ = Sj + S^_, = - e^ = ?; (the stigmatic excess).
This is a valuable formula of verification, and moreover gives a rule for finding either the maximum or minimum coeffioient-sum when the other sum is given ; for if S„ has the maximum value, S^_i_„=7;- S„; if this is not the minimum let .5„' be less than i;-S„, then S'^_,_„ will be greater than S„, contrary to hypothesis. Hence the minimum value of a coefficient-sum may be found by subtracting the maximum from the stigmatic excess and vice versa.
(I may perhaps be allowed to add that this theorem suggests a generalization of itself, which I think it is safe to anticipate may be formally deduced from it, namely :
If Oi, a^, ... , a„; o, , 02, ..., ay be any given positive qtianlitiet (integer or fractional, rational or irrational) such that Sa = 2a, and if -m, M be the least and greatest values that 2E (ax) - S£ (ax) can assume when x is any positive quantity whatever, then il -m=v -n.)
72]
On Arithmetical Series
727
scheme [1, 6, 10, 210, 231, 1155; 2, 3, 5, 7, 11, 105] with the unimportant difference that (for typographical convenience) negative coefficients are indi- cated by dots instead of by bars placed over them.
Table B.
The coefficients of the [1, 6, 10, 14,
1000000000
loiiloiooo ooiooooiio ooooooiooi
0010201000
oooooooioi liooiooooo
10101-20010
ooiolooooi loUooiooo
loooloooio iiioooooio oollooiooo loioioiiio loioooloio loooioioio loioioiool oi 10000000 oooooiiioo loooioooia
ooooiiooio loooiolooo loiiloiooo ooolooooio iloawiioo
3010101010
ooooioooii loooioiolo ilioloooio ooiooioool
loooooioil loooloilio
1010002010
lolooliooo
1011201010
ooilooliio ooioooioio
1 110201010
loiooooooo oiooooiooo
first 2068 105 ; 2, 3,
ooooloioio olioloooio loooooiolo ilooioioio ooiollooio
lOOOOOOOOl lOOOlOlOOl
ooiooolooo
1000001 loo OOlllOOOlO
oooooioiii
0000201000
1011200010 oooooooolo
1200001000
ooioUioio loioooooil
1020001000
loioioooio ooioooloil
looiooioao 1010101210 ioioooioio oooolliooo
1000201000 OllUOOOlO lOlOOlOOOO
ilioioiooi loooooooio
1000000000
ooioloioio oooiooooii loooioilol ooiolooooo ooiolooioo looooooolo ilioUioio loioUooio
lOi 1001002
lololoioio
terms of the stigmatic series to 5, 7, 11, 13, 385, 1001].
1110010010
lololiiooo
lOlUOlOll
ooioooioio liooooooio ooioloioio ooilooooio 0000001 loo ooooloioio
0010300011 10(X)000020
ilioloiloo ooiooUoio looooliool
0000201010
ooolooloii loioloiooo ililioooio loiooooloo
0000001000 1010201010
loiooooooo lUoooiool loioiiiooo ooioooiiil lolooooolo loioilooio ooioloooio
0002001000
lOioloilio
11 lot 00010
ooooiooooo loioioiolo
Ol 10000010 lOOOOllOOO OOlOlOlOl^ OOlOOOOOOl
loooiooioo loioloioio ooiloioooo
loloioiolo loioloilio ooiooolooo looioliooo
10102011 10
2011000010 1020101000 ilioloiooo ilioooloio looooliolo
loioloioil
0000000010
oooooooool loooioiooo ooiiloolio loooooiloo ooioUiooo
0010200010
loolooiolo olooioioio
oloooiooii ooooloiool loooloooio ooiooolooo
1000001000
loiiloioil loiooooloo ooloooiloo 1010.100010 ooiooilooo
Ilooioioio loiUiioio
000000001 1
olioooiooo ooooloooio ooioloooio looioooooo loioloiioo loioooooil loooloiolo
728
On Arithmetical Series
[72
loooiiioio ooioloooii loiooooool loioioooio ooiooooiio looioooooo ooioiiiiio ioooioooii looiSoiooo loooiooooo
:ioiooooooo oooooliooo
Oi 10002010
ooooooioio loiiioioio ooiooooiio ioooooiooo
1000201011
ooioioooio lioooooioo loioiloooo ooiooiooii
loioioiooo ooiiioooio
loooioiioo ooiiioioio loooioooio liioooioio oiioloioio ooiooiioio loooooioii loooioioio ooioooioii ooooooioio
looiioiiio ooioioii
ooioioiioo olooooioii loooiSiooo
1010^0001 1
oooioooooo loiooooooo liioioioio Soiiooooio loioooiioo ooioioioio
ooioioioio
In Tables I and II below, in addition to pairs of numbers — (»/)+(»? + ^),
meaning -^©+^(^^)'
and + {rt) -iv + O) meaning
- + ©-t(,-f-.)'
there will be found the unpaired numbers (15) and (66) in the one and (19), (229) and (1891) in the other; to understand how these are got, it should be observed that <S„ (the sum of the first n numbers in Table B) first becomes 0 when TO = 15, first becomes — 1 when w = 66 and first becomes 2, 3, 4 when n=19, 229, 1891 respectively*.
Table I.
+ (15)
■ (17)+ (22)
- (19)+ (21)
- (23)+ (26)
- (29)+ (35)
- (41)+ (45)
■ (47)+ (52)
- (59)+ (65) + (66)
- (67)+ (78)
- (79)+ (418)
- (107)+ (135)
- (210)+ (275)
- (289)+ (385)
- (419) + (2068)
- (521)+ (585)
- (629)+ (795)
- (839)+ (936) -(1049) + (1144) -(1717) + (1925)
Table II.
+ +
+ + + +
+ +
(15)-
(21)- (26)- (33)- (44)- (63)- (65)- (75)- + (242)- + (285). + (385)- + (385). + (440)- + (494)- + (770)- + (1155)-
(17)-
(31)
(29)
• (43) (61)
■ (73)
■ (71)
■ (103)-
■ (271)
■ (323)
■ (421)
■ (439)
■ (493)
■ (571) . (841)
(1273)-
(19)
(229)
(1891)
Call S the sum of the infinite series given by Table B : it may then easily be verified that {^(x)-2)-{^(^) + ^(^)}
72] On Arithmetical Series 729
The reasoning employed in dealing with previous schemes serves to show that superior and inferior asymptotic limits to -^ (x) -=- x, which we shall call El, Fi in order to distinguish them from the corresponding optimistic limits (E, F), may be found from the equations
E, = M+aE,-hF,\
F, = M+cFi -dEi]-
where a is the sum of the reciprocals of the numbers occurring
in Table I with the sisrn
"O^
c „ „ „ in Table II „ +
"' » » » i» n ~
and M is the stigmatic multiplier,
namely « = ! + 1 + ^ + ... + ^^^ = •;«352 ... ,
^=^+^ + A + -+rA7 = -^^^«<^-'
may be resolved into term-pairs of the form
that shall contain among them all those in Table I, and
into term-pairs of the form +^(-) -l^( — » ) that shall contain among them all those in Table II above.
The maximom value of S, is here 4: if it had been 2, then instead of 3 unpaired positive terms appended to {^(x)-2} there woald have been but 1. This is what happens for the scheme [1, 15; 2, 3, 5, 30] given in the footnote on p. [710]: and accordingly, we see that
{ ^ (x) - 2 } -I- \(' I r^ ] , for that scheme, is resoluble into paired terms of the form
-♦(f)-*(,-f.)-
So again, the minimum being 0 (instead of - 1), there will be but 1 unpaired negative term to append to \f(x)-Z), and accordingly, we see that {^(a:)-S}-f(^) in that scheme is resoluble
into term-pairs of the form -^(-) -t-^( 3 ) •
• The above values of a, b,c, d give a-|-c = -603 ... and ac-6d = -005 ... , and consequently the roots of the "characteristic" equation /^ - {a+e) p + {ac - bd)=0 satisfy the necessary condi- tion of being each less than unity in absolute value.
730
On Arithmetical Series
[72
and
Jlf = 1 log2 + I log3 + ^ logs + ^ log 7
Hence
-lioge-^^iogio-^iogu-jig
(l-c-b)M
log 105 = -98859
{l-a)(l-c)-bd
(l-a-d)M
(l-a){l-c)-db
= 1-04423..
= -95695..
E,
09120
(so that the mean of E^ and Fi is less than 0006), and p-' = 1'
Thus then (see footnote to p. [694]) by taking x sufficiently great, the number of primes not exceeding x, multiplied by log x and divided by a;, may always be made to lie between the numbers
1-04423... and -95695...,
the divergences of which from unity are
•04423 ... and 04304 ... (as against
Tchebycheff's -10555 ... and -07807...).
These divergences, there is little doubt, would become even more nearly equal than they are, if anyone should feel inclined to undertake the very laborious task of extracting the optimistic values {E, F) from the scheme employed.
In order to understand this necessarily abbreviated sketch of a method more easy to think out and apply than to find language to express, I must not conceal that a careful study of the several schemes given, and of the principles embodied in the calculations relating to them, is a sine qua non. It may somewhat lighten the burden thrown upon the reader, if I add a few words concerning one or two points, perhaps inadequately explained in what precedes.
