THE THEORY OF DETERMINANTS IN THE HISTORICAL ORDER OF DEVELOPMENT dI THE THEORY OF DETERMINANTS IN THE HISTORICAL ORDER OF DEVELOPMENT PART I. PART II. GENERAL DETERMINANTS UP TO 1841 SPECIAL DETERMINA NTS UP TO 1841 BY THOMAS MUIR, M.A., LL.D., F.R.S. SUPERINTENDENT-GENERAL OF EDUCATION IN CAPE COLONY SECOND EDITION MACMILLAN AND CO?, LIMITED NEW YORK: THE MACMILLAN i0O1PANY 1906 All lights reserved GLASGO,: PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE AND CO. LTD. PREFACE. THE main object of this work and the contents of it will be found specified in the Introductory Chapter. It is intended for the student who aims at acquiring such a knowledge as can only be got by a study of the subject in the historical order of its development, for the investigator who is specially interested in this branch of mathematics and wishes to become acquainted with the various lines of attack opened up by previous workers, and for the general working mathematician who requires guidebooks and books of reference concerning special domains. T. M. CAPETOWN, SOUTH AFRICA, 19th July, 1905. CONTENTS. PA-Rf I. CHAPTER I. INTRODUCTION,. 1-5 CHAPTER II. GENERAL DETERMINANTS, 1693-1779, 6-52 LEIBNITZ FONTAINE CRAMER BEZOUTT VANDERMONDE LAPLACE LAGRANGE BEZOUT (1693), (1748), (1 750), (1764), (1771), (1772), (1773), (1773), (1773), (1779), * pp. 6-1 0 *I 10-11 * 11-14 * 14-17 * 17-24 * 24-33 33-37 37-40 * 40-41 * 41-5 2 CHAPTER Ill. GENERAL DETERMINANTS, 1784-1812, 53-79 HINDENBURG ROTHE GAUSS (1784-), (1800), (1801), *. pp.~ 53-55 *. 55-63 *. 63-66 viii viii ~~~CONTENTS MONGE HIRSCH BINET PRASSE WRONSKIC (1809), 1. pp. 67-68 (1809), 69 (1811),.... 69-71 (1811), 71-72 (181 1), 72-78 (1 81 2), 78-79 CHAPTER IV. GENERAL DETERMINANTS, 1812, BINET CAUCHY (1812), (1812), *.. ~80-131 * pp. 80-92 * 92-131 CHAPTER V. GENERAL DETERMINANTS: RETROSPECT, ICHAPTER VI. GENERAL DETERMINANTS, 18113-1825, 132-133 with Table. 134-175 GERGONNE.GARNIER WRONSKI DESNANOT CAUCHY SCHERK SCIIWEINS (1813), (1.814), (1815), (1819), (1821), (1825), (1825), pp. 134-135.. 135-136' * 136 *136-148 *148-150 *150-159 *159-175 CHAPTER VIIL GENERAL DETERMINANTS, 1827-18t` JACOBI (1827), R EI SS (1829), CAUCHY (18-2 9)), JACOBI (1829), MINDING (1829), DRINKWATER (1831), MAINARDI (1832), JACOBI (1831-33) (1 834), (1835), 3 5, 1 *.pp. 176-178 * *. 178-187 * *. 187-188 *. * 188-193 * *. 194-197 *.. 198-199 200-206 206-212 *.. 212-213 214 176-214 CONTENTS ix PAGES CHAPTER VIII. GENERAL DETERMINANTS, 1836-1840, GRUNERT (1836), LEBESGUE (1837), REISS (1838), CATALAN (1839), SYLVESTER (1839), MoLINS (1839), SYLVESTER (1840), RICHELOT (1840), CAUCHY (1840), SYLVESTER (1841), CRAUFURD (1841),.915-246 pp. 215-219 219-220 220-224 224-226 227-235 235-236 236-238 238-240 240-243 243-245 245-246 CHAPTER IX. GENERAL DETERMINANTS, 1841, CAUCHY JACOBI CAUCHY (1841), (1841), (1841), 247-285 pp. 247-253 * 253-272 * 273-285 CHAPTER X. GENERAL DETERMINANTS: RETROSPECT, 286-288 with Table. PART HI. CHAPTER XI. AXISYMm'ETRIC DETERMINANTS, 1773-1841, 289-305 ROTHE BINET JACOBI CAUCHY JACOBI LEBESGUE JACOBI CAUCHY (1800), (181-1),. (1827), (1829), (1831), (1832), (1833), (1834), (1837), (1841), (1841), pp. 290-292 292-293 293-294 295-296 296-297 297-298 298-300 * 300-301 301-303 304 304-305 x x ~~~~CONTENTS CHAPTER XII. PAGES 306-345 ALTERNANTS, 1771-1841, PRONY CAUCHY SCHWEI NS SYLVESTER JACOBI CAUCHY (1795), (1812), (1825), (1839), (1841), (1841), pp. 306-308 *308-310 *311-322 *322-325 *325-342 *342-345 CHAPTER XIII. JACOBJANS, 1815-1841, CAUCHY JACOBI CATALAN JACOBI CAUJCHY (1815), (1822), (1829),. (1830), (1832-33), (1839), (1841), (1841), * pp. 346-349 *. ~349 * 349-352 I * 352-354 *.354-356 * * 356-358 * *358-392 * *393-394 346-394 - CHAPTER XIV. SKEW DETERMINANTS, 1827-1845, 395-406 PFAFF JACOBI (1815), (1827) (1 845), pp. 396-401 * 401-405 * 405-406 CHAPTER XV. ORTHOGONANTS, 1827-1841,.. 407-471 - JACOBI CAUCHY JACOBI LEBESGUE CATALAN CAUCHY (1827),. (1827),.I (1 829),. (1831), (1832), (1833), (1837),. (1839), (1826),: (PostScript),. pp. 410-415 *415-424 *425-435 *435-451 * 452 *453-463 *463-467 *467-470 470-471 CONTENTS xi PAGES CHAPTER XVI. MISCELLANEOUS, 1811-1841, WRONSKI (1812), (1815), (1816-17), (1819), SCHERK (1825), SCHWEINS (1825), JACOBI (1835), SYLVESTER (1840), 472-487 pp. 472-474 474-476 476-478 478 478-479 479-485 485-487 487 CHAPTER XVII. RETROSPECT ON SPECIAL FORMS, 488 INDEX TO THE NUMBERED RESULTS IN PART I., * 489 LIST OF AUTHORS WHOSE WRITINGS ARE REPORTED ON, 490-491 CHAPTER I. INTRODUCTION. THE way in which the material for a history of the theory of Determinants has been accumulated is quite similar to that which has been observed in the case of other branches of science. In the middle of the eighteenth century one of the independent discoverers of the fundamental idea, viz., CRAMER, was fortunate enough to attract attention to it, and in time it became the common property of mathematicians in France and elsewhere. As it slowly spread it naturally also received accretions and developments, and of the dozen or so of writers who thus handled it in the sixty years that followed Cramer's publication there were of course a few who by a more or less casual reference kept alive the memory of some of their predecessors. It was then taken up by CAUCHY, and, thanks to the prestige of his name and to the inherent excellence of his extensive monograph, its position as a theory of importance became more firmly assured. The thirty years that followed Cauchy's memoir resembled the sixty that preceded it, save that the number of contributors was considerably larger. Then another great analyst, JACOBI, the most noteworthy of those contributors, produced in Germany a monograph similar in extent and value to Cauchy's, and the importance of the subject in the eyes of mathematicians became still more enhanced. As a consequence, the single decade following gave rise to quite as many new contributions as the preceding three decades had done, and closed with the appearance of the first separately published elementary treatise on the subject, viz., SPOTTISWOODE'S. The M.D. A 2 HISTORY OF THE THEORY OF DETERMINANTS preface to this contains the first notable historical sketch of the theory, and includes references to the writings of twelve outstanding mathematicians, beginning with Cramer (1750) and ending with the author's own contemporaries, Cayley, Sylvester and Hermite. In the same year (1850) there also occurred something out of the ordinary, for the correspondence between Leibnitz and the Marquis de l'Hopital having been published from manuscripts in the Royal Library at Hanover, the striking discovery was made that more than half-a-century before Cramer's time the fundamental idea of determinants had been clear to LEIBNITZ, and had been expounded with considerable fulness by him in a letter to his friend. So strongly attractive had the subject now become to mathematicians that in the single year succeeding the publication of Spottiswoode's short treatise a greater number of separate contributions to the theory made their appearance than in the whole sixty-year period from Cramer to Cauchy. The wants of students everywhere had to be attended to: a second edition of Spottiswoode was consequently prepared for Crelle's Journal in 1853; a textbook by Brioschi was published at Pavia in 1854; French and German translations of Brioschi in 1856; and an elementary exposition by Bellavitis in 1857. So far as historical material is concerned, the last-mentioned work was of little account; that of Brioschi resembled Spottiswoode's, the number of references, however, being greater. Of quite a different character was the text-book by BALTZER, which was published at Leipzig the year after the German translation of Brioschi had appeared at Berlin, an important part of the new author's plan being to deal methodically with the history of the subject by means of footnotes. On the enunciation of almost every theorem a note with historical references was added at the foot of the page, the result being that in the portion (thirtyfour pages) devoted expressly to the pure theory of determinants about as many separate writings are referred to as there are pages. This was a marked advance, and although during the next twenty years the publication of text-books became more frequent —in fact, if we include those of every language and of every scope, we shall find an average of about one per INTRODUCTION 3 year-Baltzer's dominated the field; enlarged editions of it appeared in 1864, 1870, and 1875, and the historical notes grew correspondingly in number. Of the other text-books only one, Ginther's, which was published in 1875, sought to follow the historical line taken by Baltzer and to add to the supply of material. Then in 1876 another new departure took place, this being the year in which the first writings were published which dealt with the history alone, the one being an academic thesis by E. J. Mellberg printed at Helsingfors, and the other a memoir presented by F. J. Studnicka to the Bohemian Society of Sciences. About this time, while engaged in writing my own so-called "Treatise on the Theory of Determinants," I had occasion to look into the question of the authorship and history of the various theorems, and I was reluctantly forced to the conclusion that much inaccurate statement prevailed in regard to such matters and that the whole subject was worthy of serious investigation. A resolution was accordingly taken to set about collecting the titles of all the writings which had appeared on the theory up to the end of 1880. The task was not an easy one, as will readily be understood by those who know how scanty and defective are the bibliographical aids at the disposal of mathematicians, and how often the titles given by investigators to their memoirs are imperfect and even misleading in regard to the nature of the contents. The outcome of the search was published in 1881 in the October number of the Quarterly Journal of Mathematics (vol. xviii. pp. 110-149) under the title of "A List of Writings on Determinants." It contained 589 entries arranged in chronological order. Some three or four years afterwards, when there had been time to test the completeness of the earlier portion of the list, the writings included in it were taken up in historical succession and suitable abstracts or reviews of them made for publication in the Proceedings of the Royal Society of Edinburgh; the first contribution of this kind was presented to the Society in the beginning of the year 1886. At the same time there. was being prepared an additional list of writings containing omitted titles, 84 in number, belonging to the period of the 4 HISTORY OF THE THEORY OF DETERMINANTS first list, and 176 titles belonging to the further period 1881 -1885. This second list appeared in 1886 in the June number of the Quarterly Journal of Mathematics (vol. xxi. pp. 299 -320). In 1890 a collection was made of the contributions, just mentioned, which had up to that date been printed in the Edinburgh Proceedings, and with the consent of the Society was published separately. Unfortunately in that year all this train of work had to be laid aside on account of the pressure of official duties, and ten years elapsed before it could be resumed. It was thus not until March 1900 that a second series of analytic abstracts began to appear in the Edinburgh Proceedings, and that the preparation of a third list of writings was methodically undertaken. The period to be covered by this list was the fifteen years 1886-1900; and as the number of writers interested in the subject had in these years continued to increase, and as closer examination of the literature of the previous periods had led to new finds, the resulting compilation was more extensive than the first two put together. It was presented to the South African Association for the Advancement of Science at its inaugural meeting in April 1903 and was published in the Report; it is also to be found in the Quarterly Journal of Mathematics for December 1904 and February 1905 (vol. xxxvi. pp. 171-267). The number of titles in the three lists is about 1740; they furnish, it is hoped, an almost complete guide to the literature of the theory of determinants from the earliest times to the close of the nineteenth century. From these later labours it became manifest that it was undesirable in the way of separate publication to issue merely another volume as a continuation of, and similar to, that of the year 1900. The better course clearly was to reproduce the material of that volume along with the intercalations necessitated in it by the existence of subsequently discovered papers, and to follow this up in such a way as to give finally within the compass of a reasonably sized volume a full history of the subject in all its branches up to about the middle of the nineteenth century. This is what is here attempted. The plan followed is not to give one connected history of determinants as a whole, but to give separately the history of INTRODUCTION 5 each of the sections into which the subject has been divided, viz., to deal with determinants in general, and thereafter in order with the various special forms. This will not only tend to smoothness in the narrative by doing away with the necessity of frequent harkings back, but it will also be of material importance to investigators who may wish to find out what has already been done in advancing any particular department of the subject. To this end, also, each new result as it appears will be numbered in Roman figures; and if the same result be obtained in a different way, or be generalised, by a subsequent worker, it will be marked among the contributions of the latter with the same Roman figures, followed by an Arabic numeral. Thus the theorem regarding the effect of the transposition of two rows of a determinant will be found under Vandermonde. marked with the number xi., and the information intended thus to be conveyed is that in the order of discovery the said theorem was the eleventh noteworthy result obtained: while the mark xI. 2, which occurs under Laplace, is meant to show that the theorem was not then heard of for the first time, but that Laplace contributed something additional to our knowledge of it. In this way any reader who will take the trouble to look up the sequence xI., xi. 2, xi. 3, &c., may be certain, it is hoped, of obtaining the full history of the theorem in question. The early foreshadowings of a new domain of science, and tentative gropings at a theory of it, are so difficult for the historian to represent without either conveying too much or too little, that the only satisfactory way of dealing with a subject in its earliest stages seems to be to reproduce the exact words of the authors where essential parts of the theory are concerned. This I have resolved to do, although to some it may have the effect of rendering the account at the commencement somewhat dry and forbidding. CHAPTER II. DETERMINANTS IN GENERAL, FROM THE YEAR 1693 TO 1779. THE writers here to be dealt with are seven in number, viz., Leibnitz, Fontaine, Cramer, Bezout, Vandermonde, Laplace, Lagrange. Of these the first two exercised no influence on the development of the theory; the real moving spirit was Cramer; Lagrange alone of the others may have been unaffected by this particular part of Cramer's work. LEIBNITZ (1693). [Leibnizens mathematische Schriften, herausg. v. C. I. Gerhardt. 1 Abth. ii. pp. 229, 238-240, 245. Berlin, 1850.] In the fourth letter of the published correspondence between Leibnitz and De L'Hospital, the former incidentally mentions that in his algebraical investigations he occasionally uses numbers instead of letters, treating the numbers however as if they were letters. De L'Hospital, in his reply, refers to this, stating that he has some difficulty in believing that numbers can be as convenient or give as general results as letters. Thereupon Leibnitz, in his next letter (28th April 1693), proceeds with an explanation:"Puisque vous dites que vous aves de la peine a croire qu'il soit aussi general et aussi commode de se servir des nombres que des lettres, il faut que je ne me sois pas bien explique. On ne scauroit douter de la generalite en considerant qu'il est permis de se servir de 2, 3, etc., comme d' a ou de b, pour veu qu'on considere que ce ne sont pas de nombres veritables. Ainsi 2.3 ne signifie point 6 mais autant qu' ab. Pour ce qui est de la commodite, il y en a des tres DETERMINANTS IN GENERAL (LEIBNITZ, 1693) 7 7 grandes, ce qui fait que je m'en sers souvent, sur tout d'an s les c'alculs longs et difficiles ou. il est aise6 de se tromper. Car outre la commodit4" de l',preuve par des nombres, et m~me par labjection du novenaire, j'y trouve un tres grand avantage m~nme pour lavancement de l'Analyse. Comme c'est une ouverture assez extraordinaire, je n'en ay pas encor lparle' 'a d'autres, mais voicy. ce que e 'est. Lorsqu'on a besoin de beaucoup de lettres, n'est il pas vray que cIes lettres n'expriment point les rapports qu'il y a entre les grandeurs qu'elles signiflent, au lieu quen me servant des nornbres je puis exprimer ee rapport. Par exemple soyent proposeles trois equations simples pour deux ineonnues la dessein d'oster ces deux inconnues, et cela par un canon general. Je suppose 10+11x+12y = 0 (1) et 20 + 2L +22y = 0 (2) et 30 +31x +32y = 0 (3) on le nombre feint estant de deux eharacteres, le premier me marque de quelle equation il est, le second me marque 'a quelle lettre il,appartient. Ainsi en calculant on trouve par tout des harmonies qni non seulement nous servent de garans, mais encor notus font entrevoir d'abord des regles ou theoremes. Par exemple ostant premierement y par la premiere et la seconde equation, nous aurons: + 10. 22 +11.22x = 0 (4)* - 12. 20 -12.21.. et par la premiere et troisieme nous aurons: + 10. 32 +11. 32x I= 0 (5) - 12.30 -12.31.,on il est aise de connoistre que ces deux equations ne different qu'en ce que le charactere antecedent 2 est change' an charactere, antecedent 3. Du reste, dans un m~me terme d'une me'me equation les characteres antecedens sont les m~mes, et les characteres posterieurs font une m~me, somme. Il reste maintenant d'oster la lettre x par la quatriemle et cinquieme equation, et pour cet effect nous aurons t 010.21.32 10.122. 31 11. 22. 30 = 11.20.32 12. 20. 31 12. 21.30,qui est la derniere equation delivre'e des deux inconnues qn'on vouloit -oster, et qni porte sa prenve avec, soy par les harmonies qni se remarquent par tout, et qu'on auroit bien de la peine 'a decouvrir en * This is written shortly for +10.22+11.22x=0O - 12. 20 -12. 21x = O ~ The author here slightly changes his notation. What is mneant to be indi~cated is 10.21.32 + *11.22.30 + 12.20.31 = 10.22.31 + 11.20.32 + 12.21.30. 8 HISTORY OF THE THEORY OF DETERMINANTS employant des lettres a, b, c, sur tout lors que le nombre des lettres et des equations est grand. Une partie du secret de l'analyse consisto, dans la caracteristique, c'est 'a dire dans l'art de bien employer les notes dont on se sert, et vous voyes, Monsieur, par ce petit echantillon, que Viete et des Cartes n'en ont pas encor connn tons les mysteres. En poursuivant tant soit pen ce calcul on viendra 'a nn theoremie genuerat pour quelque nombre de lettres et d'equations simples qu'on puisse prendre. Le voicy comme je l'ay trouve' autres fois: "Datis aequationibus quoteunque suflicientibus ad tollendas quantitates, quae simplicem grad'nz non egrediwutur, pvro aequatione prodeunte, primao sumendcae sunt omines combinationes possibiles, quas ivgreditur una tantunt coeftici ens uniuscujusque aequationis: secundo, eae combinationes opposita, ha bent signa, si in eodem aequationis _prodeuntis latere ponantur, quae habent tot coe/jjicientes comnmunes, quot sunt unitates in numero quaentitatum& tollendar-um unitate minuto. caeterae habent eadem signa. "J'avoue que dans cc cas des degre's simples on auroit pent estre decouvert le m~me theoreme en ne se servant que de lettres 'a l'ordinaire, mais non pas si aisement, et ces adresses sont encor bien plus necessaires, pour decouvrir des theoremes qni servent 'a oster les inconnues mont~e' 'a des degre's plus hauts. Par exemple. It will be seen that what this amounts to is the formation of a rule for writing out the resultant of a set of linear equations. When the problem is presented of eliminating x and y from the, eqnations a+bxr+cy =0, d +ex +fy =0, g +hx +lcy=0, Leibnitz in effect says that first of all he prefers to write 10, for a, 11 for b, and so on; that, having done this, he can all the more readily take the -next step, viz., forming every possible product whose factors are one coefficient from each equation,* the result being 10. 21. 32, 10. 22. 31, 11. 20. 32, 11. 22. 30, 12. 20. 31, 12. 21.30; and that, then, one being the number which is less by one than. the number of unknowns, he makes those terms different in sign which have only one factor in common. The contributions, therefore, which Leibnitz here makes to algebra may be looked upon as three in number: (1) A new notation, numerical in character and appearance,, for individual members of an arranged group of magnitudes;, the two members which constitute the notation being like the, *Of course, this is not exactly what Leibnitz meant to say. DETERMINANTS IN GENERAL (LEIBNITZ, 1693) 94 Cartesian co-ordinates of a point in that they denote any one of' the said magnitudes by indicating its position in the group. (I.), (2) A rule for forming the terms of the expression which equated to zero is the result of eliminating the unknowns from a set of simple equations. (II.), (3) A rule for determining the signs of the terms in the said result. (In.) The last of these is manifestly the least satisfactory. In the first place, part of it is awkwardly stated. Making those terms different in sign which have only as many factors alike as is indicated by the number which is less by one than the number of unknown quantities is exactly the same as making those terms different in sign which have only two factors different. Secondly, in form it is very unpractical. The only methodical way of putting it in use is to select a term and make it positive; then seek out a second term, having all its factors except two the same as those of the first term, and make this second term negative; then seek out a third term, having all its factors except two the same as those of the second term,. and make this third term positive; and so on. Although there is evidence that Leibnitz continued, in his. analytical work, to use his new notation for the coefficients of an equation (see Letters xi., xii., xiii. of the said correspondence), and that he thought highly of it (see Letter viii. " chez moi c'est une des meilleures ouvertures en Analyse"), it does not appear that by using it in connection with sets of linear equations, or by any other means, he went further on the way towards the subject with which we are concerned. Moreover, it must be: remembered that the little he did effect had no influence on succeeding workers. So far as is known, the passage above quoted from his correspondence with De L'Hospital was not published until 1850. Even for some little time after the date of Gerhardt's publication it escaped observation, Lejeune Dirichlet being the first to note its historical importance. It is true that during his own lifetime, Leibnitz's use of numbers in place of letters was made known to the world in the Acta Eruditorum of Leipzig for the year 1700 (Responsio ad Dn. Nic. Fatii Duillerii imputationes, pp. 189-208); but the particular 10 HISTORY OF THE THEORY OF DETERMINANTS application of the new symbols which brings them into connection with determinants was not the re given. In a subsequent volume of Leibrtizens mcathemati-sche Schriften,-the third volume of the second Abtheilung,-published at Halle in 1863, the following equivalent of the above 'the'ore'me.general' appears (pp. 5-6): " Inveni Canonem pro tollendis incognitis quotcunque aequationes non nisi simplici gradu ingredientibus, ponendo aequationum numerum oexcedere unitate n-umerum incognitarum. Id ita habet. Fiant omnes combinationes possibiles literarum coefficientium ita ut nunquam concurrant plures coefficientes ejusdem incognitae et ejusdem aequationis. Hae combinationes affectac signis, ut mox sequetur, -componuntur simul, compositumque aequatum nihilo dabit a-equationem omnibus incognitis carentem. Lex signorum hace ist. Uni ex combinationibus assignetur signum pro arbitrio, et cacterae combinationes quae ab hac differunt, coefficientibus duabus, quatuor, sex etc. habebunt signum oppositum ipsius signo: quae vero ab hac differunt coefficientibus tribus, quinque, -septem etc. habebunt signum idem cum ipsius signo. Ex. gr. sit 10+llx~+12.y =O, 20~+21x~+22y = 0 30+31x+32y =0; fiet + 10. 21.32 -10. 22. 31 -11. 20. 32 ~ 11. 22.30 +12-.20. 31 - 12. 21.30 = 0. Coefficientibus eas literas computo, quae sunt nullius incognitorum, ut 10, 20, 30.") Although Gerhardt, the editor,. states that the original manu-script of Leibnitz, from which this is taken, bears no date, it is very probable to date farther back than 1693, and not impossible to belong to 1678.* FONTAINE (1748). j[Memoires donne's 'a l'Acade'mie Rioyale des Sciences, non imnprime's dans leurs temps. Par M. Fontaine T' de cette Acade'mie. 588 pp. Paris, 1764.] These memoirs of Fontaine's, sixteen in number, cover a con-,siderable variety. of mathematical subjects: it is the seventh of * See also GERHARDT, K. I., Leibniz fiber die Determinanten, Silzungsb. A/cad. d. Wiss. (Berlin), 1891, pp. 407-423. ~ The full name is Alexis Fontaine des Bertins. The very same collection, was issued in 1770 under the less appropriate title Trait de, ccelcul diffdrentiel e~t intelgral. Vandermonde is said to have been a pupil of Fontaine's (v. Nouv. Annceles de Math., v. p. 155). DETERMINANTS IN GENERAL (FONTAINE, 1748) 1I the. series which indirectly concerns determinants. There is not, however, even the most distant connection between it and the work of Leibnitz. The heading is "Le calcul integral. Seconde methode," the sixth memoir having given the first method. The date is indicated in the margin. The matter which concerns us appears as a lemma near the beginning of the memoir (p. 94). The passage is as follows:"Soient quatre nombres quelconques al, a2, a3, a4, et quatre autres nombres aussi quelconques al, a2, a3, a4; faites al a2 - al a2 = all, a2 a3 - a2 a3 = a2, a3 a4 - a3 a4= al3, al a3 - al a3 = a21, a2 a4 - a2 a4 = a22, al a4 - al a4 = a31, vous aurez a31 a12 - a21 a22 + all al3 = 0." Manifestly this is the identity which in later times came to be written Iab2,. l3b4 - 1a1,b3. a2b4 + Ialb4. a2b3 = 0, and which, so far as we know, appeared first in its proper connection in the writings of Bezout. (xxIII.) It is curious to note that Fontaine was not satisfied with the lemma in this form, but proceeded to take "autant de nombres quelconques que l'on voudra, al, a2,.,. a...," and wrote the identity one hundred and twenty-six times before he appended " et cetera," the 126th being c36 a17 - a26 a27 + c6 al8 = 0. CRAMER (1750). [Introduction a l'Analyse des Lignes Courbes algebriques. (Pp. 59, 60, 656-659.) Geneve, 1750.] The third chapter of Cramer's famous treatise deals with the different orders (degrees) of curves, and one of the earliest theorems of the chapter is the well-known one that the equation 12 HISTORY OF THE THEORY OF DETERMINANTS of a curve of the nth degree is determinable when ln(n + 3 points of the curve are known. In illustration of this theorem he deals (p. 59) with the case of finding the equation of the. curve of the second degree which passes through five given points. The equation is taken in the form A+By~Cx+Dy~y~Exy+xx = 0; the five equations for the determination of A, B, C, D, E are. written down; and it is pointed out that all that is necessary is the solution of the set of five equations, and the substitution of the values of A, B, C, D, E thus found, " Le calcul veritablement en seroit assez long," he says; but in a footnote there is the remark that it is to algebra we must look for the means of shortening the process, and we are directed to the appendix fora convenient general rule which he, had discovered for obtaining, the solution of a set of equations of this kind. The following is the essential part of the. passage in which the rule occurs: "Soient plusieurs inconnues z, y, x, v, &c., et autant d'6quations Al1 Z1z + Yly + X'x + VIv + &C. Al = Z2z +Yly + Xx+ V2V +&C. A3 = Z3Z + Y~y + X3X + V3v + &C. A4 = Z4 + Y4y +X4.V+V4V + &C. &C. oii les lettres Al, A2, A3, A4, &C., ne marquent pas, comme 'a lordinaire, les puissances d' A, mais le premier membre, suppos6 connu, de la, premiere, seconde, troisie'me, quatrie'me, &c. equation." [Here the solutions of the cases of 1, 2, and 3 unknowns are given,, and he then proceeds.] "Lexamen de ces Formules fournit cette IRegle generale. Le, nombre des 6quations et des inconnues h'ant n, on trouvera la valeur de chaque inconnue en formant n fractions dont le de'nominateur commun a autant de termes qu'il y a de divers arrangements de n choses, diff~rentes. Chaque ternmer est compose' des lettres ZYXV, &c., toujours e6crites dans 'le me'me ordre, mais auxquelles on distribue, comme exposants, les m premiers chiffres range's en toutes les manie're& possibles. Ainsi, lorsqu'on a trois inconnues, le de'nominateur a [1 x 2 x 3 =]6 termes, compos6s des trois lettres ZYX, qui recoivent. successivement les exposants 123, 132, 213, 231, 312, 321. On donne, A ces termes les signes + ou -, selon la Re~gle suivante. Quand un exposant est suivi dans le me'me terme, mi'diatement ou immediatement, d'un exposant plus petit que lui, j'appellerai cela un d~rangemenl. DETERMINANTS IN GENERAL (CRAMER, 1750) 13 Qu'on compte, pour chaque terme, le nombre des derangements: s'il est pair ou nul, le terme aura le signe +; s'il est impair, le terme aura le signe -. Par ex. dans le terme Z1Y2X3 il n'y a aucun derangement; ce terme aura done le signe +. Le terme Z3YIX2 a aussi le signe +, parce qu'il a deux derangements, 3 avant 1 et 3 avant 2. Mais le terme Z3Y2X1, qui a trois derangements, 3 avant 2, 3 avant 1, et 2 avant 1, aura le signe -. "Le denominateur commun etant ainsi forme, on aura la valeur de z en donnant a ce denominateur le numerateur qui se forme en changeant, dans tous ces termes, Z en A. Et la valeur d'y est la fraction qui a le meme denominateur et pour numerateur la quantit6 qui resulte quand on change Y en A, dans tous les termes du denominateur. Et on trouve d'une maniere semblable la valeur des autres inconnues." It is evident at once that the new results here given are(1) A rule for forming the terms of the common denominator of the fractions which express the values of the unknowns in a set of linear equations. (IV.) (2) A rule for determinzing the sign of any individual term in the said common denominator (and, included in the rule, the notion of a "derangement"). (III. 2) (3) A rule for obtaining the numerators from the expression for the common denominator. (v.) The problem which Cramer set himself at this point in his book was exactly that which Leibnitz had solved, viz., the elimination of n quantities from a set of n+-1 linear equations. The solution which Cramer obtained, and which, be it remarked, was the solution best adapted for his purpose, was quite distinct in character from that of Leibnitz. Leibnitz gave a rule for writing out the final result of the elimination; what Cramer gives is a rule for writing out the values of the n unknowns as determined from n of the n +1 equations, after which we have got to substitute these values in the remaining (n + l)th equation. The notable point in regard to the two solutions is, that Cramer's rule for writing the conmmon denominator, of the values of the n unknowns (an expression of the nth degree in the coefficients) is exactly Leibnitz's rule for writing the final result, which is an expression of the (n + l)th degree. Had either discoverer been aware that the same rule sufficed for obtaining both of these expressions, he could not have failed, one would think, to -14 HISTORY OF THE THEORY OF DETERMINANTS note the recurrent -law -of formation of them. The result of eliminating w, x, y, Z'from the equations, aw + brx+ Cry +drz Cr (r1l, 2, 3,4, 5) is, according to Leibnitz, if we embody his rule in a later symbolism,abcde5=0 whereas, according to Cramer, it is-.a,1 e3c4d5 + b1 2e~c4d ~,I a2b ec + di a2b3 4 5 a~3c4d5 cC4t2b3c4c5 1a2b3c4 a 5bdd5 and from the collocation of these, the one natural step is to the identity - ~~~4e5l = all C2b3c4d +ba2e3C4d51 + 2 -e a~3C4515 The fate of Cramer's rule was very different from that of Leibnitz'. It was soon taken up, and after a time found its way into the schools, where it continued for many years to be taught as the nutshell form of the theory of the solution of simultaneous linear equations. Indeed Gergonne is reported * to have said, "Cette me'thode e6tait tellement en faveur, que les examens aux ecoles des services publics ne roulaient, pour ainsi dire, que sur dile; on e'tait admis ou rejete' suivant qu'on la possedait bien. on Finally, the exact difference between Cramer's notation for the coefficients of the unknowns and the notation of Leibnitz should be noted, and in connection therewith the fact that when dealing with the subject of elimination between two equations of the mth and nth degrees in x Cramer uses a notation closely resembling that which Leibnitz employed, viz., [12] [13], &c. BILZOUT (1764). [Recherches sur le degre' des e'quations re'sultantes de le'evanouissement des inconnues, et sur les moyens qu'il convient, d'emnployer pour trouver ces e'quations.