Digitized by the Internet Archive
in 2007 with funding from
IVIicrosoft Corporation
http://www.archive.org/details/collectedmathema04sylvuoft
s
MATHEMATICAL PAPERS
CAMBRIDGE UNIVERSITY PRESS
aonHoii: FETTER LANE, E.C.
C. F. CLAY, Manager
ffininbutstl : loo, PRINCES STREET
Bftlin: A. ASHER AND CO.
leipjis: F. A. BROCKHAUS
jjtto aork: G. P. PUTNAM'S SONS
Bomtag ant ffialtutta: MACMILLAN AND CO., Ltd.
All rights reserved
|
^^p |
«5 |
|
|
^^^^^^^^^^^^B CM |
« |
l^^^^^^^^l |
|
^^^^^^^^^^^^^^^^m |
||
|
P |
vV |
|
|
Kl« |
||
|
sao^ |
■HiH=JH |
v-edi^
M2V-V
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) |
|
|
2 2 3 3 (8) |
|
|
4 4 2 (7) |
|
|
13 3 (6) |
|
|
10 8 4 (1) |
|
|
9 5 1 (2) |
|
|
111 (3) |
|
|
4 4 4 (4) |
|
|
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) |
|
|
1 (8) |
|
|
1 0 0 0 (7) |
|
|
2 1 1 1 (6) |
|
|
12 9 5 1 (1) |
|
|
10 8 4 0 (2) |
|
|
1 3 3 0 (3) |
|
|
8 8 6 4 (4) |
|
|
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
|
Ui- |
{ |
i |
|||||||||
|
HL |
K |
||||||||||
|
NF |
< |
!) |
|||||||||
|
Rr |
s |
||||||||||
|
V |
|||||||||||
|
1 |
^ |
J |
|||||||||
|
i |
i ^ |
1 |
|||||||||
|
, |
|
t— • |
||||||||||
|
E |
FD |
|||||||||
|
K |
LH |
|||||||||
|
O |
PN |
|||||||||
|
S |
^R |
|||||||||
|
Q |
||||||||||
|
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.
|
22 |
19 |
16 |
|
26 |
27 |
28 |
|
16 |
12 |
|
|
21 |
22 |
|
|
15 |
10 |
|
|
17 |
18 |
|
|
10 |
||
|
15 |
||
|
4 |
4 |
9 |
|
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
,