Let fj, be the least common multiple of the stigmata of any given harmonic scheme and Sn the sum of the coefficients of
t(^). v^d). t(f)....^o
* In Tables I and II above, the ratio is greater than 1-09120 ... for every pair of terms, J
except -(1049) + (H44) in Table I. In the case of this pair, we have |HS = 1"0905 ... , which shows that the exclusion of it from that table would have led to asymptotic limits better (but very slightly so) than those arrived at in the text.
72] On Arithmetical Series 731
I
in the corresponding stigmatic series. Then from the formula of [p. 710] combined with the equation which connects the stigmata, it follows that
O^ — ", ^^n+t^^ — ^n
Hence an infinite number of values of n will give /S„ its greatest value ; the difference of these values will be of the form kix — fi! where p! may, and in general will, besides zero have various other values less than /x, thus giving rise to the collections of terms called groups (see p. [709]) of which the period of /tt terms will be composed. The same will be true when we substitute the word least for greatest.
If now i be taken any number such that (S,- has its greatest value it may be shown that the sum of all the terms in the stigmatic series subsequent to
the one containing V^ (^) will be negative or zero, and similarly when Si has
its least value such sum will be positive or zero* ; consequently when i is properly determined we can find immediately a superior limit in the one case and an inferior limit in the other, to the sum of the first i terms of the series.
I will conclude this portion of the subject with the remark that from the values of £", and F^ it is easy to infer that if fx is equal to or less than (■95695 ...) i — (1"04423 ...), and x exceeds a certain ascertainable number whose value depends on k and fi,, then between x and kx there will be found
X
more than u, -. primes +.
log a; "^ '
* The reason of this is that the sum of all the terms beyond the tth may be separated into partial sums, each containing ^ terms, which ultimately vanish. If now yi{kn + i + l) + -f^(kii + i + 'i)+ ... + 7^ (fc/t + i + /i) be one of them, then 7, + 7j+ ... +7( will be zero when t = ii, and will have a constant algebraical sign (or else be zero) when t <: ;i ; from which it follows (see footnote p. [722] where, be it observed, a coefficient +X or -X is supposed to be represented by a sequence of \ black or X white beads) that each partial sum may be decomposed into an aggregate of quantities of the form + (1;) -(■<) + 8) or - (1;) + (i) + 9) according as the first coefficient in each such sum is positive or negative, and will therefore, if not zero, have the same algebraical sign as that coefficient has, namely - or + according as S^ has its greatest or least value.
t In order that it. may be positive (which ensures the existence of tome primes between x and kx, when x exceeds a certain limit) it is only necessary to take A > 1-09120 ... (which differs very little from |f), whereas if we limited ourselves to the results of the oft-quoted memoir of 1850 [see p. 704, above], we could not prove the existence of prime numbers between x and kx, for a given value of x, however great, unless k exceeds f .
73.
NOTE ON A NINE SCHOOLGIRLS PROBLEM.
[Messenger of Mathematics, XXII. (1893), pp. 159, 160.]
This is a parallel question to the well-known one of fifteen schoolgirls extended to the supposition of their walking for one week, three and three together, so that in any the same day no two, and at the end of the week no three, taking four walks a day, shall have walked more than once together. Let us understand by the development of the array
a, b, c, d, e, f, g, h, k, the four arrangements {ahc, def, ghk),
{adg, heh, cfk),
(aek, hfg, cdh),
{afh, bdk, ceg),
(corresponding, in fact, to the four sets of three lines through the nine
inflexions of a cubic).
If we suppose the nine girls to walk out four times a day, the same two never being together more than once in the same day, and that at the week's end each has been associated with every pair of the remaining eight, the above will serve to represent one day's walks. To find the other six, I first form the three following pairs of subsidiary arrays, by circular motion per- formed successively on the three columns of the primitive array, namely
|
9' |
b, |
c, |
d. |
b, |
c, |
|
a, |
e, |
f. |
9> |
e, |
f, |
|
d. |
h. |
k, |
a, |
h, |
k, |
|
a, |
h, |
0, |
a, |
e, |
c, |
|
d, |
b, |
/. |
d, |
h, |
/. |
|
9< |
e, |
k. |
9. |
b, |
k, |
|
a, |
b, |
k, |
a, |
b. |
A |
|
d. |
e, |
c, |
d, |
e, |
k. |
|
9> |
h, |
f. |
9. |
h, |
c. |
73] Note on a Nine Schoolgirls Problem 733
Then making any similarly placed line (I have taken the first) in each of the above six groups circulate in one direction as regards the three on the left, and in the opposite direction as regards the three on the right, we obtain six new arrays: these together with the original one furnish the following table :
6, c,
e, f, h, k,
|
a, |
|||
|
d. |
|||
|
9' |
|||
|
c, |
9. |
b. |
|
|
a, |
e, |
/. |
|
|
d, |
h. |
k, |
|
|
c. |
a, |
h. |
|
|
d, |
b, |
/. |
|
|
9' |
e, |
k, |
|
|
k, |
a. |
b, |
|
|
d, |
e. |
c, |
|
|
9. |
h, |
/. |
|
b. |
c, |
d. |
|
9. |
e, |
f. |
|
a, |
h, |
k, |
|
e. |
c, |
a. |
|
d. |
A, |
/. |
|
9' |
b. |
k. |
|
b. |
f, |
a. |
|
d, |
e. |
k, |
|
9> |
h, |
c. |
When the seven arrays in the above table are developed according to the rule previously given, the triads thus arising will be found to be all distinct or, which is the same thing, will comprise among them the whole of the eighty-four ternary combinations of the nine symbols. We have therefore in this table a solution of the proposed problem.
Of course the general problem, when n is any odd multiple of 3, is to construct sets of ^(n — 1) synthemes, each containing ^n triads with no element in common, and to distribute the whole number of triads into (n — 2) such sets.
This problem I solved very many years ago, but I believe have nowhere published, for the case where n is any power of 3, by a method of compound rhythmical displacement strictly analogous to (but of course more intricate than) the one here exhibited.
74.
ON THE GOLDBACH-EULER THEOREM REGARDING PRIME NUMBERS.
[Nature, Lv. (1896-7), pp. 196, 197; 269.]
In the published correspondence of Euler there is a note from him to Goldbach, or, the other way, from Goldbach to Euler, in which a very wonderful theorem is stated which has never been proved by Euler or any one else, which I hope I may be able to do by an entirely original method that I have applied with perfect success to the problem of partitions and to the more general problem of denumeration, that is, to determine the number of solutions in positive integers of any number of linear equations with any number of variables. In applying this method I saw that the possibility of its success depended on the theorem named being true in a stricter sense than that used by its authors, of whom Euler verified but without proving the theorem by innumerable examples. As given by him, the theorem is this : evei'y even number may be broken up in one or more ways into two primes.
My stricter theorem consists in adding the words " where, if 2n is the given number, one of the primes will be greater than ^ . and the other less
than -„- ." This theorem I have verified by innumerable examples. Such
primes as these may be called mid-primes, and the other integers between 1 and 2n — 1 extreme primes in regard to the range 1, 2, 3 ..., 2n — 1.
I have found that with the exception of the number 10, Euler's theorem is true for the resolution of 2n into two extreme primes ; but this I do not propose to consider at present, my theorem being that every even number 2n may be resolved into the sum of two mid-primes of the range
(1, 2, 3...,2«-l).
74] On the Goldbach-Eider Theorem 735
As, for example
4 =
12 =
18 =
40 =
100 =
200 =
500 =
1000 = 257 + 743 = &c. And so on.
My method of investigation is as follows. I prove that the number of ways of solving the equation x-\-y= 2n, where x and y are two mid-primes to the range 2n — 1, that is twice the number* of ways of breaking up In into two mid-primes -f- zero or unity, according as n is a composite or a prime number, is exactly equal to the coefficient of a^ in the series
|
2+ 2 6 = 3-}- 3 |
8= 3+ 5 |
10 = 3 + 7 |
|
5h- 7 14 = 7+ 7 |
16= 5 + 11 |
|
|
5+ 13= 7-1-11 |
20= 7+13 |
|
|
11-1-29= 17-1- 23 |
50=13 + 37 = |
= 19 + 31 |
|
29+71= 41+ 59 |
||
|
61 + 139= 73+127 = |
= &c. |
|
|
27 + 373 = 193 + 307 = |
= &c. |
/ 1 1 1 Y
where p, q, ..., I are the mid-primes in question. This coefficient, we know d prion, is always a positive integer, and therefore if we can show that the coefficient in question is not zero, my theorem is proved, and as a consequence the narrower one of Goldbach and Euler. By means of my general method of expressing any rational algebraical fraction, say <f>x, as a residue, by taking the distinct roots of the denominator, say p, and writing the variable equal to pe*, and taking the residue with changed sign of Sp~" e'^'<f>(pe*), we can find the coefficient of x" or (if we please to say so) of xP^ in the above square, and obtain a superior and an inferior limit to the same in terms oi p, q, ...,l; and if, as I expect (or rather, I should say, hope) may be the case, these two limits do not include zero between them, the theorems (mine, and therefore ex abundantia Euler 's) will be apodictically established.
The two limits in question will be algebraic functions of p, q,...,l, whereas the absolute value of the coefficient included within these limits would require a knowledge of the residues of each of these numbers in respect to every other as a modulus, and of 2w in respect of each of them. In a word, the limits will be algebraical, but the quantity limited is an alge- braical function of the mid-primes p, q, r, ..., I.