-Hist. dle l'A cad. Boy. des Sciences, Ann. 1764 (pp. 288-338), pp. 291-295.] The object of Be'zout's memoir is sufficiently apparent from the title; we may therefore at once give those portions of it *By Studni~ka. But see Klftgel's Worterbitch d. reinen.Math,, Suppl. II. p. 67. DETERMINANTS IN GENERAL (BE- ZOUT, 1764) 1 15. which directly concern our subject. On p. 291 is the commencement of the following passage: "M. Cramer a donne' une rebgle generale pour, les exprimer toutes, de'barrasse'es de ce facteur: j'aurois Pu m'en tenir 'a aette rebgle; mais. l'usage m'a fait connottre que quoiqu'elle soit assez simple, quant aux lettres, elle ne Fest pas de, MeMe a e'~gard des signes lorsqu'on a au-dela' d'un certain nombre d'inconnues 'a calculer; Lemme I. "Si lon a un nombre n d'e6quations du premier degre6 qui renferment chacune un pareil nombre d'inconnues, sans aucun terme absolument. connu, on trouvera par la rebgle suivante. la relation que doivent avoir les co~fficiens de ces inconnues pour que toutes ces equations aient lieu. "Soient a, b, c, d, &C.,3 les co~fficiens de ces inconnues dans la premiebre equation. a',) b',) e, d',~ &C., les co~fficiens des mbmes inconnues dlans la seconde equation. a", b" elc", " &c., ceux de la troisib'me & aiinsi de suite. "Formez les deux permutations ab & ba, & e'crivez ab - ba; avec ces. deux permutations & la lettre c formez toutes les permutations possibles, en observant de changez de signe toutes les fois que c chaugera do, place dans ab & la Mbme chose 'a e'6gard de ba; vous aurez abe - acb + cab - bac ~ bca - cba. Avec ces six permutations & la lettre d, formez toutes les permutations possibles, en observant de changer de signe 'a chaque fois que d changera de place dans un mbme termne; vous aurez abcd - abdc + adbc - dabc - acbd + acdb - adcb + dacb +,cabd - eadb + cdab - dcab -bacd + bade - bdae _+ dbae + bead - beda + bdea - dbca -cbad + ebda - edba + dcba & ainsi de suite jusqu'ah cc que vous ayez 6puise' tous les co~fficiens de la premiebre equation. "Alors conservez les lettres qui occupent la premiebre place; donuez a~ celles qui occupent la seconde, la mbme marque qu.'elles ont dans, la seconde 6quation; 'a celles qui occupent la troisiebme,5 la mbme marque qu'elles out dans la troisib'me equation, & ainsi de suite; e6galez enfin le tout 'a zero et vous aurez l'~quation de-condition cherche'e. "Ainsi si vous avez deux equations et deux inconnues comme ax + by =0 a'ce + b'y =,0 I'equation de condition sera ab' - ba' = 0. on ab',,- a'b = 0... TIn the same way the next two cases are-given; then-....mais comme ces equations -do condition- doi-vent servir de formules pour le'limination dans los equations do diffl~rens degr6s, ii -16 HISTORY OF THE THEORY OF DETERMINANTS,convient de leur donner une forme qui rende les substitutions le momns _penibles qu'il se pourra; pour cot effet, je les mets sous cette forme: ab' - a'b =O (abl - a'b) c" + (a"b - ab") c' + (a'b"l - alb') c = 0 [(ctb' - a'b) C" + (a"b - ab") C' + (a'b" - a"b') c ] di' + [(a'b - ab') c I' (a"if"b '4 a"1 '")c]d + [(a"'b - all") c" + (ab" - a"'b) c"' ~ (a"b"' - a"'b"') c ]d' + [(a'b"' - a"'b) c" + (a"'b" - a"b"') c' + (a"b' - a'lb") c"'1] d = 0. Cette nouvollo formo a deux avantages: le premier, do rendre los substitutions 'a venir, plus commodes; le deuxie'me, c'est d'offrir une regle encore plus simple pour la formation do ces formules. "Eu offet, il est facile de remarquer 10, quo le premier terme do l'une quolconque de ces equations, est forme' du premier membre do l'equation precedente, multiplie' par la premie'ro des lettres qu'olle no renferme point, cotte lettre e'tant affecte'e de la marque qui suit imm6'-,diatement la plus haute do cellos qui entrent dans cc meme membre. 11' Le deuxie'me terme so forme du premier, en changeant dans celui-ci la plus haute marque en cello qui est imme'diatement au-dessous.& r~ciproquement, & do plus en changeant los signes. "~3' Le troisierne, so forme du premier, en changeant dans celui-ci Ila plus haute marque en cello do deux nume'ros au-dessous & re'ciproquemont, & do plus en changeant los signes. "4'. Le quatrie'me, so forme du premier, en changeant dans celui-ci la plus haute marque en cello do trois nume'ros an-dessous & re'ciproquemont, & chiangeant los signes, & toujours do imeme pour los suivans. " Par exemple..... " D'apre's ces observations, il sera facile do voir quo 1'e6quation do,condition pour cinq inconnues et cinq equations, sera The latter part of this we aro drawn to at once, as it enunciates -quite clearly the Recurrent Law of Formation to which attention has above been directed. It has to be observed, however, that the three 'equations of condition' are not in the form got by merely following the 'rule,' and that by deriving each 'terme,' not from the -first but from the preceding 'terme' we should,obtain, viz.: ab' -a/b = 0, (ab' - a'b) c" - (cab" - a"b) c' ~ (a'b" - a"b') c = 0, [(cab' ct') c" - (ab" - a"b) c' + (ct'b" - a'/b) cl d"'1 - [(ab' -ac'b) c"' - (ab"'/ - a"/'b) c' + (a'b" - a"'/b') ci d"l + [(ab" -a"b) c"' - (ab"' - a~"'b) c + (a"b" - a"b'b") c] d' - Va'" - c'b') "' -(a/b"' - a"'b) c" (a"b"' - a'"b") c'] d =0. DETERMINANTS IN GENERAL (VANDERMONDE, 1771) 17 The notable point in regard to the earlier portion is, that Bezout throws his rule of term-formation and his rule of signs into one. In the case of finding the resultant of arx + bry + crz = (r =l, 2, 3) his process consists of four steps, viz.:(1) a, (2) ab -b a, (3) a b c - c b + c a b -ba c a -cb a, (4) a-tb2c - ac2bA + cla2b - ba2c3 + blc2a3 - cb2a3. The first term of (2) is got from (1) by affixing b, and the second is got from the first by advancing the b one place and changing the sign. The first term of (3) is got from the first term of (2) by affixing c, the second term is got from the first by advancing c a place and changing the sign, and the third is got from the second by advancing c a place and changing the sign; the last three are got from the second term of (2) in the same way as the first three are got from the first term of (2). It will thus be seen that while Leibnitz and Cramer direct us to find the permutations in any way whatever, and thereafter to fix the sign of each in accordance with a rule, Bezout requires the permutations to be found by a particular process, and attention given to the question of sign throughout all this process, so that when the terms have been found their signs have likewise been determined. Bezout's contributions to the subject thus are(1) A combined rule of term-formation and} (II. 2)+(. rule of signs. (2) The recurrent law of formation of the new functions. (vi.) VANDERMONDE (1771). [Memoire sur l'elimination. Hist. de t'Acad. Roy. des Sciences (Paris), Ann. 1772, 2e partie (pp. 516-532).] This important memoir of Vandermonde and that of Laplace, which is dealt with immediately afterwards, both appear in the History of the French Academy of Sciences for 1772, Laplace's M,D. B 18HISTORY OF THE THEORY OF DETERMINANTS memoir occupying pp. 267-376, and Vandermonde's pp. 516-539. There is, however, a footnote to the latter, which states that it was read for the first time to the Academy on 12th January 1771. The part of it which concerns us is the first article, which treats of elimination in the case of equations of the first degree. Vandermonde here writes: "Je suppose quo 1'on represente par 12 &c., 123 ) 3 3 &c., &c., autant de diff~rentes quantite's generales, dont l'uno 31 3~~~~~~~~~~ 3) ~ ~ ~ ~ / quolconque soit a une autre quolconque soit b &c., & que le produit a, b des deux soit de'signe' 'a lordinaire par a. b. "Des deux nombres ordinaux a & a, le premier, par exomplo, designora de quolle iquation est pris lo co~fficient a- & le second a, designora lo rang quo tient co co~fficient dans e'6quation, comme on le verra ci-apres. "Je suppose encore le syste'me suivant d'abre'viations, & que 1'on fasse aj/3a 3 afi a — b - a. b - b. a' afol/_a/31y a/3qy a/31y a Ib Ic-a.b Ic b. c Ia c. ab' a 1,1/31y _a1 -/3Iy 8 a/3 1y a/ yj-18 a /3I y 18 a IbIctId-a.b6 cd b. cId Ia c.djajb d.alblc' a /3yj8jIEj a P fy 18 E aIb I cId IeI-a.b I cI dI e~ "Le symbole sort ici do caracte'ristique. Los soules choses a obsorver sont l'ordro dos signos, et la loi dos pormutations ontro los lettres a, b, e, di &c., qui me paroissent suffisammont indique'os ci-dessus. "1Au lieu, do transposor los lettres a, 8, C, d, &c., on pouvoit los laissor dans l'ordro alphabe'tique, & transposer au contrairo los lettres a, A3 y, 8, &c., los re'sultats auroient e'te parfaitement los m~mes; ce qui a lieu aussi par rapport aux conclusions suivantes. "Promie'roment, il ost clair quo a b represonto doeux tormes diffirens, l'un positif, & lFautre negatif, re'sultans d'autant do permutations DETERMINANTS IN GENERAL (VANDERMONDE, 1771) 19 possibles de a & b;' que b en represente six, trois positifs & trois negatifs, r6sulta~ns d'autant de permutations possibles de a, b, & C; qa IP Jy1 q e a Ib Ic Id.... "Mais de plus, la formation de ces quantit~s est telle que l'unique changement que puisse re'sulter d'une permutation, quelle qu'elle soit, faite entre les lettres du me'me alphabet, dans lune de ces abre'viations, sera un changement dans le signe de la premie're valeur. "La demonstration de cette ve'rite' & la reeherche du signe r~sultant d'une permutation determin~e, dependent ge'neralement de deux propositions qui peuvent eftre 6'nonce'es ainsi qu'il suit, en se servant de nombres pour indiquer le rang des lettres. "La premiere est que 1121 31...lmjqm~1 I... n I12 3 K... In-m+1 In-m~2 n-m~3 I...1 n mlrn+l Im~2 I...I n I I 1 2 K.. rnM- I le signe - n'ayant lieu que dans le cas oft n & m sont 1'un & 1Pautre des nombres pairs. "La seeonde est que I1 213.. rnm ~1m+I.... 112131... rmn m+ 1 1...I 1 12 13 <rn.I - II m Im+1I m~ 2 **. I "Il sera facile de voir que, la premie're eiqu ation suppose'e, celle-ci n'a besoin d'e'tre prouve'e que pour tin seul cas,_ comme, par exemple, celui de rn = n - 1, c'est-a'dire, celui oft les deux lettres transpose'es sont les deux dernie'res. "Au lieu de d'montrer g ~alement ces deux 6quations, ce qui exig~eroit un calcul embarrassant pluto~t que difficile, je me contenterai de de'velopper les exemples les plus simples: cela suffira pour saisir lesprit de la demonstration. (21 pages are occupied with verifications for the case of a10 of al/3Y and of aPojyj albI 0 al b Ic al b Ic d-) "On verra qu'en general la demonstration de notre seconde 6quation pour le cas n = a, depend de cette me'me equation pour le cas n = a - 1, quel que soit a: d'oit 11 suit que puisque 1 12_ = 1)2 elle est generalement vraie. 12 2j1 20 HISTORY OF THE THEORY OF DETERMINANTS "De ce que nous avons dit jusqu'ici il suit que a b Ic d... si deux leltres quelconques du mi'me alphabet sont 6gales entr'elles; car quelque part que soient les deux lettres d'gales, on peut les transposer aux deux dernie'res places de leur rang, ce qui ne fera au plus que changer le signe de la valeur; alors, de leur permutation particulie're, il ne peut, d'une part, re'sulter aucun changement, puisqu'elles sont egales; d'autre part, selon notre seconde equation ci-dessuis, il doit en resulter un changement de signe; cette contradiction ne pent b'tre levee qu'en supposant la valeur ziro.... "Tout cela pose; puisque P'on a id entiquement, I 1 1j2 1 1 2 1 1j2 +1 1 2. 0 2I1 1 2 2 1 12 2 1 12 2 1 12 1 1213 1.213+-2.3 1+3.1 12 si l'on propose de trouver les valeurs de ~1 et de ~2 qui satisfont aux deux equations 1.~1 +2. ~2~+3=0 2 2 2 1.$1j+ 2. ~2~+3=0, on pourra comparer, & lPon aura 1 12 1 12 1 12 1 12 (Three equations with three unknowns are similarly dealt with.) "II est clair que ces valeurs n'ont point de facteurs inutiles: mais pour les rendre aussi commodes qu'il est possible dans les applications, & particulib'rement dans celles oii ion veut faire usage des logaritlimes, il sera bon d'y empl-oyer le plus qu'il se pourra, la multiplication des facteurs complexes. J'observe donc 1 que si l'on substitue dans le de'veloppement de aPI91Y718) les valeurs des a I K 9'en al~ on alIblIelId' alIb Ic ab aura, en re'duisant & ordonnant, d'aprebs les observations ci-dessus, aI/3y118 alp y/I + al /3* al abeId al c b dI ad b Ic a /lp lK/8 a lP yli alIP ^/1 alblld a ldb-da Ic a I b_ _ yl DETERMINANTS IN GENERAL (VANDERMONDE, 1771) 21 si de meme on substitue dans le developpement des a I /I Y I d ' les valeurs des acP i/3/l8 IE a /3lp valeurs des a - 8-I en a I b, on aura, en reduisant & ordona I b I c Id e aI b c Id nant, d'apres les observations ei-dessus, a I3 y yi|i ai/ 7E3 iai i i a b clld elf alcb d elf ald blecelf blc aldlelf b d alclelf ble aclcdlf c ld a b I e f c e a b dif c f a c d e aI, \/ |8 |E + I I I _ I I I I a b blc dleif d e al blc f d If ab lcle + I I.I I I elf lb cb\ d a l e — b I c dl'+ alf b lc Idle b If a | c j d e "La loi des permutations & des signes est assez manifeste dans ces exemples, pour qu'on en puisse conclure des developpemens pareils pour les cas de huit & dix lettres, &c., du meme alphabet; alors, en employant les premiers developpemens pour les cas d'un nombre impair de ces lettres, on aura les formules d'elimination du premier degre, sous la forme la plus concise qu'il soit possible. "Si l'on veut exprimer ces formules, generalement pour un nombre n d'equations 1 1 1 1 1 1 1.$1+2. 2+3.$3 +... +m.m+... +n.n+(n1+)=O 2 2 2 2 2 2 1. 1+2.2+3.3 +.. 3 +n. +. +n. +.n+(n+l)=0 &c. la valeur de l'inconnue quelconque $im, sera renfermee dans l'equation suivante, a une seule inconnue 1 12 3... In n 1 12 I 3 1... I n 1 1 2 1 3 1...... 1n-mni-nj- In-n+21n-qn+3I 3.... 1 In+1 + lin +2 1+...... 1...m-l le signe + ayant lieu seulement dans le cas oh mn & n, sont impairs l'un & lautre." 22 HISTORY OF THE THEORY OF DETERMINANTS Taking this up in order, we observe first that Vandermonde proposes for coefficients a positional notation essentially the same as that of Leibnitz, writing 2 where Leibnitz wrote 12 or I2. Then he defines a certain class of functions by means of their recurrent law of formation-a law and class of functions at once seen to be identical with those of Bezout. A special symbolism is used for the first time to denote the functions; thus, the expression 10.21'32 + 11.22.30 + 122031 - 102231 - 11r20.32 - 12'2130, which occurs in Leibnitz's letter, Vandermonde would have denoted by 1 2 3 1 2 3' and the result of eliminating x, y, z, w from the set of equations lx + 2y + 3, + 4rw =r O (r = 1,, 3, 4) by 1 2 314 1 2 134 It is next pointed out that permutation of the under row of indices produces the same result as permutation of the upper row, that the number of terms is the same as the number of permutations of either row of indices, and that half of the terms are positive and half negative. The part which follows this is a little curious. The proposition is brought forward that if in the symbolism for one of the functions a transposition of indices takes place in either row, the same function is still denoted, the only change thereby possible being a change of sign. The demonstration is affirmed to be dependent on two theorems, neither of which is proved, as the proofs are said to be troublesome to set forth. Now it will be seen that the second of these theorems is to the effect that the transposition of any two consecutive indices causes a change of sign, and that consequently this alone is sufficient for the required demonstration. The first of the auxiliary theorems, in fact, is an immediate deduction from the second, the particular permutation which it concerns being produced by (?-r-m+1)(m7-1) transpositions of pairs of consecutive indices. DETERMINANTS IN GENERAL (VANDERMONDE, 1771) 23 Passing over the illustrations of these propositions, we come next to the theorem that if any two indices of either row be equal the function vanishes identically, and we note particularly that the basis of the proof is that the interchange of the two indices in question changes the sign of the function, and yet leaves the function unaltered. Upon this theorem the solution of a set of simultaneous linear equations is then with much neatness made to depend. In more modern notation Vandermonde's process is as follows:-It is known that a I b1c2 I1, + c I c 12 I + c = abbc2 I = 0, and a2 I b1c2 I+ b2, c la + c2 alb2 = I a2bl2 I = 0, lbIc2l b Ic1a2I a. alb2 + b, 1a + c1 = and ca2alb 2 + balb + = 21 a2ib 2+a1b 2 + 2 hence, if the equations ax +by + c1 = a2x + b + b + 2 - be given us, we know that b= b1C2l y 1al2 is a solution. This result, moreover, is generalised; the solution of r + r +..+ +r + rn+ = 0 (r=l, 2,... n) being fully and accurately expressed in symbols, although the numerators of the values of xl, x2,.., x are not in so simple a form as Cramer's rule for obtaining the numerator from the denominator might have suggested. Lastly, and almost incidentally, Vandermonde makes known a case of the widely general theorem nowadays described as the theorem for expressing a determinant as an aggregate of products of complementary minors. His case is that in which the given determinant is of the order 2m, and one factor of each of the products is of order 2. Summing up, therefore, we must put the statement of our indebtedness to Vandermonde as follows: 24 HISTORY OF THE THEORY OF DETERMINANTS (1) A simple and appropriate notation for the new functions, 1121a e.g., 1 2 3 (VII.) 112f3' (2) A new mode of defining the functions, viz., using substantially Bezout's recurring law of formation. (VIII.) (3) The remark that the ordinary algebraical expression of any of the functions is obtainable by permutation of either series of indices. (IX.) (4) The remark that the positive and negative terms are equal in number. (x.) (5) The theorem regarding the effect of interchanging two consecutive indices. (xi.) (6) The theorem (with proof) regarding the effect of equality of two indices belonging to the same series. (XII.) (7) A reasoned-out solution of a set of n simultaneous linear equations, by means of the new functions as above defined. (xIII.) (8) Expression of any of the new functions of order 2m as an aggregate of products of like functions of orders 2 and 2m- 2. (XIV.) In addition to this, we must view Vandermonde's work as a whole, and note that he is the first to give a connected exposition of the theory, defining the functions apart from their connections with other matter, assigning them a notation, and thereafter logically developing their properties. After Vandermonde there could be no absolute necessity for a renovation or reconstruction on a new basis: his successors had only to extend what he had done, and, it might be, to perfect certain points of detail. Of the mathematicians whose work has thus far been passed in review, the only one fit to be viewed as the founder of the theory of determinants is Vandermonde. LAPLACE (1772). [Recherches sur le calcul integral et sur le systeme du monde. Hist. de 'Acadc. Roy. des Sciences (Paris), Ann. 1772, 20 partie (pp. 267-376) pp. 294-304. 6(evr'es, viii. pp. 365-406.] In the course of his work Laplace arrives at a set of linear equations from which n quantities have to be eliminated. ]DETERMINANTS IN GENERAL (LAPLACE, 1772) 2 25 This he says can be accomplished by means of rules which mathematicians have given: " Mais comme clles ne me paroissent avoir e't6 jusqu'ioi de'montre'es que par induction, et quo d'ailleurs elies sont impracticables, pour pen que le nombre dos equations soit 'considerable; je vais reprendre de nouveau cette matie're, et donn'er queiques proce'des plus simples que ceux qui sotit de'ja connus, pour 6'liminer entre un nombre quelconque d'equations du premier degrel" Taking n homogeneous linear equations with the coefficients 2ai 2b, 2C) he first gives Cramer's rule for writing out what hie, Laplace, calls the BResqttlartt, using in the course of the rule the term var-iation instead of Cramer's term ccderangement." Then he gives the " perhaps simpler " rule of Be'zout, and shows that of necessity it will lead to the same result as Cramer's. The theorem in regard to the effect of transposing two letters is next enunciated, and the blank left by Vandermonde is filled, f or a proof of the theorem is given. The exact words of the enunciation and proof are"Si an lieu de combiner d'abord la lettre a avec la lettre b, ensuite, ces deux-ci avec la lettre C, et ainsi de suite; c'est-a'-dire, si au lieu do combiner les lettres a, b, c, d, e, &c.,' dans l'ordre a, b, cl di e, &c., on les, efi combine'es dans l'ordre a, c, b, d, e, &C., ou a, d, b, c, e, &C., ou a, e, b, c, d, &c., on &c., je dis qu'or auroit toujours eu la Meme, quantit' 'a la ditbirence des signes preis. "'Pour de'montrer ce Tbeiore'me nommons en general, re's'allante, la quantity' qni resulte de l'une quelconque de ces combinaisons, en sortor que la premiire re'sulante soit celle qui vient de la combinaison suivant, l'ordre a, b, C, d, e, &C., que la secornde r~suta~nte soit cello qui viont de la combinaison suivant l'ordre a, C, 6, d, el &o., quo la trisire re'sultante soit cello qui vient do la combinaison suivant l'ordre a, d, b, C,5 e, &C.,5 eainsi do suite; cola pose', ii est clair quo toutos ces r'sultantes renformen le me nombro do termes, et pre'cise'ment los mmeos, puisqu'ellos renferment tous los termes.qui pouvent re'sulter do la combinaison dos ni lettres a, b, C, d, e, &C., dispose'es ontre clles do tontes los manietros possiblos; ii no pout done y avoir do diffironce entre deux re'sultantes, quo dan s los signes do chacun do leurs termes; or, il ost visible quo la promie're r~sultanto donno lk' soconde, si l'on change dans la premie're b en c, et re'ciproquoment c en b; mais ce changemont augmiente ou diminue d'une unite' le nombre des variations 26 HISTORY OF THE THEORY OF DETERMINANTS de chaque terme; d'oii' il suit que dans la seconde resultante, tous les termes dont le nombre des variations est impair, auront le signe +, et les autres le signe -;partant, cette seconde re'sultaiite n'est que la premniere, prise ne'gativement. "II est visible pareillement que. "&C. The proof is thus seen to consist in establishing (1) that the terms of the one " resultant " must, apart from sign, be the same as those of the other; and (2) that the term-s of the one resultant are either all affected with the same sign as the like terms of the other, or are all affected with the opposite sign, the comparison of sign being made by comparing the number of variations. After this, the theorem that when two letters are alike the resultant vanishes is established in a way different from Yandermonde's, but not more satisfactory, viz., by considering what Be'zont's rule would lead to in that case. Application is then made to the problem of elimination, and to the solution of a set of linear simutltaneous eqnations, the mode of treatment being again different from \Tandermonde's, but this time with better canse. He says"Je suppose mairitenant que lon ait les trois 6quations o l a.14 + lbju' + 'c.iii', o0 -a j + 2b.IA ~ 2C. [LI o 5a., + lb.j-,' + cp' je forme d'abord la resultante des trois lettres a, b, c, suivant l'ordre a, b, C, cc qui donne, 'a.2b.%c - 1a.2c. 36 + 'c.2a.5b -lb.2a.3c + lb.2c.53a - C2.3 on 1t. [2b.3c - 2c. 5b] + 2a. I'C.3b - b.3c] + 3a.P'b.2c - 'c.2b]; je multiplie ensuite la premiere des equations pr~ce'dentes par C- 2c.3b, la seconde par 'c.5b - lb.'c, la troisieune par lb.cet je les ajoute ensemble, cc qui doune, o = ~x.['a.(2b. C - cb) + 2a.('c.5 - lb.3C) + 5a('.2C - 1C.2b)] + u.['b.(2b.5C - 2C.3b) + 2b. Qc.3b - 'b.3c) + 3b (b.2c - 'C.2b)] - 2C lb) + 2c.~ - 'b.3c) + 3c ('b.2C - l.2b)]; or, il suit de ce quo nous venous do voir, que les coefficiens de IA' e t pk", sont identiquernent nuls, puisqu'ils no sont quo la resultante dos trois DETERMINANTS IN GENERAL (LAPLACE, 1772) 2 27 lettres a, b, c, dans laquelle on e'crit b, ou. c, par-tout oi'i est a; donc, oil aura pour e'~quation de condition demande'e, 0 - la.(2b.3c - 2c.5b) + 2a.('c.3b - 'b.3c) + 3a.(lb.2c - 1C.2b); c'est-a'-dire, la resultante de la combinaison des trois lettres a, b, c egalee 'a zero. On de'montreroit la m~me chose, quel que soit le nombr~e des equations. "cPour montrer l'analogie de cette matie're, avec e'~limination des equations du premier degr6, je suppose, que ion ait les trois 6quations, lp= la.1k + lb./_' + lC./L", 2p = a./j + 2bj1t + 2C.~I"I' 3 = 5a.IL + 3b.1j' + 3c.pk". Je nmultiplie, comme ci-devant, la premie're par (2b. 3c - 2C. 3b), la seconde par ('c.3b - lb.5c), et la troisie'me par ('b.2C - 1c.2b), je les ajoute ensemble,7 et j'observe que les co~fficiens de Ik' et de [k', sont indentiquement nuls dans e'~quation qui en resulte; d'oiui je conclus, I 2.c 2C.b) + 2p.(C.5 - 1b.5c) ~ 5p.'b.C- 1C.2b) -a2b3 2c.3b) + 2ca('c.3b - lb.5c) + 3a('b.2c - C.2b) on voit done que le nume'rateur de 1'expression de pa, se forme du defnominateur, en y ehangeant a en p; on aura ensuite I-' ou. 1j", en ehang~eant dans l'expression dle 4'&C. This mnode of treatment leaves nothing to be desired. It is that which is most commonly employed in the text-books of the present day. The next point taken up is the most important in the memoir, and requires special attention. It is introduced as " a very simple process for considerably abridging the calculation of the equation of condition between a, b, e," &c.-that is to say, the calculation of a resultant. It is, howvever, something of much more value than this, involving as it does a widely general expansion-theorem to which Laplace's name has been attached, but of which we have already seen special cases stated by Vandermonde. The theorem may be described as giving an expansion of a resultant in the form of an aggregate of terms each of which is a product of resultants of lower degree. Laplace's exposition is as follows: "Je suppose que vous ayez deux equations, 0 = la.1j + lb14, 0 = 2a.It + 24~t'. 6crivez + Ab et donnez l'indiee 1 'a la premib're lettre, et l'indiee 2 'a la seconde; e'~quation de condition demande'e sera + la.2b - lb. a -0. 28 HISTORY OF THE THEORY OF DETERMINANTS " Jo suppose que vous aycz trois equations; e'crivez + ab, combinez ce terme avec la lettre c do toutes los rnaniei.res possibles, en changeant le signo do chaque terme chaque fois quo c change do place, vous aurez ainsi ~ abc -acb +cab; donnoz dans chaq~ue terme l'indice 1 'a la prcmib're lettre, l'indice 2 'a-la seconde, l'indice 3 'a la troisie'mo, et vous aurez +l'a.2b5 - a2.b c2 b; cola pose', au lieu do +l'a.2bMc ecrivez + (1a.2b - lb. a).5c; an lieu do - lac.b 'crivez - (la.-b - l.a. ot au lieu do + le.2ab cie+ (2ab 2b 5a).'c; l'equation do condition demand6e sera o = (1a.2b - lb.2a).5c - (1a.3b - lb.3a).2c + (2a.3b - 2b.3a).'c. "Jo suppose quo vous ayez quatre &quations, kcrivez + abc - acb + cab), et combinez ces trois termes avec la lettre d, en observant I' do n')admettro quo los termes dans lesquels c pr'ce'de d; 2' do changer do signe dans chaque torme toutes los fois quo d change do place, ot vous aurez + abcd - acbd + acdlb ~ cabd - cadb ~ cdab; donnez onsuite l'indice 1 'a la premierc lettre, lindice 2 'a la seconde, &c., et vous aurez + 'a.2b.3c4d - la.2C.3b.4dI + la.2C.3d.4b + 'c.2a.3b.4d - 'c.2a.3d.4b + lc.2d 3a. 4b cola pose', an lieu do + 1a.2b.5c.4c1 ecrivez + ( 1a.2b - lb.2a).(5c.4d - d4) et ainsi des autres termes, et l'quation do condition sera' 0 = ('a.2b - b.2a).(3c.4d - 5c1.4c) - (1a.5b - 'b.3(a).(2c.4d 2c1.4C) + ('a.4b -1b.4a).(2c.3d - 2d.53c) + (2a.3b - 2b.3a).( 1C.4c1 -1d.4C) - (2a. 4b -2b.4a).('c.5d - 1d.c) ~ (3a.4b - 3b.4a).( 1c.2d -I'c.2c). "Jo suppose quo vous ayez cinq 6quations, 6cerivez los six termes + abcd - acbd ~.. relatifs 'a quatro equations, et combinez-les avec la lettre e, do toutes los maniebres possibles, en observant do changer do signe chaque fois quo e change do place; donnez ensuite l'indice 1, &c., &c..;.. au lion du terme + la.2. e.5 cie ('a.3b l b. a).(2C.5c1 - 2d,.5C).4e, &C..... "Lorsqu'on aura six equations, on combinera los termes + abcde - abced + &C., relatifs 'a cinq equations avec la lettre j, en observant 1' do n'admettre quo los termes dans lesquels e pr'cb'de ft 2' do changer do signe lorsque f change de place: on transformera ensuite, par la rebglo prece'dente. DETERMINANTS IN GENERAL (LAPLACE, 1772) 29 Notwithstanding the multiplicity of instances, the rule here illustrated is not made altogether clear. This is due to two causes,-first,. the linking of one case to the case before it; and, second, the want of explicit notification that the letters b, d, f... are combined in one way, and the intervening letters c, e,... in another. For the sake of additional clearness, let us see all the steps necessary in the case of the resultant of the five equations axi + bx2 + cx3+ dr.x+ ex5 =O (r = 1, 2, 3, 4, 5), and supposing, as we ought to do, that the case of four equations has not been already dealt with. These steps are1~. Combining b with a subject to the condition that a precede b: resultab. 2~. Combining c with this in every possible way, the sign being &c.: resultabc - acb + cab. 3~. Combining d with each of these terms subject to the condition that c precede d: resultabed - acbd + acdb + cabd - cadb + cdab. 4~. Combining e with each of these terms in every possible way: resultabcde - abced + abecd - aebcd + eabcd - acbde + acbed -...... 5~. Appending indices: resultab2c3d4e, -alb2,e4d,5 +... ~................... 6~. Changing ambn into (arbn- bman), Crds into (Crds-drCs), &c.: result(ab - b1a2) (cd,- d3c4)e5 - (alb2- bl1a)(3d5 - d3c)e +... This is the required resultant in the required form. It is of the utmost importance to notice what is accomplished in 1~, 2~, 3~, 4~ is simply (a) the finding of the arrangements of a, b, c, d, e subject to the conditions that a precede b, and c precede d, and obtaining each arrangement with the sign which it ought to have in accordance with Cramer's rule. The number 30 HISTORY OF THE THEORY OF DETERMINANTS of necessary directions might thus be reduced to three, viz., (a), (5), (6), in which case (1), (2), (3), (4) would take their proper places as successive steps of a methodic and expeditious way of acconiplishing (a). Laplace appends a demonstration of the accuracy of this development of the resultant of the nth degree, the line taken being that if the multiplications were performed the terms found would be exactly the 1.2.3....n term-s of the resultant, and would bear the signs proper to them as such. He then goes on to deal with a rule for obtaining a like development in which as many as possible of the factors of the terms are resultants of the th~ir'd degree. To do so succinctly he is obliged to introduce a notation for resultants. On this point his words are"Je de'signe par (abc) la quantite' abc - acb + cab - bac + bca - cba, et par (ab) la quantite' ab - ba, et ainsi de suite; par (la. 2b.5c) j'indiquerai la quantite6 (abc), dans los termies de laquelle on donne 1 pour indice 'a la premie're lettre, 2 'a la seconde, et 3 'a la troisie'me; par (la.2b), je d~signerai la quantite' (ab) dans les termes do laquelle on donne 1 pour indice 'a la premie're lettre, et 2 'a la secondo; et ainsi de suite." We can but remark that here again he leaves little room for improvement: his symbolism is essentially that which is still in common use. The exposition of the rule is as follows: " Jo suppose maintenant quo. vous ayez trois equations, e'~quation do condition sera 0 = (la. 2b.5c). "Jo suppose quo vous ayez quatro 6quations; e'crivez + abc, et combinez ce termie do toutes los manie'res possibles avec la lettre d, en observant do changer do sign e lorsque d change do place, ce qui donne + abcd - abdc + adbc - dabc; donnez l'indice 1 'a la premie're lettre, l'indice 2 'a la seconde, &c., et vous aurez + a.2b.3c.4c1 - 'a.2b.3d.4C + la.2d.3b.c -.b.C an lieu du terme +1'a.2b.3c.4d, ecrivoz + ('a.2b3)4;a iud -1a.2b.3d.4c, ecrivez - (1a.2b. 4C).5Id, et ainsi do suite, et vous formerez lIequation do condition 0=('a.2b.3c).4d - (la.2b.c).3d + (1a.3b.4)2 I (a.3b.c3d DETERMINANTS IN GENERAL (LAPLACE, 1772) 3 31 "1Je suppose que vous ayez cinq 6quations, combinez les termes + abcd - abdc ~ &c., relatifs 'a quatre equations avec la lettre e en observant I' de n'admettre que les termes dans lesquels d pr'ce'de e; 2' de changer de signe lorsque e change de place, et vous aurez + abede - abdce + abdec + &c. donnez l'indice 1 'a la premib're lettre, l'indice 2 'a la seconde, &c., et vous aurez +la. 2b.c4d. e, -l '.2 bd.e5e, + la. b..e. 5c + &c.; ensuite, au lieu de ~ 1a.2b e.3.4d.5e, 'crivez + ('a.2b.5c). (4Cd.5e); au lieu de - la.2b5.4.e, 6crivez -(a2.)