(4 — «)■ Thus the generated arrange- ment is licit and belongs still to the third species; but now [(7] + [4,] = 4 and [i4] = 0 > 4- Hence the reversing operator ^' is applicable to the new arrangement ; the remaining cases to consider (in which 4 = a for the arrangement as well before as after being operated upon) may be separated into those where C. > 0, and at the same time either 0 -(- 4j < ,4 or i4 = 0, which are amenable to the operator ^' and the complementary cases which are amenable to '^. > In the cases first considered [4,]=4i — c, [i4] = (7 = 41 ^[C*] + 0 or C (and d, fortiori >0), consequently the new arrangement is licit and still belongs to the third species, and since either [C] = 0 or else and [,4] > 0, it is one of the complementary cases and is subject to the reversing operator '^. Again, any arrangement for which 4 = a belonging to the complementary cases is defined by the conditions i4 > 0 and (7 + 4i = > i4 and is by hypothesis to be subjected to the operator '^ which will make [4i] = 4, -1- c, [i4] = 0 or > i4 [(7] = i4 - ^1 - c, and since (7 = > i4 — 4i , [(7] < (7, so that the operation leads to a licit new arrangement. Also [C] + [4i] = i4. and consequently either [,4] = 0 or [C + 4,] < [i4], which is a condition belonging to the first considered class of cases, subject to the reversing operator ^', and thus for the third as for both the antecedent species of the second genus, it has been proved that each designated operator prior to any arrangement being performed does not take away its licit character nor carry it out of the species to which it belongs, and on being repeated brings it back to its original form, and that the eflFect of any single operation is to maintain the content (or inertia) of the arrangement constant but to reverse each of its characters. This is the thing that was to be proved and brings my wearisome but indispensable task to an end. 1] three Acts, an Interact and an Exodion 77 (68) Another and perhaps somewhat clearer image of the classification of the numbers of the second Genus may be presented as follows : The com- binations of the characters XGOExcoe give rise to eight pairs of groups, say eight classes. Of these classes four belong to Species 2, and may be repre- sented by four indefinite vertical parallelograms, set side to side, and sub- divided each of them into four, (say) black, white, grey and tawny stripes, corresponding to the four varieties of the second species. The other four classes may be similarly represented by four such parallelograms as before, but separated by a transverse horizontal line into eight sub-classes, four corresponding to the first species and four to the second. The upper parallelograms may then be each divided into blue and green, the lower into yellow and red stripes to represent the respective couples of varieties of the first and third species. There will thus be in all thirty-two stripes, namely four blue, green, yellow and red, and four black, white, grey and tawny, each of which is bifid, representing two groups of opposite sexual characters, which may be fittingly represented by the upper and under sides of the sixteen unlimited single-coloured stripes of the first and the eight unlimited double-coloured stripes of the second set of parallelograms. The above logical scheme is not intended to convey any notion of the relative frequency of the three species. The general case is that of the first species. The second is conditioned by 'i?= 6 or -S =0, and the third by 5= 0. When 'B = b it is about an even chance whether the arrangement is of the second or first species, and when 5 = 0 of the second or third. Either equality is a particularization of the B series, the latter signifying that there are no B'a in the arrangement, the former that there are B'a descending in rational progression down to b : this supposition is apparently infinitely more general than the former, because there is no limit to the number of terms in the progression, and the case of a natural progression of B's of the kind men- tioned with any given number of terms as regards the probability of its occurring in an arrangement seems to be on a par with the case of the jB's being all wanting. Hence the first species is infinitely more frequent than the second, and the second than the third. According to Prof. Max MuUer's theory of the relation of thought to language (if I interpret it rightly) I ought to have thought out my divi.sions and schemes of operation in language, but I certainly had formed in my mind a dim abstract of them before I had found the language that was competent to give them expression. In conclusion, I may remark that whilst the experience of the past indi- cated the probability that there did exist (if one could find it) a method of di.stributing the arrangements of the second genus into pairs, in sUch a way that in each pair the total or partial character should be reversed in passing from the one to the other, there was nothing to induce a reasonable degree of assurance that both those characters should be found simultaneously reversed 78 A Constructive theory of Partitions, arranged in [1 in one and the same distribution ; for aught that could have been foreseen to the contrary, it might very well have happened that one mode of distribution might have been needed to prove Jacobi's theorem for the case of only negative signs appearing in the factors on the left-hand side of the equation, and a dififerent one for the other case where only every third factor contains such sign — indeed upon the principle of divide et impera or doing one thing at a time (as invaluable a maxim to the algebraist as to the politician) I had completed the proof for the former case without thinking of the latter, and only when on the point of attacking it was agreeably surprised to find that there was nothing left to be done, for that the proof found for the one extended to the other — in familiar phrase, I had hit two birds with one stone. We may now ask whether this was a happily found chance solution or was predestined by the nature of things, and that simple necessarily implies double enantiotropy of conjugation. Probably I think not, and if so, a question arises as to the number of solutions for each of the two sorts of enantiotropy and whether the number of each kind of simply-enantiotropic conjugations is the same. Viewed merely as a question of direct multiplication, I think it must be allowed that what I have here called Jacobi's theorem (including Euler's marvellous one, as the ocean a drop of water) is the most surprising revela- tion that has been made in elementary algebra since the discovery of the general binomial theorem, and that the space devoted to its independent, and so to say, materialistic proof in these pages, although considerable, is not out of proportion to its intrinsic importance. H. Intuitional Exegesis of Generalized Farey Series*. (69) The demands of the press will only admit of a rapid sketch of what appears to me to be the true underlying principles of the theory initiated by Farey, honoured by the notice of Cauchy, and to a certain extent generalized by Mr Glaisher, whose inductive method in the cases treated by him finds its full development in the method of continuous change of boundary, explained in the course of what follows. Let us start from the conception of an infinite cross-grating formed by two orthogonal systems of parallel lines in a plane, the distance between any two parallels being made equal to unity. The intersections of any two lines of the grating may, as heretofore, be termed nodes. A triangle which has nodes at its apices and at no other point on or within its periphery, may be termed an elementary triangle, and the double of the area of any such triangle will be unity. If any finite aggregate of nodes be given it must be possible to pick out a certain number of them which may be formed together by right lines so as to form a sort of ring- fence, within which all the rest are included : the area thus formed, if it * Continued from note G, Interact, Part 2. 1] three Acts, an Interact and an Exodion 79 admits of being mapped out into elementary triangles, may be termed a complete nodal aggregate. Any other contour consisting of lines of any form (curved or straight) drawn outside of this ring-fence in such a manner that no nodes occur between the two, may be termed a regular contour. If any node 0 be taken as origin and any nodal lines through 0 as axes of coordinates, and if 'A, A' are the nearest nodes to 0 in the radial lines on which they lie, and if no nodes of the given aggregate are passed over as an indefinite line rotating round 0, passes from one of these radial lines to the other, '.40.4 is an elementary triangle, and if 'p, 'q ; p, q be the coordinates of 'A, A respectively, 'pq — p'q = e where e is + 1 or — 1 but is fixed in sign when the direction of the rotation is given. When the aggregate is complete, if the values of the coordinates of the successive points passed over by the rotating line be called . . . "p, "q ; 'p, 'q ; p, q ; p', q', p", q" ; ... ,vte shall have a Farey series formed by the successive coifples p, q, that is p" 'q — p'q" = e; p'q—pq' = e; pq' —p'q = e ... . Thus we see that the Farey property is invariantive in the sense of being inde- pendent of the position of the origin. Next I say, that if any contour to a given aggregate is regular, every contour similar thereto in respect to any node of the aggregate regarded as the centre of similitude is also regular, provided the boundary is simple; meaning that there are no interior limiting lines giving rise to holes or perforations in aggregate, and no loops formed by the boundary cutting itself. In the above figure 'BOB is any triangle whose sides are bisected in 'A, A, A'. Suppose 0 to be the origin, 'A, A' two nodes of greatest proximity to 0 successively passed over by the rotating line for a given 80 A Constructive tJieory of Partitions, arranged in [1 contour. As this contour expands uniformly in all directions through 0, the line 'AA' remains parallel to itself. Since 'AOA' is an elementary triangle so also must the similar triangles 'AAA', A'AB', 'AA'B be all elementary, consequently A will be the first new node intervening between 'A, A' brought into the enlarged aggregate as 'AA' moves continuously parallel to itself, and AOA, AOA' will be elementary triangles ; it may be noticed in order to bring this method into relation with that indicated by Mr Glaisher, that the coordinates of this new node A are the sums of the coordinates of its neighbours 'A, A'. If the contour were not supposed to be simple, this condition could not be drawn ; for if there were a hole round the middle point of 'AA' the node A would be missing in the enlarged aggregate, and if the first node to intervene as the contour went on enlarging be called (4), 'AO{A) or {A)OA' or each of them would be a multiple of the elementary triangle, so that the constancy of the value of the successive determinants would no longer hold. In like manner it will be seen that on the same supposition as above made, if in consequence of the contour contracting about 0 as the centre of similitude, two points 'A, A' which originally are non- contiguous, at any moment become contiguous, at the moment previous to this taking place A (and no other point) must have intervened, and after A has disappeared from the reduced aggregate, no other point can make its appearance between 'A, A'. (70) Hence we may contract at pleasure the given contour about any node as origin, and if the contour so contracted contains at least one node besides the origin, it will suffice to determine whether the given contour is or is not regular. Thus for example in the case of a triangle limited by the axes and by the right line a; + y = n, we may make w = 1 and the trial series will then become :r Y K which possesses the Farey property. Hence this will hold good for a triangular boundary of any size and wherever the origin is situated : this includes the case of the ordinary Farey series when the origin is taken at either extremity of the hypothenuse. So again for the area contained within the axes and the hyperbola xy = n, we may take xy = \ and the trial series is the same as before. (71) It is easy to form unperforated areas of any magnitude which shall not satisfy the Farey law : for example we may as in the annexed figure draw a curve passing through the origin, the point (0, 1), and the point (2, 3), 0 2 -, - does not satisfy the Farey law, and consequently no similar contour X o obtained by treating any one of the three nodes which it contains as a centre of similitude will be a " complete contour," and the successive values of (p, q) 1] three Acts, an Interact and an Exodion 81 obtained by the rotation of a line round the origin in such contour will not constitute a Farey series. The theory will, I believe, admit of being extended to solid reticulations, formed by the intersections of three systems of equidistant parallel planes, determinants of the third order between the three coordinates of successive points, replacing the pq —p'q of the plane theory. The chief difference will consist in the introduction of a new element in the multiplicity of the " normal orders " in which a given set (of points in a plane or) of radii in solido may be taken. (Points in a plane arranged in any order of sequence, such that the successive determinants formed by their trilinear coordinates are of uniform sign, are said to be in a normal order. Rays of a conical pencil arranged in any order of sequence, such that their intersections by a plane satisfy the above condition, are also said to be in a normal order : see privately printed syllabus* of my lectures on Partitions, 1859, or M. Halphen's theory of Aspects.) But as far as I can see this will in no way militate against the existence of the laws of invariance and similitude established for the case of a plane reticulation, but will only introduce a further principle of invariance, namely that the law of unit-determinants if satisfied by one normal arrangement of the points of the solid reticulation will be satisfied by every other. APPENDIXt. LIST OF CORRECTIONS SUGGESTED BY M. JENKINS TO PROFESSOR SYLVESTER'S CONSTRUCTIVE THEORY OF PARTITIONS. Page 5, 5 lines from end, 2« — (t + 3) should be n — (t + 3). „ 6, between 2nd and 3rd rows of sinister table insert 13.2.0. 7th and 8th „ „ „ „ 11.2.2. „ in 6th row of dexter table, for 8 . 4 . 3 (2) write 8.4.3 (1). „ 11, line 8 from the end, interchange protraction and contraction so as to read "contraction could not now be applied to A' and B' nor protraction to C." „ 13, line 25. If /(a;) = (l-a;)(l -a:»)(l-a»)(l -a;')(l -a*), for the second a* read «*. [* Vol. IX. of this Reprint, p. 119.] [t These corrections have been included in those made in the text preceding.] 8. IV. 6 82 A Constructive theory of Partitions, arranged in [1 Page 13, line 29, for " latter " read " former," „ 15, line 11 from end, for l' read l\ „ 20, line 4, for 1 + 2 read i + 2. „ „ line 5, for 1 + 2 read i + 2. „ 22, line 11, for XjX « read XjX « . „ „ line 20, for " the minimum negative residue of i — \ " read i + 1. „ 25, line 7, for j-^n^ ""^^ i_^ • „ „ line 4 from the end, for " to the 5th now " read " to the 5th row now." „ 27, line 15, for 15, 7, 3 read 13,. 11, 3. „ „ line 19, for (1 + ax) (1 — aa^) (1 —ax^) ... read (1 +ax)(l + aaf)...(l+ aa^'). „ „ line 22, for ^ a read , a. 1—x l—x" „ „ line' 30, for " angle whose nodes contain i nodes " read whose sides. „ 28, line 5, for " with j — i or fewer parts " read j — 1. „ line 12, for 1 + — af + — —— ic-+i etc. 1 — ar \ — X .i. — ap , ^ l-a^-' ^,, l-a;"-'.!-**"-' ^^ read of + -, - x"+^ + — — ; — «»+* + etc. \—a? 1— ar'.l— a^ If in the expression in line 9, namely in 1 _a;^-!!?+2 . 1 _ a^"-!!?+< ... 1 _ aJit-a \-3? . 1 -a^... \-x'i-'' we put J = 3 we^obtain g.fl--i + 1X l-ar-.l-a:* 1-a^.l-ar* ' l-a;»-M-«"-' \-a?.\-a^ ' since to = 2i— 1, and similarly for other terms when we put.;" = 2 and j = 1. The correction which I offer seems to me to be right, and the expression in the paper to give a wrong result in the cai^e when n happens to be equal to w + 2 ; for then the number of parts being supposed to be exactly i, the first bend contains 1i — 1 or to nodes, and there is then no way of placing the remaining 2 nodes so as to make the partition a conjugate partition — sup- posing I have not misunderstood the article. Page 29, line 8, for 19, 7, 6, 6 read 10, 7, 6, 6. „ „ figure, either insert a node at junction of 5th column and 7th row or remove a node from junction of 7th column and 5th row. „ „ lines 7 and 8 from the bottom, if we remove a node from the figure no change is required in these two lines; but if we 1] three Acts, an Interact and an Exodion 83 insert a node in the figure, then 111111733 should be 11 11 11 7 5 3 and 5 5 5 3 1 1 should be 5 5 5 3 2 1. Page 31, line 15 from end, after ; ; -r insert "or of ° \—ax.\—aa?...\ — aa? x" ai'.' „ 34, line 7, for a) read a*. „ „ line 8, for (x» + aa;'«)} read (a? + x^). I., 36. line 8. for ^' ^^ 7 -^> read ^-(|-^). „ 37, line 4, for a:" read a;^, „ line 7, forar^+' read 0,'^+^ „ 40, line 6, a,— i is, I believe, the right final term; but it appears as if it were the first of a pair instead of the last of a pair, »ai~i being a quantity which may vanish. If the pair of expressions which in the text precede a<— i, if definitely expressed and not left to be understood, should be [ai_, + «(_,- (2i - 3)], [tti-, + a._, - (2i - 2)], and not as in the text [a.-_, + ai_, - (2i - 1)], [a.-_, + o.- - 2i], the factor which should precede Oi—i is [aj + o, — (2i — 1)]. I do not quite follow lines 9 — 13 of p. 40, possibly from the oversight in the subscripts I do not see what is intended. But it seems to me the following proof would be right : The expressions of the same form succeeding a, + a, — 1 and Ui + 0.^—2 must be continued so long as they are positive, and must be rejected when they become negative. Now from the fact of i being the content of the side of the square belong- ing to the transverse graph a,- = or > i, a, ■= or > i, therefore Oj + a< — (2t — 1 ) is positive and is therefore one of the terms of the series. Also aj+, = or < i and tti+i = ori. If the extreme inside bend degenerate, so that it consists only of a vertical I line or of a single point, then 04 = 1; and since a,+,^a; = ou < 4 ar'+2a;4-l 4 En appliquant a ces deux in^galites la m^thode d'approximation successive que j'ai appliqu^e, dans* le Memoire cit6, aux in^galitfe auxquelles est assujettie la fonction ■^(a;) (?;oiV Senet, Alghhre supdrieure, Edition de 1879, t. II. p. 233), je parviens facilement et rigoureusemont a d^montrer que, etant donn^e une quantite e aussi petite qu'on veut, on peut trouver une limite supdrieure L et une limite inf^rieure A a Jx, ou L=^(— + Tl\a^-Ax + R(\ogx) A = ('i- T)'\x'-A'x + R'(\ogx), ou R (log x), ii'(log x) sont tous les deux fonctions rationnelles et entieres de log« d'un degr^ fini, dont les coefficients aussi bien que A et A' restent toujours finis et ou r], rj' sont tous les deux plus petits que e. II s'ensuit que la fraction \^ possede une valeur asymptotique — (ce qui n'est pas demontr^ pour la fraction analogue ^-- , dans la th^orie Jx parallels de M. Tchebycheff ) et que la valeur de — approche indefiniment 3 prfes quand x est pris suffisamment grand de — , c'est-a-dire de •30396.... II est facile de voir que la quantite Jx diminu^e de I'unit^ n'est autr chose que le nombre des fractions dans les Tables pareilles a celles de M. Airy. Ainsi, pour le cas de a; = 100 selon M. Airy, Ja;=3044. Poui o ce cas —ar'= 3039-6. Avec I'aide de ces limites on peut calculer la probabilite que deux nombres dont la limite sup^rieure est tres grande soient premiers autre eux. Car si cette limite est x, le nombre total des cas qui peuvent arriver est of, et le nombre des cas pour lesquels les nombres choisis sont premiers entre eux sera 2/a; — 1. Consdquemment, la probability en question 6 sera — .. M. Franklin, I'auteur de la belle demonstration, ins^ree dans les Gomptes rendus, du thdorfeme d'Euler sur le produit (1 — x) {\ — a?) {\ — a?) ... , a bien [* Vol. in. of this Eeprint, p. 532.] 2] Fractions ordinaires in^gales 87 voulu m'adresser la remarque que cette conclusion pent etre au moins con- firmee, peut-etre meme absolument d^montr^e, de la maniere suivante : X ^tant pris trfes grand, la probability que deux nombres inf^rienrs h, X, pris au hasard, ne contiennent pas tons les deux le nombre premier p, sera 1 — ^ . Done, la probability cherch^e sera (•4)(-J.)(-|.)('4.)-. qui est la r^ciproque de 1 1,1.1,1 1 2' ^ 3' 4' 5' 6* "^ 7» "*" ■"' c'est-k-dire est ^gal a — - . ° it' II y a une suite doublement infinie d' Equations fonctionnelles exactes qu'on pent former avec les J{x). En particulier, il y a une serie simplement infinie de telles fonctions ou les signes sont altemativement positifs et n^gatifs, et consdquemment peuvent servir chacun a donner une suite infinie de limites k Jx. Ainsi, si Ton ^crit dx=Jx- -'I -f/3- -Jl + /?- 0 ■'1 + + e,x = 3Jt -4>J-,+3Jt-*Jt+^J'^-J 3 '" 4 ■ - 6 *^ 8 + '*'' 9 "^ 12 + + on aura toujours, quand X ^{le' + k) i, 6tx = 2(/fc'' + A;)' {x+\f et quand x = {k'-^k)i-l, etx=^-^^ ^ , et, quel que soit le r^sidu de x par rapport au module k^ + k,on pent calculer la valeur de O^x. Enfin, si x est une quantity positive quelconque, on trouvera rt a? —x . cc' + 1x ->r\ ^*^ = °"> 2(FTT)' ^*^=""< 2-(ifc'T)fc)' • 3. NOTE SUE LE TH^ORjfcME DE LEGENDRE CIT^ DANS UNE NOTE INS6r6e dans LES COMPTES RENDU S. [Comptes Rmdus, xcvi. (1883), pp. 463—465.] Le thdorbine de Legendre, cit^ par MM. de Jonquiferes et Lipschitz, est une consequence immediate d'un theorfeme logique bien connu, lequel, mis sous forme sensible, equivaut a dire que, si A, B, C, ... sont des corps avec la faculty de s'entrecouper, contenus dans un vase d'eau, et si a, ab, abc, ... reprdsentent symboliquement les volumes de .4, de la partie commune k A et k B, de la partie commune a A, B, C, ..., alors le volume du liquide d^place par la totality des corps sera Sa — l,ab + tahc — .... Consdquemment, ce th^orfeme admet une generalisation infinie dent je donnerai un seul exemple. Nommons les nombres premiers qui n'excedent pas 7i, nombres premiers subordonnds k n, et distinguons entre eux ceux qui sont plus grands que \/n comme sup^rieurs, Le theor^me de Legendre Equivaut k dire que, si p^, p^ pi sont les nombres premiers subordonn^s a >Jn, le nombre des nombres premiers subordonnds ^ n du genre sup^rieur augmente de Tuuit^ est egal k \pj \PipJ \PiPiPJ Or, repr^sentons la fonctiou ^x(x + l) par Ax; alors on aura le th^oreme que la somme des nombres premiers subordonn^s a « du genre superieur augment^ de I'unite sera ^gale k An-%pA{^)^tp.p.^{-^^)-.... Par exemple, si » = 11, les nombres premiers subordonn6s a 11 du genre superieur seront 5, 7, 11, et les nombres premiers subordonn^s a V'* sont 2, 3. 3] Sur im thS&reme de Legendre 89 On doit done trouver, et en effet on trouve (11 . 12) - 2 (5 . 6) - 3 (3 . 4) + 6 (1 . 2) = 2 (1 + 5 + 7 + 11). Je saisis cette occasion pour dire que j'ai fait calculer la valeur de J (n), " somme-totient de n," pour toutes les valeurs entieres de n jusqua 500, et je 3 trouve que sans aucune exception J(n) est toujours plus grand que —^(n") 3 et plus petit que — (n + 1)^ TT" II reste a demontrer que ces limites sont d'application universelle pour un nombre entier quelconque n. On peut faire une extension illimitee du th^oreme donne dans le numero prec^ent des Comptes rendus sur les sommes-totients, tout a fait analogue a I'extension ci-dessus donn^e au theoreme de Legendre sur les nombres premiers. Nommons, par exemple, u (j) la somme de tous les nombres premiers et inf^rieurs a j, et Uj la somme u(l) + u(2) + ... + u(J). On ^tablit facilement* I'identit^ £A(^^)u(i) = iJ(j+l)0-+2), oil Ax signifie le nombre triangulaire iix(x + 1), et avec ce th^or^me, en se servant, comme dans la th^orie des sommes-totients, du principef de la division harmonique et en ^crivant vj = Uj -2u^^+3u i-wi+ 5ui- ..., on en d^duit facilement ^ = fn — o q»and j est pair, ( J -)- 1 )» J -f- 1 Vj = —Ta h quand j est impair, etc. Dans ma Note J Sur le nombre des fractions ordinaires inigahs, etc., j'ai omis de dire que I'^quation lEiTr=^-^ r r 2 peut etre dcrite sous la forme ^; + ^i + ./| + /|+...=-^^. (1) [•With u(r) = Jrr(r), ti(l) = i, T{r) being the totient of r, we have 2 2 A(E^^tt(r) = J»(t + l)(2i + l).] [t Vol. rn. of this Reprint, p. 678.] [J p. 84 above.] 90 Sur un thioreme de Legendre De mfime, I'^quation [3^ 1 — a; . 1 — a;" . . . 1 — a^ De mfime, le nombre des partitions qui correspondent k la seconde hypothese sera le coefficient de af*~^cJ~^ dans le developpement de l + ax.l+aa? ...l + aa^-' ~ a; " . l-x.l-a:!'...l-x*-' En donnant a 6 toutes les valeurs depuis 1 jusqu'a I'infini, on obtiendra toutes les partitions de n dans j parties inegales. Les cas oh 6 excede j n'offrent rien d'exceptionnel, car, pour ces cas, le coefficient de a^~* dans les deux fonctions generatrices sera nul. Or le coefficient de a;"~*'a-'~* dans chacune de ces deux fonctions est le meme que le coefficient de a:"a^ dans les produits qui resultent de leur multiplication par «*"«*. En comparant les coefficients de x^a^ pour toute valeur de n et i, on trouve done (!+««) (1 +x'a){l+a^a)+ ... 1 + ax „ l+ax.l + aa? . „ 1—x 1—x.l — a^ 1 + ax. 1 + ax'. ..l + aa^ ^ e 1 + ax . „ + xa + ~ af'a^ + ... 1 — x l+ax.l+aa?...l + ax^-'^ ?^ „ H ^ 5 z a_. X ^ a» + .... 1— x.l—oc' ... 1— af^^ En mettant a = — 1, on obtient ainsi l-x.l-a?.l-a?- ... = l-x-a?-... + {-f\x ^ +x « ) + ..., ce qui est le theoreme d'Euler. En r^unissant les deux series dans une seule, on obtient, pour le cas general, {l+xa){l + a?a){l+a?a)+ ... l + aar* 1 + ax .1 +ax* ^ „ H-air.l+ oar* . 1 + aa^ 1 — a; 1 — a;.l— ar* 1 —x.l — x' .1 — x' c'est-ii-dire I'equation que j'ai donnde dans la Note prec^dente [p. 91]. Je dois dire que c'est M. Durfee, dtudiant a Baltimore, qui, le premier (dans un tout autre probleme), a fait usage du genre de decomposition d'une assemblee reguliere de poiuts dans un carr^ et deux groupes supple- men taires dont j'ai profite dans I'analyse precedente (voir Johns Hopkins Circular, [Vol. Iil. of this Reprint, pp. 661 fF.]). 7. DEMONSTRATION GRAPHIQUE* D'UN TH^OREME D'EULER CONCERNANT LES PARTITIONS DES NOMBRES. [Comptes Rendus, xcvi. (1883), pp. 1110—1112.] CoMME confirmation de la puissance de la methode graphique appliqude a la theorie des partitions, la preuve suivante d'un thdoreme que je crois etre nouveau ne sera pas, je I'espfere, tout h fait depourvue d'int^ret pour les g^ometres; car il serait, il me semble, assez difficile d'en trouver une preuve directe analytique au moyen de la comparaison de fonctions generatrices, comme on le fait ordinairement pour des theoremes de ce genre. Euler a trouve facilement, par une comparaison de telles fonctions, que le nombre de partitions de n en nombres impairs est le meme que le nombre de partitions de n en nombres inegaux ; je precise ce theoreme en ajoutant que le nombre de partitions de n en nombres impairs, qui se divisent en i groupes de nombres distincts, est egal au nombre de partitions de n en i suites tout a fait distinctes de nombres cons^cutifs. Nommons U une partition en nombres impairs et V une partition en nombres inegaux. Je dis qu'on peut passer de U k V par la methode suivante. Supposons, par exemple, que U soit la partition 11.11.7.7.7.5. Je forme deux assemblages r^guliers de points en prenant dans I'un 11+1 11 + ] 7 + 1 d'eux, sur cbaque ligne, un nombre de points ^gal k 7+1 7 + 1 5+1 2 ' 2 ' 2 ' , et I'antre assemblage en diminuant de I'unit^ chacun 2 ' 2 ' 2 de ces nombres de points. On forme ainsi ces deux assemblages : 1. 2. et, en comptant le nombre de points dans les angles successifs de chaque figure, on obtient, dans I'un, 11, 9, 5, 2, et, dans I'autre, 10, 8, 3 ; en les r^unissant, on obtient la partition 11.10.9.8.5.3.2, qui est un V. [* See p. 39 above.] 96 Le» partitions des nomhres [7 Or il est facile de voir que dans cette m^thode de transformation U devient V, et Ton d^montre (en construisant un certain systeme d'equations lindaires) que, pour un V quelconque donne, on pent trouver un et un seul U qui se transformera dans ce V, de sorte qu'il y a correspondance un k un entre la totalite des U et la totality des V, ce qui sert k d^niontrer le theoreme original d'Euler. Mais si tel ^tait le but de cette recherche, cette ra^thode de transformation serait peine perdue, car il existe una tout autre m^thode, infiniment plus simple, d'^tablir une telle correspondance : on la trouvera expliquee dans le cahier de V American Journal of Mathe- matics qui va paraitre. L'utilite de cette m^thode spe'ciale de cr^er la correspondance consiste en ceci : que le V ainsi conjugue avec un U contiendra le meme nombre de suites distinctes de nombres consecutifs que le ?7contient de nombres impairs distincts: cela veut dire que le nombre des lignes in^gales (disons i) dans I'un ou I'autre assemblage de points est toujours dgal k j, nombre de suites distinctes obtenu en operant de la maniere expliquee ci-dessus. La preuve en est facile ; car si Ton enleve Tangle ext^rieur a I'un et a I'autre des assemblages, on verra facilement que quatre cas se pr^senteront : pour un de ces cas, j ne change pas de valeur, a cause du changement opdr6 dans les deux assemblages ; dans un autre cas, j subira une diminution de deux unites, et dans les deux cas intermediaires d'une seule unite. Ces cas correspondent aux quatre suppositions qui resultent de la combinaison des hypotheses que les deux premieres lignes ou les deux premieres colonnes dans I'un ou I'autre des assemblages sent ou ne sont pas dgales entre elles : de sorte qu'on verra facilement que le j et le i seront toujours diminu^s de la meme quantity, ou 0, ou 1 ou 2, et cons^quemment on aura i — j constant; si Ton enleve I'un apres I'autre les angles des deux assemblages jusqu'a ce qu'on arrive a un assemblage qui sera de I'une ou I'autre des quatre formes suivantes : 4. pour lesquels cas i=2, j' = 2 ; i = \, j=\\ i=l, j=l; t' = l, 7=1; re- spectivement on aura toujours ainsi i =j, de sorte qu'il y a correspondance une k une entre les partitions du mSme nombre n qui contiennent justement i nombres impairs r^p^t^s (ou non) k volont^, et celles qui contiennent justement i suites distinctes de nombres consecutifs, et cons^quemment il y aura le meme nombre des unes et des autres : ce qui est le theoreme que j'ai voulu d^montrer. 8. SUR UN THEOREME DE PARTITIONS* DENOMBRES COMPLEXES CONTENU DANS UN THEOREME DE JACOBI. [Gomptes Rendus, XCVI. (1883), pp. 1276—1280.] Dans le Journal de Crelle, t. xxxii. p. 166, Jacobi fait la remarque que le developpement en s^rie de SiX donne lieu k un thdoreme que j'expriine de la maniere suivante. Soient a etb deux quantites c=a + h; alors le produit infini +00 fle+Ha-b) {lTg')(l + q>')(l-q')(lT^+'){l+q'^')(l-r)--='^ i+Yq * — oo Ce theoreme etant vrai pour un nombre infini de valeurs de t sera, par sa forme meme, n^cessairement vrai quand a et b sont de symboles absolument arbitraires, et Ton voit facilement que, pour le montier dans ce sens universel, il suflBra d'^noncer un certain theoreme sur les nombres complexes dont voici I'^nonc^ : Designons par C, B, A des nombres complexes de la forme fc, fc + b, fc + a, oil / est ou ziro ou un nombre entier et positif quelconque. Considerons un arrangement compose avec des C, des B et des A nan rip4tis ou avec des C, B, A pris seuls ou combines deux d deux, en excluant les arrangements (que je nomme exceptionnels) qui ne contiennent que des B formant vne sirie arithmetique dont b est le dernier terme et c la difference constante, ou des A formant une sirie semblable dont a est le dernier terme. Par le caractire majeur et le caract^re mineur d'un tel arrangement, je disigne la parite ou Vimpariti du nombre total des termes et du nombre des G qu'il contient. Je dis qu'd chaque arrangement {non exceptionnel) on pent en associer un autre pareil dont la somme totale des dements (les A, B,G) sera la meme, mats dont les caracteres seront tous les deux opposds. La demonstration deviendra plus claire en se servant de la notation suivante. En d^signant par X un symbole d'une s^rie de termes, je me eervirai de Z et de X pour signifier le terme le plus haut et le terme le plus [* See above, p. S9 ff.] 8. IV. 7 98 8ur un tMorertie de partitions [8 bas de la sdrie, et en me servant de Y on Z pour signifier un symbole ou simple ou affect^ de marques quelconques, j'emploie les notations F=0, Y+Z=^0, F>0, Y+Z>0, pour signifier que les Y manquent, que les Y et les Z manquent tous les deux, que les Y ne manquent pas, que les Y et les Z ne manquent pas tous les deux. Je divise les B (d'un arrangement quelconque) en deux especes, 'B et B', dont 'B repr^sente un B appartenant a la s6rie arithmetique (la plus grande qu'on puisse former) commen^ant avec le plus grand B, et B' les autres B qui se trouvent dans I'arrangement. Ainsi je divise les A en ^A et en A^ ; A^ signifie un A appartenant a la s^rie arithmetique la plus grande qu'on puisse former, dont a est le terme minimum (de sorte que, si I'arrangement ne contient pas un a, A^ manque) et ^A signifie les autres A de I'arrangement. Finalement un point au centre d'un symbole a droite ou k gauche signifiera ce symbole diminu^ ou augmente respectivement de c. On voit que dans cette notation les arrangements exceptionnels seront exprim^s ainsi: ceux qui appartiennent k I'une des deux classes par lea conditions 'B — b = 0 avec A + C=0, et les autres par les conditions B = 0 avec ,A + C=0. Je divise les arrangements non exceptionnels en trois classes, dont les conditions seront respectivement les suivantes: Premiere classe : 'B-b>0 ou ('B-b = 0 avec C-c&'B-h). Deuxifeme classe : 'B-h = 0 avec (C-o'B-b ou (7=0, mais A + G>0), ou B = 0 avec {A = 0 ou A —a > G). Troisieme classe : 5 = 0 avec A>0 et A-a 0. Toutes les hypotheses possibles se trouvent comprises dans ces tableaux des arrangements exceptionnels et non exceptionnels. A chacune des trois classes des derniers je vais assigner un ope'rateur qui pent etre appliqu^ a chaque arrangement de cette classe et qui le trans- formera dans un autre arrangement appartenant a la nieme classe; cette disposition, appliqu^e deux fois successivement, reproduira I'arrangement sur lequel on opere, lequel ne changera pas la somme des elements, mais changera chacun des deux caracteres en sens oppos^ : c'est-a-dire que chacun des trois op^rateurs que je vais d^finir, et que je nommerai , yjr, ^, doit 8] de nomhres complexes 99 satisfaire h. cinq conditions qu'on pent nommer catholicite, homoeoghihse, mutualite, inertie et enantiotropie. 1. signifie que, si C = 0 ou C — o'B — 'B, on doit former un nouveau C, en substituant, pour chaque 'B, 'B- (c'est-a-dire sa valeur diminu^e de c), et reconstituer I'inertie originale en ajoutant ensemble les c ainsi soustraits pour former un nouveau G, et que, dans le cas contraire, C doit etre decompose en simples c, dont on ajoutera un au premier 'B (le B le plus grand), un au second 'B, etc., jusqu'i ce que tous les c dont on a a disposer soient ^puises. 2. i/r signifie que, si B>0 ou C = 0, ou C>'B-\-A, on doit former un nouveau C en substituant k'B et A leur somme et que, dans le cas contraire, C doit etre decompose en 'B et .d si £ > 0 et en 6 et J. si £ = 0. 3. ^ signifie que, si C=0 ou (7 + ^4, = >4, il faut decomposer ^ en •A, et C7 ou en a et G, selon que 4, = ou > 0, et que, dans le cas contraire, pour G et A,, il faut substituer leur somme. On sera satisfait en ^tudiant les conditions des trois classes que les ^, i/r, ^ possedent tous les trois cinq attributs voulus: la preuve en est facilitee en supposant que, dans chaque s^rie des G, des B et des A, prise s^parement, on suit un ordre regulier de grandeur dans I'arrangement de ces termes respectivement au multiple de c qui entre dans chacun d'eux. Si Ton donne a a et a 6 des valeurs quantitatives (ce qui est toujours permis), et en particulier les valeurs 1 et 2 respectivement, on retombe 8ur le theoreme d'Euler, mais (chose a noter) la correspondance donnee par le proc^d6 general applique k ce cas ne sera nuUement identique h la correspondance donnee par le proced^ de Franklin. En effet, les arrange- ments exceptionnels ne seront pas les memes dans les deux m^thodes : selon le proc^d^ de Franklin, les arrangements non conjugables sont de la forme i, i+1, ..., 2t-l ou i+1, i + 2, ..., 2i, tandis que la methode actuelle donnera, comme non conjugues, les arrange- ments de la forme 1, 4 3t-2 ou 2, 5 3i-l. La methode employee ici fournira elle-meme toujotirs deux systfemes de correspondance absolument distincts, dont on obtient I'un, qui n'est pas exprim^, en ^hangeant entre eux les o, A et les 6, B, car la methode n'est pas sym^trique dans son operation sur ces deux systfemes de lettres. Ce cas est analogue k celui de la correspondance perspective entre deux triangles, laquelle pent ^tre simple ou triple, comme je I'ai raontre ailleurs. Jacobi, dans I'endroit cite, a fait la remarque que, pour a = l, 6 = 2, en se servant du signe sup«5rieur (?) dans son th^orfeme, on retombe sur le 7—2 100 Sur un tMoreme de partitions de nombres complexes [8 th^or^me d'Euler et que, pour le cas de a = 1, 6 = 1, en se- servant du signe infdrieur, sur un thdoreme donnd (il y a longtemps par Gauss). On peut ajouter que, si avec cette supposition on se sert du signe superieur, on obtient 0 = 0, mais si Ton &rit a = 1 — e, 6=1, en faisant e infinit«^simal, on tombe (chose singuli^re) sur I'^quation de Jacobi elle-mSme, (1 - qy(l-qj{l -q>f+ ... = 1 - 1 1 1 ■30 27 18 230 221-59 63 62 882 863-83 2 1 2 1-22 28 12 242 238-31 54 18 900 886-36 3 2 4 2-74 29 28 270 255-63 55 40 940 919-49 4 2 6 4-86 30 8 278 273-56 56 24 964 953-23 5 4 10 7-60 31 30 308 292-11 57 36 1000 987-58 6 2 12 10-94; 32 16 324 311-26 58 28 1028 1022-64 7 6 18 14-90 i 33 20 344 331-01 59 58 1086 1058-10 8 4 22 19-46 .34 16 360 .351-38 60 16 1102 1094-27 9 6 28 24-62 i 3a 24 384 372-35 61 60 1162 11.31-06 10 4 32 .30-40 36 12 396 393-93 62 30 1192 1168-44 11 10 42 .36-78 37 36 432 416-12 63 36 1228 1206-43 12 4 46 43-7"! .38 18 450 438-92 64 32 1260 1245-03 13 12 58 51-37 39 24 474 462-32 65 48 1308 1284-26 14 6 64 69-58 40 16 490 486-.34 66 20 1.328 132407 15 8 72 68-39 41 40 530 510-96 67 66 1394 1364-49 16 8 80 77-81 42 12 542 .536-19 68 32 1426 1405-63 17 16 96 87-84 43 42 .584 662-02 69 44 1470 1447-17 18 6 102 98-481 44 20 604 688-47 70 24 1494 1489-42 19 18 120 109-73 45 24 628 615-62 71 70 1564 15.32-28 20 8 128 121 -.58 46 22 650 64.3-19 72 24 1588 1575-76 21 12 140 1.34-06 ! 47 46 696 671-46 73 72 1660 1619-82 22 10 150 147-12 48 16 712 700-33 74 36 169C 1664-51 23 22 172 160-79 49 42 754 729-82 75 40 17.36 1709-80 24 8 180 175-08 50 20 774 759-91 76 36 1772 1755-69 25 20 200 189-98 61 32 806 790-61 77 60 1832 1802-20 26 12 212 205-48 52 24 830 821-92 78 24 1866 1849-31 104 On the Number of Fractions contained in amj Table {continued). [9 n T(n) T(n) !."• n T(n) T(n) i- n T(n) T(n) !."■ 79 78 1934 1897-04 134 66 5498 5457-97 189 108 10904 10857-88 80 32 1966 1945-37 135 72 5570 5539-74 190 72 10976 10973-09 81 54 2020 1994-31 136 64 5634 5622-11 191 190 11166 11088-90 82 40 2060 2043-85 137 136 5770 5705-09 192 64 11230 11205-31 83 82 2142 2094-01 138 44 5814 5788-68 193 192 11422 11322-34 84 24 2166 2144-77 139 138 5952 5872-88 194 96 11518 11439-97 85 64 2230 2196-14 140 48 6000 5957-69 195 96 11614 11558-21 86 42 2272 2248-12 141 92 6092 6043-10 196 84 11698 11677-06 87 56 2328 2300-70 142 70 6162 6129-12 197 196 11894 11796-52 88 40 2368 2353-90 143 120 6282 6215-75 198 60 11954 11916-59 89 88 2456 2407-70 144 48 6330 6302-99 199 198 12152 12037-26 90 24 2480 2462-10 145 112 6442 6390-83 200 80 12232 12158-54 91 72 2552 2517-12 146 72 6514 6479-29 201 132 12364 12280-43 92 44 2596 2572-75 147 84 6598 6568-35 202 100 12464 12402-93 93 60 2656 2628-98 148 72 6670 6658-02 203 168 12632 12526-03 94 46 2702 2685-82 149 148 6818 6748-29 204 64 12696 12649-75 95 72 2774 2743-27 150 40 6858 6839-18 205 160 12856 12774-07 96 32 2806 2801-33 151 150 7008 6930-67 206 102 12958 12899-00 97 96 2902 2860-00 152 72 7080 7022-77 207 132 13090 13024-54 98 42 2944 2919-27 153 96 7176 7115-48 208 96 13186 13150-68 99 60 3004 2979-15 154 60 7236 7208-80 209 180 13366 13277-43 100 40 3044 3039-64 155 120 7356 7302-72 210 48 13414 13404-79 101 100 3144 3100-73 156 48 7404 7397-26 211 210 13624 13532-76 102 32 3176 3162-44 157 156 7560 7492-40 212 104 137-28 13661-34 103 102 3278 3224-75 158 78 7638 7588-15 213 140 13868 13790-52 104 48 3326 3287-67 159 104 7742 7684-51 214 106 13974 13920-32 105 48 3374 3351-20 160 64 7806 7781-47 215 168 14142 14050-72 106 52 3426 3415-34 161 132 7938 7879-04 216 72 14214 14181-73 107 106 3532 3480-08 162 54 7992 7977-22 217 180 14394 14313-34 108 36 3568 3545-44 163 162 8154 8076-01 218 108 14502 14445-57 109 108 3676 3611-40 164 80 8234 8175-41 219 144 14646 14578-40 110 40 3716 3677-96 165 80 8314 8275-41 220 80 14726 14711-84 111 72 3788 3745-14 166 82 8396 8376-02 221 192 14918 14845-89 112 48 3836 3812-92 167 166 8562 8477-24 222 72 14990 14980-54 113 112 3948 3881-31 168 48 8610 8579-07 223 222 15212 15115-81 114 36 3984 3950-31 169 156 8766 8681-50 224 96 15308 15251-68 115 88 4072 4019-92 170 64 8830 8784-55 225 120 15428 15388-16 116 56 4128 4090-14 171 108 8938 8888-20 226 112 15540 15525-25 117 72 4200 4160-96 172 84 9022 8992-46 227 226 15766 15662-94 118 58 4258 4232-39 173 172 9194 9097-33 228 72 15838 15801-24 119 96 4354 4304-43 174 56 9250 9202-80 229 228 16066 15940-15 120 32 4386 4377-08 175 120 9370 9308-88 230 88 16154 16079-67 121 no 4496 4450-33 176 80 9450 9415-57 231 1-20 16274 16219-80 122 60 4556 4524-19 177 116 9566 9522-87 232 112 16386 16360-53 123 80 4636 4598-66 178 88 9654 9630-78 233 232 16618 16501-87 124 60 4696 4673-74 179 178 9832 9739-29 234 72 16690 16643-82 125 100 4796 4794-43 180 48 9880 9848-42 235 184 16874 16786-38 126 36 4832 4825-72 181 180 10060 9958-15 236 116 16990 16929-55 127 126 4958 4902-63 182 72 10132 10068-49 237 156 17146 17073-32 128 64 5022 4980-14 183 120 10252 10179-44 238 96 17242 17217-70 129 84 5106 5058-26 184 88 10340 10290-99 239 238 17480 17362-70 130 48 5154 5136-98 185 144 10484 10403-15 240 64 17544 17508-30 131 130 5284 5216-32 186 60 10544 10515-92 241 240 17784 17654-51 132 40 5324 5296-26 187 160 10704 10629-30 242 110 17894 17801-32 133 108 5432 5376-81 188 92 10796 10743-29 243 162 18056 17948-74 9] "Farey Series" of which the Limiting Number is given 105 Table (continued). n r{n) T{n) ^- n r{n) T{n) 5.. n r{n) T{n) i- 244 120 18176 18096-77 299 264 27318 27174-65 354 116 38174 38091-50 245 168 18344 18245-41 300 80 27398 27356-72 355 280 38454 38307-01 246 80 18424 18394-66 301 252 27650 27539-40 356 176 38630 38523-12 247 216 18640 18544-51 302 150 27800 27722-69 357 192 38822 38739-85 248 120 18760 18694-97 303 200 28000 27906-59 358 178 39000 38957-18 249 164 18924 18846-04 304 144 28144 28091-10 359 358 39358 3917513 250 100 19024 18997-72 305 240 28384 28276-21 360 96 39454 39393-68 251 250 19274 19150-01 306 96 28480 28461-93 361 342 39796 39612-83 252 72 19346 1930290 307 306 28786 28648-26 362 180 39976 39832-60 253 220 19566 19456-40 308 120 28906 28a35-20 3&3 220 40196 40052-97 254 126 19692 19610-51 309 204 29110 29022-75 364 144 40340 40273-95 255 128 19820 19765-23 310 120 29230 29210-90 365 288 40628 40495-54 256 128 19948 19920-56 311 310 29540 29399-66 366 120 40748 40717-74 257 256 20204 20076-49 312 96 29636 2958903 367 366 41114 40940-55 258 84 20288 20233-03 313 312 29948 29779-01 368 176 41290 41163-96 259 216 20504 20390-18 314 156 30104 29969-59 369 240 41530 41387-98 260 96 20600 20547-94 315 144 30248 30160-79 370 144 41674 41612-61 261 168 20768 20706-30 316 156 30404 30352-59 371 312 41986 41837-85 262 130 20898 20865-28 317 316 30720 30545-00 372 120 42106 42063-69 263 262 21160 21024-86 318 104 30824 30738-01 373 372 42478 42290-15 264 80 21240 21185-05 319 280 31104 30931-64 374 160 42638 42517-21 265 208 21448 21345-84 320 128 31232 31125-87 375 200 42838 42744-87 266 106 21556 21507-25 321 212 31444 31320-71 376 184 43022 42973-15 267 176 21732 21669-26 322 132 31576 31516-16 377 336 43358 43202-04 268 132 21864 21831-88 323 288 31864 31712-22 378 108 43466 43431-53 269 268 22132 2199511 324 108 31972 31908-88 379 378 43844 43661-63 270 72 22204 22158-95 325 240 32212 3210615 380 144 43988 43892-34 271 270 22474 22323-39 326 162 32374 32304 03 381 252 44240 44123-65 272 128 22002 22488-44 327 216 32590 32502-52 382 190 44430 44355-58 273 144 22746 22654-10 328 160 32750 32701-62 383 382 44812 44588-11 274 136 22882 22820-37 329 276 33026 32901-32 384 128 44940 44821-25 275 200 23082 22987-25 330 80 33106 33101-63 385 240 45180 4")055-00 276 88 23170 23154-73 331 330 33436 33302-55 386 192 45372 45289-35 277 276 23446 23322-82 332 164 33600 33504-08 387 252 45624 45524-32 278 138 23584 23491-52 333 216 33816 33706-22 388 192 45816 45759-89 279 180 23764 23660-83 334 166 33982 33908-96 389 388 46204 45996-07 280 96 23860 23830-75 335 264 34246 34112-31 390 96 46300 46232-86 281 280 24140 24001-27 336 96 34342 34316-27 391 352 46652 46470-25 282 92 24232 24172-40 337 336 34678 34520-84 392 168 46820 46708-25 283 282 24514 24344-14 338 156 34834 34726-01 393 260 47080 46946-87 284 140 24654 24516-49 339 224 a5058 34931-80 394 196 47276 4718609 285 144 24798 24689-44 340 128 35186 35138-19 395 312 47588 47425-91 286 120 24918 24863-00 341 300 35486 35345-19 396 120 47708 47666-35 287 240 25158 •25037-18 342 108 35594 35552-80 397 396 48104 47907-39 288 96 25254 25211-96 343 294 35888 3576101 398 198 48302 48149-04 289 272 25526 25387-34 344 168 36056 35969-83 399 216 48518 48391-30 290 112 25638 25563-34 345 176 36232 36179-26 400 160 48678 48634-17 291 192 25830 25739-94 346 172 36404 36389-30 401 400 49078 48877-64 292 144 25974 25917-15 347 346 367.W 36599-95 402 132 49210 49121-73 293 292 26266 26094-97 348 112 36862 36811-21 403 360 49570 49366-42 294 84 263r.0 26273-40 349 348 37210 1 37023-07 | 404 200 49770 49611-72 295 232 26582 26452-43 350 120 37330 37235-54 405 216 49986 49857-62 296 144 26726 2663207 351 216 37546 37448-61 406 168 50154 50104-14 297 180 26!MJ6 26812-32 352 160 37706 37662-30 407 360 50514 50351-26 298 148 27054 26993-18 353 352 38058 37876-59 408 128 50642 50598-99 106 On the Number of Fractions contained in any Table {continued). [9 n r(n) T(n) i." n r(n) T{n) > n r(n) T(n) >■ 409 408 51050 50847-33 464 224 65630 65442-14 519 344 82028 81875-93 410 160 51210 51096-27 465 240 65870 65724-52 520 192 82220 82191-75 411 272 51482 51345-83 466 232 66102 66007-51 521 520 82740 82508-18 412 204 51686 51595-99 467 466 66568 66291-11 522 168 82908 82825-21 413 348 52034 51846-76 468 144 66712 66575-31 523 522 83430 83142-85 414 132 52166 52098-14 469 396 67108 66860-13 524 260 83690 83461-10 415 328 52494 52350-12 470 184 67292 67145-55 525 240 83930 83779-95 416 192 52686 52602-72 471 312 67604 67431-58 526 262 84192 84099-42 417 276 52962 52855-92 472 232 67836 67718-22 527 480 84672 84419-49 418 180 53142 53109-73 473 420 68256 68005-46 528 160 84832 84740-17 419 418 53560 5336415 474 156 68412 68293-32 529 506 85338 85061-46 420 96 53656 53619-17 475 360 68772 68581-78 530 208 85546 85383-36 421 420 54076 53874-80 476 192 68964 68870-85 531 348 85894 85705-87 422 210 54286 54131-04 477 312 69276 69160-52 532 216 86110 86028-98 423 276 54562 54387-89 478 238 69514 69450-81 533 480 86590 86352-70 424 208 54770 54645-35 479 478 69992 69741-70 534 176 86766 86677-03 425 320 55090 54903-42, 480 128 70120 70033-20 535 424 87190 87001-97 426 140 55230 55162-09 481 432 70552 70325-31 536 264 87454 87327-51 427 360 55590 55421-39 482 240 70792 70618-03 537 356 87810 87653-66 428 212 55802 55681-26 483 264 71056 70911-35 538 268 88078 87980-42 429 240 56042 55941-76 484 220 71276 71205-29 539 420 88498 88307-79 430 168 56210 56202-86 485 384 71660 71499-83 540 144 88642 88635-77 431 430 56640 56464-57 486 162 71822 71794-98 541 540 89182 88964-35 432 144 56784 56726-89 487 486 72308 72090-73 542 270 89452 89293-54 433 432 57216 66989-82 488 240 72548 72387-10 543 360 89812 89623-34 434 180 57396 57253-36 489 324 72872 72684-07 544 256 90068 89953-75 435 224 57620 57517-50 490 168 73040 72981-65 545 432 90500 90284-77 436 216 57836 57782-26 491 490 73530 73279-84 546 144 90644 90616-39 437 396 58232 58047-62 492 160 73690 73578-63 547 546 91190 90948-62 438 144 58376 58313-58 493 448 74138 73878-04 548 272 91462 91281-46 439 438 58814 58580-.16 494 216 74354 74178-05 549 360 91822 91614-91 440 160 58974 58847-34 495 240 74594 74478-67 550 200 92022 91948-97 441 252 59226 59115-14 496 240 74834 74779-90 551 504 92526 92283-64 442 192 59418 59383-54 497 420 75254 75081-73 552 176 92702 92618-91 443 442 59860 59652-54 498 164 75418 75384-18 553 468 93170 92954-79 444 144 60004 59922-16 499 498 75916 75687-23 554 276 93446 93291-28 445 352 60356 60192-38 500 200 76116 75990-89 555 288 93734 93628-38 446 222 60578 60463-22 501 332 76448 76295-15 556 276 94010 9396608 447 296 60874 60734-66 502 250 76698 76600-03 557 556 94566 94304-39 448 192 61066 61006-70 503 502 77200 76905-52 558 180 94746 94643-31 449 448 61514 61279-36 504 144 77344 77211-61 559 504 95250 94982-84 450 120 61634 61552-62 505 400 77744 77518-31 560 192 95442 95322-98 451 400 62034 61826-49 506 220 77964 77825-62 561 320 95762 95663-72 452 224 62258 62100-97 507 312 78276 78133-54 562 280 96042 96005-07 453 300 62558 62376-06 508 252 78528 78442-06 563 562 96604 96347-03 454 226 62784 62651-75 509 508 79036 78751-19 564 184 96788 96689-60 455 288 63072 62928-05 510 128 79164 79060-93 565 448 97236 97032-77 456 144 63216 63204-97 511 432 79596 79371-28 566 282 97518 97376-55 457 456 63672 63482-48 512 256 79852 79682-23 567 324 97842 97720-94 458 228 63900 63760-61 513 324 80176 79993-79 568 280 98122 98065-94 459 288 64188 64039-35 514 256 80432 80305-96 569 568 98690 98411-55 460 176 64364 64318-69 515 408 80840 80618-74 570 144 98834 98757-76 461 460 64824 64598-64 516 168 81008 80932-13 571 570 99404 99104-58 462 120 64944 64879-20 517 460 81468 81246-12 572 240 99644 99452-01 463 462 65406 65160-36 518 216 81684 81560-72 573 380 100024 99800-05 9] "Farey Series" of which the Limiting Number is given 107 P Tablk (continued). n T(«) T{n) 5- n r(n) r(n) 3 2 n r{n) T(«) 3 „ —„ n- 7r- 574 240 100264 100148-70 629 576 120544 120260-45 684 216 142380 142211-17 575 440 100704 100497-95 630 144 120688 120643-14 685 544 142924 142627-30 576 192 100896 100847-81 631 630 121318 121026-44 686 294 143218 143044 03 577 576 101472 101198-28 632 312 121630 121410-35 687 456 143674 143461-37 578 272 101744 101549-36 633 420 122050 121794-86 688 536 144010 143879-32 579 384 102128 101901-05 634 316 122366 122179-98 689 624 144634 144297-88 580 224 102352 102253-34 635 504 122870 122565-71 690 176 144810 144717-05 581 492 102844 102606-24 636 208 123078 122952-05 691 690 145500 145136-82 582 192 103a36 102959-75 637 504 123582 123338-00 692 344 145844 145557-20 583 520 103556 103313-87 638 280 123862 123726-55 693 360 146204 14597819 584 288 103844 103668-60 639 420 124282 124114-71 694 346 146550 146399-79 585 288 104132 104023-93 640 256 124538 124503-48 695 552 147102 146821-99 586 292 104424 104379-87 641 640 125178 124892-86 696 224 147326 147244-80 587 586 105010 104736-42 642 212 125390 125282-85 697 640 147966 147668-22 588 168 105178 105093-58 643 642 126032 125673-44 698 348 148314 148092-25 589 540 105718 105451-35 644 264 126296 126064-64 699 464 148778 148516-89 590 232 105950 105809-72 645 336 126632 126456-45 700 240 149018 148942-14 591 392 106342 106168-70 646 288 126920 126848-87 701 700 149718 149367-99 592 288 106630 106628-29 647 646 127566 127241-89 702 216 149934 149794-45 593 592 107222 106888-49 648 216 127782 127635-52 703 648 150582 150221-52 594 180 107402 107249-29 649 580 128362 128029-76 704 320 150902 150649-20 595 384 107786 107610-70 650 240 128602 128424-60 705 368 151270 151077-48 596 296 108082 107972-72 651 360 128962 128820-06 706 352 151622 151506-37 597 396 108478 108335-35 652 324 129286 12921612 707 600 152222 151935-87 598 264 108742 108698-59 653 652 129938 129612-79 708 232 152454 152365-98 599 598 109340 109062-43 654 216 130154 130010-07 709 708 153162 152796-70 600 160 109500 109426-88 655 520 130674 130407-96 710 280 153442 153228-02 601 600 110100 109791-94 656 320 130994 130806-46 711 468 153910 153659-95 602 252 110352 110157-61 657 432 131426 131205-56 712 352 154262 154092-49 6a3 396 110748 110523-89 658 276 131702 131605-27 713 660 154922 154525-64 604 300 111048 110890-77 659 658 132360 132005-59 714 192 155114 154959-40 605 440 111488 111258-26 660 160 132520 132406-52 715 480 155594 155393-76 606 200 111688 111626-36 661 660 133180 132808-06 716 356 155950 155828-73 607 606 112294 111995-07 662 330 133510 133210-20 717 476 156426 156264-31 608 288 112582 112364-39 663 384 133894 133612-95 718 358 156784 156700-50 609 336 112918 112734-31 664 328 134222 134016-31 719 718 157502 157137-30 610 240 113158 113104-84 665 432 134654 134420-28 720 192 157694 157574-70 611 552 113710 113475-98 666 216 134870 134824-86 721 612 158306 158012-71 612 192 113902 113847-73 667 616 135486 135230-04 722 342 158648 158451-33 613 612 1 14514 114220-09 668 332 135818 135635-83 723 480 159128 158890-56 614 306 114820 114593-05 669 444 136262 136042-23 724 360 159488 159330-40 615 320 115140 114966-62 670 264 136526 136449-24 725 560 160048 159770-84 616 240 115380 115340-80 671 600 137126 136856-86 726 220 160268 160211-89 617 616 115996 115715-59 672 192 137318 137265-08 727 726 160994 160653-55 618 204 116200 116090-99 673 672 137990 137673-91 728 288 161282 161095-82 619 618 116818 1 16466-99 674 336 138326 138083-35 729 486 161768 161538-69 620 240 117058 116843-60 675 360 138686 138493-40 730 288 162056 161982-17 621 396 117454 117220-82 676 312 138998 138904-05 731 672 162728 162426-26 622 310 117764 117598-65 677 676 139674 139315-31 732 240 162968 162870-96 623 528 118292 117977-08 678 224 139898 139727-18 733 732 163700 163316-27 624 192 118484 118356-12 679 576 140474 140139-66 734 366 164066 163762-18 625 500 118984 118735-77 680 256 140730 140552-75 735 336 164402 164208-70 626 312 119296 119116-03 681 452 141182 140966-44 736 362 164754 164655-83 627 360 119656 119496-90 682 300 141482 141380-74 737 660 165414 165103-57 628 312 119968 119878-37 683 682 142164 141795-65 738 240 165654 165551-92 108 On the Number of Fractions contained in any . Table (continued). [9 n T(n) T(n) i- n T(n) T(n) > n r(n) T(n) h" 739 738 166392 166000-87 794 396 191870 191629-56 849 564 219340 219097-23 740 288 166680 166450-43 795 416 192286 192112-56 850 320 219660 219613-66 741 432 167112 166900-60 796 396 192682 192596-17 851 792 220452 220130-71 742 312 167424 167351-38 797 796 193478 193080-39 852 280 220732 220648-36 743 742 168166 167802-77 798 216 193694 193565-21 853 852 221584 221166-62 744 240 168406 168254-76 799 736 194430 194050-64 854 360 221944 221685-48 745 592 168998 168707-36 800 320 194750 194536-67 855 432 222376 222204-96 746 372 169370 169160-57 801 528 195278 195023-32 856 424 222800 22272504 747 492 169862 169614-39 802 400 195678 195510-57 857 856 223656 223245-73 748 320 170182 170068-82 803 720 196398 195998-43 858 240 223896 223767-03 749 636 170818 170523-85 804 264 196662 196486-90 859 858 224754 224288-93 750 200 171018 170979-50 805 528 197190 196975-98 860 336 225090 224811-44 751 750 171768 171435-75 806 360 197550 197465-66 861 480 225570 225334-56 752 368 172136 171892-61 807 536 198086 197955-96 862 430 226000 225858-29 753 500 172636 172350-07 808 400 198486 198446-86 863 862 226862 226382-62 754 336 172972 172808-14 809 808 199294 198938-37 864 288 227150 226907-57 755 600 173572 173266-82 810 216 199510 199430-48 865 688 227838 227433-12 756 216 173788 173726-11 811 810 200320 199923-21 866 432 228270 227959-28 757 756 174544 174186-01 812 336 200656 200416-54 867 544 228814 228486-05 758 378 174922 174646-52 813 540 201196 200910-48 868 360 229174 229012-43 759 440 175362 175107-63 814 360 201556 201405-03 869 780 229954 229541-41 760 288 175650 175569-35 815 648 202204 201900-19 870 224 230178 230070-01 761 760 176410 176031-68 816 256 202460 202395-95 871 792 230970 230599-21 762 252 176662 176494-62 817 756 203216 202892-32 872 432 231402 231129-02 763 648 177310 176958-16 818 408 203624 203389-30 873 576 231978 231659-43 764 380 177690 177422-31 819 432 204056 203886-89 874 396 232374 232190-46 765 384 178074 177887-07 820 320 204376 204385-09 875 600 232974 232722-09 766 382 178456 178352-44 821 820 205196 204883-89 876 288 233262 233254-33 767 696 179152 178818-42 822 272 205468 205383-30 877 876 234138 232787-18 768 256 179408 179285-00 823 822 206290 205883-32 878 438 234576 234320-64 769 768 180176 179752-19 824 408 206698 206383-95 879 584 235160 234854-70 770 240 180416 180219-99 825 400 207098 206885-19 880 320 235480 235389-37 771 512 180928 180688-40 826 348 207446 207387-03 881 880 236360 235924-65 772 384 181312 181157-42 827 826 208272 207889-48 882 252 236612 236460-54 773 772 182084 181627-04 828 264 208536 208392-54 883 882 237494 236997-04 774 252 182336 182097-27 829 828 209364 208896-21 884 384 237878 237534-14 775 600 182936 182568-11 830 328 209692 206400-49 885 464 238342 238071-85 776 384 183320 183039-56 831 552 210244 209905-37 886 442 238784 238610-17 777 432 183752 183511-61 832 384 210628 210410-86 887 886 239670 239149-10 778 388 184140 183984-28 833 672 211300 210916-96 888 288 239958 239688-64 779 720 184860 184457-55 834 276 211576 211423-67 889 756 240714 240228-78 780 192 185052 184931-43 835 664 212240 211930-98 890 352 241066 240769-53 781 700 185752 185405-92 836 360 212600 212438-91 891 540 241606 241310-89 782 352 186104 185881-01 837 540 213140 212947-44 892 444 242050 241852-86 783 504 186608 186356-71 838 418 213558 213456-58 893 828 242878 242395-43 784 336 186944 186833-02 839 838 214396 213966-32 894 296 243174 242938-62 785 624 187568 187309-94 840 192 214588 214476-68 895 712 243886 243482-41 786 260 187828 187787-47 841 812 215400 214987-64 896 384 244270 244026-81 787 786 188614 188265-60 842 420 215820 215499-21 897 528 244798 244571-81 788 392 189006 188744-34 843 560 216380 216011-39 898 448 245246 245117-43 789 524 189530 189223-69 844 420 216800 216524-18 899 840 246086 245663-65 790 312 189842 189703-65 845 624 217424 217037-57 900 240 246326 246210-48 791 672 190514 190184-22 846 276 217700 217551-58 901 832 247158 246757-91 792 240 190754 190665-39 847 660 218360 218066-19 902 400 247558 247;i05-96 793 720 191474 191147-17 848 416 218776 218581-40 903 504 248062 247854-61 9] "Farey Series" of ichich the Limiting Number is given 109 Table* (continued). n T(n) Tin) 1- n r(n) T(n) !