Postscript. The shortest way of stating my refinement on the Goldbach- Euler theorem is as follows : — " It is always possible to find two primes
* This nomber may be shown to be of the order ^ ^ , and a very fair approximate value of
(logn)'
it IS — where /i is the namber of mid-primes corresponding to the frangible number 2n.
736 On the Goldbach-Euler Theorem [74
differing by less than any given number whose sum is equal to twice that number."
Another more instructive and slightly more stringent statement of the new theorem is as follows. Any number n being given, it is pos.sible to find two primes whose sum is 2n, and whose difference is less than n, n — 1, n — 2, n — 3, according as n divided by 4 leaves the remainders 1, 0, —1, —2 respectively.
Major MacMahon, to whom and to the Council of the Mathematical Society of London I owe my renewed interest in this subject, informs me that in a very old paper in the Philosophical Magazine I stated that I was in possesision of "a subtle method, which I had communicated to Prof Cayley," of finding the number of solutions in positive integers of any number of linear equations in any number of variables. This method (never printed) must have been in essence identical with that which within the last month I have discovered and shall, I hope, shortly publish.
I have verified the new law for all the even numbers from 2 to 1000, but will not encumber the pages of Nature with the details. The approxi- mate formula hazarded for the number of resolutions of 2n into two primes,
namely — , where fj. is the number of mid-primes, does not always come n
near to the true value. I have reasons for thinking that when n is sufficiently great, ~ may possibly be an inferior limit. The generating function
I 1-xP
is subject to a singular correction when the partible number 2n is the double of a prime. In this case, since the development to be squared is
fji+x'' + ai^ + ...+xP + afP+... + &c.,
the coefficient of x^ will contain 2fx, arising from the combination of 0 with 2n, which is foreign to the question, and accordingly the result given by the generating function would be too great by 2/x.
This may be provided against by always rejecting the centre of the mid- range from the number of mid-primes. The formula will then in all cases give twice the number of ways of breaking up 2« into two unequal primes. Another method would be to take as the generating function not the square
of the sum, but the product of the fractions :; (without casting out n
1 — x^
when it is a prime), but this method would be inordinately more difficult to work with in computing series involving the roots of unity than the one
74] regarding Prime Numbers 737
chosen, which is in itself a felicitous invention*. Whether the method turns out successful or not, it at the very least gives an analytical expression for the number of ways of conjoining the mid-primes to make up 2n without trial, which in itself is a somewhat surprising result. Having lost my pre- liminary calculations, it may be some little time before I shall be able to say whether the method does or does not contain a proof of the new theorem ; but that this can be ascertained, there is no manner of doubt. This is the first serious attempt to deal with Euler's theorem, or to bring the question into line with the general theory of partitions.
It is proper to regard the range 1 to 2« — 1 as consisting of two com- plementary flank regions, two lateral mid-prime regions, and a region reduced to a single term in the middle, as for example,
1, 2. 3 : 4, 5 : 6 : 7, 8 : 9, 10, 11.
Or, again, I, 2, 3 : 4, 5, 6 : 7 : 8, 9, 10 : 11, 12, 13.
And the question of 2n being resoluble into 2 primes breaks up into three, aamely, whether 2n can be composed with two flank primes, two lateral mid- primes, or with the number in the central region repeated.
* For the generating function we may take any power greater than 2, instead of the square, and the coefficient of i** will then be the number of couples making up 2n multiplied by (r^-r) /t"^', which can be calculated by the same method as for the square, but is more difficult and must give rise to numerous theorems of great interest, arising from the multiform representa- tion of the same quantity.
8. IV.
47
75.
ON THE NUMBER OF PROPER VULGAR FRACTIONS IN THEIR LOWEST TERMS THAT CAN BE FORMED WITH INTEGERS NOT GREATER THAN A GIVEN NUMBER.
[Messenger of Mathematics, xxvii. (1898), pp. 1 — 5.]
A SLIGHT reflexion will show that the number of such fractions
( rr counting as one of them ] with the limit n is the sum of the totients of
all the numbers from 1 to n.
Let us use Ej as usual to denote the integer part of j, tEj to denote the totient (or number of numbers not exceeding and prime) to Ej, and JEj to denote the sum of such totients for all numbers from 1 to j. Then we may establish the following exact equation given by the author of this article, but without proof and with some slight inaccuracy, in the Phil. Mag. for April, 1883 [p. 102, above]. The equation is
JEj + JE (i j) + JE ( J j) + etc., or, more shortly,
ljEi = k{{Ejy + {Ej% (1)
1 ^
The proof is as follows. Remarking that E{j—l)=Ej—l,thQ right- hand side of equation (1), when j is reduced to j—\ obviously sufiFers a diminution equal to Ej.
On the left-hand side of the equation any term JE t remains unaltered,
when for j is written (^ — 1), unless Ej is divisible by i, in which case the
term undergoes a diminution JEj. Thus for example J^f- — J'^ = 0,h\xX,
Ej . Ji^ — J9^= J (20). And, as in the case supposed, -^ is a factor of Ej, the
total diminution, when j-1 replaces j, will be the sum of the totients
75] On the Number of Fractions in their Lowest Terms 739
of the factors of Ej, which by a known theorem equals Ej. Hence equation (1) is satisfied for j if it is satisfied for j — 1, and as it is true when Ej=l it is true for all values of j, as was to be proved. From equation (1) it follows that JEj is of the order {Ejy, and making
JEj = lif,{Ejy + y,
where e; is zero when j= cc , we obtain
6 . , ^. Si'
or /* = — , or approximately jj = -^,
In the tables in the Phil. Mag. for April and September, 1883*, the value
3
of Jj is computed up to _;' = 1,000 and compared with the mean value —^j^-
3 . From this table it appears that Jj is always intermediate between ^j^ and
g
~j 0 + !)'> ^°d much nearer to their mean, which to an insignificant fraction
3 pr^ is the same as —iU'+j), than it is to either extreme. The first, at
least, of these statements ought to be susceptible of proof.
As a matter of philosophical interest as embodying a principle applicable
to other cases, I will show how I originally found the value —j" for the
number of proper vulgar fractions in their lowest terms that can be formed by means of the first integers.
It is obvious that the probability of any unknown number being divisible
by a prime number i is - , and of any two numbers, being each so divisible,
I
is -^, so that the probability of two unknown numbers being each not
divisible either by 2, 3, 5, 7, n, or any other prime, will be
(-^)(-|.)(-|.)(-fO('-^.)--
which we know is equal to the sum of the reciprocal of the squares of the
natural numbers, that is, is equal to — . Hence the number of fractions in
their lowest terms that can be got by combining each of _;' integers with each of i others, found roughly by adding together the probable expectation
of any such combination consisting of two relative primes, will be —^j\ and
the number of proper fractions in their lowest terms so capable of being
3}^ formed will be the half of this or -^ . It appears incidentally from this
[• p. 103, above.]
47—2
/ /
740 On the Number of Fractwns [75
that the average or mean vahie of the totient of any number is — into, or
rather more than, ^ths of that number.
In like manner, if we define a mid-prime to the number 2n to be one which is greater than Jn. and less than f n, the range of numbers amongst which such primes are to be found will, to a unit prh, be n. Let us call the number of such mid-primes /x. Then the probability of any number and its complement in respect to 2n being each of them primes will
be ^. If now we seek the number of solutions of the equation in prime
numbers x + y= 2n, which will be an even or an odd number, according as n is a composite number or a prime, we may suppose a row of n white balls and n black balls, each series being marked with all the numbers from 1 to n inclusive. It follows from what has been said that the sum of the expecta- tion of a; being inscribed on any one of the white balls being itself a prime, and its complement 2n—x upon one of the black balls being so likewise,
will be n . — , , that is — ,* and as the same will be true when a; is a figure ri^ n
on a black ball and 2n — a; on a white, the total value of the expectation of the equation in primes x + y= 2n being satisfied will be the double of this,
or — . I have had tables constructed for determining the number of the solutions of this equation (x and y being primes) from 2n = 2 up to 2n = 500.
Call the number of solutions for any value oi n, 6 — ; on taking the
average value of 6 for all values of 2n on the 1st, 2nd, 3rd, 4th, 5th, centuries respectively, it will be found that
^0= •96344 = -99349 = 1-00603 = -98281 = -99764,
of which the sum is 4-94341 and the average is -98868, agreeing with wonder-
2u' ful nearness to the rough estimate of the number of solutions being -^ .
* >» is of the order of, and ultimately in a ratio of equality with, . , in the sense that,
however small e be taken, a limit Le can be found such that for all values of n beyond it, /i will be
limited on the two sides by (1 ± e) ; this follows demonstrably from a known theorem
proved within the last few years, and as a consequence we see that the number of solutions in
"mid-primes" of the equation x + y = 2n will necessarily be of the same order as ,-; r= and
• (logn)'
presumably in a ratio of equality with it in the sense explained above, but this, of course, awaits
demonstration.
1
75] in their Lowest Terms 741
I ought not, however, to suppress the fact that, from another point of view, this number might be expected to eventuate as — instead of — .
In equation (1) we may write F{j) for the sum of the totients of all the numbers not exceeding j, and it then takes the form
which, by the well-known formula of reversion (see Phil. Mag., December, 1884*), gives
Fj =<f>j-<f> (ii) - <!> (ij) - <f> aj) + <p a J) - etc.
Thus for example the number of terms in a Farey series with 17 as a limit should be equal to
K17 -8- 5 -3 + 2-2 + 1-1-1+1 + 1-1) + ^(289-64-25-9 + 4-4+1-1-1 + 1+1-1)
that is J(l) + ^(191) or 96, which is rightf.