(d.),et ainsi de suite; et en egalant 'a zero la somme de tous ces termes, vous formerez le'quation de condition demande'e. "Je suppose que vous ayez six equations, combinez les termes, +abede - &c., relatifs ' cin quations avec, la lettre f, nosevn V' de n'admettre que les termes oti e, pr'ce'de f; 2' de changer de signe lorsque f change de place: donnez ensuite 1 pour indice 'a la premie're lettre. " Si vous avez sept 6quations, combinez les termes + abcdef - &c. relatifs 'a six equations avec la lettre g de toutes les maniebres possibles; pour huit equations, combinez les termes relatifs 'a sept avec la lettre h, en n 'admettant que les termes dans lesquels g pr'ce'de h, et ainsi du reste." The really important point in all this is in regard to the manner in which the letters are bronght into combination. It will be seen that the set begun with is abe, consequently a precedes b, and b precedes c throughout: then d is combined in every possible way with this: e is combined subject to the condition that d precede e; f is combined subject to the condition that e precede f: g is combined in every way possible: h. is combined subject to the condition that g precede lb: and so on. It would appear therefore that the lettres are to be combined in every possible way are d and every third one afterwards, and that each of the other letters is conditioned to be preceded by the letter which immediately precedes it in the original arrangement abcdefghi....Condensing these directions after the manner of the former case, we should draft the rule as follows: (a) Find every possible arrangement of abcdefghi...subject to the conditions that in each arrangement we must hav~e a, b, c in their -natural order; d, e, f in their natural order; g, it, i in their natural order; and so on. 32 HISTORY OF THE THEORY OF DETERMINANTS (b) Prefix to each arrangement its proper sign in accordance with Cramer's rule. (c) Append in; order the indices 1, 2, 3,... to the letters of each arrangement. (d) Change ambnc,. into (arn.bn.cr), dsefy into (dc.e,.fy), &c. Without saying anything as to the verification of the developments thus obtained, Laplace concludes as follows:"On decomposeroit de la mime maniere l'equation R en termes composes de facteurs de 4, de 5, &c., dimensions." To show how this could be effected would have been a tedious matter, if the method of exposition used in the previous cases had been followed, viz., multiplying instances with wearisome iteration of language until the laws for the combination of the letters could with tolerable certainty be guessed. On the other hand, had Laplace condensed his directions in the way we have indicated, the rule for the case in which as many as possible of the factors are of the 4th degree could have been stated as simply as that for either of the two cases he has dealt with. The only changes necessary, in fact, are in parts (1) and (4), and merely amount to writing the letters in consecutive sets of four instead of two or three. Further, when the rule is condensed in this way, the problem of finding the number of terms in any one of the new developments-a problem which Laplace solves in one case by considering how many terms of the final development each such term gives rise to-is transformed into finding the number of possible arrangements referred to in part (1) of the rule. Where the highest degree of the factors of each term is 2 and the resultant which we wish to develop is of the nth degree (which is the case Laplace takes), the number of such arrangements is evidently (1.2.3.... )/(1. 2)8, s being the highest integer in n/2; if the highest degree of the factors is 3, the number of arrangements is 1.2.3.... n (1.2.3)s (l.2)t' where s is the highest integer in n/3 and t the highest integer in (n - 3s)/2; and so on. DETERMINANTS IN GENERAL (LAGRANGE, 1773) 33 The facts in reduction of the claim which Laplace has to the expansion-theorem now bearing his name are thus seen to be (1) that the case in which as many as possible of the factors of the terms of the expansion are of the 2nd degree had already been given by Vandermonde; (2) that Laplace did not give.a statement of his rule in a form suitable for application to all possible cases, and, indeed, was not sufficiently explicit in the statement of it for the first two cases to enable one readily to see what change would be necessary in applying it to the next case. Notwithstanding these drawbacks, however, there can be no doubt that if any one name is to be attached to the theorem it should be that of Laplace. The sum of his contributions may be put as follows:(1) A proof of the theorem regarding the effect of the transposition of two adjacent letters in any of the new functions. (XI. 2) (2) A proof of the theorem regarding the effect of equalizing two of the letters. (xII. 2) (3) A mode of arriving at the known solution of a set of simultaneous linear equations. (XIII. 2) (4) The name resultant for the new functions. (xv.) (5) A notation for a resultant, e.g. (laC.2C.b). (vii. 2) (6) A rule for expressing a resultant as an aggregate of terms composed of factors which are themselves resultants. (xiv. 2) (7) A mode of finding the number of terms in this aggregate. (XVI.) LAGRANGE (1773). [Nouvelle solution du probleme du mouvement de rotation d'un corps de figure quelconque qui n'est anime par aucune force acceleratrice. Nouv. Mem. de V'Acad. Roy... (de Berlin). Ann. 1773 (pp. 85-128). (Euvres, iii. pp. 577-616]. The position of Lagrange in regard to the advancement of the subject is quite different from that of any of the preceding mathematicians. All of those were explicitly dealing with the problem of elimination, and therefore directly with the functions afterwards known as determinants. Lagrange's work, on the M.D. C 34 HISTORY OF THE THEORY OF DETERMINANTS other hand, consists of a number of incidentally obtained algebraical identities which we nowadays with more or less, readiness recognise as relations between functions of the kind! referred to, but which unfortunately Lagrange himself did not view in this light, and consequently left behind him as isolated; instances. With him x, y, z and x', y', z' and x", y", z" occur primarily as co-ordinates of points in space, and not as coefficients in a triad of linear equations; so that (xy'z" + yz'x" + zx'y" X-, YX" yz" - yYIXN when it does make its appearance, comes as representing six times the bulk of a triangular pyramid and not as the result of an elimination. In days when space of four dimensions was less attempted to be thought about than at present, this circumstance might possibly account for no advance being made to like identities involving four sets of four letters x, y, z, w; x', y', z', IV'; &C. In this first memoir the algebraical identities are brought together and stated at the outset as follows "LEMME. "1. Soient neuf quantit's quelconques xi y, -, I',) yI', z', xi/, y"/, z"/ je dis qu'on aura cette equation identique (xy'z" + yz'x" + zx'y" - xz'y" - yx'z" - Xy 3 1)2 - (i2 + y2 + Z2) (in2 + y '2 + Z/2) (XI"2 + Y"2 + in"2) + 2 (xx' + yy' + xz') (xx" + yy"f + xx") (x'x" + y'y" + X'/z") - (X2 y2 + Z2) (X'XII '+y'y" +z' )2 - (X12 + y72 + Z'2) (XX"1 + yyl + in1)2 (in"2 + y"2 + "112) (xx' + yyl + XZ')2. "Corollaire 1. "2. Donc si l'on a entre les neuf quantites preedentes ces six equations i2 + y2 + i2 = a, X/X' +y y'y" + X'X" = bl x'2 + y'2 + in2 = a', XI" + yy"// + XX" = bf in"2 + y"2 + i/"2 - a" xx' + yy' + XX' = VI DETERMINANTS IN GENERAL (LAGRANGE, 1773) 35 et qu'on fasse pour abreger = ' - zy", i =' zX" - x'z, = Xy" ' yX" P = ((aa'a" + 2bb'b" -ab - a'b'2 a"b"2); on aura +z + yq + zc= p. On aura de plus les equations identiques suivantes x'f + y'v + z' = O, x"+ y"r + +Z"= 0, $2' 2 + + -.- a'a" - b2, y'C- z'~r = bx' - aIx", y"C - z" r = a"x' - bx", z'~ - x' = by' - a'y", z"ll - x"f = a"y' - by", x'l - y'1 = bz' - a'z", x"r/ - y"- = a"z' - bz", qui sont tres faciles a verifier par le calcul. "Corollaire 2. "3. Si on prend les trois equations x + + y + = /, xxz ' +yy' + zz'= b", xx" + yy" + zz" = b', et qu'on en tire les valeurs des quantites x, y, z, on aura par les formules connues p(yz" - z'y") + b'(qz' - (y') + b"(/ "- -z) (Yz" - Z'y") + 7 (z'x" - ) + ('y"- ) p (z'x" - x'z") + b'(x' - z') + b"(z" — -") y (y'Z" - zy"/) + D (z'x" - x'z" +) +( -y' )' (x'y" - y'x")+ Y(y' - vx') + b'(27x" - "y") V (y'z" - z'y") + q (z'" - x'z") + C(x'y" - y'")' done faisant les substitutions de l'Art. prec. et supposant pour abreger = a'a" -b2 on aura _/ + (a"b" - bb')z' + (a'b' - bb")x" a / 3 + (a"b" - Mb')y' + (a'b' - WI)y" y= aa /3 + (a"b" - bb)z' + (a'b' - bb") z" a In regard to the first identity here (the so-called lemma), the important and notable point is that the right-hand member is the same kind of function of the nine quantities x2+y2+z 2, 36 HISTORY OF THE THEORY OF DETERMINANTS xx' + yy' + ZZ', xx" ~ yy" ~ zz", xx' + yy' + zz', X12 + y'2 + z12, 'x" + Y',y" ~ z'z", XX" + yy ~ zZ", Xx + y'y / + z'z", X"2 ~y"2 + z2 as the left-hand member is of the nine x, Y, Z, x', y', Z', x", y", z" indeed, without this distinguishing characteristic, the identity would have been to us of comparatively little moment. Possibly Lagrange was aware of it; but, if so, it is remarkable that he did not draw attention to the fact. It is quite true that Lagrange's identity and the modern-looking identity X y z 2 x2 + y2 +z2 xx' +Iyy' + ZZI XXI" +yy"l +ZZI/ x' z' Z xx' + yy' + zz' X'2 + y12 + z'2 XIXII + y'y" + z'z" x" y" z" 1xx"+ yy" ~ ZZ" X1X + y'y" ~ Z'Z" X112 + y"2 + z"2 are essentially the same; but no one can deny that the latter contains on the face of it an all-important fact which is hid in the former, and which in Lagrange's time could be made known only by an additional statement in words. The second identity x'eS+ y'+z z' = 0 is a simple case of one of Vandermonde's, viz., that regarding the vanishing of his functions when two of the letters involved were the same. The third identity e2 f 12 +,2 =+2a'a" - b2 is in modern notation /' y2+ /' 2 x' "2 -'2 + y'2 + Z/2 x'x" + y/,'y-1- z" z x' x" y' y" -x'x"+y'y"~z'z" x"/2 +y2 +;Z"/2 and is thus seen to be a simple special instance of a very important theorem afterwards discovered. The fourth identity y'- z'1 = bx' - (t may be expressed in modern notation as follows: Z XX +Yyl~tXI +Z I 1 ZlIX x'y"I X 2 +y' 2 +z'2 x' and, quite probably, has also ere this been generalised in the like notation. The fifth identity f3e+ (a"b" - bb')x' + (a'b' - bb") a DETERMINANTS IN GENERAL (LAGRANGE, 1773) 37 is not so readily transformable, the determinantal theorem which it involves being indeed completely buried. Multiplying both sides by a; then doing away with a, which seems perversely introduced "pour abreger " when no like symbol of abridgment takes the place of a"b" - bb' or of a'b' - bb"; and transposing, we have f= x(C'a" - b2) - x'(a"b" - bb') + x"(bb" -'b'), = x b" b' t' a' b x" b a"; that is, finally, x xx ' yy' + zz xx" +yy" +Z" xy'z"' z l = ' x'2 +y2 + '2 X'x+ y'J" + z'/ x" xI'x" + 'y"y + z'z" x,"2 + y/2 + 2, which we recognise as an instance of the multiplication-theorem on putting C y Z 1 x' x" x' y' z' x 0 y'y" x" y" z' 0 ' Z" for the left-hand member. LAGRANGE (1773). [Solutions analytiques de quelques problemes sur les pyramides triangulaires. Nouv. Menm. de 'VAcad. Roy.... (de Berlin). Ann. 1773 (pp. 149-176). (Euvres, iii. pp. 659-692.] In this memoir also there is a preparatory algebraical portion, the subject being the same as before, and the author's standpoint unchanged. Indeed the two introductions differ only in that the second is a rounding off and slight natural development of the first. In addition to, ], we have now %', v', ~', s", ", /" used as abbreviations for zy" - yz", xz" - zx",...; in addition to a, we have a', a", 3, /3', /3", standing for aa" - b'2, aac-b"2, b'b" - ab, bb"-a'b', bb'- a"b"; and X, Y, Z, X', Y',...., A, A',. are introduced, having the same relation to, $, ', '.... a, a.... as these latter have to x, y, z, x, y',..... a, a',.... Lagrange then proceeds: 38 HISTORY OF THE THEORY OF DETERMINANTS "3. Or en substituant les valeurs de $, $', &c., en x, x', &c., et faisant pour abreger A = zy'z" + z'x" + 'y" - z'y" - yxz" - zy'x", on trouve X = Az, Y = A/, Z = Az, X'= A, Y' = Ay', Z' = Az', X"= Ax', Y"= Ay", Z" = Az', done mettant ces valeurs dans les dernieres equations ci-dessus, on aura en vertu des six 6quations supposees dans l'Art. 1. A = A2a, B = A2b, A' = 2a', B' = A2b', A"- A2a", B"= A2b" et de la il est facile de tirer la valeur de A2 en a, a', a", b, &c.; car on aura d'abord A2 = A =a'a" - P2 at a et substituant les valeurs de a', a" et P/ en a, a', &c. (Art. 1) A2 = aa'a" + 2bb'b" - ab2 - a'b'2 - a"b"2 on trouvera la meme valeur de A2 par les autres equations. Si on remet dans cette equation les quantit6s x, y, z, z', &c., on aura la meme equation identique que nous avons donnee dans le Lemme ci-dessus (p. 86). "4. Il est bon de remarquer que la valeur de A2 peut aussi se mettre sous cette forme _ aa + 'a' + a"a" + 2 (/b +/3'b' + /b"). 3 or si on multiplie cette equation par A2 et qu'on y substitue ensuite A a la place de A2a, A' a la place de A2a' et ainsi de suite (Art. prec.) on aura A4 Aa + A'a' + A"a" + 2(B/3 + B'3' + B"/3") 3 ou bien en mettant pour A, A', &c., leurs valeurs en a, a', &c. (Art. 2) A4 = aa'a" + 2P/3'/" - a32 - a/3/2 - a"/'"2; d'of l'on voit que la quantit6 A2 et son carre A4 sont des fonctiols semblables, l'une de a, a', a", b, b', b", l'autre de a, a', a", 3, 3', /3". "5. De plus, comme l'on a (Art. 3) xy'z" + yz'x" + zx'y " - xz'y" - 'y'z" - zy'x" = /(aa'a" + 2bb'b" - ab2 - a'b'2 - a"b"2) = A, DETERMINANTS IN GENERAL (LAGRANGE, 1773) 39 et qu'il y a entre les quantites x, y, z, x', &c., et a, a' a", b, &c., les memes relations qu'entre les quantit6s $, r,,, ', &c., et a, a' a",/, &c.,(Art. 1), on aura done aussi $7'5" + '7 + 'rt " - $f' -?$ - r = /(aa'a" + 2/33'7" - a/32 - a''2 - a"/812) = A2. Done on aura cette equation identique et tres remarquable $'(' + Y(/ + ~g"- - $t - ( - ' = (xy'z" + yz'x" + zx'y" - xz'y" - yx'z" - zX'X")2." The remaining portion is of little importance; its main contents -are four sets of nine identities each, viz.:1. x+.'e' + "T" = A, + y''+y"+"=0, &c. 2. x=a+yx +zx =A, x'$+y'r+z' =0, &c. ax +,Bf' -+ 'X & 3. =, &c. 4. = a+ b"' + b'" &c. A Besides the fact that Art. 3 contains a proof of the Lemma,of the previous memoir, we have to note the new identity X=Ax, -which in modern determinantal notation is I \ I Y " Iy ', z I I 1I = I xy j Y -a simple special instance of the theorem regarding what is nowadays known as "a minor of the determinant adjugate to another determinant.",The last two lines of Art. 4 by implication make it almost *certain that Lagrange did not look upon xy'z" + yz'x" + zx'y" - xz'y" - yz"-zy'x".and ctct'cta + 2bb'b" - cb2 - a'b'2 - t"b"2.as functions of the same kind. The new theorem in Art. 5, which Lagrange justly characterises.as " very remarkable," is in modern determinantal notation y'z" lz'c", cx'y"l x y z 2 zy" I XZcz" I yx" = x y Z' IY' zIXI IXY' y" z" 2 — y Xz YX *" ": 400 HISTORY OF THE THEORY OF DETERMINANTS -a simple instance of the theorem which gives the relation, as we now say, " between a determinant and its adjugate." In regard to the remaining identities which we have numbered(1), (2), (3), (4), we note that (1) and (3) are not new, although (3) is here given almost in the form desiderated above (pp; 36-37);. (2) involves the fact that A is the same function of x, x', x", Y, y', y", z, z', z", as it is of x, y, z, x', y', z', x", y", z"; and (4) may be transformed as follows: XZ = aC + b"( + b's", =a y z - 2 +Y2 +z2 Y z Xx + ~yy~zz'y' z' XX "+ '+ zz, ylz" so that it may be considered as another disguised instance of the multiplication-theorem, the determinant just reached beingequal to Xyz xOO x' y'Z' X y 1 0 x/ y Z// z z 0 LAGRANGE (1773). [PRecherches d'Arithmetique. NTouv. Mim. de 1'Acad. Roy. (de Ber-lin). Ann. 17'173 (pp. 265-312).] This is an extensive memoir on the numbers "qui peuvent ktre: represente'es par la formule Bt2 + Ct + DU2." At p. 285 the; expression )y2 ~ 2qgyz + rz2 is transformed into PS2 ~ 2QSx + RX2 by putting y = Ms + Nx, and z=rnMs + x, and Lagrange says-....je substitue dans la quantit6 PR_- Q2 les valeurs de P, QI et R, et je trouve en effacant cc qui se de'truit PR - Q2 = (pr - pq) (Mn - NNn)2;... ]DETERMINANTS IN GENERAL (BEZOUT, 1779) 4 41 which we at once recognise as the simplest case of the theorem connecting (as we now say) the discriminant of any quantic with the discriminant of the result of transforming the quantic by a linear substitution. Putting now in compact form all the identities obtained frofa, the three preceding memoirs of Lagrange, we have(1) (cty'z" +z zx" + zx'y - xzy" xz zyx') =actct'" ~ 2 b.b b" - ab2 _ ab 2 ca"b"12 (XVII.): where a-x 2+y2+-2, a/= (2) C2 +42+ ~2-a'a" -b 2, where =y'z'- z'y"/, i=....(xvuir.> (3) y'-zj = bxc - a/x". (xix.)> (4) CA~ axe + 03"x' + 03'x" where a - a'a" - b2 "=. and zA=xcy'z" +yz'x" +zxe'y" -xz'y" - yx'z" -zy'x". (xvii. 2) (5) X = x, where X = q'~"- ~'q". (xx.>, (6) (Xy'Z" + yz'X"I + ZX'y" - XZ'y" - yX'z"1 -yX1 = $'~ + ~' " ~ ~~" - ~'~" ~'~"-N ell. (xxi.), (7) PR-Q2-(pr- q2) (Mn- NM)2, (XXIT.). if p(Ms~ Nx) + 2q(Ms+ Nx) (ms-F x) +r(rns+ x =82 + 2Qsx + RX2 identically. BEZOUT (1779). [The'orie Generale des Equations Alge'briques, ~~195,-223, pp. 171-187; ~~ 252-270, pp. 208-223. Paris.] In his extensive treatise on algebraical equations Beizout, was bound, as a matter of course, to take up the question of' elimination; and, as he had dealt with the subject in a separate. memnoir in 1764, one might not unreasonably expect to find the treatise giving merely a reproduction of the contents of' the memoir in a form suited to a didactic work. Such, however,. is far from being the case. He merely mentions the necessary references to the work of Cramer, himself, Vandermonde, and. Laplace; and then adds - "Mais lorscqu'il a e'te question d'appliquer ces diff~rentes m~thodes an proble'me de l'd1im'ination, envisaoge dans toute son e'tendue, je me: 42 HISTORY OF THE THEORY OF DETERMINANTS suis bient6t apper~u qu'ils laissoient tous encore beaucoup 'a desirer du c6te' de la praticjue." His main objection to the said methods is that when one has to deal with a set of equations of no great generality, with 'Coefficients, it may be, expressed in figures"Ii faut construire ces formules dans toute la ge'neralite' dont les equations sont susceptibles, et faire par consequent le me'me travail,que si les equations avoient toute cette ge'neralit&' (17.Au lieu done de nous proposer pour but seulement, de donner des formules generales d'e'imination dans les 6quations du premier,deogre, nous nous proposons de donner une re'gle qui soit indiff~remnment et eigalement applicable aux equations prises dans toute leur,g neri6 et aux equations considere'es avec les simplifications qu'elles pourront offrir: une r"gle dont la marche soit la me'me pour les unes *que pour les autres, mais qui ne fasse calculer que ce qui est absolument indispensable pour avoir la valeur des inconnues que lPon cherche: une reigle qui s'applique indilfhreniment aux 6quations numlriques et aux ~equations litt~rales, sans obliger de recourir 'a aucune formule. Telle -est, si je ne me trompe, la re'gle suivante. Th Rgle g'n~rale pour calculer, toutes 'a la fois, ou sipare'ment, les valeurs des incconnmmes clams les 6quatioms dit _prem~ier degr6J, soit litterales soit rrnumjriques.. "(198). Soient u, x, y, z. &C., des inconnues dont le nombre soit m, ainisi qui celni des 6quations. "1Soient a, b, C) d, &C., les co~ifflciens respectifs de ces inconnues dans la premierre equation. "a, b',c' d', &c., les co~fficiens des me'mes inconnues dans la seconde &juation. "Ca", b", c", d",11 &C., les co~fficiens des m~mes inconnues dana3 la troisir'me 6quation: et ainsi de suite. " Supposez tacitemnent que le terme tout connu de chaque equation soit affecte' aussi d'une inconnue que je repr~sente par t. "1Formez le produit uxyzt de toutes ces inconnues 6crites dans tel ordre que vous voudrez d'abord; mais cet ordre une fois admnis, -conservez-le jusqu'a' la fin de l'ope'ration. "1Echangez successivement, chaque inconnue. contre son co~fficient dans la premierre equation, en observant de changer le signe 'a chaque echange pair: ce re'sultat sera, ce que j'appelle, utie preniie're ligne. "1Echangez dans cette _preminire ligne, chaque inconnue, contre son,co~fficient dans la seconde equation, en observant, conmme ci-devant, de,.Changer le signe 'a chaque 6change pair: et vous aurez une seconde ligne. "Echangez dans cette seconde ligne, chaque inconnue, contre son offietdans la troisir'me equation, en observant de changer le signe, 'a chaque e'change pair: et vous aurez une troishnae, ligne. DETERMINANTS IN GENERAL (IBiZOUT, 1779) 4 43 "1Continuez de la ni~me manie're jusqu'a' la dernieire equation inclusivement; et la dernie're ligne qu~e vous obtiendrez, vous donnera les valeurs des inconnues de la manie're suivante. " Chaque inconnue aura pour valeur une fraction dont le nume'rateur -sera le co~fflcient de cette me'me inconnue dans la dernie're ou ne ligne, et qui aura constamment pour de'nominateur le co~fficient que l'inconnue introduite t se trouvera avoir dans cette m~nie n e ligne." The application of this very curious rule is illustrated by a,considerable number of varied examples, of which we select the,second"(200). Soient les trois equations suivantes ax + by + cz ~ d = 0, aix+ b'y +c'z +d' = 0, a/x + bly +Cz + d" = 0. "Je les 6cris ainsi ax ~by ~ cz ~ dt = 0, a'x +b'y ~ c'z + d' = 0, ali "+Vy + c"z +d"t = 0. Je forrne le produit xyzt. Je change successivement x en a, y en b, z en c, t en d, et observant la re'ole des signes, j'ai pour prernie're ligne ayzt - bxzt + cxyt - dxyz. Je change successivement x en a', y en 1/, z en c', t en d', et observant la rebgle des signes, j'ai pour seconde ligne (ab' - a'b) zt - (ac' - a'c) yt ~ (ad' - a'd) yz + (be' - Y'e) xt - (bd,' -b'd) xz + (cd' - c') xy. Je change successivement x en a", y en b", z en c", t en d", et observant 1a rebgle des signes j'ai pour troisiebme ligne [(ab' - a'b) c" - (adc - a'e) b" + (be' - b'c) a"]I - [(ab' - a'b) d" - (ad' - a'd) b" ~ (bd' - b'd) a"] z + [ (ac' - a'c) d" - (ad' - a'cl) c" + (ed' - c'd) a"] y - [(be' - b'e) d" - (bd' - b'd) c" + (ccl' - c'd) b" ] x. D'oii (198) je tire - [(be' -- b'c) d"1 - (bd' - b'cl)c" + (cd' - c'cl) b"] (ab' - a'b) e" - (ac' - a'c) 6" + (be' - b'e) a" ~ [ (ac' - a'c) d" - (ad' - a'd) c" + (cd' - c'd) a"] 6'(ab' - a'b) c" - (ac' - a'c)b" + (be' -Yea' - [ab' - a'b) d" - (ad' - a'd) b" + (bd' - b'd) a", (ab' - a'b) c" - (ac' - a'c) 6" + (be' - b'C ) a" 44 HISTORY OF THE THEORY OF DETERMINANTS Among the other examples are included (1) one in which the coefficients in the set of equations are given in figures; (2) one in which some of the coefficients are zero; (3) one showing the simplification possible when the value of only one unknown is wanted; (4) one showing the signification of the vanishing of one of the "lignes"; (5) one showing the signification of the absence of one of the unknowns from the last " ligne "; and (6) one or.two concerned with the allied problem of elimination. Bezout nowhere gives any reason for his rule; it is used throughout as a pure rule-of-thumb: its effectiveness being manifest, he leaves on the reader the full burden of its arbitrariness. The unreal product xyzt at the very outset must have been a sore puzzle to students, and none the less so because of the certainty which many of them must have felt that a real entity underlay it. To throw light upon the process, let us compare the above solution of a set of three linear equations with the following solution, which from one point of view may be looked upon as an improvement on the ordinary determinantal modes of solution as presented to modern readers. The set of equations being ax +by +cz +d = 0 x + b'y + c'z + d' = 0 a"x + b"y + c"z + d" = 0 we know that the numerators of the values of x, y, z, and the common denominator are b c d a c d a b d a b c -b' c' d' ca' c' d - a' ' + a' c' b" c" d", a"c" cd" 1, a" b" dc", a" b" c" They are therefore the coefficients of x, y, z, t in the determinant a b c d a' b' c' d' a" b" c" d",or A say. xyzt x y z t Thus the problem of solving the set of equations is transformed into finding the development of this determinant. In doing so DETERMINANTS IN GENERAL (BEZOUT, 1779) 45 let us use [xyz] to stand for the determinant of which x, y, z is the last row, and whose other rows are the two rows immediately above x, y, z in A: similarly let [zt] stand for the determinant of which z, t is the last row, and its other row the row c", d" immediately above z, t in A; and so on in all possible cases, including even [xyzt], which of course is A itself. Then clearly we have [xyzt] = a[yzt] - b [xzt] + c [xyt] - d [xyz] (1) Developing in the same way the four determinants here on the right side, we have as our next step [xyzt] = a(b'[zt] - c'[yt] + d'[yz]) -b(a'[zt] - c'[xt] + d[xz]) + c(a'[yt]- b'[xt] + da[xy]) - d(a'[yz] - b'[xz] + c'[xy]), = (ab' - 'b)[zt] - (ac' - 'c)[yt] + (ad - a'd)[yz] + (bc' - b'c)[xt] - (bd'- b'd)[xz] + (cd'- c'd)[xy]. Again, developing the six determinants [zt], [yt],.... in the same way, and rearranging the terms, we have finally [xyzt] = {(ab' - a'b)c" -(ac' -a'c )b" + (b' -b'c )a"}t -(b - a'b)d" - (ad'- a'd)b" + (bd'- b'd)a"}z + {(ac' -a'c)d"-(cLd'- a'd) " +(cd' -c'd)ca"}y -{(b' - b'c)d"-(bd'- b'd)c" + (cd'- c'd) b"}x. But the coefficients of x, y, z, t in [x y z t] were seen on starting to be the numerators and the common denominator of the values,of x, y, z in the given set of equations: hence -{(bc' - b')d" -(bd'- b'd)c"+ (c'- c'd)b" } x {(acb'- aCb) c" - (cC/ - c)(I/c + (be'- b'c)a"}' - = y --.=............... Now it is at once manifest that the successive developments here obtained of the determinant [xyzt] are letter by letter identical with the successive " lignes " obtained by Bezout from the unreal product xyzt; but that instead of having one arbitrary step succeeding another, as in the application of Bezout's rule, there is here a fluent reasonableness characterising the whole 46 HISTORY OF THE THEORY OF DETERMINANTS process.* As for the peculiarities requiring elucidation in the series of special examples above referred to, they are seen, when looked at in this light, to be but matters of course. Not only so, but it will be found that the translation of xy into [xy], &c., is an unfailing key to much that follows in Bezout in connection with the subject. For example, let us take the wide extension of the rule which is expounded later on in the treatise, in a section headed Conside'rationrs utiles pour abreger consid'erablemnet le calcul des coefficients qui servent a l'elimination. There are in all fifteen pages (pp. 208-223, ~~ 252-270) devoted to the subject. The contents of three paragraphs will give a sufficiently clear idea of the nature of the whole. The notation used is identical with that of Laplace, e.g., (ab') = ab'- ab, (b'c") = (ab'-ab)c" - (ab"-a"b)c' + (a'b"-ab')c, Two of the three selected paragraphs stand as follows:" (264.) Cette maniere de proceder au calcul des inconnues, en les grouppant, n'est pas applicable seulement a notre objet; elle peut en general etre appliquee dans toutes les equations du premier degre. * If the fact at the basis of the process were made use of nowadays, it would be advantageous, of course, in the first instance to simplify the determinant as much as possible. For example, the equations being (Bezout, p. 178) 2x +4y+5z=22 } 3x + 5y+2z =30, 5x +6y + 4z =43J we might proceed as follows:2 4 5 -22 0 2 11 -6 3 5 2-30 1 1 -3 -8 5 6 4-43 0 -3 -3 9 x y z t x y z t 0 0 9 0 0 0 1 0 1 0 -4 -5 1 0 0 -5 0 -1 -1 3 0 -1 0 3 x y z t x y z t =27{-t+Oz-3y-5x}; whence x=5, y=3, z=0. DETERMINANTS IN GENERAL (BE~ZOUT, 1779) 4 4 "Si l'on avoit, par exemple, les quatre 6quations suivantes ax + by + cz + dt + e = 0, a'x +b'y +c'z +cl't ~el' 0, alix + 1/y + c~z + d"t + e" = 0, a"'/x + billy + c"'zz + d"it + e"' = 0. En se rappellant que chacjue inconnue a pour valeur le co~fficient qu.'elle se trouve avoir dans la dernie're lignie, divisi' constaminent par celui que l'inconnue introduite aura dans cet ~e ligne, on verra bient6t qu.'on pent ri'duire le calcul 'a chercher le co~fiicient de 1'une quelconque des inconnues dans la dernie're ligne; parce que de la meme maniere qu'on en aura calcul1' un, Onl calculera de me'me tons, les autres: on. m~me, lorsqu'on en aura calcule' un, on pourra en de'duire tons les autres, lorsque les equations auront toute la ge'neralite' possible. Or pour avoir la valeur du co~fficient d'une des inconnues, dans la dernie're ligne, la question se re'duit 'a calculer la valeur du prodnit des antres incounnues. Mais pour ne pas se tromper sur les, signes, il faudra toujours ~ne pas perdre de vue, la place que cette inconnue est cense'e occuper dans le produit de toutes les inconnues. Ainsi, dans le cas present, an lieu de calculer ge'neralement la dernie're ligne pour avoir xyzlnt, je calcule senlement cette derniere ligne pour yziu: et pour l'avoir de la manie're la plus commode, je grouppe en cette manie're yz. lit, et je proce'de conime il suit, an calcul des lignes, observant que y est cens6 'a la seconde place. Premiere ligne. - bz. tu - yz. di Seconde ligne. ~ (be').ttu- bz.d'u+ b'z. du +yz. (de'), Troisirnie ligne. -(be').dc"u +(bc"). d'u -bz. (d'e") -(b'c"). du ~b'z. (de") -b"z. (de'), Quatribrne ligne. ~ (bc). (diel") -R(e". (cl'e") ~ (bc"'). (d'e"l) + (b'c"). (dc"'/) - (bc" '). (de"l) + (bVc"'). (de'); c'est le coiifficient de x dans la dernie're ligne. "1Pour avoir celni de it, je calculerois de me'me la valeur de xyl, en le grouppant ainsi, xy. zi, et je trouverois pour valeur du co~fflcient de u dans la dernii're ligne, la quantite' (ab').(c'd"') - (ab").('d"...) + (ab"'..).(c'") + (a'b").(cd"') - ('b')(c")+(a")(c) D'olIi je conclus +(be').(Wd"&") -(ab"). (cd"'")~+(bc"). (Cde")+(a'b')..(de"') -(a-b'"'). -1-) (ab""'). (cd') et ainsi de snite. (265.) Si j'avois les cinq 6quations suivanteso1x + by ~cez +di- + el +f = 0, aix +b'y +~c'z + d'r + e'1 ~1' = 0,7 al'x + b~ly + cliz + diir + e"t +f" = 0,) a"'/X + billy + c"'lz + d"'r + e"'it +f" =10, aivx + bi'vy + Civz + divr, + evt1 +f iv 0. 48 HISTORY OF THE THEORY OF DETERMINANTS Je calculerois, par exeruple, le coefficient de x dans la dernie're ligne, en edeculant yzr. tu, ou yz. r9tat, ou yz. r. ui. Si j'avois six equations dont les inconnues fussent x, y, z, r1, s et 1, je calculerois, par exemple, le coefficient de x, en calculant ou. jz. rs. u ou yz~s. tu, on yzr. sht, et ainsi de suite. The next paragraph deals with an illustrative example. The twelve equations-.Aa + AVa'+ Aa" =0 Ab + A'b + A//b"t Ac + A'c + A"c" + Ba + B'a + B"a" =0 + Bb + B'b' + B"" =0 ~ Be +' B'c + B"c" =0 + Bd + B'd' + B /Id" ~ Ca ~ C'a' ~ C"a" =0 ~ Cb +C'b'~C Gb" =0 ~ Cc~+C'c + C"c, =0 + Cd + C'd' + C"d"i +Da + D/a/ + Dait =0 + Db ~ D'b + DIb" =0 + De + D'c + D'c" =0 Ad + A'd' + A"d" + Da + D'a' + Da" =0 are given, and what is required is the result of the elimination (e'quctions de condition) of the twelve quantities-A, A' A" B, B' B" CC', C" D, D', D". This is fon- teL ntels equation being misprints for d's) to be(ab'c"). [(bc'cd")5 (ab'c")(ab'd"')] =0. The two paragraphs quoted (~ 264, 265) show that Be'zout could obtain with considerably increased ease and certitude any one of Laplace's expansions of numerator and denominator. What it accomplished in the illustrative example is virtnally, in modern symbolism, the reduction of a a' a" b b' b" c c' c" a a' a" b b' b". ~~.. c c c d cd' d a a' a" b b' b". c C' c" ci d' di" a a' a" b b'b" c CcI' d, di cd.c ald-, DETERMINANTS IN GENERAL (BiEZOUT, 1779) 4 49.to the form I Wbe" I. Ibc'd" //1 _I abe" 11. Iab'cd" Although this can be done nowadays with ease by means of Laplace's expansion-theorem in its modern garb, it may be safely affirmed that Lap lace himself, using his own process, would not have succeeded in making the reduction. Considerable importance thus attaches from more than one point of view to Be'zout's curious " rule." The only other section with which we are concerned bears the heading M~thode pour trouver des fontons d'un nombre quelconqpue de quarntit4', qui soient ze'ro par elle's-me"mes. In the second paragraph of the section the principle is explained as follows "(216) Concevons un nonmbre n d'e'quations du premier degre renfermant un nombre n +I1 d'inconnues, et sans aueun terme absolument connu. "Ilmaginons que ion augmente le nombre de ces equations de l'une d'entr'elles; alors il est clair que cc que nous appellons la dernie're ligne sera non seulement 1e'quation de condition ne'cessaire pour que cc nombre n + I d'equations ait lieu; mais encore. que cette e6quation de condition aura lieu; en sorte qu'elle sera une fonction des co~fficiens de ces equations, laquelle sera zero par elle-me'me. "Y oila' done un moyen tre's-simple pour trouver un nombre nm + 1 de fonetions d'un. nombre n + 1 de quantite's, lesquel les fonctions soient zero par elles-me'mes." For example, the pair of equations ax +by +cz =0' a'x + b'y + c'~; = Of is taken, the first equation is repeated, and for this set of three equations the equation de condition is found to be (ab'-a'b)c - (ac'-a'c)b ~ (bc'-b'c)a = 0. "Or 11 est clair quc la troisie'me equation n'exprimant rien dc diff6rcnt dc la premie're, cette dernie're quantite' doit eftre zero par elle-me'me: done si ont a ces deux suites de quantite's a, by c on pent e~tre assure' qu'on aura toujours (ab' - a'b) c - (ac' - a'c) b + (be' - Ye) a = 0. *Should be nm. M.D. D) 50 HISTORY OF THE THEORY OF DETERMINANTS "Et si au lieu de joindre la premiere equation, c'eft ete la seconde, nous aurions trouve de meme (ab' - a'b)c' - (ac' - a'c)b' + (bc' - b')a' = O." Similarly in regard to the quantities a, b, c, d a, b', c', d' a", b", c", (d" the identity [(ab'- a'b) c" -(a' - a'c)b" + (be' -b'c) a"] d -[(b' - a'b)d" - (ad' - 'd) b" + (bd - b'd)a"] c + [(ac' - a'c) d" - (ad' - a'd)c" +(d' - c'd)"] b -[(be' -b'c)d" - (bd' -b'd)c" + (cd' - c'd) b"] = 0 and two others are established, the general theorem of course being merely referred to as easily obtainable. Thus far there is in substance nothing new. What we have obtained is simply a different aspect of Vandermonde's theorem, that when two indices of either set are alike the function vanishes, or, as we should now say, a deterninavnt with two rows identical is equal to zero. Indeed the identities are used by Vandermonde in Bezout's form when solving a set of simultaneous equations. But what follows is important. By taking two of these identities (ab'- a'b)c -(ac'- a'c)b +-(bc'-b'c)a =0 (ab'- a'b) c'- (ac'- a'c) b' + (be'- b'c) a' = 0, multiplying both sides of the first by d', both sides of the second by d, and subtracting, there is obtained in regard to the quantities a, b, c, d a', b', c', d' the identity (ab'- a'b) (cl'-c'd) - (ac'- a'c)(bd'-b'd) + (b' - b'c)(ad'- a'd) = 0. Similarly by taking the three next identities before obtained, which for shortness we may write in modern notation, ab'c" d - ab'd" c + lac'd" b - bc'd" a = 0, ab'c" d' - ab'd" c' + ac'd" b'- bc'" a' = 0, ab'c"c d"- Iab'd" c"+ Iac'd"I" - bc'd" a" = 0, DETERMINANTS IN GENERAL (BEZOUT, 1779) 5 '51 there is deduced in regard to the quantities a, b, c, ci, e a' b', c', d' el the identities ab'c" de' -jab'd" ce' J+ ac'd".