>'■ n x(«) T{n) i- 904 448 248510 248403-88 937 936 267256 266870-57 1 1 970 1 384 286076 285999-30 905 720 '249230 24895375 938 396 267652 267440-51 971 1 970 287046 286589-30 '906 300 249530 249504-22 939 624 268276 268011-05 972 1 324 287370 287179-90 907 906 250436 250055-31 940 368 268644 268582-19 973 828 288198 287771-11 908 , 452 250888 250607 00 941 940 269584 269153-95 974 i 486 288684 288362-92 909 1 600 251488 251159-31 942 312 269896 269726-31 975 1 480 , 289164 288955-35 910 288 251776 25171222 943 880 270776 270299-28 1 976 i 480 1 289644 289548-39 911 ,910 252686 252265-73 944 464 271240; 270872-861 977 976 290620 ! 290142-03 912 288 252974 252819-86 945 432 271672 271447-05 978 ! 324 1 290944 290736-28 913 820 253794 ) 253374-59 946 420 272092 272021-84 979 880 1 291824 291331-13 914 456 254250 253929-93 947 946 273038 272597-25 980 336 292160 291926-60 915 480 254730 254485-88 948 312 273350 273173-26 981 648 292808 292522-67 916 456 255186 ( 255042-44 949 864 274214 273749-88 982 490 293298 293119-35 917 780 255966 255599-61 950 360 274574 274327-10 983 ; 982 294280 293716-64 918 i 288 256254 256157-38 951 632 275206 274905-94 984 320 294600 294314-54 919 918 257172 256715-76 952 384 275590 275483-38 985 784 295384 29491304 920 352 257524 257274-75 953 952 276542 276062-43 986 448 295832 295512-15 921 612 258136 257834-34 954 312 276854 276642-09 987 552 296384 296111-87 922 460 258596 258394-55 955 760 277614 277222-36 988 432 296816 296712-20 923 840 259436 258955-36 956 476 278090 277803-23 989 924)297740 297313-14 924 240 259676 259516-78 957 560 278650 278384-71 990 240 297980 297914-68 925 720 260396 260078-81 958 478 279128 j 278966-80 991 990 298970 298516-83 926 462 260858 260641-45 959 816 279944 279549-50 992 480 299450 299119-59 927 612 261470 261204-69 960 256 280200 280132-81 993 660 300110 i 299722-96 f»28 448 261918 261768-55 961 ; 930 281130 280716-72 994 420 300530 1 300326-94 .•J9 928 262846 262333-01 962 i 432 281562 281301-24 995 792 1 301322 300931-52 j:jo 240 263086 262898-07 963 , 636 282198 281886-37 996 328 301650 301536-71 931 756 263842 263463-75 964 480 282678 28247211 997 996 302646 302142-51 932 464 264306 264030-03 965 768 283446 283058-46 998 498 303144 302748-92 933 620 264926 264596-93 966 264 283710 283645-41 999 648 1 3a3792 3a3355-93 934 466 265392 265164-43 967 966 284676 ( 284232-97 1000 400 1 304192 303963-55 935 640 266032 265732-53 968 440 285116 t 284821-14 936 288 266320 266301-25 969 576 285692 1 285409-92 * Id the extended as well as in the original Table it will be seen that the som-totient is always intermediate between 3/t' . n" and 3/t' . (n + 1)'. The formula of verification applied at every tenth step to the T colnmn preclndes the possibility of the existence of other than typographical errors or errors of transcription. Accnmolative errors are rendered impossible. 10. ON THE EQUATION TO THE SECULAR INEQUALITIES IN THE PLANETARY THEORY. [Philosophical Magazine, xvi. (1883), pp. 267 — 269.] A VERY long time ago I gave, in this Magazine*, a proof of the reality of the roots in the above equation, in which I employed a certain property of the square of a-symmetrical matrix which was left without demonstration. I will now state a more general theorem concerning the product of any two matrices of which that theorem is a particular case. In what follows it is of course to be understood that the product of two matrices means the matrix corre- sponding to the combination of two substitutions which those matrices represent. It will be convenient to introduce here a notion (which plays a conspicuous part in my new theory of multiple algebra), namely that of the latent roots of a matrix — latent in a somewhat similar sense as vapour may be said to be latent in water or smoke in a tobacco-leaf If from each term in the diagonal of a given matrix, X be subtracted, the determinant to the matrix so modified will be a rational integer function of X ; the roots of that function are the latent roots of the matrix ; and there results the important theorem that the latent roots of any function of a matrix are respectively the same functions of the latent roots of the matrix itself: for example, the latent roots of the square of a matrix are the squares of its latent roots. The latent roots of the product of two matrices, it may be added, are the same in whichever order the factors be taken. If, now, m and n be any two matrices, and M=mn or nm, I am able to show that the sum of the products of the latent roots of M taken i together in every possible way is equal to the sum of the products obtained by multiplying every minor determinant of the ith order in one of the two matrices m, n by its altruistic opposite in the other : the reflected image of any such determinant, in respect to the principal diagonal of the matrix to which it belongs, is its proper opposite, and the corresponding determinant to this in the other matrix is its altruistic opposite. [• Vol. I. of this Keprint, p. 378.] 10] On the Equation to the Secular Inequalities, etc. Ill The proof of this theorem will be given in my large forthcoming memoir on Multiple Algebra designed for the American Journal of Mathematics. Suppose, now, that m and n are transverse to one another, that is, that the lines in the one are identical with the columns in the other, and vice versd, then any determinant in m becomes identical with its altruistic opposite in n ; and furthermore, if in be a symmetrical matrix, it is its own transverse. Consequently we have the theorem (the one referred to at the outset of this paper) that the sum of the i-ary products of the latent roots of the square of a symmetrical matrix (that is, of the squares of the roots of the matrix itself) is equal to the sum of the squares of all the minor determinants of the order i in the matrix ; whence it follows, from Descartes's theorem, that when all the terms of a symmetrical matrix are real, none of its latent roots can be pure imaginaries, and, as an easy inference, cannot be any kind of imaginaries ; or, in other words, all the latent roots of a symmetrical matrix are real, which is Laplace's theorem. I may take this opportunity of stating the important theorem that if \,, \,, ... Xi are the latent roots of any matrix m, then _ (m-\,)(m-X,)...(m-X,)^^ ^"-^(\-x,)(x.-x.)...(x.-x,)'''^ This theorem of course presupposes the rule first stated by Prof. Cayley {Phil. Trans. 1857) for the addition of matrices. When any of the latent roots are equal, the formula must be replaced by another obtained from it by the usual method of infinitesimal variation. If 1 (fun = wi", it gives the expression for the wth root of the matrix ; and we see that the number of such roots is <<>•, where i is the order of the matrix. When, however, the matrix is unitary, that is, all its terms except the diagonal ones are zeros, or zeroidal, that is, when all its terms are zeros, this conclusion is no longer applicable, and a certain definite number of arbitrary quantities enter into the general expressions for the roots. The case of the extraction of any root of a unitary matrix of the second order was first considered and successfully treated by the late Mr Babbage ; it reappears in M. Serret's Cours d'Algebre superieure. This problem is of course the same as that of finding a function j of any given order of periodicity. My memoir will give the solution of the corresponding problem for a matrix of any order. Of the many unexpected results which I have obtained by my new method, not the least striking is the rapprochement which it establishes between the theory of Matrices and that of Invariants. The theory of invariance relative to associated Matrices includes and transcends that relative to algebraical functions. 11. ON THE INVOLUTION AND EVOLUTION OF QUATERNIONS. [Philosophical Magazine, xvi. (1883), pp. 894 — -396.] The subject-matter of quaternions is really nothing more nor less than that of substitutions of the second order, such as occur in the familiar theory of quadratic forms. A linear substitution of the second order is in essence identical with a square matrix of the second order, the law of multiplication between one such matrix and another being understood to be the same as that of the composition of one substitution with another, and therefore depending on the order of the factors ; but as regards the multiplication of three or more matrices, subject to the same associative law as in ordinary algebraical multiplication. Every matrix of the second order may be regarded as representing a quaternion, and vice versa; in fact if, using i to denote V(— 1), we write a matrix m of the second order under the form we have by definition, 1 0 ^ t 0 0 1 . 0 z where a = ^ ^, ^ = o_., 7 = _i o' ^=^0" Now a- = a, yS'-' = 7^ = g2 _ _ jK^ a/3 = /3a = /3, ay = 7a = 7, aS = 8a = S, ^7 = _7^ = a, 7S = -S7 = ^, 8;8 = -^S = 7; so that we may for a, /8, 7, S, substitute 1, /(, k, I, four symbols subject to the same laws of self-operation and mutual interaction as unity and the three Hamiltonian symbols. Now I have given the universal formula for expressing any given function of a matrix of any order as a rational function of that matrix and its latent roots; and consequently the 5th power or root of any a + hi, c + di, - c -f di. a- hi, m = aa -f 6^ -f- C7 -f dh. /3 = i 0 "0 -i' 7 = 0 1 -1 0' 11] On the Involution and Evolution of Quaternions 113 quadratic matrix, and therefore of any quaternion, is known. As far as I am informed, only the square root of a quaternion has been given in the text- books on quaternions, notably by Hamilton in his Lectures on Quaternions. The latent roots of m are the roots of the quadratic equation X'' -2a\ + a? +11' + cr + d" = 0. The general formula where i is the order of the matrix m, when i = 2 and ^m = mi, becomes A-x — A,2 Aj — Aj where Xj, X, are the roots of the above equation. If fi, is the modulus of the quaternion, namely is V(«' + 6* + C + d"), and /a cos ^ = a, the latent roots \, , Xj assume the form fi (cos 6 ±i sin 6). When the modulus is zero the two latent roots are equal to one another, and to o, the scalar of the quaternion ; so that in this case the ordinary theory of vanishing fractions shows that \a q J In the general case there are q' roots of the gth order to a quaternion. Calling 1 77 - = w, and writing m» = Am + B, ^ cos f - + 2^•w j + i sin f - + 2k (m, n) are any rational functions whatever of m, n, the determinant to the matrix m,n — mn is contained as a factor in the determinant to the matrix f — <^/. It may be noticed that /, need not be integer functions by stipulation, because any linear function of mn, m, n, 1, divided anteriorly or posteriorly by a second like function, can it-self be expressed as a linear function of the same four terms. As a very simple example of the theorem, observe that the determinant of m^n — mnm will contain as a factor the determinant of mn — nm. 13. SUR LES QUANTIT^S FORMANT UN GROUPE DE NONIONS ANALOGUES AUX QUATERNIONS DE HAMILTON. [Comptes Rendus, xcvii. (1883), pp. 1336—1340,] On sait qu'on peut tout a fait (et tres avantageusetnent) changer la base de la theorie des quaternions en consid^rant les trois symboles i, j, k de Hamilton comme des matrices binaires. Si h, j sont des matrices binaires qui satisfont a I'^quation hj = —jh, on deraontre faeilement que, en ecartant le cas ou hj =jh = 0, h^ et ^■^ seront de la forme c 0 7 0 0 c' 0 7 o'est-k-dire cu, yu, ou u est I'unit^ binaire 1 0 0 1 On peut ajouter, si Ton veut, les deux conditions c^ = l, 7- = !; alors, en supprimant, pour plus de brievete, le u, qui jouit de propri^tes tout a fait analogues k celles de I'unite ordinaire, on obtient faeilement les Equations connues _ _ _ /i,2=i^ j^=l_ k^ = l^ hj = —jh = k, jk = — kj = i, ki = — ik =j. De plus, en supposant que {i, j) soit un systfeme particulier qui satisfait a r^quation ij = —ji, on peut deduire les valeurs universelles de /, J qui satisfont a I'^quation IJ = — JI en termes de i, j. En efifet, on d^montre rigoureusement que, en dcartant toujours la solution mn = nm = 0, on aura I = ai + hj + cij, J=m + ^j + rij, avec la seule condition aa + b^ + cy = 0. De plus, si Ton suppose t' =j' = u et aussi I^ = J' = u, on aura a» + 6^ + c» = l, a» + ^' + 7*=l, 13] Sitr les qiiantites formant un groupe de nonions, etc. 119 de sorte que, en ecrivant ij = k, IJ = K et K = Ai + Bj+ Ck, la matrice a b c a /3 7 ABC formera une matrice orthogonale. Une solution, parnii les plus simples, des equations ij = —ji, i' = u.,j- = u, est la suivante : 1 = et cons^quemment 0 -0 J = 0 -1 1 0 k = ij = 0 -0 -0 0 ou^=V'(-l)- En dcrivant une quantity binormale quelconque (c'est-i-dire une matrice binaire) sous la forme a + b0, - c - d0, c -d0, a- b0, on voit qu'elle pent ^tre mise sous la forme au + bi + cj + dk, oil il est souveut commode de supprimer (c'est-k-dire de sous-entendre) sans ^crire I'unite binaire u. On peut construire d'une maniere tout k fait analogue un systeme de nonions en considerant I'^quation m — pn, ou to, n sont des matrices ternaires et p une racine cubique primitive de I'unitd (voir* la Circular du Johns Hopkins University qui va prochainement paraitre), en prenant pour les nonions foudameutaux u (I'unite ternaire) 10 0 0 10 0 0 1 et les huit matrices m, m'; n, n'; m'n, mn'; mn, m'n" construites avee les valeurs les plus simples de m, n qui satisfont aux equations Les valeurs nm = pmn, nr m' 1 0 0 0 p 0 et n = 0 0 p' n' = u. 0 1 0 0 0 /} p» 0 0 peuvent Stre prises pour les valeurs basiques du systfeme de nonions. Une quantity ternaire (c'est-a-dire une matrice) quelconque s'exprime alors sous la forme o + 6to + /9to' + cn + Tw' + dm'w + hmn^ + emn + em?n^ ; [* VoL m. of this Reprint, p. 647. Also below, p. 122.] 120 Sur les qtiantit^s formant un groupe de nonions [13 mais, quand cette matrice M est capable de s'associer avec une autre N dans r^uation NM = pMN, alors il devient n^cessaire que a = 0, 6/3 + C7 + dS + ee = 0. Je n'entrerai pas ici dans les details de la m^thode d'associer la solution gdndrale de I'equation NM = pMN avec une solution quelconque particuliere de cette Equation, mais je me bornerai k expliquer quelles sont les conditions auxquelles les Elements de M Qt Aq N doivent satisfaire afin que cette Equation ait lieu. M. Cayley a resolu la question analogue pour les matrices binaires dans le beau M6moire, qu'il a publid dans les Transactions of the Royal Society de 1858. En supposant que m et n sont les matrices ah a' b' c d c' d' il trouve que, afin que nm = — mn, il faut avoir a + d = 0, a' + d' = 0, aa' + be' + cb' + dd' = 0. Au lieu de cette troisieme Equation (en la combinant avec les deux precddentes), on pent ecrire ad' + a'd - be' - b'c = 0. Alors ces trois conditions equivalent a dire que le determinant de la matrice osu + my + nz {u etant I'unite binaire), qui, en general, est de la forme ar^ + -IBxy + 2Cxz + Dy^ + 2Eyz + Fz^, se r^duira k la forme a? + By'' + J^^^ car, dans le determinant de arit + my + nz, c'est-^-dire de a; + 03/ + a!z by + b'z cy + c'z x + dy + d'z les coefficients de ocy, xz, yz seront ^videmment a + d, a' + d', ad' + a'd - be' - b'c respectivement. Passons au cas de m et n, matrices ternaires qui satisfont k I'equation nm = pmn. Formons le determinant de a^ + ym + zn, ou u represente I'unite ternaire 10 0 0 1 0 0 13] aiialogues mix quaternions de Hamilton 121 Ce determinant sera de la forme a? + Wa?y + ^Ca?z + SBxf + QExyz + ^Fxz'- + Gf + 2Hfz + SKyz^ + Lz', et je trouve que, dans le cas suppose, il faut que les sept conditions souscrites soient satisfaites; B = 0, C=0, D = 0, E = 0, F=0 H=0, K = 0, de sorte que la fonction en x, y, z devient une somme de trois cubes, mais ces sept conditions, qu'on pourrait nommer conditions paramdtriques, quoique necessaires, ne sont pas suflBsantes; il faut y ajouter une huitieme condition que je nommerai Q = 0. Pour former Q, voici la manifere de proc^der : En supposant que a b c a! y et n< »»= d e f g h h on ecrit, au lieu de m, son transversal I a' d' b' e' a d' 9 e h' c r k' h' et Ton forme neuf produits en multipliant chaque d(?terrainant mineur du second ordre contenu dans m avec le determinant mineur semblablement pos^ dans le transversal de n : la somme de ces neuf produits est Q. Ces huit conditions que je d^montre sont suffisantes et necessaires (en ecartant corame auparavant le cas oil nm = mn = 0) pour que nm<=pmn. On pourrait tres bien se demander ce qui arrive dans le cas oil les sept conditions parametriques sont satisfaites, mais non pas la huitifeme condition supplementaire. Dans ce cas, je trouve* que mn et nm restent fonctions I'une et I'autre et qu'on aura nm = A + B^mn + G(mny, mn = — A + B^nm + G (nmf, oil 5,, 5, sont les racines de I'equation alg^brique 5'' + 5+1 =0, A, G etant deux quantit^s arbitraires et ind^pendantes, sauf que I'une d'elles ne pent pas s'evanouir sans I'autre, les deux s'^vanouissant ensemble pour le cas (et seulemeut pour le cas) ou Q (qui foumit la condition supplementaire) s'^vanouit. [* See footnote [f], p. 154 below.] 14. ON QUATERNIONS, NONIONS, SEDENIONS, ETC. [Johns Hopkins University Circulars, ill. (1884), pp. 7 — 9.] (1) Suppose that m and n are two matrices of the second order. Then if we call the determinant of the matrix a; + my + nz, x^ + 2bxy + 2cxz + dy" + 2eyz +fz-, the necessary and sufficient conditions for the subsistence of the equation nm = — mn is that b = 0, c = 0, e = 0, and if we superadd the equations ?n^ 4- 1 = 0, n" + 1 = 0, then d=l and /= 1, or in other words in order to satisfy the equations mn = — nm, m^ = — 1, ?i- = — 1, where it will of course be understood that in these (as in the equations m- + 1 = 0, n'' + 1 = 0) 1 is the abbreviated form of the matrix and 1 of* the form -^ , the necessary and sufficient condition is that the determinant of a; + my + nz shall be equal to X- + y* + z". i 0 The simplest mode of satisfying this condition is to write 111= ., w= A ' * meanmg v(— 1). which gives mn= . and nm= . . -I \j "~~ % yj % \j It is easy to express any matrix of the second order as a linear function of 1 ( meaning \ m,n,p, where p stands for mn. For if ' , be any such matrix it is only necessary to write a =/+ ig, h = —h— ki, d=f—ig, c = — h + ki, and then ' j =/+ ffin + /'w + kp. The most general solution of the equations MN =■ — NM, M^ = N^ = —l, must contain three arbitrary constants, namely, the difference between the number of terms in m and n, and the number of conditions 6 = 0, c = 0, e = 0, ci!=l, /=!, which are to be satisfied. [* I denotes - 1.] I 14] On Quaternions, Nonions, Sedenions, etc. 123 Suppose M, N to be the most general solution fulfilling these conditions; we may write M= f + gm + hn + kp, N =/' + g'm + h'n + k'p, where m, n is any particular solution and p = mn, and we shall have inas- much as M^ = 1, /■-g'- h- -!. To avoid prolixity I shall confine my remarks to the general case, which is, that where the determinants (or as I am used to say the contents) of m and n are each of them finite ; with this restriction, the proposed equation is impossible for general values of A; as will be at once obvious from the fact that the totalities of the latent roots of mn and of nm are always identical, but the individual latent roots are by virtue of the proposed equation in the ratio to one another of 1 : k, which, since by hypothesis no root is zero, is only possible when fr* = 1. When the above equation is satisfied the w' equations arising from the identification of nvi with kmn cease to be incompatible and (as is necessary or at all events usual in such a contingency) become mutually involved. Thus, for example, when w = l and k = \, the number of independent equations is 0, that is, 1 — 1, when w = 2 and A; = — 1 the number is 3, that is, 4 — 1, when w = 3 and k = p or p' the number is 8, that is, 9 — 1 ; it is fair therefore to presume (although the assertion requires proof) that for any value of m when A is a primitive wth root of unity the number of conditions to be satisfied when nm = kmii is w' — 1. Of these the condition that the content of a; + my + m shall be of the form x" + cy" + c'z" will supply C'" +!)(<" +1) ■ 3, that is, ^'^4^- 2, and there will therefore be '-3w , (i»-l)(a)-2) h 1 or ^^ ~ - 'I to be supplied from some other source. When A; is a non-primitive wth root of unity, the number of equations of condition is no longer the same. Thus when k=\ we know that n may be of the form A + Bm + (7m= -F . . . + Xm"-', where A, B, ... L, and all the <»' terras in m are arbitrary, and consequently the number of conditions for that case is 2(o^ — (co' + w) or -— B where 8 is 14] On Quaternions, Nonions, Sedenions, etc. 129 the greatest common measure of q and a> : but, of course, this assertion awaits confirmation. When &) = 4 besides the case of nm = mn, that is, of n being a function of m of which the solution is known, there will be two other cases to be considered, namely, nm — — mn and nm = imn : the former probably requiring 14 and the latter 15 conditions to be satisfied between the coefficients of m, the coefficients of n and the two sets of coefficients combined. It is worthy of notice that the conditions resulting from the content of X + my + nz becoming a sum of 3 powers are incompatible with the equation nm = vmn when v is other than a primitive wth root of unity (a> being of course the order of m or n). Thus suppose w = 4 ; the conditions in question applied to the middle one of the 5 identical equations give m'n' + n'm' + mn'm + nm^n + mnmn + nmnm = 0 ; when nm = imn the left-hand side of this equation becomes (1 + 1* + i* + 1» + i + 1") m'n*, that is, is zero, but when nm = — mn, the value is (1 + 1-1 -l-l-l)m'7i' which is not zero, and bo in general. Thus the pure power form of the content of x + my + nz is a condition applicable to the case of — being a mn primitive root of unity and to no other. The case of nm being a primitive root of ordinary unity is therefore the one which it is most interesting to thrash out. There are in this case, we have seen, ^(o»' + 3a) — 4) simple conditions expressible by the vanishing of that number of coefficients in the content of x + my + nz and ^(w — l)(w — 2) supplemental ones. What are these last ? I think their constitution may be guessed at with a high degree of probability. For revert to the case of a» = 3 in which there is one such found by equating to zero the second coefficient in the identical equation (mnf - 3B (mny + SDmn -G = 0. Suppose now (m*ny - 3B' {m*n*y + ZD'm'ri' -G' = 0 is the identical equation to m^n\ By virtue of the 8 conditions supposed to be satisfied we know that nm = pmn as well as m' = 1, w' » 1, and consequently that (m»ri»)'= 1. Hence fi'= 0, D' = 0, by virtue of the 7 parameters in the oft-quoted content and of D being all zero, and thus the evanescence oi B' or D' imports no new condition. 8 IV. 9 130 On Quatemiotis, Nonions, Sedenions, etc. [14 Now suppose o) = 4, and that {mn)* - 45 (mn)' + QD (mny - 4,Gmn + M = 0, {nv'n')* - iB'{ni'ny + 6D'{m''ny - 4:G'm,^n' + M' = 0. Here we know that B vanishes by virtue of b, c and e vanishing, but D = 0, G = 0, which must be satisfied if nm = imn, will be two new conditions not implied in those which precede. It seems then, although not certain, highly probable that B' = 0, D' = 0, will be implied in the satisfaction of the antecedent conditions but that G' = 0 will be an independent condition, so that Z) = 0, G = 0, G' = 0, .will be the three supplemental conditions: and again when to = 5 forming the identical equations to mn, ni'n', m^n^, and using an analogous litteration to what precedes, the supplemental conditions will be J) = 0, G =0, M =0, M" = 0, and so in general for any value of as. The functions D, G, M, etc., above equated to zero are known from the following theorem of which the proof will be given in the forthcoming memoir*. If (mn)'" + ki (rmi)--'^ + . . . + ij (mnf-' + . . . = 0 is the identical equation to mn, then ki is equal to the sum of the product of each minor of order i in m multiplied by its altruistic opposite in n. The annexed example will serve to illustrate in the case of « = 3 that unless the supplemental condition is satisfied we cannot have nni = pmn. Write m = l 0 0, n= 0 c k, 0 p 0, k 0 cp, 0 0 p\ cp' k 0, then the determinant to a; + my + nz will be easily found to be ic' + ^ + (C + ^)^; but B becomes — Spck, and does not vanish unless c = 0 or k = 0, and accordingly we find n7u =0 pc p'k, mn =0 c k, k 0 c, pk 0 p^c, p'c pk 0, pc p-k 0. When k = 0 mn = p'nm, when c = 0 nm = p^mn, but on no other supposition will — be a primitive cube root of unity. mn ^ ■' * This theorem furnishes as a Corollary the principle employed to prove the stability of the Solar System. (See Land, and Edin. Phil. Mag., October, 1883.) [Above, p. 110.] 14] On Quaternions, Nmiions, Sedenions, etc. 131 Addendum. Referring to the equation MN = — NM, and to the eight equations expressing M and N in terms of the combinations of the powers of m with those of n, in which it is to be understood that M and N are non-vacuoiis, we know that the sums of the latent roots of M and of N must each vanish and consequently, as may be proved, that a = 0, a' = 0, leaving 8 — 2 or 6 conditions to be satisfied. If we further stipulate that Jlf' = l, i\r'=], there will be 8 relations connecting the coeflScients h, c, ... k and V, c', ... k', so that the 64 coefficients in the 8 equations connecting M, M'; N, N'; MN, M'N'; WN, MN", or say rather M, M^; N, N"; p^N, p'M^N'; pM^N, pMN'^, with like combinations or multiples of combinations of powei-s of m, n * will be connected together by 56 equations ; the coefficients in the expression for any one of the above 8 terms may then be arranged in pairs /i,yi'; gi, Qi \ hi, V> ^i, ki ; and in the expression for its fellow by Fi, Fi ; Gi, Gi ; Hi, HI; Ki, Kl; so that the Matrix is resolved as it were into 4 sets of paired columns and 4 sets of paired lines : the 4 different sets of paired lines being found by writing successively i = 1, 2, 3, 4. It is then easy to see that there will be 4 equations of the form 2(/.(?.'+/.G.')=l. and 6 quaternary groups (that is, 24 equations) of the form S (/.G^' + /.(?/) = 0, with liberty to change / into F or G into g or each into each : together then the above are 28 of the 56 conditions required. But inasmuch as the 8 [m, n] arguments may be interchanged with the 8 \M, N'\ ones, we may transform the above equations by substituting for each letter/ its conjugate d log A — -P. — (where A is the content of the Matrix) and thus obtain 28 others, giving in all (if the two sets as presumably is the case are independent) the required 56 conditions : the latter 28, however, may be replaced by others of much simpler formf. * It is easy to see that the sam of the latent roots of 'MSiii mast be zero for all values of i, j 80 that it is a homogeneous linear function of the 8 quantities m, m^, ..., mn, m^v?, t I am still engaged in studying this matrix, which possesses remarkable properties. Is it orthogonal? I rather think not, but that it is allied to a system of 4 pairs of somethings drawn in four mutually perpendicular hyperplanes in space of 4 dimensions. In the general case of ilN=pNM where p is a primitive uth root of unity, there will be an analogous matrix of the order u'-l where each line and each column will consist of u+l groups of w-1 associated terms. The value of the cube of any one of the 8 matrices M, W; ... ; MN, M^N^ may be expressed as follows : It is JF* into ternary unity. Such a quantity may be termed by analogy a Scalar. To find P(_,- I imagine the 8 letters corresponding to M'N' (but without powers of p attached) to be set over 8 of the 9 points of inflexion to any cabic curve, the paired letters being made suitably 9—2 132 On Quaternions, Nonioiu, Sedenions, etc. [14 To me it seems that this vast new science of multiple quantity soars as high above ordinary or quaternion Algebra as the M^canique Celeste above the " Dynamics of a Particle " or a pair of particles, (if a new Tait and Steele should arise to write on the Dynamics of such pair,) and is as well entitled to the name of Universal Algebra as the Algebra of the past to the name of Universal Arithmetic. coUinear with the missing 9th point. Then among themselves the 8 letters may be taken in 8 different ways to form collinear triads and the product of the letters in each triad may be called a collinear product ; P,_y (which is identical with the Determinant to M*N') will be the sum of the cubes of the 8 letters less 3 times the sum of their 8 collinear products, and its 8 values will be analogous to the 3 values of the sum of 3 squares in the Quaternion Theory. Each of these 8 values is assumed equal to unity. It may be not amiss to add that the product of four squares by four is representable rationally as a sum of four squares, so if we place (not now 8 specially related but) nine perfectly arbitrary letters over the nine points of inflexion of a cubic curve the sum of their 9 cubes less three times their 12 collinear products multiplied by a similar function of 9 other letters may be expressed by a similar function of 9 quantities lineo-linear functions of the two preceding sets of 9 terms. By the 8 letters of any set as, for example, 6, ..., h' being " specialized," I mean that they are subject to the condition bb' + dd' +//' + hh' = 0. When this equation is satisfied, and not otherwise, M^ will be a Scalar, and it must be satisfied when MN=:pNM. 15. ON INVOLUTANTS AND OTHER ALLIED SPECIES OF INVARIANTS TO MATRIX SYSTEMS. [Johns Hopkins University Circulars, in. (1884), pp. 9 — 12, 34, 8.5.] To make what follows intelligible I must premise the meaning and laws of vacuity and nullity. A matrix is said to be vacuous when its content (the determinant of the matrix) is zero, but it may have various degrees of vacuity from 0 up to w the order of the matrix. If from each term in the principal diagonal of a matrix X be subtracted, the content of the resulting matrix is a function of degree o) in \ ; the to values of X which make this content vanish are called its latent roots, and if i of these roots are zero, the vacuity (treated as a number) is said to be i. This comes to the same thing as saying that the vacuity is i when the determinant, and the sums of the determinants of the principal minors of the orders w — 1, w — 2, ... (« — I'+l) are each zero. A principal minor of course means one which is divided into 2 [equal] triangles by the principal diagonal of the parent matrix. Again the nullity is said to be i when every minor of the order (w — i+l), and consequently of each superior order, is zero. It follows therefore that it means the same thing to predicate a vacuity 1 and a nullity 1 of any matrix, but for any value of i greater than 1, a nullity i implies a vacuity i but not vice versd ; the vacuity may be i, whilst the nullity may have any value from 1 up to i inclusive. The law of nullity which I am about to enunciate is one of paramount importance in the theory of matrices*. * The three cardinal laws or landmarks in the science of mnltiple quantity are (1) the law of nullity, (2) the law of latency, namely, that if X,, X,, ... X„ are the latent roots of m, then /X]> /\< ... /Xu are those ot fm, inclading as a consequence that ■^ -''■'*' (X|-X,)(Xi-X3)-(V.-M' and (3) the law of identity, namely, that the powers and combinations of powers of two matrices m, n of the order u are connected together by (u + 1) equations whose coefficients are all included among the coefficients of the determinant to the Matrix . x+ym + zn. 134 On Involutants and other allied species of [15 I The law is that the nullity of the product of two (and therefore of any number of) matrices cannot be less than the nullity of any factor nor greater than the sum of the nullities of the several factors which make up the product. Suppose now that X,, X™, ... X„ are the latent roots of any matrix with unequal latent roots of the order w. It is obvious that any such term as 7/1 — Xi will have the nullity 1, for its latent roots will be 0, X, — X,, X3 — Xi, ... X„ — Xi, and consequently its vacuity is 1. Moreover we know from Cayley's famous identical equation that the nullity of the product of all the w factors is w. - Hence it follows that if Mi contains i, and Mj the remaining w—i of these factors (so that i +j =, 2 'i(», 2 T • • ' "r" ^W, « 'w 15] Invariants to Matrix Systems 135 where the t system is the same for all matrices of the order to. If, then, we have a^ such matrices, their topical resultant is the Resultant in the ordinary sense of the w- linear forms above written, proper to each of them re- spectively. Suppose now that m, n are two independent matrices of the order «, we may form ox' matrices by taking each power of m from 0 to a> — 1 as an antecedent factor, and can combine it with similar powers of n as a con- sequent factor, and in this way obtain tu' matrices, of which the first will be the ta-ary unity, that is, a matrix of the order w in which the principal diagonal terms are all units and the other terms all zero. The topical resultant of these co- matrices I shall for brevity denote as the Involutant to m, n. In like manner, inverting the position of the powers of m and of n so as to make the latter precede instead of following the former in the w" products above referred to, we shall obtain another topical resultant which may be termed the Involutant to n, m. The reason why I speak of these topical resultants as involutants to m, n or n, mi is the following : In general if m, n are two independent matrices, any other matrix p, by means of solving w' linear equations, may obviously be expressed as a linear function of the «' products (1, TO, m', ... , TO"~')(1, 11,71", ... , n"~'). There are, however, exceptions to this fact. The most obvious exception is that which takes place when n is a function of m ; for then any m of the to' products will be linearly related, and there will be substantially only co disposable quantities to solve w^ equations. Another exception is when the m, n Involutant, that is, the topical resultant of the w' matrices, is zero ; in which case the general values of the «' disposable quantities each becomes infinite. So that m, n may be said to be in a kind of mutual involution with one another. So, again, p may in general be expressed as a linear function of the &>' matrices (1, n, n», ... , n--») (1, to, m', ... , to*^'), but when the i», to Involutant vanishes this is no longer possible. When o) = 2 the two involutants, considered as definite determinants, are absolutely equal in magnitude and iu Algebraical sign, but when to exceeds 2 this is no longer the case; the two Involutants are then entirely distinct functions of the elements of m and n. 136 On Involutants and other allied species of [15 10 0 Q p k Thus to take a simple example : if m = 0 p 0 and n = k 0 p' it will 0 0 p» 1 A 0 be found by direct calculation of two topical resultants of the 9th order, that the two involutants will be 81 {p - p^) (A* - pY and 81 (p" -p)(k'- p»)» respectively. The reason why the two involutants coincide in the case of ft) = 2 is not far to seek. It depends upon the fact of the existence of the mixed identical equation mn + nm — 2bn — 2cm + 2e = 0 ; from which it is obvious that the topical resultant of 1, m, n, mn is the negative of that of 1, m, n, nm or identical with that of 1, n, m, nm. By direct calculation it will be found that the Involutant m, n, or n, m, where m = {^ n^{,(,is -(9h'-g'kf+[{f-k)g'-{f'-k')g][{f-k)h'-{f'-k')h], which is the same thing as the content of the matrix {mn — nm). It may also be shown d priori or by direct comparison to be identical (to a numerical factor prh) with the Discriminant of the Determinant to the matrix (x + ym + zn) which is a ternary quantic of the second order. Its actual value is 4 times that discriminant. Let us consider the analogous case of Mechanical Involution of lines in a plane or in space. There are two questions to be solved. The one is to find the condition that the Involution may exist, that is, that a set of equilibrating forces admit of being found to act along the lines ; the second, to determine the relative magnitudes of the forces when the involution exists, and this is the simpler question of the two. In like manner we may consider two questions in the case of m, n being in either of the two kinds of involution; the one being to find what the condition is of such involution existing, the other what are the coefiicients of the (iP coefficients in the equation which connects the co" products, when the involution exists. This latter part of the question (surprising as the assertion may appeal- and is) admits of a very simple and absolutely general direct and almost instantaneous solution by means of the Law of Nullity, above referred to, as I will proceed to show. The determination of the Involutants, or at all events of their product, will then be seen to follow as an immediate consequence from this prior determination of the form of the equations which express the involutions of the two kinds respectively. 15] Invariants to Matrix Systems 137 But first it may be well to explain why and in what sense I refer in the title to Involutants as belonging to a class of invariants. I say, then, that universally involutants are invariants in this sense, that if for m and for n, any function of m, or any function of n be substituted, the ratio of the two Involutants, say / and J, remains unaltered. By virtue of the Identical Equation (m)' will be of the form of Ai + Bi + CiVi' + ... + Liin"-^ and as a consequence it is easy to see that when m» is substituted for m, / and J will become respectively PI, PJ where P is the Q>th power of the determinant to the matrix formed by writing under one another the (&>— 1) lines of terms, of which the line £,-, C,-, ... ; Z,- is the general expression. Moreover, in the particular case where w = 2 and I=J*, besides being an Invariant in this modified sense, / will be an invariant in a sense including but transcending the more ordinary conception of an Invariant ; for if when, for m and n, f{m, n) and(m, n) are substituted, / becomes /', then /' will contain 7 as a factor ; this is a consequence of the fact that when m and n are in involution f{m, n) and <^{m, n) will also be in involution, for in consequence of the identical equation mn + nm — 2bn — 2cm + 2e = 0 /and ^ andy^ will each be reducible to the form A + Bm + (7n + Dmn and it is obvious from the ordinary theory of the determinants that the topical resultant of 1, (meaning ^ ,). and three linear functions of 1, m, n, nm, will contain as a factor the topical resultant of 1, m, n, mn. Nor must it be supposed that Involutants are the only species of invariants in the modified sen.se first described which appertain to the * I for some time had imagined, and indeed thought I had proved, that the two involutants were always identical. When crossing the Atlantic last month on board the "Arizona," having hit npon a pair of matrices of the third order, for which the two topical resultants admitted of easy calculation, I found, to my surprise, that they were perfectly distinct. The cause of the tailnre of the supposed proof constitutes a paradox which will form the subject of a communication to a future meeting of the Johns Hopkins Mathematical Society. I will here only premise that the seeming contradiction between the logical conclusion and the facts of the case takes its rise in a sort of mirage with which invariantistg are familiar, namely : the apparent a priori establishment of algebraical forms as the result of perfectly valid processes, which forms have no more real existence in nature than the Corona of the Sun under oar Or Hastings' scrutinizing gaze : the contradiction between the logical inference and the truth being accounted for by the circumstance that any such supposed form on actual per- formance of the operations indicated, turns out to be a congeries of terms, each affected with a nnll coefficient ; we are thus taught the lesson that all a priori reasoning until submitted to the test of experience, is liable to be fallacious, and it is impossible to prove that a proof may not be erroneous by any other method than that of actoal trial of the results which it is supposed to yield. 138 On Involutants and other allied species of [15 system m and n. Thus, for example, when w = 2 it is not only true that the determinant of the matrix mn — nni is such a kind of Invariant (which for greater clearness it may be desirable to denote by the term Perpetuitant*), but each element of that matrix will also be a perpetuitant, and these 4 per- petuitants, when for m, n pm, 0, 0 0 0 0 0 3p and the three values of (w — p-i) (n — /x,) are p^k + 6^ k' +pd 1 +ek + ek -k+ ff^ k^ +p^0 + e p +ek -pk+ &' p^k + p^d k + p-'ek + p'e The general value of (vi — \,) (m — X,) (n — /li) (n — /ju) will (to a numerical factor pres) be a matrix consisting of a single column accompanied by two columns of zeros, the non-zero column being some one of the 9 columns found in the above 3 matrices. Now by direct calculation we know that the n, m Involutaut in this case is a numerical multiple of (P — p'^y and vanishes when k^ = p-, which gives 6 = ^{\ + p-), that is, — p = 6^, and if we please k=-6^. Hence not merely one but three of the products of (m - X') (m - X") (n - jji') {n - p,") will in this case vanish, for the above equations will cause the 2nd, 4th and 9th columns all to become columns of nulls. If now instead of the factor (m — X') (m — X") we substitute the factor (m — X)', the three values of (m — X)' will become 00 0 -3 00 -3 00 0-8/9 0 00 0 0-3/3 0 0 0 -3/D^ 0 0 -3p» 0 0 0 1 k' -pk + pd' 15] Invariants to Matrix Systems 141 so that if {m - Xf (n - /x') (n - fj.") is to vanish, it will readily be seen that each of two columns of one or the other of the two matrices representing (n — /uf) (n — jx") will have to vanish simultaneously, and that this cannot be brought to pass when 6' = — p and p = p- =. $^ whether we make A; = ^ or — ^^ or ^. Hence {m-Xf{n-^l){n-^^.") = 0 is not an admissible general involution form of equation. Similarly by interchanging the above special values assigned to m and n, it may be shown that (m - V) (to - X") (n - fj.y = 0 is not an admissible form, and consequently that the one universal form of the involution equation is expressed by saying that (to - X') (m - X") (n - fj.') (n - m") is an absolute null. If no connexion exists between the elements of to and n, we know from the law of nullity that the above matrix has a nullity 2, that is, that all its minors except the elements themselves have zero contents. The effect of the vanishing of / is to make the elements themselves one and all vanish when the two sets of latent roots are duly selected. So in general if jP = X- - ^,X--' + ^,X"-» - ^sX--' . .. = 0, and (? = /!-- 5,M"-' + jB.M""' - B.^l'^'' . . . = 0, are the two equations to the latent roots of to, n matrices of order w, and if M= m"^' - (Ai - X)m"-» + (A^ - ^,X + X')to"-» ... and i\r = 71— ' - (5. - /i) n"-* + (5, - B,X + X») w^' ... , MN = 0 for some value of X and of /* is the one equation of involution, and NM = 0 for some value of X and some value of /* ia the other such equation. I will now show how to deduce from the above statement the following marvellous theorem. Let H represent the sum of the product of each term in the matrix M by its altruistic opposite in N (so that H is a. function of X and /j. and of degree B - 1 in each of them) then will the ordinary Algebraical Resultant of* F, G, H* be exactly equal (in magnitude as well as form) to the product of the two involutants to the corpus m, nf. * The system of equations whose resultant expresses the undifiFerentiated condition of tnTolation, may be written under the form (x,y)" = 0; {z,t)" — 0; (x, y)"-'=0. Qu(Bre whether CDch a resultant may not be written nnder the form of a determinant by an application of the Dialytie Method? + If / and J be the two involutants, /=0 will be the condition of left-handed involution of m, n or right-handed of n, m, and J=0 of right-handed involution of 711, n or left-handed of n, 711, for Involution, like light, " has sides." But IJ = 0 will be the condition of cme or the other kind, or so to say of undifferentiated Involution. 142 On Involutants and other allied species of [15 By the theorem proved at the beginning of this note, the nullity of M and that of N are each w— 1, hence the nullity of MN and consequently a fortiori its vacuity cannot be less than « — 1, and accordingly the identical equation to MN may be written under the form {MN)''-H{MN)'^' = Q, where H is the sum of the product of each element in the Matrix M or the Matrix N multiplied by its altruistic opposite in the other. Suppose now that 7=0 then for some one system of X, /x out of the w' systems given by the equations F=0, G = 0, H must vanish (for the nullity and a fortiori the vacuity of MN in that case becomes w) ; hence the double norm of H, that is, the product of the w^ values of H, or, which comes to the same thing, the resultant of F, G, H, must vanish when / vanishes and must therefore contain 7; in like manner because the nullity of NM and cb fortiori its vacuity is to when 7=0, it follows that the same resultant, say R, must contain also J\ R will therefore contain IJ, from which it may readily be concluded that it can differ from IJ, if it differ at all, only by a numerical factor. I need hardly pause to defend the assumption that 7, J have no common factor, and that it is the first and not necessarily any higher power of R which contains 77; the single instance, when 10 0 0 p k m = 0 p 0, n = k 0 p", 0 0 p' 1 k 0 of 7, J being respectively (to a numerical factor prh) the cubes of ^ — p and k' — p^ which have no common factor, settles the first part of this assumption at all events for the case of o) = 3, and as regards the second, it is only necessary to show that neither 7 nor J is equal to, or contains a square or higher power of a function of the letters in m and n as may be done easily enough when w = 3 by another simple instance*. We may then at once proceed to compare the dimensions of R with those of 7 and J. * Limiting ourselves to the case of matrices of the third order, if we take for m, n the matrices 0 fc 0 0 B 0 d 0 f, D 0 F, it may be shown by direct computation that one of the Involutants 0 ft 0 0 H 0 becomes (bH-hBf (fD - dFf (bd+fh) {BD - FH) {dB -fH) .{(hF+ bDf - (bd +fh) (BD + FH)\, and consequently if there were any square factor in either involutant such factor would contain the elements belonging to the two sets indecomposably blended, but on the other hand, if we 10 0 Q f F take for m, n the matrices 0 p 0, O 0 g, either involutant to m, n may easily be shown 0 0 p» ft il 0 (also by direct computation) to be made up of three factors, each of which is an indecomposable cubic function of /, g, ft, F, G, H. Hence it follows that neither Involutant can in its general 15] Invariants to Matrix Systems 143 R being the product of tu- values of \"~' ^"^^ + etc., where \, ^ are codimensional with the elements in m and n respectively, is obviously of the degree &>-. (tu — 1) in regard to each set of elements, that is, of the degree 2 to make it valid, it must be demonstrated as a Lemma that the two general twin involutants (even were they decomposable forms, which they un- doubtedly are not) could not have any common factor, nor either of them contain any square factor. The Resultant of F, G, H may be compared to a cradle just large enough to contain the twin forms in question, so as to give assurance that no other form is mixed up with them ; and the proof given above shows that this must be the case if neither twin is doubled form contain any square factor. As a matter of fact, not only for ternary matrices but for mAtrices of any order, there can be no reasonable doubt whatever in any sane mind that every Involutant is abiolutely indecomposable. One must try, however, to obtain a strict proof of this npon the general principle of crushing every logical difficulty regarded as a challenge to the human reason, which falls in our way; it is in overcoming the difficulties attendant upon the proof of negative propositions that the mind acquires new strength and accumulates the materials for future and more significant conquests. To prove that involutants in their general form are indecomposable may possibly, I imagine, prove to be a hard nut to crack, or it may be exceedingly euy. 144 On Tnvolutants and other allied species of [15 up upon itself, and if the two do not grow into one another, but like such creatures each possesses a perfectly distinct organization. A single instance will serve to establish the fact that the Resultant of F, G, H is the very product IJ itself, without any numerical multiplier. I have made this verification for binary and ternary matrices, and as the point is not one of an essential importance need not dwell here further upon it. To pass to a much more important subject, I am inclined to anticipate as the result of a long and interesting investigation into the relations of the involutants of a certain particular corpus of the third order that the sum of the two involutants of any corpus admits of being represented by means of invariants similar in kind to that which expresses the single involutant to a binary corpus (m, n), namely, the content of (that is, the determinant to) the matrix mn — nin, which itself (as previously observed) may be written as the {Til, 71 1 ^ , or say (m, n\ ; and in some similar way it is, I think, not unlikely that the product also of the two involutants (the resultant of F, 0, H) is capable of being expressed; but I must for the present content myself with exhibiting the bare fact of the existence of invariants of the kind referred to for matrices of any order. Suppose then that m, n is a corpus of the third order. Form the deter- minant 771 71 m' nr m n m' n" m n m? n" m n tn? n' say (»n, n, ni", n'^. The number of terms, half of them positive and half of them negative, in such determinant is 24 ; but of these, all but 8 will obviously appear as pairs of equal terms affected with opposite signs and so cancel one another : the 8 excepted ones are those in which no m and n come together, to wit : mnm^n' + nmn^ni' + ni'n'mn + n^m^nm — rn'nmn' — nui'n^m — mn'm^n — n'mnni^. The determinant to this matrix will be of the total degree 18 in the two sets of elements belonging to ?/i and n respectively, that is, of the degree 9 in respect to eacli set of elements per se. And so in general if m, n be of the order w the determinant (m, m', ... m"~S n, »i", ... n"~^)^ will contain only 2 (tto))' effective terms, of which half will bear the positive and the others the negative sign. 15] Invariants to Matrix Systems 145 The determinant to this matrix will be of the order a>[2{l + 2+...+(w-l)}], that is, (a)-l)a)>, in regard to the combined elements in m, and n, that is, equi-dimensional with either involutant to the corpus m, n. Whatever else may be its properties (on which I do not dare yet to pronounce), it is certain that such determinant (and over and above that, every term in the matrix of which it is the content) will be an Invariant to the corptis in the same sense in which either Involutant has been previously shown to be entitled to bear that name. And here for the present it becomes necessary for me to break oflF, bidding au revoir to any reader who may peruse this sketch, and trusting to meet him again in the broader field of the American Journal of Mathematics, where I hope to be spared to set out this portion of the theory with more certainty, and the whole doctrine of multiple quantity with much greater completeness and in more ample detail than is possible within the limits of the Circulars and in the short interval re- maining between the present time and the date of my intended departure for Europe. 8 IV. 10 16. ON THE THREE LAWS OF MOTION IN THE WORLD OF UNIVERSAL ALGEBRA. [Johns Hopkins University Circulars, III. (1884), pp. 33, 34, 57.] In the preceding Circular allusion was made to tbe three cardinal prin- ciples or conspicuous landmarks in Universal Algebra ; these may be called, it seems to me (without impropriety), its Laws of Motion, on the ground that as motion is operation in the world of pure space, so operation is motion in the world of pure order, and without claiming any exact analogy between these and Newton's laws, it will be seen that there is an element in each of the former which matches with a similar element in the latter, so that there is no difficulty in pairing off the two sets of laws and determining which in one set is to be regarded as related by affinity with which in the other. They may be termed the law of concomitance or congruity, the law of consentaneity and the law of mutuality or community. The law of congruity is that which affirms that the latent roots of a matrix follow the march of any functional operation performed upon the matrix, not involving the action of any foreign matrix ; it is the law which asserts that any function of a latent root to a matrix is a latent root to that same function of the matrix ; in so far as it regards a matrix per se, or with reference solely to its environment, it obviously pairs off with Newton's first law. The law of consentaneity, which is an immediate inference from the rule for combining or multiplying substitutions or matrices, is that which affirms that a given line (or parallel of latitude) can be followed out in the matrices resulting from the continued action of a matrix upon a fixed matrix of the same order, that is, in the series M, mM, m^M, m'M, ... (which may be regarded as so many modified states of the original matrix) without reference to any other of the lines or parallels of latitude in the series, or again any column or parallel of longitude in the correlated series M, Mm, Mm'', ... without reference to any other such column or parallel of longitude. 