* I do not know whether the annexed important case of reversion has been noticed or not : »■ being greater than unity, let o-j denote the sum of the negative t'th powers of the prime numbers 2, 3, 5, 7, etc., and «,- the logarithm of the sum of the negative ith powers of the natural numbers 1, 2, 3, 4, etc. (which, when i is an even integer, is a known quantity), then it is easily shown that
»<=(ri + 4(r« + J(r«+Jff« + J<rM+eto., and therefore by reversion
'r<=»i-4»3*-J»«-t««+i«M-f»7<+A»iM+eto.
A very general case for reversion arises when <t>i = 'Z—^<t>(n' . i). In this last application of
the formula r=l, » = 1 ; in the case considered in the text relating to Farey series r = 0, 8= - 1. t And so in general, since by a well-known theorem
Ej - E (ii) - E (i j) + E (iJ) + etc is always equal to nnity, bo that
{Ejy-iJEj + l = Elijy+E(^j)'-E{ijr + eto., we have always
2JEj-i={Ejy-E(ijy-E{ijy+E{U)'+6to.
a very convenient, and, I believe, new formula for calculating the number of fractions in their lowest terms where neither numerator nor denominator exceeds j.
To this E theorem there exists a pendant which may be called the H theorem, namely let Hx mean the nearest integer (when there is one) to x, but when x is midway between two integers Hx is to denote the first integer above x; let p, 5, r, ... be the primes not exceeding the integer n, and call
H, = n-2fl'- + 2H — -2ff — +6tc.;
p pq pqr
then B^ will be the namber of primes greater than n and less than 2n, so that H„ is always greater than zero ; and if e (x) means unity or zero according as x is a prime or not, we shall always have
^»--ff»-i = M2n-l)-e(n).
I do not know whether this theorem has been previously noticed. It may be obtained by the Eratosthenes sieve process applied to the progression n + 1, n + 2, »! + 3, ..., 2n, replacing therein every prime namber by unity.
742 On the Number of Fractions in their Lowest Terms [75
If not already known, it may be worth while to register the two following additional theorems concerning £,n and &,n, by which I mean what £, and H^ become when the even prime 2 does not ooont among the primes p, q, r, which are less than n, namely
£.n=^(^)-Z£j..£,-^^.etc.=£(JMP). ' 2 2p 2pj
This paper was sent by Professor Sylvester to the editor on Feb. 12th, 1897, with a letter in which he wrote " I could subsequently send you the valuable table referred to in the text, giving the number of solutions of the equation x + y=2n in prime numbers for all values of n up to 500." In subsequent letters he made several slight additions to the paper. He corrected the proof sheets about the end of the month, and then added the first footnote and the last paragraph of the third note. His death took place on March 15th.
INDEX TO PKOFESSOR SYLVESTER'S CONTRIBUTIONS TO "MATHEMATICAL QUESTIONS... FROM THE EDUCATIONAL TIMES."
In the following index, prepared by Mr C. C. Scott, of the Library of St John's College, Cambridge, the date is that of the volume of the " Mathe- matical Questions. ..from the Educational Times," the page is that of the volume, the number is that of the question, and the word 'Solution' indicates that a solution of a question is given by Professor Sylvester. It will be noticed that in some cases a question occurs several times, in different volumes (as indicated by the number of the question being the same).
|
VoL I. (1864) |
VoLV. |
(1866) |
|||||
|
pp. 19, 25 |
Q. 1402 |
(Solution) |
p. xvi |
Q. 1832 |
|||
|
37 |
1439 |
(Solution) |
xvi |
1833 |
|||
|
45 |
1421 |
20 |
1818 |
(Solution) |
|||
|
51 |
1443 |
21 |
1840 |
||||
|
77 |
1416 |
35 67 |
1887 1798 |
(Solution) (Solution) |
|||
|
Vol. II. (1866) |
84 |
1480 |
|||||
|
52 |
1532 |
105 |
1950 |
||||
|
54 |
1538 |
See also pp |
81 and 108 |
||||
|
65 |
1439 |
VoL VI |
(1866) |
||||
|
64 |
1502 |
p. xiv |
Q. 1849 |
||||
|
70 |
1561 |
(Solution) |
xiv |
1850 |
|||
|
91 |
1584 |
(Solution) |
XV 35 |
1910 1990 |
|||
|
VoL III. (1866) |
53 |
1503 |
|||||
|
78 |
1667 |
66 67 |
1892 1969 |
||||
|
VoL IV. (1866) |
70 |
1990 |
|||||
|
xiv |
1480 |
88 |
1990 |
||||
|
XV |
1503 |
100 |
1990 |
||||
|
IV |
1504 |
VoL |
VII. (1867) |
||||
|
XV |
1511 |
XV |
2317 |
||||
|
xvi |
1604 |
XV |
2325 |
||||
|
29 |
1711 |
37 |
2254 |
||||
|
64 |
1752 |
38 |
2271 |
||||
|
77 |
1829 |
(Solution) |
49 |
2232 |
(Solution) |
||
|
81 |
1790 |
(Solution) |
72 |
2291 |
|||
|
101 |
1229 |
(Solution) |
74 |
2332 |
(Solution) |
||
|
111 |
1811 |
106 |
2246 |
(Solution) |
744 Index to Contributions to "Mathematical Questions
|
Vol. VIII. (1868) |
|
|
p. xii |
Q. 2285 |
|
xiii |
2337 |
|
xiv |
2391 |
|
XV |
2492 |
|
18 |
2249 |
|
36 |
2371 |
|
43 |
1910 |
|
59 |
2447 |
|
61 |
2452 |
|
88 |
1805 |
|
92 |
1849 |
|
106 |
2473 |
|
Vol IX. (1868) |
|
|
xiv |
2531 |
|
72 |
2411 |
|
Vol. X. (1868) |
|
|
35 |
2675 |
|
53 |
2697 |
|
74 |
2511 |
|
112 |
2552 |
|
Vol XL (1869) |
|
|
18 |
2736 |
|
21 |
2779 |
|
38 |
2778 |
|
43 |
2636 |
|
49 |
2823 |
|
81 |
2877 |
|
Vol XII. (1869) |
|
|
17 |
2802 |
|
29 |
2864 |
|
48 |
2930 |
|
56 |
2950 |
|
70 |
2296 |
|
83 |
2977 |
|
92 |
3003 |
|
VoL XIIL (1870) |
|
|
21 |
2975 |
|
42 |
3041 |
|
50 |
2845 |
|
94 |
3127 |
|
Vol. XIV. (1871) |
|
|
68 |
3207 |
|
VoL XV. (1871) |
|
|
22 |
3279 |
|
36 |
1950 |
|
54 |
3330 |
(Solution) (Solution)
(Solution)
(Solution)
(Solution)
(Solution)
(Solution) (Solution)
(Solution)
|
Vol. XV. {cont.) |
||
|
p. 57 Q |
3355 |
|
|
99 |
3401 |
|
|
Vol. XVI. (1872) |
||
|
VoL XVII. (1872; |
||
|
19 |
3230 |
(Solution) |
|
33 |
3590 |
|
|
VoL XVIII. (1873) |
||
|
VoL XIX. (1873) |
||
|
17 |
3928 |
|
|
25 |
3279 |
(Solution) |
|
38 |
3979 |
|
|
46 |
3876 |
|
|
87 |
3959 |
|
|
VoL XX. (1874) |
||
|
17 |
4092 |
|
|
VoL XXT (1874) |
||
|
57 |
4231 |
|
|
58 |
4320 |
|
|
111 |
4320 |
|
|
VoL XXII. (1875) |
||
|
31 |
4386 |
(Solution) |
|
VoL XXIII. (1875) |
||
|
43 |
4591 |
|
|
48 |
4615 |
(Solution) |
|
59 |
4637 |
|
|
71 |
4660 |
|
|
VoL XXIV. (1876) |
||
|
40 |
4693 |
|
|
57 |
4751 |
(Solution) |
|
78 |
3067 |
(Solution) |
|
VoL XXV. (1876) |
||
|
27 |
4869 |
|
|
68 |
4922 |
(Solution) |
|
VoL XXVI. (1876) |
||
|
32 |
4869 |
|
|
83 |
5080 |
|
|
VoL XXVII. (1877) |
||
|
43 |
5131 |
|
|
81 |
5208 |
|
|
92 |
5131 |
|
|
VoL XXVIIL (1878 |
||
|
24 |
5327 |
(Solution) |
|
106 |
from the Educational Times"
745
|
Vol. XXIX. (1878) |
Vol. XXXVII. (1882) |
||||
|
p. 52 Q |
5563 |
p. 22 Q |
.5052 |
||
|
93 |
5563 |
42 |
6919 |
||
|
Vol. XXX. (1879) |
90 |
5688 |
|||
|
52 |
145 |
98 |
6978 |
||
|
81 |
573 |
101 |
7002 |
(Solution) |
|
|
Vol. XXXI. (1879) |
Vol. XXXVIII. (1883) |
||||
|
36 |
5793 |
37 |
7073 |
||
|
87 |
5493 |
Vol. XXXIX. (1883) |
|||
|
Vol. XXXII. (1880) |
34 |
4971 |
|||
|
98 |
5521 |
50 |
7249 |
||
|
Vol. XXXIII. (1880) |
74 |
7277 |
(Solution) |
||
|
20 |
.5713 |
85 |
7219 |
(Solution) |
|
|
34 |
5762 |
109 |
5820 |
||
|
53 |
4817 |
(Solution) |
122 |
7322 |
|
|
63 |
5763 |
Vol. XL. (1884) |
|||
|
^ |
1628 |
(Solution) |
21 |
7143 |
|
|
■ 92 |
5762 |
25 |
5850 |
||
|
■ |
4031 |
(Solution) |
32 |
7189 |
(Solution) |
|
V 97 |
5357 |
54 |
7403 |
(Solution) |
|
|
115 |
1588 |
(Solution) |
77 |
7428 |
|
|
Vol. XXXIV. (1881) |
112 |
7454 |
|||
|
21 |
6243 |
(Solution) |
VoL XLI. (1884) |
||
|
34 |
5901 |
21 |
7382 |
||
|
34 |
6094 |
53 |
7567 |
||
|
46 |
6339 |
58 |
7508 |
(Solution) |
|
|
55 |
5624 |
(Solution) |
66 |
7351 |
|
|
55 |
6188 |
67 |
7377 |
||
|
64 |
5926 |
69 |
75.36 |
||
|
71 |
6034 |
||||
|
79 |
5983 |
Vol. XLII. (1885) |
|||
|
99 |
6218 |
29 |
6218 7454 |
||