~be' - bc'd" ae' =0, W be".de" - ab'd".ce" + ac'd" Hbe" - bc'd" {ae" 0, ab'c" d'e" - acb'd" c.! ce" ac'd".1b'e" be'cd" * a'e" =0. Finally these last three identities are taken, both sides of the first multiplied by f',. both sides of the second by -fboth sides of the third by f, and then by addition there is obtained in regard to the quantities a, b, C, ci, e, f a", b", c", d1 e", f the identity I able" I- del'f" - lab'd" 1.Ice'f"I + Iac'd" j.Ibe'f"I - I bc'd"1"I aef'fI =0. The subject of what may appropriately be called vanishing aggregates of determinant-productts is not pursued farther, the concluding paragraph being cc(2'23) En voilav assez pour faire connoitre la route qu'on doit tenir, pour trouver ces sortes de the'ore'mes. On voit qu'il y a une infinite' d'autres combinaisons 'a faire, et qui donneront chacune de nouvelles fonctions, qui seront zero par elles-mrames: mais cela est facile 'a trouver actuellement." % -* It is very curious to observe, in passing, that although Be'zout does not obtain all his vanishing aggregates directly by means of the principle which he so carefully states at the commencement, nevertheless every one of them can be so obtained. He does not extend the principle beyond the case where only one of the original equations is repeated. If, howvever, we take the equations ax ~ by + cz ~ dw = 0, a'x +b'y +c'z +d'w =0, repeat both of them. so as to have a set of 'four, andJ then proceed by the mt'thode, pour abrYger to find the dquation de condition, we obtain I WI.lIcd'I - I ae'l. Ibd' I + I1ad'HbcI'W + IbcI'HIad'I - IdWI. I ac' + Icd'HabI'I = 0, i.e. 2{ Iab'I.cd'I - I ae' I.Ibd'I + I ad'1.lIbWI } = 0. This is the identity near the foot of p. 51, and others are readily seen to be obtainable in the same way. 52 HISTORY OF THE THEORY OF DETERMINANTS Our second list of Bezout's contributions thus is: (1) An unexplained artificial process for finding the numerators and denominators of fractions which express the values of the unknowns in a set of linear equations, or for finding the resultant of the elimination of n quantities from n +1 linear equations,a process especially useful when the coefficients have particular values. (I.:3 + III. 4 + v. 2) (2) An improved mode of finding Laplace's expansions, especially (but not exclusively) useful when the coefficients have particular values. (xiv. 3) (3) A proof of Vandermonde's theorem regarding the effect of the equality of two indices belonging to the same set. (xii. 3) (4) A series of identities regarding vanishing aggregates of products. (xxiii. 2) CHAPTER III. DETERMINANTS IN GENERAL, FROM THE YEAR 1784 TO 1812. THE writers of this period are eight in number, viz., Hindenburg, Rothe, Gauss, Monge, Hirsch, Binet, Prasse, Wronski. Of these the first two and Prasse, belonging as they did to the so-called Combinatorial School, were not independent of one another; Hirsch was a mere expositor; and the others were authors who had not specially studied the subject, but who had attained results in it in the course of other investigations. HINDENBURG, C. F. (1784). [Specimen analyticvmn de lineis curvis secundi ordinis, in delucidationem An alyseos Finitorum Kaestneriance. Auctore Christiano Friderico Riudigero. Cam praefatione Caroli Friderici Hindenburgii, professoris Lipsiensis. (pp. xiv-xlviii.) xlviii+74 pp. Lipsice.] One of the problems dealt with by Riidiger being the finding of the equation of the conic passing through five given points (" coefficientium determinatio Traiectoriae secundi ordinis per data quinque puncta"), Hindenburg, in his preface, takes occasion to show how the generalised problem for -n(n+3) points has been treated, pointing out that it is, of course, immediately dependent on the solution of a set of simultaneous linear equations. He directs attention to the labours of Cramer and Bezout, specially lauding the method of the latter given in the treatise of 1779. Then he says-" Haec de Opere Bezoldino in universam, quod plurimis adhuc Lectoribus nostris ignotum 54 HISTORY OF THE THEORY OF DETERMINANTS erit, dicta 8uffl ci ant. Nune Regulctm ipsam propona~m." The seventeen pages which follow, contain a tolerably close Latin translation of the Re'gle generale pour calculer. -- and the Me'tode pour trouver...., pp. 172-187, ~~ 198-223, which have been expounded above. Cramer's rule is next given, the second mode of putting it being in words, and the first as follows: " Sint plures Incognita-, z, y, xi W, &c. totidemque Aequationes, simiplices indeterminat~eAl = Z1z + Y'y + XIx + W'w + &C. A2 = Z2z + Y2y + X2x ~ W2w + &C. A3 = ZIZ + Y~y + X3IX + W3W + &C. A4 = Z4Z + Y4y + X4X + W4w + &C. &C. &C. &C. &C. &C. &C. Erit,.,.. positis terminorum signis, ut praccipitur in fine Tabule, pag. seq. AY'XWVUT Permut (1, 2, 3, 4, 5, 6, 7. )vi.3 ZPermut (1, 2, 3, 4, 5, 6, 7. )(,.3 Z YXWVU T.. The similar expressions for 'y, x, w, v, m, t are given, and then the " regula signorum." After an illustrative example, the question of the sequence of the signs is taken up. "Quod si itaque + sg(l, 2, 3, n,i) denotet signorurn vicissitudines, quibus hic afficiuntur Permutationum a nurneris 1,7 2, 3,... n singulhe species, et - sg(l, 2, 3,... n) signa contraria vel opposita: appatet fore sg (1, 2) = +sg (l). -Sg(1) sg (l, 2, 3) = + sq(l, 2) - sg (l, 2) + sg (I, 2) sg (l, 2, 3, 4) = ~ sg(1, 2, 3) -sg (l, 2, 3) +sg (l, 2, 3) -sg(l, 2, 3) unde, quia Sg( I) es +, facile eruitur sg(l, 2) esse + - sg(1, 2,3).. sg(l, 2,3, 4).. -+ -. and it is pointed out that the first sign is always +,and the last + or - according as the number 1 +2 +3 +..+ (n-I) is even or odd. DETERMINANTS IN GENERAL (HINDENBURG, 1784) 55 Bearing in mind that Hindenburg wrote his permutations in a definite order, this remark regarding the sequence of signs entitles us to view him as the author of a combined rule of term-formation and rule of signs, which may be formulated as follows:Write the permutations of 1, 2, 3,..., n in ascending order of magnitude as if they were numbers; make the first sign +, the second -, the next pair contrary in sign to the first pair, the third pair contrary in sign to the second pair, the next six (1. 2.3) contrary in sign to the first six, the third six contrary in sign to the second six, the fourth six contrary in sign to the third six, the next twenty-four (1.2.3.4) contrary in sign to the first twenty-four, and so on. (I. 4 + III. 5) ROTHE, H. A. (1800). [Ueber Permutationen, in Beziehung auf die Stellen ihrer Elemente. Anwendung der daraus abgeleiteten Satze auf das Eliminationsproblem. Sammlung combinatorischancalytischer Abhandlungen, herausg. v. C. F. Hindenburg, ii. pp. 263-305.] Rothe was a follower of Hindenburg, knew Hindenburg's preface to Ridiger's Specimen Analyticum, and was familiar with what had been done by Cramer and Bezout (see his words at p. 305). His memoir is very explicit and formal, proposition following definition, and corollary following proposition, in the most methodical manner. The idea which is made the basis of it, that of place-index (" Stellenexponent"), is an ill-advised and purposeless modification of Cramer's idea of a "derangement." The definition is as follows:-In any permutation of the first n integers, the pilace-index of any integer is got by counting the integer itself and all the elements after it which are less than it. For example, in the permutation 6, 4, 3, 9, 8, 10, 1, 7, 2, 5,of the first ten integers, the place-index of 9 is 6, and that of 7 is 3. The counting of the integer itself makes the placeindex always one more than the number of " derangements" 56 HISTORY OF THE THEORY OF DETERMINANTS connected with the integer. This necessitates the introduction of a corresponding modification of Cramer's " rule of signs," viz. "3. Willkiihrlicher Satz. Jede Permutation der Elemente 1, 2, 3,.., r, werde mit dem Zeichen + versehen, wenn entweder gar keine, oder eine gerade Menge gerader Zahlen, unter ihren Stellenexponenten vorkommt; mit dem Zeichen - hingegen, wenn die Menge der geraden Zahlen, unter den Stellenexponenten ungerade ist." (III. 6) It is difficult to suggest any justification for the changes here introduced. The author himself refers to none. Indeed, in the very next paragraph he points out that to ascertain whether there be an even number of even integers among the placeindices is the same as to diminish each of the place-indices by 1, and ascertain whether there be an even number of odd integers, that is, whether the sum of the odd integers be even. He then concludes"Man kann also auch die Regel so ausdriicken: Jede Permutation bekommt das Zeichen + wenn die Summe der um 1 verminderten Stellenexponenten gerade, - hingegen, wenn sie ungerade ist." This is simply Cramer's rule, and it is the only rule of signs employed henceforward in the memoir, the expression "die Summe der um 1 verminderten Stellenexponenten," occurring over and over again as a periphrasis for "the number of derangements." The next four pages are occupied with a very lengthy but thorough investigation of the theorem that two permutations differ in sign if they be so related that either is got from the other by the interchange of two of the elements of the latter. Strictly speaking, however, the proposition proved is something more definite than this, viz.If in c permutation of the integers 1.2,... r there be d integers intermediate in place and value between any two, A and B, of the integers, the interchanging of the said two would increase or diminish the number of inversions of order by 2d+1. (III. 7) The proof consists in finding the sum of the place-indices for the given permutation in terms of d as just defined, c the number of elements less than both A and B and situated between them, f the number of such elements situated to the right of B, and DETERMINANTS IN GENERAL (ROTHE, 1800) 57 e the number of elements between A and B in value and situated to the right of B; then finding in like manner the sum of the place-indices for the new permutation; and finally comparing the two sums. The concluding sentence is as follows:"Denn da...., so ist die Summe der Stellenexponenten der zweyten Permutation um d + e + 1 - e + d oder um 2d + 1 grosser als bey der ersten Permutation; folglich gilt das auch bey der Summe der um 1 verminderten Stellenexponenten, da bey beyden Permutationen r einerley ist. Also ist die eine Summe gerade, die andere ungerade, folglich haben nach (4) beyde Permutationen verschiedene Zeichen." As immediate deductions from this, it is pointed out that The sign of any one permutation may be determined when the sign of any other is known, by counting the number of interchanges necessary to transform the one permutation into the other; (III. 8) and that If one element of a permutation be made to take up a new place, by being, as it were, passed over m other elements, the sign of the new permutation is the same as, or different from, that of the original according as m is even or odd. (III. 9) A third corollary is given, but it is, strictly speaking, a self-evident corollary to the second corollary, and is quite unimportant. Rothe's next theorem isThe permutations of 1, 2, 3,...., n being arranged after the manner in which numbers are arranged in ascending order of magnitude, any two consecutive permutations will have the same sign, if the first place in which they differ be the (4n+3)th or (4n +4)th from the end, and will be of opposite sign if the said place be the (4n+1)th or (4n+2)th from the end. (III. 10) Thus if the permutations of 1, 2, 3,..., 10 be taken, and arranged as specified, two which will occur consecutively are 8,4, 9, 3, 10, 7, 6, 5, 2, 1 8,4,9,5, 1,2,3,6,7,10; and as the first place in which these differ is the 7th from the end, it is affirmed that the signs preceding them must be alike. The mode of proving the theorem will be readily understood by 58 HISTORY OF THE THEORY OF DETERMINANTS seeing it applied to this illustrative example. Taking the permutation 8, 4, 9, 3, 10, 7, 6, 5, 2, 1, and interchanging 3 and 5 we have the permutation 8, 4, 9, 5, 10, 7, 6, 3, 2, 1, and thence by cyclical changes the permutation 8, 4, 9, 5, 1, 2, 3, 6, 7, 10, the number of alterations of sign thus being 1+(5+4+3+2+1) i.e. 1+1(5x6), -an even number. Annexed to the theorem is the following corollary, which is not essentially different from Hindenburg's proposition regarding the sequence of signs,If the permutations of 1, 2, 3,..., n-1 be arranged after the manner in which numbers are arranged in ascending order of magnitude, and also in like manner the permutations of 1, 2, 3,...., n-l, n, then those permutations of the latter arranged set which begin with r, say, have in order the same signs as the permutations of the former arranged set, or different signs, according as r is odd or even. (ITI. 11) For example, arranging the permutations of 1, 2, 3, each with its proper sign in front, we have +1, 2, 3 --1, 3, 2 -2, 1,3 (A) +2, 3, 1 +3, 1, 2 -3, 2, 1; then arranging those permutations of 1, 2, 3, 4 which begin with 3 say, each with its proper sign, we have +3, 1, 2, 4 -3, 1, 4, 2 -3, 2, 1,4 (B) +3, 2, 4, 1 +3, 4, 1, 2 -3, 4, 2, 1; DETERMINANTS IN GENERAL (ROTHE, 1800) 59 - and the two series of signs are seen to be identical, 3 being an odd number. Viewing this quite independently of the theorem to which it is annexed, it is evident that a change of sign at any point in the series (A) implies a change at the corresponding point in the other series, and consequently attention need only be paid to the first sign of (B) as compared with the first sign of (A). Now the first sign of (A) must necessarily be always plus, there being no inversions; and the first sign of (B) depends on the, changes necessary for the transformation of the natural order 1, 2, 3, 4, into 3, 1, 2, 4. The truth of the corollary is thus apparent. A second corollary is given, but it is of still less consequence, the difference between it and the first being that in the arranged set (B) the place whose occupant remains unchanged may be any one of the n places. (III. 12) The next few paragraphs concern the subject of "conjugate permutations" (verwandte Permutationen),-apparently a fresh,conception. The definition isTwuo permutations of the numbers 1, 2, 3,..., n are called CONJUGATE when each number and the number of the place which it occupies in the one permutation are interchanged in the case of the other permutation. (xxIV.) For example, the permutations 3, 8,5,10,9,4,6,1,7,2 (A) 8,10,1, 6,3,7,9,2,5,4 (B) are conjugate, because 3 is in the 1st place of (A) and 1 is in the 3rd place of (B), 8 is in the 2nd place of (A), and 2 is in the 8th place of B, and so on in every case. The first theorem obtained isConjugate permutations have the same sign. (III. 13) This is proved in a curious and interesting way, a special 'conjugate pair being considered, viz., the pair just given as an example. To commence with, a square divided into 10 x 10 'equal squares is drawn, the vertical rows of small squares being numbered 1, 2, 3, &c. from left to right, and the horizontal rows 1, 2, 3, &c. from the top downwards. The permutation 3, 8, 5, 10, 9, 4, 6, 1, 7, 2 60 HISTORY OF THE THEORY OF DETERMINANTS is then represented by putting a dot in each of the horizontal rows, in the first under 3, in the second under 8, and so on; so that if the rows be taken in order, and the number above each dot read, the given permutation is obtained. For the representation of the conjugate permutation nothing further is necessary: we obtain it at once if we only turn the paper round clockwise until the vertical rows are horizontal, and read off in order the numbers above the dots. In the next place the number of " derangements" belonging to the permutation 3, 8, 5,.... is indicated by inserting a cross in every small square which is to the left of one dot and above another; thus the two crosses in the first horizontal row correspond to the two "derangements" 1 2 3 4 5 6 7 8 9 10 32, 31; the six crosses in the second 1 x x_ horizontal row to the six " derange- 2x x x x x x_ 3 -. - ments" 85, 84, 86, 81, 87, 82; and 4 x- x X-x x - so on. Then it is observed that if --. 5 xx x xx we turn the paper and try to indi- 6 X x cate the "derangements" of the 7 ix _conjugate permutation by inserting 8 _ a cross in every small square which 9 x is to the right of one dot and above 10 another, we obtain exactly the same crosses as before. The signs of the two permutations must thus be alike. Immediately following this, the 24 permutations of 1, 2, 3, 4 are given in a column, each one having opposite it, in a parallel column, its conjugate permutation. The existence of selfconjugate permutations, e.g., the permutation 3, 4, 1 2, is thus brought to notice, and the substance of the following theorem in regard to them is given:If Un be the number of self-conjutgate permutations of the first n integers, then Un = Un- +(n-l)Un-2 (XXV.) where U= 1 and U = 2. This, however, is the only one of his results which Rothe does not attempt to prove. DETERMINANTS IN GENERAL (ROTHE, 1800) 61 In the second part of the memoir, which contains the application of the theorems of the first part to the solution of a set of linear equations, there is not so much that is noteworthy. Methods previously known are followed, the new features being formality and rigour of demonstration. The coefficients of the equations being 11, 12, 13,...., 1r 21, 22, 23..., 2r rl, r2, r3,..., rr it is noted, as Vandermonde had remarked, that the common denominator of the values of the unknown may be got in two ways, viz., by permuting either all the second integers of the couples, 11, 22, 33,....., rr, or all the first integers: but this is supplemented by a proof, that if any term be taken, e.g., 16.24.33.47.51.68.79.82.95 with the couples so arranged that the first integers are in ascending order, and the sign be determined from the number of inversions in the series of second integers, then the sign obtained vill be the same as would be got by arranging the couples so as to have the second integers in ascending order, and determining the sign from the inversions in the series of first integers. The proof rests entirely on the previous theorem, that conjugate permutations have the same sign; indeed the new proposition is little else than another form of this theorem. (III. 14) The desirability of an appropriate notation for the cofactor, which any one of the coefficients has in the common denominator, is recognised,* and the want supplied by prefixing f to the coefficient in question; for example, the cofactor of 32 is denoted by f32. It is thus at once seen that the denominator itself is equal to ln..fln +- 2n.f2n +... + rn.frn, or nl.fnl + n2.fn2 +....+ nr. fnnr. (vi. 2) Also by this means one of Bezout's (or Vandermonde's) general theorems becomes easily expressible in symbols, viz., In. flm + 2n.f2m +.... + rn. frm = 0, (xI. 4) * Lagrange's use of a corresponding letter from a different alphabet must not be forgotten. 62 HISTORY OF THE THEORY OF DETERMINANTS the proof of which is given as follows. In all the terms of f l;m, every one of the integers except one occurs as the first integer of a couple, and every one of the integers except m occurs as the second integer of a couple: consequently, in every term of I,.fl m the first places of the couples are occupied by the integers from 1 to r inclusive, while in the second places m is still the only integer awanting and n occurs twice. Suppose then all the terms of In.flm + 2n. f2m +~.... + mn.frin so written that the first integers of the couples are in ascending order of magnitude, and let us attend to a single term......~pn....... qn..... in which the two couples, having n for second integer, are the pth and qth. If we inquire from which of the expressions In.flm, 2n.f2m,... this term comes, we see that it is a term of both pn. fpm and qn. qm, and must, therefore, occur twice. Further, we see that inpn.fqm it has the sign of the term.. pm...... qn...... of the common denominator, and that in qn.fpm, it has the sign of the term..... pn'.......q*...... of the common denominator. But these two terms of the common denominator have different signs: consequently ln. f lm + 2n. f2m +... + rn. frm consists of pairs of equal terms with unlike signs, and thus vanishes identically. (xII. 4) These preparations having been attended to, the set of r equations with r unknowns is solved by Laplace's method; and a verification made after the manner of Vandermonde. It is also pointed out, that if the solution of a set of equations, say the four ax + bx2 + C3 + dx4 = s ex1 +Pfx2~ + g3 + hx4 = 2 ix~ + kx,2 + 1X3 + zX4 = 83 nft + O2 + +p + qx4 = 84 -be DETERMINANTS IN GENERAL (GAUSS, 1801).x =As,+ Bs2~ Cs~ +Ds4 x2= Es,+ Fs2+ Gs93+ Hs4 X3= Is1+Ks.2 +Ls3+Ms4 X4- Nsl ~082~+Ps3~+Qs4j 63 then the solution of the set aYl~ ey2+ 73~flY4-V I' by,~ fY2+ 7'Y/3 + 0y4 V2 cy1 +gY2 + 1y3 +PY4=V3 dy + hy2 +my3 +qY4=v4j, which has the same coefficients differently dispose d, will be Y1=Avl+Ev2+ Iv3+Nv4 /2 = Bvl~ FV2~ Kv3~ Or4 Y= CV, + GV2 ~ Lv + Pv4f y4Dv ~12Mv+v;(xxvi.) and hence, that the solution of a set having the special forni ax, + bce2 ~+cx, + dX4 = s1j bx1 + eX2 ~ fx3 +g9x4 =892 cx1 + fc2 + hx, + icc4 =53, dcx1 + gcc2 ~ ice3 + jc484) will itself take the form, viz. As1 ~ BS2 + Cs3 +Ds4 =x Bs, ~ Es2 + Fs3 + Gs4=2 Cs, + F82 + Hs3+ Is4= X Ds1~Gs2 + 13 + Js4 =e4J (xxvi. 2) GAUSS (1801). [Di~squisitiones Arithmeticce. Auctore D. Carolo Friderico Gauss. 167 pp. Lips. Werkee, I. (1863) Gbttingen.] The connection of Gauss with our theory was very similar to that of Lagrange, and:doubtless was due to the fact that Lagrange had preceded him. The fifth chapter of his famous work, which is the only chapter we are concerned with, bears the title "De formis amquationibusque indetermirnatis secvuldi, grctchs," and its subject may ~be described in exactly -the same words as Lagurangye used in regard to his memoir Recherches 64 HISTORY OF THE THEORY OF DETERMINANTS d'Arithmdtique (1773: see above), viz. "les nombres qui peuvent etre representes par la formule Bt2 + Ctu +Du2." Gauss writes his form of the second degree thusaxx +2bxy + cyy; and for shortness speaks of it as the form (a, b, c). The function of the coefficients a, b, c, which was found by Lagrange to be of notable importance in the discussion of the form, Gauss calls the " determirnant of the form," the exact words of his definition being "Numerum bb-ac, a cuius indole proprietates forme (a, b, c) imprimis pendere in sequentibus docebimus, determinantem huius formms uocabimus." (xv. 2) Here then we have the first use of the term which with an extended signification has in our day come to be so familiar. It must be carefully noted that the more general functions, to which the name came afterwards to be given, also repeatedly occur in the course of Gauss' work, e.g., the function aS-,/y in his statement of Lagrange's theorem (xxII.) b'b'- a'c' = (bb - ac)(aS - /3y)2. But such functions are not spoken of as belonging to the same category as bb-ac. In fact the new term introduced by Gauss was not "determinant" but "determinant of a form," being thus perfectly identical in meaning and usage with the modern term "discriminant." Notwithstanding the title of the chapter Gauss did not confine himself to forms of two variables. A digression is made for the purpose of considering the ternary quadratic form ("formam ternariam secundi gradus"), cxx + a'x'x' + c"x'"x" + 2bx'x" + 2b'xx" + 2b"xx', or as he shortly denotes it (a, ab', \b, b\ bV. In the matter of nomenclature the following paragraph of this digression is interesting,"Ponendo bb - a'a" = A, b'b' - aa" = A', b"b" - ac' = A", ab - b'b"= B, a'b' - bb" = B', a"b" - bb' = B", DETERMINANTS IN GENERAL (GAUSS,. 1801) 6 65 oritur alia forma quam. formic adjudamdiceus.Hinc rursus inuenitur, (X1. denotando breuitatis caussa numerum abb + a'b'b' + a"b"b" - aa'a" - 2bb'b" per D, BB - A'A" = aD, B'B' - AA" = a'D, B"/B" - AA' = aD AB - BIB" = bD, A'B' - BB" = b'D, A"'B" - BB' =b/D unde patet, forma-, F adjunctam. esse formam. (aD, a'D, alID kbD, b'D, b"D Numerum. D a cuuns indole proprietates formie ternarire f imprimis pendent, delerminantern huius formaic uocabimus; (xv. 2) hoc modo determunans formic F sit =DD, sive a-equalis quadrato determiviantis formic f, cuii adjuncta est." In this there is no advance so far as the theory of modern determinants is concerned, the identities given being those nnmbered (xx) and (xxi) under Lagrange. On the same page, however, an extension is given of Lagrange's theorem (xxii), regarding the determinant of the new form obtained by effecting a linear substitution on a given form. Gauss' words in regard to this are"Si forma aliqua tennaria f determinantis D, cuius indeterminatic sunt x, x, ~x" (puta prima = x, &c.) in formam. ternariam g determinantis E, cuius indeterminatic, sunt y, y', y", transmnutatur per substitutionem. talern x a + j3y' ~ n",I = a'y ~/'y' +y'y", ubi niouem coefficientes a, /3, &c. omnes supponuntur esse numeri integri, breuitatis caussa neglectis indeterminatis sirmpliciter dicenmus, f transire in g per substitutionem. (5) a,/3 atquef imnplicare ipsam. g, siue sub f contenlaim esse. Ex tali itaque suppositione sponte sequuntur sex equationes pro sex co~fficientibtis IM.D. E 66 HISTORY OF THE THEORY OF DETERMINANTS in g, quas apponere non erit necessarium: hinc autem per calculum facilem sequentes conclusiones euoluuntur: "I. Designato breuitatis caussa numero a/3py" + Py/ya + ya'/3" y/3'a" - ay/3 - 3a'y" per k inuenitur post debitas reductiones E=kkD,..... (. 2) When freed from its connection with ternary quadratic forms the theorem in determinants here involved is If Ao = a02 ao2 + ala +a2 aa2 + 2bo0la2 + 2bl1 a%2 + 2b2a0a1, A1 = aoPo2 + acpl2 + a2sp2 + 2boj132 + 2bo1p28 + 2b2fopl, A2 = y2 + + a2 + ay + a 2bo072 + 2blYoY2 + 2b27071, Bo = ao03o7o + alPly + a2f272 + bo (!172 + 27Y1) + bl (0o72 + 27o) + b2 (P3oY1 + /1Yo), B1 = ao7o ao aal1yl + a272 a2 + bo (yia2 + 7a) +l (Ta + ) + b (y2(7o al + ylao), B2 = ao/o + a, all1 + a2a2p32+ bo (alP2 + a2/3l) + bl (aoP32+ a20/3) + b2(a0oP + alPO), then AoBo2 + A1B12 + A,B,2 AA1A2- 2BoBB2 = (a0obo + ab1a2 cb,2 - aoaa - 2bo0lb2) X (a/Oy72 + /07la2 + YOa/l2 - Y031a2 - aoYi 2 - /o'al'/)2. As thus viewed it is an instance of the multiplication-theorem, the product of three determinants (in the modern sense) being expressed as a single determinant. The multiplication-theorem is also not very distantly connected with the following other statement of Gauss:"Si forma ternaria f formam ternariam f' implicat atque haec formam f": implicabit etiam f ipsam f". Facillime enim perspicietur, si transeat f in f' per substitutionem f' in f" per substitutionem a, P3,, 8, C, a', Y',', 8 E, ~' a", /3" y,,, f transmutatum iri per substitutionem a8 +/' +y' 8", ac +/ 3E' + E", a' +/3' +fy"' a'8 + 3 + y'", a'E + P/3" ' +3 yE ', a 4- /3'' + y'" a"8 + /3"/ +^ y"", a' E + PIIEIC+y a +13 +y 7"". (XVII. 3) DETERMINANTS IN GENERAL (MONGE, 1809) 67 MONGE (1809). [Essai d'application de 1'analyse a qiuelques parties de la geometrie e&ementaire. Jotrrn. de I'E~c. Po1yt., viii. pp. 107-109.] Lagrange, as we have already seen, was led to certain identities regarding the expression ey'z" + yz'x" zx'" - xz'y" - yx - zy'x" in the course of investigations on the subject of triangular pyramids. The position of Monge is that of Lagrange reversed. From the theory of equations he derives identities connecting such expressions, and translates them into geometrical theorems. The simpler of these identities, as being already chronicled, we pass over. At p. 107 he takes the three equations azm + t x, + c1y + diz + e1 =0 a2U + b2X + c2y + d2z + e2 = 0 a3u + b3x ~ c3y ~ d3z + e3 = 0, and eliminating every pair of the letters m, x, y, z, obtains the six equations Ott+ ax+ P=0 (1) x+/3y+Q=0 (2) 8y+ yxz+M=0 (3) az~ uLT+N=0 (4) y/- ay~+ S=O (5) /3z- c~ R=0 (6); the ten letters t, $, ->, 8, M, iN, P, Q, R, S being used to stand for the lengthy expressions which we nowadays denote by b 1C~d3, Ic2d3 a, I albd, I ab2C3, I ab2e3, blczeA, cld2e3 aide oAe3, bcd/e3. Then, taking triads of these six equations, e.g., the triads (1), (2), (5) he derives the identities -68 HISTORY OF THE THEORY OF DETERMINANTS aQ+/3S - y/P aP+ aR-3N =0 -YN+ 63S+aM= -/3M+yR+ 6Q= or -b1C2d3l. lad2e3J + IalC2d31.1 lbld2e3l - Ialb2d3. Jc1d2e.3 = 0' I alb~~~~~c.,I. I c~~~~~~~d~~e~~l + I b~~~~lCA I 3 a1C2e'31 - IalCAd I blC2e3l X II lalb dcl3 b cde3 +lbd a (xii.3 -b2d. bC2e31 ~ a~b2C81. Jb1d2e31 + bC2d31.la~b e3 2 0 - lalC2d3. lab2e3l + ja~b~d3 lalC2e.1 - Iab2C31. lad2e'3 = 0) which in their turn, he says, by processes of elimination, may be the source of many others. For example, each of the four being linear and homogeneous in a, /3, y, (5, these letters may all be eliminated with the result RS+~QN-PM =0, or Iaic~e3.b~d2e3 - ald2e3.b1C2e3 - cld2e3.a~b2e3 = 0 Also, eliminating P f rom the, first and second, S from the first and third, Q from the first and fourth, and so on, we have - 0yN +8aQ + /38S +ayR =0, a/3M + y8P- 3yN- 5'aQ =0, a/3M1-73P+ 136S-ayR=0, -a~c2d3 1. Iab2d3l. bIC2'31 - I a1b2c3..l' b cd3 1. Iaicle3 + IaC2d3 a1b2C3 1. Ibld2e3l + Ib1C2d3 1. Ialb2d, 1. Ia1C2e, I &C. &C. (xxviii.) Mouge does not pursue the subject further. His method, however, is seen to be quite general; and w e can readily believe that he possessed numerous other identities of the same kind. This is borne out by a statement in Biiiet's important memoir of 1812. Binet, who was familiar with what had been done by Yandermonde, Laplace, and Gauss, says (p. 286):- M. Monge m'a communiqu6, depuis la lecture de ce me'moire, d'autres the5'ore'mes tre's-remarquables sur ces re'sultantes; mais ils ne sont pas du genre de ceux que nous nous proposons de donner ici." DETERMINANTS IN GENERAL (HIRSCH, 1809). 6 69 HIRSCH (1809). [Sammiung von Aufgaben aus der algebraischen Gleichungen, von Meier Hirsch (pp. 103-107). xvi+ 360 pp. Berlin.] The 4th Chapter Von der Elimination u. 8. w., contains five pages on the subject of the solution of simultaneous linear equations. These embrace nothing more noteworthy than a statement, without proof, of Cramer's rule, separated into three parts (iv., iii. 2, v.), and carefully worded. BINET (May 1811). [Me'moire sur la the'orie des axes con] ugu~s et des momens d'inertie des corps. Journ. de l'fi'cole Polytechnique, ix. (pp. 41-67), pp. 45, 46.]' In this well-known memoir, in which the conception of the moment of inertia of a body with respect to a plane was first made known, there repeatedly occur expressions, which at the present day would appear in the notation of determinants. There is only one paragraph, however, containing anything new in regard to these functions. It stands as follows: "1Le moment d'inertie minimum pris par rapport au plan (C) a pour valeur:~,?k2 f2 X ABC - AF 2 - BE2 - CD2~+2DEF g 2(BC - F2) +h h(AC - E2) + i2(AB -D2 + 2gh(EF -CD) +2gi(DF -BE) +2hi(DE -AF Si, dans le nume'rateur, ABC - AF2 - BE2- CD' +2DEF on remplace A, B, C, &c. par `5rnX2, J:My2, &c. que ces lettres representent, on a 2w:My2EpnZ2 - EinX2 (jMyZ)2 -~y ~x) - YrnZ2 (linxy) 2 + 2 1mxY~mxz`4Emyz, et l'on peut s'assurer que cette expression est identique 'a 2Minn'm"(xy'z" ~ yz'cY' + zx',y" - xz'y" - yX'z"1 - Zy'X"1)2; *Ani abstract of this is given in the Nouv. Bull. des Sciences par ila Sociltd Philomaetique, ii. pp. 312-316. 70 HISTORY OF THE THEORY OF DETERMINANTS par une transformation analogue, on peut ramener la quantit6 g2(BC-F2) +lh2 (AC-E2) +2 (AB-D2) + 2gh(EF - CD) + 2gi(DF - BE) + 2hi(DE - AF), k celle-ci 'mm [g(yz' - zy') + h(zx' - xz') + i(xy' - yX')]2. Now the numerator referred to would at the present day be written A D E D B F E F C, and since mxa2, &c. stand for mx2 + m1Zl2 + mx2 +..., &., the first identity may be put in the form mx2 + mlx2 + m22 x+.. mxy + 11X]y1 - + m.X2y2 +. mxz + mlxll + m2XZ2 +.. mxy + mlxly1 + m2x2y2 +.. my2 + mlyl2 + m2y22 +.. rmyz + mlyll'+ m2y2Z2 +. mxz + m1xSxZ + m2x.z2 +.. yz + m1ylzl + my7z2y +-.. mz2 + m?112 + mz22 +.. X1 X2 x 1 X32 m= m1n Y Yi Y2 + ral0 3 ny Y Y Y3 + (XVIII. 2) Z Z1 Z2 Z Z1 Z3 where, Y2,... are for convenience written instead of ', y",... It will been seen that this is an important extension of a theorem of Lagrange, the latter theorem being the very special case of the present obtained by putting m = n-r = m21, and m=m4=... =0,-a fact which is brought still more clearly into evidence if, instead of the left-hand member of the identity, we write the modern contraction for it, viz. x m xi m n22 VsI 33. * x xi x2 x3... my mY1c1 m2Y2 c n3my... xy Y1 y2 y3 qnz mrz 1 m2z2 m2 mz3 *. Z Z1 z2 Z3 ' * Again the denominator y2(BC-F2) +/2 AC-E2) + i2 (AB-D2) + 2gh(EF- CD) + 2i (DF - BE) + 2hi(DE - AF) being in modern notation.,q h i y A D E h D B F i E F C DETERMINANTS IN GENERAL (BINET, 1811) 71 the second identity may be written g h i g mx2 + m^xz2 +.. mxy+mxcy + -... mxz+m-xlz... h inxy + mlxy+.1.. y2 + ml2 +. y.. myz+ lylzl +. i rxz +mxz +... myz+myz+n1-1.... mz2 +m12 +.. g x x 2 g x x2 2 g x x2 = mMn1 h y yl +A-mm2 h Y Y2 '+-m9n2 h Y y2 +.. (xxIX.) i z z i z z2 i z2 z2 This is also an important theorem, and is not so much an extension of previous work as a breaking of fresh ground. BINET (November 1811). [Sur quelques formules d'algebre, et sur leur application a des expressions qui ont rapport aux axes conjugues des corps. Nouv. Bull. des Sciences pac) la Societe Philomatique, ii. pp. 389-392.] In this paper Binet returns to the consideration of the first of the two identities which have just been referred to, writing it now in the form (xy'z" - xz'y" + yz'x" - yx'z" + zxy" - zy'X1) = Ex2Ey2z2 _ x2 (Jyz)2 - jy2 (7xz)2 - Ez2(xy)2+ 2 xyYxzyz. He puts it in the same category as the identity (y' - zy')2 = -y2z2 - (yz)2, which he speaks of as being then known. Further, he says "Ces deux formules sont du meme genre que la suivante 2 ( ux 'yZX/-txX'zXy//X+Zty//z'/-tyI+/x//z/+- z +x'y/-zyt/zX/+gxxy'u/z///-xy /Zu/'+ 2 - + xz'y"t'" - XZ'uyt" + xu/z'"y/"' -xu 'y z"' +yz'u?"x"' -yz'x"ui"' +yu'x"z'/ -yiuz"x"' x t+ yxz u'" -yx'u" z +zu' y" '-zux"l y +zx'y y -zx'uty" +zy x"u" -zyI" x" 2= -2xJ2xy2ZZ2 - 22 (z)2 -?U2>y2 (3xz)2 - t2G,26 x (:Xy)2 - 25yf2(Si1z)2 -_ fx22 (x uLy)2 - Ey2 2 (Ltux)2 + 2Et2ZxyfxzEyz 2 2yZ + 2xyuzEyz + 2E uxuzGxz + 2:z2~uxzUyYxy + (zitX)2 (:yz)2 + (Y:y)2 (Xz)2 + (2uz)2 (Xy)2 - 2Eux2xylyzzu -- 2zuyEyzEzxxxu - 2Euy1yxExzEzl," 72 HISTORY OF THE THEORY OF DETERMINANTS -a result which in modern notation would take the form /1 U U/ U 2 U U UC U 2 U 1 '2 3 1 ~-i '2 4 i Xi X2 X + X xi X2 X + Y Y1 Y2 Y3 Y YI Y2 Y4 Z Z1 Z2 Z8 Z Z1 Z2 Z4 n 2+ n12 ~. X 'aX+I1t1+,A Uy +'y1y1+. uz+u1z1+.. Ui+m Ax1V. x2 i 2+. c y+Xjy1 XZ +X1Z1+.