16] On the Three Laws of Motion, etc. 147 An immediate consequence of this obvious fact (a direct consequence for the rule of multiplication) obtained by dealing at will with either of the systems of parallels referred to, is that a system of simultaneous linear equations in differences may be formed for finding each term in any given line or in any given column at any point in the series, and the integration of these equations leads at once to the conclusion that any term of given latitude and longitude in the I'th term of either series is a syzygetic function of the tth powers of the latent roots of m. If, then, M be made equal to multinomial unity, this at once shows that supposing &) to be the order of m, on substituting rti for the carrier (or latent variable) in the latent function to m, and multiplying the last term by the proper multinomial unit, the matrix so formed is an absolute null, which proves the proposition concerning the " identical equation " first enunciated by Professor Cayley in his great paper on Matrices in the Philosophical Transactions for 1858. This proposition admits of augmentation, (1), from within, as shown in a former note, by applying to it the limiting law of the nullity of a product (a branch of the 3rd law), which leads to the very important conclusion that the nullity of any factor of the function of a matrix which is an absolute null, or more generally of any product of powers of its linear factors, is exactly equal to the number of distinct linear factors which such factor or product contains, at all events, in the general case where the latent roots are all unequal ; and (2), from without, by substituting for m,m-'ren where n is any second matrix whatever and e is an infinitesimal. This leads to the catena of identities, to which allusion has been made in the preceding Circular. Then, again, the endogenous growth of the theorem (that which determines the exact nullity of any factor of the left-hand side of the identical equation) in its turn seems to lead to a remarkable theorem concerning the form of the general term of any power of m into M. Observe that every such term is expressed as a syzygetic function of powers of the constants, so that the total number of syzygetic multipliers is w'; but the number of variables in m and M together is 26)' ; and, consequently, apart from the to arbitrary latent roots the number of independent constants in m'M should be 2«i)' — w. The ft)' syzygetic multipliers ought then to contain only ft)(2ft) — 1) arbitrary constants, and such will be found to be the case by virtue of the following hypothetical theorem : Calling \ any one of the latent roots, the multipliers of X' in m'M will form a square of w' quantities; the theorem in question* is that every minor of the second order in such square is zero, so that the o)' terms in the square is given when the bounding angle containing * I have not had leisure of mind, being mach occupied in preparing for my departure, to reduce thia theorem to apodictic certainty. I state it therefore with all due reserve. 10—2 148 On the Three Laws of Motion in [16 2ft) — 1 terms is given ; and the same being true for the multipliers of each latent root (which resolve themselves into o> squares) the number of arbitrary quantities in all is m{2m — 1) as has to be shown. The law of consentaneity in so far as it relates to the decomposition of the motion of a matrix into a set of parallel motions, has an evident affinity with Newton's second law*. Remains the law of mutuality, which is concerned with the effect of the mutual action upon one another of two matrices, and so claims kindred with Newton's third law. This law branches off into two, one of which may be termed the law of reversibility, the other that of co-occupancy or permeability. The law of reversibility affirms that the latent function of the product of two matrices is independent of the sense in which either of them operates upon the other, that is, is the same for mn as for nm, just as the kinetic energy developed by the mutual action of two bodies is not affected by their being supposed to change places. As regards the second branch of the third law, the word co-occupancy refers to the fact that although the space occupied by two similarly shaped figures (say two spheres) is not absolutely determined (in the absence of other data) by the spaces occupied by them each separately (for they may intersect or one of them coincide with or contain the other), a superior as well as an inferior limit to such joint occupation is so determined ; the inferior limit being the space occupied by either such figure, that is, the dominant of these two given spaces, and the superior limit their arithmetical sum. So the nullity resulting from the action in either sense of two matrices upon one another is not given when their separate nullities are assigned, but has for an inferior limit the dominant of these two nullities and for a superior limit their sum ; the nullities of the two component matrices may also be conceived under the figure of two gases or other fluids which are mutually permeable and capable of occupying each other's pores. Although the limits spoken of are independent of the sense in which the two matrices act on one another, it must not however be supposed that the actual resultant nullity is unaffected by that circumstance ; thus, for example, if the latent roots of a ternary matrix m are X, \', X.", the nullity resulting from {m — \) (m — \') acting sinistrally upon {m — \") n, that is, of (m — X) (m — X') (m — X") n is 3, but from the same acting dextrally upon the same, that is, of (7?i — X") n (m — X) (m — X'), need not necessarily exceed 2. * For another and closer bond of affinity between the two laws see concluding paragraph of this note. 16] the World of Universal Algebra 149 Snch then are the three primary Laws of Algebraical Motion ; but as Conservation of areas, Vis viva, D'Alembert's Principle, the principle of Synchronous Vibrations, of Least action, and various other general laws may be deduced from Newton's three ground laws, so, of course, various subordinate but very general laws may be deduced from the interaction of the above stated three ground laws, namely, the law of Congruity, the law of Consentaneity, and the law of Mutuality. The deduction of the catena of identical equations connecting two matrices m and n from the second and third laws combined, atfords an instance of such derivative general laws. Another instance of the same is the theorem that when the product resulting from the action upon one another of two matrices, is the same in whichever of the two senses the action takes place, the matrices must be functionally related, unless one of them is a scalar, that is, a multiple of multinomial unity, at all events when neither m nor n possesses a pair of equal latent roots. This very important and almost fundamental law (seemingly so simple and yet so hard to prove) may be obtained as an immediate inference from that identical equation in the catena of such equations connecting the matrices m and n, in which one of the two enters only singly at most in any term. As for example if m and n are of the 3rd order, beside the identical equation w»* — 36to* + Sdm — gr = 0 we have* the identity nihi + mnm + nm* — 36 (mn + nm) — 3cm' + 2dn + 6em — 3/t = 0. But if nm = win then mnm = m'», nm* = mnm — m'n, so that this equation becomes mhi — 2bmn + dn = m'c — 2em + h, or n = — ; — —. ; f, m' — 26m + d ' unless m* — 26m + d is vacuous. The first branch of the third law, namely, the law of reversibility, is an almost immediate inference from the rule for the multiplication of matrices, and becomes intuitively evident when the process of multiplication in each of the two senses between m and n is actually set out. The second branch, namely, the law of co-occupancy or permeability, as it is the most far-reaching 80 it is the most deep seated (the most cach^ of all the primary laws of [• See p. 126 above.] t Whence it follows that n mast be a fnnction of m convertible into an integral polynomial form, unless the namerator and denominator of the fraction to which n is equated vanish simul- taneously, which is what happens when m is scalar. If the numerator exactly contains the denominator n becomes a scalar. Seeing that a constant c is a specialized case of a function of • variable x althongh the converse is not true, we may say that whenever nm = rim, one at least of the two matrices m and n is a function of the other, and that each is a function of the other aniess that other is a scalar. Compare Clifford's " Fragment on Matrices '' in the posthumous edition of his collected works. 150 On the Three Laim of Motion in [16 motion. I found my proof of it upon the fact that the value of any minor determinant, say of the ith order, in either product of m and n (two matrices of the order m) may be expressed as the quantitative product of a certain couple of rectangular matrices (in Cauchy's sense of the term), of which one is formed by i columns and the other by t lines in the two given matrices respectively. Such rectangle as shown by Cauchy (and as may be intuitively demonstrated by the simplest of my umbral theorems on compound deter- minants) is the sum of the 7r( — i) TTl complete determinants of the one rectangle multiplied respectively by the corresponding complete determinants of the other rectangle. This shows at once the truth of the proposition in so far as relates to the lower limit, that is, that if mn=p, and m, n have the nullities e, f, and p the nullity 0, then d must be at least as great as e and at least as great as ^. As regards the superior limit the proof is also founded on the theorem in deter- minants already cited, and the form of it is as follows. If e be any number r, it may be shown that ^ must be at least as great as 9 — r; hence giving r all values successively from 0 to f — 1, it follows that e+ ^ cannot be less than 6, that is, that 0 cannot be greater than e + f . The proof of the first law, that of concomitance or congruity, I ought to have stated antecedently, is a deduction from the theory of resultants and the well-known fact that the determinant of a product of matrices is the product of their determinants. Thus each of the three laws of motion is deduced independently of the two others. As another example of a derivative law of motion, I may quote the very notable one which results from the interaction of the first and second funda- mental laws upon one another, and which gives the general expression for any function whatever of a matrix in the form of a rational polynomial function of the same and of its latent roots, to wit, the magnificent theorem that whatever the form of the functional symbol <^, and whether it be a single or many valued function, if Xj, Xj, ... \„ be the latent roots of 7n, •^"^ = ^"^^^ (X,-X,)(X,-X3)...(X.-XJ • p p As for example if = 0, and that conse- quently X cannot be a function of y. 17. EQUATIONS IN MATRICES. [Johns Hopkins University Circulars, III. (1884), p. 122.] I HAVE been lately considering the subject of equations in matrices. Sir William Hamilton in his Lectures on Quaternions has treated the case of what I call unilateral equations of the form x'+px + q = 0, or x' + xp + q^O, where we may, if we please, regard x, p, q as general matrices of the second order. He has found there are six solutions, which may be obtained by the solution of an ordinary cubic equation. In a paper now in print and which will probably appear in the May number of the Philosophical Magazine, I have discussed by my own methods the general unilateral equation, say af+paf^^ + qocf-'' + ...+l = 0, where x,p, q ...I, are quaternions or matrices of the second order, and have shown, by a method satisfactory if not absolutely rigorous, that the number of solutions is w' — o)' + to, that is to say, the nearest superior integer to the general maximum number of roots {m*) divided by the augmented degree (a, + 1). But after I had done this it occurred to me that there were multitudinous failing cases of which neither Hamilton nor myself had taken account, as for example x" + px = 0, besides the solutions x = 0, x = — p, will admit of a solution containing an arbitrary constant, I think ; but that is a matter which I shall have to look further into before committing myself to a positive assertion about it. I have only had time to pass in review the more elementary case of a unilateral simple equation, say px = q, where p, q are matrices of any order to. If p is non-vacuous there is one solution, namely, x = p~^q; but suppose p is vacuous : what is the condition that the equation may be soluble ? (1) Suppose q — O,]) being vacuous has for its identical equation pP = 0, and consequently we may make x = \P where \ is an arbitrary constant. (2) Suppose q is finite and that a; = r is one solution, then obviously the general solution is a; = r + XP. 17] Equations in Matrices 153 We have now to inquire what is the condition that r may exist. I find from the mere fact of x being indeterminate (and confirm the result by another order of considerations) that the determinant of ? + Xp must vanish h' c identically ; so that for instance when p, q are of the second order and , . are the parameters to the coi'pus (p, q), we must have when d = 0, which is implied in the vacuity o{p,f=0 and e = 0. The first of these conditions is known d priori immediately from my third law of motion ; but not so, without introducing a slight intervening step, the intermediate one (I mean the con- nective to d and /, namely) e = 0. So in general in order that px + q = 0 may be soluble, that is, in order that p~^q where p is simply vacuous may be Actual and not Ideal, q must satisfy as many conditions as there are units in the order of p or q, all implied in the fact that the determinant to p+\q, where \ is an arbitrary constant, vanishes identically. When these conditions are satisfied p~'q becomes actual but indeterminate. (This, by the way, shows the disadvantage of calling a vacuotis matrix indeterminate, as was done in the infancy of the theory by Cayley and Clifford — for we want this word as you see to signify a combination of the inverse of a vacuous matrix with another which takes the combination out of the ideal sphere and makes it actual.) So in general in order that p~^q where jo is a null of the ith order (that is where all the (i + l)th but not all the tth minors of p are zero) shall be an actual (although indeterminate) matrix, it is necessary and sufficient that p + \q, where \ is arbitrary, shall be a null of the .same (ith) order. What will be the degree of indeterminateness in p~^q, that is, how many arbitrary constants are contained in the value of x which satisfies the equation px = 0 remains to be considered. The law as to the conditions is an immediate corollary to my third law of motion, for if px = q then p + X^ = p (1 + \x) ; consequently p + \q, what- ever X may be, must have at least as high a degree of nullity as p. q.e.d. 18. SUR LES QUANTITES FORMANT UN GROUPE DE NONIONS ANALOGUES AUX QUATERNIONS DE HAMILTON. [Comptes Rendus, xcvm. (1884), pp. 273—276, 471—475.] Dans une Note prec^dente*, j'ai fait allusion au cas oil le determinant de X + ym + zn devient une fonction lineaire de a?, y, 2? sans que la quantity noram^e Q s'^vanouisse. . Dans ce cas, on aura (r)inf + Q{mn)-R=Q, (1) R ^tant le determinant de mn. C'est bien la peine, comme on va le voir, de donner plus de precision aux Equations qui lient ensemble mn et nm pour ce cas. En suivant la meme marche que pour le cas particulier oil Q = 0, on trouvera sans difficult^ les r^sultats suivants : «m = - -^ {mny - ^^— mn--^, (2) 3Q, „ ?-9iJ 2Q^ ,„, mn= -^ {nmy - ^ nm + -^, (3) f ^tant le produit des differences des racines de la fonction \' + Q\ — iJ, de sorte que ^ = - (4Q» + 27i?»). Cons^querament on peut ^crire nm= A{mnf + Bmn+G, (4) mn=-A {nmf + B'nm - G, (5) ou A et 0 peuvent etre tons les deux z4to, ou tous les deux des quantity finies quelconques, mais non pas I'un d'entre eux une quantity finie et I'autre zero, et B, B' les deux racines par rapport a B de I'^quation An B' + B+l + ^ = Oi. (6) * Comptes rendus, t. xcvii. p. 1336. [tit follows from n {mn -f- 9) = {nm + 6) n that M, =mn and N, = nm both satisfy equation (1) ; further MN=NM (footnote * p. 127 above), so that (p. 149 above) there exists an equation N=pM' + qM + r; from (1), if \M-N\*0, follows M^ + MN + N^ + Q = 0. Hence (2), (3) can be deduced.] 18] Sur les quantites formant uu groupe de nonions 155 On peut verifier, comme je I'ai fait, par un calcul alg^brique direct, que les eqiiations (4) et (5), en vertu des Equations (1) et (6), sont compatibles. Or une chose digne de remarqiie, c'est ce qui arrive quand ?=0, car cela servira a reveler un phenomeue d'Algebre universelle d'un genre que personne n'avait encore ineme soup9onne. Dans ce cas, les deux equations (4) et (5) changent leur caractere et deviennent Qimnf + ^RmnJrlQ'^O, Q (nm)"- + SRnm + ^Q' = 0, de sorte que mn et nm cessent d'etre fonctions I'un de I'autre. Nommons, pour le moment, mn = u, nm = v; on aura, comme auparavant, uv = vu, sans que v et u soient fonctionnellement li^s ensemble. Dans le Johns Hopkins Circular de Janvier 1884 (dans I'article intitult^ Ox the three laws of motion in the world of universal Algebra, [above p. 146]), on trouvera le moyeii d'etablir qu'en general cette Equation amene k la conclusion que ou COO u doit etre un scalar, c'est-i-dire de la forme 0 C 0, ou bien v un scalar, ou 0 0 C sinon que nm, mn doivent etre fonctions I'un de I'autre ; mais on remarquera (ce qui m'avait alors ^chapp^) que, si Fu = 0 est I'^quation identique en u et que la deriv«?e fonctionnelle F'u est une matrice vide (vacuous), c'est-a-dire dont le determinant est z^ro, le raisonnement est en d^faut ; cette vacuity a lieu dans le cas, et seulement dans le cas, ou deux des racines latentes (lambdaiques) de m sont egales. On peut g^ndraliser celte conclusion et r^tendre a deux matrices u et v d'un ordre quelconqne au-dessus du deuxieme ; c'est-^-dire qtiand les racines latentes de u (ou bien de v) ne sont pas toutes in^gales, il est des cas ob. uv = vu, sans que u ou v soient des scalars et sans que v et w soient fonctions I'un de I'autre. Par exemple, si Ton fait u = 0 p P' 1 0 1 p" p 0 - p uv = p v = 0 1 1 p 0 p' p p' 0 on trouvera P 1 I — p 1 ' = tm. P' P' -p\ Mais on d^montrera sans difficult^ que v ne peut pas s'exprimer comme sorame de puissances de u, ni vice versa v comme somme de puissances de u. On n'a pas besoin de remarquer que la seule condition de I'existence de racines latentes Egales en u ou en v ne peut pas suffire en elle-mSme pour 156 Siir lea quantiUs formant un groupe de nonions [18 assurer que m = vu, mais il faut r^server pour une autre occasion la pleine discussion de la totality des solutions de cette equation importante. J'ajouterai seulement cette remarque, qui est essentielle. En supposant I'existence des Equations TO*n + mnm + nm* = 0, n'm + nmn + mn* = 0, (m»)» + Qmn - i2 = 0, {nmf + Qnm - R =0, qui ont lieu necessairement quand le determinant de x-\-ym-\-zn devient une fonction lineaire de a?, f, s?, et en regardant nm comme fonction de mn (en vertu de I'^quation mn . nm = nm . mn), alors, en additionnant aux deux valeurs de nm (exprimd comme fonction de mn) donn^es ci-dessus, qui corre- spondent aux deux valeurs de f, c'est-^-dire V-(4Q'+ 27^),on a k consid^rer quatre autres valeurs, le nombre total en ^tant six. Car si Ton suppose nm = A (mny + Bmn + C et si \,\^, X, sont les trois racines de X' + Q\ - iJ = 0, les valeurs d^ A, B, G sont determin^es en mettant ^Xi» + -BX. + C=X.-, A\^^ + B\, + C = \k, oh i,j, k sont respectivement 13 2 2 3 1 1 2 3 ou ou bien 3 2 1 3 12 2 13 Les valeurs de A,B,C donn^es ci-dessus correspondent au deuxifeme de ces groupes de valeurs de i,j, k. Si Ton 6crit i = 1, j = 2, k = 3, on trouvera nm = mn. Si Ton ecrit i =l,j = 3, k = 2, en faisant X, = A, on trouvera 2Aimny-Qmn + 2AQ 3A' + Q Dans le cas critique ou f = 0, de sorte que 3A^ + Q = 0, I'equation devient (mny + Amn - 2A» = 0, comme dans le cas d^ja traite. Quand on suppose Q egal k z^ro et R (c'est-^-dire le determinant de mn) fini, les seules solutions possibles avec ces conditions sont celles fournies en ^crivant i,j, A = 2, 3, 1, ou 3, 1, 2 ; mais, pour le cas gdn^ral, il n'y a pas de raison (au moins trh Mdente) pour exclure aucune des trois classes de solution. Si Ton admet la Mgitimit^ des solutions de la troisieme classe, en dcrivant nm = A (mny + Bmn + C, A r on trouvera B^ + B + — - = 0 18] mmlogues aiix quaternions de Hamilton 157 au lieu de 1 equation AC 5^ + 5 + 1+^ = 0, qui est applicable aux solutions de la deuxieme classe. Avant de coiisiderer I'equation xy = yx, il importe d'avoir une id^e nette d'une certaine classe de matrices que je nomme priviUgUes ou derogatoires, en tant qu'elles d^rogent a la loi g^nerale que toute matrice est assujettie a satisfaire a une Equation identique dent le degre ne pent pas etre moindre que I'ordre de la matrice. Les matrices derogatoires sont justement celles qui satisfont a une Equation d'un ordre inf^rieur k leur ordre propre; on pent les nommer simplement, doublement, triplement, . . . derogatoires, selon que le degre de I'equation identique a laquelle elles satisfont differe par une, deux, trois, ... unites du degre minimum ordinaire. Pour le cas des matrices du deuxifeme ordre, il n'y a que les scalars „ 0 a qui soient derogatoires. Pour le cas des matrices du troisifeme ordre, en dcartant les scalars de la a 0 0 forme 0 a 0, toute matrice x derogatoii-e peut 6tre ramen^e ou k la forme 0 0a a + 6(6 + €»), oh. e est une matrice qui satisfait a I'equation e* = 1, c'est-i-dire une matrice dont les racines latentes sont 1, p, p', ou a la forme a + 6(l + e + r')?, oil 6»=1, f=l et ^€ = pe^, p signifiant une racine cubique primitive de I'unit^ Dans le premier cas, ar" - (2a + 6) a; + (a» + o6 - 26») = 0, ' et dans le second sd'-2ax + d' = 0, car on trouvera facilement que (l + e + e')C(l+€ + €»)?=0. Pour le cas du quatrifeme ordre, en ^cartant les scalars et en se bornant au cas ou I'equation identique d^rog^e (vue pour le moment comme une equation ordinaire en x) ne contient pas des racines egales, toute matrice X peut 6tre ramen^e k I'une ou k I'autre des deux formes suivantes : a + b{U+U',) oubien a + 6 (tr+ ^^ f7^ + /fcf/' 158 Sur leg qvantit^a formant un grmipe de nonions [18 oh U est une matrice du qiiatri^me ordre telle que U*+l =0; a,b,k sont des scalars arbitraires et i est une racine primitive biquadratique de I'unitd ; quand, pour la seconde forme A;= 1, on trouvera qu'il y aura une derogation double de I'ordre de I'equation satisfaite par x, I'^quation identique pour x ne sera que du deuxieme degre. En r^servant les details du calcul, voici le resultat g^ndral que j'ai demontr^ rigourensement (en m'aidant de la notation des nonions) pour les matrices du troisieme degr^ qui satisfont a I'equation xy = yx. A moins que x ne soit une matrice privil^giee ou derogatoire, y sera toujours une fonction rationnelle et entiere quadratique de x, et de meme, a moins que y ne soit privil^giee, x sera une fonction pareille de y. II est bien entendu que le caractfere derogatoire d'une seule des deux matrices n'emp^che pas quelle ne soit une fonction entiere et rationnelle quadratique de I'autre. Dans le cas oil a; et y sont tous les deux d^rogatoires, ni I'un ni I'autre ne peut etre exprim^ comme fonction explicite I'un de I'autre, mais ils seront li^s ensemble par une Equation lineo-lineaire. II parait peu douteux qu'une regie semblable doive #tre applicable a requation xy = yx, quel que soit I'ordre des matrices x et y, sauf quand I'equation qui lie ensemble x et y pourra etre d'un degr^ moindre que I'ordre de chacune d'elles. II est bon de remarquer que nulle matrice ne peut etre derogatoire, sauf pour le cas ou il existe des egalites entre ses racines latentes; mais ces egalites peuvent parfaitement subsister sans que la matrice a laquelle elles appartiennent soit derogatoire. En general, si x = a + by + cy*, on peut, par une formule generale que j'ai d4jk donnee, exprimer y sous la fonne a + ^x + yx' ; avec I'aide des racines latentes de x, cette formule ne cesse pas en general d'etre valable, meme pour le cas ou x contient des racines egales, en regardant leur difference comme une quantite infinitesimale ; seulement le nombre des racines finies subira dans ce cas une diminution ; mais, dans le cas ou I'equation xy = yx (x etant derogatoire) menerait a I'equation x = a + hy + cy'', on trouverait que nulle fonction explicite de x avec des coefficients finis ne peut exprimer le y cherche. II est a peine necessaire d'ajouter que rien n'empeche, dans le cas ou I'un ou I'autre de x et y ou tous les deux sont derogatoires, qu'on puisse satisfaire k xy = yx, en supposant que x et y soient des fonctions explicites chacune I'une de I'autre : tout ce qu'on affirme, c'est que, dans le cas admis, cette supposition cesse d'etre obligatoire ; c'est un cas tres semblable a ce qui arrive dans le cas de defaut (failing case) du theoreme de Maclaurin : c'est I 18] analogues aux quaternions de Hamilton 159 celui ou une variable est une fonction sans pouvoir etre ddvelopp^e dans une serie de puissances d'line autre variable. Dans ce qui precede, on a vu un example du fait general que, m. ^tant ime matrice donnee, I'^quation ^ (a;, m) = 0, pour certaines valeurs de m, cesse Ld'admettre la solution ordinaire x = Fm. Mais il existe encore une classe assez etendue d'equations entre x ei m pour lesquelles, quand m prend certaines valeurs, x ii'a aucune existence actuelle ; par exeniple, m dtant une matrice vide d'un ordre quelconque, si mx = 1, la matrice x devient inexprimable et n'a, pour ainsi dire, qu'une existence ideale. Je citerai encore I'exemple a? = m, m 6tant une matrice du deuxieme ordre ; si les racines latentes de in sont inegales, on trouvera, par la formule I g^n^rale, quatre valeurs de x. Si les deux racines latentes sont ^gales et finies, ces quatre valeurs se r^uisent a deux ; mais, si les deux racines sont ' toutes les deux egales a zero, il n'y aura aucune valeur de x qui satisfasse k a [I'^quation donnee, c'est-^-dire si m= k; I'^quatiota devient absolument ka —a [insoluble, ou, si Ton peut s'exprimer ainsi, les quatre racines carrees de m [sont toutes ideales. Dans le cas suppose, on v^rifiera ais^ment que m' = 0 et, vice versa, toute a racine carrde du zero binomial est de la forme k , de sorte que Ton peut ka — a dire qu'une racine carree quelconque du zero binomial ne poss^de pas elle- meme des racines aig^briques quelconques, ou, en d'autres termes, une racine alg^brique quelconque du quaternion i + V(— l)i est puremeut ideale et n'adniet pas d'etre representee sous la forme d'un quaternion. Finalement je remarque que toute matrice est d'un certain ordre et d'une certaine classe; I'ordre, c'est le nombre total de ses racines latentes ; la classe, c'est le degre minimum de I'equation latente (c'est-k-dire de I'^quation identique k laquelle la matrice satisfait), lequel ne peut etre plus petit que le nombre des racines latentes in«?gales. Je dois ajouter (ce que j'aurais Ah dire auparavant) que, quand x est une matrice temaire ddrogatoire dont toutes les racines latentes sont Egales, r^quation xy = yx peut subsister sans que ni a; ni y ne soit une fonction explicite Tun de I'autre, meme quand y n'est pas une matrice privile'git^e ; c'est le cas ou, e et f fai.sant partie d'un groupe de nonions ^l^meataires, on a x= a + b{l + e+ f')^. Les caiculs sont un peu compliques pour ce cas special, mais je crois ne pas me tromper en faisant cette correction. Le champ de la theorie de la quantity multiple est tellement nouveau et inex- ploite que, sans les plus grandes precautions, on est toujours en danger de se beurter centre quelque cause impr^vue d'incertitude ou meme d'erreur. 19. SUR UNE NOTE R^CENTE DE M. D. ANDRE*. [Comptes Rendus, xcviii. (1884), pp. 550, 551.] Le the'oreme de M. Andr^ est une consequence immediate de la gene- ralisation que j'ai donnee du thdorfeme de Newton {Arithmitique universelle, 1" Partie, Ch. ll.) sur les racines imaginaires des Equations. On verra, eu consultant mon travail f sur ce sujet (Proceedings of the London Mathematical Society, No. 2), que si Mo, w,, «,,... «,„ sont les co- efficients d'une Equation du degrd m et si Gr = rUr' - (r -h 1) 7r M,_, M^+l V + r—l ou 7»- = ; . 7r ^tant une quantity r^elle quelconque qui n'est pas interm^diaire entre 0 et — m, I'dquation aura n^cessairement au moins autant de racines imaginaires qu'il y a de variations de signes dans la s^rie G^, Gi, G.^ G,„. En faisant i) -= — m, on a le th^oreme de Newton ; en faisant t; = 1, on voit qu'on peut prendre G, = m^'' — Mr-i^^+i. Cons^quemment le theoreme de M. Andr^ subsiste, quel que soit le signe de la quantity qu'il nomme a et quels que soient les signes des quantit^s qu'il nomme u^, v^, ..., «„. De plus, le theoreme subsistera encore quand, outre ces modifications, au lieu de I'^quation M„ = aM„_i-|-/3u„_2, on ecrit i)„ = o[u„_i + /3y„_2 ou ^0, fi, V.2, ... , Vm, identiques avec M„ Ml Wo m' 1 ^(m.m-l) ^--^-^Tn(m-l)(in-2) * Comptes rendus, stance du 18 Kvrier 1884. [+ Vol. II. of this Eeprint, pp. 501, 507.] 19] Sur line Note recente de M. D. Andre 161 II y a encore une autre extension importante a ajouter, en considdrant r^quation M„_, !/„+, -Un' = Aa'' + B/a"^ + Crf, dont j'ai donn^ une solution particuliere dans Y American Mathematical Journal, Vol. IV. [Vol. lii. of this Reprint, pp. 546, 633.] II est peut-^tre digne de remarque que si, dans la formule dtablie pour 7,, on fait V infini, la regie calquee sur celle de Newton (mais plus gen^rale) enseigne que, quels que soient o, b, c ou m, I'^quation f, , , a? a? a;"" \ "V^'^^ir^^r^r^-'-'TY-rn) +H'-" + T-r:2T3 + -±r2r:^) + ^=o ne peut jamais avoir plus de deux racines reelles. IT. 11 20. SUR LA SOLUTION D'UNE CLASSE TRfiS ETENDUE D'^QUATIONS EN QUATERNIONS. [Comptes Rendus, xcviii. (1884), pp. 651, 652.] L'^QUATION parfaitement gen^rale du deuxifeme degre en quaternions sera de la forme 2 {axhxc + dxe) +/ = 0 et admettra seize solutions, qu'on pourrait obtenir d'une manifere directe au moyen de quatre Equations, chacune du deuxifeme degr6, contenant les quatre elements de x comme inconnus. De meme, I'^quation en quaternions ou en matrices du deuxifeme ordre du degre a admettra m* solutions. Parmi ces formes g^n^rales, on peut distinguer celles dans lesquelles tous les quaternions donnes se trouvent du meme c6te du quaternion cherch^, par exemple aa? + 6a; + c = 0. On peut nommer de telles Equations equations uni- laterales. Hamilton a considdre le seul cas de I'^quation quadratique {voir Lectures on Quaternions, art. 636, pp. 631 — 2), et a determine le nombre (6) des racines. Or, je trouve que ma m^thode g^n^rale de traiter les matrices amfene directement k la solution d'une equation unilat^rale d'un ordre quelconque (o (c'est-a-dire la fait d^pendre de la solution d'une equation alg^brique ordinaire) et donne sans la moindre difficult^ et sans aucun eflfort d'in- vention le nombre des racines. Ce nombre est exprim6 par la fonction ft)» — 0)2 4- o), de sorte que le nombre des racines, pour ainsi dire ^vanouies par suite de I'unilateralisme de la forme, est w* — «' + w' — <», c'est-a-dire (o)'— o)) (o)" -f 1). On comprend bien qu'en certains cas le nombre des racines subit une reduction ; par exemple, le nombre des racines de a;" + i = 0 est co" et celui de x" + kx + l = 0 est 2a)' — to. II semble que le nombre, pour I'^quation doit etre (0 + l)(o^— Ow, lequel, quand 0 = co — l, devient le nombre g^n^ral to'— ay' + CO. Les details de ce petit travail seront donnes dans un prochain num^ro du London and Edinburgh Philosophical Magazine. 21. SUR LA CORRESPONDANCE ENTRE DEUX ESPfeCES DIF- F^RENTES DE FONCTIONS DE DEUX SYST^MES DE QUANTITES, CORR^LATIFS ET ^GALEMENT NOMBREUX. [Comptes Rendus, xcviii. (1884), pp. 779—781.] Voici le theor^me a demontrer, dans lequel, par somme-puissance, on sous-entend une somme de puissances de quantit^s donndes: A i quantites on pent en associer i autres telles, que chaque fonction syme- trique (qui est une fonction des differences) des premieres sera une fonction des sommes-puissances du 2', du 3', ... , du i**"" ordre des demieres. Faisons, pour plus de clart^, t = 3. Soient r,, r^, r, les racines de I'^uation fr = aT^ + br' + cr+d = 0. En prenant b,c,d; ri,r,,r, com me deux systfemes correlatifs de variables ind^pendants, on trouve Done ZaZb + 26S, + cS^ = - S S,, oBi, + bBc + cSa= dt -^ Sr. rj r Soient a = a, 6 = 3/3, c = 3 . 2 . 7, d = 3 . 2 . 1 . 8, et soient p,, p,, /», les racines de lequation ap'+/3p'' + 7p+5=0. Alors, si 2 8, <^ = 0, on aura (a.1^ + /38^ + 784) <^ = 0. C.Q.F.D. L'integrale g^n^rale de la premiere ^uation est ^ = S (''i - n, n - n), et celle de la demifere est = JF. (/>>' + />.' + P*', Pi* + pi + />.')• 11—2 164 Sur la correspondaiice entre deux especes [21 Ces deux int^grales sont done identiques, et, le raisonnement ^tant g^n^ral pour une valeur quelconque de i, on voit que chaque fonction des difiPSrencea des r doit pouvoir s'exprimer comrae une fonction de i— 1 sommes-puissances cons^cutives des p (commen9ant avec la seconde), les r et les p dtant li^s ensemble par les Equations ar^ + 6r*-' + ct^-* + dr^' + ... = 0, api+ ~.p'-'+ .,.^ ..p^-'+ ... .f,. „■/)<-'+ ■■■ = 0, I "^ t(t — 1) i(i— l)(t — 2) '^ et cons^quemment une fonction symAtrique des differences des r sera une fonction rationnelle et entifere des i — 1 puissances cons^cutives (dont on a d^ja fait mention) des p. En prenant i = oo , on voit que le th^orfeme ^quivaut h, dire que tous les sous-invariants, sources des covariants de (a, h, c\x, y)', (a, h, c, d^x, yf, ... (a I'infini), seront des fonctions des sommes-puissances prises a I'infini, avec la seule exception de la somme lineaire, des racines de I'equation a + bx + -^^x'+ ^ , „ af+...(k I'infini). Tel est le thdorfeme capital d^couvert par M. le capita! ne Mac-Mahon, de rArtillerie royale anglaise, dont il a fait le plus heureux usage en d^veloppant la th^orie des perp^tuants (voir America7i Journal of Mathematics). II est Evident que le meme principe peut etre appliqu^ aux invariants de toute espfece, de sorte que, grace a la belle d^converte de M. Mac-Mahon, avec la generalisation (qui en sort presque intuitivement) que j'ai donnee, on est aujourd'hui en etat de traiter les parties les plus difficiles et les plus essentielles de la thdorie des formes alg^briques, comme M. Schubert I'a fait avec sa Zahl-Geometrie pour les figures dans I'espace, en faisant abs- traction, pour ainsi dire, de toute question de substance (de matiere contenue dans les formes), et en se bornant a un calcul purement arithmetique. Je dois avertir que le theorfeme de correspondance, tel que M. Mac-Mahon I'a donne, a paru dans YAvierican Journal of Mathematics (Vol. VI. p. 131). M. Mac-Mahon affirme (mais sans aucune preuve) que, si (o, /3, 7, ... etant des nombres entiers plus grands chacun que I'unite) est de la forme l,r^s^P, ... ,ou r,s, t, ... sont les racines de I'equation («»'"- 0'nf:3'-)^^'^>"=«' alors (ttoSa, -f- a,Saj -f- a^Sa^ ■+ ...) <^ = 0, et il donne Et ^ le nom de fonction symetrique nan unitaire des racines. Ce theoreme est vrai seulement pour le cas ou n est infini (ce que M. Mac- n 21] differ entes de fonctions 165 Mahon a oublid de dire), et dans ce cas il conduit k la consequence que les differentiants (c'est-a-dire les sous-invariants) de sent Ae^ fonctions symetriques non unitaires des racines de I'equation «„ + Oi a;- + ^ x-^ + j-|^ a;-» + ... = 0 et vice versa. Or il est Evident que chaque fonction symdtrique non unitaire d'un nombre infini de quantites n'est autre chose qu'une fonction des sommes • de toutes les puissances de ces quantites au del^ de la premiere. Voil^ pourquoi j'ai attribud k M. Mac-Mahon, dans ce qui pr^cfede (pour le cas d'une Equation dont le degre est infini), la connaissance du th^oreme que j'ai d^montre dans toute sa gendralit4 22. SUR LE THEORJ^ME DE M. BRIOSCHI, RELATIF AUX FONCTIONS SYMETRIQUES. [Comptes Rendus, xcviii. (1884), pp. 858—862.] Dans la demonstration du th^oreme sur une correspondance alg^brique, ins^rd dans les Comptes rendus de la semaine dernifere [p. 163 above], j'ai eu occasion de consid^rer I'int^graie de I'^quation ( d _ d d \ ^ . Je me suis aper5u depuis que cette int^grale pent se d^duire imm^dia- tement du beau theoreme de M. Brioschi, sur les fonctions syndtriques, k savoir que : d^ d^ d est une fonction des n premiferes sommes-puissances des racines de I'equation ao a;" + a, «"-' + ... =0, avec exclusion de la puissance r'^"", on aura dd> dd> _ aar aa„ et consdquemment F(si, s^, ... , Sr-i, Sr+i, ... , s„) sera I'^quivalent complet de I'expression f d , d . d\-' f. V aar aar+i (ia„/ Dans le cas que j'ai consider^, r = 1, et nous avons trouv6 On pent trouver aussi facilement I'integrale complete de I'equation / d d d \*\ . 22] Sur le theoreme de M. Brioschi 169" ou I'asterisque signifie qu'on doit prendre le produit complet de Taction de la forme lineaire agissant i — \ fois sur elle-meme. Ainai, par exemple, ('^ i + ^ zT''^'''^' "' [S + ^"^ 11+^^ (I) » d + a dc ' On trouvera sans difficulte que la valeur de cette int^grale est F+s,F, + s,^F, + ... + s,'-' Fi^^ , oil chaque F est une fonction exclusivement de s^, Sj, ... ,s„. Consequemment le t**"* coefficient d'un covariant quelconque de (a„a,, ...,a„){x,yy* pent etre mis sous cette forme, si Ton se sert de s„ pour exprimer la somme des o)''""** puissances des racines de «" + a,a;»- + ~ a;"-» + f-f-g*""' + ••• = 0. En efifet, en ecrivant — = », tout covariant de degre arbitraire v apparte- nant a ce quantic sera de la forme [Mo, (m«, Mi$S, 1), (Mo. «i . Wal*, l)*, ("o. "i, "2. Mj"5», 1)'. • ••] (.'"< !/)', oh, en general, dug rf", dt/g v„ ^tant une fonction exclusivement de (o,ii; Sj, s,, ...,«„ du poids « + 1. J'ajoute encore cette observation que tout differentiant (c'est-^-dire sous- invariant ou seminvariant) d'un systeme de i quantics des degr^s m, fi, ... , M sera fonction exclusivement de s^, a,, .... Sm', a-,, a,, ..., a-^, ..., 82,83, ...,Sx et de I — 1 fonctions lin^aires ind*?pendantes de la forme Isi + Xo-i + . . . + LSi , soumises a la condition que l + \+...+L = 0. Je ne sais s'il vaut la peine de dire, comme conclusion, qu'en combinant le theoreme de M. Brioschi avec le mien sur les puissances (avec astdrisque) on trouve, pour I'equation (ou le i est sans asterisque), I'int^grale partielle 4> = F+F,s, + F,s,' + ...+ Fi^, sr\ oil chaque F est une fonction arbitraire de s,.j.,, «i+s, ..., s„. Ed effet, cette expression est I'integrale complete du systfeme form^ par I'equation supposee conjointe avec les Equations 168 Sur le tMoreme de M. Brioschi [22 On voit aussi facilement que I'int^grale de est ^ - JJ, + UtSr + U^Sr' + ... + Ur-i8r*-\ oil chaque U est une fonction arbitraire de Si, s„ ... , »,_i, «r+ii ... s». On peut former un nombre infini de syst^mes construits au moyen des op^rateurs (flo 3 — H ••• ) dont on connaitra d'avance les int^grales ; ainsi, par exemple, le systfeme de r Equations aura pour int^grale complfete = [/o + «,£/,+«j''I/j+...+Sj*-'Cr,_,, oil chaque U repr^sente une fonction arbitraire de (sjSjSj ... Sji-iSj,- ... s„), en omettant celles des quantit^s s,, S3, ..., Sji_i dont les sous-indices excfedent n. Pour indiquer le moyen de justifier ces ^nonc^s, prenons comme exemple le cas des Equations simultan^es (ooSai + . . . + a„_i Sanf (f) = 0, ou Ei'tf) = 0, (ucBtti + . . . + a„_a5a„) = 0, ou E^tf) = 0, (aoSa,-!- ... +a„_sSa„) ^ = 0, ou Es = 0. On trouvera facilement qu'en general ^i' = ^V— 2£'V^a + -^j. de sorte que le syst^me donn^ ^quivaut au systfeme EV = 0, E, = 0. Pour que ces equations soient satisfaites s^parement, il faut et il suffit que soit respectivement de la forme F («aSjS4 . . . S„) + SiFi (S2S3S4 • • • «n) + «i'-f2 (SsSjSi • ■ • ««), G (S1S3S4 . . . S„), H (SxSaSi . . . S„). Cons^quemment, afin que les trois Equations soient toutes satisfaites simultan^ment, la condition suflSsante et n^cessaire sera que = 0, (aoSaj+... + an-iBa„y - o), de sorte que M' + r' = ^^(a'-^o-5) + §^(/3'-^/9-5) a — p /3— a = (a + fi)p-a^-Ap-B=f-Ap-B. On a aussi uv = 0 et cons^quetnment (u + vy=u'' + v'' = (u — v)\ Done (a; — r,) (« - r,) = (a; - 1))» - (« + v)" = a,-' - 2px + Ap + B, {x - rj) (a; - r^) = (a; -p)''-(M -vy = a^- 2px + Ap + B. Or consid^rons le cas g^nc^ral d'une Equation monothetique du degr^ n en matrices de I'ordre w. Cette Equation (que j'6crirai fx = 0), en vertu de ce que j'ai nomm^ la seconde loi de mouvement alg^brique (c'est-a-dire la formule .„ _ y (m - h) (m - c)...{m-l) {a — b){a — c) ...{a-l) ^ oi a,h,c,...,l sent les racines latentes de la matrice m), aura n" racines qu'on peut repr^senter par les symboles composes '*l > '*2 1 • • • > ^01 ) oil chaque r parcourt les valeurs 1, 2, 3 n. En refl^chissant sur la manifere de d^montrer le principe de Harriot, on arrivera facilement a la conclusion suivante : en prenaut une combinaison quelconque de n symboles r^.r^ r„, de telle mani^re que chaque r parcoure toutes ses n valeurs, R^, R^, ...,Rn, on aura /x = {x- J?,) (x -R,)...{x- A.). Ainsi on arrive au theoreme suivant : Toute fonction monothetique rationnelle et entiere de x du degre n en matrices de I'ordre a petit etre reprhentde de (1 . 2 . 3 ... n)"~' manieres differentes comme un prodxdt de n facteurs lineaires dont chacun sera la difference entre x et une des racines de la fonction donnee. Telle est la loi de Harriot, ^tendue au cas des quantit^s multiirration- nelles. Dans le cas de I'Algfebre ordinaire, w = 1, et le nombre des decompositions de/x en facteurs, selon la formule, devient unique, comme il doit etre. De mSrae, pour les quaternions, le nombre des decompositions d'une ' fonction monothetique du degr6 n en facteurs lineaires sera im. Par 41 I 23F] relative aiix dquations algebriques 171 exemple, si w = 3, les racines de fx peuvent etre exprimees par les neuf symboles 0.0 0.1 0.2 1.0 1.1 1.2 I 2.0 2.1 2.2 La fonction (comme on le demontrera facilement) pent §tre mise sous la forme a; — 0 . 0 multipliee par une fonction quadratique dont les racines seront des racines de fx, et consequemment, par raison de sym^trie, seront les quatre racines 1.1 1.2, 2.1 2.2; done la fonction quadratique dont j'ai parl^ sera ^gale h, (a;-l.l)(a;-2.2) eti (a;-1.2)(a;-2.1). Ainsi il y aura deux decompositions de fx qui correspondent aux deux diagonalea 0 . 0, 1 . 1, 2 . 2 ; 0. 0, 1 . 2, 2 . 1, et de meme il y aura des decom- positions qui r^pondent aux diagonalea 0.1,1.2, 2.0; 0.1,1.0, 2.2; 0.2, 1 . 0, 2 . 1 ; 0.2, 1 . 1, 2 . 0, de sorte que le nombre total est 6gal &, 1 . 2 . 3. De mSme, quand fx est monothetique et matrice da troisieme ordre, on peut prendre les diagonales d'un cube. Par exemple, les racines de I'^quation monothetique du second degre en matrices du troisieme ordre peuvent etre representees par 0.0.0 0.0.1 0.1.0 0.1.1 1.1.1 1.1.0 1.0.1 1.0.0 et Ton aura les quatre decompositions {x-0.(\.O^x-\.\.\); {x-0.0.\\x-\.\.Qi); (a; -0.1. 05a; -1.0.1); {x -0 A A\x-\ .0 .0); et de m^me, en general, pour le degr^ n, le nombre des diagonales (en se servant de ce mot dans le sens analytique, bien entendu) sera (1.2. 3. ..«)». C'est ainsi qu'on trouve I'expression g^nerale que j'ai donn^e (ttti)""' pour le nombre des decompositions quand le degre est n et que I'ordre des matrices est 10. En multipliant ensemble toutes les Equations de decomposition, et en nommant v chacune des n" racines, on parvient k 1' equation ^(^x- «)'("-■>'•-' = ifxy"" ; done, quoiqu'on ne pui-sse pas en general conclure que, si X' = F* (X et Y 172 Sur une extension de la loi de Harriot [23 6tant des matrices), X est n^essaireraent ^gal k Y,'\\ y & toute raison de croire qu'on pourra d^montrer que, dans le cas actuel, on aura ,r(a;-t;) = (/x)»— . Ainsi la regie de Harriot se reproduira de nouveau sous la forme trfes peu modifi^e qu'un polyn6me (monothdtique) en x (dlev^ a une puissance convenable) est ^gal au produit des differences eutre x et toutes les racines en succession de ce polyndme. On aura remarqu^, dans ce qui precede, qu'en appliquant la seconde des trois lois du mouvement alg^brique aux Equations monoth^tiques, on a trouv6 que le nombre des racines est n", et cons^quemment est w' dans le cas des quaternions, tandis que le nombre des racines pour la classe des Equations en quaternions unilatdrales (h laquelle les formes monoth^tiques appartiennent) est en g^n^ral n^ — n^ + n {voir le num^ro d'avril 1884 du London and Edinburgh Phil. Mag.), de sorte qu'il y a une Elimination n(n — 1)' de racines en passant du cas general au cas particulier. II reste k examiner s'il n'est pas possible d etendre la loi de Harriot aux Equations unilat^rales polyth^tiques. C'est ce que je vais Etudier, mais sans cela, et en me bornant au cas monoth^tique, il me semble qu'en attribuant aux elements des matrices des valeurs entiferes (simples ou complexes), comme le fait M. le professeur Lipschitz pour les quaternions, on voit s'ouvrir un nouveau champ immense de recherches arithm^tiques fondles sur la loi fondamentale de Harriot gen^ralis^e de la mani^re indiqude dans ce qui pr^cfede. 24. SUR LEs Equations monoth^tiques. [Gomptes Rendus, xcix. (1884), pp. 13 — 15.] Dans une Note pr^cddente sur une extension de la loi de Harriot, j'ai eu occasion de consid^rer les Equations dites monothetiques dont tous les coeffi- cients sont des fouctions d'une seule matrice. Or il y a une circonstance tres int^ressante et importante relative aux Equations de cette forme qu'il est essential de faire connaitre ; car, k defaut d'une telle explication, le lecteur de la Note cit^e pourrait facilement ^tre induit dans une erreur tres grave. Voici en quoi consiste I'addition k faire. Supposons que tous les coefficients d'une equation donn^e soient des fonctioDS d'une seule matrice m. En appelant x I'inconnue, on pent r&oudre r^quation en regardant x comme fonction de m, et Ton trouvera ainsi n" racines, en supposant que n soit le degr^ de I'^quation et (o I'ordre de m. Ces racines seront parfaitement d6termin6es: mais on n'a nullement le droit de supposer qu'il n'y a pas d'autres racines qui ne sont pas des fonctions de m, qu'on peut nommer racines aberrantes, et un exemple, des plus simples qu'on puisse imaginer, suffira a d^montrer que de telles racines, en effet, existent ; je me servirai, pour cet objet, de I'dquation en quaternions (ou matrices binaires) se' — px=0. En effet, on connait d^jk, a priori, la possibility de I'existence des racines aberrantes, car I'&juation en matrices a^ + q = 0, quand q est une matrice / . f^ ^ ^ scalar I comme si, par exemple, q^iO q 0 I, possede, on le sait, bien des \ [O 0 q/ racines qui ne sont pas scalars et consequemment ne sont pas des fonctions de q, et, de plus, ces racines contiennent des constantes arbitraires. Comme ou va le voir, c'est aussi le cas pour I'^quation a? —px = 0, qui possfede une seule constante. Si Ton veut trouver ses racines normales (ou non aberrantes), on n'a qu'k r^oudre cette equation comme une Equation ordinaire, et Ton trouve ainsi 174 Sur les ^qtiations monolMtiqiiea [24 En nommant r et s les racines latentes de p, on obtient par ma formule d'interpolation (pour ainsi dire), r^cemment cit^e par M. Weyr, 1 / .p-s ^p-r \ IT i ti ^ s\ S i Ti ~~ 7* I c'est-Jl-dire a; = 0, p, —^ , — — , et il n'y a pas d'autres racines de ^ r — 8 8 — r ■' ^ ce caractfere. Mais sortons de cette restriction arbitraire (produit de la paresse de I'esprit humain, qui se fatigue enfin en voyant sans cesse se reproduire des horizons nouveaux et inattendus), et posons hardiment a B a b 70 ^ c a oil a, /9, 7, 8 sont les quantit^s a determiner. Puisqu'on fait abstraction des solutions x = 0, x=p, on sent, en vertu de la troisieme lot du mouvement algebrique, que x et x—p auront chacun un degr^ de nullity (car leur produit possede deux degr^s) ; ainsi, si a + S = 0, on aura a!'=0, done aussi px = 0, et p sera aussi une matrice vide, c'est-k-dire qu'on aura ad — he — 0. La solution pour ce cas (dont, dans ce qui suit, je veux faire abstraction) 8er» , {ac — a? x = X\ , (a' — ac \ ^tant arbitraire. Dans tout autre cas, en ^galant la raison du second au troisifeme membre de a? avec la meme pour px, on trouve sans difficulte que x sera de la forme -\{d-r) \b fiC —fi{a — r) oil r et « sont les racines latentes de p, c'est-^-dire les racines de I'^quation i^-{a + d)r + ad-bc = 0. Alors, en calculant a^ et px, et en les ^galant terrae a terme, on obtient les quatre Equations suivantes : \(d- r)' +fibc =bc-a{d- r), b [\ {d - r) + n (a - r)]= - br, c [\ (d — r) + /x (a — r)] = — cr, \bc + /M(a — ry =bc — d(a — r). 24] &ur les equations monothetiques 175 En dcartant le cas special pour lequel 6 = 0 et c = 0, on voit (et c'est M. Franklin, de Baltimore, qui le premier s'est aper9u de cette conclusion capitale) que toutes ces Equations seront satisfaites avec la seule supposition \ (d - r) + /* (a - r) + r = 0, de sorte qu'une constante reste parfaitement libre dans la solution aberrante de I'equation iv'—px = 0. Dans le cas ou p = . , on trouvera facilemeut les deux solutions deter- ■^ 0 a minees a 0 , 0 0 ^=0 0 '' ^ = 0 d- Dans ses Lectures sur les quaternions, Hamilton n'a pas mis le doigt sur les cas v^ritablement singuliers des Equations quadratiques unilat^rales. La condition de singularite, c'est-^-dire de la presence de I'un ou de I'autre des cas oil une ou plusieurs des trois paires de racines de I'equation pa^+qx+r=Q disparaissent ou deviennent indetermiu6es (c'est-a-dire affect^es de constantes arbitraires), peut se r^sumer dans la seule Equation 1=0, oil I est Tin variant quartique temaire quadratique (en u, v, w) qui exprime le determinant d'une matrice up-\-vq-\- wr. 25. SUR L':6QUATI0N en matrices px^xq. [Comptes Rendua, xcix. (1884), pp. 67—71 ; 115, 116.] SoiENT p et q deux matrices de I'ordre a>. Pour r^soudre I'^quation px = xq, on obtiendra m^ Equations homogfenes lin^aires entre les w" elements de I'inconnue x et les ^l^ments de p et de q, de sorte que, afin que I'^quation donn^e soit r&oluble, les dl^ments de p et de q doivent etre lies ensemble par une et une seule equation. Mais, si Veqiuition identique en p est dcrite sous la forme p" + Bp"-' + Gp"-^ + . . . + i = 0, on aura apparemment, en vertu de I'^quation p = xqx~^, x(fx-^ + Exq^-^x-' + Cxq-'-'x-^ + ...+L = 0 ou bien ^ + Bq"~^+ Cq-^^ + ... + L = 0; done les m racines de q seront identiques avec celles de p et, au lieu d'une seule Equation, on aura en apparence (au mains) a equations entre les i\i- ments dep et de q. Pour faire disparaitre ce paradoxe, il n'y a qu'une seule supposition k faire : c'est que x, sous les suppositions faites, devient une matrice vide, car alors x~^ n'a plus une existence actuelle, et I'^quation p=xqx~^ n'aura pas lieu ; c'est ce qu'on va voir arriver dans le cos gdndral, ou px = xq. Pour fixer les idees, supposons « = 1 et faisons p: a b c d 1 = a B \ \ X a I \ V IT En 6galant px k xq, on obtient les quatre Equations simultandes et homo- g^nes entre X, ^, v, tr suivantes : (a - a) \ + c/i - /Sv + Ott = 0, iX + (d - a) /i + 0 J/ - /Stt = 0, -yX + Ofj, + {a-S)v+ C7r = 0, 0X + yfi + bv + {d - B) -TT = 0, 25] Sur V Equation en matrices px = xq 177 et consequemment on aura* 6»c» + ^-'f - 2hc^-f - 2abcd - 2a^yB + (be + ^y) (a + d}(a + B) - be {oi' + B') - ^y (a' + d') + aS (a' + #) + ad (a= + 8') + 2adaS + a^d^ + a=8' - (a + d) (a + S) {ad + aB) = 0, ou, en dcrivant a + d= B, ad—bc=D, a + 8 = C, aS — ^y = F, (^D-Ff + iB- C){BF- CD) = 0; c'est-^-dire, si R est le resultant de X- - Bx + D, X''-Cx + F, R = 0 sera la condition g^nerale de la possibility de satisfaire k I'equation px = xq. II est facile de faire voir que ce r^sultat peut etre etendu au cas general lOU p et q sont des matrices de I'ordre w : on n'a qu'a demontrer que si une ies racines latentes de p est egale k une de q, I'equation px = xq est resoluble; let de plus, sans que cette condition soit satisfaite, I'equation est irr^soluble. Solent done X,, X,, ..., X„ Ies racines latentes de p et /Xj, (i^, ..., /t„ de q et Bupposons que Xj = /^ , alors {p-\-^)x = x{q- ixi), Bt Ton peut satisfaire k cette equation en ecrivant x={p-X,){p-\,)...(p-\^){q-^l^){q-|M3)...{q-^i„). Consequemment, si Ies racines latentes de p et de g^ sont Ies racines des deux formes alg^briques X" + BX"'^ + ... +L, X" -h GX"-^ + ... + M, quand R (le resultant de ces deux formes) s'evanouit, le resultant des oj" Equations homogenes lineaires obtenues en ^galant px = xq s'^vanouira ; mais R est indecomposable et du mSme degr6 (&>') que ce dernier resultant dans Ies Elements de p et q. Consequemment Ies deux resultants (a un facteur num^rique pres) sont identiques : ce qui d^montre que la condition R = 0 est non pas seulement n^cessaire, mais de plus suffisante afin que px = xq soit resoluble. Pour ce qui regarde la valeur de x, posons x= UV, oil U=(p-\)(p-X,)...{p-X„); V = {q - fj^) {q - fi^) ... {q - fj,J, le seul fait que x contient U comme facteur ou que x contient Fcomme facteur suffit a constater que x n'est pas seulement vide, mais de plus possede au moins ta — 1 degrds de nullity, c'est-a-dire que tous ses determinants mineurs du second ordre sont des z^ros. Cela est la consequence d'un th^oreme que j'ai d^montr^ dans le Johns Hopkins Circular^ relatif au degr6 de nullity des combinaisons des facteurs latents d'une matrice, dont le th^orfeme relatif k I'equation dite identique de Cayley ou de Hamilton n'est qu'un cas particulier, ou pour mieux dire le cas extreme; seulement il faut y ajouter un theoreme qui fait partie de ma troisieme loi de mouvement algebrique, c'est-a-dire que le degr6 de nullity d'un facteur ne peut jamais exc^der le degre de nullity du produit auquel il appartient. [' The expressions for p, q in line 7 from the bottom of p. 176 should be interchanged; in the last line of p. 176, for +yfi read -7/1.] [t p. 134 above.] 