|
99 |
6008 |
||||
|
102 |
5452 |
61 |
6118 |
||
|
108 |
6154 |
(Solution) |
86 |
7705 |
|
|
110 |
5493 |
89 |
7769 |
(Solution) |
|
|
101 |
7836 |
(Solution) |
|||
|
Vol. XXXV. (1881) |
|||||
|
32 |
6532 |
VoL XLIII. (1885) |
|||
|
6405 6531 |
21 |
4118 |
|||
|
46 |
38 |
7805 |
|||
|
55 |
6563 |
51 |
4569 |
||
|
79 |
6596 |
53 |
4481 |
||
|
93 |
5080 |
85 |
4139 |
||
|
109 |
4994 |
93 |
8042 8078 |
||
|
VoL XXXVI. (1881) |
105 |
7922 |
|||
|
24 |
6469 |
1 110 |
4266 |
||
|
39 |
6373 |
(Solution) |
|||
|
97 |
(6795 (6826 |
VoL XLIV. (1886) |
|||
|
21 |
8115 |
47—5
746 Index to Contributions to "Mathematical Questions
|
Vol. XLV. (1886) |
Vol. XLVIII. {cont.) |
||
|
p. 21 Q |
. 8216 (Solution) |
p. 164 C |
\. 8042 |
|
29 |
81.^6 |
164 |
8078 |
|
70 |
8389 18275 |
Vol. XLIX. (1888) |
|
|
85 |
1 8.321 |
21 |
2352 |
|
18.394 |
j9229 |
||
|
94 |
8306 |
37 |
^9259 |
|
pp. 125—145 inclusive |
(9301 |
||
|
64 |
9381 |
||
|
Vol XLVI. (1887) |
69 |
9418 |
|
|
p. 48 Q |
. 2231 |
86 |
9449 |
|
61 |
5178 |
127 |
4721 |
|
93 |
8710 |
||
|
112 |
1453 |
Vol. L. (1889) |
|
|
Vol. XLVII. (1887) |
33 |
2853 |
|
|
21 |
2810 |
54 |
9571 |
|
37 |
2866 |
Vol. LI. (1889) |
|
|
63 |
8242 |
49 |
9892 |
|
69 |
59.'J5 |
66 |
9892 |
|
85 |
2935 |
81 |
9609 |
|
90 |
2832 |
97 |
10025 |
|
101 |
2934 |
||
|
118 |
8631 |
Vol. LIT. (1890) |
|
|
137 |
2391 |
26 |
8184 |
|
137 |
.3651 |
30 |
10103 |
|
138 |
3535 |
76 |
10180 |
|
140 |
,3427 |
97 |
10219 |
|
144 |
5420 |
134 |
4169 |
|
166 |
8631 |
135 |
4195 |
|
167 |
8978 |
Vol. LIII. (1890) |
|
|
160 |
9024 |
||
|
26 |
2853 |
||
|
163 |
1856 |
||
|
33 |
7668 |
||
|
165 |
5305 |
||
|
41 |
2883 |
||
|
Vol. XLVIII. (1888) |
67 |
2827 |
|
|
23 |
7740 |
90 |
10476 |
|
37 |
7889 |
100 |
10257 |
|
46 48 |
8822 (Solution) 9112 |
Vol. LIV. (1891) |
|
|
8864 9004 |
89 |
10621 |
|
|
69 |
Vol. LV. (1891) |
||
|
76 |
(9071 (9024 |
25 |
10554 |
|
77 |
10408 |
||
|
85 |
2903 |
89 |
10914 |
|
100 |
7189 |
2906 2997 |
|
|
101 |
2906 |
106 |
|
|
157 |
2926 6.306 |
Vol. LVI. (1892) |
|
|
163 |
8586 |
26 |
10951 |
|
164 |
8242 |
67 |
11084 |
|
164 |
8216 |
97 |
2552 |
from the Educational Times'
747
|
Vol. LVII. (1892) |
|
|
p. 54 |
Q. 8509 |
|
64 |
1850 |
|
^ 97 |
11437 |
|
r ^^^ |
11480 |
|
Vol. LVIII. (1893) |
|
|
25 |
11512 |
|
97 |
11648 |
|
VoL LIX. (1893) |
|
|
98 |
11851 |
|
133 |
2572 |
|
134 |
2589 |
|
135 |
2610 |
|
136 |
2758 |
|
Vol. LX. (1894) |
|
|
79 |
11988 |
|
K 97 |
12020 |
|
■ 129 |
2792 |
|
■ 129 |
2824 |
|
W 1.30 |
2859 |
|
130 |
2890 |
|
132 |
2921 |
|
133 |
2941 |
|
1.34 |
2958 |
|
134 |
2959 |
|
134 |
3013 |
|
135 |
3014 |
|
1.35 |
.3019 |
|
Vol. LXI. (1894) |
|
|
39 |
12088 |
|
121 |
3085 |
|
121 |
3106 |
|
123 |
3163 |
|
125 |
32.50 |
|
126 |
3305 |
|
Vol. LXII. (1895) |
|
|
76 |
9878 |
|
121 |
3454 |
|
123 |
3480 |
|
124 |
3506 |
|
Vol. LXII. (fiont.) |
|
|
p. 127 Q. 3676 |
|
|
128 |
3708 |
|
Vol. LXIII. (1895) |
|
|
125 |
3907 |
|
128 |
4065 |
|
Vol. LXIV. (1896) |
|
|
121 |
4290 |
|
122 |
4.338 |
|
123 |
4365 |
|
125 |
4411 |
|
126 |
4437 |
|
126 |
4459 |
|
127 |
4506 |
|
127 |
4529 |
|
128 |
4551 |
|
VoL LXV. (1896) |
|
|
124 |
4792 |
|
125 |
4841 |
|
127 |
4896 |
|
128 |
4945 |
|
Vol. LXVI. (1897) |
|
|
73 |
11154 |
|
122 |
5012 |
|
128 |
5152 |
|
VoL LXVII. (1897) |
|
|
57 |
13375 |
|
108 |
13430 |
|
VoL LXVIIl. (1898) |
|
|
55 |
13461 |
|
121 |
5537 |
|
125 |
5659 |
|
VoL LXIX. (1898) |
|
|
25 |
1.3604 |
|
40 |
13631 |
|
Vol. LXX. (1899) |
|
|
56 |
13660 |
|
Vol. LXXI. (1899) |
GENERAL INDEX TO THE PAGES OF THE FOUR VOLUMES
Actions mutuelles des formes invariantives
d^riv^es, sur les, in 218 D'Alembert-Carnot, geometrical paradox, rv
238 Algebra, universal, iv 146, 208 Alliance, or Ueberschiebung, in 132, 217 AUineation, theory of, iii 390 Allotrious factor, i 438, 580 Alternants, iv 416 Amphigenous surface, ii 436, 478 Anakolouthic sum, ii 40 Annihilator, iv 288, 451 Apocopated, i 580
Approximation, Poncelct's, to a square root, 11 181, 200
to a linear function of two irrationalities, HI 635, 644 Arborescent functions, ii 49 Arithmetic, see Numbers
addition to the vocabulary of, iv 588
theorems in, ii 40, 484, 485 Arithmetical progression containing an in- finite number of primes, ii 712; iv 620 Arithmetical series. On, iv 687 Arrangement, compound, ii 325 Arrangements, a theorem of Cauchy for,
II 245, 290
Associated algebraical forms, i 198 Astronomical prolusions, ii 519, 546 Asymptotic limits for number of primes,
III 530; IV 704
Asyzygetio invariants, extension of theorem,
IV 515
Atomic tlieory and theory of concomitants,
III 148 Axis of rotation of a rigid body, i 157
Barycentrio perspective, ii 342, 358 Bernoulli's numbers, ii 254 Bezoutians, i 430, 444, 548, 557, 580 Bezoutio square, i 430, 444 Bezoutoid, i 555 Bicorn, ii 469, 478 ; iii 214
Binariants, in 571 ; iv 294
Binary system of cubics compared with ter- nary system of quadratics, ii 15
Binomial extractor, in 14
Biorthogonal reduction of lineo-linear form, IV 638, 650, 654
Bipartition, iv 34
Bipotential, in 38
Biquadratic, see Quartic
Bismarck, in 32
Bisyntheme, i 92
Boole-Mongian, iv 283, 380
Bring, Jerrard and Hamilton on quintic equa- tion, IV 531, 553
Brioschi's equation for symmetric functions, IV 166
Buffon's problem of the needle, iv 663
Burman's law for inversion of independent variable, n 44, 50, 65
Caesura, n 146
Calculus of forms, On the, i 284, 328, 402, 411 ;
n 11 Canonic roots, n 331 Canonical forms, i 184, 190, 202, 208
a memoir on elimination, transformation, and canonical forms, i 184
for binary forms, i 190, 202
of cubic surface, i 195
of ternary cubic, i 201
Essay on, i 203
for odd degrees, i 208, 265
for even degrees, i 216, 271, 279, 293
a discovery in, i 265
of binary sextic, ii 18
of quartic and octavic, ii 18
for several variables, iv 527 Cartesian ovals, ii 527, 550 Catalectic, i 211 Catalecticant, i 293 Cayley's theorem for number of invariants,
in 55 ; iv 458, 519 Central force, ii 547
General Index
749
Centre of gravity of figures in homography, n 323 of a qaadrilateral, II 338 of a tmncated pyramid, n 342 Characteristic, i 580 Chemistry and algebra, ni 103, 150 Circle of convergence, n 301 Circle, successive involutes to, ii 629, 630,
641, 663 Cissoid, in 16
Clausen and von Standt's law for factors of de- nominators of Bemoullt's numbers, n 254 Clebsch, a theorem for curves of the fourth
order, iv 527 Coexistence, rational derivation from equa- tions of, I 40, 47, 54 Cogredience and contragredience, i 285
compound, i 287 Cogredient and Contragredient, i 581
systems, i 290 Colligation, in 23 Combinant, i 402, 411, 554, 580 Combinatorial aggregation, i 91 Commemoration-day address at Baltimore, ni
72 Commutants, i 201, 255, 305 Compound cogredience, i 287 Compound partitions, ii 113 Computing products without logarithms, n 34 Concomitance, complex, i 291 Concomitant, i 200, 286, 681 plexus, I 291
of given order and degree for any sygtem of forms, in 67, 113, 241 Concomitants, derivation of one from another,
I 287, 290 Cone projecting intersection of two snrlaceB,
I 169 Congruences, the resultant of two, ni 475 Conies
intersections and contacts, i 119 having contact of third order, i 15a porismatic property of, i 155 intersections of two, t 162 meeting cubic curve in six conseoatiTe points,
n 59 differential equation of, nr 282, 380 Conjugate equations, n 399 Conjugate system of regular substitutions,
u 623 Conjunctive, i 581 Connumerant, u 133 Conoid, I 228 CoDseentive points, four upon a tangent line