* (xvi.3) my u+ y1~.. xy+x y1~.. y + y12 yQ - yBY1j1S- uz+m1z1~.. vz+x 1z ~.. yZ +yyz+.. Z2 + z 2 +.. It is thus clear that, in November 1811, Binet was well on the way towards a great generalisation. He even says that the three identities may be looked upon "comme les trois premieres d'une suite de formules construites d'apres une meme loi facile 'a saisir." He merely indicates, however, the mode of proof he would adopt for the results obtained, and refers to possible applications of them in investigations regarding the Method of Least Squares (Laplace, Uonnaissance des Terns, 1813) and the Centre of Gravity (Lagrange, AP'm. de Berlin, 1783). The mode of proof' need not be given here, as it turns up again in the far more: important memoir in which the theorem in all its generality falls to be considered. PRASSE (1811). [Commentationes Mathematicm. Auctore Mauricio de Prasse. 120 pp. Lips., 1804,1812. (Pp. 89-402; Commentatio vii.*: Demonstratio eliminationis Cramerianme.)] Of previous writings the one which Prasse's most resembles is Rothe's. There is less of it, and it shows less freshness; but there is the same stiff formality of arrangement, and the same effort at rigour of demonstration. * Separate copies of the Demonstrrtio eliminationis Crameriance are also to be found, bearing the invitation title-page: Ad mernoriamn Kregelio-Sternbachianam in auditorio philosophoronm die xviii Jedji Mucccxi. h. ix celebrandaam invitant ordinum Academim Lips. Decani seniores cceterique od-sessores.... Demnonstratio eliminationis Crameriance. It is these copies which fix the date. See Nature, xxxvii. pp. 246, 247. DETERMINANTS IN GENERAL (PRASSE, 1811) 73: The definition of a permutation (variatio) being given, the: first problem (which, however, is called a theorem) is propounded,. viz., to tabulate the permutations of a, y,, 8,... ("Variationumn ex elementis a,,B,,... constuctcarum et in Classes combinatorias digestarum Tabulam parare"). The result is a 0 Y ly aP ay aS ya r7 yS &a s3y l38 a4p ac8y I y/37 a3yS ca(3 a7-y /3ay /3a5?c yci 7y8 y8a 63y oyac -ya5 yBa -ypS -y6a 'y&13 6ap day 5/3a 5&3Y 3-ya 6-yp apy6 aj3y pa/ya a/ya f3Syc P6/3y Pya/3 ayo3a /3oya &3yap &-ypa (3yvI af357o 74 HISTORY OF THE THEORY OF DETERMINANTS The first row of the permutations involving two letters is got by taking the first letter of the previous row and annexing each of the others to it in succession and in the order of their occurrence; the second row is got in like manner from the second letter; and so on. Similarly the first row of permutations involving three letters is got from a/3 the first obtained permutation of two letters, the second row from ay the next obtained permutation of two letters, and so on.* The second problem (and on this occasion actually so designated) is somewhat quaint in its indefiniteness, viz., to prefix to each permutation the sign + or the sign -, so that the sum of all the permutations involving the same number of letters (>1]) may vanish (" Singulis Variationibus, omissis repetitionibus, signa + et - ita praefigere, ut summa secundce et cujuslibet classis insequentis evanescat"). There is no indefiniteness or multiplicity about the solution, which in substance is:-Make the permutations in every row of the preceding table alternately + and -, the first sign of all being +, and the first permutation of every other row having the same sign as the permutation from which it was derived. In this way the table becomes +a, -p, +7-, -} +a/3, -ay, +a6 -/ia, +/37, -b38 + yt, - 73, + 78 -pa, + /3, - 8y + apy, -a86 - ayf3, + a-y + a/i, - a8y -P3ay, +/ pa +/3ya, - ay6 - 3/a, + (35y + yas+, -ya6 - y3a, +y/35 + y6a, - y7/3 - a/3, + aay + 3a, - Ay - 5-ya, + y7/3 * It will be seen that the order in which the permutations come to hand in this process of tabulation is the order in which they would be arranged according DETERMINANTS IN GENERAL (PRASSE, 1811) 75 - a3-y - ay/3 + ITa +a6py - aopsy/3 - DayS + Pya38 - yyacp - ypaa -y+f3a -y/3&a - yapy + /3ay + 0ayI - a/3ya - S^ya3 + 6ya/3 A proof by the method of mathematical induction (so-called) is given that with these signs the sum of all the permutations of any group vanishes. Up to this point the essence of what has been furnished is a combined rule of term-formation and rule of signs. (II. 5 + III. 15) In connection with it Bezout's rule of the year 1764 may be recalled. The third problem is to determine the sign of any single permutation from consideration of the permutation itself. The solution is:-Under each letter of the given permutation put all the letters which precede it in the natural arrangement and which are not found to precede it in the given permutation; and make the sum + or - according as the total number of such letters is even or odd. to magnitude if each permutation were viewed as a number of which a, 3, y, u were the digits, a being -</3 -y -< (" ordo lexicographicus," "lexicographische Anordnung" of Hindenburg). 76 HISTORY OF THE THEORY OF DETERMINANTS "EXEMP. Datae complexiones sint hse: Ey83, 8 yE7, E8,, 8/3Ey. Literse secundum I subjiciantur a a a a a. 3PP3 aaa. aaaa // P3 / Pr 3 P.y 7 7 7Y 7 3 8 quarum numeri sunt 9 6 9 7 qui complexionibus datis prsefigi jubent signa The proof that this rule of signs, which is manifestly nothing else than Cramer's, leads to the same results as the previous. rule, is quite easily understood if a particular permutation be first considered. For example, let the sign of the particular permutation 683ay be wanted. Following the first rule, we should require to note four different members, viz., (1) the no. of the column in which 8/3ay occurs in the 4th group, (2),,, 8/a,, 3rd (3),,,, S3,, 2nd (4),,,,, lst,, The first of these numbers being 1, we should infer that in fixing the sign of ~63ay in the fourth group there had been no change from the sign of 8/Sa in the third group; the second number being also 1, we should make a like inference; the third number being 2, we should infer that in fixing the sign of 8/3 in the second group there had been 1 change from the sign of 6 in the first group; and finally, the fourth number being 4, we should infer that in fixing the sign of 6 in the first group there had been 3 changes from the sign of a in that group. The total number of changes from the sign of a in the first group being thus 3+1+0+0, i.e., 4, the sign would be made +. Now the 3. in this aggregate is simply the number of letters in the first group which precede 8, the 1 is simply the number of letters taken along with 8 before /3 comes to be taken along with it. to form 8/3 in the second group, and the two zeros correspond DETERMINANTS IN GENERAL (PRASSE, 1811) 77 to the fact that 6/3a on the third group and 8o3ay on the fourth group have no permutation standing to the left of them. Consequently to count the number of changes (3+1+0+0) from the sign of a in accordance with the first rule is the same as to count the number of letters placed under the given permutation, thus, S/3ay a a.. 7 in accordance with the second rule. Another point of resemblance between Rothe and Prasse is thus made manifest, viz., that they both refused to accept Cramer's rule of signs as fundamental, preferring to base their work on a rule equally arbitrary, and then to deduce Cramer's from it. In case it may have escaped the reader, attention may likewise be drawn to the fact that Prasse prefixes a sign not only to permutations involving all the letters dealt with, but also to any permutation whatever involving a less number; so that in reckoning the sign of ad/3, say, the full number of letters from which a, 8, 3 are chosen must be known. A theorem like Hindenburg's is next given, viz., If the permutations of any group be separated into sub-groups (1) those which begin with a, (2) those which begin with /, and so on, then the series of signs of the 3rd, 5th, and other odd sub-groups is identical with the series of signs of the 1st sub-group, and the signs of any one of the even sub-groups is got by changing each sign of the first sub-group into the opposite sign. (III. 16) It is more extensive than Hindenburg's in that it is true of permutations which involve less than all the letters, provided such permutations have had their signs fixed in accordance with Prasse's rule. The proof depends, of course, on the first rule of signs, and consists in showing that if the theorem be true for any group it must, by the said rule, be true for the next group. It will be remembered that Hindenburg gave no proof. Following this is Rothe's theorem regarding the interchange of two elements of a permutation, or rather an extension of the 78 HISTORY OF THE THEORY OF DETERMINANTS theorem to signed permutations involving less than the whole number of letters. The proof is as lengthy as Rothe's, even more unnecessary letters than Rothe's c, f, e being introduced. (ill. 17) The last theorem is Vandermonde's (xII.); and this is followed by two pages of application to the solution of simultaneous linear equations. No reference is made by Prasse to Hindenburg, Rothe, or Vandermonde. WRONSKI (1812). [Refutation de la Thdorie des Fonctions Analytiques de Lagrange. Par H6ene Wronski. (pp. 14, 15,..., 132, 133.) 136 pp. Paris.] In 1810 Wronski presented to the Institute of France a memoir on the so-called Technie de I'Algorithmie, which with his usual sanguine enthusiasm he viewed as the essential part of a new branch of Mathematics. It contained a very general theorem, now known as " Wronski's theorem," for the expansion of functions,a theorem requiring for its expression the use of a notation for what Wronski styled combinctory sums. The memoir consisted merely of a statement of results, and probably on this account, although favourably reported on by Lagrange and Lacroix, was not printed. The subject of it, however, turns up repeatedly in the Refutation printed two years later; and from the indications there given we can so far form an idea of the grasp which Wronski had of the theory of the said sums. At page 14 the following passage occurs:"Soient Xl, X2, X, &c. plusieurs fonctions d'uno quantite variable. Nommons somme combinatoire, et designons par la lettre hebraique sin, de la maniere que voici = [A X. I X. A 3x... a PX], (xv. 3) (vir. 4) la somme des produits des differences de ces fonctions, composes de la maniere suivante: Formez, avec les exposans a, b, c,..., p des diff6rences dont il est question, toutes les permutations possibles; donnez ces exposans, dans chaque ordre de leurs permutations, aux differences consecutives qui composent le produit AX AX2.AX... AX,; donnez de plus, aux produits s6pares, formes de cette maniere, le signe positif lorsque le nombre de variations des exposans a, b, c, etc., DETERMINANTS IN GENERAL (WRONSKI, 1812) 79 consideres dans leur ordre alphab6tique, est nul ou pair, et le signe negatif lorsque ce nombre de variations est impair; enfin, prenez la somme de tous ces produits s6par6s.-Vous aurez ainsi, par exemple, w[AaX1] = AaX,, X1 [AX1. AbX2] = AXI. Ab2 - A bX AX2, The new name, combinatory sum, and the new notation, did not originate in ignorance of the work of previous investigators, for memoirs of Vandermonde and Laplace are referred to. The only fresh and real point of interest lies in the fact that the first index of every pair of indices is not attached to the same letter as the second index, but belongs to an operational symbol preceding this letter, and is used for the purpose of denoting repetition of the operation. This and the allied fact that the elements are not all independent of each other, AIX1 and A2X1, for example, being connected by the equation 2X1 = A (alX1), indicate that Wronski's combinatory sums form a special class with properties peculiar to themselves. CHAPTER IV. DETERMINANTS IN GENERAL IN THE YEAR 1812. HERE we have the record of only one year and of only two authors to deal with; but the authors, Binet and Cauchy, are of supreme importance, and the product of the year probably exceeded that of all the years that had gone before. BINET (November 1812). [Memoire sur un systeme de formules analytiques, et leur application a des considerations gdometriques. Journ. de l'Ec. Polyt., ix. cah. 16, pp. 280-302,...] It would seem as if the above-noted frequent recurrence of functions of the same kind had led Binet to a special study of them. In the memoir we have now come to, his standpoint towards them is changed. They are viewed as functions having.a history: for information regarding them, the writings of Vandermonde, Laplace, Lagrange, and Gauss are referred to: they are spoken of by Laplace's name for them, re'sultantes a deux lettres, a trois lettres, a quatre lettres, &c.; and the first twenty-three pages of the memoir are devoted expressly to establishing new theorems regarding them. Of these the fundamental, and by far the most notable, is the afterwards well-known multiplication-theorem. It is enunciated.at the outset as follows:"Lorsqu'on a deux systemes de n lettres chacun, et nous supposerons chaque systeme ecrit avec une seule lettre portant divers accens, qui,serviront a ranger dans le meme ordre les deux systemes; on peut former avec ces lettres un nombre n — - de resultantes a deux lettres, DETEIRMINANTS IN -GENERAL (BINET, 1812) 8 81 en ne prenant dans le second terme de chacune que des lettres portant les me'mes accens que celles du premier. Si, avec, deux autres syste'mes de lettres, on forme encore des re'snltantes 'a deux lettres, et qu'on les multiplie chacune par sa correspondante obtenne des deux premiers syste'mes, c'est-ah-dire, par celle dont les lettres portent les ine'mes accens~ la somme des produits de toutes ces r~sultantes correspondantes sera elle-me'me une resultante 'a deux lettres, dont les ternies ou lettres seront des sommes de produits des 616mens des deux syste'mes portant les me'mes accens. Avec deux groupes de trois syste'mes de n lettres chacun, on peut former semblablemnent deux series de re'sultantes a trois lettres; faisant ensuite la somme des produits de celles qui 'se correspondent par les accens de leurs lettres, on aura encore une resultante 'a trois lettres. Pareille chose ayant lieu pour des r~sultantes a quatre lettres, &c., on pent conclure cc the'ore'me: Le produit d'un nombre quelconque de sommes de produits * de deux r~sultantes correspoudantes de m~me ordre, est encore une resultante de cet ordre." (xvii. 4~ xviii. 4) The mode, of proof adopted is lengthy, laborions, and not very satisfactory, except as affording a verification of the theorem for the cases of " resultantes " of low orders. It rests too on certain identities, the demonstration of which is open to similar criticism. All that Binet says regarding these absolutely essential identities is (p. 284)"1Je repre'senterai par la la somme a' + a"/ + a"' + &c., des quantite's a', a"/, a"'/, &c.; par >Jab la somme des produits ab +a'b' +a"b" ~&c., dans chacun desquels les lettres a et b out le mebme accent; par lab' la somme a'b" ~ b'a" + a'b"' + &c., la' tons les prodnits d'un des a par un des b, portent un accent diffbrent de celui de a; par lab'd" la somme a'b"c'" + b'clia"' + c'a"b"' + &C., et ainsi de suite. Cela pos6, on ve'rifie aisbment les formules suivantes: Ylab' a'fb - lab, labictl = a~bc + 21abc - fal~bc - Eb>Jca - Ecfab, =a'dd la>2b~c' - 60abcd - IaIO~cd - ~~a',c~bd - Ia~d >2bc - 12c>2d>ab - >2b>d>2ac - M2>c>2ad + >2ab>2cd + >2ac>2bd + >2abt>2be + 22a>2bcd + 2>2b>2cda + 2>2c>2dab + 2>2d>abc, >2ab'C"d"'eiv = >2a>2blc>d>2e + &c., *There is an extension here which one is scarcely prepared for, viz., "le produit d'un nombre, quelconque de, somrnes de produits,," instead of lae somme d'un nombre de, produits. ~ Meant for Maed. M.D. F 82 HISTORY OF THE THEORY OF DETERMINANTS It is thus seen that not only is no general proof of the identitie's given, but that even the law of formation of the right-hand members of the identities themselves is left undivulged. The exact words employed in the demonstration of the first case of the multiplicatio-n-theorem are (p. 286)"Avec un nombre n de lettres y', y", y"', &c. et un m~ne nombre de z',Z",r", &c. on peut former n -lr'sultantes ' deux lettres (y', ) 2 (y' z"'/), &c. (y"l, z"') &c.; ayant forms pareillement avec ics lettres, (Vs") &c., conside'rons la somme 1(y, z')(v, (') des produits des resultantes qui se correspondent par les accens dans les deux systemes. On voit, en de/veloppant, par 'la multiplication, chacun des termes de cette somme, qu')elle revient AA ly V. -, >2ZV. A ces deux dernie'res int~grales, on pelt appliquer la transformation indiqu6e par la premie're des formules de l'art. 1: on parvient ainsi 'a 1;(y, Z') (V, l' yvIzC - Izv1yC Ce dernier memrbre pouvant eftre assimil6 a la' forme (y, z'), il en r6sulte que le produit d'un nombre quelconque de fonctions, telles que The application here of the identity lab'= Ea>lb- lab requires a little attention. The result of multiplication and, classification of the terms is or, as it might preferably be written, and this we know from the said identity IY.l~ - ZZ. ly~ The inherent weak points, however, of the mode of demonstration stand out more clearly'when the next case comes to be considered, viz., the case for resultants of the third order. From the three sets of n letters DETERMINANTS IN GENERAL (BINET, 1812) 83 x, xi, x". Y, yl y1l. z, z', z"., all possible "resultantes ' trois lettres" are formed, and each resultant is multiplied by the corresponding resultant formed from other three sets of n letters, U, U, U. Each of these -in(n -1)(n -2) products consists of 36 terms, there being thus 6n(n -1)(n-2) terms in all. But these 6n (n - l)(n - 2) terms are found to be separable into six groups, viz. +1xe.y'U "V/.Z"}, ~/{Y/.z X " "}. so that the result which we are able to register at this point is ~(X, y, z) ( u' c") = I. y'v'. z"/" + f C. Z. XI + 1Z $. x'u'. y"S-" - Ll. ZuVY. - Eye. X7Vu. Z"/ E" - ~2. yU. X/ To the right-hand member of this the substitution flab'd'= lalbL o + 2I'abc - I;a'cbc - EbC'ca - Eclab is now applied six times in succession; that is to say, for and the five other term-aggregates which follow, we substitute Ixe'yvuz~ ~ 21 (x4. yv. zx) - ExC5E(yv. z5 - ~y (4. xe)- I(x e qu) and five other like expressions. By this means we arrive, "toute reduction faite," at 1(X, y', z") Z u', c") xelyulzx + Xyefzv1xS ~ ~YZexvUy Y - IXCIZVUE /$- XveZx - Z$e~yu4X4 which is the result desired. It is easy to imagine the troubles in store for any one who might have the hardihood to attempt to establish the next case in the same manner. 84 HISTORY OF THE THLEORY OF DETERMINANTS If Binet's multiplication-theorem~ be described as expressing a sum of products of resultants as a single resultant, his next theorem may be said to give a sum of products of sums of resultants as a sum of resultants. The paragraph in regard to it is a little too much condensed to be perfectly clear, and must therefore be given verbatim. It is (p. 288)"cDe'signons par S(y', z") une somme de r~sultantes, telle que et continuous d'employer la caracte'ristique I pour les int6grales relatives aux accens supe'rieurs des lettres. L'expression devient par le de'veloppement de chacun de ses termes, et en vertu de la premie're formule de lPart. 1 ou de celle du no. 4, 1-y'V, '~' - 1z'v1:2Y,, + >2yI/V, lz/ ni- ylc + &C. + //Z// - IZ +' /,,v,,,(, Z1)1y,>+Sc + &iC. En indiquant done par S, des integrales qui supposent, dans chaque terme, les me'mes accens inf~rieurs aux lettres du me'me alphabet, ces accens pouvant 6tre on. non les m~mes pour celles des alphabets diffirens, on pourra e6crire la precedente suite, en faisant usage de ce signe, ce qui donne I [S (y, z') S (V, a']=SI [Iyvlz~ - >Jzv2y~fl. Cette nouvelle quantite' est encore de la forme S (y', z"), en sorte qu'on pent dire que le produit de fonctions, telles que sera lui-me'me de la forme S (y', z"). " This, if I understand it correctly, may be paraphrased and expanded as follows: Take the product of two sums of s resultants, viz. { y1'Z1 +Y~2'Z22+ Y31Z3 2 1+. + y81Z.821) x V11 ~12 I + v2~22 + V. + 2. + 211 where, it will be observed, all the resultants in the first factor are obtained from the first resultant I y1Z2I by merely changing the lower indices into 2, 3,..., s in succession, and that the DETERMINANTS IN GENERAL (BINET, 1812) 85 second factor is got from the first by writing v for y and 5 for z. Then form all the like products whose first factors are Yl1 y \> | 1Zl4 |, ' y' n - lZ l; these being along with I yl12 [ the -in(n -1) resultants derivable from the two sets of n quantities Y1, 'Y, Y,...., Zll) Z12, z13.... Z l The sum of these -in( (-1) products may be represented, if we choose, by _= s=s s=s qt=2 s=l s=l1 m7z< n Now if the multiplications be performed, there will be s2 terms in each product, and the theorem we are concerned with has its origin in the fact that the sum of all the first terms of the products is expressible as a resultant by applying the multiplication-theorem, likewise the sum of all the second terms, and so on, the result being an aggregate of 82 resultants. For if we fix upon a particular term of the first product, say the term iI 21 which arises from the multiplication of the hth term of the first factor by the kth term of the second factor, then take the corresponding term of the other products, and write down their sum yh Zh 2 I | 2k J |21 + Yhh3. * V1 k3 I +.... + I yf,-1. I. |- i, it is manifest that this sum is by the multiplication-theorem y j1b1 + yh V +...+/ * ZhVk + Z A +V+....+ Z,?V + - 7Ih 2 +. +y.jk.+ +.l+Z2...+ Z. |hl~ TYh + + -2 yh827;n NhlV,1 A ^+ h &hn|* Consequently since h may be any integer from 1 to s, and k likewise any integer from 1 to s, the theorem arrived at is accurately expressed in modern notation as follows:[n=n r s=.s s8s E I I YsZsn I|. I | VsI m, n=2 s=-1 s- 1 2m < n k=s h /k~lkl+ y_2lyn,22n J+...2_ 2.+^ n E 1 VkI +h Uk ** Zhk V 7 k **+ Zh k=l h=xl |- yt6 ax + 871 * * * + y_ yn7n Z ~nl + Z2 _kv_ + Z n |n or E = 1 2 * * 2 ' |k * * * k=J =....z^ * k C....^. 86 HISTORY OF THE THEORY OF DETERMINANTS It is easily seen to be true of resultants of any order, as Binet himself points out. (xxx.) When s is put equal to 1, it degenerates into the extended multiplication-theorem. The theorem which follows upon this, but which is quite unconnected with it, may be at once stated in modern notation. It isIf 1 xy2z3 1 denote the sum of the resultants obtainable from the three sets of n quantities X1 X2 X3... Xn Y1 Y2 Y3.. Yn z1 Z2 Z3.... Zn, and E xy 2 I denote the like sum obtainable from the first two sets, then yI x1lY2z3I = xA. y12 I+ 1.x + Y. z + I. Z X YI. (xxxI.) This is arrived at by writing out the terms of 2 y1'z2, of I zlx2 |, and of Z1 xxy2 in parallel columns, thus I2Yl 1 I 1 X2 I x1 Y2 I 1 3 3 I Z X31 I xI Y31 I n-lZn Z | xn-1Xn X | Zn-lYn |; then deriving n results from the members of the first row by multiplying by x,, y, z1 respectively and adding, multiplying by x2, y2, z2, and adding, and so on; then treating the second and remaining rows in the same way; and then finally adding all the n. ~n(n-1) results together. Each of these results is a vanishing or non-vanishing resultant of the 3rd order, and it will be found that each non-vanishing resultant occurs twice with the sign + and once with the sign -. This process is readily seen to be simply the same as performing the multiplications indicated in the right-hand member of (XxxI.), i.e., (X1+X2+...+xn) (\y1Z\ + ly1z1 +...+ \y~1n ) +(Y1+Y2+ * +Yny) (I X2 I + I Z3 +. ~ + | n-l I) +(Z1+ Z2+. * +Zn) ( xlY21 + XIY3 I +. * + Xn-lYn 1), DETERMINANTS IN GENERAL (BINET, 1812) 87 summing every three corresponding terms in the products, and writing the sum as a vanishing or non-vanishing resultant. There would be n. In (n - I) resultants in all; but as each suffix occurs n -1 times in the second factors and once in the first factors, there must be in each product n -1 terms having the said suffix occurring twice: consequently there must be n — 1 resultants vanishing on acconnt of this recurrence, and therefore altogether n (n -1) vanishing resultants. Of the non-vanishing resultants,-in number equal to n. In(n-1)-n(n-1), or ~n (n - 1) (n - 2),-each one of the f orm XAI k-' where h<k<l must be accompanied by two others, I XkYYhA Z and XYh-Zk and the sum of these is XhYA- XhYkZ'l I XhY~ ixe., I XhYkZ'I 'The final result is thus the sum of the resultants of the form I XhYkZZ I where h <k<l, and [=3, 4,..., n the number of them, as we may see from two different standpoints, being 67n(n- 1)(n- 2). Returning to the series of identities, X3 Y1Z2 + Y3 1 IX2! + Z3 XlY2 =XlY2 -X4 Y12 + Y14 -' + -4 1 X1Y2 = X111Y2z, &c. &c. which by addition give the result 1XF1YIZ21 + YYIZ1X21 + Z"JX1Y2J = 1IX1Y231,Z IBinet next raises both sides of all of them to the second power, and obtains 3 Ix IX12zj2 = JX2Y Iy1z212 + EY21 1 1Z2 2 + IZ2 XY2 12 +21yzZ(IzIx2H xly2 )+ 21zxEx(lyl2. 1y'zl2) (xxxJI.) + 2Y~xyY(jyZ21. IZAD 88 HISTORY OF THE THEORY OF DETERMINANTS Substituting for ly|z22, zxxl2,...., their equivalents as given by the multiplication-theorem, he then deduces Xly2zs 2 = -SX2Jy2Iz2 + 2IyzZzx xy - X2(2yz)2 } - Zy2(Zzx)2 - _ 2(Zxy)2 not failing to note that this is not a fresh result, but merely a case of the multiplication-theorem in which the factors are equal. By putting the right-hand member here into the form Jy2 {^z2JX2 _ (yZ)2} + {Z2 I{ ZXy2 - (xy)2} - {X2{Zy22_z-(Zyz)2} + 22yz{2zx2xy- -yzXS2}, there is next arrived at the first identity of the set xzY2Z3 12 = 1y2 ZX 2 + 2 1Y2 12 _ X2~ I Y1Z2 I2 2+ yz Y I Z1X2 1I X1y2 I1 = 22 X1Y21 2+ 2I1Z2 2 _ Y 12 12 22zx I X1Y2 I YIZ21, j(XXXIII.) = x2 Y1Z 12 + y2 z 2 12 z2_ I X1Y2 12+2 2tyzi YZ 1 I,12 1, J and immediately from these the set 1 XY2Z312 = X2 I Y1Z2 12 + ZX I X1Y2 'I Y1Z21 + ':XY I Y1Z2 I I Z1X2I, =y2 z zs22 + ExyZ Y1Z2 IZ12 I + YZZ I ZX2 I 1Y2 (xxxiv.) = 221 X11y2l + /z2 I z1X2 I. I X1Y2 + 2ZX I X1Y21I YZ2 J We may note in passing that either of these sets leads at once to the initial theorem 38lxly2z3|2 _= 2 212|y l2 + y22jlsx 2 + z2l2xy1 2 + 22yz I z \x2I.* Xy21 + 22|zx I X1'y2. | YlJ2 + 22|xyYIY1|z2. zI Z1x, and that with the multiplication-theorem already established this reverse order would be the more natural. The next step taken is the formation of resultants of the 2nd order from elements which are themselves resultants of the 2nd order; viz., just as from the three rows of n quantities X1 X2.3.... Xn Y1 Y2 Y3. * Yn ZI Z2 Z3. * Zn DETERMINANTS IN GENERAL (BINET, 1812) 8\ there were formed the three other rows of in?(n-1) quantities Y12 I, YYyn, Y22 Z3, ~.. IY n-1,n, 11X2 I Ix..1., X1*Zln I| X Z231 * * *., I n-li^n I X1Y21, Xl1Y31.. * *, XIYn, |2Y3...., Xn-iYn | so from the latter three other rows of quantities I z1x2 1 1 Z3 I X1 2 I I X1 3 I/ I > 2I | 1 Ix1y3 1 I 1 21 1Z3 I I 11 1Z2 1 IY1iZ31 I Z 2 1 I 1I 3 11 ' * |I Zn-2n I I Zn-lXn I| |.............., | Xn-2yn I I Xn-lYn |> I| Xn-2yn I I Xn- ln |.............. *, n | |Z*n-2Xn | | I n-lz ln > are formed, the number in each new row being clearly {2-n(n- 1)}{( -(n 1)- } i.e., 1(n~+ l)n(n -I )(n- 2). The new quantities are, of course, not written by Binet in the form [ I I but the fact that they are resultants of the 2nd order is carefully noted. Each of them is shown to be transformable, by a theorem which may be viewed as an extension of a result given by Lagrange, so as to have two of the elements resultants of the 3rd order, and the other resultants of the 1st order. This is done by taking, for example, the identities h I yzj I + y | zixj I + z^ xiyj I = xhyizj, Xk YiZjI + yk ZiXj + zk X|iYj = kyiZj multiplying both sides of the first by x, and both sides of the second by x,, subtracting, and writing the result in the form i xyh I | iXj + XkZh XiYj Xk XhYij - Xh I XYiZj = | Xkj Xhj |xkyij |xy izj 90 HISTORY OF THE THEORY OF DETERMINANTS where of course it has to be noted that in many cases one of the resultants of the 3rd order will vanish. The quantities, therefore, to be dealt with, are X1 X1Y2 I, ~., XYiSZj - X | i h X kYZj ~ ~ I. -n-1Zn i Yl XY223, * I Yh ZiXj - Yh | YkZiXj | *, n |. n-2Yn-lzn |; 1 1lYZ.., Zk ZhXiyj h Xiyj, *, Z k I Z n-n- ln X. By raising each of the elements of the first row to the second power, taking the sum and simplifying, we could, we are told, show that the result would be EX1l2Z1Xly23 12. Very prudently, however, another process is chosen. It is recalled that the quantities in the third triad of rows are related to those in the second as those in the second are related to those in the first, and that consequently the required sum of squares of resultants is, by the multiplication-theorem itself, expressible as a resultant, viz., 1x [ X2, I 3 12 = Z- Z I 2. I Y22 _2 (S IZ2 I XlY2 I)2, where the elements of the resultant on the right are sums of products of quantities in the second triad of rows. Then the same theorem is used to make a further step backwards, viz., to express each of these three sums of products of resultants as a resultant whose elements are sums of products of the quantities in the first triad of rows, the effect of the substitution being Z lZ I|, | xy, 112 = {1zl2x2 - (zm)2} {x12zy2 - (X1y1)2} - {YZXlxyly — 2ylzixia2}2. Simple multiplication transforms this into E - 2 y~ 2(z- IZIX()2 - }Zj2(jXy 1)2 X+ 21y +2yzlzlxxyl - ~X12(y/Z1)2 ' which, by still another use of the multiplication-theorem, we know is equal to '12 I yXSX123l2. DETERMINANTS IN GENERAL (BINET, 1812) 91 The set of six results of which this is one, is X1Y2 -ES12 xly2zX2312, 2 ='y Yjxly2zs]2, yZ,2 1 2yz2 YIX1Y2z312, ~ - 1 llY2-3> ~ (xxxv.) 2YA Y2Y2-|1 E 3 I lY2 312 1 YZiXi = Iyz A Ix2y23 12, 2XiY1 = ExlylE x1y23 12, if, for shortness, we denote the quantities of the third triad of rows by XY1, XY2,.... Z 21.... Following these, and deduced by means of them, is an equally noteworthy theorem regarding the sums of squares of all the resultants of the third order, which can be formed from the quantities of the second triad of rows. Denoting these quantities temporarily by el 2 ', 2),.... i?12 ]'.... we know (xxxII.) that 3 1 t ^ ~ 12 = 2X12y$12 + SY121 12 + EZ12Y12 d- 22YA1Z.2^il + 27Z1X1.Ill + 2EX1Y1. E2ll; whence, by using the set of six results just obtained, we have 3 1 I123 12 -= xI XY23 2 { J121I2 + J112JYI2 + 2IZ1 2 -+ 22,1.iylz~ + 221e.2zlx1 + 22,11.Y2xy1 J and therefore, again by (xxxIII.) z I I232 = ( I XIY2Z3 12}2. (xxxvi.) It is finally pointed out that from the third triad of rows there might, in like manner, be formed a fourth triad, and 92 HISTORY OF THE THEORY OF DETERMINANTS analogous identities obtained; also that, instead of starting with three rows, we might start with four, tl, t2, t3, ' *., tn x1, X2, X3,.... Xn Yi, Y2, Y3,. * 'Yn Z1, Z2, Z3, *.. Zn, form from them other four lY231,......... I YZ2t3........ I zlt2X3 1, tlx2Y3.......... thence in the same way a third four, and in connection therewith establish the identity (xxxi. 2) 2tl2 xyY2z3 I - 2^x2 ] y1z2t3 Y + M Zlt2z3 -z1 t1X2Y3 I = 0 and other analogues. (xxxII. 