8. IV. 12 178 Sur r^qitation en matrices px = xq [25 Nous avoDS douc compl^tement r^solu le paradoxe qui ^tait k expliquer. Mais, sur-le-champ, une nouvelle contradiction surgit, car il semble que nous avons d^montr^ que, dans tout cas sans exception, si px = xq, x est ndcessaire- ment une matrice vide, ce qui est evidemment faux, car on sait bien que, si, CD dtant de I'ordre de p et de q, q=^ yfi\)p, alors, afin que I'^quation px = xq soit resoluble, il n'est jamais n^cessaire que x soit vide. Ainsi, par exemple, pour les matrices binaires, I'equation qx = xq est satisfaite quand x est une fonction quelconque de q, et I'equation qx——xq est resoluble, pourvu que g' soit scalar, en imposant deux conditions (dont une que son carre soit scalar) sur x. Pour lever cette contradiction, revenoas au cas oil w = 2 et aux Equations fondamentales (a — a) \ + c/x — /9i' = 0, 6\ + (d-a)/x-/S7r = 0, — 7\ + (a — S) 1/ + CTT = 0, - 7/i + ti/ + (d - S) TT = 0. Certes, si ces Equations donnent des valours d^termiii^es aux rapports X, /t, V, v, le raisonnement precedent rend certain que x doit etre vide, c'est-k-dire que Xir — fiv = 0, mais cette conclusion devient fausse aussitdt que p et q sont pris tels que ces rapports deviennent ind^termin^s, ce qui arrive quand tous les premiers determinants mineurs de la matrice (a -a) c -/3 0 b (d-a) 0 -/3 -7 0 (a-B) c 0 -ry b (d-B) s'^vanouissent simultan^ment. Dans ce cas, quoique la solution gendrale qui donne x vide tienne bon, rien n'empeche qu'il n'existe d'autres valeurs de x, c'est-k-dire de ^ , pour lequels cela n'est pas vrai. La matrice ecrite en haut doit posseder et possede, en effet, la propri^te remarquable que, en supprimant une ligne horizontale quelconque et en nommant A, B, C, D les quatre determinants mineurs de la matrice rect- angulaire qui survient, affectes de signes convenables, la quantity AD — BG contiendra le determinant complet comme facteur. II sera peut-etre utile, avant de conclure, de donner un exemple d'un genre nouveau de subsistance de I'equation pa; = a;g avec une valeur finie du determinant de a;. Faisons done a - 8 = 0, d-a = 0, be- fiy = 0, on aura (a — c?) \ + c/t — /Sv = 0, 6\ - /Stt = 0, -yX + CTr = 0, — 7/x + Jv + (d — a) TT = 0, i 25] Sur r equation en matrices px = xq 179 Equations qui n'equivalent qu'k deux, 6\ - /Stt = 0, {a-d)\ + (cfj, - ^v) = 0, et le determinant de x, c'est-a-dire Xir — /xp, aura en g^n^ral une valeur finie. Dans la demifere Note (insdr^e dans les Comptes rendus*) qui roule sur r^quation en matrices binaires or' —px = 0, j'ai remarqu^ qu'en addition aux solutions normales p-s p-r x= — 6 degr^s de nullity, c'est-k-dire tons ses determinants mineurs de I'ordre ^ + 1 s'dva- nouiront. Mais on sait, pour le cas oh 6 = a> (et Ton a toute raison de croire pour le cas oil ^ a une valeur quelconque au-dessus de I'unit^), qu'il existe pour des valeurs sp^ciales de /) et de q des solutions singulieres de I'^qua- tion px=:xq, lesquelles (comme dans le cas de I'equation de Riccati) sont bien autrement int^ressantes et beaucoup plus importantes que la solution g^n^rale. On remarquera que, quand d — o>, la solution g^n^rale disparait, tandis que les solutions singulieres pour des valeurs particulieres de p et de q, ayant tontes les racines latentes de I'un identiques avec celles de I'autre, forment la base de la presentation des matrices sous la forme de quaternions, nonions, etc. n i 26. SUR LA SOLUTION DU CAS LE PLUS GENERAL DES EQUA- TIONS LINEAIRES EN QUANTITES BINAIRES, C'EST-A- DIRE EN QUATERNIONS OU EN MATRICES DU SECOND ORDRE. [Comptes Rendus, xcix. (1884), pp. 117, 118.] SoiENT p, q deux matrices d'un ordre donnd et servons-nous du symbole p( )q pour signifier I'operateur, lequel, applique a nne autre matrice x du mSme ordre, donne pscq. Alors, si Ton pose pA )qi+pA )q2+ ■■■+Pn{ )qn = 4>, x sera une matrice dont chaque element sera une fonction lin^aire des ^l^ments de x ; cons^uemment, en supposant que les matrices p, q sont de I'ordre co, on parvient ainsi k une matrice de I'ordre to', et cons^quemment (f> sera asaujetti a une Equation identique de I'ordre to'^; disons F=0. Je vais donner la valeur de F pour le cas ou w = 2, c'est-a-dire oii F sera une fonction du quatri^me degrd Suppo.sons que P et P' sont deux quantics du second ordre dans les deux systemesde variables a;i,a:2, ...,a;„; fi, fa, ...,fn contragredients. Alors, si Ton represente par P' ce que devient P' quand on ^crit Sx,, 8«,, ..., S^ au lieu de fi, fa fn. (-P*)'- P* sera un invariant du systfeme donn^ pour toute valeur de i. Considerons le cas ou P = ax' + bxy + cy- et P' = af + /Jf?; + 77;^. Dans ce cas, on trouvera que \ \jJ^Y ^ ~ ^{^ • -^ W sera identique avec le r^sultat de ax' + bxy + cy'', '^a? — ^xy + ay^, de sorte qu'on peut le nommer le contra- risultant des formes (a, b, c), (a, /3, 7). Je nommerai done, en g^n^ral, I'invariant ^ [(P')' P* — 4 (P' . P)'] le quasi contra-rdsultant des deux formes P, P quand elles contiennent un nombre quelconque de variables. Or, en revenant k I'expression , nommons P le determinant de U^pi + U^p.i + . .. + UnPn + 4>-'U et Q le determinant de "igi + Ma?'^ + . . . + u„qn - V, 182 Le cos le plus gSn^ral des equations lin^aires [26 oh. ^, pour le moment, est traits comme une quantity ordinaire. J'ai trouv^ que le quasi contra-resultant de P, Q, quand <^ appartient k des matrices du second ordre (lequel sera une fonction biquadratique de ^), dgal^ k zero, est r^quation identique cherchde en ^. II est probable, mais je n'en suis pas encore absolument convaincu, qu'une m^thode analogue donnera I'^quation identique de <^ pour des matrices d'un ordre quelconque. Si Ton suppose que les p et les q sont des quaternions, rien ne change avec I'exception que P et Q seront d^finis comme dtant les modules (les tensors carr^s) au lieu d'etre les determinants de ^v + 2p«, — v + 'S.qu respectivement. Connaissant ainsi I'^quation identique de , on a I'dquation connue * + B^' + Cif}' + J) + E = 0, et, consequemment, en exceptant toujours le cas oil E=0 (dans lequel cas I'equation devient on impossible ou indetermin^e), on trouve Par exemple, si I'equation donn^e est pxq + rxs = T, T = pTq + rTs, (fy'T = p^Tq^ + prTsq + rpTqs + r'Ts', 4>^T = fTf -irp^Tsq^ + prpTqsq + rp^Tq^s + pi^'Ts'^q + rprTsqs + r'pTqs' + r»2V, et, ^ventuellement, en ne se servant que des coeflScients qui entreat dans les fonctions P et Q par le moyen de formules connues, on r^duit x a une somme de multiples de termes de la forme pT, rT, prT; pTq, rTq, prTq; pTqs, rTqs, prTqs, et ainsi en g^n^ral. Done le problfeme de la resolution des Equations lin^aires est completement r^solu ; seulement il reste k traiter en detail le cas singulier oil la matrice appartenant kest vide. 27. SUR LES DEUX METHODES, CELLE DE HAMILTON ET CELLE DE L'AUTEUR, POUR RESOUDRE L':6qUATI0N LIN^AIRE EN QUATERNIONS. [Comptes Rendus, xcix. (1884), pp. 473—476, 502—505.] Un celebre quaternioniste m'ayant demande de lui expliquer la portee de ma solution de I'^quation lineaire en matrices sur la solution du meme probleme en quaternions, il me semble desirable de donner explicitement le moyen de passer d'une solution a I'autre. Prealablement, il sera bon cependant de remarquer que, faute d'un examen suffisamment attentif de la forme du resultat obtenu ou plutot indique par Hamilton {Lectures on Quaternions, pp. 559 — 561), on pourrait attribuer a sa solution une pro- pridt^ qu'elle ne poss^de pas, celle de fournir le moyen de trouver la solution de I'^quation lineaire en quaternions sous une forme reduite semblable a celle que fournit ma m^thode : mais, en effet, I'examen d'un seul terme de m (voir au bas de la page 561), par exemple Srj7'^, suffit k montrer que le d^nominateur to de Hamilton est du douzieme degr^ dans lea Elements des quaternions (b et a) de son Equation 'S.bqa = c (p. 559), tandis que le degr6 pour la forme rdduite n'est que huit. II s'ensuit que le num^rateur (si Ton avait la patience de le deduire des formules de Hamilton), aussi bien que le denominateur obtenu par ce moyen, serait affectd d'un facteur Stranger k la question, du quatrieme degr^, dans les elements nommes. J'ajoute qu'il est parfaitement possible de donner la valeur de x dans I'equation ^pxp = T comme fonction seulement des p et p' et des coefficients des deux formes assoaides sans aucune irrationnalite. Car le determinant du nivellateur '2p{ )p', disons N, etant obtenu sous la forme il^ + \/(^4). le determinant du nivellateur -10 NO ^P()P'+ () 0-1 ON (disons FN) sera aussi exprim^ sous une forme semblable k celle-la, disons *a + s/{ ,). 184 Resolution de Equation lin^aire en quaternions [27 Or, au lieu de I'^quation identique FN = 0, on peut se servir d'un multiple quelconque de cette Equation pour obtenir I'inverse de N comme fonction de puissances positives de JV^ Ainsi Ton peut, dans ce but, se servir de I'equation *j' — 4>4* = 0, au lieu de FN= 0, et, avec I'aide de cette Equation, on obtiendra x exprim^ en fonction des p et p' et de fonctions rationnelles des coefficients des deux formes associ^es ; mais alors, au lieu d'etre obtenu sous sa forme la plus simple, son num^rateur et son d^nominateur con- tiendront un facteur commun qui sera une fonction du huitifeme degr^ des ^l^ments des p et des p'. Je passe k la regie pour traduire ma solution de I'equation en matrices ^pscp' = T en solution de cette m6me Equation quand les p, les p' et le T, au lieu d'etre matrices, sont donnas comme quaternions. Evidemment tout ce qui est n^cessaire, c'est de connaitre I'equation qui serait identique pour ^P( )p'; J6 v^'is donner la rfegle pour I'obtenir. Sous le signe S, je suppose compris p, q, r, ..., p, q', r' Ecrivons la forme symbolique [Nx + {p)y + (q)z + ...]', disons X; les co- efficients de xy, xz, ..., symboliquement ecrits, sont 2(p)N, 2iq)N, ...; a (p), (q), ... il /aut svhstituer Sp, Sq, ... ; le coefficient de y^ est (p)' auquel il faut substituer Tp'; finalement le coefficient de yz est (p)(q), auquel il faut substituer S(Vp. Vq)*. De meme, on construit et Ton interprete la forme [-x' + (p')y+(q') / + ...]' (disons X'). On calculef la valeur de J^'^X- — ^ {XXy. Ce rdsultat (une fonction du quatrieme degrd en N) (disons UN) sera une partie de la fonction qui doit Stre identiquement zero. Le reste de cette fonction (disons 64fl,iV) sera [■2S{VpVqVr) S{Vp'Vq'Vr')] N-lSpSp' S(VpVqVr) S (Vp Vq' Vr'), etjedisque niV+64n,iV = 0 sera I'equation identique en N, et servira pour trouver la valeur de x, c'est- ^-dire N-^T comme fonction du quaternion T, des quaternions p, q, .... p', q, ... et des symboles S, V, T; de plus la valeur ainsi obtenue sera x sous sa forme reduite. II y a encore une petite observation a ajouter k mes remarques sur la solution de Hamilton de I'equation ^bqa = c (Lectures, p. 559). II divise q en deux parties, le scalar w et le vecteur p. C'est cette derniere quantite (p) qu'il exprime sous la forme — ; alors 1^ ~ — ^CY/ t\ . de sorte que, k d^faut d'avoir recours k des reductions 2,0 (ao) [* See first note on p. 191 below.] [+ See p. 181 above and p. 202 below.] 27] Resolution cle T equation lineaire en quaternions 185 ulterieures, le d^nominateur de q contiendra, non sevdement le facteur etranger du quatrieme degre dans les elements des a et des b dont j'ai d^ja parld, mais encore le facteur etranger XS{ab). On remarquera que, dans cette solution, on aura des combinaisons des b avec des a et des fonctions quatemionistiques de ces combinaisons, tandis que, dans la solution infiniment plus simple que je donne du probleme, il ne se trouve nulle part des melanges de cette nature, mais seulement des fonctions quatemionistiques de combinaisons des a entre eux-memes et des b entre eux-memes. Le vice fondameutal de la m^thode de Hamilton, c'est la reduction du problfeme donne a un autre, ou, au lieu de q, il n'entre que sa partie vectorielle. Neanmoins le travail de Hamilton (quoique sa raison d'etre ne subsiste plus) ra^ritera toujours d'etre regard^ comme un monument du g^nie de son grand et admirable auteur. C'est la, pour la premiere fois dans I'histoire des Math^matiques, qu'on rencontre la conception de i'^quation identique (voir Lectures, pp. 566, 567) qui est la base de tout ce qu'on a fait depuis et de tout ce qui reste k faire dans revolution de la Science vivante et remuante de la quantity multiple, c'est-a-dire I'Algebre universelle, nee a peu pres 250 ans apres I'organisation definitive de sa sceur ain^e I'Arithmetique universelle, dans le Memoire de M. Cayley sur les matrice-s, daus les Philosophical Transactions, vol. 148. Dans une Note pr^c^dente, on a vu que dans la rioiivelle et seule bonne m^thode pour r^soudre, par rapport a x, I'equation en quaternions pxp' + qxq' -(- rxr' + sxs' + ...=V, on fait trois operations. La premiere, k laquelle on peut donner le nom de nivellation, consiste a trouver le nivellant, c'est-a-dire le determinant de la matrice du quatrieme ordre appartenant k un nivellateur donne du second ordre. La seconde, qu'on peut appeler deduction, consiste a obtenir I'dquation identique, k laquelle un nivellateur correspond au moyen d'un autre nivel- lateur qu'on obtient du nivellateur donn6 en y adjoignant un couple de plus de la forme— i\r( )S«, ou,ce qui revientau mdme.le couple v'(—-^)( )V(— -^). ou N est consid^rd comme un scalar. Finalement, on arrive h. la demi^re operation, que je nommerai substitution et reduction, et qui consiste k sub- stituer a I'inverse du nivellateur sa valeur en fonction rationnelle du troisieme ordre de lui-m6me, puis k faire des reductions dont je parlerai tout k I'heure. Au moyen de ces operations, on arrive a la valeur de I'inconnue de I'equation sous sa forme reduite la plus simple qu'elle puisse prendre. Pour obtenir la forme de I'dquation identique, voici ce que j'ai trouv^ en appliquant la methode indiqu^e dans la Note pr^cedente. 186 absolution de Viquation lin4aire en quaternions [27 Pour plus de simplicite, je me sers de la notation suivante, qui s'applique k des lettres quelconques, accentu^es ou non, repr^sentant des quaternions. Je pose Sp = ip), Tf=p,, S(VpVq) = (j>q), S{VpVqVr) = (pqr). Alors, en ^crivant P( )p' + q( )q' + r{ )r' + s( )s' + ...=N, on aura N* - 4S (p)(p')N' + 2 [HpYp, + 4 {pjp, - 2p,p\] N" - S (4 {p){p')p^p\ + 8 [.ip){q'){n)-p'^ + ip'Kq}p'q -Pi] - ^{p){p')q.q\ + 4 [{q){p')p.q\ + (p) {q')p\q.] - 8pp'(qr) {q'r) + 8 \.<.P){q)iqr)p'r + (p'KqKq'r'Kpr)] + St (pqr}(p'qV)] N + t[p^'p\''-2p^p\.q^q\ + 4 [p^qi (p'q'y + p'2q\ (pqY] - ip^p'tipq ■ p'q + 4p,p\qr . qV + 8 [p, (qr)(pq)(p'r') + p\ (q'r'){pq)(pr)] + 8[pq.rs. p'r . qs +p'q' . r's' .pr.qs\-8 {p){p'){qrs){q'r's')] = 0, oh. le dernier terme de la partie fonctionnelle de I'^quation est le nivellant deK Quant a la substitution, si, dans I'dquation prdcedente on remplace N~^ F par la fraction NT - AN'T + BNr - Cr D tons les termes du numdrateur de cette fraction seront des multiples connus de la forme FTP', ou P est de I'une des formes suivantes : p' ; p''q, pqp, qp? ; P', pq; p; ..., et oil de meme P' a des types semblables avec des lettres accentuees. II ne reste plus qu a reduire chaque P a sa forme la plus simple, c'est-a-dire a I'exprimer comme fonction lindaire de ^,p, q,pq —qp, et de meme pour P'. Alors le num^rateur de a; ne contiendra plus que des termes dont les arguments seront- tous d'un des types suivants (je remplace la moiti^ de pq — qp par [pq]) : r, pV, Tp, pTp', pTq\ [pq-\v, np'q'l pnp'q^ ipqWp, [pqWlv'q'V, il faut y ajouter le type pqrTr'q'p', qui est d^jk sous sa forme la plus simple et n'exige aucune formule de reduction. • D est le determinant de la matrice qui appartient au nivellement N. Qnand D = 0, la solution de I'^quation Nx = T devient ou idiale (ce qui a lieu en g^n^ral), ou (ce qui a lieu pour des cas particuliers) actuelle, mais iuditermin^e. H 27] Resolution de T equation lin^aire en quaternions 187 Je n'entreprendrai pas pour le moment de calculer les coefficients de ces guments, mais j'indiquerai du moins les formules de reduction qui seules nt necessaires pour effectuer ce calcul. Ce travail, bien digne d'attirer I'attention de quelque jeune gdomfetre, peut tres probablement amener k des resultats qui, a I'aide d'une notation symbolique, pourront Stre presentds sous une forme d'une simplicity tout a fait inattendue et pour ainsi dire pro- videntielle. J'en ai eu I'experience pareille dans d'autres recherches du meme genre, dans la solution de certains cas d'^quations quaternionistiques du second degre. Voici toutes les formules de reduction dont on aura besoin : p' = 2(p)p-p„ p> = [4,{py-p,]p-2{p)p„ pq = [pq] + (p) ? + (3) p - ( pq), qp = - [pq] + {p)q + (q)p - (pq), p'q = 2 (p)[pq] + 2 (p)(q)p + (2p» -p,) q-2 ip)(pq), pqp = *(p)[pq] + [8 (p)(q) - 2 ipq)]p - [4 (py +p.]q- [2 (q) p. + 4 {p)(pq)] ; dans les formules on peut, au lieu de [pq], ecrire V(VpVq). Remarque. — Quand un nivellateur devient symetrique, c'est-a-dire quand p = p', q = q', ... , alors les deux formes associ^es coincident en une seule dont le nivellant devient un invariant orthogonal. Qu'il me soit permis, avant de conclure, d'ajouter encore une petite reflexion sur I'importance de la question traitde ici. Elle constitue, pour ainsi dire, un canal qui, comme celui de Panama, sert a unir deux grands oceans, celui de la theorie des invariants et celui des quantity complexes ou multiples : dans I'une de ces theories, en efifet, on considere Taction des substitutions sur elles-m^mes, et dans I'autre, leur action sur les formes; de plus, on voit que la theorie analytique dea quaternions, ^tant un cas particulier de celle des matrices, cesse d'exister comme une science ind^- pendante ; ainsi, de trois branches d'analyse autrefois regardees comme ^nt ind^pendantes, en voila une abolie ou absorbee, et les deux autres i^anies en une seule de substitution alg^brique. 28. SUE LA SOLUTION EXPLICITE DE L'^QUATION QUADRATIQUE DE HAMILTON EN QUATERNIONS OU EN MATRICES DU SECOND ORDRE. [Comptes Rendus, xcix. (1884), pp. 555—558, 621—631.] Hamilton, dans ses Lectures on quaternions (p. 632), a fourni un moyen de resoudre I'^quation (en quaternions ou en matrices binaires) de la forme of - 2px + q = 0; mais les circonstances les plus int^ressantes de la solution ne se font pas voir dans sa m^thode de traiter la question. Voici la maniere analytique directe que nous employons pour obteuir x sous sa forme explicite. On suppose a.- — 2Bx + D = 0 r^quation identique pour x, oh B et D sont des scalars k trouver. En combinant ces deux Equations en x, on obtient 2x=(p-B)-\q-D), et, en supposant que la forme associSe a [1], p, q, c'est-^-dire le determinant de X + /ji,p + vq, soit X- + 26X/i, + IcX-v + dfj? + 2eii,v -\-fv-, on aura* 4>{d- 2hB + &) x,^ - A>{e-bD - cB -^ BD)x,J^f - 2cD + D-- = Q. Cons6quemment, en ecrivant u=B — h,v = D—c, d-b^ = a, e-bc = ^, f-c'=y, et, en comparant cette Equation avec I'equation donnee, on voit qu'on pent " 6crire M^ + a = \, uv + ^= 2X(i(, + b), v- + y = 4X (v + c). De plus, puisque p^ — 2bp + d = 0, on aura _ {p — b + u)(q — c — v) _ _ {p — b + u)(q — c — v) ^~ 2(b'-d-u') ~ 2\ • [* The determinant of 2Bx^ -D- 2x,y + q being zero, if Xg is a latent root of x.] i 28] Solution de V equation quadratique de Hamilton 189 En ^liminant m, v entre les trois equations qui les lient avec h, c, a, yS, y, on trouvera I'equation bien remarquable e^(-«c-«d' .1 = 0, ou I est le discriminant de la forme associ^e donnde plus haut, c'est-^-dire 16c 1= b d e =d/+2bce-dd'-e^-fb-, c e f de sorte que la quantity exponentielle symbolique repr^sente une fonction cubique et donne lieu a une equation cubique en \. A chaque valeur de X correspondent les deux valeurs + i\/{\ —a) de w et k chaque valeur de u (autre que u = 0) correspondra la seule valeur „^ , {2\ + c)b-e , 2X + ^^ de V, u Quand m = 0, X = a = d — 6^ et I'equation »» - 4Xi; + 7 - 4\c = 0 a ses deux racines finies. Done, quand u = 0, il faut que prenne la forme ^, et i cette valeur de n (qu'on peut envisager comme deux valeurs de u r^unies en une) correspondront pour v les deux valeurs donn^es par I'equation quadratique ci-dessus. Ainsi Ton voit qu'en general x a trois paires de valeurs d^termin^es et qa'aucune de ces valeuis ne cesse d'etre actuelle et determinde que pour le seul cas ou I'une des trois valeurs de \ est 6gale a z^ro, c'est-k-dire ou /, Tin variant de \&pleine* forme associ^e h, (p, q), s'^vanouit. Cela revient k dire que / est le crit^rium de la normality de I'equation doDD^e. Si Ton regarde p et q comme des quaternions, on aura b=Vp, c=Vq, d = Tp\ e = SpSq-S{VpVq), f=Tq\ II est bien digne de remarque que 4/ est identique avec i^pq — qp^. On peut d^montrer que, si p et g sont des matrices d'un ordre quelconque, les racines de I'equation a? — 2px + 3 = 0 seront toujours (comme icl) associe'es en paires ; car, si Ton ^crit x + x, = 2p, on aura x^-2a:ip + q = 0, et consequemment, si p" — (obp"^^ + ... = 0 est I'equation identique connue en p et x"— a)Baf~^ + . . . = 0 I'equation identique k trouver en x, k chaque valeur * Noug avoDg diji d^fini la formt ataoeUe au corps p, q, r, .... Par la pleine forme, on peat sons-entendre oe que devient la forme aB8ocite quand on adjoint an corps one matrice nnitaire. 190 Solution de Tdqimtion [28 de 5 — 6 correspondra une valeur ^gale de b — B, c'est-k-dire que I'dquation pour trouver B sera de la forme F{B — b)* = 0. En se servant de I'^quation conjugu^e (c'est-k-dire en a;,) dont la somme des raciues sera ^videmment la meme que pour I'equation en x, on obtient immediatement, dans le cas oil p et q sont dii second ordre, par le moyen de la formula (p + b-v)iq-c-v) * 2k et de I'dquation en \, la valeur de 2a;*. Cette valeur sera 6[p + (28c — Sd) -^^1, de sorte que la valeur moyenne d'une racine de I'dquation a;" — 2px + q = 0 est j) (la valeur moyenne pour le cas ou p et g' sont scalars), augment^e de (2Be — Bg)I^, ou I^ doit avoir le signe qui le rend dgal k ^{pq — qp). De mSme on trouve 2a;' = 2j32a; - 6q, et ainsi la valeur moyenne de a;* sera 2p'-q + {iB,-2Sa)I^p, et Ton peut trouver successivement, par la meme m^thode, la valeur moyenne d'une puissance quelconque de x. Les details du calcul pr^c^dent, et encore d'autres propriet^s de I'equation en x, seront donnas prochainement dans le Quarterly mathematical Journal ou quelque autre recueil math^matique. Ici on n'a voulu que produire les rdsultats principaux obtenus par notre m^thode. L'^quation de Hamilton en quaternions ou en matrices binaires est celle que nous avons traitee dans une Note precedente. C'est I'dquation ar" + 2g'a! + r = 0. Nous avons trouv^ que la solution de cette Equation depend d'une equation cubique ordinaire en X, a chaque valeur de laquelle correspondent deux valours de x, et qu'elle est normale ou rt^guliere quand le dernier terme de cette Equation differe de zero. L'^quation est dite reguliere ou normale quand sa solution depend du nombre maximum de racines determinees, c'est- k-dire de trois paires de racines d^termint^es ; chaque paire est alors connue comme fonction de \, q, r et des parametres b, c, d, e,f qui dependent de q On aura 2x= -2 'P-'' + "^^^-^-''). H On retranche une Equation de I'autre, on substitue pour 2 — sa valeur tir^e de I'equation A cubique en X, et on 6orit pq -qp='2I', i\ 28] qiiadratique de Hamilton 191 et r et sont d^finis au moyen du determinant de u-\-vq-\- wr* qu'on a sup- pose etre mis sous la forme u- + 2lmv + 2ciiw + dv'' + 2evw + fw", d'oh b = Sq, c = Sr, d=Tq\ f=Tr^e = SqSr- S{Vq.Vr)*. Dans ce cas, on pent dire que la solution elle-meme est r^gulifere. En nommant / I'invariant de la forme ternaire, ^crite plus haut, c'est-a- dire en posant I = df+ 2bce - l^f-cFd - e^ nous avons trouv^ que I'^quation en \ pent etre mise sous la forme oii n = 2S, - Srf, c'est-^-dire qu'on aura 4V + (4c-4d)X» + (46e-4cd+c--/)X-L/=0. Ainsi, afin que la solution soit rdgulifere, il faut et il suffit que / difffere de z6ro+. De la il suit que, dans le cas d'une equation r^guli^re, deux x ne peuvent dtre egaux, h. moins qu'ils n'appartiennent k la meme paire ou bien que deux \ ne deviennent dgaux ; car x pent Itre exprimd comme une fonction lineaire de qr, q, r, 1, dans laquelle le coefficient de qr est — ^ . Done, si deux des x sont egaux sans que deux \ le soient, une Equation lineaire subsistera entre pq, p, q, 1, mais dans ce cas nous avons trouvd ailleurs que / = 0, et la solution cesse d'etre rdguli^re. Nous allons pour le moment nous borner au cas ou I'^quation est r^- guli^re, et cons^quemment nous n'aurons qu'^ consid^rer les cas oti il y a ^galit^ ou entre deux racines de \ ou bien entre deux valours de a; qui corre- spondent a la m^rae valeur de X. Si Ton suppose que deux valeurs de \ soient egales, il en rdsultera que deux des paires de valeurs de x deviendront identiques, de sorte qu'une seule condition suffira k r^duire le nombre des racines distinctes de 6 a 4, c'est-k- * Par an oubli trAs regrettable nous avons pris, dang une Note pr^c^ente, pour le coeflBcient de 2xi/ dans la forme associ^e & S(Vp Vq) an lien de sa vraie valenr, Sp Sq - S {VpVq), et de mtime ponr les autres coefficients des temies mixtes, de sorte que le calcol du determinant dn nivellateuT "Lp ( )p' dans la Note snr I'aofadvement de la solution de I'^quation lineaire en qnatemions est erron^ et a besoin d'etre fait de nouveau. t Consiquemment, quand I'^qnation est r^guli&re, ni 7 ni u ne pent devenir z<5ro ; car, dans I'an et I'antre de ces denz cas, 1=^; aussi, pour la mdme raison, r ne peut pas Stre une fonction deg. 192 Solution de Tiqiiation [28 dire que lea valeurs de x, qui, en general, sont de la forme m, m; n, n; p,p', deviendrorit de la forme m, vi'; n, n'; n, n. Au lieu de calculer directement le discriminant de I'^quation en X, qui donnera un r^t^ultat trfes compliqu^ nous allons montrer qu'on pent substituer le discriminant de la forme tr^s simple biquadratique c+2d l.h,'^,e.f)(r,8r. Mais pr^alablement il sera utile d'op^rer une transformation lindaire sur r^quation en \. Ecrivons \ = fi + d; I'i^quation en fi sera V + 4 (c + 2d)/x2 + [(c + Uf + 46e - /]/i + 26 (c + 2d)e - 6=/ - e' = 0. On voit done que le discriminant qu'on veut calculer est une fonction complete de 6, c + 2d, e, f. Nous avons trouv6 m'' = \ — d + 6", c'est-^-dire /a + fr*. On aura done 4M» + 4(c + 2d-3&')u* + [126* - 8 (c + 2d)}P + (c + 2df + (46e -/)]«» -[26^-6(c + 2ci) + e]"-». Dans r^quation donn^e, substituons x + e, oil e est un infinitesimal . e 0 . {scalar si Ton parle de quaternions ou repr^sentant la matrice si Ton 0 e parle de matrices) ; alors p sera augmente par e et g par 26j9, et ainsi (\ + /ip + vq) deviendra (X + e/ii) + (/i + 2ev)p + vq, de sorte qu'en designant le discriminant cherch^ par D, I'accroissement de D est nul quand \ et ft deviennent \ + e/i, jx + ev simultandment, c'est-k-dire quand la forme ternaire en u, V, w devient m" + 2 (6 + e) Mt) + 2 (c + 2e6) mv + {d + 2eh) v' + (2e + 26C + 4ed) m) + (f + 4-ee) w'. Done [aBb + 2bS, + 2cSa + (c + 2d) 3, + 4e8/]Z) = 0. Ecrivons c+2d= Sm. On sait que B est une fonction complete de b, m, e,f, de sorte que, par rapport k D (comme op^rande), Bc+Ba = Bm', aiusi, en I ^crivant 1 = a, on aura (aBb + 2bB,n + 2mB, + 4eSy) D = 0. D sera done ou un invariant ou un sous-invariant de la forme biqua- dratique (a, b, m, e, f). * u sera la partie scalar de x si I'^quation est donn^e sons la forme quaternionique, ou bien la moiti6 de la somme du premier et du quatrieme element de x ei I'^quation est donnce entre des matrices. Hamilton a trouv6 I'equation ^quivalente k celle donnfie pour « dans le texte; mais, dans sa formule, les coefficients sont exprim^s sous une forme compliqu^e et assez difficile k d^brouiller. 28] quadratique de Hamilton 193 Mais, en faisant attention a I'equation en fi, on voit que D sera de I'ordre 6 dans les coefficients et du poids 12; il est done un invariant et une fonction lineaire de s' et <' (ou s et t sont les deux invariants irr^ductibles) de la forme biquadratique. En nommant A le discriminant de cette forme, on a A = s'-27<^ dont une partie sera /' — 27b*/' ; mais on voit, par I'examen de I'equation en /*, qu'une partie de D sera et, consequemment, i) = — ^ A. II s'ensuit que la condition n^cessaire et suffisante pour I'^galite de deux des racines de I'equation donnde avec deux autres est tout simplement A = 0, comme nous I'avons d^ja ^nonce. Cherchons la condition pour laquelle les trois paires coincideront toutes dans une seule paire ; alors les trois racines de /jl deviennent toutes ^gales, et Ton a noa seulement A = 0, mais encore (1 2m') — (9m' + 46e — /) = 0, e'est-a-dire /— 46e + 3w' = 0 ou « = 0. Done les conditions necessaires et suffisantes, pour qu'il n'y ait que deux racines distinctes chacune, prises trois fois dans la solution de I'equation donn6e, seront s=0, <=0. On pent aussi demander quelle est la condition ou plutdt quelles sont les Equations de condition pour que deux racines de la meme paire soient ^gales. Dans ce cas, nous avons trouv^ que m = 0; cela exige que le dernier terme dans I'equation k u' devienne z^ro. On aura done, en vertu de I'equa- tion en u\ ae - Sbm + 26^ = 0, c'est-i-dire que le sous-invariant gauche ou bien le premier coefficient du Hessien k la forme biquadratique s'^vanouit. Mais cela ne suffit pas pour que les deux x d'une paire deviennent parfaitement identiques. II faut aussi que les deux valeurs de v, qui correspondent k la valeur z^ro de u, ou que les deux racines de I'equation ■t^-*\{v + c) + y= 0, oji \ = a = d - 6', deviennent egales, c'est-k-dire que 7+c»-(2a-)-c») = 0, a IV. 13 194 Solution de liquation [28 ou bien, puisque y=f — c^, que f-{Sm-2b'y = 0; k cette Equation il faut joindre I'^quation d4jk trouv^ ae-3bm + 2b' = 0; le systfeme de ces deux Equations exprime la condition de la coincidence des deux a; d'une paire. Quoique/— (3wi — 26*)* = 0 ne soit pas en elle-meme un sous-invariant, les deux Equations ci-dessus constituent (comme elles doivent le faire) un plextis sous-invariantif ; car on trouvera (aSft + 26S,„ + 3mSe + ^eB/) [af- (3am - 26*)'] = 4 (a« - 36m + 26») = 0. En effet, puisque / - (3»» - 2b-)- ne difffere de / - 9m» + 2a6e + 66*m (le second coefficient du Hessien) que par — 26 (ae — 36m. + 26'), on peut sub- stituer, pour le plexus ^crit plus haut, le plexus H^^O, Hi = 0, oh Hi, H^ sont le premier et le second coefficient du Hessien de la forme quadratique. Or il est facile de demontrer que, quand dans la forme (a, 6, m, e, / ) {x, y) a n'est pas zdro, mais que les deux premiers coefficients du covariant irr^- ductible gauche le sont, le covariant s'evanouit compl^tement*, et la forme biquadratique a deux paires de racines egales. On sait aussi que, quand les deux invariants irr^ductibles s'^vanouissent, il y a trois racines egales, et, quand en meme temps les deux invariants et le covariant gauche s'^vanouissent, toutes les racines de la biquadratique sont Egales. Ainsi on voit que les seuls cas d'^galite possibles entre les racines de r^quation quadratique donn^e, quand sa solution est regulifere, correspondent aux quatre cas d'^galit^ entre les racines de la biquadratique ordinaire qui s'y est associee. En prenant les quatre cas : 1° ou la quadratique a deux racines Egales ; 2° ou elle a deux paires de racines egales ; 3° trois racines Egales ; 4° toutes ses racines egales ; alors la quadratique donn^e aura, dans le premier cas, deux paires de racines Egales ; dans le deuxieme, quatre racines Egales ; dans le troisifeme, trois paires de racines Egales, et dans le dernier cas toutes ses racines seront Egales. Quant au rapport de la biquadratique binaire a la forme ternaire quadra- tique, on passe de la seconde a la premiere, en se servant de la substitution dont s'est servi notre trfes honor^ collegue, M. Darboux, dans sa belle Note sur la resolution de I'equation biquadratique {Journal de Liouville, t. xviil. p. 220). On n'a qu a faire x = «*, y = 2uv, z = x^, et la forme ternaire passe dans la forme binaire biquadratique. On voit ainsi que les genres de solutions r^gulieres de I'equation en quaternions donn^e dependent ex- * Quand les deux premiers coefficients du covariant irreductible gauche d'une biquadratique binaire s'^vanouissent, le discriminant s'evanouit a6cessairement : nous avons trouv^ que ce discriminant pris ndgativement ^gale 16 fois le produit des coefficients extrSmes, moins le produit du second et I'avant-dernier coefficient du covariant gauche. 1 28] quadratique de Hamilton 196 clusivement de la relation entre la conique qui s'y est associee avec la conique absolue y"^ — 4r^. Dans le cas le plus g^n^ral, les deux courbes se coupent en quatre points ; dans les quatre autres cas, il y aura I'une ou I'autre des quatre especes de contact entre les deux coniques. Mais, de plus, on voit ^videmment que cette id^e des deux coniques pent etre dtendue h, I'^quation de Hamilton, meme pour le cas ou la solution devient irreguliere. Dans ce cas, la forme ternaire, associee h. I'^quation a? +qx + r, perdra sa forme de conique et deviendra un systeme de deux lignes droites qui se croisent ou de deux lignes coincidentes. Dans la premiere supposition, il y aura le cas ou les deux droites toutes les deux coupent et les cas ou I'une ou toutes les deux touchent la conique fixe ; il y aura aussi les cas ou la conique fixe passe par le point d'intersection des deux droites en les coupant » toutes les deux ou en touchant une. Dans la seconde supposition, il y aura les deux cas oil les droites coltncidentes coupent ou touchent la conique fixe. Ainsi done il nous parait qu'on pent aflSrmer avec pleine confiance que, dans I'equation de Hamilton*, il y a exactement douze cas, ou au moins douze cas principaux, k consid^rerf. Nous devons cette m^thode si simple * Qaant k I'equation plus gin^rale pa;^ + gx + r = 0, dans le cas oil le discriminant ou le tensenr de p devient z^ro et que, par consequent, la forme ne rentre pas dans celle de Hamilton (puisqu'on ne pent plus diviser I'equation par p), il pent se presenter encore un grand nombre de cas singuliers que nous n'avons pas encore studies a fond. t Cela donne lieu k une reflexion curiense. Si Ton consid^re tous les genres de rapports qui peavent avoir lieu entre une vraie conique et une conique variable et capable de d^gen^rer en n'excluant pas les deux cas oti la conique variable coincide avec I'autre ou s'^vanouit tout H fait, le nombre de ces genres sera 14, qui est le nombre de doubles decompositiona du nombre 4, Bavoir: 4: 3,1: 2,2: 2,1,1: 1,1,1,1: 3:1 2,1:1 1,1,1:1 2:2 1,1:2 1,1:1,1 2:1:1 1, 1:1:1 1:1:1:1. De m^rae on trouvera faeilement que, pour le cas de formes binaires, le nombre de genres semblables sera 6, car, ayant sur une ligne droite deux points fixes et deux points variables, ces demiers peuvent itre distincts entre eux-mSmes en coiucidant avec un ou tous les deux ou avec ni I'un ni I'autre des deux premiers, on bien ils peuvent 6tre r^unis dans un seul point qui peut coincider ou ne pas co'incider aveo un des points fixes, et finalement ils peuvent disparaitre ; or le nombre de decompositions doubles du nombre 3, c'est-i-dire 3: 2,1: 1,1,1: 2:1 1,1:1 1:1:1, est aussi 6. Mais nous avons demontri autrefois, dana le Philotophical Magazine, que pour le cas de deux formes qnadratiqnes de n variables dont chacane restc g^nerale, c'est-i-dire n'a pas le dis- criminant zdro, le nombre des genres de rapport est exactement le nombre de doubles decom- positions du nombre n. C'est une question qui merite d'etre examinee, si cette identite entre le Dombre de genres pour n variables dans le second cas avec celni pour le nombre n-1 dans le premier, reste vraie pour toute valeur de n. Une consideration qui s'y oppose, c'est que, dans le premier cas, qnand (n - 1 = 1) le nombre de genres, au lieu d'etre 3 (le nombre de decompositions doubles de 2), n'est que 2, mais il peut arriver que pour ce cas (le cas d'une seule variable), la forme generale etant la mSme que la forme de coincidence parfaite, ce genre doit compter pour deux, et ainsi la loi se maintiendra. 13—2 |96 Solution de Vequation [28 de d^Dombrement k la connaissance que nous avons acquiae du M^moire ci- dessus cit^ de M. Darboux*. Mais ce qui plus est, on peut beaucoup simplifier, corame on va voir, la solution de I'^quation quadratique fx^pa^ + qx + r = 0. En regardant pour le moment x comme une quantity ordinaire, soient Fx le determinant de la matrice a?p + ceq+r ot <^ un quelconque des six facteurs quadratiques de Fx ; alors ^ = 0 sera I'equation identique d'une des racines de fx — O, et ces deux Equations, en ^liminant ar", donneront la valeur precise de cette racine^f". De meme nous ferons voir qu'en g^n^ral, quel que soit le degr^ (n) de fx (fonction rationnelle entiere et unilat6rale de x), lequel, comme aussi chaque coefficient, est une matrice d'un ordre donnd (o)) quelconque, en prenant le determinant Fx defx (od pour le moment on regarde x comme une quantity ordinaire), chaque facteur du degr^ x = 0 sera I'equation identique de x. Or soit fx = 0, alors y(— \) =/(— \) — f{x) et consequemment contiendra le facteur x + \. Done le determinant de /(— \) contiendra le determinant de (\ + x), c'est-^-dire contiendra X, oix ) sent egales, on a — -. — — — ^r- i- -^ paires de TT (&) — 1) . TT L(n — 1) a) — IJ racines Egales dans I'equation du degre n qui est k resoudre. * Dan8 le cas le plas g^n^ral d'ane Equation en x da degr^ n et de I'ordre u par rapport auz matrices, on pent sapposer nn nombre ind^fini de termes dans I'equation. Chacun de ces termes sera compost d'un nombre pas plus grand que n dea x dont chacun sera saivi et pr^c^d^ par une matrice multiplicatrice. En appliquant la m^thode algebrique directe pour resoudre cette Equation, on sera amen^ k un syst^me de to' Equations du degr^ n chacune. Ainsi le nombre des racines sera en general n . + Cela dimontre que le nombre 21 que nous avions trouv4 pour le cas de n = 3 dans le Philosophical Magazine, (mai 1884) [p. 229 below] et la formula g^nerale que nous avons bas^e 14-de68us sont erron^s ; la raison en est ^videmment que Tordre apparent da syst^me d'iquations qui nous a fourni ce r^sultat surpasse I'ordre actuel de 6 unites. Nous n'avions pas discut^ en detail ces Equations, et ainsi cet abaissement da degr^ nous a ichapp^. C'est un point curieuz qui reste & discuter. 198 Solution de T Equation quadratique de Hamilton [28 Prenons comme exemple de I'application de la m^thode I'^quation en quaternions ?««* + gaa^ + qiic + ^0 = 0. La fonction r^solvante sera (3 . 3)a;« + (3 . 2K + (3 . 1+ 2 . 2) a;* (3 . 0 + 2 . l)a;» + (2.0 + l.l)ar'(1.0)a; + (0.0) = 0, oh en g^n^ral i . i et i .j signifient Tq^, HSq,qj-S{Vq,Vqj)] respectivement. Les quinze facteurs quadratiques de cette fonction 6gales k z6to don- neront chacun une Equation quadratique a laquelle doit satisfaire une des quinze racines de I'^quation donn^e, et, en combinant s^par^ment chacune de ces Equations avec la cubique donnee, on pent 61iminer a;* et a;" et obtenir ainsi quinze equations lin^aires pour determiner les quinze racines voulues. 4 29. SUR LA RESOLUTION G^N^RALE DE L'^QUATION LIN^AIRE EN MATRICES D'UN ORDRE QUELCONQUE. [Comptes Bendus, xcix. (1884), pp. 409—412, 432—436.] Ce qui interesse le plus dans les r^sultats nouvellement acquis que j'ai I'honneur de presenter a I'Acad^mie, c'est I'union ou bien I'anastomose dont lis oflFrent un exemple frappant et tout a fait inattendu entre les deux grandes theories de YAlgebre moderne et de VAlgebre nouvelle, dont I'une s'occupe des transformations lin^aires, et I'autre de la quantite g^neralisde, de sorte qu'au mSme titre que Newton d^finit I'Algebre ordinaire comrae dtant I'Arithm^tique universelle, on pourrait trfes bien caract^riser cette Algfebre-ci comme dtant I'Algebre universelle, ou au moins une de ses branches les plus importantes. En general, un invariant de deux formes signifie une fonction de deux systemes de coefficients qui reste invariable, k un facteur pres, quand les deux systfemes des variables sont ou identiques ou assujettis k des substitu- tions semblables; mais rien n'empeche qu'on n'applique ce meme mot au cas ou les substitutions sont reciproques : ainsi, sans parler du cas de deux formes mixtes, on aura des invariants de deux formes donnees k mouvement semblable et des invariants k mouvement contraire; on peut tr^s bien noramer ces demiers (comme titre distinctif) contrariants. C'est a une classe sp^ciale de contrariants que nous aurons affaire dans la solution de I'^quation generale lin^aire en matrices d'un ordre quelconque. En supposant que chaque p et p' soit une matrice de I'ordre cd, I'operateur qui contient i couples Pii )p'i + pA )p'2+--+Pi( )p'i peut fitre noram^ provisoirement un nivellateur de I'ordre a> et de I'dtendue i, et on peut le caract^ri.ser par le symbole fl„j. Servons-nous toujours du symbole 0 pour signifier une matrice dont tons les 616ments sont des z^ros, et d^signons par 1 (ou bien par v indiff^remment) une matrice dont tons les I. 200 Sur la resolution g6n4rale de I'dquation linSaire [29 dl^ments sont z^ro, k I'exception des ^l^ments de la diagonale qui seront des unites: ce sont les matrices nommdes matrice nulle et matrice unitaire respectivement. J'ai d^j^ expliqu^ comment un nivellateur g^n^ral, de I'ordre w, donne naissance k one matrice de I'ordre cu': je nomme le determinant de cette matrice le determinant du nivellateur*. Ces determinants poss^dent des propri^tds tout k fait analogues k celles des determinants des matrices simples; ainsi, par exemple, je demontre la propriety dont je me suis servi avee grand avantage dans les recherches actuelles, que le determinant du produit de deux nivellateurs est egal au produit de leurs determinants separes, et que le determinant d'une fonction rationnelle d'un nivellateur, disons FH, est egal au resultant (par rapport k fl regarde comme une quantite ordinaire) de F£l et IH, oh. lil = 0 represente I'equation identique du degre a>^ k laquelle fl est assujetti. En general, a un systerae ou corps de matrices pi, p.^, ...,pi de I'ordre to correspond un quantic de I'ordre m, c'est-a-dire le determinant de <«iP\ + iHiPi + ... + a;ipi. Je nomme les coefficients de ce quantic les parametres du corps. Ces paramfetres doivent etre regardes comme des quantites connues. Ainsi, par exemple, si au corps p, q (deux matrices binaires) on adjoint la matrice unitaire v, et qu'on forme le determinant de la matrice x + yp + zq, on obtiendra un quantic a^ + Bxy + Cxz + Dy'^ + Eyz + Fz'^, ovi, si Ton regarde p, q comme des quaternions, on aura, dans le langage du grand Hamilton, 5 = Sp, G = Sq, D=T'p, F=T% E^SiVp.Vq). II resulte de cette definition qu'k chaque nivellateur ri„,,- appartiennent deux quantics de I'ordre w et avec i variables, dont I'un appartient au corps Pi, Pa, •••,Pi et I'autre au corps p\, p\ p'i. Si Ton connait I'equation identique /Q = 0 a laquelle le nivellateur fl obeit, on peut immediatement, comme je I'ai dejk montre, resoudre I'equation nx = T. Mais il est tres facile de voir que lil n'est autre chose que le determinant du nivellateur il — \v{ )v, quand dans ce resultat on substitue XI k X. Done la question de la solution lineaire la plus generale est raraenee k ce seul probleme: Exprimer le determinant d'un nivellateur en termes de quantites connues. Or la premiere conclusion et la plus difficile k etablir dans cette recherche, mais que j'ai enfin reussi a demontrer, c'est que ce determinant est toujours * Quelqnefois ce determinant sera nomm^ un nivelUint. I 29] en matrices d^un ordre quelconque 201 une fonction entiere, mais pas ndcessairement rationnelle, des coefficients des deux qtiantics qui sont associ^s au nivellateur. Cela ^tant convenu, on demontre avec une extreme facility que ce determinant est un contrariant du degr^ <» dans chaque syst^me de coeffi- cients des deux quantics associ6s. Cela ne suffit pas ou pent ue pas suffire en soi-meme k d^finir com- pletement le contrariant cherche; nommons, en general, ce contrariant le nivellant des deux quantics. Supposons que Nx, y, ... z, t soit le nivellant ponr deux quantics d'un ordre donne «d, et representons par N^, y, ... z, o ce que ce nivellant devient quand on r^duit a ze'ro tons les coefficients qui appartiennent aux termes dans les deux quantics qui contiennent t; alors il est facile de voir que ■"X, y, ..., z, 0 '^ ■" X, y, ... z- Cette propriete seule est suffisante (avec I'aide d'un quelconque des op^- rateurs differentiels qui servent pour annuler un contrariant) pour preciser le contrariant (nivellant) dans le cas de deux quantics du second ordre, et c'est ainsi que j'ai obtenu la solution de I'^quation lindaire pour le cas des matrices binaires donn^ dans la Note prec^dente. Or il est bieu con- cevable que cette loi ne pent pas suffire ^ determiner les parametres arbitraires qui entreut dans le contrariant d'ordre (w, ai) appartenant k deux quantics de I'ordre &>. Mais il y a encore une autre loi (constituant par elle-ineme un trfes beau th^oreme) qui doit suffire surabondamment a cette fin. Cest une loi qui ^tablit une liaison entre les nivellants de deux systfemes de quantics contenant chacun le meme nombre de variables, mais dont I'un est d'un ordre plus grand par unite que I'ordre de I'autre. Supposons que N soit le nivellant de deux quantics de I'ordre w, F{x,y 2) et 0{x,y,...,z); soit N' ce que devient N quand F{x,y,...,z) = {lx + my+...->rnz)Fi{x,y,...,z) et Q(,x,y z) = {\x + fiy+ ...+vz)Qiix, y,...,z); alors je dis que, quand l\ + ni/i, + ... +nv = 0, le nivellant de (Fi, 6i) sera contenu comme tacteur dans le nivellant modifie N'. A I'aide de ces principes, je me propose de calculer les nivellants pour les degi-es sup^rieurs au second. On voit par ce qui precede que la solution de I'equation lineaire 'S.pxp' = T sera alors connue en termes des p, des p', de T et des paramfetres des deux corps Pi, pa, .... Pi, p'l, p'n, ..., p'i,&\igment6s I'un et I'autre d'une matrice unitaire. 202 Sur la resolution gSnSrale de V^quation linSaire [29 C'est dans les Lectures, publi^es en 1844, que pour la premiere fois a paru la belle conception de I'equation identique appliqu^e aux matrices du troisifeme ordre, envelopp^e dans un langage propre k Hamilton, apres lui mise k nu par M. Cayley dans un tres important M^moire sur les matrices dans les Philosophical Transactions pour 1857 ou 1858, et ^tendue par lui aux matrices d'un ordre quelconque, mais sans demonstration; cette demonstra- tion a 6t^ donn6e plus tard per feu M. Clifford {voir ses ceuvres posthumes), par M. Buchheim dans le Mathematical Messenger (marchant, comme il I'avoue, sur les traces de M. Tait, d'Edimbourg), par M. Ed. Weyr, par nous- meme, et probablement par d'autres; mais les quatre m^thodes cities plus haut paraissent Stre tout k fait distinctes I'une de I'autre. Par le moyen d'une chaine de matrices coupleos (disons N), operant non pas sur une matrice gen^rale, mais sur une niatrice x (disons du degre a>) d'une forme sp^ciale suivie par un autre op^rateur V qui aura I'effet de reduire la matrice du degr^ to de Nx (dont les elements sont des fonctions lindaires des ^l^ments de x) a une forme identique a celle de x, il est facile de voir qu'k I'op^rateur compose VN on peut faire correspondre une matrice d'un ordre quelconque non sup^rieur k to", et c'est ainsi virtuellement que Hamilton, a cause d'une transformation qu'il effectue sur I'equation lin^aire g^nerdle, est tombe dans ses Lectures sur la matrice du troisieme ordre, et ce n'est que dans les Elements publics en 1866 (apres sa mort) qu'on trouve quelque allusion k I'equation identique pour les matrices du quatrieme ordre. On pourrait nommer I'operateur composd VN, pour lequel I'equation identique est d'un degr^ moindre que ay', nivellateur qualifie, mais il est essentiel de remarquer que ces operateurs ne poss^deront pas les propriet^s analogues k celles des matrices que possfedent ces nivellateurs purs dont il est question dans ma m^thode. Comme exemple d'un nivellateur qualifid, on pourrait admettre que le x (matrice du deuxifeme ordre), sur lequel opfere le N, aura son quatrieme element zdro, et que I'effet du V sera d'abolir le quatrieme Element dans Nx, ou. Ton peut supposer (et cette supposition est, dans son essence, a peu pres identique a la mdthode des vecteurs de Hamilton) que le premier et le quatrieme Element de x sont ^gaux, mais de signes contraires, et que I'effet de V est de substituer dans la matrice du second ordre N (x) la moitie de la difference entre le premier et le quatrieme element au lieu du premier et, au lieu du quatrifeme, cette meme quantity avec le signe algebrique contraire. Evidemment un tel operateur donnera naissance a une matrice et sera assiijetti k une equation identique du troisieme ordre. Avant de conclure, pour convaincre de la justesse de la formule importante i[{Pyp'-i{P'.py]-^^(i.ir, * Pour rendre intelligible cette formule, il est nicessaire de dire que I'expression 29] en matrices d^un otdre quekonque 203 applicable au cas d'un nivellateur du second ordre a quatre couples de matrices, il sera bon d'en donner une demonstration parfaite a posteriori, ce qu'une transformation legitime rend trfes facile a faire. Remarquons que le determinant du nivellateur du second ordre 2 , ( ) \> est le determinant c d^ ^7 0 de la matrice suivante: 2aa 2ca 2a/3 2c/3 26a %dcL 26/8 2d/8 2a7 Icy 2a8 2c8 267 2^7 26S Xdh laquelle contiendra dans le cas suppose 144 termes, puisque chaque 2 com- prend 4 produits : mais, sans perdre en gdneralite, on peut prendre une forme de nivellateur dont le determinant ne comprendra pas plus de 24 termes ; car il est facile de demontrer que, si aux 4 matrices de gauche on substitue 4 fonctions lin^aires quelconques, pourvu que sur les 4 de droite on opere une substitution contragrediente k la substitution pr^ce- dente, la valeur du determinant ne subira nul changement. On peut done supposer que les 4 matrices de gauche sent 10 01 00 00 00 00 10 01 respectivement, et, si la formula est verifiee dans cette supposition (vu que les contravariants des deux quantics associes ne sent pas afifectes par les sub- stitutions contragredientes operees sur les deux s^-stemes de matrices), elle donn^ dans la Note da 21 joiilet [pp. 181, 184 above], a besoin d'nne correction (dont je pensaia avoir fait mention dans le texte) : il fait lui ajoater la racine carrfe d'un contrariant eonnue da qnatridme degr£ (appartenant aax denx Jormea aifocUea), laqaelle sera ane fonction rationnelle des ^l^ments des matrices da nivellatear. Poor le cas d'nn nivellateur k quatre couples de matrices, c'est la racine carr^e du prodoit de / et /', les discriminants des denx formes a88oci^es prises s^par^ment ; en nommant les quatre matrices a gauche a b a' b' c d c' d' la racine carr^ de / sera igale an determinant a b a' V a" b" a'" b"' qn'on peut nommer le developpant de ces quatre matrices ; de mgme la racine carrce de I' sera igale au developpant des quatre matrices correspondantes i, droite, de sorte que le terme irration- nel dans la formale poor le nivellant k quatre couples de matrices est ^gal an produit de ces deux d^veloppants ; dans le cas g^n^ral, la partie relativement irrationnelle de la formule pour an nivellant sera ^gale a la somme de tons les produits de d^veloppants accoupl^s qu'on peut former en combinant qaatre k quatre, ensemble, les couples de matrices qui en dependent. Dans le cas o£i le nivellateur contient moins de qnatre couples, la racine carrce disparalt enti^re- ment de la formale pour le nivellant. Je nommerai P . P et (P')'P', 3-, et ^j respectivement. a" b" a'" b'" c" d" c'" d!" c d e d' c" d" • e'" d'" 204 Sur la rholution ginfyrale de V^uation linfyiire [29 sera noD pas seulement verifide, mais absolument dimontrie pour les valeurs parfaitement g^ndrales des deux syst^mes. Avec ces valeurs des matrices gauches, la matrice ^crite plus haut, en prenant a /8 a' /S' 78 7' S' pour les matrices k droite, devieut a «! a' a 7 7i 7' 7 dont je nommerai le determinant Q. De plus, le quantic k gauche deviendra xt — yz, et le quantic k droite (aS - ^7) ^ + (5 S - yS 7) «' + (a'8' - ySV) f + («i8> - A7.) ^' + (1 . 2)xy + (3 . 4)z« + (1 . 3)a;2 + (2 . 4) y< + (1 . 4) ««+ (2 . 3) yz, oh (1.2) = oS'+Sa'-^7'-/3'7, (3 . 4) = a,S + ^.S - /9,7 - 7,^8, a. A ayS 7i Si 7S iS A /3' y8 S 8. s' a Done a. = («S + fiS - /37 - /87) - (o'S, + a,S' - ;8'7i - ^W). \% = (aS + aS - ^7 _ ;S7)^+ (a'8, + a,8' - /3'7i - /8.7')' + 2 (aS - ^7) {al - 0y) + 2(a'B' - /3 '7') (a:8, - ^i7:) - (aS' + Sa' - ,87' - /3'7) (a J + g^a - ^7 - /871) - (aS, + Sa. - y37i -_A7) (a'S + B'a - ^'7 - ^y') - (aS + fiS - /37 - /87) (a'S. + a,S' - /3'7i - ^i7'). et V(-^ • ^') (pris avec le signe convenable) sera le determinant de la matrice a ^ y B a /3' y B' tti ;8, 7, Si a y9 7 S. En faisant les multiplications necessaires, on trouvera que i^,-V-V(/-/') = 2Q. ce qui d^montre I'exactitude de la formule donn^e pour un nivellateur du deuxieme ordre k quatre couples de matrices. D'ici a pea de temps, j'espere avoir I'honneur de soumettre k I'Aca- d^mie la valeur du determinant du nivellateur du troisifeme ordre a trois couples de matrices. Pour presenter I'expression g^n^rale de ce determi- nant pour une matrice d'un ordre et d'une etendue quelconques*, il faudrait avoir une connaissance des proprietes des formes qui va beaucoup au deli • C'est-i-dire pour r^soudre I'^quation lin^aire en matrices dans toute sa g6n^ralit6. 29] en matrices (Tun ordre quelconque 205 des limites des facultes humaines, telles qu'elles ne sont manifestoes jusqu'au temps actuel et qui, dans mon jugement, ne pent appartenir qu'a I'intelli- gence supreme. Post-scriptum. — Qu'on me peimette d'ajouter une petite observation qui fournit, il me semble, une raison suffisante a priori pour le signe ambigu du terme v'(-^ ■ ^ ) qui entre dans la formule donnee pour un nivel- lant (c'est-a-dire determinant d'un nivellateur) du deuxieme ordre. Les determinants d'un nivellateur et de son conjugue ^tant identiques en signe alg^brique tout autant qu'en grandeur, ce n'est pas dans cette direction qu'on peut chercher I'origine de I'ambiguitd Mais, si, en se bornant aux matrices correspondantes d'un nivellateur de la menie esphce, c'est-a-dire a main droite ou a main gauche du symbole ( ), on ^change entre eux, dans chacune de ces matrices, le premier terme avec le quatrieme et le deuxieme avec le troisieme, on verra f'acilement que le nivellant et en meme temps les deux qualities associes restent absolument sans alteration; mais, si Ton execute I'une ou I'autre de ces substitutions s^par^ment, alors, tandis que les deux quantics associes restent constants, le nivellant (quand son nivellateur possede plus de trois couples) subira un changement de valeur (et, pour I'une et I'autre substitution, le vieme change- ment), de sorte que pour les qnatre positions qu'on peut assignor simul- tan^ment aux elements des matrices de la meme espece sans changer en rien les quantics associes, le nivellant aura deux valeurs distinctes. Voila, il me semble, I'explication suffisante et la veritable origine de I'ambiguitO dont il est question. A peine est-il n^cessaire de remarquer qu'on peut faire 4 autres dis- positions semblables et simultan^es des matrices a I'un ou I'autre c6t6 du symbole ( ), dispositions qui donneront naissance k des nivellants identi- ques en valeur avec les deux dont j'ai parlO (c'est-k-dire deux k une valeur et deux a I'autre), et pour lesquelles les deux quantics associes seront sans autre changement que celui du signe alg^brique. En combinant les 24 dispositions semblables des matrices d'un c6t6 d'un nivellateur donne avec les 24 de I'autrd c6te, on obtiendra un systfeme de 576 nivellateurs correlatifs dont les determinants ne prendront que 3 paires le valeurs; de plus, les deux valeurs d'une quelconque de ces paires seront es racines d'une Equation quadratique dont les coefficients seront des con- trariants rationnels et entiers d'une des trois paires de formes quadratiques; mais le discriminant de ces trois Equations sera le meme certainement quand les nivellateurs du systeme seront formes avec quatre couples de matrices et probablement quel que soit le nombre de ces couples. Quand ce nombre est moindre que 4, le discriminant de ces trois quadratiques devient nul pour toutes les trois. le 30. SUE L'jfeQUATION LIN^AIRE TRINOME EN MATRICES D'UN ORDRE QUELCONQUE. [Coniptes Rendus, xcix. (1884), pp. 527 — 529.] Pour r^soudre I'^quation trindme pa^' + qxq' + r = 0 (oil toutes les lettres li^signent des matrices du mStne ordre to) sous sa forme sym^trique, on a besoin de coimaitre I'^quation identique k un nivellateur de cet ordre k deux couples de matrices, ce qui equivaut virtuellement k connaitre le determinant d'un nivellateur k trois de ces couples. Mais, sans avoir recours k cette mdthode g^n^rale, il existe, comme on va le voir, un raoyen plus court et plus direct pour r^soudre I'equation et exprimer x sous la forme essentiellement bonne d'une fraction reduite, si Ton est d'accord k se dispenser de la condition que le numdrateur soit sym^trique. A cet effet, on pent multiplier I'equation, a volont^, ou par q~^ (, )p'~^ ou par j3~'( )q'~^. Choisissons le premier de ces deux multiplicateurs et ^crivons q~^p= , q'p'~^ = — yfr, — a; — ooyjr = /ji (mais d^ja avec une breclie de sym^trie, par la raison du choix d'une entre deux choses pareilles). En multipliant cette Equation par le nivellateur 0'( ) + (^'-'( )-«/r + ^'-2( )^2 + ...