of a surface, i 177 Constmctive Theory of Partitions, tv 1
Contacts of conies, i 119, 223 of lines and surfaces of the second order, I 219, 227; u 30 Contents of polygons, von Staudt's theorems
for, I 382 Continuants, ni 249 Continued fractions, i 641
for the quadrature of the circle, ii 691 improper, i 583 arithmetic theorem, rv 659 expressing the roots of a quadratic, rv 641, 645, 647 Contrary, reciprocal or complementary substi- tutions, I 200 Contravariant, i 200, 581
changing to a covariant, i 200 Contravector, n 19 Convergence circle of, n 301 corona of, ii 301 Coreciprocants, iv 419 Corpus of matrices, rv 222 Correlations, of two conies, i 119 Correspondence between arrangements of com- plex numbers, iv 59 Correspondence of partitions, rv 24, 38 Covariant, i 200, 581 Crocchi's theorem, in 653 Cross-gratings used to prove formulae in
elliptic functions, in 667 Crystals, Fresnel's theory of, i 1 Cube root extracting machine, in 18 Cubes, sum and difference, numbers so re- soluble, ni 347 Cabic and linear form, concomitants of, ni 97,
393 Cubic and quadratic, concomitants of, m 97, 394 syzygies, in 505 Cubic and quartio, concomitants of, m 127, 398 reconciliation of two enumerations of con- comitants, in 132, 136 Cobic, binary, concomitants of, m 283, 579 generating function for covariants, in 113 Cubic curve and conic of sextactio contact, ii 59 polygons inscribed and circumscribed, in
841 rational derivation of points, ii 107 ; in
351 law of squares, in 359 triangles inscribed and circumscribed, in 474 Cubic Form, in integers, i 107, 110, 114 ; n 63,
107 ; III 312 Cubic, quadratic and quartic, ni 625
750
General Index
Cubic residues, 2 and 3 as, m 345 Cubic surface, expressed by five cubes, i 195 polar reciprocal, i 302 twenty-seven lines of, n 242, 451 Cubic, ternary, degeneracies of, i 335
concomitants of, i 192, 308, 327, 331, 599 ; n 13, 387 Cubics, two binary, concomitants of, iii 97, 395 reconciliation of two enumerations of con- comitants, III 258 Cumulant, i 504, 580
Cnrsality or genus of a plane curve, in 14 Curve of any order, differential equation of,
IV 495, 524, 529 Curves in space analogous to Cartesian ovals,
II 555, 559 Cyclodes, ii 629, 641, 663 Cyclotheme, i 93
Cyclotomy, iii 317, 326, 381, 428, 437, 446, 477, 479; iv 607, 626
Declmic, binary, concomitants of, in 256,
302 Definite integrals, two new, n 208, 298 Degree of a symmetrical function in the oo-
efScients, i 595 Denumerant, n 120; m 609, 614 of a diptych, ii 668 for invariants of octavie, in 52 Derivation, rational, from equations of co- existence, I 40, 47, 54 Derivative, of two equations, of specified de- gree, I 41 points of curves of third order, ii 107; in 351 Determinants
diminished, i 126, 136 compound, i 126 and quadratic forms, i 129, 147 minor, condition of all vanishing, 1 147, 221 relative, i 183, 188
of two quadratic forms, summary of possi- bilities, I 236 relation of minor determinants of equivalent
quadratics, i 241, 647 a fundamental theorem, i 252 combination of, i 399 definition, i 581 Sylvester's theorems in the First Volume,
1 647 double, n 326, 331, 836 and polar umbrae, ii 327 of parallel motion, in 35 and duadic disynthemes, in 264 comprising the secular determinant, in 453 determinants composes, snr les, in 456
Dialytic elimination, i 133, 256, 581 for ternary forms, i 62, 76 restatement, i 86 extensions of, i 256 origin of, in 77 Difference and differential equations, n 689 ;
in 546, 551 ; iv 630 Differential equation of conies, iv 282, 380 Differential equation of a curve of any order,
IV 495, 524, 529 Differential equations of a concomitant, i 352 Differential invariants, iv 245, 520 Differential transformation, n 50, 65 Diflerentiants, in 113, 118, 124, 151, 232;
IV 165 Diploidal contact, i 225 Diplotheme, i 92 Diptych, II 665 Discriminant, i 581
of the canonizant, u 418 Discriminatrix, ii 395, 478 Disjunctive, i 582 Ditheme, i 175 Divisors of cyclotomio functions, in 428, 437,
446, 479 Divisors of the sum of a geometrical series,
IV 607, 625 Double integration, i 36 Double six of lines, ii 243, 451 Duadic disynthemes and determinants, m
264 Duadic syntheme, in 170 Duodecimic, binary, concomitants of, in 489 Dyadism, ni 23
E(x), the function, n 177, 178, 179 Edticational Times, index to occurrence of Author's name in mathematical questions from, IV 743 Eduction, ii 147 Elimination
a new theory of, i 40
by inspection, i 54
note on, I 58
Dialytic method, i 61, 86 extensions of, i 256
linear method of, i 75
between quadratic functions, i 139, 145
Sketch of a Memoir on, i 184
from ternary forms, i 62, 76, 298 Elliptic integral of the first kind, u 203, 211 Elliptic motion, ii 496 Emanant, i 288, 431, 582 Endoscopic and exoscopio, i 431, 582 I'Entrelacement d'une fonction par rapport k
une autre, m 449 Equal roots and multiple points, i 367
General Index
751
I'Eqaation qni sert k determiner les in^galit^s
s^culaires des planetes, i 366 Equation, roots of, difference functions in cor- respondence with power sums, iv 163 Equations of which all the roots are real,
III 411 Eqaatrix, n 395, 478 Euler's numbers, ii 254
Euler's theorem of reciprocity in partitions,
I 597 ; II 120
Euler's theorem for parabolic motion, n 522 Euler's theorem for the partition of pentagonal numbers, in 664, 685; iv 93, 95 Frankhn's proof, iv 11 Euler's theorem for perfect numbers, iv 589 Evectants, i 329, 367 Even number, partition into two primes,
II 709
Expansion of first negative power of a power
series, ii 103 Extent and content of a partition, rv 2 Eyes, ears, nose, lips and chin as singularities,
IT 293 Ezekiel's valley of dry bones, iv 282
Facultative point, in theory of qnintic, i
436 Farey series, m 672, 687; rv 55, 78, 101,
603 Fermat's theorem, n 239, 232, 234, 241, 263;
IV 591 Fermatian, iv 607, 625
Finite differences, ii 307, 308, 313, 318;
ni 262, 633 Fluids, on the motion and rest of, i 28 Form, definition of, i 582 Forms, Calculus of, i 284, 828, 402, 411;
n 11 Formes-adjointes, i 200 Formes-associ^s, ni 108, 199 Fractions
with limited numbers, m 672, 689 ; iv 84,
738 ; lee Farey series vulgar, a point in the theory of, in 440 Functional relations among the roots of a
qnartic, i 192 Fundamental theorem of the theory of in-
varianto, in 117, 232; iv 458 Funionlar solution of Buffon's problem, rv
663
Generating function
for invariants of octavic, ni 52
for invariants of binary forms, ni 58
for covariants, in 113
in partitions, iv 21
for reciprocants, iv 402
Geometrical notions and determinations,
IV 259 Geometrical problem, on a simple, i 392 Geometry, descriptive and metrical, n 8
Lecture before the Gresham Committee, n 2 Germany, the classical laud of learning,
ni 79 Goldbach-Euler theorem for primes, rv 734 Graphical conversion of a continued product
into a series, iv 26, 91 Graphical dissection, H. J. S. Smith on,
rv 49 Groups, n 269 intransitive, n 275 continaons, iv 422
Halphen, on reciprocants, iv 290 Hamilton's numbers, iv 553, 585
Hammond's theorem on, iv 550, 557, 586 Hammond, benefit of intercourse with, iv 300 Hessian, or Hessean, i 583
of a cubic surface, i 195 Homaloid, i 175
Homaloidal Law, i 129, 150 ; u 717 Homogeneous functions, general properties,
I 165 Homonomial resolubility, n 289 Homonymous, n 122 Huxley, on Mathematics, ii 653, 654 Hyperdeterminants, i 185, 583
a discovery in, i 265
Imaginaries, the eight square, in 642 Imaginary roots, Newton's rule for, ii 376 Inaugural lecture at Oxford, iv 278 Independent variable, change of, n 44, 50, 65 ;