2+ xxxv. 2) The rest of the memoir, 52 pages, consists of geometrical applications of the series of theorems thus obtained. CAUCHY (1812). [Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs egales et de signes contraires par suite des transpositions operees entre les variables qu'elles renferment. Journ. de l'Ec. Polyt., x. Cah. 17, pp. 29-112. (Euvres (2) i.] This masterly memoir of 84 pages was read to the Institute on the same day (30th November) as Binet's memoir, of which we have just given an account. The coincidence of date has to be carefully noted, because the memoirs have in part a common ground, and because there is a presumption that the authors, knowing this beforehand, had, in a friendly way, arranged for simultaneous publicity. Binet's words on the matter are (IX. p. 281) DETERMINANTS IN GENERAL (CAUCHY, 1812) 9 93 "Ayant eu derni'rement occasion de parler 'a M. Cauchy, inge'nieur des ponts et chausse'es, du the'ore'me ge'n~ral que j'ai 4'nonce' ci-dessus, il mne dit 6tre parvenu, dans des recherches analogues 'a celles de M. Gauss, 'a des th~ore'mes d'analyse qui devaient avoir rapport aux miens. Je m'en suis assure, en jetant les yeux sur ces formules: mais j'ignore si elles ont la Meme generalit6 que les. miennes: nous y sommes arrives, je crois, par des voies tre's-diffhrentes." And Cauchy's corroboration is (p. 111)"Javais rencontr6 1'e'te dernier, 'a Cherbourg, ofi j'~tais fixe' par les travaux de mon 6tat, ce the'ore'me et queiques autres du m~me genre, en cherchant 'a ge'neraliser les formules de M. Gauss. M. Binet, dont je me f~licite d'e~tre l'ami, avait W conduit aux M emes re'sultats par des recherches diffl~rentes. De retour 'a Paris, j'etais occup6 de poursuivre mon travail, lorsque j'allai le voir. Ill me montra son th~ore'me qui e'tait semblable au mien. Seulement ii de'signait sous le nom, de re'sultante, ce que j'avais appell6 d~terminanI." Cauchy prefaces his memoir by another, entitled Sur le nombre des valeurs mu'une fonction peut acque'rir lorsqu'on yq permumte de tomtes les manie'res possibles les quantite's gvu'elle venferme. This latter must to a certain extent be taken into account, because it serves to show the point of view which he considered most natural for examining the subject, and also the exact position held by the functions now called determinants, when functions in general come to be classified according to the number of values they are able to assume in certain circumstances. At the' outset of it the writings of Lagrange, Vandermonde, and Ruflini are referred to; the fact is recalled that the maximum number of values which a function can acquire by interchanges among its n variables is 1.2.3.... n; also that when the maximum is not obtained, the actual number must be a factor of the maximum; and then proof is given of the very notable theorem that the number of values cannot be less than the greatest prime contained in n without being equal to 2. It is. pointed out likewise that functions capable of having only two values are known from Vandermonde to be constructible for any number of variables, For example, the number of 94 HISTORY OF THE THEORY OF DETERMINANTS variables being three, a1, a2, as, all that is needed is to form their difference-product (a3 - a) (a3 - a) (a2- a) or a32a2 + a22al + 12a3 - (aC32al + a22a + al2a2), when it is found that either of the parts a32a2 + a22a1 + a12a, or a32a1 + a22a + a12a2, is an instance of a function capable of only two values by permutation of the variables; the result indeed of any permutation being merely that the one function passes into the other. Further, the whole expression a32Ca + a2a1 + 2a3C - (a32al + a22aca + a2a2) is another example, the difference between the two values which it can assume being however a difference of sign merely. As a. reference to the title of the memoir of November 1812 will show,, it is functions of this latter class which Cauchy there considers. At the commencement he contrasts them with functions which suffer no change whatever by permutation of variables, that is to say, symmetric functions: and, noting the fact, afterwards ascertained, that the new functions consist of terms alternately + and -, and that were it not for this alternation of sign they would be symmetric functions, he decides to extend the term "symmetric" to them, and having done so, seeks to distinguish them from ordinary symmetric functions by calling them "fonctions symetriques alternees," and calling the other "fonctions symetriques permanentes." Cauchy's view of determinants may therefore now be described by saying that he considered them as a special class of alternating symmetric functions. To include them, however, either the adoption of a convention is necessary, or an extension of the definition must be made. For example, alb2-a2b1 is not an alternating function, unless the elements be so related that the interchange of a1 and a2 necessitates the interchange of b1 and b2 at the same time; or unless the definition be so worded that interchange shall refer DETERMINANTS IN GENERAL (CAUCHY, 1812) 9 95 to suffixes, not to letters. Cauchy selects the former course his words being (p. 30)... concevons les diverses suites de quantite's a1, a2).,.. a., C1, C2.,C, tellement li~es entre elies, que l a transposition de deux indices prig dans l'une des suites, necessite la m6me transposition dans toutes les autres; alors, les quantite's bp ell,...,) b2, c2,..., b3 C 3).... pourront 6tre conside'rees comme des fonctions semblables de al, a2, a,,.... et par suite, les fonctions de apb,c1...,aD b2 C~ 2). *.. an) ).n qui ne changeront pas de valeur, mais tout au plus de signe, en vertu de transpositions op~r~es entre les indices 1, 2, 3. n, devront. eterages parmi les fonctions syme'triques de a,, a2..., a, ou, ce qui revient au me'me, des indices 1, 2, 3,..., n. Ainsi a12 + a22 + 4a a, aAb + a 2b2 + a~b, + 2c 1CAc3 aAb + a bA + a~bl + a 2b1 + a3b2 + a~b3, cos (a, - a2)cos (a1 - a,5) cos (a 2- aA) seront des fonctions syme'triques permanentes, la premie're du second ordre et les antres du troisie'me; et au contraire, alb 2+ a bA + a 3b1 - a2b1 - alb 3 - ab2 sin (a1 - a 2) sin (a1 - a3) sin (a2 - a3) seront des fonctions syme'triques alterne'es du troisie'me ordre." The question of nomenclature being settled there next arise,% the question of notation. This also is decided on the ground of' the resemblance of the functions to symmetric functions. It, being known that any symmetric function is representable by a, typical term preceded by a symbol indicating permutation of the. variables, e.g. S (alb2) or 82 (alb 2) standing for alb2 + a2b1 and S3 (alb2) standing for a1b2 + a2b, + a3b, + a2b, + a3b 2+ alb3; 96 HISTORY OF THE THEORY OF DETERMINANTS also, that any non-symmetric function may be taken as the typical term of a symmetric function, the question arises whether the like may not be true of alternating functions. A lengthy examination of the latter point leads to the conclusion that any non-symmetric function K cannot be the originating or typical term of an alternating function unless it satisfies a certain condition, viz., that it be such that any value of it obtained by an even number of interchanges of indices will be different from any other value obtained by an odd number of interchanges. Should, however, this condition be satisfied, and Ka, K, K,.... be all the values of the former kind, and KA, K/, K,.... all the values of the latter kind, then (Ka+K+K,+.... )-(Kx+~K+K,+....) is an alternating function and is appropriately representable by S(+K) if the indices appearing in K alone are to be permuted, and by S(_+K) if the indices to be permuted be 1, 2, 3,..., n. For example, taking the typical term alb2 we have S(~ alb2) = alb2-ab,, and S3( alb2) = alb2 + a2b3 + ab1 - a2b - ab -alb, = S3(a2bl) = S3(Tal) b3= S4(+_alb2) is an impossibility, as when there are four indices alb2 does not satisfy the condition required of a typical term; indeed, Cauchy notes that the number of indices in any term must either be the total number or 1 less. The number of permutations being even, it is clear that the numnber of + terms Ka, Ki,.... is the same as the number of,negative terms KA, KY, (x. 2) a generalisation of a remark of Vandermonde's. Further, since Ka, Kg,.... are all the terms that arise from an even number of transpositions, and KA, K/,.... all those that arise from an odd number of transpositions, it is plain that DETERMINANTS IN GENERAL (CAUCHY, 1812) 97 any single transposition performed upon each of the terms of the function (K,+Kp+K,+.... )-(K,+K,+K,+.. ) must change it into (K,+K.+Kv+.... )- (Ka+Kp+KY+.....) -this is, in fact, the proof that it is an alternating functionconsequently each of the parts K,+K+K, +... Kx+K +K+..... belongs to the class of functions which have only two different values. Also it is evident that if throughout the function any particular index be changed into another and no further alteration made, the resulting expression must be equal to zero, (xiI. 5) a theorem regarding alternating functions which is the generalisation of a theorem of Vandermonde's. We have lastly to note, that the criterion which determines whether a particular K belongs to the class Ka, Kp,.... or to the class K,, KM,.... is incidentally shown to be reducible to a more practical form. For example, if the term be K0, and it be derivable from K, say, by the change of the suffixes 1, 2, 3, 4, 5, 6, 7 into 3, 2, 6, 5, 4, 1, 7, that is to say, in Cauchy's language by means of the substitution 1, 2, 3, 4, 5, 6, 7\ \3, 2, 6, 5, 4, 1, 7, we transform this substitution into a "product" of "circular" substitutions, viz., into (1,3, 6 /45\ 5 2\ (7\ \3, 6, \54/ 4 7 and subtracting the number of "factors," 4, from the total number of suffixes 7, make the sign + or - according as this difference is even or odd. Here the subject of general alternating functions may be left for the present. What remains of the first part of the memoir, refers to special cases, which naturally fall to be considered M.D. G 98 HISTORY OF THE THEORY OF DETERMINANTS in another chapter of our history. At the close of the part Cauchy says (p. 51)"Je vais maintenant examiner particulie'rement une certaine esp'ce de fonctions syme'triques altern~es qui s'offrent d'elles-me'mes -dans uin grand nombre de recherches analytiques. C'est an moyen de ces fonctions qu'on exprime les valeurs ge'ne4rales des inconnues que reniferment plusieurs' equations du premier degre' Elles se- repre'sentent toutes les fois qu'on a des equations 'a former, ainsi que dans la th~orie generale de le'limination." The writings of Laplace, Vandermonde, Bezout, and Gauss are referred to, and from the latter the name "determinant" is adopted. The second part bears the titleDes fonctions syme' triques alterne'es de'sign~ees sous le nom de d~terminans. (xv. 4) and opens with the following explanatory definition (p. 51)"Soient a,, a2,. a,, plusieurs quantite's diffhrentes en nombre egal 'a n. On a fait voir ci-dessus qu'en multipliant le produit de ces quantite's, oul a a a a.. par le produit de leurs difibrences respectives, ou par (a2 )(a.3 - a,)... (a8,- aj)(a3 - a2)...(a,- a2) (a -1) on obtenait pour re'sultat la fonction syme'trique altern~ee S(~ala 2a.3.. a,,,'), qui par consequent se trouve toujours 6'gale au produit a~a2a3... a-, Supposons maintenant que l'on de'veloppe cc dernier produit, et que dans chaque terme du -developpement on remplace l'exposant de chaque lettre par un second indice 6gal 'a l'exposant dont il s'agit, en eicrivant par exemple a,., an lieu de a,.s, et a,,,. an lieu de a8', on obtiendra pou'r r~sultat une nouvelle fonction syme'trique altern'e, qui, au lieu d'~tre repr6sente'e par S (1 a, 1a22a..'.... a,,8,) sera repr~sente'e par S(~a.,a2.2a3..3.... a,,.n), le signe S 6tant relatif aux premiers indices de chaque lettre. Telle DETERMINANTS IN GENERAL (CAUCHY, 1812) 99 est la forme la plus gen'rale des fonctions que je d~signerai dans la suite sous le nom de dtterminans. Si ion suppose successivement " n=1, n=2, &c. on trouvera S(~a,.1a2.2) = a1.1a22. - q2-16'1-21 S~ a,.162.2a3.3) = a,.ja2-.2a3., + a2.1a,32a13 + a.1a,'2aT.3 - a1.1a3.2a2'3 - a3.1a2.2a1.3, a2'1-1'2a,1,, pour les d~termiuans du second, du troisie'me ordre, &c. In regard to this it is important to notice that there are really two definitions given us. The latter, viz., that involved in the symbolism of alternating functions, S(4 aj.ja 212a3'3.... an-n) contains nothing more than Leibnitz's rule of formation'and an improved rule of signs. The former is new and may be paraphrased as follows: If the multiplications indicated in the expression aCa2a, - -. an x (a - a,)(a. — a,). (a - a,)(a3 -a2)... (a. - a2). (a. abe performed, and in the result every index of a power~ be changed into a second suffix, e.g., a,.s into a,.,,, the expression so obtained is called a determinant, (iii. 18), (viii. 2) and is denoted by S( ~ a1.12.2a3.3.... a,,) (vII. 5) In this definition the rule of signs and the rule of termformation are inseparable-a peculiarity already observed in the case of Bezout's rule of 1764. After the definitions various technical terms are introduced. The n2 different quantities involved in S(+~a,.,a2.2a3.3... a,,.,) are arranged thus Ia,,, al.2, al.3. al.n a2.1,a2.2, 23, a2'. a3.1, a3.2, a3.3. a3., &c... a.., an'2' a,-.a,, a.., *n=2, n=3, &c. is meant. 100 HISTORY OF THE THEORY OF DETERMINANTS "sur un nombre egal a n de lignes horizontales et sur autant de colonnes verticales," and as thus arranged are said to form a symmetric system of order n. The individual quantities a1., &c., are called the terms of the system, and the letter a when free of suffixes the characteristic. The " terms" in a horizontal line are said to form a suite horizontale, in a vertical column a suite verticale. Conjugate terms are defined as those whose suffixes (" indices ") differ in order, e.g., a2.3 and c3.2; and terms which are self-conjugate, e.g., a,,, a,.2,... are called principal terms. The determinant is said to belong to the system, or to be the determinant of the system. The parts of the expanded determinant which are connected by the signs + and - are called symmetric products, and the product a1'1a2'2a'33.'.. an n of the principal "terms" is called the principal product. The " principal product," however, is also called the terme indicatif of the determinant, and thus an awkward double use of the word " terme " is brought into prominence. The system Ia11 a2.1 a3.1.... an a1'2 a2'2 a3'2 * * * n'2 al.3 a2.3 a3.3. an3 I.......... al'n a2 'n a3'n * * * n'n derived from the previous system by interchanging the suffixes of each "terme" is said to be conjugate to the previous system. A symbol for each of these systems is got by taking the last "terme" of its first "suite horizontale," and enclosing the "terme" in brackets: in this way we are enabled to say that (ac.,) and (an,.) are conjugate systems. In the course of these explanations a modification of the rule of term-formation is incidentally noted, the form taken being specially applicable when the quantities of the system have been disposed in a square. Cauchy's wording of this now familiar rule is (p. 55)-.... "pour former chacun des termes dont il s'agit, il suffira de multiplier entre elles n quantit6s diff6rentes prises respectivement dans DETERMINANTS IN GENERAL (CAUCHY, 1812) 101 les diff6rentes colonnes verticales du systeme, et situees en meme temps dans les diverses lignes horizontales de ce systeme." (ii. 6) Here 'we may note in passing that the disposal of the " termes" in a square might have enabled Cauchy to point out (which he did not do) the difference between Gauss' use of the word "determinant" and his own, by saying that the " determinant of a form " had its conjugate " termes " equal. The rule of signs applicable to alternating functions in general is modified for the special case of determinants, and takes the following form (p. 56):"1tant donn6 un produit sym6trique quelconque, pour obtenir le signe dont il esb affecte dans le determinant S( 4al.2.2a3.3 8... a-'-) il suffira d'appliquer la regle qui sert a determiner le signe d'un terme pris a volonte dans une fonction symetrique alternee. Soit aa,. a.2.... a., le produit symetrique dont il s'agit, et designons par g le nombre des substitutions circulaires 6quivalentes a la substitution /1 2 3..... n\ a p y..... C/. Ce produit devra 6tre affect6 du signe + si n - g est un nombre pair, et du signe - dans le cas contraire." (III. 19) Thus if the sign of the term a6.' a. a.8 a.4 a9.5 a2.6 a5.? a4.8 a7.9 in the determinant S( al.1 a2.2 a3..... a9.9), be wanted, we write the series of first suffixes 6, 8,... under the corresponding suffixes of the "principal product," that is to say, under the series 1, 2, 3,..., 9, obtaining the substitution /1 2 3 4 5 6 7 8 9\ \6 8 3 1 9 2 5 4 7/; this we separate into circular substitutions, finding them three in number, viz., 3 5 7 9\ 1 2 4 6 81 \3, \9 5 7/, \6 8 1 2 4/; 102 HISTORY OF THE THEORY OF DETERMINANTS and the determinant being of the 9th order, we thence conclude that the desired sign is (-)9-3, i.e., +. In connection with this subject a modification of Cramer's rule is given, no reference being made to "derangements" at all. Put into the fewest possible words it is-The sign of the term aa,. ap.2.... a.n is the same as the sign of the difference-product of the first suffixes, that is, the sign of (/3-a)(y-a).... (a -)(Y-)..... (III. 20) For example, the sign of al a8'2 L3'3 a14 a'5 a2'6 a5'7 4'8 a7'9 above sought, is the sign of the difference-product of 6,8,3,1,9, 2,5,4,7 i.e., the sign of (7 -4)(7 - 5)(7 - 2)(7 - 9)(7 - 1)(7-3)(7 -8)(7 - 6) x(4-5)(4-2)..(4-6) x(5-2)....(5-6) x(8-6) The object which Cauchy had in view in stating the rule in this unnecessarily complex form was doubtless to show its essential identity with the rule implied in his new definition. He says (p. 58)"On demontre facilement cette regle par ce qui precede, attendu qu'une transposition oper6e entre deux indices change toujours, comme on l'a fait voir, le signe du produit (a, - a)(a, - aa)... (a, - a,)(aa - a,)...., et par consequent celui du produit (/- a)( y- a) a)... ( y- (y- /). * * " The way having thus been prepared, the propositions of determinants are entered on. Those known to his predecessors we may dispose of rapidly, giving little, if anything, more than the enunciation of them, in order that the new: garb in which they appear may be seen. DETERMINANTS IN GENERAL (CAUCHY, 1812)10 103 *... "le determinant du syste'me (a,,.,) est e'gal 'a celui du syste'me (a,.,),).... En cons6quence, dans l'expressilon on peut supposer indiffldremment, on que le signe S se rapporte aux premiers indices, ou qu'il se rapporte aux seconds. (ix. 2) Si l'on e'change entre elles deux suites- horizontales on deux suites verticales du syste'me (a,.,,) de manie're 'a faire passer dans une des suites tons les termes de l'autre et 'reciproquement on obtiendra un nouveau syst~me syme'trique, dont le determinant sera e'videmnient egal mais de signe contraire 'a celui du syste'me (a,.,,,). Si 1'on r~pe'te la m~me opiration plusieurs. fois de suite, on obtiendra divers syst~mes syme'triques dont les d~terminans seront 6ganx entre eux, mais alternativement positifs et n~gatifs. On pent faire la me'me remarque a l'6gard du syste'me (a,,.). (xi. 3).si l'on de'veloppe la fonction syrn,1trique altern~ee S8[ ~a,.,,S ( +a1.1a2.2.... tons les termes du d~veloppement seront des prodnits syme'triques de 1'ordre n, qui auront 1'unit6 pour coefficient. Ces termes seront done respectivement e'ganx 'a ceux qu'on obtient en d6veloppant le determinant Dt= S(~a,.,a2.2 a.'.) et comme le prodnit principal a1.1al2.2. a.,.,, est positif de part et d'autre, on aura ne'cessairement D S a,.S(4a,.,a2. 2.... a,..)](vi. 3) En g6ne'ral, si l'on de'signe par I l'un des indices 1, 2, 3,...,n on trouvera de lameme manie're = D S [~a,,.,,S(~+a1.la2.2.... aAM.,,_ja,L~1j.,L~1.... a.. )]. (vi. 4)....Cette dernie're equation 0 == al.vb1,.IL + aJ2.,,2. +...+ a,,.vb1,m, (xii. 6) sera satisfaite toutes les fois que v et 1- seront deux nombres diff6rens l'un de l'autre....on aura donc aussi D. = a.,~b,.1 + a,2,- +.... b + n (vi. 4) 0 = a,bt.jb, + av.2b,-2 +... + av,. nb/4I (xii. 6) les indices u et v e'tant cense's ine'gaux." 104 HISTORY OF THE THEORY OF DETERMINANTS The expressions here denoted by b.l, b1g,.... are spoken of as adjugate (" adjointes ") to a,., a,.2..; and the system bl. br2.....b }2.x b 2.2.... b2.1 as adjugate to the system (a,.). Similarly the system (b,.) is said to be adjugate to the system (an.l); and, on the other hand, it is said to be adjugate and conjugate to the system (a.,). (xxvII. 2) Up to this point no new property has been brought forward. The following paragraph (p. 68), however, opens new ground, the formula given in it being of some considerable importance in the after development of the theory. "Si dans le systeme de quantit6s (a.,,) on supprime la derniere suite horizontale et la derniere suite verticale, on aura le systeme suivant, 1a,.,,1 21 a, * * * alna2.1, a22 a2.n-1, &c... a-ll, can-1a -2n-ln-l que je designerai a l'ordinaire par (a,.,_,). "Soit maintenant (el.,,_) le systeme adjoint au pr6c6dent. Si dans l'6quation (13) on change b en e et n en n - 1, on aura en general Dn-l = b,., = a,.le,,.l + a,.2e,.2... + a...,,e-,,-l Pour d6duire de cette derniere equation la valeur de b6., il suffira en vertu des regles Btablies, de changer a,., en a,., dans l'expression precedente de b,,., et de changer en outre le signe du second membre: ou aura done g6n6ralement bn., = -(a.le -l + a-.2e,,.2 +..... + an.,-le,.,n-l) Si dans cette 6quation on donne successivement a p toutes les valeurs entieres depuis 1 jusqu'a n - 1, et que l'on substitue les valeurs qui en resulteront pour b.,,, b2-,..., b,_,., dans l'equation D = a.l.bl. + a2.,b2.n +.... + + a.,b.,, on obtiendra la formule suivante, '^al.nanlel-l + a2-.,a..2e2.2 +... + a_-l.A.-_en-1.n-l + al.,(a.2e1.2 + a.1.3 +... + an.-len-l) D= an.,b., - + a2.,(an,. le2. + a,.e2.3 +... + a,.e2n.l) + &........ + an-.,,(anlen-l, + a.2e,,-1.2 + - + a,.n-2e,,-l-.,2). DETERMINANTS IN GENERAL (CIAUCEHY, 1812) 105 Cette equation peut Atre mise sous la forme Dn = aiz~nD*~n~l - S"1n-S (arv.,a,.yL (xxxvII.) les deux signes S etant relatifs le premier a l'indice 1L et le second ' l'indice v." This is the well-known formula nowadays described as giving the development of a determinant according to binary products of a row and column. The special row here used is the nth and the special column the nlth likewise. The four pages regarding the application of determinants to the solution of a set of simultaneous equations may be passed over with the remark that they give evidence of the importance attached by Cauchy to his new definition of determinants, the solution in the case of the example aix,+b1x2 = ml~ a2xl +b2x2 = m2f being first put in the form mb(b-m) am(m - a). X-ab (b -a) ab n(b -a) and similarly in the case of the example arx, + box2 + rX3 = mr (r = 1, 2, 3). The determinant solution of a set of simultaneous equations is put to good use by Cauchy to obtain new properties of the functions. Taking the set of equations allA + a1,.22 -+.. + cx = afl1 (20) j2i1 i. a2'2X2 + -. + a2.nxn = 2 &C..a,'lx + a.2x. +. + a,', = ma and solving for x1, x2,... he obtains of course the set m1b1.l + m2b2'. +. + mnbD. l = Dx, rnAb1.2 + m2b.2 +. + mnb.2 = Dx2,1 M~b1'n + m~b2'n+.+ mb, = Dwx,J *Misprint in original, for D -,. 106' HISTORY OF THE THEORY OF DETERMINANTS wherb b2.1)......have the signification aboeidct, and D. stands for S ( ~a1.1a2.2...a,,). This second set may be treated in Ithe same way as the first set, the quantities in1, 'i?2,....,m, being viewed as the unknowns. To express the- result the system of quantities adjugate to (b1.,) is denoted by (c.) and the determinant of the system (bl.n) is denoted by BW, the new set thus being ~cl'AD 1 + c1.2Dx2 +......+ Cl.nDnxn = B nWl1, (27) c21D CT X + C2 2DAX2 +.-4 ---+ C2.'IDnXn = n2 K..1DnX1 + C..2D~x2 + + Cn n~ = BnMn Cauchy then proceeds (p. 77)"Les equations (27) peuvent encore 6tre mises sous la forme suivante, c1. Dxi 12 nX + + cln,= ml)t D'nX + Dn~ + cl2- D.X =M2 D D n D = C"1n,+ Cn.2FX2 + Cn. nff~n n M et comme celles-ci doivent avoir lieu en m~me temps que les equations (20), sans que l'on suppose d'ailleurs entre les termes de la suite XI'X2..,x,, et ceux du syste'me (a,..,) aucune relation particulie'e, il faudra ne'cessairement que l'on ait, quels que soient IA et v, ou ck.,. a, (xxxviii.) Cette equation 6tablit un rapport constant entre les termes du syst~me (a,.,,) et les termes du syste'me adjoint du second ordre (c1.n,)." More definitely, and in more.modern nomenclature, the theorem is The ratio of any element of a determi'nant to -the corresponding element of the second adjugate determinant is eq ual to the ratio of the determinant itself to iltsfirs8t adjugate. (xxxviii.) Attention is next directed to the group of equations DETERMINANTS IN GENERAL (CAUCHY, 1812) 107. II II II 11 II + ~ ++ *+ + -+? ~ + + <c, ~ t ' & & + + + II + + + t;$ a 3 ti r( v: 6Q & II II II ++ 4~F ++ +J H3 H H Hl H H H3 H~ H H$4 H 108 HISTORY OF THE THEORY OF DETERMINANTS Here there are three symmetric systems of quantities (al&) (al.), (.ln), the first appearing in every column of equations, the second in every row, and the third only once. The determinants of these systems are denoted by Dn, J, M,, respectively: that is to say D, = S(~t a,.a a,.. a^) Dn= S(~ all a2.2. a.n) Mn = S ( m.m2.2*.... nn). If now in S ( ~mi rm2.2.....n) there be substituted for mn. m,.... their values as given by the group of equations, there will be obtained a function of all the a's and a's, which must be an alternating function with respect to the first indices of the a's and also with respect to the first indices of the a's. Further, since each of the m's is of the first degree in the a's and of the first degree also in the a's, each term of the development of S(~m.m1r2.2... mnn) must evidently be of the form ~ al.j(l2. V.... a.* * n7r.tl.l a2 *. * * Can. But the development by reason of its double alternating character cannot contain such a term without containing all the terms of the product ~ S ( ~ a1.la2.... a.7r)S( + a1.a2.v.. an.). Consequently it must equal one or more products of this kind. But again the indices /u, v,..., 7r are either all different or not. If they be different, we have S( ~ al.,a2.v. an.,) =: S(f a.11a2.2-2 * * * an-n) = ~ s; and if any two of them be equal S( ~al..a2.... anr) = 0. The like is true in regard to S~(a.ra2.v... a.,). This DETERMINANTS IN GENERAL (CAUCHY, 1812) 109 enables us to conclude that the development of M. is equal to one or more products of the form ~ A; in other words, that Mn= cDn,n where c is a constant. But if we take the very special case where ad. A=1, a,^. = 1, av = O, av =0, and where consequently m. =l, m1.v= 0, we see that M,=l, Dn=l, n,=1, and that therefore c=l. Hence the final result is Mn= DAn. (xvII. 5) This, the now well-known multiplication-theorem of determinants, Cauchy puts in words as follows (p. 82):Lorsqu'un systeme de quantitys est determie' sym'triquement au moyen de deux autres systemnes, le determinant du systeme resultant est toujours egal au produit des dMterminans des deux systemes composans. (xvII. 5) It is quite clear, from what has been said above, that it was discovered independently, and about the same time, by Binet and Cauchy, and ought to bear the names of both. Binet has the further merit of having reached a theorem of which Cauchy's is a special case, and then made an additional generalisation in a different direction; and Cauchy has the advantage over Binet of having produced, along with his special case, a satisfactory proof of it. From the theorem Cauchy goes on to deduce several results equally important. Substituting for the system (al,.) the system (b.) adjugate to (am.) so that 8n = S(+ i.b2.'2.. b,.n) = Bn, we know that then me. = Dn and m,. =0; 110 HISTORY OF THE THEORY OF DETERMINANTS that consequently M% Iconsists of but a single term, viz. and that therefore by the theorem whence B~ =D n-l (xxi. 2).This result, afterwards so well known, Cauchy translates into words as follows (p. 82): —...le de'terminant du systime (bl.,,) adjoint au systeme (1.n ) est 6gal ' la (n - i)'me puissance du de'terminant de, ce dernier syste'ne. (xxi. 2) Again, by returning to the identity n and substituting the value -of B n just obtained, there is deduced the result C1111P = n-2 a,.,;(xxxviii. 2) or, in words, e tant donn6 un terme quelconque- a,, di% syste'me (a,1.), pour obtenirr le terme correspondant du syste'me a'djoint du second ordr-e (c,..) il suffira de multiplier le terme donne' par la (n - 2)me puissance du determinant du premier syste'me. A considerable amount of space (pp. 82-92) is devoted to the consideration of the adjugate systems of and the adjugates of these adjugates; but nothing new is elicited. The section closes with the manifest identity (a1.1 + a2.1 +. + a,..,) (a,., +a a2.1+ + a., + (a1.2+ a2.2 +. + a,.2) (a1.2 +a 2-2 + +a. + (alin + a 2.n +. + a n') (a,,, + a 2n +~ +a nn) = rn 1.1 + Mn2.1 +... + Mn.2 + mn1.,n + M 2.n +... + mn', DETERMINANTS IN GENERAL (CAUCIY, 1812) 111 which, using later technical terms, we may express as follows:If there be two determinants, and the sum of the elements of one first column be multiplied by the sum of the elements of the other first column, the sum of the elements of one second column by the sum of the elements of the other second column, and so on, then the sum of these products is equal to the sum of the elements of the product of the two determinants. (xxxIx.) The third section breaks entirely fresh ground, its heading being Des Systemes de Quantites derivees et de leurs Determinans. Of the integers 1, 2, 3,..., n all the possible sets of p integers are supposed to be taken, and arranged in order on the principle that any one has precedence of any other if the product of the members of the former be less than the product of the members of the latter. The number n(n-l1).... (n-p+1)/ 1.2.3.... p of the said sets being denoted by P, the pth and last set would thus be n-p+l, n-p+2,...... n. Now, any two of the sets being fixed upon, say the /th and vth, the system of quantities (a.,) is returned to, and from it are deleted (1) all the "termes" whose first index is not found in the Mth set, and (2) all the " termes" whose second index is not found in the vth set. What is left after this action is clearly "un systeme de quantites symetriques de lordre p," the determinant of which may be denoted by l)., For example, if,= == 1, all the a's would be deleted whose first or second index was not included in the set 1, 2, 3,..., p, and there would be left the system ainl a1-2.. al.p Ia.1 a.2 a~... a2.l a22.... a aP.-^ 1 pes A.... topp of which the determinant would be denoted by a(,1. 112 HISTORY OF THE THEORY OF DETERMINANTS As any one of the P sets could be taken along with any other, preparatory to forming such a determinant, there would necessarily be in all P x P possible determinants. Arranged in a square as follows: Ia(P a(P. a.. il1 P2 PP they manifestly form "un syst'eme syme'trique de l'ordre P," which, in strict accordance with previons convention, is denoted by "(4P Cauchy then proceeds (p. 96)Si l'on donne successivement 'ajp toutes les valeurs 1, 2, 3....,n-3, n- 2,n-lI P prendra les valeurs suivantes, n(n-l) n(n-1)(m-2) n('n -1) n 1.2 ' 1. 2. 3 ' ' 1. 2' et l'on obtiendra par suite un nombre elgal 'a n -1 de syste'mes syne'triques dilfhrens les uns des autres, dont le premier sera le syst~me donne' (a,.,,). Ces diffhrens syste'mes seront de'sign6s respectivement par 3 (n - 3) ~ (qz-2) [a:w1~2, [1I a, I.n- (a~" ~;(XL.) je les appellerai systmres de'rives de (a.,.Parmi ces syste'mes, ceux qui correspondent 'a des valeurs de p dont la somme est e'gale 'a n sont toujours de me'me ordre; je les appellerai sys times de'rivgs comple'mentaires. Ainsi en g6n6ral (a(P)) et (a n;P) sont deux systbmes derive's compl~mentaires lFun de l'autre, dont l'ordre est e'gal a? nn-1..(npl p == (n1).2.3.. (n-+1 Up to this point a thorough understanding of the notation (ar) is the one essential. Taking the particular instance (a (2)0 DETERMINANTS IN GENERAL (CAUCHY, 1812) 113 we first call to mind that it is an abbreviation for the " systeme symetrique" whose first row has for its last " terme" the determinant 1'10 -that is to say, an abbreviation for the system whose determinant we should nowadays write in the form at(2) (2) ( ) a(2) 1.1 1.2 ' ' ' C1.10 a(2) (2) a (2) 2.1 62.2 2.10 a(2) a(2) a(2) C10.1 610.2 ' ' 10.10 The next point is to realise what determinants are denoted by a (2) a(2) 1.1, 2......... Now the number 10 being of necessity a combinatorial, and, as the figure in brackets above it indicates, of the form n(n - 1) 1.2 ' we see that n must be 5, and that the said determinants are all derived from a1.l a1.2 a.'3 a1.4 a'5 2'.1 a22 (t23 a2'4 a2'5 a,.41 4'2 a4.3 c4.,4 a4'. a5.1 a5.2 a5.3 a5.4 a5.5s The details of the process of derivation are recalled in connection with the interpretation of the pairs of suffixes. A requisite preliminary is to form all the different pairs of the numbers 1, 2, 3, 4, 5; arrange them in the order 12, 13, 14, 15, 23, 24, 25, 34, 35, 45; and then number them 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. These last are the numbers from which the suffixes are taken, and what each one as a suffix refers to, is the combination under which it is here placed. For example, the first suffix in a(2 refers to the combination 1 2, and implies the deletion of all the rows of the above determinant of the fifth order except the 1st and M.D. H 114 HISTORY OF THE THEORY OF DETERMINANTS 2nd; the second suffix refers to the same combination, and implies the deletion of all the columns except the 1st and 2nd; and the symbol as a whole thus comes to stand for al.1 a1'2 a2. a22 (XL. 2) Interpreting a(12, a (2.. in the same way, we see that -(a (2) 1(a.10) is a compact notation for the system of which the determinant is It (6-lall 2 -2 n * * n 1 Ct3-3665-4 1 X | as 3ff5*5 1 al4a2'5 I a3.4a4.5 a,] aa |.., a3.3c54, a3a5\5, a.34a5.5 Ia4~5'21,. a,a |3a 41 I a 4 3Ct5 1, a4.4a55 1. Similarly (al0o) stands for the system of which the determinant is I a1C2'2a3'.3 a, l3a2-4a',5 I......... a..... I 2.3 3. 4a5.5..?...... * * * * *, I ~ 23a34ass 1 and which is called the "complementary derived system." (XL. 3) To every "terme" of the latter there corresponds a "terme" of the former, the one "terme" consisting exactly of those a's of the original determinant which are awanting in the other. This relationship Cauchy goes on to mark by means of a name and a notation. He calls two such "termes," la3.a42.a.s3l and la1.4a2.5 for example, "termes complementaires des deux systemes;" (XL. 4) and if the symbol for the one be by previous agreement at(P) the symbol for the other is made a( - P) (XL. 5) P-[t+I.P-r+lI * If Cauchy had adopted a slightly different principle for determining the order of combinations, the /th combination of p things and the (P - + 1)th combination of n -p things would have been mutually exclusive, and the convention here made in regard to notation would have been unnecessary. DETERMINANTS IN GENERAL (CAUCHY, 1812) 11 115 As for the signs of the " terines " in " derived systems," Cauchy's words are (p. 98)"En ge'ne'ral, il est facile de voir 'que. le produit de deux termes complementaires pris 'a volonte' est toujours, an signe pres, une portion de cc m~me determinant (D,,). Cela pose', e'tant donne' le signe de l'un de ces deux termes, on d~terminera celni de l'antre par la condition que leur prodnit soit affecte' de me'me signe que la portion correspondante du determinant DR." All these preliminaries having been settled, the weighty matters of the section are entered on. The first of these is a complete and perfectly accnrate statement of the expansiontheorem, known by the name of Laplace, but which, as we have seen, Laplace and even Bezout who followed him were- very far from fully formulating. The passage is of the greatest interest. No better example could be chosen to illustrate the powerful grasp which Cauchy had of the subject. What Laplace, and Bezout laboured at, lengthily expounding one special case after another, Cauchy sets forth with ease and in all its generality in the space of a page. His words are (p. 99)"Onl a fait voir dans le ~ 30 que la fonction syme'trique altern~ee S (~-a. ia2.2a3..3... a. = n 6tait. 6quivalente 'a celle-ci S[+~S(~-al.,a2.2.. a,-,1,,1).a,.] On f era voir de m~me qu'elle est enc-ore equivalente 'a les operations ilndique'es par" le s~igne S pouvant 6tre consid~re'es comme relatives, soit aux premiers, soit aux seconds indices. On a d'ailleurs par ce qui pre'cede S(-~,+,-Pl.a. a..) a —p Enfin les signes des quantite's de la forme (YI(P) a~ P doivent 6tre tels que les produits semblables 'a 116 HISTORY OF THE THEORY OF DETERMINANTS soient das le d~teminant D~affect's du signe +.Cela pose, ii r~sulte de l'iquation que D,, est la somme de plusieurs produits de la forme a(p a(n-P) a11 P. Scion que pour obtenir ces diff~rens produits on e'changera entre eux les premiers ou les seconds indices du syste'me (a,.,,), on trouvera oun l'equation D ap a (nP + ()a~-~ + +.a.. +. nou celle-ci Dn= ap~' a n- + a()an-P +. +(P On aura de me'me en g~n~ral les deux equations (p) (n -p) (p) (n-) (p) (n-p) D21== a 1.7rtP. — r+l + a 2. apIj~p_7~i +..