+( )-^< (disons Ui) et en dcrivant Uifi = /u.i+i, on obtient la suite d'^quations X — xyfr = fi, *x — x-\Jr^ = fj,3, (fy^x — X'y^ = fji^, . . . , '-'x— xyfr'' = /i„. Soient B^, £,,..., 5„ et Go, Cj, ...,(7„ les coefficients des deux formes associ^es aux deux systemes p, q et p', q' respectivement; alors, en vertu d'un theor^me general en matrices*, on aura C'^t" + C„-it""' + . . . + Co = 0, 5„ - A0 + . . . + (-yB^cfy = 0. Avec I'aide de ces deux Equations et de la suite prec^dente, on pent deduire une equation de I'une ou de I'autre des deux formes Mx = N ou xM=N. Faisons le choix (qui amfene encore une fois une brfeche de sym^trie) de la premifere. On aura (C„^" + C„_i^"-i + . . . + C,^ + C„) a; = C„/4„ + C„_,/i„_i + . . . + C,yn. Or, selon la th^orie ordinaire d'elimination, on pent determiner 'is et H deux fonctions chacune du degr^ (w — 1) en <^ (traitd comme une quantity ordi- naire), telles que ^ [£o - ^a + • • • + {-YB^r] + SiC. " + C„-i <^"-' + . . . + Co) * Ainsi, par exemple, si p, q sont des qaaternious, on a 2>2 (p-i q)^-2S{rpVq){p-^q) + Tq^ = 0. I 30] Sur Tequation lineaire trindme en matrices 207 sera egal k R, le contre-rdsultant des deux formes associees a (p, q)et (p', q')* respectivement, et Ton aura _CiHiJ. + C^Hfi^ + ... + G„Hfi„ '^~ R ' et ainsi x sera determine. Si ft est zero, alors, afin que x ne soit pas zero, le R doit devenir zdro, comme nous avons deja trouvd dans une Note pr^c^dente. En g^n^ral, si R (le contre-r^sultant des deux formes adjointes a |), 5' et p', 5' dans Tequation pxp' + qx(i + r = 0) s'^vanouit, Tequation ne peut pas admettre une solution en ineme temps actuelle et ddterminee; sans autres conditions, la solution deviendra ideale; avec conditions convenables, elle peut redevenir actuelle, mais contiendra (selon les circonstances) une ou plusieurs constantes arbitraires. Hamilton, dans ses Lectures, a considere I'equation trindme pour les quaternions, mais il n'en a pas pousse la solution, e'est-^-dire la valeur de I'inconnue, a sa forme finale dans laquelle le d^nominateur doit 6tre un scalar (je dis doit etre), parce que, ici comme dans toutes les Equations en matrices, c'est le denominateur de I'iuconnue convenablement exprim6 dont I'evanouissement est le cntinuvi pour distinguer le cas 011 la solution est actuelle et ddterminee d'avec les cas ou elle devient ou ideale ou inde- terminde. En combinant le rdsultat ici obtenu avec celui de notre Note prec^dente, on voit qu'on est entre en pleine possession de la solution de I'equation Nx = r dans les deux cas ovi le nivellateur N est de I'ordre 2 et d'une ^tendue quelconque ou bien de I'^tendue 2 et d'un ordre quelconque. Remarque. — On peut objecter que le num«5rateur de I'expression trouv6e pour X dans I'equation trin6me contient des combinaisons de q'^-p, q'p'~^, q~^rp'~^ et que, cons^quemment, x pourrait devenir id^al k. cause de I'eva- nouissement du determinant de p' ou de q sans que le contre-resultant R s'evanouisse. Pour r^pondre k cette objection, soient D', A les determinants de p' et de g~'; alors, en se servant des equations identiques k p' et k q, on peut substituer pour leurs inverses des fonctions rationnelles de I'un et de Tautre divisees respectivement par Z)' et A, et alors le numerateur de x sera une quantite incapable de devenir iniinie, tandis que son denominateur sera R multiplie par des puissances de Z)' et de A ; mais, vu qu'on peut repre- senter x tout aussi bien par une autre fraction dont le numerateur sera aussi incapable de devenir infini et dont le denominateur sera R multiplie par des puissances de D' et de A (les determinants de p et de q'), il est evident que ces deux fractions doivent toutes les deux admettre d'etre simplifiees et que dans leurs formes reduites le denominateur sera tout simplement R et qu'ainsi ce contre-resultant est le seul criterium pour distinguer le cas de Tactuel et determine d'avec le cas de I'ideal ou indetermine. * C'e8t-a-dire le r£>ultant dea fonctions multipli^es par ^ et if ci-dessus. 31. LECTURES ON THE PRINCIPLES OF UNIVERSAL ALGEBRA. [American Journal of Mathematics, vi. (1884), pp. 270 — 286.] LECTURE I. Preliminary Conceptions and Definitions. Apotheosis of Algebraical Quantity. A MATRIX of a quadrate form historically takes its rise in the notion of a linear substitution performed upon a system of variables or carriers; regarded apart from the determinant which it may be and at one time was almost exclusively used to represent, it becomes an empty schema of operation, but in conformity with Hegel's principle that the Negative is the course through which thought arrives at another and a fuller positive, only for a moment loses the attribute of quantity to emerge again as quantity, if it be allowed that that term is properly applied to whatever is the subject of functional operation, of a higher and unthought of kind, and so to say, in a glorified shape, — as an organism composed of discrete parts, but having an essential and undivisible unity as a whole of its own. Naturam, expellas furcd, tamen tisque recurret*. The conception of multiple quantity thus rises upon the field of vision. At first undifferentiated from their content, matrices came to be regarded as susceptible of being multiplied together; the word multiplication, strictly applicable at that stage of evolution to the content alone, getting transferred by a fortunate confusion of language to the schema, and superseding, to some extent, the use of the more appropriate word composition applied to the reiteration of substitution in the Theory of Numbers. Thus there came into view a process of multiplication which the mind, almost at a glance, is able to recognize must be subject to the associative law of ordinary * Choitez le naturel, il revient au galop, a familiar quotation which I thought was from Boilean, but my friend Prof. Babillon informs me is from a comedy of Destonches (bom in 1680, died 1754). 31] Lectures on the Principles of Universal Algebra 209 multiplication, although not so to the commutative law; but the full signi- ficance of this fact lay hidden until the subject-matter of such operations had dropped its provisional mantle, its aspect as a mere schema, and stood revealed as bona-fide multiple quantity subject to all the affections and lending itself to all the operations of ordinary numerical quantity. This revolution was effected by a forcible injection into the subject of the concept of addition, that is, by choosing to regard matrices as susceptible of being added to one another ; a notion, as it seems to me, quite foreign to the idea of substitution, the nidus in which that of multiple quantity was laid, hatched and reared. This step was, as far as I know, first made by Cayley in his Memoir on Matrices, in the Phil. Trans. 1858, wherein he may be said to have laid the foundation-stone of the science of multiple quantity. That memoir indeed (it seems to me) may with truth be affirmed to have ushered in the reign of Algebra the 2nd; just as Algebra the 1st, in its character, not as mere art or mystery, but as a science and philosophy, took its rise in Harriot's Artis Analyticae Praxis, published in 1631, ten years after his death, and exactly 250 years before I gave the first course of lectures ever delivered on Multinomial Quantity, in 1881, at the Johns Hopkin.s University. Much as I owe in the way of fruitful suggestion to Cayley's immortal memoir, the idea of subjecting matrices to the additive process and of their consequent amenability to the laws of functional operation was not taken from it, but occurred to me independently before I had seen the .memoir or was acquainted with its contents; and indeed forced itself upon my attention as a means of giving simplicity and gene- rality to my formula for the powers or roots of matrices, published in the Comptes Rendus of the Institute for 1882 (Vol. xciv. pp. 55, 396). My memoir on Tchebycheff's method concerning the totality of prime numbers within certain limits, was the indirect cause of turning my attention to the subject, as (through the systems of difference-equations therein employed to contract Tchebycheff's limits) I was led to the discovery of the properties of the latent roots of matrices, and had made considerable progress in developing the theory of matrices considered as quantities, when on writing to Prof. Cayley upon the subject he referred me to the memoir in question: all this only proves how far the discovery of the quantitative nature of matrices is removed from being artificial or factitious, but, on the contrary, was bound to be evolved, in the fulness of time, as a necessary sequel to previously acquired cognitions. Already in Quaternions (which, as will presently be seen, are but the simplest order of matrices viewed under a particular aspect) the example had been given of Algebra released from the yoke of the commutative principle of multiplication — an emancipation somewhat akin to Lobat- chewsky's of Geometry from Euclid's noted empirical axiom; and later on, 8. IV. 14 210 Lectures on the Principles [31 the Peirces, father and son (but subsequently to 1858) had prefigured the universalization of Hamilton's theory, and had emitted an opinion to the effect that probably all systems of algebraical symbols subject to the associative law of multiplication would be eventually found to be identical with lineai' transformations of schemata susceptible of matricular representation. That such must be the case it would be rash to assert; but it is very difficult to conceive how the contrary can be true, or where to seek, outside of the concept of substitution, for matter affording pabulum to the principle of free consociation of successive actions or operations. Multiplication of Matrices. A matrix written in the usual form may be regarded as made up of parallels of latitude and of longitude, so that to every term in one matrix corresponds a term of the same latitude and longitude in any other of the same order. Every matrix possesses a principal axis, namely, the diagonal drawn from the intersection of the first two parallels to the intersection of the last two of latitude and longitude ; and bj' a symmetrical matrix is always to be understood one in which the principal diagonal is the axis of symmetry. If there were ever occasion to consider a symmetrical matrix in which this coincidence does not exist, it might be called improperly symmetrical. This designation might and probably ought to be extended to matrices symmetrical, not merely in regard to the second visible diagonal, but to all the (ft) — 1) rational diagonals of a matrix of the order &>, a rational diagonal being understood to mean any line straight or broken, drawn through ^ components separate into a group of ft)' — 1 terms analogous to the vector and to a single term analogous to the scalar of a quaternion. This outstanding single term is of an invariable form, namely, its principal diagonal consists of elements having the same value, which may be called its parameter, and all the other elements are zeros. A matrix of such form I shall call a scalar. When the parameter is unity it may be termed a multinomial unity and denoted by T *, or in place of m we may write to dots over T, or for greater simplicity when desirable write simply T. Any scalar, by virtue of what precedes, is a mere monomial multiplier of some such T. Let kT be any scalar of order m. It will readily be seen, by applying the laws of multiplication and addition previously laid down, that (k).T, and that kT.m==m .kT = km. Thus a scalar possesses all the essential properties of a monomial quantity, and a multinomial unity of ordinary unity; in particular, the faculty of being absorbed in any other coordinate matrix with which it comes in contact. A scalar whose parameter vanishes of course becomes a zero-matrix. The properties stated of a scalar ^T serve to show that in all operations into which it enters the T may be dropped, and supplied or understood to be supplied at the end of the operations when needed to give homogeneity to expression. Thus, for example, (m + AT) (wi + kT) = rn' + (h + k) Tm + hkV = m^+(h + k)m + hkT; but this result may be obtained by the multiplication of (m+ h)(m-{-k), and supplying T (or imagining it to be supplied) to the final term in order to preserve the homogeneity of the form. In like manner, 0^ or 0 with « points over it may be used to denote the absolute zero of the order a ; but it will be more convenient to use the ordinary 0, having only recourse to the addi- tional notation when thought necessary or desirable in order to make obvious the homogeneity of the terms in any equation or expression. Thus, for example, such an expression as ni' + 2bm + d = 0, where to is a matrix, say of the 2nd * Perhaps more advantageously by !„. I shall hold myself at liberty in what follows to use whichever of these two notations may appear most convenient in any case as it arises. 214 Lectures on the Principles [31 order, and h and d monomials, set out in full would read m^ + 26m + dT = 0, meaning m.m + 2 6?n+,= . On the Inverse and Negative Powers of a Matrix. The inverse of a matrix, denoted by m~S means the matrix which multi- plied by m on either side produces multinomial unity. It is a matter of demonstration that when a matrix is non-vacuous (that is, has a finite content or determinant appertaining to it), an inverse to it fulfilling this double condition can always be found, and that if the product of mn is unity, so also must be that of nm. It is a well-known fact, proved in the ordinary theory of determinants, that if every element in the first of two matrices is the logarithmic dif- ferential derivative, in respect to its correspondent in the second, of the content of that second, so conversely, every element of the second is the logarithmic derivative, in respect to its correspondent in the first, of the content of the first. But two such matrices multiplied together in either sense would not give for their product multinomial unity; to obtain this product either matrix must be multiplied indifferently into or by the transverse of the other (meaning by the transverse of a matrix, the new matrix obtained by rotating the original one through 180° about its principal diagonal). In other words, if m be a given matrix and n be obtained from it by substituting for each element the logarithmic derivatives of its content in respect to its opposite, then mn = 'T and nin = T, where w means (as will always be the case throughout these lectures) the order of the matrices concerned. The n which satisfies these two equations (and it cannot satisfy the one without satisfying the other) will be called the inverse of m and be denoted by m~K For brevity and suggestiveness it will be advantageous to write in general 1 for T as we write 0 for 0„, so that mn = 1 will imply nm = 1 = mn and n = m~^. We may define in general (as in monomial algebra) m~* to mean the inverse of m*, that is, (m')~^ We shall then have (m~')' = m~*, for mn . mn = 1 implies m . mn . n — mn = 1 or mV = 1. Hence n'' = m~'\ that is, (»ft~')' = m~'. Also since mVi' = 1, m'n' = m,n = 1 or w' = m~', that is, (mr^f = m~', and so in general for all positive integer values of i, (m~')' = wi~*. And, as in monomial algebra, it may now be proved and taken as proved that, for all real values of i and j, whether positive or negative, m* . W = m'+-', and the same relation may be assumed to continue when i, j become general quantities. The elements in the inverse to any matrix m all involving the reciprocal of the ( 31] of Universal Algebra 215 determinant to m, if D be the content of m we may write mr^ = j^ fj,, where ft, is a, matrix all of whose elements are always finite. Hence we come to the important conclusion that for vacuous matrices inverses only exist in idea and are incapable of being realized so as to have an actual existence. In the sequel it will be shown that the inverse is only a single instance of an infinite class of matrices which exist ideally as functions of actual matrices, but are incapable of realization. Suppose now that M, N are any two matrices such that MN = 0 or that NM=0; multiplying each side of the equation by if-' if such expression ha-s an actual existence (that is, if M is non- vacuous), we obtain, from the known properties of zero, iV^ = 0, but if M is vacuous no such conclusion can be drawn. So further if »n' = 0 (i being any positive integer), it will be seen under the third law of motion that m is necessarily vacuous. Hence from this equation it cannot be inferred that any lower power than the ith of m is necessarily zero. On the Latent Roots and Different Degrees of Vacuity of Matrices. If m be any matrix, the augmented matrix m — \T or m — \ . 1„ or m — \ will be found simply by subtracting \ from each element in the principal diagonal of m. The content of this matrix or the same multiplied by — 1 or any other constant, I term the latent function to m, which will be an algebraical function of the degree &> in \ (which may be termed the latent variable or carrier); and the w roots of this function (that is, the to values of the carrier which annihilate the latent function) I call the latent roots of the unaugmented matrix m. It is obvious from this definition that if \i be any latent root of m, the content of m — \i will vanish, that is, m — \, will be vacuous, and conversely that if rre — X, is vacuous, \, must be one of the latent roots to m. Thu.s if m is vacuous, one of the latent roots must be zero ; if only one of them is zero I call m simply vacuous and say that its vacuity is 1 : thus zero vacuity and simple vacuity mean the same thing as zero nullity and simple nullity respectively. More generally if any number i, but not i+1, of the latent roots of m are all of them zero, m will be said to have the vacuity i. By a principal minor determinant to any matrix I mean any minor deter- minant whose matrix is divided by the f)rincipal diagonal into two triangles. It will then easily be seen that if «< means in general the sum of the principal tth minors to nt, and «„ means the complete determinant, the assertion of m having the vacuity i is exactly coextensive with the a.ssertion that s„ = 0, «! = 0, s, = 0, . . . s,_, = 0. If the nullity of m is t, every gth minor of m is zero when q < i. Hence the vacuity cannot fell short of the nullity, but the converse is not true. 216 Lectures on the Principles [31 A matrix may not have any vacuity up to — I being here unity) that (m — X,) (m — X,) = 0. Understanding by m the matrix «., ti — K, the latent equation to w is 'J. Tj— X = 0, 31] of Universal Algebra 217 that is, >.'-(«, + T,) X + ((jTj - t,T,) = 0, so that ni' - (<, + r.) m + (ttT, — t^Tj) = 0, or, using the literation applied to the parametric triangle, m'-2bm + d = 0; (1) for since the content of x+ym + zn is supposed to be a? + "ibxy + 2cxz + dy^ + 2eyz +fz\ that of - \ + m will be found by making z = 0,x = — \,y=\. The variation of equation (1) obtained by taking en for the increment of m (remembering that the variation of m' is {m + €n){m + en) - in", that is, e{mn + nm)) gives rise to the identical equation win + nm - 2bn — 2cm + 2e = 0, (2) and the variation of this again gives n» + n' - 2cn - 2cn + 2/= 0, or n' — 2cn+/=0, as of course will be obtained immediately from (1) by substituting n, c, f in place of m, 6, d. The parameters c, /, if n represents ' ' are the sum of the principal diagonal elements and the content of u, just a&b,d are such sum and content in respect to m. The parameter e (the connective to d and /) or rather its double 2e is obviously the emanant of d in respect to the operator or, if we please, of/" in respect to the inverse operator that is, tiVt + WiTj — Uvx — MjT,. With the aid of the catena of equations in m, in m and n, and in n, any combination of functions of vi and n may be reduced to the standard form Amn+Bm + Cn + D. For, in the first place, ^ = P (m' — 26m + d) + rm + « «■ rm + «, and similarly ■^n = pn + a. Hence the most general combination referred to is expressible as the product of alternating linear functions of m and n, and may therefore be reduced to a sum of terms of which each is a product of alternate powers of m and of n, each of which powers may again be reduced to the form of linear functions, and this process admits of being continually repeated. Suppose then, at any stage of it, that the greatest number of occurrences of linear functions of jn and n in the aggregate of terms is i; then at the 218 Lectures on the Principles [31 next stage of the process the new aggregate will consist of monomial multi- pliers of one or more simple successions of m and n, and of terms in which the number of alternating linear functions never exceeds i — 1; hence, eventually we must arrive at a stage when the aggregate will be reduced to a sum of monomial multipliers of simple successions of m and n, every such succession being of the form (7n»)« or w~' (?«»)« or (mn)''n~^ or m~' (mn)^n~\ But {mnf = TO . nm ,n = —m (mn — 2bn — 2cm + 2c) n = — m'n' + 2bmn' + 2cm?n — 2evin = - (26m - d) (2cM -/) + 26m {2cn -/) + 2c (26m -d)n- 2emn = — (2e — 46c) mn - df. Hence {mnf + 2 (e - 26c) mn + df= 0. Hence (mn)« = P {{mnf + 2 (e - 26c) mn + df}-\-A mn + B=Amn + B, where A and B are known functions of (e — 26c) and/ ; and therefore ,/ x» ^ r> , A B 2Bb m~' (mn)« = An + Bm~^ = An ? m H j- . d d _,..,, y ^„ , J -S 2Bc Similarly (mn)'^n~^ = Am — ^n H — ^ , ^ ' 2e 46c\ and ' m~' (mn)in-^ =A+B (mn)~' = — -rj.mn + (A—B — -j^ — ] . And this being true (mutatis mutandis) for all values q, it follows that the function expressed by any succession of products of functions of m and n is reducible to the form of a linear expression in m, n, m?i, in which the 4 monomial coefficients are known or determinable functions of the parameters to the corpus m, n. The latent function to any such linear expression, say Amn + Bm + Gn + D, may be found in the same way as the latent function to mn has been found, namely, as follows: (Amn + Bm + On + Dy = A^ (mnf + AB (mnm + mmn) + AC (mnn + nmn) + 2ADmn + B'm^ + BG (mn + nm) + (7»n» + 2BDm + 2GDn + Z>^ = A'(-2e+ 46c) mn - AHf+ ABm (2bn + 2cm - 2e) + AG(2bn + 2cm -2e)n + 2ADmn + B^m^ + BG(2bn + 2cm - 2e) + C7W + 2BDm + 2GDn + J>. Let (Amn + Bm + Gn + Df - 2P(Amn + Bm+Gn + B) + Q=0 be the identical equation to Amn + Bm + Gn+D. The coefficient of mn in the development of the first term being (46c - 2e) A^ + 2bAB + 2c.4a + 2AD, m 31] of Universal Algebra 219 and m?, n" being reducible to linear functions of m, n respectively, it follows that P = A{nc-e)^Bh + Gc + D. To find Q it is only needful to fasten the attention upon the constant terms in the before named development reduced to the standard form. These will be - AHf- 2ABcd - 2ACbf- BH - 2BCe - G'f+ D\ say K, and the constant part in —2P{Amn+Bm+Cn+D) being— 2Z>P, it follows that Q = 2AD{2hc -e) + 2BDb + 2GDc + IP - K = A'df+ 2ABcd + 2ACbf+ 2AD (26c - e) + B^d + 2BCe + C^/+ 2BDb + 2CDc, and consequently the latent function A'' — 2PA + Q, of which the algebraical roots are the latent roots of Amn + Bm + Cn + D, is completely determined. Thus, for example, if the latent function ofm + n is required, making A=D = 0, £ = (7=1, its value will be seen to be A''-2 (6 +c) A +d + 2e +/=0, so that the roots will be 6 + c ± '/{{b + cf -{d + 2e +/)}. On Involution. In general, if m and n be two given binary matrices, and p any third matrix, say m= ... n= ' % p = nm = 10 U U ' Tj T4 ' jT g J4 p may be expressed as a linear function of T, m, n, mn or of T, m, n, nm. For in order that p may be expressible under the form A + Bm + On + Dnm, observing that ^1X3 + ^3X4 ^jTj + ^4X4 and that T = ^ " it is only necessary to write A+Bt^ + Cr^ + D (<,T, + <3T,) = Ti, || jB<,+ CT,+ 2)(<,Ti + <4T,)=r„ 5«, + Ct, + Z> («, T, + <,T4) = ^3 , f ! and then A, B, C, D may be found by the solution of these four linear equations : and this solution must always be capable of being effected unless Lthe determinant 1, U, Ti, ) m + (0 + 2bD) n - Dnm = 0. The above determinant then will be called the involutant to m, n or n, m, indifferently, for it will be seen, and indeed may be shown, a priori, that its value remains absolutely unaltered (not merely to a numerical factor pres, but in sign and in arithmetical magnitude as well) when the Latin and Greek letters, or which is the same thing, when the matrices m and n are interchanged. On the lAnearform or Summatory Representation of Matrices, and the Multiplication Table to which it gives rise. This method by which a matrix is robbed as it were of its areal dimensions and represented as a linear sum, first came under my notice incidentally in a communication made some time in the course of the last two years to the Mathematical Society of the Johns Hopkins University, by Mr C. S. Peirce, who, I presume, had been long familiar with its use. Each element of a matrix in this method is regarded as composed of an ordinary quantity and a symbol denoting its place, just as 1883 may be read 1^ + 8A + 8« + 3m, where 6, h, t, u, mean thousands, hundreds, tens, units, or rather, the places occupied by thousands, hundreds, tens, units, respectively. Take as an example matrices of the second order, as a /3 a b 78 c d. These may be denoted respectively by a\ + 0lj.+ yv + Btt, aX + bfi + cv + d-rr; their product by (aa + c/3) \ + (6a + d/3) fi + (ay + cS) p ■¥ (by + dS) tt, which therefore must be capable of being made identical with aa\' + a/3X/i + arfKv + aSkir + boifi\ +b^fi'' +byfj,p +bSfnr + cavX + c^v/j. + cyv^ + c^mr + dcL-rrX + d^irix + dyirv + dhir'. % 31] of Universal Algebra 221 when a proper system of relations is established between the quadric com- binations and the simple powers of \. The arguments of like coeflScients in the two sums being equated together, there result the equations V = X, Xl' = V, fjX = /J,, flV = TT, Vfl = \, VTT^V, ■Trfl = H, 77^ = TT, and again, the arguments to the 8 coefficients in the second sum which are not included among the coeflScients of the first, being equated to zero, there result the equations X/i = 0, XtT = 0, ft- = 0, /iTT = 0, i,\ = 0, 1/2 = 0, 7rX = 0, ■7rv = 0. These 16 equalities may be brought under a single coup d'oeil by the follow- ing multiplication table : \ V fl IT X X I. 0 0 1^ 0 0 X v /» fl n 0 0 TT 0 0 /i »r a b c In like manner it will be found that any rnatrix of the 3rd order as ci e f, g h k regarded as a quantity, may be expressed linearformly by the sum aX -f i^ -f- cv + dTT -I- e/a +/<7 + gfT + Aw + i<^, where the topical symbols are subject to the multiplication table below written : X IT r p- p t; V a 0 0 0 times, and the remaining &>* —

= 0, and , = Jl+25^,. Hence [{^B^ - Df - 2h {'^B' - D)-\- d] x" -2 [ii^B" - D) BD - dA.B'- D)- 2bBD + e} X + 4,&]>-2cBD+f=0. Hence we may write ('iB' - Dy - 2b{'iB' - D) + d = \, 2{'tB'-D)BD-c{'iB'-D)-2bBD + e = \B, 4,B'D'-2cBD+f=\D; from which equations B and D are to be determined. Eliminating \ between the first and second and between the first and third of these equations, we obtain two equations, of which the arguments are If; B'D',1)'; B*D, BB, BD, D ; 1 for the one, Biy; B'D.BD.D; B',B',B',B; 1 for the other. Eliminating D by the Dialytic method between these two equations, we shall have (using points to signify unexpressed coefficients) the following three linear equations in B', D, 1, namely: ■BI> +(-5' + &c.)Z)+(-5» + &c.) = 0, ■ B'D' + (■ B> + &c.) D + (-B' + &c.) = 0, • B'D' + (-B' + &;c.)D +(-B> + &c.) = 0. Hence in the final equation B rises to the 15th power; and by com- bining any two of the above equations, D is given linearly in terms of B; and, finally, a; is known from the equation _:(p + D-'iB'-2b)(q + 2BD) * - _(45i_ />)2_ 2(4/^ - i>) + d' and has 15 values. A like process may be extended to a unilateral equation (of the Jerrardian form) of any degree, say af + qx + r = 0. Introducing the auxiliary equation with scalar coefficients as before, namely x'-2Bx-\-D = 0, X may be expressed as a function of q, r, B, D; and the term containing the • I ose^ and -v^ to signify M~^L and LM~^ Tespectively. 15—2 228 On the Solution of a CUm of [32 highest power of B in the equation for determining B (of which D is a one- valued function), when a, = 4, will be found to be the determinant -B B' •£" •B' .£• .£» .£' 'B> .B" •B' ■B> -B" .£' •£• -B" -B"* and a similar determinant will fix the degree of B in the resolving equation for any value of to. Hence the number of solutions of the unilateral equation in quaternions of the Jerrardian form of the degree o) is o) (2a, - 1) or 2a,' - ' + «■>, or {(O" -to) (to' + CO- 1). Moreover the same method with a slight addition will serve to de- termine the roots of the general unilateral equation in quaternions, the number of which will be a cubic function of «, as I propose to show and to give its precise value in some future communication, either in this Journal, or at all events in the memoir on Universal Algebra now in the course of publication, under the form of lectures, in the American Jourmil of Mathematics^. I very much question whether the old method of Hamilton, as taught by its most consummate masters, Tait in this country, or the late Prof. Benjamin Peirce in America, would be found sufficiently plastic to deal effectually with an analytical investigation in quaternions of this degree of complexity, so as to lead to the formula for the number of solutions of the unilateral equation of the Jerrardian form above given. I invite my much esteemed and most capable former colleague and former pupil DrStory.of the Johns Hopkins, and Prof Stringham, of the University of California, who carry on the traditions of the Harvard School, to put the power of the old method as compared with the new to this practical test. Postscript— U a^-5pa?+5qa;-r = 0, (where p, q, r are perfectly general matrices of the second order which satisfy the general equations q'-2bq + d = 0. qr+rq-2bq-2b,q + 2e = 0, r'-2b,q + d, = 0, pq + qj)-2bp-2fiq-^2e = 0, p»-2^p + S = 0, pr + rp- 2bip - 2^r + 2ei = 0), • It may readily be seen that the highest term in the equation for finding B is identical with the resultant of ^^ _ ^^^^ ^ ^^^^^ ^^^ ^^^ _ ^^^,^, ^ g^,^ _ g^^^, that is will be 2'8.8.7.19iJ=«; and that the last term (at aU events to the sign prlt) will be b*S\ which is of 4 . 3 + 2 . 2 . 4 {that is of 28) dimensions in *, and is therefore codimensional (as it ought to be) with B^. t It is given in the Postscript below. I 32] Equations in Quaternions 229 : r + .SDp px= — 2Bx + D = 0, BD and if we write a? ■ and I find by perfectly easy and straightforward work that B, D may be determined by means of the following equations: (B'-BY B^ — D ^ — 9—^+2 {h - PB) ~^~ + {d- 2eB + 485=) = 9\, B'D — BD' B^ — D g + ih + 3/3i)) -^ + (e-e,B + 3eD- 6BBD) = 3B\, B'D^ - 2 (6, + 3/3i)) BD + d, + QDe, + 981)' = D\. The order (by which I mean the number of solutions of this system of equations) is readily seen to be the same as that of D' +B'D+B*D = 0 ■BBp+B^D+R =0; that is, is the same as the degree in 5 of fi' (£»)' . R, where R is the resultant of • D^ + ■ 5» + ■ 5^ and • D^ + ■ Br-D + ■ B\ Hence* the number of solutions is 3 + 10 + 8, that is, is 21. Practically, therefore, we have now suflBcient data to determine the number of solutions of a unilateral equation in quaternions of any order w; for it is morally certain that such number is a rational function of m ; and as it cannot but be of a lower order than &>*, we have only to determine a cubic function of o) whose values for w = 0, 1, 2, 3 are 0, 1, 6, 21, which is easily found to be (o' — (o' + m; so that the evaporation is tu* — o)* + w' — >+l)(a>'-a>). Practically also we can solve (subject to hardly needful verification) the number of roots of a unilateral equation of the special form sr + q^a^ + qt-ia^~' + ... + qo = 0. For when ^ = &>, we know the number is a)'; and when ^ = 1, the number is «*' + <»* — <»; consequently if the second differences of the function of (&>, 6) which ejtpresses the number of roots are constant, the value of this function when ^= &) — 1 is ft)' — a>' + ft), which we have found to be the actual number; and consequently, if the second differences are not constant, they must be sometimes positive and sometimes negative, which is in the highest degree improbable. Hence in all probability it will be found that the required number of solutions in the form supposed is (1 + 6)0}' — 6m. I need hardly add that the nine quantities 26, 26,, 2/3; 2e, 26,, 26; d.h.d^, which occur in the discussion above given of the general unilateral cubic, or, say, rather the ten quantities obtained by adding on to these unity, are the [* See footnote t p. 197 above.] 230 On, the Solution of a Class of Equations, etc. [32 ten coefficients of the determinant to the binary matrix {x ■\- py -^^ qz + rt), which of course there is not the slightest difficulty in expressing in terras of scalar and vector affections of p, q, r and their combinations, if any one chooses to regard them as given in quaternion form. Scholium. In what precedes it is very requisite to notice that only general cases are considered; and that there are multitudinous others which escape the direct application of this method, and do not conform to the rule which assigns the number of solutions. Thus, for example, the equation a^ + px = 0, besides the solutions x = 0,x = —p, will have two others which will require the method of the text to be modified in order to determine. Or take the most elementary case of all, the simple equation pa; = q. If p is not vacuous (that is, if its determinant when regarded as a matrix, or its modulus when regarded as a quaternion, is finite), there is the one solution x=p~^q. But if p is vacuous, then, unless q is also vacuous, the equation is insoluble. If g = 0, there will be two solutions ; one of theni x = 0, the other x = con- jugate of p in quaternion terminology; or — d; b , a; b x= , when «= ' , c; — a ^ c; a in the language of matrices. If, p still remaining vacuous, q is vacuous but not zero, a further condition must be satisfied, namely, if a; b , o; /9 p- , and q= j, ^ c; a ^ y; 6 the condition is aB + ad — by — c^ = 0; or if p'=a + bi + cj + dk and q=a + ^i+yj + Sk, the condition is aa + b^ + cy + dB= 0. When this condition (besides that of q being vacuous) is satisfied, the equation px = q is soluble, and p~^q becomes finite but indeterminate, containing two arbitrary constants*. * So in general if p, q be two simply vscnons matrices of any order, the condition that the equation px=q may be soluble, or, in other words, that p~'g (a combination of an ideal with a vacuous matrix) may be non-ideal, may be shown to be that the determinant to the matrix \p + nq (where X, fi are scalar quantities) shall vanish identically— which (p being supposed already to be vacuous) involves just as many additional conditions as there are units in the order of the matrix. 4 33. [. ON HAMILTON'S QUADRATIC EQUATION AND THE GENERAL UNILATERAL EQUATION IN MATRICES. [Philosophical Magazine, xviii. (1884), pp. 454 — 458.] In the Philosophical Magazine of May last I gave a purely algebraical method of solving Hamilton's equation in Quaternions, but did not carry out the calculations to the full extent that I have since found is desirable. The completed solution presents some such very beautiful features, that I think no apology will be required for occupying a short space of the Magazine with a succinct account of it. Hamilton was led to this equation as a means of calculating a continued fraction in quaternions, and there is every reason for believing that the Gaussian theory of Quadratic Forms in the theory of numbers may be extended to quaternions or binary matrices, in which case the properties of the equation with which I am about to deal will form an essential part of such extended theory*. Let us take a form slightly more general than that before considered, namely, the form pa^ + qx + r = 0, with the understanding that the determinant of p (if we are dealing with matrices), or its tensor if with quaternions, dififers from zero. Let us construct the ternary quadratic aw' + 2buv + 2cuw + dir" + 2ei>«; +fu/', defined as the determinant of up + vq+ wr, on the one supposition, or by means of the equations a = Tp', d=T x be any quadratic factor of this biquadratic function in a;: I have found that ^ = 0 will be the identical equation to one of the roots of the given equation fx = 0, where fx = px^ + qx + r. Between the two equations yic = 0, (f)x=0, a^ may be eliminated and x found in terms of known quantities : x will have six different values, which will give the six roots of/a;=0. It is far from improbable that a similar solution applies to a unilateral equation /ic = 0 of any degree n in matrices of any order «a. Call Fx the determinant offx when x is regarded as an ordinary quantity; then, if x is an algebraical factor of the degree o) in a; contained in Fx, it would seem to be in all probability true that (f)x = 0 is the identical equation to one of the roots of fx = 0 ; and, vice versa, that the function identically zero of any such root is a factor of Fx. By combining the equations fx = 0, ^ = 0, all the powers of x except the first may be eliminated, and thus every root of X determined. The solution of the given equation will depead upon the solution of an ordinary equation of the degree n may be, and for the case of n = 2, w = 2-f-. * From the number 21 above referred to, now known to be erroneous, the general value was inferred to be n^ -n- + n, whereas it is demonstrably 2n^-n only for the general unilateral equation of degree n in quaternions, as I proved it to be for the Jerrardian form of that equation. t I have since obtained an easy proof of the truth of the conjectural theorem for all values of n and a; see the Comptes Rendus of the Institute of France for October 20th last [p. 197 above]. 34. NOTE ON CAPTAIN MACMAHON'S TRANSFORMATION OF THE THEORY OF INVARIANTS. [Messenger of Mathematics, xiii. (1884), pp. 163 — 165.] The whole question as is well known consists in finding the free forms of fi~'0, where il = aoSa, + 2aiSa2 + ... + iof-iSoj; but, as long ago noticed by me* in the Atti. Math. Journal, il~'0 is only a deformation of V~^0, where F= ajSai — a^Sa^ + ... ± ai_iBai, n~'0 being deducible from F~'0 by altering the dimensions of the a elements which it contains in known numerical proportions, so that X1~*0 may be said to be V~^0 subjected to a known strainf. To fix the ideas let i = 3 and call the a's by the names a, b, c, d or, for greater simplicity, 1, b, c, d. Let b=r + s + t, c = rs + rt + St, d = rst. Then the matrix so that D{b,c,d) \ ^ rtV \l = S-ift t + r r + s, D(r,s,t) st tr rs , (r-s){r-t) {s-r)(s-t) (t-r)(t-s)' D {r, s, t) r s t D{b,c,d)~ {r-s){r-t) {s-r){s-t) (t-r)it-s)' 1 1 1 (r-s)(r-t) (s-r){s-t) (t-r){t-s)' [* Vol. III. of this Reprint, p. 570.] t In fact the numerical multipliers of the terms in 0 may be taken perfectly arbitrary without prodaoing any effect upon the form Q-'O than what may be represented by a strain. I 34] Note on Captain MacMahon's Transformation, etc. 237 Consequently y _^r' — jr -^^ a + t)r + {rs ■¥ rt + st) ^ _ ^ st - {r-s){r-t) ^'--^(r-sXr-O In like manner in general for \, a^, a^, ... at we shall find, on writing ai = ri + r2+ ... +ri, 0-3 = n^j + r^rs + . . . + n^iVi, ai = rir^...ri, V= Sui - OiSOi + ... ± Oi^iBoi = 2 , .,"'"'* r Sn. (ri-r^){r,-r,)...{n-ri) Hence V-'O = i'(«i, s„ ... Si), ■ where, in general, s^ = r," + rj" . . . + Vi" ; and consequently the theory of invariants, which endoscopically treated in the ordinary way hinges upon symmetrical functions of the differences of a set of letters, is made to depend upon functions of the simple sums of powers com- mencing with the second power and ending with a power whose index is the order of any given finite quantic, but in the case of perpetuants taking in all the powers except the first. It goes without saying that the same method applied to the constrained V will show that it is equal to SSr,, so that Fo~' is an arbitrary function of the differences of the r's corresponding to that hypothesis, as we know ought to be the case. What has been established in the foregoing investigation is a principle of correspondence whose importance as a simplifying agent recalls Ivory's use of such principle in Attractions, namely, the remarkable algebraical law that any symmetrical function of the differences of a set of i quantities is a symmetrical function of the sums of the 2nd, 3rd, ..., ith powers of another equi-numerous set. By virtue of this principle the numerical part of the Calculus of Invariants is capable of being entirely divorced from all question of algebraical content and a Zahl-Invariant theory comes into being, in its fundamental conception analogous to the Zahl-Geometrie of Schubert. Further remarks on this subject will be found in the Comptes Rendus de I'Institut presumably for March 31 and April 7 of this year [p. 163 above]. 35. ON THE D'ALEMBERT-CARNOT GEOMETRICAL PARADOX AND ITS RESOLUTION. {Messenger of Mathematics, xiv. (1885), pp. 92 — 96.] I WILL presently state the simple geometrical problem which led D'Alem- bert to call into question the validity of the received Cartesian doctrine of positive and negative geometrical magnitudes, and which, according to Camot, furnishes an unanswerable argument against it. See Mouchot, La refwrme Cartesienne, pp. 74, 75. Against this doctrine, presented in its crude form, the objections of these illustrious impugners of it are unquestionably well founded and unanswerable; but the inference to be drawn from this is not that no such or such-like doctrine reposing on an unassailable logical basis exists or is capable of being established (woe worth the day! when such a conclusion should be admitted), but that the doctrine as usually stated is incomplete and requires a supplement. This has been anticipatively furnished by me many years ago in this very Journal, and in conjunction with the substitution of positive and negative indefinite rotation in lieu of Euclid's positive and limited angular magnitude, made the basis of a strictly logical deduction (which was before wanting) of the trigonometrical canon. It consists in the notion of a line having, so to say, sides (returning upon itself at its two semi-points at infinity), or to put the matter in a more practical form, in regarding an Euclidean indefinite straight line as repre- senting two distinct lines locally coincident, but running in contrary directions, and in referring the algebraical sign of any rectilinear segment to the con- currence or discordance of its flow (which is represented by the order in which its two extremities are named or written down) with that of the indefinite line, upon which it is supposed to be carried. « 35] On the D'Alembert-Camot Geometrical Paradox, etc. 239 Thus, for example, AB taken on the upper side of a line or line-pair will be the negative of AB taken on the same side, but the same as BA taken on the under side. I will now state the D'Alembert-Caraot problem. "Voici" says Carnot, "un exemple aussi simple que frappant, qui seul suffit pour renverser toute cette doctrine" of positive and negative magnitudes. "D'un point K, pris hors d'un cercle donn^, soit propose de mener une droite Kmm', telle que la portion mm', intercept^e dans le cercle, soit egale k une droite donnee. K "Du point K, et par le centre du cercle menons une droite KAB qui rencontre la circonfdrence en A et B. Supposons KA = a, KB = h, mm' = c, Km = X. On aura done par les propri^tfe du cercle ab = x{c + x)=:cx + a? done a? + ex — ab = 0 ou x= — \c±*J{\(i' + ah). X a deux valeurs: la premiere, qui est positive, satisfait sans difficult^ k la question; mais que signifie la seconde, qui est negative? II parait qu'elle ne peut repondre qu'au point m', qui est le second de ceux oil Km coupe la circonf^rence ; et, en eflFet, si Ton cherche directement Km', en prenant cette droite pour I'inconnue x, on aura x(jc — c) = ab on x = \c ± >J{\(? + ah) dont la valeur positive est pr^is^ment la mSme que celle qui s'dtait pr^sent^e dans le premier ca.s avec le signe n^gatif Done, quoique les deux racines de liquation soient I'une positive et I'autre negative, elles doivent etre prises toutes les deux dans le mSme sens par rapport au point fixe K. Ainsi, la rfegle qui veut que ces racines soient prises en sens opposes porte k faux. Si au contraire le point fixe K ^tait pris sur le diamfetre meme AB et non sur le prolongement, 240 On the D'Alembert-Carnot Geometrical [35 on trouverait pour x deux valeurs positives et cependant elles devraient dtre prises en sens contraires i'une de I'autre. La regie est done encore fausse pour ce cas. " Si Ton dit que ce n'est pas ainsi qu'il faut entendre ce principe, que les racines positives et negatives doivent dtre prises en sens opposes, je de- manderai comment il faut I'entendre ? et j'en conclurai par la mfeme qu'il faut une explication pour empecher qu'il ne soit pris dans I'acceptation la plus naturelle. II suit que ce principe est obscur et vague." The answer has been already given to the question, "comment il faut entendre ce principe," and it will be seen in such a way as to remove all grounds for the charge of its being any longer "obscur et vague." This is how the problem set out in full ought to be enunciated : A complete line (that is, a line-pair or two-sided line) drawn from K cuts the circle in the points m, m'; mm' measured on either side of the line (and of course denoted quantitatively by the number of units of given length which it contains) is to be equal to c a given positive or negative number. Required the value of Avi. (1) Suppose K to be exterior to the circle as in the diagram above. I distinguish the two sides of the complete line, as the under and upper line, and suppose the flow of the under one to make an acute Euclidean angle with the flow from K to the centre of the circle. In all cases Km' = Km + mm', and consequently the equation for finding x remains always sc' + cx = ah, of which the two roots are —hc + n/(\c^ + ab) and — ^c — V(ic'-t- a6). Adhering to the letters of the diagram, if c is positive the two values of a; will correspond to Am on the under line and Am' on the upper line of the line-pair. If, again, c is negative, the two values of x will correspond to Am on the upper and Am' on the under one. (2) Suppose K to be within the circle. It will still be true (paying attention to the signs) that Km' = Km + mm' (that being a universal identity in algebraical geometry), but the algebraical values of KA, KB being contrary, we may regard KA as positive and equal to a, KB as negative and equal to — b, and shall have the equation x' + cx = — ab, of which the two roots are -hc + »/(i(^-abj, --^c-VdC-ai). Understand by the two segments Km and Km'. I ■ It^ 35] Paradox and its Resolution 241 We may suppose the indefinite line-pair mKm to swing round K, its under-side in the position of coincidence with the diameter having the same flow as KA\ then, if c is positive, until the swinging line revolving with the sun has described a right angle, the first root will be the in/rd-diametral segment taken on the lower line (or side), and the second root the suprd- diametral segment taken on the upper line (or side) of the line-pair (or complete line); in the next quadrant of rotation the first root will be the «M/w«-diametral segment on the under and the second root the infrd- diametral segment on the upper side of the complete line. When c is negative a similar statement may be made it only the words under and upper are interchanged. In the critical position, when the swinger is at right angles to the diameter, the two roots become equal and undistinguishable ; but throughout and subject to no exception, the complex of the two roots contains the complete solution of the problem, and the complete solution of the problem necessitates the retention of the complex of the two roots. Thus, then, as in the preceding case, it has been shown that the Cartesian view of the equipollence of positive and negative roots (the latter Descartes influenced by hereditary prepossessions calls radices falsae) is made exact through the intermediation of the conception of sides to a line. D'Alembert and Carnot are entitled to the gratitude of Geometers and all lovers of truth for raising objections so perfectly well founded to the then, and even now, too prevalent interpretation of the meaning of the geometrical positive and negative, but the difficulty which they so justly appreciated and so clearly expressed is overcome and exists no longer. P.S. I am informed that M. Laguerre has emitted the same view as that I have set forth relative to the sign to be given to geometrical distances, and made use of the same conception of the double or complete line-carrier. My note on the subject appeared before my exodus across the Atlantic, probably nine or ten years ago. M. Laguerre 's publication must have been many years posterior to this. The references to the reappearance of the theory on the other side of the Channel, obligingly furnished to me by M. Mannheim in Paris, have unfortunately got mislaid. I believe the com- munication containing it was made by M. Laguerre within the last three or four years, but it has already had time to find its way into some of the most esteemed French text-books. Being not only true but the truth, it must eventually find universal acceptance. It is not without interest (it seems me) that we may regard a double or complete right line as a sort of embryonic embodiment of the idea of a Riemann Surface. 8 IV. 16 36. SUR UNE NOUVELLE TH^ORIE DE FORMES ALG^BRIQUES*. [Comptes Rendus, ci. (1885), pp. 104.2—1046, 1110—1111, 1225—1229, 1461—1464.] Si Ton imagine une fonction de d^rivees diffdrentielles (toutes d'un ordre sup^rieur k I'unit^) de y par rapport k x, qui, sauf I'introduction d'un facteur multiple num^rique, d'une puissance de t^ , ne change pas sa valeur quand on remplace x par y ei y par x, il est evident qu'une telle fonction restera invariable (sauf Tintroduction d'une constante comme facteur) quand pour x et y on substitue des fonctions lin^aires quelconques, homogenes ou non homogenes de y et x. Ainsi une telle fonction couduira immMiatement k la connaissance d'un point singulier d'une courbe d'un degre quelconque. Le seul exemple d'une telle fonction, traits jusqu'4 ce jour, est la simple fonction dJ'v -7^ qui, par cette seule propri^td, sans aucune autre consideration, sert k d^montrer I'existence d'une propri^t^ projective de courbes dont la condition est -T^ = 0. II nous parait done tr^s utile de chercher un moyen de produire toutes les fonctions de cette espece auxquelles nous donnerous le nom de rdciprocants purs ou simplement r6ciprocants. On verra qu'il existe des r^ciprocants mixtes, c'est-a-dire contenant des puissances dcT^f comme la dy d'v 3 d^v d''y\ forme bien connue de M. Schwarz, -^ -^ — ^ -~_, -~] qui possfedent la m6me facultf^ d'invariance par rapport a I'echange de y avec x, comme les rdciprocants purs, mais qui dvidemment ne peuvent pas indiquer I'existence de points singuliers dans les courbes. Nous ^crirons, au lieu de 8xy, Bx'y, ^xy, Bx*y,..., les lettres t, a, b, c, ..., et pour leurs r^ciproques B^x, By'x, S/x, ..., t, a, /3, 7, On verra facilement que, pour que F(t, a, b, c, ...) soit un reciprocant pur, F doit etre d'un degr^ et d'un poids constant dans les lettres de chaque terme ; de plus (pour uu [* See the lectures, below p. 303.] 36] Sur une nouvelle tMorie de formes algehriques 243 r^ciprocant ^d'une nature quelconque), on aura i^(a ...)/^ (a ...) = (— 1)*<^, oii 6 sera le plus petit nombre des lettres a,b, c, ... dans un terme quelconque de F, et X sera la moyenne arithmetique entre le poids et trois fois le degr^ de F, en comptant le poids de t, a, b, c, ... comme etant — 1, 0, 1, 2 Cela donne lieu a une remarque importante par rapport aux re'ciprocants rrdxtes: pour qu'on puisse additionner deux formes mixtes afin de former un nouveau reciprocant, il faut non seulement que le degr6 et le poids soient les memes pour tous les deux, mais aussi le caractire qui depend de la valeur de 6 et que Ton peut qualifier comme caractere pair ou impair selon la parite de 6. Ainsi, par exemple, Ith — 3a' et a? sent tous deux r^ciprocants, mais 2i6 ne le sera pas, parce que les caracteres des deux donnees sont contraires. II est facile de d^montrer que, si R est un reciprocant quelconque, {2tb - 3a') BaR + {2tc - 4a6) Sji? + {2td - 5ac) S^iJ + . . . sera aussi un reciprocant de mfeme caractere que R. Ainsi, en commen9ant avec le reciprocant a, on peut obtenir une suite infinie de reciprocants mixtes : ces reciprocants ainsi obtenus ne seront pas en general irreduc- tibles ; mais, sans les reduire, leur forme fait voir imm^diatement que tout , reciprocant, qu'il soit pur ou mixte, peut etre exprim^ comme une fonction rationnelle et aussi (si Ton regarde t comme unit^) entifere de combinaisons ' legitimes* de ces quantity. Pour obtenir tous les reciprocants purs de poids, degr^ et ordre (c'est-k- dire nombre de lettres) donnes, lin^airement independants les uns des autres, on peut former une Equation partielle differentielle, lin^aire, oii R est la variable d^pendante, et a, b, c, ... les variables ind^pendantes ; elle exprimera la condition n^cessaire et suffisante pour que R soit un tel reciprocant et foumira un moyen sftr de resoudre le probleme propose. Voici la maniere de demontrer ce theorfeme fondamental. Si, dans I'equation F{a.b,c....) = (-l)*t^F(a,ff,y,...), on donne k y \& variation ea;, on voit que a, i, c, ...,et consequemment F, restent in variables. Les variations de a, /S, 7, ... sont faciles a determiner, et la variation de t est donnee. Ainsi, apres quelques calculs faciles, en egalant k zero, separement, dans la variation de t^F{a, 0, ...), les termes qui contiennent t et ceux qui ne le contiennent pas, on arrive a deux equations dont I'une sera Je nomme Ugitime nne combinaison qaelconqae de reciprocants oi Ton evite d' additionner eenx dont le poids, le degr^, I'ordre et le earactire ne sont pas les mSmes pour tous. 16—2 244 Sur une nouvelle theorie [36 qui exprime la valeur num^rique de X, comme fonction du poids et du degr^ de F; I'autre equation, eu ^crivant F= Sa'Sft + lOaftS, + (15ac + 106') 8^ + (21arf + 356c) S, + (28ae + 566d + SSc") S/+ .. ., sera VR=0. Pour voir la loi des chiffres arithmetiques dans V, formons les suites des coefficients de (1 + «)' en commen^ant avee i= 4 ; divisons chaque coefficient central en deux parties ^gales, et suppriiiions la derniere moiti^ des series num^riques ainsi forraees ; on obtiendra ainsi la Table : 14 3 1 5 10 1 6 15 10 1 7 21 35 1 8 28 5G 35 En n^gligeant les deux premieres colonnes, on trouve les nombres qui paraissent dans la formule. On demontre ainsi que Fi? = 0 est une condition n^cessaire pour que R soit un reciprocant. Mais il faut aussi d^montrer que cette condition est suffisante. Soit done D la valeur de i^(a, 6, ...)— t"^^(a, yS, ...), exprim^e comme une fonction de a,h,c,... seulement. D sera done une fonction de la meme forme que F(a, 6, ...). On suppose que AD = 0 ; c'est-a-dire que la variation de D produite par la substitution dQ x+ ey k x est dgale a z(5ro, en vertu de I'^quation VR = 0. Donnons k y une variation arbitraire y + rju; alors, si I) devient £>', la variation de D' sera nuUe, quand on substitue, pour as, x+ey + er)U, et, con- sdquemment, quand on substitue x -\-2y pour x ; on aura done AD' = 0, et, en prenant la difference des variations de D et D', on obtient a(u'^ R + u"~R + u"'^R + ...)