IV 445 Indicatrix, n 398 Induction and verification in Mathematics,
n 714 Inertia, law of, for quadratic forms, i 381,
511, 583; IV 532 Infinitesimal variation, i 33, 326 ; ii 385 Integers, successions that cannot be indefi- nitely continued, in 656 Integration, double, i 36 Interaction of covariants and invariants,
ni 207 Intercalations, theory of, i 511, 545, 583
effective scale of, i 582 Intermutants, i 201, 317 Interpolation, Lagrange's rule, i 645 Interpositions, theory of, i 614 Intuitional exegesis of generalised Farey series,
IV 78 Invariant factors of a determinant [nnnamed],
I 221
752
General Index
InTariants, i 583
of a function of even degree, i 273 nmnber of, theorem of reciprocity, i 606 general theorem for number of, in 93, 117,
232 ; IV 458, 515 limits to the order and degree of, iii 101 Involutants, iv 133, 134 Involutes to a circle, ii 629, 630, 641, 663 Involution of lines, ii 236, 240, 304 ; iii 657,
651 ; IV 136 Irrationality, value of linear function of,
II 250, 305; m 635, 644 Irrationality of ir, iv 680, 682 Ivory's theorem for potential, in 45
Jacobi, at Trinity College, Cambridge, ni 77 theorem of, proved by partitions, iv 60, 97 Jacobian, i 583
Jerrard's form for a quintic is singular, I 211
Kant's Doctrine of Space and Time, u 719 Kenotheme, i 176, 683
Lady's fan, in 35
Lagrange's theorem for linear function of an
irrationality, ii 250, 305 theorem of interpolation, i 645 Lambert's theorem for elliptic motion, ii 496,
519 Latent integer, n 100 Latent roots of a matrix, iv 110 Law of succession, i 287 Law of synthesis, i 292, 348 Laws of verse, iii 123 Lemniscate, va 14 Limits for the real roots of an equation,
I 423, 424, 620, 627, 630
for number of concomitants of binary forms,
m 110 to the order and degree of concomitants,
III 113 for prime numbers, in 630; iv 704 Linear complex, construction from five lines,
II 237
Linear and cubic forms, concomitants of,
III 97
and quadratic forms, concomitants of, in
392 functions, two, concomitants of, ni 392 substitutions, powers and roots of, ni 662 Lineo-Unear form reduced to canonical shape,
IV 638, 650, 654
Lines in space, involution of, n 236, 240, 304 ;
m 557, 651 ; iv 136 Lines on a cubic surface, ii 242, 451
Linkwork and linkage, in 9 Logarithmic waves, rectifiable compound, n 694
MacMahon's transformation of snbinvariants,
IV 164, 236 Malfatti's problem of inscribed circles, i 153 Mathematical questions in the Educational
Times, iv 748 Mathematics and Observation, il 665, 714 Mathematics, philosophy not calculation, iv
329 Matrices, i 247, 683
orthogonal, n 616
inversely orthogonal, ii 615
powers and roots of, m 562, 565
properties of split, in 645
latent roots of, iv 110
involution of, iv 115, 219
systems, rv 133
vacuity, nullity and latency, iv 133
equations in, iv 152, 176, 181, 199, 206, 231, 272
and the law of Harriot, iv 169
multiplicity of, rv 210
zero, nullity and content of, iv 211
latent roots and vacuity of, iv 215
biorthogonally reduced to canonical form, IV 638, 650, 654 Mean value of coefficients in an infinite deter- minant, ni 253, 257, 277 Mechanical conversion of motion, in 7 Meicatalecticizant, i 293 Minor determinants, i 147, 584
and linearly equivalent quadratic functions, I 241, 647
conditions for all to vanish, i 147, 221 Mixed reciprocants, iv 289, 312 Monadelphic, in 153 Mongian, iv 283, 380 Monotheme, i 175, 584 Monothetio equations, iv 169, 173 Motion, mechanical conversion of, in 7 Multipartite system of equations, resultant
of, II 329 Multiple quantity, iv 133 Multiple roots, i 66, 69, 370
and evectant of discriminant, i 367, 370 Mutual action of concomitants, in 218
Napier and Briggs, anecdote of, rv 279 Newton's rule for imaginary roots, ii 376, 489,
491, 493, 495, 498, 514, 615, 623, 704 ;
HI 414 ; IV 160 Nomes, n 272, 288 Nonic, binary, table of concomitants, m 281,
293
General Index
763
Nonions, ni 647 ; iv 118, 122, 154
Notation for loci in space, i 175
Null system [unnamed], ii 237
Nullity of a matrix, iv 133
Number, Space and Order, cardinal notions of
Mathematics, ii 5 Numbers
theory of, i 107, 110, 114 ; n 177, 178, 225 ;
HI 252, 438, 440, 446 ; iv 88 Wilson's theorem, i 39; n 10, 249, 293 expressed as four squares, ii 101 of primes, ii 225 ; in 530 ; iv 592, 696 of Bernoulli and Euler, ii 254 cubic ternary form, i 107, 110, 114 ; ii 63,
107; III 312 as sums of cubes, ui 347 law of reciprocity, in 433 resultant of two congruences, in 475 snccessions of integers not indefinitely con-
tinuable, in 580, 656 geometrical proof of a theorem in, in 635,
644 fractions with limited, iii 672, 687 ; iv 738 ;
gee Farey series Ely's proof of a theorem for residues, iv 50 Hamilton's, it 550, 553, 585 vocabulary for, it 588 diTiding the sum of a geometrical series,
IT 607, 625 ; see Cyclotomy perfect, it 611, 615, 626 arithmetical series, it 687 irrationality of jr, iv 680, 682 Numbers, Partition of, tee Partitions
OctaTic
canonical form, n 18
inTariants of, in 52
table of concomitants, in 115, 290
irreducible covariants, in 480
reconciliation of two enumerations of con- comitants, m 509 Octopus, in 34
Operations, calculus of, n 567, 608 Optical theory of crystals, i 2 Orbit, under attraction of a circular plate,
n 539, 550 Orders
theory of, i 145, 170, 221, 549, 584, 587
loss of, deduced from discriminant, i 139 Orthogonal invariants, i 351 Orthogonal reciprocants, it 249, 338 Osculants, n 364, 368
Oyster, twin-soul to the mathematician, in 73
Pantigraph, ni 12, 26 Paradox, m 20, 36
Partial differential operators, ii 567, 608 Partitions
Euler's theorem of reciprocity for, i 597; IT 2
of numbers, n 86, 90, 176, 701; in 634, 680, 683 ; IV 92
symmetrical functions of, ii 110
compound, n 113
Seven Lectures on, ii 119
a theorem of Cauchy for, n 245, 290
of an even number into two primes, ii 709
and rational fractions, in 605
fundamental theorem of the new method, in 658
Durfee's theorem, in 659
Franklin's proof of Euler's theorem, in 664
expression of a certain product as a series, m 677
Euler's theorem for pentagonal numbers, in 685; it 93, 95
a ConstructiTe Theory of, it 1
proof of a formula of elliptic functions, IT 34
table of, IT 391 Pascal's theorem, i 138, 145, 151 Peaucellier's bar motion, in 7 Perfect numbers, iv 589, 604, 611, 615, 626 Permutants, i 201, 210, 214, 318 Perpetuant, m 592 ; it 237 Perpetuitant, it 138 PerspectiTe, barycentrie, n 342, 358 Persjrmmetrical, i 584 Pertactile point on a cubic curve, in 367 Plagiogonal inTariants, i 351 Plagiograph, ni 26 Plexus of forms, t 291, 346 Poinsot's representation of the motion of a
rigid body, n 517, 577, 602 Polar reciprocal, i 303, 363, 377
of a cubic surface, i 302 Polar umbrae, n 327 Poles, in the theory of potential, m 49 Polhods, in 4 Polynomial functions, expressed by fewer linear
functions of Tariables, i 587 Poncelet's approximation for radicals, u 181 Post-Schwarzian, it 321 Potential, theory of, m 49 Presidential Address to British Association,
II 650 Pressure of earth on revetment walls, n 215 Prime numbers between giTen limits, ni 530 ; IT 704, 711
inequalities for, it 592
Goldbach-Euler theorem, it 734 Prime radical circulator, n 97
754
General Index
Prinoipiant, tv 882
expressed as an invariant, it 465 Probabilities, a class of qaestions, n 480 Probability, Bu£fon'B problem, rv 663 Probationary Lecture on Oeometty, n 2 Problem of least circle enclosing given points,
n 190 Product, a oontinned, expressed as a series,
m 677 Projectiles, a triile on, u 5S