+ a raLP-7+ D=a(p) (n -P) (P) (in-) +a (P) a(n- P) D x= a, pa- 1 + 1.p + +,, ap,+.... +k1. Ces deux 6quations sont comprises dans ia suivante qui a lieu 6'galement, soit que l'on considere le signe S comme relatif 'a l'indiceu~, soit qu'on le considere comme relatif 'a 1'idice wr." Taking as an illustration the case where n =5, p ==2, and -=7r (that is, the ordinal number corresponding to the pair 2 5, of the suffixes 1,~ 2, 3, 4, 5), and translating literally from Cauchy's notation into our own, we have a11a,22 a33 a44a 5.5 = la12a25 1.1 a 4 - la12a35 Ct1 21a,43a54 I+..+ la42a55 -Ial2341 With the same certainty of touch and with still greater conciseness, all the identities directly obtainable by Bezout's JBlikthode pour trouver des fonctions.... qui soient zero par elles-me'rnes, are f ormulated as one general identity, and established on a proper basis. The paragraph is (p. 100)"ia tant une fonctio 'yntrique altern'e des indices du syste'me (a1l.,,) doit se re'duire 'a z~ro, Iorsqu'on y remplace unl de ces indices par un autre. Si l'on ope're de semblables remplacemens 'a e'~gard des indices qui occupent la premie're place dans le -syste'me (a1.,), et qui entrent dans la combinaison ([z), cette me'me combinaison se DETERMINANTS IN GENERAL (CAUCHY, 1812) 117 trouvera transformee en une autre que je designerai par (v), et ac() sera change en a(.. D'ailleurs, en supposant le signe S relatif a 7r, on a D, = Sp(c,(p) (n -p) D= S [%.7t rP-/t+L.P-7r+l); on aura done par suite o = S (p ).p - r+lX) (XII. 7; XXIII. 4) On aurait de meme, en supposant le signe S relatif a l'indice /u, et en designant par (r) une nouvelle combinaison diff6rente de (7r) 0 = S (a(P.) a(' -_:+i.pr+i) (XII. 7; XXIII. 4) As this theorem is twin with the preceding, it is best to illustrate it by the same special case. By so doing, indeed, both theorems become more readily grasped and their details better understood. Taking then as before t = 5, p =2 and -r = 7, we first form the determinants which Cauchy would have denoted by a(2), (2) a(2) 1.7 ' t2.7 10.7' and which we denote by la12251 ' al12a 35.1 ' ^' ' ', 42a55 1 Next, for cofactors, we form the determinants which are complementary, not of these, as in the preceding theorem, but of the members of one of the nine other groups corresponding to the values 1, 2, 3, 4, 5, 6, 8, 9, 10 of 7r,-say the group (2) (2) a(2) 1.6 t2.6 '.10.6 These complementaries being la31a43,551 ' I 21, 43.... | al 23a35, 1 we have the desired identity 0 = l 2a25l 1la3,a43a55l - I 12a351 1.la,43551...+ la42C551- lla23 351 the right-hand side of which is nothing more than an expansion of the zero determinant which arises from the determinant alla22C%3a44ca5 "lorsqu'on y remplace un des indices par un autre," viz., the second 4 by 5. With the help of these two theorems a third theorem of almost equal importance is derived, viz., regarding the product of the determinants of two complementary systems. Denoting the determinant of the system (PC)) by D(), (L P/ 118 HISTORY OF THE THEORY OF DETERMINANTS and that of the complementary system (a(-P)) by Dn-p), and multiplying the two determinants together, we see with Cauchy that by (xiv. 4) the principal "termes " of the resulting determinant are each equal to D,, and by (xII. 7) all the other " termes " are equal to zero. Consequently D( ). DM-)= (D) (XLI.) As an example of this theorem, it may be added that the product of the two determinants printed above (p. 114) to illustrate the notation ( P) \('l. P/ that is to say, the determinants of the systems (a(.2), ((130) is equal to a11a2.'2Ca33as4.'4a'55 10 In connection with all the three theorems, the special case p =1, is given, so that their relation to previously well-known theorems (vI., XII., xxI.) may be noted. It is also pointed out, that when in the third theorem n is even and p = In, the result takes the interesting form InN P D ) (D), (XLI. 2) This brings us to the last section of the memoir, the fourth, bearing the heading Des Systemes d'Equations derivees et de leur Determninans. What it is concerned with is the relations subsisting between a " derived system" of the product-determinant M12.1 n22.* * * n2 n'1 9mn.2... Tmn DETERMINANTS IN GENERAL (CAUCHY, 1812) 119 and the corresponding " derived systems" of the factors al1 a. 2 ~... a~ - al' a aln a2'1 C2'2 * *. 2'a1 a2l 2 * * a2 an'l an'2 n'n * n'l an I in other words, the relations which must connect the systems (a(1p), (a(vl), (m(p) 1'.P] 1 \ 1.P \ 1.P] by reason of the relations Z[Sn(Ca.1.1)= a. v] (given in full above on p. 107) which connect the systems (al'-) ' (aiJn) I (ml n)First of all, attention is concentrated on a single "terme" of -the system (p) ( IWT1. P.A or, as we should nowadays say, on a minor of the productdeterminant. The process of reasoning, which occupies about four quarto pages, is exactly analogous to that previously followed in dealing with the product-determinant itself; and the result obtained is MW = SP(a[(.P (P), (xvIII. 5) where SP is meant to indicate that the terms on the right-hand.side are got by changing the second suffixes into 2, 3, 4,., P in succession. Speaking roughly and in modern phraseology, we may say that this means that Any minror of a product-determicnant is expressible as a suzm of products of minors of the two factors. (xvIII. 5) Cauchy then proceeds (p. 107)"Si dans cette equation [xvIII. 5] on donne successivement &a u et: v toutes les valeurs entieres depuis 1 jusqu'a P, on aura un systeme d'tequations symetriques de lordre P, que l'on pourra representer par le symbole (63) V{SP(a[), a. =.,) =m( }, 120 HISTORY OF THE THEORY OF DETERMINANTS P 6tant toujours 6gal h (n -1) ('n- p1) 1.2.3... Pour deduire des equations ~ [S"(a, a,.,) = mv] les 6quations (63), ii suffit 'videmment de remplacer les trois systemes de quantit6s par les systemes d'riv6s de mnme ordre (avp (m/p)1.P Je dirai pour cette raison que le second systeme d'equations est d~riv6 du premier." (XL. 6) The close outward resemblance here noted between the original and the derived system of connecting equations is due of course to the choice of the notation 1.P for the minors of the determinant S ~- (ca1_1c. a22 an.. and is so far a recommendation of that notation. From the system of equations (63S) two deductions follow immediately. In regard to the first Cauchy's words are (p. l08) — "Designons par 8(p) T\CP) M(P)~ P UP IP les d~terminans des trois systemes (a(23) ) a() (MP on aura en vertu des equations (63) (65) - = ID " (XLI.) The enunciation of this in modern phraseology would beAny compound of a product-determninant is equal to the product of the corresponding compounds of the two factors. (XLII.)) DETERMINANTS IN GENERAL (CAUCHY, 1812) 121 The next deduction is stated equally succinctly (p. 109)"Si l'on ajoute entre elles les equations (63) on aura la suivante, (66) SP1 SP(a(p)SP(a(C ) = S ( (xxx. 2), le premier signe S, c'est-a-dire le signe exterieur, etant relatif a l'indice v, et les autres, c'est-a-dire les signes interieurs, etant relatifs a l'indice,." This (66) corresponds to (xxxix.) as (65) corresponds to the multiplication-theorem Mn = Dn6n, the transition from the general to the particular being effected in both cases by putting p = 1. With these deductions, the 4th Section practically comes to an end; but one or two results, intentionally omitted in the account of the 2nd Section because they seemed to belong naturally to the 4th, fall now to be noted. The first is very simple. It arises (p. 91) from observing that (D)n - x (6,)n - = (D,,jn -1, and.. = (MI) by the multiplication-theorem. The result (xxI. 2) above (p. 110),. is then thrice applied, and a theorem at once takes shape, which in later times we find enunciated as follows:The adjugate of the product-determinant is equal to the product of the adjugates of the two factors. (XLII. 2) It is not noted, however, by Cauchy that this is but a case of' XLIII., viz., where p = n- 1. The next is z[S"(,21.v b.l) = Dr)V], or [S (m,.1 ' ) = L.J. (XLIII.) It is nothing more than the result of solving the n.n equations (33) J[S.(a. a,.1) =m,..] first, in columns, for all the a's, and secondly, in rows, for all the c's. 122 HISTORY OF THE THEORY OF DETERMINANTS The last is YE[Sn(al.,L ~1 or ~ [9(~~'1)=(XL1II. 2) -where (r1.,,) is the system adjugate to (m1.n). It is obtained -from the n.n equations (XLIII.) just as they were obtained from -the n.n equations (3-3), use being made of the theorem In concluding, Cauchy refers to Binet's researches on similar matters. Most of what he says in regard to them has already been given (see p. 9:3 above). The rest of it is as follows '(P.11 - "I [iEhnet] me dit en outre qu'il avait ge'n6ralise' le the'ore'me dont il s'agit [M,,, = DU6,8J, en. substituant an produit de deux re'sultantes -des sommes de produits de me'ine espece. J'avais de's lors d~ja' d&montre' le the4orenme suivant: D'nm syslmre quelconque d'e'qualions syn~lriques on Pent diduire cinq autres syst~nmes du mone ordre; mais on n en sautrait dcleuire Un -Plus ~,g)andl nomibre. J'ai dernontre6 depuis 'a laide des ni'thodes prbce'dentes cet antre the'ore'me: D'un syste'me quelcooque d'6quations syrnitriques delordre n, on peztt toujus d~duire deux systb~nes d'6~qtations symn'riques de l'ordre n(n- 1 2 &lemx systnies cl'equations symdtriques (le ['orcire n (n -) (n -2)&. En ajontant entre elles les equations syme'triques comprises dans -Un rnbre systebme, on obtient, comme on l'a vn, les formules (50), (51) vet (70) qui me paraissent devoir etre semblables 'a celles dont M. Binet in a parl&'" The last sentence here raises an important question for the historian to settle, viz., whether Cauchy is to share with Binet -the credit of the generalisation of the multiplication-theorem. 'The identities on.which the, claim is based areSn-noLv Sn(b,~.X) S,,Sn(r,~.v) (51) Sp S ( () S ( ()) SPSPmP) (70) DETERMINANTS IN GENERAL (CAUCHY, 1812) 123 The first of these, given formerly (p. 110) in the uncontracted form (am.~+a,.i +..~ ~ +,.,)(ca.,l+a;.'+. + 1,,.) +(al.2+a21.+. +,a.2) (Ct+a2. +... +a,.) +:......:. +(a+, + a(- 2. +. ~ + al~~)(al' + (a2 +... + an'~) = mrnl+ 2.-1 + + +m.' ++m12 + 12 2 + + m.2 +........... + n.,l +. + nn'. where r,. v= C.*la.l + a.2av.2 +.. + *aC.nv.n, may be at once left out of consideration; it is not even a case of the multiplication-theorem. Cauchy, we may be sure, mentioned it only because it is the first of the series to which (51) and (70) belong. The next concerns the systems (ln, (bl,') (rl)n) adjugate to the systems (a0'9) I (a 1-n) 7 (MAln) dealt with in (50). It indeed is comparable with Binet's theorem; but as it is only a case of (70),-the minors in (70) being of any order whatever, whereas in (51) they are the principal minors, —we may without loss pass it over. Directing our attention, then, to (70) let us for-the sake of greater definiteness take the case where n =5 and p =2, and where consequently P = 4 — = 10. The theorem then becomes 1.2 S4(10(a (2) S10o(a(2) )} S1oSo(gn (2).) For the purpose of comparison with Binet's result, it is absolutely necessary, however, to depart from this exceedingly condensed mode of statement. Remembering that the inner S's refer always to the first suffix, and the outer to the second suffix, we obtain the more developed form 124 HISTORY OF THE THEORY OF DETERMINANTS (a(2) a2(2) (2 ).(2) 1.1+. a 2.1 + + o.. +10.1) 12 { 2... 10. 1.2 2.2 10.2 + o1.2 + a 2.2 + s u r(l02) 1(a2 n + a22.10 + to(2) i /N(2)-t- ~(x~;. + a~..~ +. + ~0........ ) ( + (................ (2).(2) +m(2) al1. + f21 0 + 2,, +,0510- f all0 yr C2. + for.) 1. + " 'l2.12 ~ " +n10.1 + " +.2 ++ 2.2 +,,. +.(2) { ^+m+. +.. C. ++ q{1.120 ~+ Kioa2 + 2 1 * + (10).10~ Interpreting now the suffixes and superfixes of the a's, a's, and m's, after the manner already described,-any suffix r signifying along with the superfix (2) the rth combination of two numbers taken from 1, 2, 3, 4, 5, —we finally reach the suitable form ~ {al.laa2.l + lal.la3.2 +.. + a3.a4. + \a3.4a(~. + 1la4-.a52} {la,1a22.l + la,.jal + *31..+ 1 24.. + -2l + +a4,a'1}, +(GL~31.,~. IY~C35+/....~IY3......... /1.'+ I2. 1.4 '21 +{|al3a2.5l + lal-.a3.-5 +.* a + a3-a451 + la3'3a55l + ICt443a-51.} {\I.1ma251 + 1.13,51 +* *... *+...3..+4.51 + a3.13a5.2l + la4.3a5.2I} +{la.l4a2-51 + la~4a3.5l + * *. + la3-a4.5l + la3m4a5.l- + mla4.345.5l}.{Ila 1.442.51 + ma.4rn3.5 +... + IaT44'451h + la3.4a.51 + lCa4.4a4.51} = Im.lm.21 + ImnLM3'2 +. + 1m3~lm4'21 + 1MT31"N21 + Im7 mlsaVM21 where mM.. = a,-.lav + aM.2O,.2 +... +q a/.55.5. The series of suffixes for the a's, a's, and m's are seen to be the same, the series of pairs of first suffixes in every row and the series of pairs of second suffixes in every column being 12, 13, 14, 15, 23, 24, 25, 34, 35, 45; that is to say, the combinations arranged in ascending order, of the numbers 1, 2, 3, 4, 5, taken two at a time. On the first side of the identity are 10 products, and as both factors of each DETERMINANTS IN GENERAL (CAUCHY, 1812) 1 2 0 product contain 10 terms, the result of the multiplication would be to produce 1000 terms of the form larGpusqd. a mpanq, the whole expansion in fact being q=5 s=5 n=5 _ E aE p asjp. ampCManq. q=2 s=2 n=2 p<q r<s nz<n On the right-hand side are 100 terms of the form Vq,and if a proof of the identity were wanted, we should only have to show that each of the 100 terms of the latter kind gives rise to a particular 10 terms of the former kind. This, too, it is interesting to note, Cauchy himself could have done. For,example, the last of the 100 terms, a41 a4, + Ct42a42 + ~ a4,a45 a41a5, + Ct42a52 + ~a45"55 a.51a41+ a52a42 +. + a55a45 a51a51 + a52a52 +... + Ca55a55 CL41 a42 a443 a4445 a4.1 a42 a43 a44 a45 a.51 at52 a53 a.54 a5 a51 a52 a53 a154 a55 =41 a49 a41 cx42 + a4,a43 a41 a45 +.......+ a44 a45 44 a45 a51 a52 a51 a52 a51 a53 a51 a5 a54 55 a54 a55 which is nothing more than Cauchy's formula (62) M(P) = SP (a(P) aEA(.1) when we put /A =10 = P, and p = 2. Instead of 1000 terms on the left-hand side and 100 on the right, we should clearly have for the general theorem P3 terms on the left and P2 terms on the right, P be it remembered being the combinatorial n(n-l)(n-2).. (n-p+l) 1.2: 3... Leaving Cauchy, let us now return to Binet, and in order that the comparison between the two may be complete, let us formally 126 HISTORY OF THE THEORY OF DETERMINANTS enunciate in 'all its generality the latter's theorem also. Binet himself did not do this. After dealing with the case in which the determinants involved are of the 2nd order, he merely added (~p. 289)"On aura encore pour les integrales des re'sultats semblables, savoir, = Si { - c ~ lyzvxvlxz + lz`'yl =, f 17X~YI + fXTyIVZ + &c. } With the help of modern phraseology, the general theorem thus intended to be indicated can be made sufficiently clear. Binet in effect says: Take s rectangular arrays each with mn 616ments in the row and n elements in the column, in being greater than ni, viz.and other s rectangular arrays of the same kind, viz.From each array, by taking' every- set of n columns, form Cm~n determinants, arranging them in any order, provided it be the same for all the arrays. Add together all the 1st determinants DETERMINANTS IN GENERAL (CAUCHY, 1812) 127 formed from the first s arrays, and multiply the sum by the corresponding sum for the second s arrays; obtain the like product involving all the 2nd determinants, the like product involving all the 3rd determinants, and so on. Then, the sum of these products is equal to the sum of the products obtained by multiplying each array of the first set by each array of the second set. Or we may put it alternatively as a formal proposition, thus:If s rectangular arrays be taken, each with m elements in the row and n elements in the column, m being greater than n, viz. X1, X2....., Xs and other s rectangular arrays of the same kind, viz., Ag n 25 ** ) S s and if the minror determinants of the nt" order formed from XI, X2,..., -1, ZS be Xll X12 '' XeC 11 ~C12 ' '' 10 X21 X22.... X20 21 22 * *. 2C os2.. esC $sl s... C then (ll + x21-. + sl) (1 + 21 +. + s1) +(12 +X22+ *. +X,2) (12 +2+ 2.. + s2) + (o+2+ ~ ~ + + osC) ( IG+ o2a + +o) (X +X2 +... + X,) (1 + +.2 +. +), where C stands for Cm,n i.e., m(m —1).... (m —n+ 1)/1.2.3...... Now, counting the terms here as we did in the case of Cauchy's theorem, we see that on the left-hand side there are C multiplications to be performed, each giving rise to sxs terms, and that therefore the full number of terms in the development of this side is s2C; also that on the right-hand side the number is s2. 128 HISTORY OF THE THEORY OF DETERMINANTS In Cauchy's theorem the corresponding numbers were found to be PI and P2, P being not aly whole number as s is, but like C a combinatorial. Without further investigation, we might consequently assert that, supposing the two theorems to be alike in other respects, Binet's must be the more general, the passage from it to Cauchy's being effected by taking s=C. A closer examination, however, will show that this is not the full measure of the difference between the two theorems as to generality. Not only must we specialise by putting s = C, but s must become C in a very special way. In order to make this clear, let us take the particular case of Binet's theorem which approximates as nearly as possible to the particular case of Cauchy's given above. In the latter the determinants were of the 2nd order; therefore to get the comparable case of Binet's theorem we must put n =2. Again, since P in the particular case of Cauchy's theorem was 10, we must for the same purpose put 8=10 and Cmn,=lO, and.. rn=5. The result is 4' c31la32 + a21a22 + + aa al12 +. + a210,2 it CCG, ~ib, bz b a10,2 iiTaal aa10,1~ l, 11 12 21 22 101110,2 011012 P10,1010,2 + a1113l + a2la23 +dCt10a1 10,3 {ll 1a13 a+lo + alO,3 + b b b b b b I I 11 131 21 23 101 10,31 3110131 I10S1101 3 f a(C iat a!a a aa 1,., a10,5 14 15 2425 11047105 14 ]15 4 ~. b14b15 + b24 25 + iC10,4 "SOS -10, ( - 14$15 /010,4/10,5 I Cll a12... al+ a21a22... a25 ~a11i..2. a10,5 11 b12. b15 b21 22 2 10,1 10,2. 1O,5 1 2 15 + + 10,51,11,2' 4' a110a12... a55 + a211a22... a25 + + a10,1a10,2... a10,5 the elements involved b eing 200 in number, and disposable in two sets of arraysa11, a12... a 15 a21 a22. 125.... al0,1 a10,2... a,5 b1 b12... b15, b21 b22.b25).... b10,1 b10,2...10 DETERMINANTS IN GENERAL (CA'UCHY, 1812) 129 and all a12 a5 a21 a22 ' 25 aO 0,2 * a P11 P12 ' * P15' P2l22 ' * 25'.... Plo1.0,2 1 P10,5' In the corresponding identity of Cauchy there are only 50 different elements, viz., the elements of the two square arrays11 a12 ' ' ' 15 all a *2.. al5 21 a22 *.. a25 a21 a22... a25 a51 a52. a55, a51 a52. ~ ~ a55 Indeed,-and it is this which brings the comparison to a point,if from the first of these square arrays we form 10 rectangular arrays by taking every possible pair of rows, thus using each row 4 times over, viz., a, a12 * ~. a a 41 a42.. a5 a21 a22 *. a2, a231 32... a35,, a5 52 * *. a55 and similarly from the a's form a second set of 10 arrays, viz., a11 a12. * * a15 all al2... a5 a41 a42 * * * a45 a21 a22.' ' a25, al3 a2 ' ' a35,, a5 a52 * * a55' and then to these two special sets of arrays apply Binet's theorem, we obtain Cauchy's theorem. Regarding the two theorems in all their generality, the decision we have reached may therefore be expressed by saying that Binet's is a theorem concerning 2smn quantities, where s, m, n are any positive integers, and Cauchy's is a case of it in which s = n(rmn- 1)... (m-r+l)/1.2. 3...i, and in which, further, the number of different quantities involved is not m(m - 1)... (mA- n +l) 1.2... n xmn but by reason of repetitions is only 2m2. Although this decision is against Cauchy's claim as put by himself, it deserves to be noticed that, apparently by oversight, 3 1,D(.. I 130 HISTORY OF THE THEORY OF DETERMINANTS he failed to make his case as strong as he might have done. It will be remembered that Binet made two advances in the generalisation of the multiplication-theorem. In the first place, he gave the generalisation from which the multiplication-theorem is got by putting m = n, or, as we nowadays say, by substituting two square matrices for two rectangular matrices, and then he gave the theorem which we have been comparing with Cauchy's and which degenerates into his own first theorem when s is put equal to 1. Now the first of these generalisations Cauchy could justly have laid claim to. His identity (xvIIi. 5) is not indeed stated or viewed as a generalisation of the multiplication-theorem, but it is unquestionably so in reality. Ostensibly the identity concerns any minor of a product-determinant, but every such minor is obtained by multiplying together two rectangular matrices, and, conversely, every determinant which is the product of two rectangular matrices may be viewed as a minor of the product of two determinants. On looking back, however, at Cauchy's memoir as a whole, one cannot but be struck with admiration both at the quality and the quantity of its contents. Supposing that none of its theorems had been new, and that it had not even presented a single old theorem in a fresh light, the memoir would have been most valuable, furnishing, as it did, to the mathematicians of the time an almost exhaustive treatise on the theory of general determinants. It is not too much to say, although it may come to many as a surprise, that the ordinary text-books of determinants supplied to university students of the present day do not contain much more of the general theory than is to be found in Cauchy's memoir of about eighty years ago. One apparently trivial instrument, which Cauchy had not received from his predecessors and which he did not make for himself, viz., a notation for determinants whose elements had special values, is at the foundation of the whole difference between his treatise and those at present employed. When this want came to be supplied later on, the functions crept steadily into everyday use, and a fresh impetus was consequently given to the study of them. But if from the work of the said eighty years all researches regarding special forms of determinants be left out, DETERMINANTS IN GENERAL (CAUCHY, 1812) 131 and all investigations which ended in mere rediscoveries or in rehabilitations of old ideas, there is a surprisingly small proportion left. If one bears this in mind, and recalls the fact, temporarily set aside, that Cauchy, instead of being a compiler, presented the entire subject from a perfectly new point of view, added many results previously unthought of, and opened up a whole avenue of fresh investigation, one cannot but assign to him the place of highest honour among all the workers from 1693 to 1812. It is, no doubt, impossible to call him, as some have done, the formal founder of the theory. This honour is certainly due to Vandermonde, who, however, erected on the foundation comparatively little of a superstructure. Those who followed Vandermonde contributed, knowingly or unknowingly, only a stone or two, larger or smaller, to the building. Cauchy relaid the foundation, rebuilt the whole, and initiated new enlargements; the result being an edifice which the architects of to-day may still admire and find worthy of study. CHAPTER V. DETERMINANTS IN GENERAL, FROM 1693 TO 1812; A RETROSPECT. FROM what has just been said by way of estimate of Cauchy's memoir, it will readily appear that a suitable opportunity has now presented itself for taking a general retrospect of the work done from the date at which the history commences. The system which has been pursued, of numbering the new advances made by each writer, enables us to do this very conveniently, and with a tolerable approximation to accuracy by means of a tabular form. The table, herewith annexed, so far explains itself. The authors' names, it will be seen, are arranged both vertically and horizontally in chronological order; and vertical and horizontal lines of separation are drawn so as to apportion to each name a gnomon-shaped space. The crediting of any entirely new result to an author is done by giving its number in Roman figures after his name in the vertical list. On the other hand, any mere modification, fresh presentment, or extension of a previously known result, is notified to the right of the original number of the result, and under the new writer's name in the horizontal series. Instead of the Arabic figures placed in the gnomon-shaped spaces, a cross or other uniform mark would have sufficed, but in order to increase the usefulness of the table, a number has been inserted, telling the page at which the result is to be found. For example, if we look to the space allotted to Bezout (1779), we find him credited with one entirely new result, numbered xxIII., and with some contribution to each of five previously known results, whose numbers are II., III., IV., XII., XIV.; and we likewise see that information regarding them TABLE —SHOWING THE ADVANCE OF THE THEORY OF DETERMINANTS FROM 1693 TO 1812. cc_ c I — tr — -1 i N I a) IS '& a 1-1 I Fi CD ct C -~ i. oi n: cc i K- K- I-. K- ~ — K — 5l e7 bf -~' Q Kb Cl eC C 17 52 55 1'7 52 55 1693. Leibnitz, I. 9 II. 9 III. 9 13 1748. Fontaine, 1750. Cramer, IV. 13 V. 13 6 c6 00 o 56(1), 57(o), 58, 59(2), 61 61 61-2 cccc 1-4 r-4 c I cc "7, ani* T —( Clo 17 Ce CLU Ce 0 C cr i i... 75 101 75, 77, 78 99, 101, 102 o1o 00 Ce 0a 1764. Bezout, VI. 52 54 1771. Vandermonde, VII. VIII. IX. X. X. XI. XIII. XIV. 1772. Laplace, XV. XVI. I 24 } 33 24 24 24 24 33 24 33 24 33 24 33 33 33 78 103, 103 99 99 103 96 103 97, 103, 117 116 98 52 52 64-5 78 1773. Lagrange, 1779. Bezout, XVII. XVIII. XIX. XX. XXI. XXII. XXIII. 41, 41 41 41 41 41 41 66 66 65 70, 72 68 1784. Hindenburg, 1800. Rothe, XXIV. XXV. XXVI. 1801. Gauss, XXVII. 59 60 63, 63 68 1809. Monge, Hirsch, 1811. Binet. XXVIII. 81 1 81 86 86, 92 87, 92 88 88 91, 92 91 109 119 110 117 104 121 XXIX. Prasse, 1812. Wronski, 71 Binet, XXX. XXXI. XXXII. XXXIV. XXXV. XXXVI. Cauchy, XXXVII. XXXVIII. XXXIX. XL. XLI. XLII. XLIII. j 105 106, 110 111 112, 114, 114, 114, 114, 120 118, 118 120, 121 121, 122 Facing page 132. DETERMINANTS IN GENERAL 133 all will be got at p. 52 of the History.* Speaking generally, more importance ought to be attached to the existence of numbers at the corner of a gnomon than elsewhere, because these indicate fresh departures in the theory. Sometimes, however, a fresh departure may have been very trivial, the real advance being indicated by a number well removed from the corner of a subsequent gnomon. Thus if we examine the history of the multiplication-theorem (Nos. xvII., xvIII.), we find the first step in the direction of it credited by the table to Lagrange, and subsequent steps to Gauss, Binet, and Cauchy; whereas careful investigation at the pages mentioned shows that what Lagrange accomplished was of exceedingly little moment, in comparison with the magnificent generalisation of Binet and Cauchy. Again, it must be borne in mind that all the results numbered in Roman figures are not of equal importance, it being well known that one theorem in any mathematical subject may have vastly more influence on the after development of the subject than half a dozen others. Such imperfections, however, being allowed for, the table will be found to afford a very ready means of estimating with considerable accuracy the proportionate importance to be assigned to the various early investigators of the theory. If we look for a moment, in conclusion, at the nationality of the authors, one outstanding fact immediately arrests attention, viz., that almost every important advance is due to the mathematicians of France. Were the contributions of Bezout, Vandermonde, Laplace, Lagrange, Monge, Binet, and Cauchy left out, there would be exceedingly little left to any one else, and even that little would be of minor interest. *As regards the newness of xxIII. the table is not quite in accord with the text, an earlier writer's work having been.duly noted in the latter (p. 11). CHAPTER VI. DETERMINANTS IN GENERAL, FROM THE YEAR 1813 TO 1825. THE writers of this period are seven in number, viz., Gergonne, Garnier, Wronski, Desnanot, Cauchy, Scherk, Schweins. Of these Gergonne, Garnier and Cauchy are merely expository; Wronski only recalls an earlier communication; Desnanot is a follower of Bezout; Scherk is a follower of Hindenburg; Schweins alone stands prominently forward as being well read in the subject, fit to give a full exposition, and fruitful in new results. GERGONNE (1813). [Developpement de la theorie donnee par M. Laplace pour l'6limination au premier degre. Annales de Mathematiques, iv. pp. 148-155.] This is such an exposition of the primary elements of the theory of determinants and their application to the solution of a set of simultaneous linear equations as might be given in the course of an hour's lecture. It is confessedly founded on Laplace's memoir of 1772; but, though the matter of it is thus not original, it is nevertheless noteworthy on account of its brevity, clearness, and elegance. The word "inversion" is introduced to denote (Ii. 21) what Cramer called a "derangement," and then by easy steps the reader is led up to the theorem regarding the interchange of two non-contiguous letters. "(9) Done, si l'on permute entre elles deux lettres non consecutives, on changera necessairement l'espece du nombre des inversions. Soit en effet n le nombre des lettres intermediaires a ces deux-la; on pourra d'abord porter la lettre la plus a gauche imm6diatement a DETERMINANTS IN GENERAL (GERGONNE, 1813) 135 gauche de l'autre, ce qui lui f era parcourir n places; puis remettre cette dernie're 'a la place de la premie're; et, cornme cule sera oblig~ee de passer par-dessus celle-ci, elle se trouvera avoir parcouru n +1 places. Le nombre total des places parcourues par les deux lettres sera donc 2n + 1, et conseiquemment l'espe~ce du nombre des inversions se trouvera changee." (iii. 22) This, it must be noted, is not identical with Rothe's proposition on the same subject, Gergonne's n being different from Rothe's d. The proof, that a determinant vanishes if two of the letters bearing suffixes be the same, proceeds on the same lines as Rothe's, but is put very shortly and not less convincingly as follows: " Supposons, en effet, que lou change h en g, sans toucher 'a g ni aux indices. Soient, pour un terme pris an hasard dans le polyn6me, _p et q les indices respectifs de g et h; cc polyn6me, renfermant toutes les permutations, doit avoir un autre terme ne diff~rant uniquement de celui-la' qu'en ce que c'est h qui y porte l'indice p et g lPindice q; et de plus (9) ces deux termes doivent Atre affecte's de signes contraires; ils se de'truiront donc, lorsqu'on changera h en g; et il en sera de meme de tous les autres termes pris deux 'a denx." (xii. 8) On putting "le polynO~me D," i.e. the determinant Ja~b2C3... f, in the form Ala, +A2a2~ Aa3+'.. + Amam, this theorem of course leads at once to the identities AA b~A b+ A3b3+... +Ambm = 0 Ale, +A2C2 +A 3C3+..