=0. \ da do ac I Done, h. cause de la forme arbitraire de u, il faut que ^l;^-"' '^k"'"- ■■■■■ et, en raisonnant sur ^- B, -r. D, ... comme on a raisonn^ sur D, on voit que da do ^ le A de chacune des deriv^es secondes diff^rentielles de D sera z^ro; en I 36] de formes algebriqnes 245 poursuivant le meme calcul, on trouve ^videmment que le A d'une d^riv^e de D d'un ordre quelconque par rapport a a, 6, c, ... sera nul. Done D est nul ; car, dans le cas contraire, s'il contient un terme quel- conque, dent les lettres peuvent etre distinctes ou identiques, en isolant une seule de ces lettres et prenant la d^riv^e de D par rapport k toutes les autres lettres, on aura le A de la lettre isolee, c'est-a-dire de S^y, h^y,..., zero quand on substitue x + ey pour x, ce qui est absurde. Ainsi Ton voit que, quand AD = 0, c'est-^-dire quand VR =0, D = 0, ce qui 6tait a d^montrer. Soient a>, i,j le poids, le degre et I'ordre d'un r^ciprocant quelconque : de meme que pour les sous-invariants, le nombre de fonnes lineairement inde- pendantes s'exprime par(a); i,j) — (w — 1; i,j), oh, en general, (to; i, j) signifie le nombre de partitions de a en i parties dont nuUe n'excede j; ainsi Ton voit que, en vertu de lequation VR = 0, on aura, pour le nombre des r^ciprocants lineairement independants, la formule (w; t,j)- (<--!; i+l,j). Mon long exil en Amerique expliquera, je I'espere, comment j'ai pu ignorer I'identitd des invariants differentiels de M. Halphen avec les formes que j'ai aomm^ea rSciprocants purs. Les travaux vraiment remarquables de M. Halphen n'ont pas besoin de mes ^loges et auront ^te couronnes par I'admiration de toua les gdometres dignes de ce nom. Je crois cependant qu'il y a assez de difference entre le but et la marche de mes recherches sur ce terrain et ceux de M. Halphen pour justifier I'in- sertion dans les Comptes rendus de ma discussion de la th^orie regardee comme une theorie de formes algebriques. Si je ne me trompe pas, M. Halphen, s'il I'a d^couverte, n'a fait nul usage de I'equation partielle difffirentielle que j'ai donnee et qui sert h. dtablir le parall^lisme mer- veilleux entre les invariants differentiels et les semi-invariants ordinaires. De plus, il n'a pas eu occasion de faire allusion atix formes que j'appelle rddprocants mixtes orthogonaiuc, qui ne sont point compris dans la definition des invariants differentiels, et qui sont essentiels pour expliquer les singu- larit^s quasi-metriques des courbes. Nous rappelons que par le mot redprocant (sans qualification) il a ^te convenu de sous-entendre une forme de cette espfece qui ne contient pas t Ic'eat-k-dire j^j et nous avons trouv^ que le nombre de ces reciprocants lineairement independants, du degr^ i, de I'etendue j (c'est-k-dire contenant j+1 lettres distinctes) et du poids <», s'exprime par la formule («; hjy-ioi- 1 ; i + l,j), oh en general {I; m, n) signifie le nombre de partitions de i en m ou un plus 246 Sur une nouvelle thdorie [36 petit Dombre que m de parties dont aucune n'excede n en grandeur ; de sorte que (l; m, n), quand m est plus grand que I, signifie la m^me chose que {I; I, n), car tous les deux sont equivalents ^ (/; oo , n). Cons^quemment (i; i, i) - (i - 1 ; i + 1, j) = (i; i, j) - (i - 1 ; i. j). lequel sera toujours positif quand i et j sont tous les deux plus grands que I'unit^ ; et, puisque a, qui est du degr^ 1, est un r^ciprocant, il s'ensuit que, pour un degre quelconque donn^, il existe toujours des reciprocants (car on pent faire a> = i), mais en nombre fini, car, en faisant croitre to, (oi — 1 ; t'+l, oo), au dela d'une certaine valeur de to, deviendra n^cessairement plus grand que (w; i, x ). On peut exprimer par (I : m) ce que devient (/; m, n) quand n = oo , et alors (« : t) — (w — 1 : i + 1) exprimera le nombre de reciprocants lin^airement ind^pendants du poids « et du degr^ i sans autre limitation. Ainsi on trouvera que du degr^ 1 il n'existe qu'un seul r^cipro- cant du poids 0; pour le degr6 2, un seul du poids 2 ; pour le degr^ 3, deux qui seront respectivement du poids 3 et du poids 4 ; etc. On trouvera qu'etant donnd _;' il existe toujours, sauf pour le cas ou j = 1, un r^ciprocant qui contient toutes les j +\ lettres et qui de plus contiendra un terme qui est un produit de la derniere lettre par une puissance de a. Ces formes, qu'on peut nommer les protomorphes, sont les analogues des formes a, ac — b', a'd + ..., ae + ..., qu'on connait dans la th^orie des sous- invariants. Dans le cas des reciprocants, ces protomorphes seront a, ac, ..., a'd a'e, .... a'/, ..., a'g, ..., etc. Evidemment une fonction rationnelle quelconque des lettres peut, au moyen de substitutions successives, Stre exprim^e comme une fonction rationnelle des protomorphes et de h divis^e par une puissance de a. Soit done R un rdciprocant quelconque ; on aura a»i2 + P+Q6+... + Ji* = 0, ou P, Q, ..., J" sont eux-memes des reciprocants. En operant i fois sur cette Equation avec notre op^rateur V, on voit qu'on obtient a^J = 0 ; done J est nul, et Ton voit ainsi que tous les termes Q, ..., J disparaissent et que R (en faisant a=l) devient une fonction rationnelle et entifere des protomorphes. Nous allons appiiquer ce principe fondamental, commun aux deux theories des sous-invariants et des reciprocants, pour obtenir les formes irr^ductibles (les Grundformen) des reciprocants pour les ordres 2, 3, 4. Faisons j = 2,i = 2, a = 2 et supposons que le r^ciprocant R soit Xac + ^; on obtient VR = (Sa^Bb + lOabBc) R = {6fj.+ lOX) a'b = 0. Done — \ : /t :: 3 : 5 et nous obtenons le r^ciprocant Sac — 5b'*. * II est bon de remarquer que Sac - 56^ = 0, c'est-A-dire 3j|g_5(gy=0, ax' ax* \ax'J indique que le point (x, y), quand cette Equation est satisfaite par telles coordonn^s d'nne courbe quelconque, est un point supra-parabolique, c'est-i-dire oH une parabole passe par 5 au lieu de 4 points cons^cutifs seulement. I 36] de foj'nies algebriques 247 PassoDs au cas _;' = 3, i = 3, o) = 3, et posons R = Xa'd + jMihc + vb\ On aura VR = {^a'hi + 10a68c + 15ac + 106^8^) R = (3/i + 15\) a^c + (9«' + 10/x + lOX) a-6= = 0. On aura done A' = — 5\, Oj^ = 40X, de sorte qu'on peut ecrire R = 9a^d - 45a6c + 406'. On reconnaitra immediatement que R = 0 est I'^quation diff^rentielle donn^e par Monge et retrouvee par M. Halphen a une conique et que 9 (B^'yf (S^'y) - VoK'yh^'yK*y + 40 {K^y? = 0 exprime la condition que le point {x, y) d'une courbe quelconque sera nn point d'inflexion du second ordre, c'est-a-dire un point oil une conique passe par six points cons^cutifs. Le nombre de ces points peut etre trouv^ en fonction lin^aire de n, ordre d'une courbe donnee, en operant sur cette Equation une transformation analogue k celle au moyen de laquelle on passe du systfeme y = 0, -j^ =0 au systeme Equivalent, mais dpure, 0 = 0; ^0 = 0*. Passons au cas ou j = 4, i = 3, m = 4, et ^rivons R = Xa'e + /xabd + vac' + ■rrb'c. On aura F=3a=86 + 10a6S, + (15ac + 106")8d + (21ad + 3o6c)S., et, en posant RV=0, on obtient, en Egalant s^par^ment k z6to les coefficients de a'd, d'bc, cib', les Equations 21\ + 3m=0, 35\+15/i. + 20i/ + 67r=0, 10/i4-107r = 0, • Pour le caa d'une cubiqae, le nombre de ces points d'inflexion du second ordre est vingt-sept; on d^montre facilement que ce sont les intersections de la courbe avec son covariant du degri- ordre 12 . 9. On voit immediatement, an moyen de notre thtorie connne de riiddus giomitriques, que cei vingt-sept points sont les points de la cubique oil elle est rencontr^e par les nenf faisceanx des tangentea qu'on peut mener des nenf points d'inflexion ordinaire. Car un quelconque de ces points doit ^tre tel que sa d^riv6e k I'indice 5 sera coincidente avec le point lui-m6me. On aura done 1, 1 = 1,5, c'est-i-dire 2 = 4, ce qui veut dire que le tangentiel du point est un point d'inflexion ; ce qui ^tait k d^mootrer. Soit dit, par parentb^se, que la m6me th^orie de r^siduation enseigne que le point fixe Q, oil nne cubiqae donnie sera coup^ par une autre cubique quelconque qui a en coramun avec la premiere 8 points consecutifs k nn point donn^ P, sera le troisi^me tangentiel de P et pent fitre nornm^ son tateUite ; quand le satellite coincide avee son primaire, en se servant pour le moment de la forme canoniqne pour exprimer la cubiqae donnee, et en nommant x, y, z les coordonn^es du primaire, cellea da satellite seront (d'apr^s notre th^orie exposie dans \' American Journal of Malhematict) x, y, z multiplies respectivement par des fonctions rationnelles de x', y', «', cbacune du degr6 21. [Vol. ni. of this Reprint, p. 339.] C'est nn fait depnis longtemps connu que les points primaires qui coincident avec leurs satellites (en ne tenant pas compte des neuf inflexions) sont en nombre 72. 248 Sur une nouvelle thSorie [36 et ainsi on peut ^rire R = 5a«e - 'S5abd + lac* + 356'c. Voici done le systfeme de protomorphes pour tous les ordres jusqu'au quatrieme inchisivement : «, (1) 3ac - 66», (2) 9a'd-45a6c + 406', (3) 5o'e-35a6d+7ac' + 356''c. (4) En corabinant le cube du deuxieme avec le carr^ du troisieme, et en divisant par a, on obtient la forme (analogue au discriminant) de la cu- bique, mais d'un degre plus ^levd, 405a'd» - 4050a»6cd + 1728a»c» + 1585a6'cr' + 3600a6'rf-180006«c*. f ^^^ En combinant le produit de (2) et de (4), lineairement, avec (5), on obtient 4800a'ce - 8000a6^e - 2835a^d= - 5376ac' ) ,„. - o'IbOabcd + 308()06'd + \\^Qo¥c\ j ^ ' Si Ton se borne aux lettres a, b, c, d, les formes (1), (2), (3), (5) formeront un systfeme complet de Gh~und/ormen : si on laisse entrer la nouvelle lettre e, (5) n'est plus irr^ductible, et le systeme complet de Grundformen est con- stitud par les formes (1), (2), (3), (4), (6). Tout cela se passe precisf^ment com me avec les sous-invariants avec les memes lettres : les poids des formes sont les memes pour les deux systfemes, et la seule diffi^rence essentielle entre les deux consiste en ce fait, que les trois derni^res formes subissent chacune une Elevation d'une unit6 de degr^ en passant du systeme des sous-invariants a celui des reciprocants. II est n^cessaire d'ajouter quelques mots sur les reciprocants mixtes, qui se distinguent en deux espfeces, homog^nes et h^t^rogenes. Comme exemple des premiers, on a la d^riv^e Schwarzienne 2tb — Sa^, laquelle, ^gal^e a z^ro, ne dontie aucune espfece de singularity, mais signifie seulement qu'au point (x, y) on peut mener une conique qui passera par cinq points consecutifs, en ayant ses deux asymptotes paralleles aux axes, ou bien la forme tc — bah. Comme exemple de I'autre classe, on a la forme counue (1 + t^)b — 3to^ dont I'evanouissement (pourvu que x, y soient des coordonn^es rectangulaires) signifie que le point {x, y) est un point de courbure maximum ou minimum. * Cette fonction, 6gal^e k z^ro, exprime que x, y sont les coordonn^es d'un point par od I'on peut {aire passer une parabole cubique ayant 5 points consecutifs communs k la coorbe dont x, y sont les coordonn6es. I 36] de formes algebriques 249 Nous avons remarqud, par parenthese, que lequation {\ + t')b-Ua? = Q indique I'existence d'une singularite au point dont les coordonn^es sont les X, y sous-entendus dans t, a, b de I'^quation. Mais, pour que cela soit vrai, 11 faut introduire la restriction que x, y sont des coordonnees rectangulaires. On peut donner le nom de rAxiprocant orthogonal k tout reciprocant mixte qui jouit de la propri^te de rester invariable (sauf I'introduction d'une puis- sance de i) quaud on opfere sur x et y une transformation lineaire ortho- gonale. Cela etant convenu, on peut demoatrer t'aeileiuent que le coefficient difFerentiel par rapport a t d'un reciprocant est lui-meme un reciprocant ou pur ou mixte. La proposition reciproque est aussi vraie, de sorte qu'on a le , beau theoreme suivant : Si R et -J- sont tons les deux reciprocants, alors R est un reciprocant ortho- gonal. Par exemple, le reciprocant que nous avons cit^ plus haut a pour co- efficient difFerentiel par rapport a i la Schwarzienne 2tb — 3a' ; done c'est un reciprocant orthogonal ; et, en effet, il exprime qu'au point (x, y), ou I'^quation 2tb — 3a= = 0 est satisfaite, on peut appliquer un cercle qui aura nn contact du troisieme ordre avec la courbe dont xety sont les coordonnees; au contraire, la Schwarzienne elle-nieme ne correspond pas a une singularity quelconque, car sa derivee par rapport a t, c'est-a-dire 26, n'est pas un reciprocant. J)e raSme nous avons trouv6 qu'en integrant le reciprocant 2tc — lOab par rapport a t, entre les limites t et — c — 1.5a', la forme resultante {t''+l)c-l0abt + l5a' sera un reciprocant et consequemment un reciprocant orthogonal, de sorte que I'equation (1 +t^)c-lOabt+loa'=0 sera la condition d'une singularity de la courhe f(y, x) =0 qui se rapporte aux points circulaires k I'infini*. Peut-Stre trouvera-t-on que I'int^grale, par rapport k t, d'un reciprocant mixte quelconque, prise entre des limites convenablea, conduira necessairement k un reciprocant orthogonal. Les singularites d'une courbe peuvent etre partagees en trois classes : celles de la premifere classe seront projectives et peuvent etre d^finies indifferemment au moyen de covariants de formes ternaires ou par des reciprocants purs ; * M. James Hammond, dont on oonnalt les belles et importantes dcSeouvertes dans la th^orie invariantive des formes binaires, a trouvi I'int^grale de cette Equation, que nous avons donnie dans un discours inaugural, prononcfi devant I'Unirersit^ d'Oxford, lequel va 4tre public dans le joamal anglais Nature, [p. 278 below.] 250 Sur une nouvdle thiorie [36 celles de la deuxi^me claase seront non projectives, mais n'aurout affaire qu'avec la ligne h. I'infini ; les singularit^s de cette classe seront exprimables au moyen de r^ciprocants purs, mais non pas au moyen de covariants de formes temaires. Restent celles de la troisifeme classe qui non seulement ne sont pas projectives, inais sont quasi m^triques en caractere, c'est-a-dire ont des rapports avec les points circulaires a I'infini ; les singularites de cette classe sont signalees par I'^vanouissement de reciprocants orthogonaux. Les reciprocants mixtes, qui ne sont ni purs ni orthogonaux, comme celui, par exemple, de M. Schwarz, ne r^pondront a aucime de ces trois espfeces de singularites ; mais, quoique ne servant pas a repr^senter une propriety invariable d'une courbe, ils serviront souvent, peut-dtre toujours, comme bases des reciprocants orthogonaux, c'e8t-&,-dire qu'ils seront les coefficients diff^rentieis par rapport a < de ces derniers. L'echelle des protomorphes, aussi bien dans la th^orie des reciprocants purs que dans celle des sous-invariants, joue un role si capital, en ce qui conceme la determination des formes irreductibles, qu'il nous semble in- dispensable de donner une demonstration rigoureuse de son existence dans I'une et I'autre theorie. 1° Quant aux sous-invariants, soit j I'ordre (c'est-a-dire j + 1 le nombre des lettres que Ton considere). Si j est pair, on connait les formes inva- rianti ves ac+ ..., ae+ ..., ag + ..., et Ton pent passer au cas ou j est impair. Dans ce cas, le nombre de sous-invariants du poids_; et du degre 3 sera 0';3,i)-(i-l;3,;). Mais il faut demontrer qu'il existe une forme de ce type, dans laquelle le coefficient du produit de a' et de la derniere lettre n'est pas nul. Or je dis que le nombre des formes du type suppose, qui ne contiennent pas cette lettre, sera (j;S,j-l)-(j-l;S,j-l). Mais 0'-l;3,i) = 0-l;3,i-l) et, evidemment, {j; 3, j) - (j; 3, j - 1) = 1 ; car les partitions dont le nombre est (j; 3, _;') contiendront toutes les partitions dont le nombre est (j; S,j — 1) et en plus la partition constituee par j com- bine avec des z^ros. Consequemment il existe un sous-invariant dont un terme sera le produit de a" par la derniere des lettres que Ton considere. 2° Quant aux reciprocants purs de I'ordre j, nous avons deja demontr^ qu'on pent satisfaire a I'inegalite (j;*,j)-(i-i;a;-i-i,i)>o en donnant k x une certaine valeur pas plus grande que j —\\ et, pour de- montrer qu'il y aura un reciprocant pur qui contient actuellement un terme I 36] de formes algebriques 251 a^~' multiplie par la derniere lettre, on pourrait faire pr^cisement le meme raisonnement que nous avons fait ci-dessus pour le cas precedent, et, puisque (j;a^.j)-0"-i;« + i.i) exeede de I'unite la valeur de {j\ x,j — 1) — {j; x + \, j — 1), on conclura avec certitude I'existence d'un protomorphe pour I'ordre j. On pent, en general, trouver plusieurs valeurs de x qui rendent 0; *. i) — (i — 1 ; ^ + 1. i) positif; parmi ces valeurs, il est commode d'adopter, comme protomorphe par excellence, une quelconque de celles pour lesquelles la valeur de x qui satisfait a cette inegalite est un mini- mum. Quand la lettre la plus avanc^e est inf^rieure a h, il n'y en a qu'un seul qui reponde a cette definition. Ainsi, par exeraple, si _; = 5, I'in^galite (5:a;)-(4 :a; + l) >1 donne pour x la valeur minimum a; = 4 et, avec I'aide de I'aneantisseur Sa'Sj +10a68e + (15ac + 106') S^ + (21ad + 356c) 8, + (28ae + h (x) where e may be regarded as infinitesimal*. For then this must remain true by successive accumulation when x becomes any function whatever of itself, and accordingly making x = z we obtain (y, z) as the value of the invariable quotient as was to be shown. Calif eh<^x<= ^, then using dashes to denote differentiation qud X, and a parenthesis to signify the augmented value of the derivatives, we obtain iy') = y'-ey', il (y") = y"-26y"-0'y', {y"') = y"'-'3ey"'-2e'y"-e"y'. | • It is easy to see It priori that if the theorem is true, it can only be so in virtue of {y, x) when X receives an infinitesimal, becoming of the form {l-2$){y, x) + \e", as is subsequently shown to be the case in the text. [+ Cf. p. 306 below.] } 37] Note oil Schwarzian Derivatives 253 Hence and Hence i^y'") = y'y'" - Wy" - ^&y'y" - &'y'\ f(y'") = iy'"-6%"'-80y/, {{y, a;)) = (1 + -26) {{y, x) -^0 (y, x)} - 6" = {\-W){2i,x)-e", ((y, x) - {z. x)) = (1 - 20) [{y, x) - (z, x)]. r{y^c)-(z^\ _ (y, x) - (z, x) Ay, ^) - {^' ^)\ \dx) J [dij that is, the right-hand expression does not change, when y, z remaining fixed forms of function, x passes from one form of function of the independent variable to another ; as was to be shown. From what precedes, it appears that if y,z,x be regarded as functions of <^ (dxY -J- J is a constant function in the sense that it remains unaltered, whatever function x may be of t, or which is the same thing if y and z functions of a- when expressed as functions of a;' (any function of x) are written y', z , then {y , x) — {z', x') is identical with (y, x) — (z, x), save as to a factor which depends only on the form of the substitution of x for x. Hence to all intents and purposes, any function of the differences of the Schwarzian derivatives of any system of functions of the same variable, in respect thereto, is (in a sense comprising, but infinitely transcending the sense in which that word is used in Algebra) a covariant of the system. Addendum. — Let us for the moment call functions of x, y which either remain unaltered or only change their sign when x and y are interchanged self-reciprocating functions. // ' /// //» The first case of the kind is — , , the next is '-^ — ,, '^ , and obviously a y'\ y' very general one of this sort will be the function For greater simplicity, let us call the numerator of any such function when expanded and brought to the lowest possible common denominator, a reciprocant, the highest index of differentiation wliich such recipiocant contains its order, and the number of factors in each term its degree. Then in any reciprocant so formed the degree is always just one unit less than the order: but as a matter of fact the function so obtained is in general not irreducible, so that its degree may be depressed, and it becomes a question of much interest to form the scale of degrees of reciprocants of this sort. For the 254 Note on Schwarzian Derivatives [37 orders 2, 3, 4, 5, 6 the degrees in question are respectively 1, 2, 2, 3, 3. Calling the successive derivatives of y, a, b, c, d, ..., they will be found to be a, 2ac - 36=, ad — bbc, 2a'e- loacd - lOad' + 356'c, 20"/- 2 laic - Soacd + GOab'd + 1106c', where each form is obtained by operating upon the preceding one with the operator a (bBc + cSd + dSe + ...)—\b{\ meaning half the weight + the degree of the operand), combining the result of this operation in each alternate case with a legitimate combination of those that precede, and in that case dividing out by a. I have proved that in this way can be obtained an infinite pro- gression of reciprocants, of which the leading terms (substituting numbers for letters), will be alternately of the forms lV(2i + l) and l\(2i+2). Every other reciprocant can be formed algebraically from these primordial forms, as every seminvariant can be obtained from the primordial forms a, ac — b', a'd — 3a6c + 26', The two theories run in parallel courses, but their relationship is that which naturalists call homoplazy as distinguished from honiogeny; I propose to give further developments of this new algebraical theory in a subsequent Note. M 38. ON RECIPROCANTS. [Messenger of Mathematics, xv. (1886), pp. 88 — 92.] In a note on Invariant Derivatives in the September number of the Messenger I have given a definition and examples of reciprocants. If in any of the forms at the end of the postscript to the note we restore to a, b, c, ... their values Sxy, Bx'y, S/y, ... any such function divided by a certain power of S^y will change its sign, but otherwise remain unaltered when X and y are i iterchanged. The index of that power is the degree added to half the weight and will be called the index of the reciprocant. Any product of i of such reciprocants will be a reciprocant of the same kind or contrary kind to those in the table (subsequent to a) according as i is odd or even. In the latter case the interchange of x and y will leave the function absolutely unaltered. Reciprocants which cause a change of sign will be said to be of an odd, those which cause no change of sign of an even character. Any linear function of reciprocants of the same weight, degree, and charactei' will be itself a reciprocant of that character, but reciprocants of opposite characters cannot be combined to form a new reciprocant : those of an odd character may be regarded as analogous to skew, those of an even character to non-skew seminvariants ; the rule against combining forms of opposite characters becomes superfluous in the case of seminvariants, because those that offer themselves for combination as having the same weight and degree must of neces.sity be of like character. Any reciprocant being given there is a simple ex post facto rule for assigning its character without any knowledge of the mode of its genesis, namely its character is odd or even according as the smallest number of letters other than a in any of its terms is odd or even. Thus the character of a reciprocant whose leading term is a'e, or ab'e, or abce is odd ; that of one whose leading term is abe or ahf is even, as is also that of the remarkable reciprocant hd—o(? in which no power of a appears. A further important distinction between the two theories* is that there are two linear reciprocants a and 6 but only one linear seminvariant. As an illustration of the combinatorial law of like character it will be seen that if we operate upon 2ac — 36^ with the operator a (bBa + cSb) - 3b, * That is of reciprocants and invariants. 256 On Reciprocants [38 we obtain a new reciprocant 2ad-106c + 96', of which the character is the same as that of 6', namely both are odd ; we may therefore add — 96" to the latter expression, and then dividing out by 2a there results the reciprocant ad — 56c, but we cannot combine 2ac — 36' with 6' because these two reciprocants are of opposite characters. Again, remembering that a is of an even and b of an odd character, the three reciprocants - i^\ 5 {ac - f 6')', 3a6 (ad - 56c) are all of an even order, hence we may add them together and divide the sura by a", which gives the new reciprocant Sbd — 5c' a form not containing the first letter a. No seminvariant exists, nor, except the one just given bd — oc", have I been able to discover any other reciprocant in which the first letter does not make its appearance "f. The infinite progression of odd reciprocants with the leading terms ac, ad, a.a.e, a.a.f, a.a.a.g, a.a.a.h, ... will easily be seen to exist by virtue of the general theorem that any reci- procant of degree, extent, and weight (say brietly of dew i, j, w) gives birth to two others of the same character as its own, one of dew i+l,j + l, w+ 2, the other of dew { + 1, j + 2, w + 3. For let ^w-ti = \, then denoting the operator bSa + cSb+... by n, and the result of the action of D, upon itself (n*)^ which is in fact Ct' + fia (lij meaning c8„ + d8j+ ...) ; (ali — \6)ii will obviously be a reciprocant of dew i+l,j + l,w + 2, and will give rise to a second reciprocant {an-{\ + ^)b}{an-Xb)R, which is a' (n*)' - (2X, + i) abUR - XacR + {X^ + |X) b^R; the last term of this being a reciprocant of the same character as the entire expression may be omitted, and dividing out the residue by a we obtain the second new reciprocant (a(n*)''-(2X+i)6-Xc]ii, which will be of dew i + 1, j + 2, w + 3, as was to be shown. It is easy to see that every reciprocant must be a rational integral function of the forms above stated commencing with a, 6, 2ac — 36^ (whose dew's are alternately of the form i, 2i — 1, 3i — 2 ; i, i — 2, 3t — 1) divided by some power of a.' For if any reciprocant contains only the letters a, 6, ... t Since the above went to press I have made the capital discovery that there are an infinite number of such reciprocants, and that all those of a given weight, extent and degree may be obtained by aid of a certain quadratico-Uuear partial differential equation. 38] On Reciprocants 257 h,k, I, it may be expressed as a rational integral function of the protomorph in which I first appears and of the letters «, 6, ...k divided by a power of a, and consequently the reciprocant may be so expressed, and continually repeating this process of substitution it follows that the reciprocant will be a rational integer of the protomorphs exclusively divided by a power of a*: this of course will necessarily be tound only to contain combinations of like character; we already know the converse that the sum of all combinations of like character of the protomorphs is a reciprocant"^. If any homogeneous reciprocant consists of portions of unlike degree (although of the same index) it is obvious that each portion must be itself a reciprocant, for if P, P', P" ... be such portions, P + P' + P" .. . must be identical with 11 -t- II' + 11" + ... when n, n', n" ... are the same functions of a, /3, 7 ... (that is, S^a;, Sy'a;, S^'a; ...) \ that P, P', P" ... are of a, 6, c If then we make p-a■^Ii=^, P'-o^n' = A'..., Pwe have A + A' + A" -> ... identically zero. But P, P" ... being of the same index but diflFerent degrees must be of [diflferent weights, and consequently A, A',... are of different weights. Hence I we must have A = 0, A' = 0, &c., as was to be shown. It follows from this that every reciprocating function whatever may be [obtained by an algebraical combination of the protomorphs, and consequently by an algebraical combination of the forms (^S^Jlogy, * The proof that every seminvariant is a rational integral function of the protomorphs is very similar : any proposed seminvariant is by the method employed in the text shown to be at worst a function of the protomorphs and of b ; but the terms involving any power of b must disappear becaase no identical equation can connect seminvariants with a non-seminvariant b. In the text we see in like manner that any given reciprocant may be reduced to the form H + K, where H and K are protomorphic combinations of opposite character, so that one of them will disappear. t Another general mode of generating a class of reciprocants would be to express any function of o,6, c, ... say 0(a, 6, c, .,.) under the form ^(a, j3, 7, ...). The product 0 (a, 6,c, .,.)^(a, 6, c,.,.), or its numerator, will then obviously be a reciprocant. To take a simple example. H d»i/_ dy_dy^_%l£l_ '-d^- AJTT =-«7 + S Hence, by the rule laid down, c(ac-SV), that is, ac''-36*c ought to be a reciprocant, which U right, for it is equal to {2ac - 362)» - 9li* divided by a multiple of a. The law that the factors of neminvariants must be seminvariants cannot be extended to the theory of reciprocants. In this case the factors may some or none of them be reciprocants, and the others on reciprocation exchabge forms monocyclioally or polycyclically with one another. / add the remark that this is not true of pure reciprocants, that is, those in which -^ does not appear. Every factor of a pure reciprocant must be itself a reciprocant. 8. IV. 17 258 On Reciprocanta [38 and that we should gain nothing in generality by operating with successive operators of the form '^7*^''^'')' (74^'*'')'- where ^1, ^j, ... are arbitrary functions of y' + —, instead of with the simple V operator —= B^ continually repeated. The results of using the more general operators would only amount to algebraical combinations of the results obtained from the simple forms (^jS^yiogy, where i may take all values from zero to infinity*. As in the case of seminvariants so also reciprocants would in extent contain only a finite number of ground-forms ; but furthermore for re- ciprocants limited in degree the number of ground-forms will also be finite. Whether reciprocants which are irreducible for a given extent ever cease to be so and become reducible when the order is increased, as is the case with seminvariants, remains to be seenf. In order to facilitate the verification of the results obtained and to be obtained it may be well to express the successive derivatives of a; in regard to y in terms of those of y in regard to x, that is, of o, y8, 7, ... in terms of a,b,c, ... as shown in the following short table. a = a a", b = -^ ^. c = -ay{- 3;8' a', d=-a^S-fl0a/37- 1.5/3' oJ, e = - a»6 + 15a=/SS + lOa'^^" - 105a^^7 + 105/3* a', / = _ ar"^ + 21a»/3e -I- S5ac>yh - 210a'^/3--8 - 2800^,87= + 1260a/3'7 - 945/3" a", g = -a.*7) + 28a*/3f -1- 56a'76 + 35a*8^ - SlSa'^-e - 1260a»^78 + 31.50a=^»8| „ - 280a»y -I- 6300a''/S^7= - 17325a/3*7 -I- 10395^ J " ' where a, b, c, d, e,f, ... represent the successive derivatives of y with respect to X ; and a, /3, 7, S, e, f, ... of a; with respect to y. In any subsequent paper on reciprocants in this Journal, I shall make the absolutely necessary transliteration referred to in a preceding footnote, re- placing the present letters a, b, c, d, ... by the letters t, a,b,c,... or possibly, for reasons which carry great weight, by the expressions t, 2a, 2.36, 2.3.4c,... * This is not true of homogeneous reciprocants. + I have since found that this is true for reciprocants, as for seminvariants. 39. NOTE ON CERTAIN ELEMENTARY GEOMETRICAL NOTIONS AND DETERMINATIONS. [Proceedings of the London Mathematical Society, xvi. (1885), pp. 201—215.] A CURVE, as every one knows, may be regarded as a locus of points or as %u assembly of directions, every point being common to two consecutive irections of the assembly, and every direction to two consecutive points of [the locus ; the locus is called the envelop of the a.ssembly (that is part of the ccepted language of geometry), and, conversely, the assembly may be called the environment of the locus. So we may regard a surface as an assembly of mgent planes or as a locus of points standing to each other in the relation of envelop and environment, and extend these definitions to apace of any number of dimensions. By a plasm, waiting a better word, we may understand a figure analogous to a point-pair in a line, a triangle in a plane, a pyramid in space, etc.; and an n-gonal plasm or n-gon will signify a plasm having n vertices and n faces themselves {n— l)-gons. It is easy and desirable to find the general value of the content of a regular »-gon, say abcde, all whose edges we may call unity. If h^ = \ab, C'i = lcp, dS = idy..., it is easily seen by an elementary process of integration that fi,y, B ... are the centres of figure to the successive plasms ab, abc, abed, ..., and, making ba=pi, c^=p.„ dy=p,..., each term in p,, /)j, p, ... will be perpendicular to the one which precedes it, 80 that, if F„ is the content of the plasm, (1, 2, 3 ...n)' Vn=PiPi...pn' 17—2 260 Note on Certain Elementary [39 Moreover, we shall have of which the general integral is in the present case, since j^i = 1, (7=0, so that tr,_ '» + ! (1.2...7i)»2'» If a, b, c be the angles of a fixed triangle, and A, B, C are proportional to the distances of a variable line from a, b, c, respectively, we may denote the line hy A: B:C; as regards a variable point, it will presently be seen to be advantageous to denote its proportional coordinates, not, as is rather more usually done, by equimultiples of its distances from the three sides, but as equimultiples of these distances multiplied by the sides of the triangle from which they are measured*; so that, calling these coordinates a, b, c, the image t of the line at infinity becomes a + 6 + c. Consider now the universal mixed concomitant (which it will be convenient to call a mutuant) Aa + Bb + Cc (where a, b,c,A,B,G are used in lieu of the more usual letters x, y, z, f, 17, 0; it will readily be seen that, when a, 6, c vary, and A, B, C are fixed, the mutuant images the line ^ : J8 : C, and that, when A, B, C vary and a, b, c are fixed, the mutuant images the radiant point a:b:c; that is to say, Aa + Bb + Cc=0 is true for every point in the point- containing line A: B:C in the one case, and to every line through the radiant point a : 6 : c in the other. Supposing, then, that the two kinds of coordinates are chosen in this manner, we see (what would not be the case if the simple distances were taken) that a form F and its " polar-reciprocal " (/> image the self-same curve referred to the self-same fundamental triangle. These consequences would moreover continue to subsist if, calling the distances of a line from the vertices P, Q, R, and of a point from the sides p, q, r, we took AP -.MQ-.NR.Xp-.nq.vr for the two sets of coordinates, provided only that XAP = /iM G = vNZf ; F, G, H being the distances of the sides from the vertices of the fundamental triangle, in which case the line at infinity would no longer be imaged by a -I- 6 -t- c. I shall, however, adhere in what follows to the convention above laid down. I need hardly add that in like manner, in space taking A:B:C:D (the distances of a plane from the * Or rather divided by the distances of these sides from the opposite angles of the funda- mental triangle, whose coordinates thus become 1, 0, 0, 0, 1, 0, 0, 0, 1. t If F=0 is the equation to any locus or atsembly, I call J^ the image, and such locus or assembly the object; to a given image responds in general an absolutely definite object, but, when the object is given, the image is only determined to a constant factor pr'es. I 39] Geometrical Notions and Determinations 261 vertices of a fundamental pyramid) as the coordinate-representation of such plane, and a:b:c:d (the contents of the volumes which any variable point makes with the respective faces) as the coordinate-representation of such point, the mutuant aA+bB + cC+dD will be the image of the radiant point a:b:c:d when the capital letters are the variables, and of the plane A : B :C : D when the small letters are the variables, meaning of course that Aa + Bb+ Cc+ Dd = 0 will be true of every point in the plane A: B.C-.D and of every plane through the point a:b:c:d, and, as before, F and ^ polar- reciprocals to each other will image the self-same surface (referred to the self-same fundamental pyramid) viewed as a locus or envelop on the one hand, as an assembly or environment on the other. If a, b, c, d be used to signify the actual as distinguished from the pro- portional coordinates of a point, a linear function of these is constant, whereas it is a quadratic function o{ A, B, C, D ..., when used to signify the actual distances of a variable line, plane, &c., from the vertices of the funda- mental plasm which is constant ; and it is the principal object of this note to determine the form of this quadratic function, which, as Prof Cayley was the first to show, may be expressed by the determinant to a matrix standing in close relation to the well-known "invertebrate symmetrical matrix," the determinant to which represents a numerical multiple of any plasm in terms of its edges, as, for example : ab ac ad 1 ba . be bd 1 ca cb . cd 1 da db dc .1 1111. where ab, ac, be... are used for brevity to signify the measure of absolute distance between a,b, a,c, b,c ..., that is, stand for what in ordinary notation would be denoted by (abf, (ac)', (bcf This may be quoted as the mutual-distance matrix ; its determinant, besides representing a numerical multiplier of the squared content of the pyramid when equated to zero, expresses the conditions of the four points a, b, c, d lying in a plane, the former property being a consequence immediately deducible by strict alge- braical reasoning from the latter. That this determinant does image the condition of the plasm to which the points a, b, c, d ... are the vertices, losing one dimension of space, may be shown in a somewhat striking manner as follows. If for a moment we use X, y, z, the distances of any point in the plane of abc from be, ca, ab as coordinates, the equation to a circle circumscribed about abc will be of the iorm fyz + gzx + hxy, and, calling the sides of the triangle a, b, c respectively, 262 Note on Certain Elementary [39 cw; + 6y + c^ is constant. Hence, substituting for z its value in terms of x and y, the image of the circle may be put under a form in which fb and ga will be the coefficients of y' and a? respectively ; but, since x and y are proportional to the Cartesian coordinates y and x respectively, the coefficients of x' and y' must be equal. Hence f:g:h::a:b:c, and if now ax, by, cz, instead of a;, y, z, be used as the coordinates of the variable point, the image to the circumscribing circle becomes 2 -f- , or if we please 'S.a^yz, that is, 'S.bcyz, where 6c stands as convened for (be)'. Hence, if a, b, c, d be the vertices of a pyramid, labyz will be the image of the circumscribing sphere, for when any coordinate t is made zero the image becomes that of a circle; and so universally for a plasm of any number of dimensions. Consider the case of a circle, and suppose that ab ac 1 ac be ba ca cb 1 1 1 vanishes ; this means that the line x + y + z touches the circle abxy + bcyz + cazx. But, '\ix+y-\-z images the line at infinity, it must cut this (as it cuts any other circle) in two distinct points, namely, the so-called circular points at infinity. Hence x + y +z must, when the above determinant vanishes, cease to be the line at infinity, which can only come to pass by the triangle abc losing a dimension of space, and a, b, c coming into a straight line, in which case x + y + z = Q, instead of being true of a particular line, is true of every point in the plane. Just in like manner, if vanishes, unless x-\-y + z + t ceases to image the plane at infinity, this plane would touch the sphere 'Eabxy, that is, would cut it in a pair of straight lines, whereas it intersects it in a circle. Consequently the plasm abed must, as before, lose one dimension, and so in general. The content of a plasm vanishes when the mutual-distance determinant does so, and the latter as • ab ac ad ba . be bd ca cb . cd da db dc . 1 1 1 1 39] Geometrical Notions and Determinations 263 which is easily transfonnable into 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 to 0 1 1 1 1 I 0 0 1 0 1 0 1 0 0 I well as the former may be expressed rationally in terms of ordinary Cartesian coordinates; but the expression for the content (being linear in each set of coordinates) is obviously indecomposable, and must therefore be a numerical multiple of some power of the mutual-distance determinant ; a comparison of dimensions shows at once that this power is the square root. I As regards the numerical multiplier, when the plasm has all its edges equal to unity (say a triangle, for example), the mutual-distance determinant becomes of which the value is - 3 ; and so in general for a regular plasm with (n -I- 1) vertices; that is, in space of n dimensions the mutual-distance determinant, say Z>„, becomes (—)"+'(» -j- 1), whereas the (volume)^, say F„^ has been 8howDtobe^,^^"+ ^^,. Hence, universally, i)„ = (_)n+i 2» (1 . 2 . . . n)^ F„». It may be here noticed that, if p be the perpendicular from any vertex on an opposite face of the plasm whose content is F„_,, we shall have L 1V,P = «F„. Consequently, Dn-if = (-)" 2»-' {1 . 2 . . . (n - 1))' FVi f = (-)» 2»-' (1 . 2 ... nf F„» = - ii)„. I now pass on to the leading motive of this note, namely, the determin- ation of the connection between the coordinates A, B, C ... drawn from a, b, c .... It is clear d priori that the form of the condition will be in all cases that a homogeneous quadratic function of the distances must be constant. Thus, for example, when there are four point.s, if A, B, C be assumed, we may describe three spheres with these quantities as radii, and the fourth point will be determined by means of one of the pairs of tangent planes drawn to them, the particular pair depending on the relative signs attributed to H 264 Note on Certain Elementary [39 A, B, C. Hence, if i''(^, B, C, D)= be the general equation, each of the quantities must enter in the second and no higher degree; moreover, since by transporting the plane from which the distances are measured parallel to itself, A, B, C, D will be all increased by the same quantity, F must express a function of their differences, and consequently, since any two distances may be interchanged, F can contain no terms of the first order in the variables, so that jP=0 must amount to the predication of a homogeneous quadratic function of the distances being constant. Thus, for example, in the case of three points, we have the well-known equation 2 {ah) (A-C)iB-G) = i (obey. Suppose now that A, B, C are taken in proportions consistent with making 2(a6)»(^-C)(£-C) = 0. Let S (aby (A — C)(B—C) = P.Q, where P, Q are two linear functions of A, B, C; then P, Q image two radiant points, each of which will have the property that any of its rays is at an infinite distance from a, b, c, or at all events, if it should pass through one of them, from the other two, and it is easy to anticipate that these two points must be the circular points at infinity. That such is the fact is obvious, because (using Cartesian co- ordinates) the perpendicular distance from any point upon x ± ij—l .y contains zero in its denominator; so that the two points of the absolute may be regarded as the centres of two points of rays, all of them infinitely distant from the finite region. But these two points are the intersections of the circumscribing circle with the line at infinity, and consequently their collective ecjuation will be found by taking the resultant of "^abxy, Sa;, ^Ax, which is well known to be the determinant of the quadratic function bordered by the coeflScients of the two linear ones. Hence the constant quadratic function \n A, B, G, namely, l.ab(A — B){A — C), ought to be a numerical multiple of the determinant .ABC. A . ab ac 1 B ba . be \ G ca cb .1 . 111. as is the case, the value of this determinant being -2lab(A-G)(B-G). The same thing may be shown in a more elementary manner as follows- Combining X + y + z = 0, abiBy + bcyz + cazx = 0, I I I 39] Geometrical Notioiu and Determinations 265 we have aca^ + (6c + ca — ah) xy + icy' = 0, at each point of the absolute. And, taking x-^y^z-i, x^y^z^ as the coordinates at these two points, it follows that a^i^Jj : y,ya : z^z^ : x^y^ + x^yi : y^z^ + y^z^ -.z^x^Jr z^x^ ::bc:ca:ab: — bc — ca + ab: — ca — ab + bc: — ab—bc + ca. And, as the two points will be imaged by XjA +yiB + ZiG, x^A+y^B + z^G, respectively, it follows that their collective image will be ^{bcA' + (bc-ab-ac)BC], which is easily seen to be identical with 26c (A -B){A- C). The universal algebraical theorem upon which the first method of proof depends is the well-known one that, if Q is a quadratic function and Z,, Zj, ... Li i linear functions of _;' variables, and if Q' (where j is not less than i+ 1) is what Q becomes when i of its variables are expressed in terms of the rest, then the necessary and suflBcient condition of the discriminant of every such Q' vanishing is that the determinant to Q bordered by the coeffi- cients of the i linear functions shall vanish. When j is equal to i + 1, the theorem shows that the resultant of the quadratic and its i attendant linear functions will be the bordered determinant in question. In the above example we bad j = 3, t = 2. Let us now proceed to apply a similar principle to the case of four points a,b,c,din space. If we take the case a^ -|- y* + 2' -K' = 0, any tangent plane to it at x', y', z', i! will be x'x -f y'y -t- zz -H i!t, and , as a;'» + y'^ + /» + P = 0, it follows that every tangent plane will be at infinite distance from any point external to it; and, as this is true wherever the centre of the cone be placed, and all the cones so obtained have the "circle at infinity" in common, — it follows that every tangent plane to the circle at infinity is infinitely distant from any external point in the finite region, — the infinitely-infinite system of planes thus obtained one may regard, if one pleases, as consisting of sheaves of planes whose axes form the environment to the circle at infinity, and will be the correlative to the infinitely-infinite system of points in the plane at infinity, which are infinitely distant from all external planes in the finite region. We see, then, that the coordinates to each such plane must satisfy the condition that, on making 1x=0 and l.Ax = 0, and expressing any two of the variables x, y, z, t in terms of the two others, the discriminant U 266 Note on Certain Elementari/ [39 A B C D A , ah ac ad B ha , be bd G c« c6 cd D da dh dc , 1 1 1 1 of the form then assumed by 'S.abxy must vanish, and consequently, as before, the mutual-distance determinant to the points a, b, c, d, bordered with a row and column of units and a row and column consisting of the letters A, B,C, D, will represent to a numerical factor pres the constant quadratic function of distances, that is, this function will be and obviously a similar algebraical conclusion will continue to apply, what- ever may be the number of points n in a space of n — 1 dimensions. As regards the value of the constant, in any case, that may be obtained by taking a face of the plasm as the term (line, plane, etc.) from which the distances A, B, G ..., are measured; that is, we may make 5 = 0, (7=0, Z) = 0..., provided we make A equal to the perpendicular from a on the opposite face. The value of the bordered determinant then becomes the negative of the squared perpendicular from a on bed ... multiplied by the mutual-distance determinant to bed...; that is, by virtue of what has previously been shown, will be half of the mutual-distance determinant of abed Hence the complete relation between A, B,G, D may be exhibited by making -h A B G D A ab ac ad B ba . be bd G ea cb , cd D da db dc 1 1 1 1 0, and similarly for any number of points. Professor Cayley has obtained the same result by a more direct but not more instructive process, as follows. Taking, by way of example, three points, A+k, B+ k, C+ k, (where k is infinite,) may be regarded as the distances of a, b, c from a fourth point at an infinite distance, and accordingly we may write = 0. • ab ac {A + ky 1 ba . be (B+ky 1 ca eb . (G+ky 1 {A + ky {B + ky {G+ky 1 1 1 1 1 I n 39] Geometrical Notions and Determinations 267 For the gnomon bordering the squaxe formed by the small letters and dots, we may substitute 2lcA + A'' 1 2kB + & 1 2kC+C^ 1 2it^ + ^= 2kB + B' -IkC+C -2k' 1 I 1 1 1 1 . without altering the value of the determinant, which therefore, remembering that k is infinite, is in a ratio of equality to (2A)" multiplied into the deter- minant ab ac A 1 ba be B 1 ca cb . C 1 A B C -i . 1 1 1 This last determinant therefore must vanish, agreeing with what has been shown above by a more purely geometrical method*. I will now proceed to develop this determinant deprived of its constant term, expressing it as a function of the dififerences of the capital letters. It is obvious that it may be expressed as a sum of terms of which each variable part will be of one or the other of these three forms {A -By. iA-B)(A-C), {A-B)(C-D); and accordingly we may distribute the totality of the terms of the constant function of difference into three families depending on the form of the variable argument. In general, if we consider any invertebrate symmetrical determinant » ex pressed by the umbral notation aa ab ac ... al ^- ba bb be ... bl II la lb Ic • As a corollary, we may infer, from the vanishing of this determinant, that, using the notation previonsly employed, n and consequently that D,= - (2)"(1 . 2 ... n)» F„», and that thus the content of a regular plasm with unit edges and (n + 1) vertices is n + 1 for triangle, pyramid, plu-pyramid, etc. I 3 - namely, r^, =s 16 ' 72 ' 9 . 2»o ■ 268 Note on Certain Elementary [39 where aa=hb = cc — ll...=0 and -pq = qp, we have this simple rule of pro- ceeding : Divide the letters a . . . Hn every possible manner into cyclical sets, each set containing at least two letters. Any cycle OjOs ... a< is to be interpreted as meaning 0,0,. OjOs ... ai^xOi .ata-i, which, by virtue of the supposed condition ab = ba, will be the same in whichever direction the cycle i.s read, the effect of the inversion of the cycle being merely to give the same product over again, written under the form OiOf.OsOi ... OiOi—i. The cycle of two letters a^a^ must be interpreted to mean (a,a,)'. If now 0,(7, ... Gi are cycles of two letters each, and x>X2 ••• Xi cycles of three or more letters, the total value of the determinant will be 2(-)n+i+J2iC,C,...GiX.X^...Xj- If, the principal diagonal terms remaining zero, the other terms were general, then the expression of the value of the determinant, calling the cycles CiCj ... C^, and making no distinction between the case of their being binary or super-binary, would be 2 (— )"+'' Cj Cj . . . C,; only it would have to be understood that each cycle of two letters, as (ab), would mean (aby, but a cycle of three or more letters, as (abc), would mean ab .be .ca + ac .cb. ba. This being premised, it is easy to deduce the following rule for the deter- mination of the three different families of terms belonging to the constant determinant of distances, which, to avoid prolixity, must be left to the reader to verify. Family I. — Omitting any two letters, and forming all possible cyclical products with the remaining (n — 2) letters, if (7, Ca ■ • ■ C, be any set thereof, and v the number of them containing more than two letters, the general term will be ll(-y'+-2'' G^. C^ ...c/(A- By, a, b being the two letters which do not occur in the cycles GyGi...G,. Family II. — Omitting any one letter, and forming with the remaining n — \ letters, in every possible way, a chain x containing two or more letters, and cycles G^G, ... Gy, then, supposing the chain to be bed ... kl, and under- standing by (x) the product bc.cd ... H, the general term will be 22 (- )»+■' 2-'+' G,G,...Gy(x){A- B) {A - L), a being the letter which does not appear in the chain or any of the cycles, and v meaning as before the number of the cycles which contain 'at least three elements. Family III. — Form all the letters in every possible way into two chains (each containing two or more letters) X' %• ^^^ ^^^'^ cycles C,, C„ ... G,; i 39] Geometrical Notions and Determinations 269 then, supposing the initial and final letters of ;^ to be a, h, and of ■x^ to be k, I, the general terra of this family will be 22 i-r^^ 2--+' C,C,...C. (x) (%') {{A -K)(H-L) + iA- L) {H - K)]. I subjoin in the following table the types of the coefficients of the several families for all the values of n from 2 up to 7 ; the vacant cycle ( ) of course means unity, and a cycle {ah) means {ahy ; that is, the fourth power of the length ah. Every cycle enclosed in a parenthesis of three or more letters, will be understood to be affected with a coefficient 2, and for greater brevity the variable part of each term is left to he supplied. A round parenthesis indi- cates a cycle, a square parenthesis a chain. Number of Letters 2 3 4 Types Name of Family lat 2Dd 1st 2nd 3rd Ist 2nd 3rd 1st 2nd 3rd Ist () {he) -{cd) 2 [6cd] 2[a6].[cd] 5 {cde) -2[6ca!e]:2(6c)[de] - 2 [a6] [cde] 6 -{cdef):{cd){ef) - 2 {hcd) [ef] : - 2 (6c) [def] : - 2 [bcdef] - 2 {ab) [cd] [ef] : [ahc] [def] : [ab] [cdef] 7 {cdefg)-{cd){efg) „ 2 {bcde) [fg] : 2 {bed) [efg] -.-2 {be) {de) [ fg] 2 {be) [defg] : - 2 [hcdefg] 2nd „ 2 {ahc) [de] [fg] : 2 {ab) [cd] [efg] : - 2 [ahc] [defg] : - 2 [ahcdefg] 3rd Thus, for example, the constant function of distances for three points in a plane is 226c {A — B) {A — C) ; for four points in space is -l,cd{A-By+21bc.ed{A -B){A-D) + 2Sa6 . cd \{A -C){B - D) + {A - D){B- C)} ; for five points in hyper-space is 21{cd.de.ec){A - Bf -21 {be .cd . de) {A - B){A - E) + 2{hcy{de){A-D){B-E) - 2Sa6 .cd.de.ec{{A- C) {B-E) + {A- E) {B - C)]. The part of the constant function of distances for seven points belonging to the 2iid family of terms will be 426c .cd.de. eh .fg {A -B){A-E) + 4-lhc . cd . db . ef.fg {A -E){A- 0) - 2 {hcf {deffg {A-F){A-G)^2 {bcf {de . ef.fg) {A -D){A- G) -2hc.cd.de.ef.fg{A-B){A-0). 270 Note on Certain Etementarj/ [39 The number of types in each family for n points is easily expressible by a generating function. Obviously in the Ist family this number is the number of ways of resolving n into parts none less than 2 ; that is, it is the coefficient of ic""' in 1 l-x».l-«».l-a;*...' In the 2nd family, it is the sum of the number of ways of decomposing n — 3, n — 4, ... into parts none less than 2; tiiat is, it is the coefficient of x"~* in l+x+x' + ... ^, ^ . . 1 , that is, in (l-x')(l-x>)...' ' (l-x){l-a!')il-af)...' In the 3rd family, if the number of ways of dividing r into two parts, neither of them less than 2, is called (r), and of dividing (n — r) into any number of parts, none less than 2, is called [n — r], the number of types is 2 (r) [» — r] ; that is, it is the coefficient of x^~* in that IS, in (l-ai'){l-x'){l-x*)... ' ' (l-x)(l-x'y{l-ai')(l-x')... ' Hence the total number of types in all three families combined will be the coefficient of x"~' in (l-x)(l-a^) + x (1 - x^) + a,'' . 