constmction for, ii 61 Projective Beciprocants, iv 382 Protomorphs, iv 250, 289 Pure Beciprocauts, n 257, 289, 312, 341, 891, 403, 514
Quadratic (and quadrio)
loci, contacts, i 119, 236; ti 30
elimination, i 139
functions, solution of a system of, i 152
functions, relation between the minor deter- minants of, I 241, 647
polynomial, reducible to squares, t 378
forms, law of inertia, i 381, 511, 612; IV 532
functions, resultant of three, i 402, 415
form indicating number of real roots of an equation, i 402
radicals, linear representation of, ii 118
residues, fundamental theorem, n 180
and cubic, concomitants of, ni 97 syzygies, in 505
generating function for covariants, in 113
concomitants of, ni 283
two quadratics, concomitants of, in 394
two quadratics and one quartic, concomi- tants of, III 622
cubic and quartic, concomitants of, in 625
and two quartics, m 627 Qnadrinvariant, i 584 Quadruplane, in 28 Qaantics
to order eight, generating functions for co- variants, in 113
of unlimited order, seminvariants of, in 568 Quartic (and Quartics)
invariants of, i 329, 599 ; m 283, 579 generating function, in 113
canonical form, i 269
ternary, i 334
two binary, reconciliation of two enumera- tions of concomitants, in 61, 63, 95
two binary, concomitants of, in 402
and cubic, concomitants of, in 127, 132, 136
and linear form, in 393
and quadratic, in 395
and two quadratics, in 622
Quartic continued —
cubic and quadratic, in 625
two and quadratic, ni 627
three binary, iii 630 Quasi-catalecticant, tv 400 Quasi-covariant, iv 411 Quaternions, iv 112, 122, 162, 183, 188, 225 Quintic, binary
concomitants, i 196, 204, 207; ni 210, 284, 580
canonical form, i 193
condition for three equal roots, i 348
a syzygy, i 362
reality of roots in terms of invariants n 371, 376, 418, 482
generating function for concomitants, in 59 113
sextic and nonic, skew invariants of, in 195
germ table for, in 577 table of deduction, in 591
Tschirnhausen transformation, it 531, 553 Quot-additant, ii 87, 92 Quot-undulant, n 87 Quotients, Sturmian, i 396, 495 Quotity, u 86, 90
Radicals, approximate linear evaluation, ii
118, 181, 202 Bamification, in 23 Bational derivation of points of a cubic curve,
II 107 ; ui 351 Beciprocauts, iv 242, 249, 255, 281, 301
Lectures on the Theory of, iv 301 Eeciprocity
method of, i 339
law of, for forms, i 403, 606; in 105, 174, 189
law of, in the Theory of Numbers, in 433 - theorem of, in partitions, u 703 Ecduced-resultant, i 188 Eeducible oyclodes, ii 663 Beduction in number of variables, i 587 ; »ee
Orders Eelative determinants, i 188 Besiduation, geometrical theory of, m 317, 352 Besidues, Sturmian, i 438 Respondent, inverse of concomitant of, i 340 Resultant
of a system of equations, i 259, 584 ; ii 329, 363, 369, 694; in 426
of three forces, approximate linear repre- sentation, II 188
of a matrix, n 334 Revenants, in 593 Reversion of series, ii 50, 65 Reversor, rv 451
General Index
755
Kevetment walls, pressure of earth on, n 215
Bhizoristic series, i 516, 584
Riemann surface, iv 241
Bigid bodies, on the motion and rest of, i 33
Rigid body, rotation of, i 157, 217; n 517,
577, 588, 602; in 1 Roots
of an equation, i 66, 69
of numerical equations, rational or not, i 103
equaUty of, i 367, 370
Stnrmian functions, i 45
multiple, I 69
limits to real, : 623, 627, 630
of a particular form of equation, n 360, 374, 378, 401
and Newton's rale, ii 361
rule for separating, n 542
of the secular equation, in 451
of two polynomials, intercalation, i 517
of matrices, in 565
of unity, »ee Cyclotomy
Schlafli, double six of lines, n 243, 451 School girls problem for fifteen, ii 266, 276 for nine, rv 732 Schwarzian derivative, rv 252, 284, 304 Secular inequality equation, i 634 ; in 451 ;
IV 110 Seminvariants to quantics of unlimited order,
m 568 Septimic, binary generating function for covariants, in 113,
140, 144 covariants of, iii 146, 286 Series reversion of, n 50, 65 for a certain product, in 677 Sextic, binary geometrical form of reduction, i 176 canonical form, i 280, 283; ii 18 generating function for covariants, ni 60,
113 equation connecting three absolute invari- ants, ni 214 concomitants, in 285 germ table for, in 578 Sign successions, n 615 Signaletic, i 584 Sines and cosines of multiple arcs, expansion
of, n 294 Six-valued function of six letters, i 92 ; u 264 Sorites, in 440 Sources of covariants, iv 164 Space of four dimensions, ii 716 Spherical Harmonics, Note on, in 37 Square root extractor, ui 18
Squares, four, expression of any number by,
n 101 von Staudt's theorems for polygons and poly-
hedra, i 382 Stigmatio multiplier, rv 707 Straight lines on the Hessian of a cubic surface, i 195 on a cubic surface, ii 243, 451 Sturm's theorem, i 45, 57, 59, 396, 429, 513,
609, 620, 637; see Syzygetio Subinrariants, in 568
as functions of power sums, iv 164 Subresultant, i 188 Substitution, i 585 representable by a given number of cycles,
n 247, 292 regular conjugate system, n 623 Superlinear equations, ii 378, 401, 482 Surd forms, approximate linear evaluation,
n 181 Surfaces of second order, contacts and inter- sections, I 227, 237 Symmetrical functions degree in terms of the coefficients, i 695 Brioschi's equation for, iv 166 Syntax, n 269
Synthemes, i 91 ; n 265, 277, 286, 288 Syrrhizoristic, i 585 Syzygetic on a Theory of the Syzygetic relations of two rational integral functions, etc., i 429 functions and multipliers, i 132, 585 equations in terms of the roots, i 458 Syzygies, in 489, 603
Tactic, II 269, 277, 286 Tactinvariant, n 363 Tamisage, Tamisement, in 59, 99 Tangential on a cubic curve, in 352 Taylor's theorem, generalisation of, m 88 Tchebycheff, on primes, in 530; iv 711 Ternary cubic (««« aUo Cubic curve) concomitants of, i 192, 308, 327, 331, 599;
n 13, 387 breaking into linear and quadric factors,
1 333 sextactic points, n 59 solution by integers, i 107, 109, 114 ; n 63,
107 ; HI 312 Ternary denominational system of coinage,
in 476 Ternary quadric functions, resultant of three,
I 415 ; tee Quadratic Ternary system of quadratics compared with
binary system of cubics, ii IS Ternary systems of equations, dialytic elimi- nation from, I 61, 83
/I a
766
General Index
Tessellated, n 616
Tetrahedra, metrical relations, i 390, 404
Theorem of iiiTariaiits, proof of the fanda-
mental, in 117, 232 Three binary forms, concomitants of, ni
622 Totitives and Totient, ii 225 ; in 337 ; iv 89,
102, 689 Transformation, Memoir on Elimination,
Transformation and Canonical forms, i
184 Transformation of partitions by the cord rule, iv 48 Tscbimhausen, iv 531 Trees, the geometrical forms called, m 640 Triangles inscribed and circumscribed to a
cubic curve, in 474 Trigonometry, spherical, Delambre's theorems,
n 564 Trisection and quartisection of the roots of
unity, ni 881 Tritheme, i 175 of third degree, has six right lines at
every point, i 176
Tschirnhausen transformation, iv 631 Types, I 585; n 276, 283
Ueberschiebnng, or Alliance, m 182, 217 Umbral, i 685
Unilateral equations, iv 162, 169, 226 Unity, roots of, in 438; tee Cyclotomy Unravelment, i 322, 360
Vacuity of a matrix, iv 133 Valency, in 28, 103, 151 Vermicular, iv 294 Versors, in 30 Virgins, problem of, ii 113
Wave surface, i 1
Waves, in calculation of qnotity, n 91 Weight, I 585
Wilson's theorem in the Theory of Numbers, I 39; n 10, 249, 298
Zeta, for squared product of differences, i 59,
586; n 29 Zetft-ic multiplication, i 47, 49
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