1 l-x.l-x^.l-af... ■ ■ *''^' "■ '" l-x.il-x^y.l-a^A-x^... ' Consequently, the indefinite partitions of 0, 1, 2, 8, 4, 5, 6, 7, ... being 1, 1, 2, 3, 5, 7, 11, 15, ..., the series for the type-number will be found by summing all the terms in the odd and even places successively. We thus obtain the series 1, 1, 3, 4, 8, 11, 19, 26, ... for the number of types in the constant-distance function for 2, 3, 4, 5, 6, 7, 8, 9, ... points respectively. It may be worth while to exhibit the rule for the formation of the constant function of distances under a slightly different aspect. As before, by the reading of any cycle, understand the product of its successive duads affected with the multiplier — 1 or — 2, according as the number of letters in the cycle is two or more than two. By a modified reading of a cycle, understand what the reading becomes on substituting for any two duads pq, rs the product (P — Q){R — S), as for instance (.4 - -BJC — B) in lieu of ab . cd, (A — B^B — G) in lieu of ab . be, and (which can only happen in the case of a cycle of two letters), {A -B){B-A), that is, -{A- B)' in lieu of ab . ba. Then, to find the constant function of distances to any given set of letters, we must begin with distributing the letters in every possible way into cycles containing between them two or more letters. Each such combination of cycles we may call a distribution. I L 39] Geometrical Notions and Determinations 271 In each distribution the cycle is to be taken (each in its turn), and the sum of its modified readings is to be multiplied by the product of the read- ings of the remaining cycles, if there are any. The sum of these sums (or the single sum, if there is but one cycle) is the portion of the quadratic function sought, due to the particular distribution dealt with ; and the sum of these double sums, taken for each distribution in succession, is the total value of the function, and will be equal exactly to its representative deter- minant when the number of letters is odd, and to the same with its sign changed when that number is even. As an example for five letters a, b, c, d, e, there will be ten distributions of the form (ab) (cde), and twelve distributions of the form (abcde). From any one of the first ten distributions, as (ah) (cde), by modifying first (ab) and then (cde), we obtain (1) 2(cd.de.ec)(A-B)(B-A), (2) 2(aby [ce (C-D)(D - E)+dc(B - E)(E-C) + ed(E - C)(C - D)]. And from a distribution of the form (abcde) we obtain, by operating on con- secutive duads, 5 terms of the form —2[cd.de. ea (A —B)(B — G)], and, by operating on non-consecutive duads, 5 terms of the form -2 [bc.de. ea (A -B)(C- D)} *. The sum of all the sums of terms due to the twenty-two distributions is the constant function of distances for the five given letters. In the case of six letters the distributions into cycles will be of four kinds, corresponding to the partitions 6; 4, 2; 3, 3; 2, 2, 2. The first kind will contain two types of the 3rd family and one of the 2nd family; the second kind will contain one type of each of the three families, and the third and fourth kinds single types of the 2nd and 1st families respectively, thus giving eight distinct types of terms in all, as should be the case according to the rule. * It will be observed that the distribution {acbde) will give a term -2 {cb . de . ea{A - C) (B - D)], in which the literal part cb.de. ea is equal to the literal part bc.de .fa in the term above expressed. This is bow it comes to pass that the terms of the 3rd family may be grouped in pairs, as stated in the prior mode of arranging the result according to families instead of according to cycles. 40. ON THE TRINOMIAL UNILATERAL QUADRATIC EQUATION IN MATRICES OF THE SECOND ORDER. [Quarterly Journal of Mathematics, xx. (1885), pp. 305 — 312.] In the May number [p. 225 above] of the present year of the London and Edinburgh Philosophical Magazine (disfigured by numerous errors or inaccuracies) I investigated the number of the solutions of an equation in quaternions or matrices of the second order, belonging to what I term the unilateral class, meaning one in which the coefficients of any actual power of the unknown quantity lie on the same side of it ; this number for the Jerrardian Trinomial form I proved strictly is 2t' — i {i being the degree of the equation) and with evidence little short of moral certainty i^ — i'' + i* in the general case where none of the terms are wantingf . But it must be well borne in mind that these numbers only apply when the coefficients are left general, and that for special relations between them some or all of the roots may become either ideal or indeterminate, or some the one and some the other. In all cases of equations in matrices one principal feature of the investigation is, or should be, to determine the equation of condition between the coefficients, in order that the solution may lose or retain its normal form ; if we wish to avoid being compelled to enter upon a complicated consideration of exceptions piled upon ex- ceptions, it is necessary to presuppose a certain criterion function to be other than zero ; otherwise it is like the opening of Pandora's box, letting loose an almost incalculable train of vexatious inquiries scarcely worth the trouble they give to answer correctly. * This article was written and sent to the press many mouths ago. I have since shown that the number of roots of a general unilateral equation of degree i in matrices of the order u is the number of combinations of iu things taken u and w together, and consequently for the case of quaternions is 2t^-i for the general and not merely for the Jerrardian form. See [above, pp. 197, 233. Also] Nature, Nov. 13, 1884. t I made the assumption that the required number is an analytical function of u. I I 40] On the Trinomial Unilateral Quadratic Equation, etc. 273 Take as an instance the subject of monothetic equations. I have defined a monothetic equation to be one in which all the coetficients are functions of a single matrix, which may be called the base. In such an equation of the degree i and of the order - cfi + £i)) .T + (/ - 2cZ) + DO = 0. Hence, writing B-b=u, D-c = v, d - b- = a, e-bc = 0, /- c" = 7. and comparing with each other the two quadratic equations in x, we may write u'+a = \, Mt) + /3 = 2X(M+6), v^ + y = 'iX{v + c). Eliminating v from the latter two equations there results - (2\w + 2bX -^y + 4t\ (2Xm + 26\ - /3) m - (7 - 4cX) u' = 0, and finally writing \- a for «', we obtain (4V + 4cX. - 7) (\ - a) - (26\ - y3)= = 0. There are thus 3 pairs of roots, for to each of the three values of \ corre- spond two values of v, namely ±{\-d + b')\ and to each value of \ and m one value of v, namely 26\ + bc-e 2\ u We have also x = ^{{p-b-u)-'(q - D)}, consequently, since p' - 2bp + d = Q, _ (p - b-\-u)iq-c-v) ^ _ (p-b+u)(q-c-v) '" 2(6»-rf-wO 2X Thus then we see that x can only cease to have 6 determinate values when \ = 0, and consequently the Criterion of Normality is the last term in the equation to X. This equation, written out at length, is 4X3 + 4 (c _ 62 _ a) x' + (- 4ca + 46^ - 7) X + 07 - /3- = 0, that is, 4V + 4 (c - d) X^ + (- 4crf + 46e - / + c^ X + (d - b') (/- C) - (e - 6c)». Hence the Criterion in question is (d - ¥) (f - c^ - (e - 60^ or (//- c=rf - 6=/- e' + 2bce, which is the discriminant to the quadratic form X» + 2b\p + 2cfi,v + dfj? + 2efiv + fv^; this, as I have elsewhere shown, is the Criterion of the matrices p.q* being in involntion-l, that is, of a linear equation existing between the matrices 1, p. q, pq; or ifp, q are regarded as quaternions, it is the condition of the square of • When p, q are regarded as matrices, then p^-2bp + d = 0, q'-2cq+f=0, i(pq + qp) -bq-cp + e = 0, ^vhere \^ + 2h>a> + 2cny + d^L* + ie^p■^■fy' is the determinant toX + fip + vq. [t Above p. 116.] I 40] Equation in Matrices of the Second Order 275 the sine of the angle between the vectors oip and q vanishing; a condition which of course does not imply the coincidence of the vectors unless accom- panied by the futile limitation of such vectors being real. It admits of easy demonstration by virtue of the foregoing that in the case of the more general equation pa? + qx + r = 0, the Criterion of Normality will be the discriminant of the ternary quadratic, which is the determinant of pu + qv + rw ; this seems to me a very remarkable and noteworthy theorem. When this Criterion does not vanish, the quadratic equation above written must have 3 pairs of determinate roots. Why they go in pairs and can be found by solving only a cubic instead of [ a sextic is best seen d priori by reverting to the original form a^ — 2px + q = 0. It follows from the nature of the process for finding B and D that they will be the same for that equation as for the equation y' — 2yp 4-g = 0. But on writing x + y = 2p these two equations pass into one another. Hence each value of B, say Bj , will be associated with another value, say E, where Bj + R = 2b*, that is to say, if Ui, namely B — b is one value of u, then b — B, that is, — !<, will be another value of u, so that the equation in u^ ought to be (as it has been shown to be) a cubic. It might for a moment be supposed that X = a = d — 6" would lead to a breach of normality on account of the equation i; — 2\ = , where m'=0. This, however, is not the case. For the equation «» + 7 = 4X(t;+ c) becomes, when \= a, v^ - 4 (d - 6») « + / - c^ - 4cci + 46^c = 0, so that V remains finite ; consequently 26\ + 6c — e, that is, 2bd — 2b'+bc—e, must vanish when \ = d — b', and v — 2X assumes the form - . Obviously then in this case, to the one value m = 0 will be associated the two values of v, say Vi and v„ given by the above quadratic, and to \ = a will still correspond two values of (m, v), namely (0, v,), (0, v^); where, ideally speaking, the two zeros may be regarded as the same infinitesimal affected with opposite signs. * In qnatemion phrase, it x + y = 2p, Sx + Sy = 2Sp. It should be observed, in order to understand what follows in the text, that b-B^ = B' -b, and that the values of B mast obviously be the same in the equation x'-2px + q = 0 as in the equation x^ - 2jj> + q = 0. 18—2 276 On the Trinomial Unilateral Quadratic [40 The equation in X may be made to undergo a useful linear transformation. Let \= n + a, so that fi = m'. Then fj. (V + (8a + 4c) M + 4a« + 4ca - 7} - {2bfi + 2ba - y3)= = 0, that is 4fji' + {4 (c + 2d) - 1 2b'] fj.' + ((c + 2d)' - 8 (c + 2d) b' + 126* + ibe-/}fi- {b (c + 2d) - 2i'-e)' = 0, where it is noticeable that the number of parameters is reduced from 5 to 4, c and d only appearing together in the linear combination c + 2d. This is tantamount to the form obtained by Hamilton. Let us make another linear transformation suggested by the preceding remark. Write c + 2d = ^r, and fi — b' = y = \ — d, the equation becomes 4y + 4:gy^ + (^= + 46e -/) 7 + 2beg -¥f-e' = 0. But obviously, notwithstanding this reduction of the parameters, X itself is the most natural quantity to employ as the base of the solution, or, so to say, as the independent variable, and this admits of being determined by an equation of extraordinary simplicity. For, let / be the discriminant of det. (\ + /xp + 1/9) = 7 = d/" + 2bce - c-d - b'f - ef. Then it will be seen by actual inspection that the equation found for \ takes the following form that is (the terms in the exponential function subsequent to the fourth term adding nothing to the value of the series). If in the equation of — 2px + q=0, p and 5' be regarded as quaternions, then X = Sx' + Ip' - {Spy, c = 8q, d^Ip', and I=i(pg-qpy, which is a scalar quantity, and is to be regarded as an explicit function of Sp, Sq ; Tp', S{VpVq), Tq^; it is in fact the discriminant of the form Z' + 2SpZ F + 2SqXZ + Tp" F» + 2S ( Vp Vq) YZ + Tq'Z\ an identity unknown I believe to the geometrical quatemionists. [As an example of it, let p = i, q =j, then 8p = 0, 8q = 0, S{VpVq) = 0, Tp' 1, Tq'=-\, \ (P? - iPf = 1 = the discriminant of X'-Y-- Z-.] With these definitions e^^^*'"*"/ becomes identically zero. 40] Equation in Matrices of the Second Order 277 The equation a? — 2px+q = 0 having six roots it is natural to inquire as to the value of their sum. This may be readily found as follows. We have found (p-b + u){q-c-v) '^~ 2\ Also, if x + x'= 1p, af^ - 2x'p + g = 0, and obviously Sa; = Sa;'. Hence ^^^_^{p -h ^u^q- o-v) ^ and 12;. - S. = - S (3jlA:it|(Pzi±^) . 3 Therefore Sa; = 6p — S ^ {pq — qp) where the sign of /4 must be so taken that it shall be equal to \ ipq — qp)- So again 'Ex^ = 2p'S.x — 67 = I2p' -6q + l2 (28, - Ba)I^-p. Thus the mean value of each root is e in excess, and that of each square root ep in excess, of what these means would be if p and q were nominal quantities, e denoting (28^ — 8^) I^p. Of course Sa:* may be found by the formula of derivation lx^+' = 2p'S.x'-9lx'-\ In conclusion it may be observed in regard to the equation a^ — 2px + q = 0, (since in writing a; + ar, = 2p, we have x^^ — 2x,p + g = 0) it follows that (what- ever be the order of the quantities p and q) the roots of either equation must be associated in pairs; because, if the identical equation to p is jj"— a)6p"~' + ... and to a; is af — coBoT'^ + ..., the equation for finding B must be of the form T(B-by = 0. P.S. — Since the above was sent to press I have discovered the general solution of the unilateral equation of any degree in matrices of any order ; see the Comptes Rendtis of the Institute for Oct. 20, 1884 [pp. 197, 233 above], and Nature for Nov. 13, 1884*. [• This paper contains the Theorem " Every latent root of every root of a given unilateral function in matrices of any order, is an algebraical root of the determinant of that function taken as if the unknown were an ordinary quantity, and conversely every algebraical root of the determinant so taken is a latent root of one of the roots of the given function."] 41. INAUGURAL LECTURE at Oxford 12 December 1885. ON THE METHOD OF RECIPROCANTS AS CONTAINING AN EXHAUSTIVE THEORY OF THE SINGULARITIES OF CURVES*. [Nature, xxxiii. (1886), pp. 222—231.] It is now two years and seven days since a message by the Atlantic cable containing the single word " Elected " reached me in Baltimore informing me that I had been appointed Savilian Profes.sor of Geometry in Oxford, so that for three weeks I was in the unique position of filling the post and drawing the pay of Professor of Mathematics in each of two Universities: one, the oldest and most renowned, the other — an infant Hercules — the most active and prolific in the world, and which realises what only existed as a dream in the miad of Bacon — the House of Solomon in the New Atlantis. To Johns Hopkins, who endowed the latter, and in conjunction with it a great Hospital and Medical School, between which he divided a vast fortune accumulated during a lifetime of integrity and public usefulness, I might address the words familiarly applied to one dear to all Wykeliamists : — "Qui condis Iseva, condis collegia dextra, Nemo tuanim unam vicit utraque manvi." The chair which I have the honour to occupy in this University is made illustrious by the names and labours of its munificent and enlightened founder, Sir Henry Saville ; of Thomas Briggs, the second inventor of logarithms; of Dr Wallis, who, like Leibnitz, drove three abreast to the temple of fame — being eminent as a theologian, and as a philologer, in addition to being illus- trious as the discoverer of the theorem connected with the quadrature of the circle named after him, with which every schoolboy is supposed to be familiar, and as the author of the Arithmetica Infinitorum, the precursor of Newton's Fluxions; of Edmund Halley, the trusted friend and counsellor of Newton, whose work marks an epoch in the history of astronomy, the reviver of the study of Greek geometry and discoverer of the proper motions of the so- [* The tables referred to in the text are given pp. 301, 302 below.] 41] On the Method of Reciprocants 279 called fixed stars ; and by one in later times not unworthy to be mentioned in connection with these great names, my immediate predecessor, the mere allusion to whom will, I know, send a sympathetic thrill through the hearts of all here present, to whom he was no less endeared by his lovable nature than an object of admiration for his vast and varied intellectual acquirements, whose untimely removal, at the very moment when his fame was beginning to culminate, cannot but be regarded as a loss, not only to his friends and to the University for which he laboured so strenuously, but to science and the whole world of letters. As I have mentioned, the first to occupy this chair was that remarkable man Thomas Briggs, concerning whose relation to the great Napier of Merchiston, the fertile nursery of heroes of the pen and the sword, an anecdote, taken from the Life of Lilly, the astrologer, has lately fallen under my eyes, which, with your permission, I will venture to repeat: — " I will acquaint you (says Lilly) with one memorable story related unto me by John Marr, an excellent mathematician and geometrician, whom I conceive you remember. He was servant to King James and Charles the First. At first, when the lord Napier, or Marchiston, made public his logarithms, Mr Briggs, then reader of the astronomy lectures at Gresham College, in London, was so surprised with admiration of them, that he could have no quietness in himself until he had seen that noble person the lord Marchiston, whose only invention they were : he acquaints John Marr here- with, who went into Scotland before Mr Briggs, purposely to be tliere when those two 80 learned persons should meet. Mr Briggs appoints a certain day when to meet at Edinburgh; but failing thereof, the lord Napier was doubtful he would not come. It happened one day as John Marr and the lord Napier were speaking of Mr Briggs : ' Ah John (said Marchiston), Mr Briggs will not now come.' At the very moment one knocks at the gate; John Marr hastens down, and it proved Mr Briggs to his great contentment. He brings Mr Brigg.s up into my lord's chamber, where almost one quarter of an hour was spent, each beholding other almost with admiration before one word was spoke. At last Mr Briggs began : ' My lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto astronomy, namely, the logarithms ; but, my lord, being by you found out, I wonder nobody else found it out before, when now known it is so easy.' He was nobly entertained by the lord Napier ; and every summer after that, during the lord's being alive, this venerable man Mr Briggs went purposely into Scotland to visit him*." * A very gimilar story is told of the meeting of Leopardi and Niebnhr in Bome. What Briggs ■aid of logarithms may be said almost in the same words of the subject of this lecture: — " This most excellent help to geometry which, being found out, one wonders nobody else found it ont 280 On the Method of Reciprocants [41 Some apology may be needed, and many valid reasons might be assigned, for the departure, in my case, from the usual course, which is that every professor on his appointment should deliver an inaugural lecture before commencing his regular work of teaching in the University. I hope that my remissness, in this respect, may be condoned if it shall eventually be recog- nised that I have waited, before addressing a public audience, until I felt prompted to do so by the spirit within me craving to find utterance, and by the consciousness of having something of real and more than ordinary weight to impart, so that those who are qualified by a moderate amount of mathe- matical culttire to comprehend the drift of my discourse, may go away with the satisfactory feeling that their mental vision has been extended and their eyes opened, like my own, to the perception of a world of intellectual beauty, of whose existence they were previously unaware. This is not the first occasion on which I i)ave appeared before a general mathematical audience, as the messenger of good tidings, to announce some important discovery. In the year 1859 I gave a course of seven or eight lectures at King's College, London, at each of which I was honoured by the attendance of ray lamented predecessor, on the subject of " The Partitions of Numbers and the Solution of Simultaneous Equations in Integers," in which it fell to my lot to show how the difficulties might be overcome which had previously baffled the efforts of mathematicians, and especially of one bearing no less venerable a name than tfiat of Leonard Euler, and also laid the basis of a method which has since been carried out to a much greater extent in my "Constructive Theory of Partitions," published in the American Journal of Mathematics, in writing which I received much valuable co-operation and material contributions from many of my own pupils in the Johns Hopkins University*. Several years later, in the same place, I delivered a lecture on the well-known theorem of Newton, which fills a chapter in the Arithmetica Universalis, where it was stated without proof, and of which many celebrated mathematicians, including again the name of Euler, had sought for a proof in vain. In that lecture I supplied the missing demonstration, and owed my success, I believe, chiefly to merging the theorem to be proved, in one of before ; when, now known, it is so easy." I quite entered into Briggs's feelings at his interview with Napier when I recently paid a visit to Poincare in his airy perch in the Eue Gay-Lussac in Paris (will our grandchildren live to see an Alexander Williamson Street in the north-west quarter of London, or an Arthur Cayley Court in Lincoln's Inn, where he once abode?). In the presence of that mighty reservoir of pent-up intellectual force my tongue at first refused its office, my eyes wandered, and it was not until I had taken some time (it may be two or three minutes) to peruse and absorb as it were the idea of his external youthful lineaments that I found myself in a condition to speak. * In one of those lectures, two hundred copies of the notes for which were printed off and distributed among my auditors, I founded and developed to a considerable extent the subject since rediscovered by M. Halphen under the name of the Theory of Aspects. I 41] On the Method of Reciprocants 281 greater scope and generality. In mathematical research, reversing the axiom of Euclid, and converting the proposition of Hesiod, it is a continual matter of experience, as I have found myself over and over again, that the whole is less than its part. On a later occasion, taking my stand on the wonderful discovery of Peaucellier, in which he had realised that exact parallel motion which James Watt had believed to be impossible, and exhausted himself in contrivances to find an imperfect substitute for, in the steam-engine, I think I may venture to say that I brought into being a new branch of mechanico- geometrical science, which has been, since then, carried to a much higher point by the brilliant inventions of Messrs Kempe and Hart. I remember that my late lamented friend, the Lord Almoner's Reader of Arabic in this University, subsequently editor of the Times, Mr Chenery, who was present on that occasion in an unofficial capacity, remarked to me after the lecture, which was delivered before a crowded auditory at the Royal Institution, that when they saw two suspended opposite Peaucellier cells, coupled toe-and-toe together, swing into motion, which would have been impossible had not the two connected moving points each described an accurate straight line, " the house rose at you." (The lecture merely illustrated experimentally two or three simple propositions of Euclid, Book III.) The matter that I have to bring before your notice this afternoon is one far bigger and greater, and of infinitely more importance to the progress of mathematical science, than any of those to which I have just referred. No subject during the last thirty years has more occupied the minds of mathe- maticians, or lent itself to a greater variety of applications, than the great theory of Invariants. The theory I am about to expound, or whose birth I am about to announce, stands to this in the relation not of a younger sister, but of a brother, who, though of later birth, on the principle that the masculine is more worthy than the feminine, or at all events, according to the regulations of the Salic law, is entitled to take precedence over his elder sister, and exercise supreme sway over their united realms. Metaphor apart, I do not hesitate to say that this theory, minor natu potestnte major, infinitely transcends in the extent of its subject-matter, and in the range of its applications, the allied theory to which it stands in so close a relation. The very same letters of the alphabet which may be employed in the two theories, in the one may be compared to the dried seeds in a botanical cabinet, in the other to buds on the living branch ready to burst out into blossom, flower and fruit, and in their turn supply fresh seed for the main- tenance of a continually self-perpetuating cycle of living forms. In order that I may not be considered to have lost myself in the clouds in making such a statement, let me so far anticipate what I shall have to say on the meaning of Reciprocants and their relation to the ordinary Invariantive or Covariantive forms by taking an instance which happens to be common 282 On the Method of Beciprocants [41 (or at least, by a slight geometrical adjustment, may be made so) to the two theories. I ask you to compare the form a'd - Sabc + 26* as it is read in the light of the one and in that of the other. In the one case the a, b, c, d stand for the coefficients of a so-called Binary Quantic, and its evanescence serves to express some particular relation between three points lying in a right line. In the other case the letters are interpreted* to mean the successive differential derivatives of the 2nd, 3rd, 4th, 5th orders of one Cartesian co-ordinate of a curve in respect to the other. The equation expressing this evanescence is capable of being integrated, and this integral will serve to denote a relation between the two co-ordinates which furnishes the necessary and sufficient condition in order that the point of the curve of any or no specified order (for it may be transcendental) to which the co- ordinates may refer, may admit of having, at the point where the condition is satisfied, a contact with a conic of a higher order than the common. In the one case the letters employed are dead and inert atoms ; in the other they are germs instinct with motion, life, and energy. A curious history is attached to the form which I have just cited, one of the simplest in the theory, of which the narrative may not be without interest to many of my hearers, even to those whose mathematical ambition is limited to taking a high place in the schools. At pp. 19 and 20 of Boole's Differential Equations (edition of 1859) the author cites this form as the left-hand side of an equation which he calls the "Differential Equation of lines of the second order," and attributes it to Monge, adding the words, "But here our powers of geometrical interpretation fail, and results such as this can scarcely be otherwise useful than as a registry of integrable forms." In this vaticination, which was quite uncalled for, the eminent author, now unfortunately deceased, proved himself a false prophet, for the form referred to is among the first that attracts notice in crossing the threshold of the subject of Reciprocants, and is but one of a crowd of similar and much more complicated expressions, no less than it susceptible of geometrical interpretation and of taking their place on the register of integrable forms. A friend, with whom I was in communication on the subject, and whom I see by my side, remarked to me, in reference to this passage : — " I cannot help comparing a certain passage in Boole to Ezekiel's valley of the dry bones: 'The valley was full of bones, and lo, they were very dry.' The answer to the question, ' Can these bones live V is supplied by the advent of the glorious idea of the Reciprocants ; and the grand invocation, ' Come from the four winds, 0 breath, and breathe upon these slain, that they may live,' may well be used here. That they will L 2!. are pure reciprocants. Those from which the first derivative -— is not excluded may be called Mixed Reciprocants. An example of such kind of Reciprocants is afforded by the Schwarzian above referred to. This dis- tinction is one of great moment, for a little attention will serve to make it clear that every pure reciprocant expressed in terms of x and y marks an intrinsic feature or singularity in the curve, whatever its nature may be, of which X and y are the co-ordinates ; for if in place of the variables (x, y) any two linear functions of these variables be substituted, a pure reciprocant, by virtue of its reciprocantive character, mu.st remain unaltered save as to the immaterial fact of its acquiring a factor containing merely the constants of substitution *. The consequence is that every pure reciprocant corresponds to, and indicates, some singularity or characteristic feature of a curve, and vice versa every such singularity of a general nature and of a descriptive (although not necessarily of a projective) kind, points to a pure reci- procant. Such is not the case with mixed reciprocants. They will not in general remain unaltered when linear substitutions are impressed upon the variables. Is it then neces.sary, it may be asked, to pay any attention to mixed reci- procants; or may they not be formally excluded at the very threshold of the inquiry ? Were I disposed to put the answer to this question on mere personal grounds, I feel that I should be guilty of the blackest ingratitude, that 1 should be kicking down the ladder by which I have risen to my present commanding point of view, if I were to turn my back on these humble mixed reciprocants, to which I have reason to feel so deeply indel)ted ; for it was the putting together of the two facts of the sub- stantial permanence under linear substitutions impressed upon the variables of the Schwarzian form and the simpler one which marks the inflexions of a curve — it was, if I may so say, the collision in my mind of these two facts — that kindled the spark and fired the train which set my imagination in a blaze by the light of which the whole horizon of Reciprocants is now illumined. * The form as it stands shows that for y a linear function of x and y may be substituted; and the form reciprocated (by the interchanRe of x and y) shows that a similar substitution may be made for x. Hence arbitrary linear substitutions may be simultaneously impressed on x and y vithont inducing any change of form. 286 Oil the Method of Reciprocanta [41 But it is not necessary for me to defend the retention of mixed reciprocants on any such narrow ground of personal predilection. The whole body of Reciprocants, pure and mixed, form one complete system, a single garment without rent or seam, a complex whole in which all the parts are inextricably interwoven with each other. It is a living organism, the action of no part of which can be thoroughly understood if dissevered from connection with the rest. It was in fact by combining and interweaving mixed reciprocants that I was led to the discovery of the pure binomial reciprocant, which comes im- mediately after the trivial monomial one, — the earliest with which I became acquainted, and of the existence of compeers to which I was for some time in doubt, and only became convinced of the fact after the discovery of the Partial Differential Equation, the master-key to this portion of the subject, which gives the means of producing them ad libitum and ascertaining all that exist of any prescribed type. Of this partial differential equation I shall have occasion hereafter to speak ; but this is not all, for, as we shall presently see, mixed reciprocants are well worthy of study on their own account, and lead to conclusions of the highest moment, whether as regards their applications to geometry or to the theory of transcendental functions and of ordinary differential equations. The singularities of curves, taking the word in its widest acceptation, may be divided into three classes : those which are independent of homographic deformation and which remain unaltered in any perspective picture of the curve ; those whicii, having an express or tacit reference to the line at infinity, are not indelible under perspective projection, but using the word descriptive with some little latitude may, in so far as they only involve a reference to the line at infinity as a line, be said to be of a purely descriptive character ; and, lastly, those which are neither projective nor purely descriptive, having relation to the points termed, in ordinary parlance, "circular points at infinity" — for which the proper name is "centres of infinitely distant pencils of rays," that is, pencils, every ray of which is infinitely distant from every point external to it. Such, for instance, would be the character of points of maximum or minimum curvative, which, as we shall see, indicate, or are indicated by, that particular class of Mixed to which I give the name of " Orthogonal Reciprocants." All purely descriptive singularities alike, whether projective or non-projective, are indicated by pure reciprocants, and are subject to the same Partial Differential Equation; just as, in the Theory of Binary Quantics, Invariants, although under one aspect they may be regarded as a self-contained special class, admit of being and are most advantageou.sly studied in connection with, and as forming a part of, the whole family of forms commonly known by the name of "semi-, or subinvariants," but which I find it conduces to much ^: 41] On the Method of Reciprocants 287 I greater clearness of expression and avoidance of ambiguity or periphrasis to designate as Binariants. The question may here be asked, How, then, are projective and non- projective pure reciprocants to be discriminated by their external characters? I believe that I know the answer to this question, which is, that the former are subject to satisfy a second partial differential equation of a certain simple and familiar type, but this is a matter upon which it is not necessary for me to enter on the present occasion*. It is enough for our present purpose to remark that every projective pure reciprocant must, so to say, be in essence a masked ternary covariant. For instance, if we take the simplest dhi of all such, namely, a, that is -r^ , we have, if <^ {x, y) = 0, daf fd4\^_ \dy) ~ d^ daf d^ dxdy d^ dx d'(l> dxdy d<\> dx d' d4> dy d dy • Obviously we might instead And now if we writeas <»-)-(|)' * = 0. which, for facility of reference, let me call M. of a = 0 substitute M = 0 to mark an inflexion. the completed form of, when made homogeneous by the substitution of z for unity ; and if we suppose it to be of n dimensions in x, y, z, and call its Hessian H, we shall obtain the syzygy 4.H+\— — _/'-*!5-V \da? ' dy^ \dxdy) Hence the system «1> = 0, a = 0, will be in effect the same as the system = 0, H = 0, and in this sense a may be said to carry // as it were in its bosom. And so in general every pure projective reciprocant may, in the language of insect transformation, be regarded as passing, so to say, first from the grub to the pupa or chrysalis, and from this again, divested of all superfluous integuments, to the butterfly or imago state. Non-projective pure reciprocants undergo only one such change. There is no possibility of their ever emerging into the imago — their development being finally arrested at the chrysalis stage. It would, I think, be an interesting and instructive task to obtain the imago or Hessianised transformation of the Mongian, but I am not aware * In Paris, from which I correct the proofg, I have succeeded in reducing this conjecture to a certainty and in establishing the marvellous fact that every Projective Reciprocant, or, which is the same thing, every Differential Invariant, is, at the same time, an Ordinary Subinvariant. Thus a di£FereDtial invariant (or projective reciprocant) may be regarded as a single personality clothed with two diitinct natures — that of a reciprocant and that of a subinvariant. 288 On the Method of Reciprocants [41 that anyone has yet done, or thought of doing, this*. It seems to me that by substituting Reciprocants in lieu of Ternary Covariants we are as it were stealing a dimension from space, inasmuch as Reciprocants, that is. Ternary Covariants in their undeveloped state, are closely allied to, and march pari passu with, the fiarailiar forms which appertain to merely binary quantics. I will now proceed to bring before your notice the general partial differential equation which supplies the necessary and sufiScient condition to which all pure reciprocants are subject. It is highly convenient to denote the successive derivatives dy d'y d*y da^' d^' dx"' ■■■ by the simple letters a,b,c, The first derivative -^ plays so peculiar a part in this theory that it is necessary to denote it by a letter standing aloof from the rest, and I call it t. This last letter, I need not say, does not make its appearance in any pure reciprocant. This being premised, I invite your attention to the equation in question, in which you will perceive the symbols of operation are separated from the object to be operated upon. Writing F= 3a% + 10abBc+ (l-5ac + lOb") Ba + and calling any pure reciprocant R, VE = 0 is the equation referred to. I cannot undertake, within the brief limits of time allotted to this lecture, to explain how this operation, or, as it may be termed, this annihilator Fis arrived at. The table of binomial coefficients, or rather half series of binomial coefficients, shown f in Chart 4, will enable you to see what is the law of the numerical coefficients of its several terms. Let the words weight, degree, extent (extent, you will remember, means the number of places by which the most remote letter in the form is separated from the first letter in the alphabet) of a pure reciprocant signify the same things as they would do if the letters a,b,c, ... referred, according to the ordinary notation, to Binariants instead of to Reciprocants. The number of binariants linearly independent of each other whose weight, degree and extent or order are w, i,j is given by the partition formula (w; i,j) -(w- 1; i,j) where in general (w; i,j) means the number of ways of partitioning w into i or fewer parts none greater than j. * M. Halphen informs me that this has been done by Cayley in the Phil. Trans, for 1865, and subsequently in a somewhat simplified form by Painvin, Coniptes Rendus, 1874. But neither of these authors seems to have had the Boole-Mongian objectively before him, so that a slight supplemental computation is wanting to establish the equation between it and the function which either of them finds to vanish at a sextactic point. [t p. 302 below.] 41] On the Method of Reciprocants 289 It follows immediately from the mere form of V that the corresponding formula in the case of Reciprocants of a given type w.i.j will be (w; i,j)-{w-\; t' + l, j) the augmentation of i in the second term of the formula being due to the fact that, whereas in the partial dififerential equation for Binariants it is the letters themselves which appear as coefficients, it is quadratic functions of these in the case of Reciprocants. From the form of V we may also deduce a rigorous demonstration of the existence of Reciprocants strictly analogous to those with which you are familiar in the Binariant Theory, which are pictured in Chart 2, and are now usually designated as Protomorphs, as being the forms by the interweaving of which with one another (or rather by a sort of combined process of mixture and precipitation), all others, even the irreducible ones, are capable of being pro- duced. The corresponding forms for Reciprocants you will see exhibited in the same table. Each series of Protomorphs may of course be indefinitely extended as more and more letters are introduced. In the table I have not thought it necessary to go beyond the letter g. You also know that besides Protomorphs there are other irreducible forms, the organic radicals, so to say, into which every compound form may be resolved, always limited in number, whatever the number of letters or primal elements we may be dealing with. The same thing happens to Reciprocants as you will notice in the comparative table in Chart 2. Without going into particulars, I will ask you to take from me upon faitli the assurance that there is no single feature in the old familiar theory, whether it relates to Protomorphs, to Ground-forms, to Perpetuants, to Factorial constitution, to Generating Functions, or whatever else sets its stamp upon the one, which is not counterfeited by and reproduced in the parallel theory. So much — for time will not admit of more — concerning pure reciprocants. Let me now say a few words en passant on Mixed Reciprocants. Pure Reciprocants, we have seen, are the analogues of Invariants, or else of the leading terms, for that is what are Semi- or Subin variants, of Covariantive expansions; each is subject to its own proper linear partial differential equation. Mixed Reciprocants are the exact analogues of the coefficients in such expansions other than those of the leading terms. Starting from the leading terms as the unit point, the coefficients of rank a> are subject to a partial differential equation of order to; and just so, mixed reciprocant-s, if involving t up to the power to, are subject to a partial differential equation of that same order. I have alluded to a peculiar class of mixed under the name of "Orthogonal Reciprocants." They are distinguished, as I have proved, by the beautiful property that, if differentiated with respect to t, the result must be itself a Reciprocant. In Chart 1 you will see this illustrated in the case of a mixed & IV. 19 I 290 On the Method of Reciprocants [41 reciprocant (1 + <')6 — 3ta*, which serves to indicate the existence of points of maximum and minimum curvature. Its differential coefficient with respect to t is the oft-alluded-to Schwarzian, transliterated into the simpler notation. Proceeding in the inverse order — of Integration instead of Differentiation — I call your attention to a mixed reciprocant, of a very simple character, one which presents itself at the very outset of the theory, namely tc — bah, which, integrated in respect to t between proper limits, yields the elegant orthogonal reciprocant {t- + \)c-\Oabt+loa?. Expressed in the ordinary notation, this, equated to zero, takes the form \(dy\\Ad^_.Qdy d?y d'y fd^yV _ \\dx)^^]da^ ^"d^-d^-flt^ + ^H^/" Mr Hammond has integrated this, treated as an ordinary differential equation, and has obtained the complete primitive expressed through the medium of two related Hyper-Elliptic Functions connecting the variables x and y (see* Chart 3). It may possibly turn out to be the case that every mixed reciprocant is either itself an Orthogonal Reciprocant, or by inte- gration, in respect to t, leads to one. It will of course be understood that, in interpreting equations obtained by equating to zero an Orthogonal Reciprocant, the variables must be regarded as representing not general but rectangular Cartesian co-ordinates. Here seems to me to be the proper place for pointing out to what extent I have been anticipated by M. Halphen in the discovery of this new world of Algebraical Forms. When the subject first dawned upon my mind, about the end of October or the beginning of November last, I was not aware that it had been approached on any side by any one before me, and believed that I was digging into absolutely virgin soil. It was only when I received M. Halphen's letter, dated November 25, in relation to the Mongian business already referred to, accompanied by a presentation of his memoirs on Differential Invariants, that I became aware of there existing any link of connection between his work and my own. A Differential Invariant, in the sense in which the term is used by M. Halphen, is not what at first blush I supposed it to be, and as in my haste to repair what seemed to me an omission to be without loss of time supplied, I wrote to M. Hermite it was, in a letter which has been or is about to be inserted in the Comptes Rendus of the Institute of France ; it is not, I say, identical with what I have termed a general pure reciprocant, but only with that peculiar species of Pure Reciprocants to which I have in a preceding part of this lecture referred as corresponding and pointing to Projective Singularities. In his [* p. 302 below.] I 41] (hi the Method of Reciprocants 291 splendid labours in this field Halphen has had no occasion to construct or concern himself with that new universe of forms viewed as a whole, whether of Pure or Mixed Reciprocants, which it has been the avowed and principal object of this lecture to bring under your notice. I anticipate deriving much valuable assistance in the vast explorations remaining to be made in my own subject from the new and luminous views of M. Halphen, and possibly he may derive some advantage in his turn from the larger outlook brought within the field of vision by my allied investigations. Let me return for a moment to that simplest class of pure reciprocants which I have called protomorphs. Each of these will be found (as may be shown either by a direct process of elimination, or by integrating the equations obtained by equating them severally to zero, regarded as ordinary dififerential equations between x and y) each of these, I say, will be found to represent some simple kind of singularity at the point {x,y) of the curve to which these co-ordinates are supposed to refer. Thus, for instance, No. 1 marks a single point of inflexion ; No. 2, points of closest contact with a common parabola ; No. 3, what our Cayley has called sextactic points, referring to a general conic ; No. 4, points of closest contact with a common cubical parabola ; and 80 on. The first and third, it will be noticed, represent projective singularities, and as such, in M. Halphen's language, would take the name of Differential Invariants. The second and fourth, having reference to the line at infinity in the plane of the curve, are of a non-projective character, and as such would not appear in M. Halphen's system of Differential Invariants. It is an interesting fact that every simple p.arabola, meaning one whose equation m can be brought under the form y = a;", corresponds to a linear function of a square of the third, and the cube of the second protomorph, and con- sequently will in general be of the sixth degree. In the particular case of the cubical parabola, the numerical parameter of this equation is such that the highest powers of b cancel each other so that the form sinks one degree, and becomes represented by the Quasi- Discriminant, No. 4. This simple instance will serve to illustrate the intimate connection which exists between the projective and non-projective reciprocants, and the advantage, not to say necessity, of regarding them as parts of one organic whole. It would take me too far to do more than make the most cursory allusion to an extension of this theory similar to that which happens when in the ordinary theory of invariants we pass from the consideration of a single Quantic to that of two or more. There is no difficulty in finding the partial differential equation to double reciprocants which, as far as I have 19—2 \ 292 On the Method of Reciprocants [41 as yet pursued the investigation, appear to be functions of a, b, c, ... ; a', b', c, ...; and of {t — t'). The theory of double reciprocants will then include as a particular case the question of determining the singularities of paired points of two curves at which their tangents are parallel, and consequently the theory of common tangents to two curves and of bi-tangents to a single one. I think I may venture to say that a general pure multiple reciprocant which marks ofif relative singularities, whether projective or non-projective, of a group of curves, is a function of the second and higher differential derivatives appertaining to the several curves of the group, and of the differences of the. first derivatives, whereas in a mixed multiple reciprocant these last-named differences are replaced by the first derivatives themselves. As a particular case, when the group dwindles to an individual and there is only one t, this letter disappears altogether from the form, for there are no differences of a single quantity. In the chart (marked No. 2) you will see the table of Protomorphs carried on as far as the letter g inclusive, and will not fail to notice what may be termed the higher organisation of Reciprocantive as compared with ordinary Invariaiitive Protomorphs ; the degrees of the latter oscillate or librate between the numbers 2 and 3, whereas in the former the degree is variable according to a certain transcendental law dependent on the solution of a problem in the Partition of Numbers. Another interesting difference between general Invariants and general Pure Reciprocants consists in the fact that, whilst the number of the former ultimately (that is, when the extent is indefinitely increased) becomes indefinitely great, that of the latter is determinate for any given degree even for an infinite number of letters. ^M In carrying on the table of protomorphs up to the letter h (see Chart 6) a new phenomenon presents itself, to which, however, there is a perfect parallel in tlie allied theory. An arbitrary constant enters into the form, its general value being a linear function of U and W (for which see Chart 6). But this is not all. If you examine the terms in both U and W (there are in all twelve such) you will find that these twelve do not comprise all of the same type to which they belong. There is a Thirteentii (a banished Judas), equally (I priori entitled to admission to the group, but which does not make its appearance among them, namely, h*d. I rather believe that a similar phenomenon of one or more terms, whose presence might be expected, but which do not appear, presents itself in the allied invariantive theory, but cannot speak with certainty as to this point, as the circumstance has not received, and possibly does not merit, any very particular attention. 41] On the Method of Reciprocants 298 Still, in the case before us, this unexpected absence of a member of the family, whose appearance might have been looked for, made an impression on my mind, and even went to the extent of acting on my emotions. I began to think of it as a sort of lost Pleiad in an Algebraical Constellation, and in the end, brooding over the subject, my feelings found vent, or sought relief, in a rhymed effusion, a jeu de sottise, which, not without some apprehension of appearing singular or extravagant, I will venture to rehearse. It will at least serve as an interlude, and give some relief to the strain upon your attention before I proceed to make my final remarks on the general theory. TO A MISSING MEMBER Of a Family Group of Terms in an Algebraical Formula. Lone and discarded one I divorced by fate, Far from thy wished-for fellows — whither art flown ? Where lingerest thou in thy bereaved estate, Like some lost star, or buried meteor stone ? Thou mindst me much of that pi-esumptuous one Who loth, aught less than greatest, to be great, From Heaven's immensity fell headlong down To live forlorn, self-centred, desolate : Or who, new Heraklid, hard exile bore, Now buoyed by hope, now stretched on rack of fear, Till throned Astraea, wafting to his ear Words of dim portent through the Atlantic roar, Bade him "the sanctuary of the Muse revere And strew with flame the dust of Isis' shore." Having now refreshed ourselves and bathed the tips of our fingers in the Pierian spring, let us turn back for a few brief moments to a light banquet of the reason, and entertain ourselves as a sort of after-course with some general reflections arising naturally out of the previous matter of my discourse. It seems to me that the discovery of Reciprocants must awaken a feeling of surprise akin to that which was felt when the galvanic current astonished the world previously accustomed only to the phenomena of machine or frictional electricity. The new theory is a ganglionic one : it stands in immediate and central relation to almost every branch of pure mathematics — to Invariants, to DiflFerential Equations, ordinary and partial, to Elliptic and Transcendental Functions, to Partitions of Numbers, to the Calculus of Variations, and above all to Geometry (alike of figures and of complexes), upon whose inmost recesses it throws a new and wholly unexpected light. The geometrical singularities which the present portion of the theory professes to discuss are in fact the distinguishing features of curves; their technical name, if applied to the human countenance, would lead us to call a man's eyes, ears, nose, lips, and chin his singularities ; but V 294 On the Method of Reciproeants [41 these singularities make up the character and expression, and serve to distinguish one individual from another. And so it is with the so-called singularities of curves. Comparing the system of ground-forms which it supplies with those of the allied theory, it seems to me clear that some common method, some yet undiscovered, deep-lying, Algebraical principle remains to be discovered, which shall in each case alike serve to demonstrate the finite number of these forms (these organic radicals) for any specified number of letters. The road to it, I believe, lies in the Algebraical Deduction of ground- forms from the Protomorphs*. Gordan's method of demonstration, so difficult and so complicated, requiring the devotion of a whole University semester to master, is inapplicable to reciprocants, which, as far as we can at present see, do not lend themselves to symbolic treatment. How greatly must we feel indebted to our Cayley, who while he was, to say the least, the joint founder of the symbolic method, set the first, and out of England little if at all followed, example of using as an engine that mightiest instrument of research ever yet invented by the mind of man — a Partial Differential Equation, to define and generate invariantive forms. With the growth of our knowledge, and higher views now taken of invariantive forms, the old nomenclature has not altogether kept pace, and is in one or two points in need of a reform not difficult to indicate. I think that we ought to give a general name — I propose that of Binariants — to every rational integral form which is nullified by the general operator \aBb + fibSc + vcBa +... , where \, /i, v, ... are arbitrary numbers. This operator, I think, having regard to the way in which its segments link on to one another, may be called the Vermicular. Binariants corresponding to unit values of \, /i, v, ... may be termed standard binariants. Those for which these numbers are the terms of the natural arithmetical series 1, 2, 3, ... Invariantive binariants, which may be either complete or incomplete invariants ; these latter are what are usually termed semi- or sub-invariants. I may presently have to speak of a third class of binariants for which the arbitrary multipliers are the numbers 3, 8, 15, 24 ... (the squares of the natural numbers each diminished by unity) which, if the theorem I have in view is supported by the event, will have to be termed Reciprocantive Binariants. But first let me call attention to what seems a breach of the asserted parallelism between the Invariantive and the * See the section on the Algebraical Deduction of the Ground-forms of the Quintic in my memoir on Subinvariants in the American Journal of Matliematics. [Vol. iii. of this Reprint, p. 580.] I 41] On the Method of Reciprocants 295 Reciprocantive theories. In the former we have complete and incomplete invariants, but we have drawn no snch distinction between one set of pure reciprocants and another. A parallel distinction does however exist. If we use IV, i, j to signify the weight, degree, and extent of an invariantive form, w is never less than the half product of ij ; when equal to it the form is complete. In the case of reciprocants certain observed facts seem to indicate that there exists an analogous but less simple inequality. If this conjecture is verified it is not merely ~ — w, but ~ — (j — 2) — w, which is never negative : and when this is zero, the form may be said to be complete*. There would then be thus complete forms in each of the two theories ; in the earlier one they take a special name : this is the only difference. We have spoken of Pure Reciprocants as being either projective or non- projective, but so far have abstained from particularising the external characters by which the former may be distinguished from the latter. I have good reason to suspect that the former are distinguished from the latter by being Binariants; that, in addition to being subject to annihilation by the operator V, they are also subject to annihilation by the Vermicular operator when made special by the use of the numerical multipliers y, 8, 15 ... above alluded to, or in other words (as previotisly mentioned incidentally) are subject to satisfy two simultaneous partial differential equations instead of only onef. * If thiB should turn out to be true, the "crude generating function" for reciprocants would be almost identical with that of in- and co-variants of the same extent j. The denominators would be absolutely identical ; as regards the numerators, while that for invariantive forms is l-o"ix~' the numerator for reciprocants would be l-a~^x~^. As I write abroad and from memory there is just a chance that the index of a here given may be erroneous. t As already stated in a previous footnote this conjecture is fully confirmed, my own proof having been corroborated (if it needed corroboration) by another entirely different one invented by M. Halphen, who fully shares my own astonishment at the fact of there being forms (half- horse, half-alligator) at once reciprocants and sub-invariants, and as such satisfying two simultaneous partial differential equations. If instead of denoting the successive differential derivatives (starting from the second) a,b,c, ... we call them 1.2. a, 1.2.3.b, 1.2.3. 4. c, ... the two Annihilators will be a3j -I- 2i«, + Scig + ids, + ... a' »nd * 2 ** "*" ***'' + ^ (*" ■'■**'' *'« + '' ("''■*" **) '« ■*■ ••• the latter being my new operator, the Reciprocator V, accommodated to the above-stated change of notatiou for the successive differential derivatives. Hardly necessary is it for me to point out in explanation of the semi-sums }b^, ... that we may write the MacMahonised V under the form 4a% + o {ab + ba) d^+&(ac + lj' + ca)S^ + T {ad + bc + cb + da)St+ .... It is to be presumed that in addition to mixed reciprocants (the ocean into which flows the sea of pore reciprocants, as into that again empties itself the river of projective reciprocants) there may ^1 dx reciprocants, the most general of all, in which case we must speak of the content of these as the exist a theory of forms in which w as well as —- will appear, or, so to say, doubly mixed dx 296 On the Method of Reciprocants [41 Projective Reciprocants we have seen are disguised or maaked Ternary Covariants — Covariants in the grub, the first undeveloped state. Now ternary covariants are capable, it may or may not be generally known, of satisfying 6 reducible to 2 simultaneous Partial Diiferential Equations, and at first sight it might be surmised that nothing would be gained by the substitution of the two new for the two old simultaneous partial differential equations. But the fact is not so, for the old partial differential equations are perfectly unmanageable, or at least have never, as far as I know, been handled by any one, for they have to do with a triangular heap, whereas the new ones are solely concerned with a linear series of coefficients. I have alluded to there being a particular form common to the two theories. In the one theory it is the Mongian alluded to in the correspondence, which has been read, with M. Halphen. In the other it is the source of the skew covariant to the cubic. If the latter be subjected to a sort of MacMahonic numerical adjustment, it becomes absolutely identical with the former. Let ns imagine that before the invention of Reciprocants an Algebraist happened to have had both forms present to his mind, and had thought of some contrivance for lowering the coefficients of the Mongian written out with the larger coefficients, and had thus stumbled upon this striking fact. It could not have failed to vehemently arouse his curiosity, and he would have set to work to discover, if possible, the cause of this coincidence. He would in all probability have addressed himself to the form which precedes the source alluded to in the natural order of genesis, and have applied a similar adjustment to the much simpler form, ac — b": having done so he would have tried to discover to what singularity it pointed — but his efforts to do so we know must have been fruitless, and he would have felt disposed to throw down his work in despair, for the intermediate ideas necessary to make out the parallelism would not have been present to his mind. So long as we confine ourselves to Differential Invariants, that is, to projective pure reciprocants, we are like men walking on those elevated ridges, those more than Alpine summits, such as I am told* exist in Thibet, where it may be the labour of days for two men who can see and speak to each other to come together. Reciprocants supply the bridge to span the yawning ravine and to bring allied forms into direct proximity. ocean and of the others as sea, riTer, and brook. Curious is it to reflect that in the theory which as it exists comprises Invariantives, Reciprocants, and Invariantive Beciproeants or Reciprooant Invariantives, the order of discovery was (1) Invariantives (Eisenstein, Boole,ba'e -Zbabd Jr lac^ + Zbb-c iba?f- Alfkfibe -iia'cd + \\20a}^d-Z\bahc* -112063c a*g - 12a6/- 4b0ace+ 7926% + b68ade^ -27726crf+1925c3 302 On the Method of Reciprocanta [41 Chart 3. No. 1. a No. 2. Zac-bV* No. 3. 9a2rf - 45a6c + 406' No. 4. 45aSflP- 450a«ic+ 192aV+400a6Srf+ 165a6V - 4006*c ^ I dt infe '-/. ^|« ( 1 - 15 d^cji d({> d^* oP* cP* d^ dx^ dvdy dx dx^ dxdy dxdz CP0 flP<^ dd^ d'* (P* flterfy dy' dy dxdy dy' dydz d d4> d'* d^ cP* dx dy • dxdz dydz dz* (a) (M) (B) dy d?y 3 /flPw\« . ^, „ u • .u • •.. r 3a« "3^ —^ " o V ^7^ ) ^^ Schwarzian, otherwise written to . Chart 6. — The H Reciprocantive PROTOMOKrH. u ir 65a (x), it is clear that 6, & , 6" ... are each of them infinitesimals of the first order, and consequently that all products, and all powers higher than the first of these quantities, may be neglected. We have therefore r^q = ^y = -^ = v' - ^V ^y^ dx+dd i+ff y "^ un - ^^^ dx^'~^^^ y"{\-&)-e"^ ^y^~dx+d0 1 + 6' 1 + 6' = y"-Wy"-6"y r,/"i- ^^-^ ^ if - Wy") - 6"y' ^ y,„ ^^ _ ^g,^ _ ^^,y„ _ ^.y, ^y ^~dx + d6 1 + 6' 1 + 6' = y'" - S6'y"' - Z6"y" - 6"'y'. Hence [yV"] = yy'" - ^6'yy'" - W'y'f - r'y" W"\ =y'^-(^ffy"'-Z6"y'f [y^] =y'^-26'y\ And since by definition * As originally given in the Messenger of Mathematics, Vol. xv., this was defaced by so many errata as to render expedient its reproduction in a corrected form. I 42] Lectures on the Theory of Reciprocants 307 we readily obtain V(y,x)-\ = ^^,-^6'{y,x)-e"'^{y,x){\-2d-)-e"'. So also [{z, x)] = (z, x) (1 - 2d') - ff". Whence by subtraction \iy, X) - {z, x)-\ = (1 - 2^) [{y, x) - (z, x)]. Dividing the left-hand side of this by [/^], and the right-hand side by /* (1 — 20') which is the equivalent of [z''], our final result is [{y, x) - (z, x)'\ _ (y, x) - (z, x) Thus, then, we have seen that the expression (y, x) - (z, x) does not vary when x receives an infinitesimal variation €x, x)} (dxy = {(/a)-'Z, X) - ( (x), -^ (x) For every such invariant will be a function of (y, x) - (z, x), {y, x) - (u, x), (z, x) - (u, x), ... and will therefore remain a persistent factor of its altered self, taking on a power of ^— as its extraneous factor. ax Calling (fx, x) the Schwarzian Derivative of /(x), our theorem may be stated in general terms as follows : All invariants of a Binary Quantic whose roots are the Schwarzian Deri- vatives of a given set of functions of the same variable are Covariants (in an extended sense) of that set of functions. The theory of the Schwarzian derivative originates in that of the linear differential equation of the second order, u"+2Pu+Qu = 0, which becomes, when we write u = ve~!^^, t;"+/y = 0, where I=Q-F'-P'. Now, suppose that ttj and u^ are any two particular solutions of the first of these equations, and let z denote their mutual ratio; so that, when v, and v.i are the cori'esponding particular solutions of the second equation, we readily obtain ^ = ^ = -' M, v^' and therefore, z = z . A second difiFerentiation gives ,, vX'-%v" 2y,'(«,<-t;X) IP V,' V, . ■ »l" ^«" T But since — - = — = — i, the first term of the expression just found vanishes identically, and we have z = , 310 Lectures on the Theory of Reciprocanta [42 or. "'=--27- DiflFerentiating this again, we find g"' 3 «"« Hence Y-|^ = 2^' where the left-hand side of the equation is "the Schwarzian Derivative " with z written in the place of y. S LECTURE II. The expression "^y'y"' — 3y"^ which we have called the Schwarzian, may be termed a reciprocant, meaning thereby that on interchanging y, y", y'" with x^, x^^, a;,,, its form remains unaltered, save as to the acquisition of what may be called an extraneous factor, which, in the case before us, is a power of y (with a multiplier — 1). Before we proceed to consider other examples of reciprocants it will be useful to give formulae by means of which the variables may be readily interchanged in any differential expression. We shall write t for y and t for its reciprocal x^ , using the letters a,b,c,... to denote the second, third, fourth, etc., differential derivatives of y with respect to x, and a, /8, 7, ... to denote those of x with respect to y. The advantage of this notation will be seen in the sequel. The values of a, 0,y, ... in terms of t, a,b,c,... are given by the formulae o = - a -r i', 7 = - cf + lOabt - I5a' ^ f, B = -dt' + {loac + 106') t^ - lOoa^bt + 105a* -=- f, e=-ef+ {21ad + 356c) t^ - (210a'c + 280ab') C + 1260a'bt - 94:5a' h- t", If, in these equations, we write a = 1.2.ao, 6=1.2.3.01, c= 1 . 2 . 3 . 4.0,, ... and a=1.2.a„, /3 = 1.2.3.ai, 7 = 1.2.3.4.02,... 42] Lectures on the Theory of Reciprocants 311 they become Oo = - tto -=- <'. ai = - ait + 2a^-r-1?, fltj = — a^t^ + 5a„ait — 50o' -r V, a, = - a3<^ + (GooO, + 3ai») i^ - 21a„=ffli< + 14a„* ^ <», a, = - a^i^ + (TkoOs + 7a,aa) f - (28a„»a3 + 28oh>=) t^ + 84ao»ai< - 42a„» ■¥ i", Any one of the formulae iu either set may be deduced from the formula immediately preceding it by a simple process of differentiation. Thus,s,nce ^ = j— '^''^ dy = 1 ' di' . dB \ d (-htJr So'^N we have — ' ' dy _1 d /-bt + 3a'\ ~ f dx\ f / ■ But T^ = 7 and T- = a3{ + 69o + c96+ ..., ay aa; 80 that y = j(adt + bda+cdb + ...)( j = -, (- c«' + lOabt - 15a'). By continually operating with - (ddt + bda + cdi, + ...) the table may be z extended as far as we please, the expressions on the right-hand side being the successive values of |J(aae + Wa+ca»+...)}"(-P) found by giving to n the values 0, 1, 2, 3 Precisely similar reasoning shows that, when the modified letters Og, O] , a,, ■ ■ ■ are used, (n + 2) a„ = - (2o,3t + 3a, 3a, + 4a,9a, + • • •) a»-i , z 1 1 " ' OoN A proof of the formula r a„ = -«-»-3(e"«)a„, obtained by Mr Hammond, in which F = 4 . -^ 3a, + oaoOiSo, + 6 f ajO, + y 1 3a, + 7 (a.aj + aiOj) Sa. + • •• , will be given later on, when we treat of this operator, which, in the theory of Reciprocants, is the analogue of the operator 03^ + 263c + 3c3d+ ..., with which we are familiarly acquainted in the theory of Invariants. and that a„ = ii^, . , „^ " 3.4.5 ... (n + 2) 312 Lectures on the Theory of Reciprocants [42 Consider the expression ct — 5ab. If, in 7T — 5a/8, which may be called its transform, we write 1 a a -bt + Sa* -cf + 10aht-15a* '■=<■ *" — ?' ^ if- ' 'y= e • this becomes a fraction whose denominator is <*, while its numerator is - c<' + lOabt -lba' + ba (- bt + 3a') = - c^' + haht. Removing the common factor t from the numerator and denominator of this fraction, we have . - ct — bob 7T - oa/3 = ^ — . Here, then, as in the case of the well-known monomial for which < a = -Va, and the Schwarzian for which 2bt -^a' = -i? (2y3T - Sa"), the expression ct — bah = — V (yr — 5a;S) changes its sign on reciprocation. That reciprocation is not always accompanied with a change of sign will be clear if we consider the product of any pair of the three expressions given above. Or we may take, as an example of a reciprocant in which this change of sign does not occur, the form Zac-hb\ ^ o .a, Sa(ct^-10aht + 15a')-5(bt-3a'y Here 3a7 - .5/3^ = — ^^ — — - — — ^ . z In the fraction on the right-hand side the only surviving terms of the numerator are those containing the highest power of t, the rest destroying one another. Thus Say - 5/3^ = i-(3ac - 56»). Reciprocants which change their sign when the variables x and y are interchanged, will be said to be of odd character; those, on the contrary, which keep their sign unchanged will be said to be of even character. The distinction is an important one, and will be observed in what follows. Forms such as the one just considered, whei-e t does not appear in the form itself, but only in the extraneous factor, will be called Pure Reciprocants, in order to distinguish them from those forms (of which the Schwarzian 2tb—3a' is an example) into which t enters, which will be called Mixed Reciprocants. It will be seen hereafter that Pure Reciprocants are the analogues of the invariants of Binary Quantics. 42] Lectures on the Theory of Reciprocants 313 With modified letters (that is, writing a = 2ao, b = Ga^, and c = 24a2) Sac — 56^ becomes IMaaO^ — ISOai" = 36 (4iaa(ii — Soi"). Operating on this with F= 2a^da^ + 5a„aj3a2+ •••' we have F(4aoa2 — S^i") = 0. We shall prove subsequently that all Pure Reciprocants are, in like manner, subject to annihilation by the operator V. Hitherto we have only considered homogeneous forms ; let us now take as an example of a non-homogeneous reciprocant the expression (1 +t')b- 3aH. Here ^i + .^^ ff - s.^r = [l + -^ [^^—) —^ ^ (1 -I- <') (- bt + 3a') - 3a' f In the numerator of this fraction the terms + 3a' and — 3a' cancel, a factor t divides out, and we have finally r In general, a Reciprocant may be defined to be a function F of such a kind that F(t, a, /3, 7, ...) contains F(t, a,b, c, ...) as a factor. An import- ant special case is that in which the other factor is merely numerical ; the function F is then said to be an Absolute Reciprocant. When we limit ourselves to the case where .F is a rational integral func- tion of the letters, it may be proved that F(t,a,b,c,...)=±t^F(T,a,0,y, ...). For, in the first place, since any one of the letters 0, /S, 7, ... is a rational function of t, a,b,c, ... and integral with respect to all of them except t, containing only a power of this letter in the denominator, it is clear that any rational integral function of t, a, /9, 7, ... such a,a F(t, a, >3, 7, ...) is supposed to be, must be a rational integral function of t, a,b,c, ... divided by some power of t. But since ^ is a reciprocant, F(r, a, 0, 7, ...) must contain F(t, a,b, c, ...) as a factor; and if we suppose the other factor to be {t, a, b, c, ...) we must have where is rational and integral with respect to all the letters. I 314 Lectures on the Theory of Reciprocants [42 Moreover, Fit, a, b, c, ...) ^'^(•^' «-^' 7, -) ^(^_ ^ ^_ ^_ ), Hence we must have identically ^F(r, a, A y, ...), ■ or we may say, = (-)««" F(t, a, /3, 7, . . .), where k=1 or 0 according as the reciprocant is of odd or even character. It obviously follows that the product or quotient of any two rational integral reciprocants is itself a reciprocant ; but it must be carefully observed that this is not true of their sum or difference unless certain conditions are fulfilled. For if we write F,(t,a,...) = {-y't^'F(T,a,...) * and F,(t,a,...)= (-)"tf^F, (t, a, . . .), we see that pF,{t. a, ...) + qF,{t, a, ...) = {-)"t'^'pF,{T, a, ...) + (_)«. i^-gF,(T, a, ...), I 42] Lectures on the Theory of Reeiprocants 315 and consequently this expression will be a reciprocant if k, = k^ and /Zi = fi^, but not otherwise. If we call the index of t in the extraneous factor the characteristic, what we have proved is that no linear function of two reeipro- cants can be a reciprocant, unless they have the same characteristic and are of the same character. In dealing with Absolute Reeiprocants, since the characteristic of these is always zero, we need only attend to their character. I propose for the present to confine myself to homogeneous and isobaric reeiprocants*, that is, to such as are homogeneous and isobaric when the letters t, a, b, c, ... are considered to be each of degree 1, their respective wetghts being —1,0,1,2,.... The letter w will be used to denote the weight of such a reciprocant, i its degree, and j its extent, that is, the weight of the most advanced letter which it contains. Let any such reciprocant F(t,a,b,c, ...) contain a term .4<"'a'6'"c"..., then v+l + m + n+... = i, and —v + m + 2n+ ... =w. The corresponding term in F(t, a, y8, 7, ...) will be AT^a'^y" ... where 1 a _ 6 c ^=?' " = -?• ^ = - {t, a, b, c, ...) is an absolute reciprocant of odd character. An absolute reciprocant may be obtained from any pair of rational integral reciprocants in the same waj' that an absolute invariant is found from two ordinary invariants. For let F,{t, a, b, c, ...) = (-)'.<^.f.(T, a, A 7. •••). and F, (t, a,h,c, ...) = (-y^tr'F.ir, a, /9, 7. — ). {FAt.a,b,c,...)}^' _ ,.,,,_,,, (i^,(T,«,A7.-)h . {F,{t, a, b,c. ...)}'-'- ^ ^ {^,(t,«,/9, 7, ...))"■' or we may say that jPj*^ -r- F/' is an absolute reciprocant of even or odd character according to the parity of k^ii^ — K^fi^. I 42] Lectures on the Theory of Reciprocants 317 Thus, for example, from a = — fa and Zac -5¥ = f (Say - 5/3=) we form —, an absolute reciprocant of even character. a' From a reciprocant F whose characteristic is /x we obtain an absolute reciprocant of the same character as F by dividing it by i*. For if we only remember that t = - , it obviously follows that F(t,a,b.c,...)^±t>^F{T,a,l3,y. ...) can be written in the form ~ F{t,a,b,c,...)_ ^ F{T,a,^,y,...) where the original character of the reciprocant F is preserved. It may be noticed that a reciprocant of odd character cannot be divided ty ^(_ 1)<2 80 as to give an absolute reciprocant of even character; for, the i» /^ reciprocal of i" being -«''i^', that of /"-=- V(-l)«* will still be -F' ^'^i-l)T^. The character of a reciprocant is thus seen to be one of its indelible attributes. As simple examples of absolute reciprocants we may take , which becomes on reciprocation — , and -z, which reciprocates into — -. The character of the former is even, that of the latter odd. Observing that 1 . 1 , 1 d 1 d log< = -logTand^^.^ = ^.^. From this, in like manner, we obtain and so, in general. 318 Lectures on the Theory of Reciprocants [42 Hence l-r . ^ j log t is an absolute reciprocant, and of an odd character, for all positive integral values oft. We thus obtain a series of fractions with rational integral homogeneous reciprocants in their numerators and powers of t^ in their denominators. It will be sufficient, before proceeding to the more general theory of Eduction, as it may be called, to examine, by way of illustration, the cases in which i = 1, 2 and 3. Let i = 1 ; then So that, in the case where i = 2, we have Wt ' dco) ^^ Wf dx) ti f 2 • <« _ 2bt - 3a' The numerator of this fraction is the Schwarzian. In like manner, when t = 3, /'I ^Y] f-f^ d\f2bt-Sa''\_2ct-4,ab eabt-da" _2cf-l0aht+9a!' But here a reduction may be effected, for ( -^ 1 , as well as -; itself, is an absolute reciprocant of the same character as the whole of the expression just 9 a'. found. Hence we may reject the term ^ . -j without thereby affecting the reeiprocantive property of the form, and thus obtain ct — 5ab an absolute reciprocant of odd character. The corresponding rational integral reciprocant is ct — 5ab. We have found that -, and -^ — are each of them reciprocants. t^ t 2bt Why, then, by parity of reasoning, is not — , and therefore b, a reciprocant ? It is because — , the square of -j , is of even character, while is of t t^ " an odd character, so that no linear combination of the two would be legitimate. I 42] Lectures on the Theory of Reciprocants 319 If we differentiate any absolute reciprocant with respect to x, we shall obtain another reciprocant of the same character. For let R be any absolute reciprocant and R its transform, then R=±R'; and since t- = * t- may be written in the equivalent but more symmetrical dec ^y form 1 i_ = J^ A i\/t' dx \Jt' dy' Q On one side of this identical equation is a function of the differential derivatives of y with respect to a; ; on the other, a precisely similar function of those of X with respect to y. Hence -^ . -7— is an absolute reciprocant, and therefore --j- is a reciprocant, the character of each being the same as that of R. I will avail myself of the conclusion just obtained, which is the cardinal property of absolute reciprocants, to give a general method of generating from any given Rational Integral Reciprocant an infinity of others — rational integral educts of it, we may say. Let F be such a reciprocant, and /j, its charac- teristic ; then — is an absolute reciprocant, and consequently t- ( ~ j is a reciprocant, both of them of the same character as F; that is ^dF fl „r, *^_-£_^. or we may say 2t -j fiaF is a reciprocant of the same character as F. This is even true for non-homogeneous reciprocants, for the only assump- tion made at present as to the nature of F is that it is a rational integral reciprocant. But if we further assume that it is homogeneous and isobaric*, we know that ^ = 3t + w. Now, Euler's equation gives 3i = 3(0« -♦- a3a + Mj + c3„-H ...), * It will sabseqaently be proved that every rational integral reciprocant which is homo- geneons is also isobaric. I 320 Lectures on the Theory of Reciprocants [42 and from the similar equation for isobaric functions (remembering that the weights of*, a, 6, c, ... are - 1, 0, 1, 2, ...) we obtain so that /t = 2 ET F^GFi — FfiF,i we have GF =» — ^rnr-* 21—2 324 Lectures on the Theory of Reciprocants [42 In order to prove that this is a reciprocant, we have to show that the character and characteristic are the same for both terms of the numerator. But OF^ is a reciprocant of the same character as F^, and GF^ is one of the same character as F^; thus the two terms of the numerator are of the same character as F^F^. As regards the characteristic, it should be noticed that G, that is, the operator (26<— 3a')3o + (2c<— 4a6)94+ ..., increases the degree by unity, but does not alter the weight, so that it increases the characteristic of any rational integral homogeneous reciprocant by 3. Thus the characteristic of each term in the numerator exceeds by 3 that of F^F^. Hence GF is a reciprocant, and, taking account of its denominator as well as its numerator, we see that the operation of (? on a rational homogeneous reciprocant, whether fractional or integral, produces another in which the original character is preserved while the characteristic is increased by three units. More generally, let JP*,, F^, F,, ... be any rational homogeneous recipro- cants whose extraneous factors are (—)"'<''', (—)"'<''', {—Y'V^', ■■■ respectively; and supposeto consist of a series of terms of the form AFj"F2'^F3'' ..., such that the extraneous factor for each term is (-)*t^. Thenis a recipro- cant, but not necessarily a rational one; for the indices \,, Xj, \,, ... may be supposed fractional, provided only that they satisfy the conditions «i\i + «aXa + /CjXj + ... — K= a. positive or negative even integer, and Jh\ + /^^ + Ms^s + ■ ■ ■ — /J- = 0. We proceed to show that G^ is also a reciprocant, and that its extraneous factor is (—)"<''+'. Since we have to prove not only that each term of this expression is a reciprocant, but also that all of them have the same extraneous factor ; otherwise their sum would not be a reciprocant. Now, in * = lAFi'^'F^'^'F,'"' ..., the extraneous factor for each term is by hypothesis (—)'<'*, so that the extraneous factor for each term of ^ = 1A\F,''-^F^^FJ^..., is (—)"""'<''"'", and therefore -rrr is a reciprocant. Also, GFi is a reciprocant whose extraneous factor is (—)"■<''■+•'. Hence -rw.GFi is a reciprocant having (_)«^+s for extraneous factor, and in exactly the same way we see that every other term of G

which, as is well known, annihilates any satisfied invariant. I 42] Lectures on the Theorjf of Reciprocants 329 LECTURE V. It will be desirable to fill up some of the previous investigations by discussing some points in them that have not yet received our consideration. There may be some to whom it may appear tedious to watch the com- plete exposition of the algebraical part of the Theory, who are impatient to rush on to its applications. But it is my duty to consider what may be expected to be most useful to the great majority of the class, and for that purpose to make the ground sure under our feet as I proceed. To the greater number it will, I think, be of advantage to have their memories refreshed on the kindred subject of invariants, and probably made acquainted with some important points of that theory which are new to them. I confess that, to myself, the contemplation of this relationship — the spectacle of a new continent rising from the waters, resembling yet different from the old, familiar one — is a principal source of interest arising out of the new theory. I do not regard Mathematics as a science purely of calculation, but one of ideas, and as the embodiment of a Philosophy. An eminent colleague of mine, in a public lecture in this University, magnifying the importance of classical over mathematical studies, referred to a great mathe- matician as one who might possibly know every foot of distance between the earth and the moon ; and when I was a member, at Woolwich, of the Government Committee of Inventions, one of my colleagues, appealing to me to answer some question as to the number of cubic inches in a pipe, e.xpressed his siirprise that I was not prepared with an immediate answer, and said he had supposed that I had all the tables of weights and measures at my fingers' ends. I hope that in any class which I may have the pleasure of conducting in this University, other ideas will prevail as to the true scope of mathematical science as a branch of liberal learning; and it will be my endeavour to regulate the pace in a manner which seems to me most conducive to real progress in the order of ideas and philosophical contemplation, thus bringing our noble .science into harmony and in a line with the prevailing tone and studies of this University. Faraday, at the end of his experimental lectures, was accustomed to say — I have myself heard him do so — "We will now leave that to the calculators." So long as we are content to be regarded as mere calculators we shall be the Pariahs of the University, living here on sufferance, instead of being regarded, as is our right and privilege, as the real leaders and pioneers of thought in it. 330 Lectures on the Theory of Reciprocants [42 That Cayley's two operators, which have been called P and Q, are in fact generators, may be proved as follows •{• : Let a=adf>f 2bdc + Scdd + '^d^e + •••, and ^ = aQdida+ fjicdb+fdde +...) — Kb, ■where k, \, /j,, v, ... are numbers. When ic is the degree of the operand, and \ = ij.— v= ... =1, the operator 0 is identical with P ; but 0 is identical with Q when k is twice the weight of the operand and \ = 0, fi=l,v = 2, .... If now we use * to signify the act of pure differential operation, it is obvious that n0=(nx 0) + (n*0), 0n=(n X 0)+(0*n), so that fi0-0il = (fl*0)-(0*n). But since Ha = 0, Hi = a, He = 26, ... we have D,¥fB = a {\ada + 2fi,bdb + Svcd^ + ...-«) and €>* il = a (\hdi+2ficdc + Svddci + ...). Hence 120 - 0n = a [Xada + {2/jl - X) bdb + (3i/ - 2^) c9, + ...-«) ; now if the operand I be any invariant (satisfied or unsatisfied), we have 0,1=0, and therefore 012/ = 0 ; so that we find il^I = a {\ada + i2/jL-\)bdb + {Sv - 2fi)cde+ ... - ic] I. If in this we write \ = fi = v = ... =1, and k = i, where i is the degree of the operand, 0 becomes P and we have D.PI==a(ada + bdb + cdc+...-i)I. • But, by Euler's theorem, the right-hand side of this vanishes, and therefore npi=o. Similarly, by means of the corresponding theorem for isobaric functions, we may prove that nQi=o. For if, in the general formula, we write \ = 0, /:i = l, v = 2, ... and k = 2w, where w is the weight of the operand, we find nQI = a{2bdb + 4:cdc + 6ddi + ... - 2w) 1=0. Thus, when 0 stands either for P or for Q, it is either an annihiiator or a generator (that is, 0/ is either identically zero or else an invariant). But if I be the most advanced, or say the radical letter of /, no term of mdj can cancel with any other term of 0/; and since, for this reason, 0/ cannot vanish identically, it must be an invariant, and the operators P and Q must be generators. t In the Quarterly Journal (Vol. xx. p. 212) Prof. Cayley only considers a special example, and has not given the proof of the general theorem. I 42] Lectures on the Theory of Reciprocants 381 The generators previously given for reciprocants also possess this property of introducing a fresh radical letter at each step. The radical letter, on its first introduction, enters in the first degree only, and in the case of the educts of log*, whose values have been calculated, its multiplier is seen to be a power of t. The form of the generator for mixed reciprocants 3 («]< - a^) d„„ + 4 (fflj* - ffloai) 8a, + ..•+(»+ 3) (a„+, t - a„a„) da„ shows this, or it may be seen by considering the successive values of For let — — — " "^' '"^ denote this expression, and let its radical letter be a„ ; then, on differentiating again with respect to x, the new letter intro- duced arises solely from a term in the numerator ^^((,a„.a„a„...a„).^^. But a„ = 5^^2.3...«+2; so that ^ = (n + 3) a„+i . aar dx Hence, if when a„ is the radical letter, it occurs in the first degree only ' and multiplied by a power of t, it follows that, since -,— will be a power of t, the derived expression which contains the radical letter a„+, will contain it in the first degree only and multiplied by a power of t. And since this is true for the case i = 1, when —r. . -;- log < = -^ , it is true universalh*. Observe that for t = l, 2, 3, ... the radical letter is ao, Oi, aj, ... respec- tively. It will be remembered that (71 • j-j log i is an absolute reciprocant. It may be called the I'th absolute educt, to distinguish it from the rational integral educts E^, E^, E„ ... whose values have already been calculated. Let R(t, a,, a,, Oj, ... a„) be any homogeneous rational integral recipro- cant, and let the educts be A^, -4,, A^, ... An] then obviously a„ may be expressed rationally in terms of .4„ and a„_,, a„_2, ... a^, t, (iji-i n n j> » -"n-i 8.nd a„_2, ... ttoi f> «i „ „ „ „ -4,, a„ and t, «o „ „ „ „ A^ and t, where observe that the denominators in these expressions are all powers of t. Hence, by successive substitutions, ii(<, ao, a,, ... a„) may be expressed 332 Lectures on the Tlieory of Reciprocants [42 rationally in terms of A^, ... A^, A,,, and t. Thus any rational integral homogeneous reciprocant is a rational function of educts, and is of the form E — , where E \»& rational integral function of the educts. v Does not this prove too much, it may be asked, namely, that any function F of the letters is a rational function of the educts, which are themselves reciprocants, and will therefore be a reciprocant? But this is not so; for observe that although F will be expressed as a sum of products of educts, such products will not in general be all of the same character, and their linear combination will be an illicit one, such as is seen in the illicit com- bination of Oo' with the Schwarzian (a,t — a^). We have seen that by differentiating an absolute reciprocant, or by the use of a generator, we obtain a fresh reciprocant. But there are other methods of finding reciprocants ; as, for example, if the transform of ^{t, a, b, c, ...) is yjr{T, a, /3, 7, ...), that is, if ^ (t, a, b, c, . . .) = -^Ir (t, a, /9, 7, . . .), then yjrit, a, b, c, ...)=.yjr is a reciprocant, and, moreover, an absolute one of even character, although neither , which is a perfectly arbitrary function, nor 1^, its transform, is a reciprocant. Herein a mixed reciprocant differs from an invariant, which cannot be resolved into non-invariantive factors. It is worth while to give a proof of this proposition; but first I prove its converse, that if p, q, r, ... are all invariants, their product must be so too. This is an immediate consequence of the well-known theorem that a/=o is the necessary and sufficient condition that / may be an invariant where, as usual, O is the operator adi, + 2bde + Scdd+ ■■., and the word invariant has been used in the same extended sense as formerly. For D,(pqrs...) = (-^ + —3 -^ ^ -I- ... \ p q r But since p, q, r, ... are all invariants, we have np = 0, nq = 0, flj- = 0, ., and therefore fi {pqrs . . .) = 0. Next, suppose that I = P^Q^, where / is but Qi is not an invariant. pqrs I 42] Lectures on the Theory of Reciprocants 333 To meet the case in which P, and Qi are not prime to one another, Q^, if resolved into its factors, must contain one Q' where Q is not an invariant. Suppose that Pj contains Q\ and let i +j—k; then we may write / = PQ*, where P is prime to Q. But since / is an invariant by hypothesis, 12/ =0, and therefore, Q* HP + kPQ'-' flQ = 0 ; p=-*ap- Now P is prime to Q, so that the fraction ^ is in its lowest terms; there- fore flQ contains Q ; but this is impossible, for the weight of fiQ is less than that of Q. Hence / cannot contain any non-invariantive factor Qi. All this will be equally true for a general function J annihilated by any operator fl which is linear in the differential operators 3a,3t, 9c, •••no matter what its degree in the letters a,h,c, ... themselves ; that is, we shall still have /=PQ* and 9-_k^ and p- Ic^^, where P and Q are piime to each other, and, as before, flQ will contain Q as a factor. But if fl is an operator which diminishes either the degree or the weight, HQ is either of lower degree or of lower weight than Q, and so cannot contain it as a factor. Hence J cannot contain a factor Q not subject to annihilation by fl. If, however, il does not diminish either the degree or the weight, it may be objected that ilQ might conceivably contain the factor Q ; and were it so, there would be nothing to show the impossibility, in this case, of a function J subject to annihilation by il containing a factor Q, which is not so^ But quaere: Is it possible, when .b)R, is a reciprocant. The general value of -r- has been seen to be aSt + bda-¥ c3(, + ddc+ ..., but, since R is supposed to be pure, dtR — 0. 336 Lectures on the Theory of Reciprocants [42 We may therefore, in Za -j- — /jh, replace -j- by Mo + c9» + f^» + c9(i + ••• • Now, remembering that fj, = 3i + w, and that by Euler's theorem and the similar one for isobaric functions and w= 69i, + 2c9c + 3d9 (a, b, ...). In order that this may be a reciprocant, we must have ^(o, b, ...) = c + loa\ In this way the Orthogonal Reciprocant (1 +P)c- lOabt + 15a» was obtained originally. It will be easy to verify that this is a reciprocant by means of the identical relations 1 t=-, T a « = -;i. ^-^- -3a> 7T^-10a/3T+loo^ c = -, . We shall find that (1 +f)c- lOabt +l5a' = -(' {(1 + r=) 7 - lOaySr + 15a»), and comparing this with ct — oab = — {' (7T — oaS), it will be noticed that both forms have the same character and the same characteristic. The complete primitive of the differential equation c{l + tr)-\Oabt+lba^ = 0 has been found by Mr Hammond and Prof Greenhill. The solution may be written in the following forms : *"~j V{«(l-15<»+15<^-<«) + X(6«-20<»+6t'')}"''^ _ r tdt_ ^ ~ J Vff (1 - 15«' + 15<* - <•) + X (6« - IW+Qt')] cos(e-A)de + const. + 1' r coi _ r_sin^(^^^)d0_ '~J ^J[BcosQ{e-A)] + const. 42] Lectures on the Theory of Reciprocants 341 k'Hn\{X, k) = mn'(Y,k'), where i=sinl5°, A' = sin 75°, and X = lx + my -\-ni, Y='mx — ly + n^, I, m, «i, Tia being arbitrary constants. The last two forms of solution are due to Prof. Greenhill. LECTURE VII. I have frequently referred to, and occasionally dilated on, the analogy between pure reciprocants and invariants. A new bond of connection between the two theories has been established by Capt. MacMahon, which I will now explain. Let me, by way of preface, so far anticipate what I shall have to say on the Theorem of Aggregation in Invariants (that is, the theorem concerning the number of linearly independent invariants of a given type) as to remark that the proof of this theorem, first given by me in Crelle's Jmimal and subsequently in the Phil. Mag. for March, 1878, depends on the fact that if we take two operators, namely, the Annihilator, say n = Uodai + 20,80, + 3a,aa, + . . . +jaj^idaj and its opposite, say 0 = ajdaj_, + 2aj_tdaj_„ + 3a,_,9a^._j + . . . +j(hdao, then (no - Oil) / is a multiple of I. Thus, if/ stands for any invariant (that is, if 117=0), it follows imme- diately that nOI is a multiple of I, and consequently n"'0"'/ is also a multiple of /. We may call il and 0, which are exact opposites to each other, reversing operators. Now, MacMahon has found out the reversor to V, the Annihilator of pure reciprocants. His reversing operator is no longer of a similar, though opposite, form to F, as 0 is to il, but is simply -=- ; nor is the effect of operating with V -j- on any pure reciprocant R equivalent to multiplication by a merely numerical factor, as was the case with HOI, but {Vj-\R is a numerical multiple of aR, and as a consequence of this {^'",7";^)^ ^^ ^ numerical multiple of a!^R. Thus the parallelism is like that between the two sexes, the same with a difference, as is usually the case in comparing the two theories. 342 Lectures on, the Theory of Reciprocants [42 This remarkable relation between the operators V and -j- may be seen a priori if we assume that (as we shall hereafter prove) to each pure recipro- cant R there is an annihilator V of the form 3a»84 + (...)9. + (...)Sd + (...) 3. + -., not containing da and linear in the remaining differential operators 3^ , 3c. 98a'bd' - SGa'c'd' - S2ab-cd^ - 646^d» + 2iabc'd -i- 9%'c-d - 866'^c'. It will be found on trial that this is divisible by the invariant 4 (ae - 4,bd + 3c=), the quotient being 4a^ce - 4,a¥e - Sa^d' + 2abcd + 46»ci - 36V = 3a {ace - b'e - ad- + 2bcd - c^ + {ac - b') {ae - 46d + 3c^). |l I 42] Lectures on the Theory of Heciprocants 347 Thus the discriminant of the quadratic in f, that is, of the invariant ay - 2/ {babe - lacd + B,¥d - 66c=) + . . ., is shown to be an invariant. It will further illustrate the proof of the theorem if we remark that precisely the same invariant is obtained by eliminating y between the above form and the protomorph a"/- babe + 2acd + 86=^ - 66c''. Ex. 2. If we take the pure reciprocant 45a'd=- 450a^6cd + 400a6'd + 192a^c' + 16oa6=c= - 4006*c, which, from its similarity to the Discriminant of the Cubic, I have called the Quasi-Discriminant, and form its discriminant, when regarded as a quadratic in d, we find 45a' (192a^c' + l^bab^'c^ - 4006^) - (225a'6c - 200a6»>'. If, in this expression, we write P = 3ac — 56-, so that 3ac = P + o¥, it becomes 5 . 64a» (P + 56=^)' + 5 . 165a"-6« (P + obj - 15 . 400a='6« (P + 56») -625a-b^(SP+7b^y. On performing the calculation it will be found that all the terms involv- ing b will disappear from this result, and there will remain the single term 320a»P», that is, 320a» (3ac - ob'f, which is a reciprocant. LECTURE VIII. In my last lecture the complete expression, both in terms of the modified and unmodified letters, was obtained for V, the annihilator for pure recipro- "l^iants assuming its existence and its form. These assumptions I shall now IHoake good by proving, from first principles, the fundamental theorem that IBbe satisfaction of the equation I' VR = 0 is a necessary and suflBcient condition in order that R may be a pure reciprocant. It will be advantageous to use the modified system of letters, in which , . dv 1 d'v 1 d'y 1 d*v (, flj, a,, a.,, ... stand for /- , zr-^ . -f£, . ^ „ . -~, ^-.r-^—. ■ -A, ■■■ dx 1.2 fte" 1.2.3 da? 1.2.3.4 oar* J r- 1 d^x 1 d^x 1 d*x and. .„,«„,„... for —.^„j_^._.j-2-3-^.^,... respectively. Let the variation due to the change of x into x + ey, where e 348 Lectures on the Theory of Reciprocants [42 is an infinitesimal number, be denoted by A. Obviously this change leaves the value of each of the quantities o,, a,, a^, ... unaltered, and therefore KR(ao, a,, a,, ...) = 0, whatever the nature of R may be. But when iJ is a pure reciprocant, R{ao,a„ Os, ...)=±<^-K(ao, «!, Oj. •••). whence it immediately follows that ^t-i'Ria,, Ui, a,, ...) = 0*. Before proceeding to determine the values of At, Aoo, Aa,, Aoj, ... it will be useful to remark that since ^-t ^-1 2 a ^-1 2 3 a, ^^^*^^ ^^^''^ :£°=^^'-' { and generally -r-^ = (« + 3) an+i ■ Now let [t] denote the augmented value of t, and in general let [ ] be used to signify that the augmented value of the quantity enclosed in it is to be taken. Then so or -. ro n <^ W d[t] d[t] ,, ..d[t] also 2 [ao] = [2ao] = jM = . , V \ = ^ /i , >\ = (^~^*)^ L oj L "J ^Tg.-! d(x + €y) da;(l + €t) dx [x]~ d{x + ey) dx{l + et) = (1 - eO ^ (< - ei") = (1 - eO (2a, - 4etoo) = 2ao — 6etoo ; that is [flo] = Oo — ^eta„. Reasoning precisely similar to that which gave 2K] = (i-60^^W, leads to the formula (n + 3)[a„+,] = (l-eO^K]. \ * It has been'suggested by Mr J. Chevallier that the proof might be simpliBed by considering _M _ the variation A0(, S R {a^, a^, a„, ...) instead of At ** i? (Oj, a^, a^, ...)• 42] Lectures on the Theory of Reciprocants 349 from which the augmented values of a,, a^,a,, ... may be found by giving to n the values 0, 1, 2, ... in succession. Thus, writing w = 0, we have 3 [aj = (1 - et) ^ [aj = (1 - e«) ^ («„ - Seta,) = il-€t) (3a, - 9eta, - 6ea„0 = Sa,-e (12ta, + 6a„=), or [oi] = a,-e (ita, + ia,"). Similarly, when n = 1, 4 K] = (1 - et) ^ [a,] = (1 - et) ^(ch- 46to, - 2ea„^) = (1 - €<) (4ai, - 166^2 - 20ea„a,) = 4a, - 206^2 - 20ea„a„ [oj] = aj _ oe (taj + aoO,). and k Again. 5 [a,] = (1 - rf) A [„,] = (i _ ,^) ^ (^^ _ 5^^^ _ -^^^^^ iKdi = (1 - eO (5a, - 25eta3 - 30ea„a, - loea,-) = 5a, - aOetoj - SOeaoOj - 1 5eai', 80 that [a,] = a, - 6 (6^3 + 6a<,a, + Soi'). In like manner we shall find [a*] =a,-7e (ta^ + a^a^ + a^a^). lese results may be written in a more symmetrical form ; thus : 2[<] =2t-2et', 2 [a„] = 2a„ - 3e (to„ + a„«), 2 [a,] = 2ai - 4e (to, + a,^ + a^t), 2 [aJ = 20, - 5e (to, + OoOj + Oja, + a^t), 2 [a,] = 2a3 - 6e (to, + 0,0, + a.' + 0,0, + a,0. 2 [a<] = 2a, - 7e (to, + a„a, + 0,0, + OjOi + a,a, + aJ). The general law 2 [a„] = 2a„ - (n + 3) e (to,, + a„ o^, + . . . + a,_, a„ + a„t), or, as it may also be \vritten, A w+3 . Imits of an easy inductive proof Assuming the truth of the theorem for [a,J, and writing for brevitv in what follows, S„ = tan + a^a^-, + a^a^^ + . . . + Un^^a, + a„_iao + a„t, 360 Now, Lectures on the Theory of Reciprocants dS„ [42 dx ^ [a„] = (n + 3) a„+, - ^ Hence But, as we have already seen, = (n + 3) to„+, + 2a,a„ + (n + 2) aottn + 3a,a«_, + (n + 1) o,a„_i + ia^On-t + + + 4a,na, + (»+ 1) On-iO, + 3a„_,a, + (n + 2) a„a(, + 2a„ao + (n + 3) a,n.i« = (n + 4) (te„+i + a^an + aia„_i + . . . + an-ia, + Unaa + ttn+it) — 2ten+i = (n + 4) Sn+i - 2tan+i. n + S e {(n + 4) Sn+i - 2to„+,}. consequently, (« + 3) [a„+,] = (1 - eO ^ [a„] ; L«n+iJ = (1 - 6<) a„+, g— ebn+i + eton+i = a„+, ~ ebn+i ; that is, the theorem holds for [a„+i] when it holds for [a„]. But we know that it is true for the cases n=0, 1, 2, 3, 4, and therefore it is true universally. Resuming the proof of the main theorem, it has been shown that Ar''i2(a„, a,, a„ ...)=0; that is - fit-'At + R-'AR = 0, r>j 1 * J . dR . dR . dR . „ or - fiRt-^/^t + T— Aao + :t- Aa, + J— Aoj + . . . = 0. otcio dOi dcUi But At =-e«', Atto = — 36to„, Aai = - 6 (4to, + 2a(,'), Attj = — e (Sfaj + 5aoO,), Aa, = - e (Gtas + Gaottj + 3a,'), Aa4 = - e (7 y'". •••> whose weights are reckoned as -2,-1, 0, 1,... then, when x becomes x + hy, where h is any constant quantity, F becomes {l + hty^e i+^i?*. where t = y', F, = - t^dt + F, and /* = 3i + w, i being the degree and w the weight of F. •»» 42] Lectures on the Theor}/ of Reciprocants 353 From this, by an obvious course of reasoning, could be deduced as a particular case the condition of F{ai,, a^, a«, ...) remaining a factor of its altered self when any linear substitutions are impressed on x and y ; namely, the necessary and sufiBcient condition is that F has V for its annihilator. LECTURE IX. The prerogative of a Pure Reciprocant is that it continues a factor of its altered self when the variables x and y are subjected to any linear substitu- tion. Its form, like that of any other reciprocant, is of course persistent when the variables are interchanged ; that is, when in the general substitution, in which y is changed into fy+gx + h I and X into f y+ 9'^ + '*'> we give the particular values h—0, h' = 0, f=0, g' =0,/' =1, g = 1, to the constants. Stated geometrically, the theorem is that the evanescence of any pure reciprocant R indicates a property independent of transformation of axes in a plane. We suppose R to be homogeneous and isobaric in a, b, c, .... (If it were not, the theorem could not hold, for either the change of y into Ky or that of x into \x would destroy the form.) The persistence, under any linear substitution, of the form of pure recipro- cants may be easily established as follows : By a semi-substitution understand one where one of the variables remains unaltered. There are two such semi-substitutions, namely, where x remains unaltered, and where y does. (1) Let X remain unaltered and y become fy+gx + h; then a, b, c, ... become fa, fbj/c, ... respectively, and therefore R(a, b, c, ...) becomes f^R (a, b, c, ...), IBvhere i is the degree of R. ' (2) Let ?/ remain unchanged and a; become y^-f-^r' a; -(- A'. Then, instead of R, I look to its equal qt>^R{a,&.y....)(q = ±l); that is, to qr-^R (a, /3, 7, . . .), which becomes q (/' + g'r)-''g'^R (o, /3, 7, . . .). Since iiJ is a reciprocant, this is equal to or, replacing t by its equivalent - , V U't + 9')-<^g'R{a,b,c, ...). 8. IV. 23 354 Lectures on the Theory of Reciprocants [42 Thus we see that the proposition is true for a semi-substitution of either kind. Consider now the complete substitution made by changing y into fy-\-gx + h and X into Fy + Gx+H. d'u d*v , 9 d'x g d'x If/=0 and (? = 0, then 5J, ^. - become Jk- d^.' J,- df' ■'' ^"^ that ie(o, b,c,...) becomes pfr^ . R(a, 0, y, ...); and since this is equal to ^.qt-'^R{a.b,c,...). the proposition is true. But if either of the two letters /, 0 (say/) is not zero, we may combine two semi-substitutions so as to obtain the complete substitution, in which y changes into fy+gx + h, and X changes into Fy + Gx+ H. (1) Substitute yi{=fy +gi>: + h) for y, and x,{=x) for x. (2) Then substitute yi{=yj) fw y^ and Xii=f'yi+ g'x^ + h') for a^. By the first of these semi-substitutions R(a, b, c, ...) takes up an extraneous factor/'. By the second it acquires the factor Hence we see that the extraneous factor is a negative power of a linear function of t, which we shall presently particularize, though it is not essential to the present demonstration to do so. It only remains to show how the combination of these two semi-substi- tutions can be made to give the complete one in question. We have and x^ =/','/! + g's^i + h' =/' {fy + gx + h) + g'x + h' ==fry+if'9+9')'^ + (f'h + h'). In order that this may be equal to Fy + Gx + H, we must be able to satisfy the equations f =->, 9 =G-y , h - tl-~j , which is always possible, since by hypothesis / is not zero. Similarly it may be shown that when/ vanishes, but G does not, by substituting (1) a;,(=i^y+ Ga; + //)for «, and y,(=y) for 2/, (2) X, (= X,) for x^, and y, (=/'Vi + 9"<^i + ^") for yi. ^ we may so determine/", g", h" as to get the complete substitution as before. I « I 42] Lectures on the Theory of Reciprocmits 355 In every case, therefore, any linear substitution impressed upon the variables x and y will leave R{a, b, c, ...) unaltered, barring the acquisition of an extraneous factor which is a negative power of a linear function of t. Now, the first semi-substitution introduces, as we have seen, the constant factor /•; the second introduces the factor where ^'=/* + 5'- The complete extraneous factor is the product of these two, and is therefore f'g'Hfft+f'g+g')-''. To express /' and g' in terms of the constants of the complete substitu- tion we have Writing these values for/' and g' in the expression just found, we obtain {fG-gFf{Ft+G)->^, which is the extraneous factor acquired by R when the complete substitution is made. For example, if x becomes Fy+Ox+H, and y becomes fy+gie + h, the altered value of oUhat is, of -j^\ is {fG-gF){Ft-^G)-'a. Corresponding to the simple interchange of the variables, we have -^=1, (? = 0, H=0;f=0, g=l, h = 0, so that fO — gF= — \, and the altered value of a is -r- , or dy^ a which is right. In this case the general value of the acquired extraneous factor (/G-gF)*(Ft + G)~'' becomes (-)'r^ thus showing, what we have already proved from other considerations, that the character of a pure reciprocant is odd or even according as its degree is odd or even. 23—2 356 Lectures an tlie Theory of Redprocants [42 We saw in the last lecture that every pure reciprocant necessarily satisfied the two conditions (where /i is the characteristic), and VR = 0. We also saw that Ffl = 0 was a sufficient as well as necessary condition that any iMvwgeneom function R of a„ a,, a,, ... should be a pure reciprocant. It will now be shown that every pure reciprocant is either homogeneous and isobaric, or else resoluble into a sum of homogeneous and isobaric recipro- cants. Non-homogeneous mixed ones, it may be observed, are not so resoluble, so that the theorem only holds for pure reciprocants. (1) Let us suppose that R (a pure reciprocant) is homogeneous in o„. a,, o,... ; then it must be isobaric also. For, if i is the degree of R, Euler's theorem shows that (3a.9„, + 3a,aa, + 3a,8a, + 3a,9„, + ...)R = 3iR; and since 72 is a pure reciprocant, the condition (3ao9a« + ^^ ^j = i, X, + 2X^ + 3\, + . . . + jA, = w. In other words, the number of terms in E is equal to the number of ways in which w can be made up of i or fewer parts, none greater than j. This number will be denoted by (w; if)- Since the function R is the sum of every possible term of the form Aoo'^ch''' ... af'i, each multiplied by an arbitrary constant, the number of these arbitrary constants is also Now, suppose iZ to be a reciprocant ; this imposes the condition VR = 0. Consider the effect produced by the operation of any term of F = 4 ("I) da, + 5aoa,9«, + 6 (a„a, + y) a«. + • ■ • , Bay (aoa, + ^)9a, (rejecting the numerical coefficient 6). Operating on R with da, decreases its weight by 3 and its degree by 1 unit. The subsequent multiplication by a„(h + y , on t^e other hand, in- creases the weight by 2 and the degree by 2 units. Hence the total effect 358 Lectures on the Theory of ReciproA. '- of \a„a^ + -J- j a^j is to increase the degree by 1 and to diminish the weight by 1 unit. The same is evidently true for any other term of V. Thus the total effect of V operating on the general homogeneous isobaric function B of weight w, degree i, extent j, is t(f change it into another homogr' isobaric function whose weight, degree and extent are respectively ? i + l,j. Observe that the extent is not altered by the operation of V. It is easily seen that the coefficients of VR are linear functions Vt coefficients of i? ; for example, if '^ '" VR = ao'oj {QA + 2 J5) + a,^a^ (34 + 55 + 6(7). "^ Hence the condition VR = 0 gives us {w-\; i+ljyimear'eqm between the (w; i,j} coefficients of i? ; so that, assuming that these equl of condition are all independent, after they have been satisfied the nu of arbitrary constants remaining in R (that is, the number of linearly :^^ pendent reciprocants of the type w ; i,j) is equal to (w; i,j)-{w-\; i + l,j), when this difference is positive ; but when it is zero or negative there r reciprocants of the given type. If, however, any r of the (w-1; i+l,j) equations of condition s not be independent of the rest, these equations would be equivale. (w 1; i + l,j)-r independent conditions, and therefore the numbe linearly independent reciprocants of the type w ; i, j would be (w; i, j)-(w-l; i+ij)+ r. It is therefore certain that this number cannot he less than (w; i,j)-{iu~l; i+l,j). We shall assume provisionally that r = 0, or in other words thar the above partition formula is exact, instead of merely giving an inferior limit. Ihough It would be unsafe to rely on its accuracy, no positive grounds for doubting Its exactitude have been revealed by calculation. Such attempts as I have hitherto made to demonstrate the theorem' have proved infructuous,but it must be remembernmodifieH system of letters is employed. . Instead of a pure reciprocant, let us consider any homogeneous isobaric notion F of degree i and weight w; and [for the sake of simplicity writing ^frr T-] -'nstead of the operator F"3^™ let us consider F^a,"- aa»F''». e have identically ( F'-a," - 3,»F») F= F"-' ( FS, - 9xF) a,»-'Jf ° + F-"-* {Vd,-d, V) F'a,»-'F + Bi + F ( Fa, - 9, F) F-^a,"-' F anc + ( Fa, - a, F) F"->a,*-' F + a, ( v^d,^' - a.»-' F") F. Now, the operation of (Fa» — a,F) on any homogeneous isobaric function ^ aose characteristic is ni is equivalent, as we have seen in Lecture VII, to multiplication by /tiy"; so that if the characteristics of a,"-'i', Fa,"-'f, F»ax»-'/'. ... F'»-'a,»-'^ are /t, , /*> , M» . ■■• Mm> it follows that ( F^a." - a," F") = (At, + A*, + A*, + • . • + /i») y" F—a^-'F + a« ( F"*ax"-' - a,"-' f™) i^. Observe that F— > ( FBx - a,F) a,»-'^= F— 'Ai,y"9«"-'^= Miy">^"^'a»""'^. where the transptosition of the y" is permissible because Fdoes not act on it ; but if y" were preceded by dx it could not be similarly transposed. The numbers a*i. /*». /^».--- form an arithmetical progression, for each operation of F increases the degree by unity and diminishes the weight by nnity, so that All = 3t, + tffj becomes a»« = 3 (*, + 1) + (w, — 1) = a^i + 2. Similarly a*i = /^i + *. a*4 = /*i + 6, . . . Atm = AH + 2ot — 2. 360 Lectures on the Theory of Reciprocants [42 The characteristic of F being /* = 3i + w, that of d./'-^F is /ii = /t + n-l; for each operatioa of dx leaves the degree unaltered, but adds an unit to the weight ; hence /*, + /*, + /i, + ... + /i,„ = m (/i + TO + » - 2) ; 80 that When F= R, a pure reciprocant, so that VR = 0, our formula becomes V'^dx^R = m (/i + m + n - 2) y" F'»-'a,»-'iJ + d^ F-^a^^-'E. (!) Suppose that in (2) m>n, then F"'3a:"-R = 0. This is obviously true when n = 0, and when n = 1. When n = 2 we find V^'d^^R = m{fi + m) y" V^-'d^R + 3* V^d^R = 0 if TO > 2. Similarly the case n = 3, to > 3 can be made to depend on n = 2, m > 2, ard in general each case depends on the one immediately preceding it. Next l«t n = wi in (2) ; then, remembering that V^dx^^^R = 0, we have V'^dx'^R = TO (/x + 2to - 2) y" V^-'d^'"- 'R, from which MacMahon's theorem that V"'dx'"R = 1 . 2 . 3 ... TO {/i (/i + 2) (/x + 4) . . . (/t + 2to - 2)} (y'^R is an immediate consequence. Another special case of Formula (1) is worthy of notice, namely, that in which we take n = 1, when we obtain the simple formula ( F^a^ -dxV'^)F = m(fi + m-l)y" V'-'F. (3) If in this we write a„ in the place of F, and (the modified system of letters being used) 2ao for y", fi becomes 3 + n, and we have ( V'^d^ - dx F">) a„ = 2to (to + K + 2) a.V^-^On, or, as it may also be written, V^d^an ^ dxV^an 2 (m + n + 2) a,V«>-'an ,,. 1.2. 3... TO 1.2. 3. ..to"*" 1.2.3...(to-1) " ^ '' Mr Hammond remarks that this last formula may be used to prove the theorem _Y «« = -r''-»(e ()a„, which was given without proof in Lecture II. Assuming that a„ = - t-^-^a^ + t-»-* Va„ - r"-» ^ + . . . , 42] Lectures on the Theory of Reciprocants 361 we have to prove that the theorem is also true when n is increased by unity. Differentiating both sides of the assumed identity with respect to x, we find = - t-^-'d^an + r"-* {d^Vun + 2 (« + 3) a„an} (? Vfi ) - ir'^ ^'~ + 2 (n + 4) a„ Fa„J + ; the general terra being ._.„+, ._„_„_3 [ a»F"-a. 2(m + » + 2)a,F">-aJ ^ ' (1.2.3... wi 1.2.3...(m-l) J which, by means of (4), reduces to ^ ^ '^ 1.2.3...m' V'd a Hence 3,a„ = - «-»-'a,o„ + 1r"-*Vd^a„ - f^-' -^^ +■■■> or, more concisely, r axan = -<-'-'(e"0axO». But dxUn = (n + 3) a„+,, and dxO^ = OyO„ = (n + 3) 0 (I shall sometimes call this positive), then (w'; i, j) - (w'- 1; i, j) = > 0 for all values of w' less than w; the condition that this difference, say A(w;t,j) shall be positive being simply that ij—2w is positive (that is, ij—2w = > 0). This is not the case with the differe