+AmCm= Oj and these to the solution of m linear equations in m unknowns. GARNIER (1 814). [Analyse Alge'brique, faisant suite 'a la premi~ere section de 1'alge'bre. 2e e'dition, revue et conside'rablement augmente'e. xvi +668 pp. Paris.] The title of Garnier's chapter xxvii. (pp. 541-555) is "Th'veloppement de la the'orie donnee par M. Laplace pour e'6limination au premier degrei." It consists, however, of nothing but a simple exposition, confessedly borrowed from Gergonne's paper of 1813, and six pages of extracts from 136 HISTORY OF THE T.HEORY OF DETERMINANTS Laplace's original memoir of 1772. As forming part of a popular text-book, it probably did more service in bringing the theory to the notice of mathematicians than a memoir in a recondite serial publication could have done; and we certainly know that Sylvester, who afterwards did so much to advance the theory, expresses himself indebted to it. WRONSKI (1815). [Philosophie de la Technie Algorithmique. Premiere Section, contenant la loi supreme et universelle des Mathematiques. Par Hoene Wronski. (pp. 175-181, &c.) Paris.] Here as in the Rgfumtation of 1812 "combinatory sums" make their appearance, as being necessary for the expression of the "loi supreme." Wronski's point of view is unaltered toward them. He now, however, calls them Schin functions, (xv. 5) from the letter formerly introduced to denote them, " et pour ne pas introduire de noms nouveaux"! Two or three pages are occupied with the statement of the recurrent law of formation (Bezout, 1764). DESNANOT, P. (1819). [Complement de la Theorie des ]lquations du Premier Degre, contenant...... Par P. Desnanot, Censeur au College Royal de Nancy,..... Paris.] As far as can be gathered, Desnanot was acquainted with the writings of very few of his predecessors in the investigation of determinants. The only one he himself mentions is Bezout, and the first part of his work is in direct continuation of a topic which the latter had begun. His book is a marvel of laboured detail. No expositor could take more pains with his reader, space being held of no moment if clearness had to be secured. As might be expected, therefore, all that is really worth preserving of his work is but a small fraction of the 264 pages which he occupies in exposition. The first chapter bears the heading DETERMINANTS IN GENERAL (DESNANOT, 1819)13 1 s7 IRecherche des Rel ations qui ont lieu entre he, de'nominateur et hes numre'rateuirs des vahemrs generahes des inconnues dans chaque syste'me d~equations du pregn~ier degre'; and) after a reference to the impossibility of obtaining any result in the case of one equation with one unknown, proceeds as follows: "Si l'on a les deux equations ax +by =c, a'x +b'y =c', elles donnent cb'- bc' ac' -ca' ab'- ba" ~ab' -ba'' nommant D le de'nominateur commun, N et N' les nume'rateurs des valeurs de x et de y, nous aurons D =ab'-lba', N =cb'- bc', N'=ac'-ca'. Multiplious N par a, N' par b et ajoutons, nous trouverous aN + 6N' = c (ab' - ba') = cD; done aN + bN'-= cD. Nous aurions de mbme, en multipliant N par a' et N' par b', cette autre equation a'N + I/N' = C'D." With this may be compared Bezout's ]iH'thode pour trouver des fonctions.... qui soient ze'ro par elles-me'mes (see p. 49). Exactly the same method is followed with the set of equations axe + by + cz =d ale + b'y + e'z = d a x+ b"y + c"z = d"J. Here fifteen relations are obtained, only seven of which are viewed as necessary, viz., (ab' - ba') N'+ (ac' - ca') N" = (ad' - da')D~ (ab" - ba") N' + (ac" - ca") N" = (ad" - da")D J (da' - ad' )N + (db' - bd' )N' ~ (dc' - ed' )N" = 0 (da" - ad")N + (db" - bd")N' + (do" - ed")N" = 0 a N + b N'+ c N"= d D~ a'N + b'IN'+ C'N"1= d'D a"'N + b"N'1 + c"N" = d"/D 138.HISTORY OF THE THEORY OF DETERMINANTS From a modern point of view there are but two which are really different, viz., (ab'fJ.ac'd"I - Iac'I.Iab'd"( + jad'j.jab'c"( = 0 and ajbc'd"I - blac'd"I + clab'd"I - djab'c"j = 0, the twelve quantities concerned being abcd a' b' c' d' a' b" c/ d1l The former is obtainable fromnBezout's identity JabVC1.de'f"I - lab'd"j.jce'f"I + jac'd"j.)be'f'j - lbc'd".jae'f"I 0 by putting f1 fV1 = 0, 0, 1 and e, eC"= a, a', a" The other, as is well known, comes from Vandermonde. Before proceeding to the case of four unknowns, a notation is introduced in the following words (p. 6):"1 Soient a, b, c, d, f, g, h, etc. des lettres repr'sentant des quantit~s quelconques; k, 1, Mn, p, q, r, etc. des indices d'accens qui doivent ftre places 'a la droite des lettres. Au lieu de mettre ces indices comme des exposans, pla~ons-les au-dessus des lettres qu'ils doivent affecter, de maniere que k d6signe a affect6 du nombre i d'accens; que b indique le produit de a par b; ainsi de suite. Reprisentons ke I kG I __ k 1\ I1I I 1 la quantit6 d - par (n h) de sorte que nous ayous cette iquation a b - b a Par a b aZ T,I This being settled, the similar quantities of higher orders are defined by the equations k I m M k,7 1 1 k m km I ml a b =c c - b e b+ c ab / Z MP L./k I in m k p Z k mv I _ a bcd) dkabc) - d abc) + d abc d abc), &ci &c. &c. It is thus seen that IDesnanot's definition is almost exactly the :DETERMINANTS IN GENERAL (DESNANOT, 1819) 139 same as Vandermonde's, and his notation essentially the same as Laplace's. To this definition and the proof of the theorem regarding the effect of the interchange of two indices or two letters seven pages are, devoted, and then a fres~h step is taken. The exact words of the original (pp. 13, 14) must be given, as they distinctly foreshadow a great theorem of later times. "C14. Si nous de'veloppons cette expression fk \I p /Pkp\ ml\ ~ab kab) _ ab) ab) le re'sultat sera ab,)ab) done nous avons cette equation (A) k I MP\_ / p\/n I /, ni\/Ip (A ab ) ab) -ab}Kab) =kab,)ab 15. De cette formule je vais en d6duire d'autres. Je dis que si j'introduis la lettre c dans les -seconds facteurs de chaque terme et en meme temps lindice k, 1'6quation subsistera encore, et que j'aurai () ki \/ k 7np\ kp\/ kml \) km\(ki I XLIV.) ~ab a bej ab kabc) =1ab ka be) ( L'egaliti serait prouve'e si en di'veloppant les deux membres, les quantit6s multiplieles par, la mebme lettre c, affecte'e d'indices egaux, 6taient e'gales dans chaque membre; or j'ai,n/ k I\(k (k IC ab)'ab)\a bkaaQ!)ja { in kI k _ kp k Les quantit/s multiplie'es par P' M et ' dan s chaque membre sont 6gales entr'elles, c'est evident; et la 'formule (A) rend les coefficiens de C 6gaux; donc puisque dans (B), il n'y a que des termes multiplie's par el m kl C je conclus que heiquation (B) est exacte." Having thus shown that, if in each of the second factors of the identity a~b2Ila3b4I - Ib3a2 4I + lalb 4I1ca2b3.= () 140 HISTORY OF THE THEORY OF DETERMINANTS a new letter c be added and the index 1 be prefixed, the sign of equality may still be retained, so that we have a new identity Ica1b21 alb3c4 - jajb31 alb2c4 + jab41Kajb2c31 = 0 (B); he then goes on to prove in the same fashion that the first factors of this derived identity may be treated in a similar way with impunity, viz., that they may be extended by the appending of the letter c with a new index 5, so that we have a further derived identity I aCb2c5 a~lb3C4 - Ijalb3c5 11alb2C41 + 1alb4C511 C b2C31 = 0 (C), already known to us from Monge. And this is not all, for the next paragraph shows that these two extensions may be repeated in order as often as we please, the opening of the paragraph being as follows (p. 15):"17. Gtnnralisons et prouvons que si la formule k I q) k ni. p) (k m. b. CI kP b. k.p v k m ab..ckab..c ab.. c kab. c) = cab.. c ab..c) est vraie dans le cas oii ii y aurait n lettres comprises dans chaque facteur, elle sera encore vraie en ajoutant une nouvelle lettre d dans les seconds facteurs de chaque terme avec P'indice I qui n'y entre pas; et qu'ensuite, si ion ajoute la meme lettre d dans les premiers facteurs de chaque terme avec un nouvel indice r, 1'Bgalite ne sera pas trouble'e. Ii s'agit done de demontrer que ces deux formules sont exactes: ~ab..cjab..cd (ab..c (ab..c)dJ =(l (XLIV.2) k I.q 2,N I km..Z p k..Z r\ k p. Z k I.. p?- l k?n. Zq\ X ) (xxii. 5 ab.. c d)ca b..cd) - d ab.. cd)\ab..cd) = a b.. c d)Ka b.. cd). The line of proof is still the same, and may be shortly indicated by treating the case (D) I alb2csI a~b2cAd4 - aIb2c411ab c2dAI + Ialb2c311 ajb2c4d, = 0, which comes immediately after (C), and is derived from it by extending the factors in which alb2 does not occur. Since by definition Ia b2c~d41 = d4 Ialbc2Cj - d3l alb2c4 + d2la~b3c4 j- d1 a2b3c41, and I a,b2c3dl = d5 a~b~c3 - d3 ajb2e51 + d2 a,b3c.1 - d1 a2b3c I, DETERMINANTS IN GENERAL (DESNANOT, 1819) 141 it follows that lab2c5 la b2c~d41- laKb2C4jab2cd51 = {d4laLb25| - d5salb2C41}al\b2CA1 + {!alb2c ja1b3e4I - 1jb2C4j alb3c5 l} -d - {kIb2e5jjab3e4J - Calb2C4jla2b3C5} d1 J But the cofactor here of d2 is by (C) equal to - lab4cs lab2c31; and the cofactor of d, = la2bcl11a2b3C4 - Ia2blc4 la2b3c5l, and therefore by (C) = -a2blc311a2b4C5l, = lab2c3labcl. Making these substitutions, we have Ia1b2c5 alb2c3d4 - lab24 llab2C 3d5 = - alb2c31(d5lab,2cl - Cd4lb2AC l + d2lalb4cl - dlla2b4cl} = - lalb2c3jlajb2c4d,lj as was to be shown. The next three cases are a1b2c5d6llajb2c3d41 - lalb2c4dalab2c3cd51 + cajb2c3d^611b2c4d5I = 0 (E) \a1b2c3d4\llab2c3d5e6j - la1b2c3d5lla1b2c3d4e6| + lacb2c3d6l1ab2c3d4e51 = 0 (F) a b2c3d4e7llabcd5el - lalb2c3d5e7liajb2c3d4e61 + jalcb2c3d6e7lalb2c3d4e5j = 0 (G). When the factors of each product are of the same order, as in (C), (E), (G), the identity is, in modern phraseology, an " extensional" of (A); that is to say, there is a part common to every factor of the identity, e.g., a, in (C), acb2 in (E), alb2c3 in (G), and this common part being deleted, the result is simply the identity (A). When the factors of each product are of different orders, as in (B), (D), (F), the identity is an "extensional" of something still simpler than (A), viz., acla2b3l - a2laab31 + a3lalb21 = 0, 142 HISTORY OF THE THEORY OF DETERMINANTS In exactly the same manner and at quite as great length the identity ~a i~g) - (a)(l7) = -already known to us from Lagrange-is made the source of a numerous progeny. By putting figures for k, 4,... and at the same time writing them as suffixes, these identities, original and derived, take the form laf2la 1g96 - 1a1g21 Jaf61 = lalf2g6l aj, (A') alf2 llalb29g6 - alg2laAb f6 = la~f2gjajb2l, (B') Ia1b2f3l ajb29g6 - 1a1b2,gY31ajb2f = IaAbAf9g6la~b2l, (C') 1a1b2f3 Jajb2e3g96 - 1a1b29g3 ajb2c3f61 = IaAbAf3g6 jb2C31, (D') lcab2c3f4l Jajb2c3g96 - cLab2c3g941ctab2c3f61 = jcab2C3fXg61a 1b2c31, (E') lCtb2c3f4 Jalb c23d4g - aqbcc g4lblb2cCd3f64 = jajb2e03f49jg6 alb2cAd, (F') clb2c3dJf1j a,b2c3d49g6 - jajb2c~d495 jajb2c3d4fA = Jalb 2e3dJ596f1g alcjb'3 dj. (G') Of these (C'), (E'), (G') deserve to be noted, being along with the original (A') extensionals of the manifest identity AN6 - g2A -= Ag961. (xLIv. 3), (XXIII. 6) On the other hand (B'), (D'), (F') are essentially the same as (B), (D), (F) already obtained-a fact which Desnanot overlooks. As the source of a third series of results, obtained in still the same way, the identity ah ( f i)( f-h (A") is next taken. In reality, however, this does not differ from the first identity so treated, viz., k Z I / P k p MZ mI A) aa b abe (~a)( ab) (ab)b) (A). In (A) the letters ab remain unchanged throughout, and the indices vary; while in (A") the indices remain the same, and the letters vary. As we should now say, the difference is a mere matter of rows and columns. The derived identities (B"), (C"), and (D"),... are consequently found to be quite the same as (B), (C), (D). The fourth and last source made use of is the well-known DETERMINANTS IN GENERAL (DESNANOT, 1819) 143 theorem regarding the aggregate of products whose first factors constitute what Cauchy would have called a "suite verticale," and whose second factors are the cofactors, in the determinant of the system, of another "suite verticale." Desnanot however, viewing the theorem from a different stand-point, enunciates it as follows (p. 26): "Si oron a n lettrcs ab.... cdf, et qt'on les combine n -- I & n - 1, on aura n arrangemnens ab.... cd, ab.... ef, ab.... df. a.... cdf, b.... cdf; qu'on applique dans chaque arrangement les n -1 indices ki.... mp, ce qui donnera ces quantite's k Z. m p\ Ik ZI..mp k7 I.. M P k Z.. MP k I..mp a b..c d) i a.. cj a b.. df)..... a..cdf)) b..cdf (~~elf) et qu'ensuiic on les multiplic chacune par la lettre qui Wn'ctrc pas danns l'arrangement en l'affctcant d'un monmc indice et donnant au produit le signe plus ou le signe moins, suivant que la lettre multiplicateur occupe un rang impair ou pair dans les n lettres, en partant de la droite, la somme des produits scra sc'ro." (XII. 9) Before proceeding to deduce others from it, he gives a proof of it for the case p k ZI. m p p k~ I. P p k I.. m p (B"') )2(k a b i ) P.. of)+(bab..d)f p k I.. MP p k Z. mp 0 b al.. c d a Pb..cdfJ=0. The method of proof is interesting, because it depends almost entirely on the definition which Desnanot follows Vandermonde in using. It will be readily understood by seeing it applied to the simple case blib2c3cl4 - b2lblc3djl + b3lb cl2dj - b4ib c2dst = 0. Expanding each of the determinants Ib cbc14, 4 blcd4l. in terms of the b's and their cofactors, we have b,Ib2c3d4 - b2jbc3d4j + b3 bIc 2d4 - bjblbe2d3l b I b2Jc3d4 - b3JcAd4J + b4JcAd } - b, {blc 3dJl - bccld4j + b4 Icc,d1 + Nb3{bIeCAI - b91 eld4l + bjcied2 } - b4{b, 1c02 3 - b2lc d3j + bNfcd2j = 0, for the terms in the expanded form destroy each other in pairs. 144 HISTORY OF THE THEORY OF DETERMINANTS The derived identities are obtained exactly in the manner followed by Bezout in 1779 (see pp. 51, 52). The fundamental identity is taken, say in the form alb2c3d4e51 - e5lab2c3d45l + d5l ab2c3e4f5l - c5 b2d3e4f51 + b51 a,2d3e4f5 - a51 bc2d3e4/5l = o, and another instance is put alongside of it, in which the same letters and suffixes are involved, say f I alb2c3d4e5 - ela b2c3d4f5 + dlajb2c3e4f5l - c lab2d3e4f5 + b lac2d3e4f51 - cl b1c23e4f51 = 0. One of the constituent determinants, say the last, b1c2d3e451l is then eliminated by equalisation of coefficients and subtraction, the result being I lf5 l. clbb23d4e5l - I e5 lllaab2c3d45/ + lald5l alb2c3e4f51 - a1c5sll alb2d3e4f5 + lab5l ac2d3e45l = 0 (C"') In the next place, two additional instances of this derived identity are taken along with it, the first differing from it in having a 2 instead of a 5 in all the first factors, and the second in having a 2 instead of a 1; viz., Ialf2lalb2c3d4e5I - Iale2 alcL b2c3c4f5 I + laId2 l ab2c3e4f51 I - alce2ll ab2d 3e4f5 + lab2l^ c2c123e4f5l = 0. and l2fsa l b2c3d4esl - I2e5 1 alb2c3d4f5 + J a2d5 l cb2c3e4f5 l - Ia2c5 lalb2d3e4f51 + la2b5lalcc2d3e4J5 = 0. Multiplication by b2, -b, -bl is then effected and addition performed, when by reason of such identities as b2lalfsl - b5la1f21 - b1la2f5l = lajb2/f1, and b2la1b5l - b5la1b2I - blja2b5I = 0, elimination of lalc2d3e4f, is produced, and the result takes the form (a1b2f5 l ab2c3d4e5 - atb2e5 | ab2c3d4f5 A + I a12d5 11 b2c3e4f5 (D"') - Ia\b2callAlbd3e4,\I = 0. DETERMINANTS IN GENERAL (DESNANOT, 1819) 145 The process of derivation may be pursued further, giving next an identity in which the first factors are all of the fourth order. Desnanot says (pp. 31, 32)"Pour ne pas nous repeter constamment, nous dirons que cette formule s'etendrait a un nombre quelconque de lettres placees dans les premiers facteurs, et que /ck...p\... p\ /.c I... p\ /J... mp a b... f)a b... cd) - [a b... d)[a b... cf) (;;(k Z}...p\f; IZ...mp\ +a b...c)a b.. df) -. = 0. Les termes sont alternativement positifs et negatifs, les indices sont les memes dans les premiers facteurs de chaque terme, ils font partie des indices qui se trouvent dans les autres facteurs et sont places dans le meme ordre; quant aux lettres, il y a ou une, ou deux, ou trois, etc. lettres communes aux seconds facteurs 6crites toujours dans le meme ordre et suivies de la nieme lettre qui n'entre pas dans les seconds facteurs; de sorte que s'il y a n' lettres communes a tous les facteurs, le nombre des termes de (H"') sera n - n'." (xxIII. 7) (XLIV. 4) The general result (H"') is simply what would now be called the 'extensional' of the identity of Vandermonde from which Desnanot derives it. Co-ordinate, in a sense, with the said identity, is that other which Desnanot uses as a definition; and this latter is the next of which the extensional is found. The process, so far as indicated, is exactly similar to that employed in the preceding case. The results obtained are /Ip 2 k I...m p \ p \ 7. I n p \ p re\ I... r p \ a f)ab.cd- a d)(ab... cf) + a ) ab.. ) - (B"") ) ~(B k^~7 r \.....p np p mp....... + df abj)...cdf)b... cf and /c p r\ k I... mp\ / S )\ k I...m p\ "* / 7 P r k I1... p / p\ / m I...m m p r + \abc aab...f -...b) ab... cdf; and the general result including them is referred to. (vi. 4) (XLV.) M.D. K 146 HISTORY OF THE THEORY OF DETERMINANTS That they are extensionals of the definition is evident from the fact that the index p may be moved to the left so as to make a common to every factor of (B""), and ( b common to every factor of (C""). Still another series of results is obtained, but they are essentially the same as the foregoing, the difference again being merely a matter of rows and columns. All these preparations having been made, Desnanot returns to the subject of the relations between the numerators and denominators of the values of the unknowns in a set of linear equations. Thirteen pages are occupied with the case of four unknowns, the number of relations found being 74, of which, after scrutiny, 14 are retained. The case of five unknowns, and the case of six unknowns are gone into with about as much detail, and then, lastly, the general set of n equations with n unknowns is dealt with. None of the relations obtained need be given, as they are all included in the identities which have been spoken of above as extensionals. The second chapter (p. 94) bears the heading Simplification des forrules gene'rales qui donnent les valeurs des inconnues dans les equations du premier degre, lorsqu'on veut les evaluer en nombres. Here again the cases of three, four, five, six unknowns are dwelt upon with equal fulness in succession. The consideration of one of them will suffice to show the nature of the method, and will enable the reader to judge of the amount of labour saved by employing it. Choosing the case of four unknowns, we find at the outset the equations stated and the solution condensed as follows (p. 104):"EQUATIONS DONNEES. ax +by +cz +dt =f a'x +b'y + c'z +d't = f a"x + b"y + c"z + d"t = f" a'%x + b'"y + c'"z + d"'t = f"'. DETERMINANTS IN GENERAL (DESNANOT, 1819) 147 CALCUL. ab' - ba' = a, ab" - b"a = /3, a'b" - b"' = 7, ab"' - ba"'=, a'b"' - b'a"' =; m = c" a-c'/3 + cy, = C"' a — c'8 +c, m'= f" a -f'/p +fy, ' = f"'a-f'8 +fe; D = 1 m (ad"' - d' + cz) - n (ad" - /3d' + ) N"'= (mn- nm') N = armn(ad' - d' + Ed) - n'(ad" - fPd' +yd); f D - cN" - dN"' = S, f'D- c'N"- d'N"' = S'; aS' - Sa' a Sb' - bS' N a N N' N" N"'. X D, y D Z - DD The explanation of the mode of procedure is not difficult to see:(1) The determinants Icab', cab"l, Ia'b"l, lab"'l, Ica'b"'l are calculated. (2) With the help of these are next got four of a higher order, viz. ac"b'c"l, lb b'f" l, l afb'/"' l. (3) Two others of the same order, viz. ad"' - d' + ed, ad" - 3d' + yd, i.e. clab'd"'j, lab'd"l, 148 HISTORY OF THE THEORY OF DETERMINANTS having been calculated, the identity lab'. D = lab'c". Jab'dC'l Ij ab'C"'I. ab'd" I is used to find D. (4) A similar identity Jab'l. N"' = lab'C"J. Jab'f"' - I cab'c"'l. Iab'f"l is used to find N"'. (5) A similar identity Jab'. N" = Jab'f". ab'd"'I - Jab'f"'. lab'd" is used to find N". (6) Two subsidiary quantities S, S' are calculated, the first being = fIa b'dc/d/"I e C abf'fd"'l I dlab'caf"'l, and the second = f'Iab'cC d"'l. CI 'labj'd"'l - Jably"J (7) From these N' and N are readily got. For evidently aS' - Sa' I Cf I - I ab'c"c"'. I - lac'l lab'f"t"'di - Jad'l. Jab'c"f"' I and this by a previous theorem = 1ab'J I af/'c"d"' K = I ab'J. N'. (xiii. 3) The third chapter consists of a lengthy examination (pp. 1 57 -264) of the singular cases met with in the solution of linear equations, and does not concern us. CAUCHY (1821). [Cours d'Analyse de l'Ecole Royale Polytechnique. I. Analyse Algebrique.* xvi+576 pp. Paris. TU~vres, 2e ser. iii. pp. 73 - 82, 426 - 428.] When Cauchy came to write his Course of Analysis, afterwards so well known, he did not fail to assign a position in it to the subject of his memoir of 1812. The third chapter bears the heading, "Des Fonctions Syrn-triques et des Fonctions Alterne'es." *No more published. DETERMINANTS IN GENERAL (CAUCHY, 1821) 149 It occupies, however, only fifteen pages (pp. 70-84), and of these only nine are devoted to alternating functions and the solution of simultaneous linear equations. Of course, in so limited a space, the merest sketch of a theory is all that is possible. An alternating function is first defined, the word "alternee" being now set in contrast with " symetrique," and not, as formerly, with "permanente." Functions other than those that are rational and integral being left aside, the latter, if alternating, are shown (1) to consist of as many positive as negative terms, in each of which all the variables occur with different indices, and (2) to be divisible by the simplest of all alternating functions of the variables, viz., the difference-product. The set of equations a,+ bry + cr... +gr + hr = kr (r=, 1,..., 1) is then attacked, the method being-to take the difference-product of a, b,..., h,-denote by D what the expansion of this becomes when exponents are changed into suffixes,-denote by Ar the co-factor of ar in D,-then obtain the equations Aoao Aa+A +A2... +An-1n-l = D, Aob +Alb + A2 +... A,lbn-1 = 0, Aco0 + Ale +A2c2 +. + An_-_c,_ = 0, Aoh + Alh + Ah2 +.. +Alh-1 = 0, -and thereafter proceed as Laplace had taught. As in the memoir of 1812, the " symbolic" form of the values of x, y,. is unfailingly given. A note is added (pp. 521-524) on the development of the difference-product, showing how all the terms may be got from one by interchanging one exponent with another, how the signs depend on the number of said interchanges, and how, by counting the number of cycles (here called groups), it may be ascertained whether any two given terms have like or unlike signs. It will thus be seen that not only is the name " determinant" never mentioned in the chapter, and the notation S + aob c2... h,_never used, but that the subject is scarcely so much as touched upon. Although, therefore, Cauchy's text-book went through 150 HISTORY OF THE THEORY OF DETERMINANTS a considerable number of editions, and had a widespread influence, it gave no such impulse as it might have done to the study of the theory of determinants. SCHERK (1825). [Mathematische Abhandlungen. Von Dr. Heinrich Ferdinand Scherk,.... iv+116 pp. Berlin. (Pp. 31-66. Zweite Abhandlung: Allgemeine Auflosung der Gleichungen des ersten Grades mit jeder beliebigen Anzahl von unbekannten Grbssen, und einige dahin gehorige analytische Untersuchungen.)] The only previous writings of importance known to Scherk were, according to his own statement, those of Cramer, Bezout (1764), Vandermonde, Bezout (1779), Hindenburg, and Rothe. His style bears most resemblance to Rothe's, whose paper, however, he does not speak of with unmixed eulogy, characterising it as containing " eine strenge aber ziemlich weitlaufige Auflisung der Aufgabe." The main part of the memoir consists of a lengthy demonstration, extending, indeed, to 17 pages quarto, of Cramer's rule, or rather of Cramer's set of three rules (iv., v., iii. 2), by the method of so-called mathematical induction. The peculiarity of the demonstration is that it is entered upon without any previous examination of the properties of Cramer's functions (determinants); and it is noteworthy on two grounds-(1) as being new, and (2) because the properties, which it really if not explicitly employs, had also not been previously referred to. The cases of one equation with one unknown, two equations with two unknowns, three equations with three unknowns, are dealt with in succession, the solution of one case being used in obtaining the solution of the next. All three solutions are noted as being in accordance with Cramer's rules, and the said rules being formulated, and upposed to hold for n equations with n unknowns, it is sought to establish their validity for n+1 equations with n +l unknowns. In other words, the set of n equations being DETERMINANTS IN GENERAL (SCHERK, 1825) 15 151 lb n n-i n-i n-2 n-2 ax + a x + a x +.. 1 1 1 n n n7-i n-1 n-2 n-2 ax + a x + a x +. 2 2 2 n n n-1 n-i n-2 n-2 ax + a x + a x +.. n n A and the corresponding values of 1 2 n x, x., being P a;a, a) P a;a, a) it is required to show that the solution equations 1 1 + ax = 81 1 ii 1 1 + ax = s P a; s, of the set of n+1 a x +-ax+... 1 1 n+ n+2 Inn a x + ax~... 2 2 n+1ln~1 nfl a x + ax +... n+1 n+1 ax+.. ax+.. 2/. +ax. +ax 2 I1I 2 n+1 + ax+... +ax n+1 n+1 is 1 P(:a;;s ~) n+1 PI a;a,a). Before proceeding, the notation n+1 n+1 a; s, a - n+1 h 7b _. (xiii. 4) n+1 n+1 n+1 a; a, a n+1 h h P a; s,a) reqnires attention. It is meant to be an epitome of Cramer's rules; the -first half of the group of symbols, viz. P a implying 1 23 n permutation of the under-indices of the product a a a... a and aggregation of the different products thus obtained, each taken) 152 HISTORY OF THE THEORY OF DETERMINANTS with its proper sign: and the second half implying that in every term of this aggregate s is to be substituted for a. A modern h h writer would denote the same thing by 2 3 S a a.. a 1, 1 1 - 1 2 3 n s a a... a 2 2 2 2 2 3 n s8 a... a n n n n (nb 1 only it must be noted that in using P; s, a) at this stage, we leave out of account the signs of the terms composing it, the rule of signs being the subject of a separate investigation. Any one of the forms /n 1 1\ /n 2 2\ Pa; a, a) P; a,- a..... n A h/ \n h 7l/ it need scarcely be added, will thus stand for the common denominator. Of the nq+1 equations the first n are taken, written in the form nn ax + 1 n-1 n-1 k k 11 a x +... + ax +..+ ax 1 1 1 nn n —1 n-1 kk 11 ax + a +. ax+... + ax 2 2 2 2 nn n-1 n —1 7kk 11 ax + ax +... + ax +. + ax bb n n n and solved, the results being by hypothesis b/ n+1 n+ 1 \ P; s- x, a x 7~ A 7 - 1 - cfn a1 1 ) P( a; a, C) \n h 7/ n+1 n+1' = s - ax 1 1 b+1l n+1 = 8s- a x 2 2 n+1 +l1 = 8 - ax n n DETERMINANTS IN GENERAL (SCHERK, 1825) 153: (n 1+1 n+l k \ k P a; s- a x1, X) Pa; a, a),\ n h h h h ( n9+1 n+l fl ~ / n xn n\ P(;. a 1 a) \n h h/ ra x + a n r s 7 + ps a a h 1 t equati obn, which thu7sbomes +1 n+l P (a; a,) n+1 P (a; a, a) n h h n h hI? n+9 n+1 1 \ n$+l and as this manifestly involves none of the unknowns but x value obtainer ede is tramu tonsformable into. The way in which this is effectedl is well worthy of attention. Scherk's own words in regard to the first steps are (p. 40)9n9+l n+l + a < \12 ~ h n1 P//a;a, 9+ ~~~~~\h h h 99+1 dass in jede der in I. beschriebenen Permutationsformen fbur erst s, n+lnl +1 h h dann a gesetz, un a x in jeder i Resultate on einander abgezogen h I w erden s oll en f olgrkommt so bedeutet das Zich ist /Pn n+l n+l \ / \ / n+l n+l k P;s- a z,) = P(;s,6) -P(a; a x, ). \n h h h \n h hJ n 71 h 154 HISTORY OF THE THEORY OF DETERMINANTS In dem letzten Gliede dieser Gleichung kommt aber in jeder Form n+1 n+l x, und zwar zur ersten Potenz, vor; x ist also gemeinschaftlicher Factor aller Formen, und folglich ist n n+l 1 /n n+l /n n+l J\) P a; s - a, a) = Pa;s, ) - xP(a; a, a. n h I h/ \ h h/ n h h Macht man diese Substitution fir k=l, 2,.., n, in der letzten Gleichung, und bemerkt, dass (n 1 1\ n 2 2\ /n k J\ P a;a, a) = P(aa,a)=....= P(a; a, a n 7 h/ \n h \n h so geht diese in folgende Gleichung fiber n+1 /n k k\ n+l aP a;a,a x n+l \,n h h rn T/ n\ 7 n k\ 1 /n 1\ +;P(a;s, c +...+ a Pa;s, a +... + a Pa;s, a n+I-1 \n 7h h/ z-+ \n h / n+1 \n h / C t /n nl n\ 7G /n n+b k\ 1 / nnl 1\ ni+l - aP(a;,a) +...+ aP(a;a,a) +...+ a P(a;a, af x ln+l \n 71 h/ n+1 \n h h/ n+l \n h h/j n k 7G\ s P a, a); n1+ \n h h folglich 1 /n 1\ 2 n 2 n /n n /n k k\ n - a P(a;s, a) P (;s, )-...- aP(a;s,a)+ sP(a; a,a) n+l / n+l h / n- \n i ni+ \n h 7h/ n+l \n h h/ 77 1 n 1+l1 1\ 2 n+1 2\ n /n n +1 n\ n+l /n k \ - a P(a; a, a)- P;a,a)-...- a P a;a, a+ aP(a;a, n+l \n h l n1 n+l n h h/ n+l nl h/ The first theorem here made use of and formulated, viz., (n n+l n+l C\ /n s\ n n+l n+l \ P a;s- a x, a) = Pa; s,a) - P(a; a x,a (XLVI.) h h h / \n hJ/ \n 7h h is the now familiar rule for the partition of a determinant with a row or column of binomial elements into two determinants, or for the addition of two determinants which are identical except in one row or one column. The second theorem, viz., /n n+l n+l \ n+l n n+l k P (; a x, ) = x Pa; a, a XLVII.) \n h h n h h is the now equally familiar theorem regarding the multiplication of a determinant by means of the multiplication of all the DETERMINANTS IN GENERAL (SCHERK, 1825) 155 elements of a row or column. That these two very elementary theorems should not have been noted until the time of Scherk is rather remarkable. The consideration of the constitution of /n+l hk k P a;a,a) is next entered upon, with the object of showing that the terms are exactly the terms of the denominator 1 Xn wul 1\ 2 Un n+1 2 n+1 n k k \ - P(a; a, )- P a; a, ).....+ a P(;;a, a). n+l n h 7 / h nl \n h h /n+l n h h More than two pages are occupied with this part proof of Bezout's recurrent law of formation. The identity of the terms of /n+1 n+l\ P a; s, a ) nl+1 7 lh with the terms of the numerator then follows at once; and the n+1 desired form for the value of x, so far as the magnitude of the terms is concerned, is thus obtained. The corresponding forms for xl, x2... are of course immediately deducible. The rules for obtaining the terms of the numerator and denominator having been thus established in all their generality, the rule of signs is next dealt with. The treatment is cumbersome, but fresh and interesting. It is pointed out, to start with, that the counting of the inversions of order of a permutation, is equivalent to subtracting separately from each element all the elements which follow it, reckoning +1 as a sign-factor when the difference is positive, and -1 when the difference is negative, and then taking the product of all the said factors. This, it will be recalled, is essentially identical with an observation of Cauchy's. Scherk, however, goes on to remark that these sign-factors may be viewed as functions of the differences which give rise to them, and may be so represented. Whether there actually be a function which equals +1 for all positive values of the argument and equals -1 for all negative values is left for future consideration. Cramer's rule of signs is thus made to take the following form (p. 45): 156 HISTORY OF THE THEORY OF DETERMINANTS "Wenn 9(P) eine solche Function der ganzen Zahl P ist, welche = + 1 ist wenn P positiv, und -1 wenn n/ negativ ist, so ist das Yorzeichen Z irgend eines in dem Aggregate P a; a, ) enthaltenen Gliedes ln k k/ 1 2 3 kc-1 kc ak- n aC C... a t a.. C a a a' a(k-1) a(k)a(kfl) a(n) folgendes: Z = (a" - a' ). (a, - a' )... (a(k-') -,a'). (a() -a' ). (a(+l) - a )... (a()- a' 9 (a/ '- a"). (a"" - a")... 5 (a(k) - a"). (a(k+l) - a"). (a(~+2)- a")... (a()- a") 5(a(k-) - a(k-2)). 5(a(k - a(-2)).. 9 5(an)- a(k-2)) 9(a() - a(k-l)). (a(k+l) - a(k-l)).. (a(n)- a(-l)) (a(k+l) - a (k). (a(k+2) _ a(,)). (a(")- a(-)) (a(n)- a(n-1))." (II. 23) And it is this form which Scherk seeks to establish. The mode of proof is again the so-called inductive mode. In the case of two permutable indices the law is readily seen to hold. We thus have, preparatory for the next case, P( a; a, = (2-1) a + 0(1 - 2) a a, 2 h h 1 2 2 1 P a; a, a) -0(2 -1) a a + (1 - 2) a a, 2 h h 1 2 2 1 2 3 2\ 1 3 1 3 Pt; a, c = 56(2-1) ac + 0(1-2) aa. 2 h h 1 2 2 1 But "nach dem Obigem " 3 k I 1 2 3 1 2 2 3 2 3 2 k k P(a, =; ac = - P;, - P( t; a, ) + aP a; a, \3 h 3 \2 h h 3 2 h h 3 2 h h Consequently P(c-; o, C)= -a\{ 0(2 - )a a + 0(1 -2)a a 3 h / 3 { 1 2 1 2 2 r 1 1 3L -a o0(2 - l)o o + 0(1 -2)o oa 3 1 2 2 1 3 1 2 1 2 +oC 0(2-l)c o + 0(1-2)0 a. 3 1 2 2 1J DETERMINANTS IN GENERAL (SCHERK, 1825) 157 But as -j952-1) = and -1 = p(2 - 3) and -1 = we may substitute O(1 -2)0(2 - 3~(1 -3) for - 0(2 -1). This and five other similar substitutions give us 3I c k 1323 123 P a; a,a) = (2-1)0(3-1)0(3-2)caaa c+ (3 - 1)k(2 - 1)k(2 - 3) ca aa \3 hL h/ 1 2 3 1 32 123 123 ~ O(l - 2)>(3 - 2)5(3 - i) a aa ~ 0(3 - 2)0(1 - 2)0(1 - 3) a a a 213 231 323 123 ~b(I - 3)95(2 - 3)95(2 - 1) a ct (2 - 3)(1-3(1-2) a a a; 312 321 so that the law is seen to hold also for the ease of three permutable indices. The completion of the proof, giving the transition from n to n ~1 pernmutable indices, occupies three pages. This is followed by two pages devoted to the subjects temporarily set aside at the outset, viz., the possible existence of functions having the peculiar properties of p. Two amusing instances of such functions are given,(1) P(/3) =,-.-2+-3 0 0 03 (2) si(/3) = in + sin 207jr + sin 20v +* (P-1)2T (/3-)7 (0-3,)27 sill 2,r7 sin 2,37r sin 207w + 1)27>~r (3 + 2)27( + )27 r = /% 1 )sin 2,3 where Po. stands for the hth coefficient in the expansion of (a ~ b)0. Success, however far from brilliant it nay be, in thus expressing the rule of signs by means of the symbols of analysis, led Scherk to try to do the same for the rule of formation of the terms. Nothing came of the attempt, however. "Bald aber," he says, "zeigte es sich dass Permutationen niemals durch andere analytische Zeichen ersetzt werden ki$nnten." 158 HISTORY OF THE THEORY OF DETERMINANTS Such speculations are not altogether uninteresting when later work like Hankel's comes to be considered. In an Appendix dealing (1) with the case of a set of linear equations which are not all independent, (2) with the solution of particular sets of equations, there is given at the outset a proof of the theorem regarding the sign of a permutation which is got from another permutation by the interchange of two elements. If the under-indices of the one term whose sign is z be aa a".... -a(a... a)aa.... a~, and of the other whose sign is Z * be a' a".... a )a ( a.... a - ai l.... an it is shown that _ (a+_ a- ) 0( i+2 _- a).... 0(ak_ a) Z 5 (ai-a t) * (at- ai+2) (Ai _t at) 0(aia+ ) - (ak _- ai+l) 0((_ -al-)a q(a - al) (a)i+l - at) 1(a7G-1 - a-) and there being here 2k - 2i - 1 quotients each =- 1, the result arrived at is z 2- -1 or z= -Z, Z as was to be proved. (II. 24) The body of the Appendix contains, along with other matter which falls to be considered later, the statement and proof of propositions identical in essence but not in form with the following:1 2 — 1? Ca a... a. 1 1 1 1 1 2 n —1 i a a.... a a 2 2 2 2 (1).......... = 0, (XLVIII.) 1 2 n-1 n t a.... a a - n —1 n- -1 1 2 n-1 n T T... T T * More than a page is occupied in writing the expressions for z and Z. DETERMINANTS IN GENERAL (SCHERK, 1825) 159 1 1 1 1 where T=ma+ma+....+m a 1 1 2 2 n-i n — 2 2 2 2 T = na + ma +.... + a 1 1 2 2 n-1 n-l a 1 1 2 a a 2 2 (2) a a = aaa...... a. 33 1 2 3 3 1.2123. 1 2 3 n a a a. a Xn l n it The first of these is proved from first principles, and not by the immediate use of theorems XLVI., XLVII. above. The second is proved by noting that any other term is got from the first, 123 123 n a ac (t... Ca, 1 2 3?n by permutation of the under-indices, that any such permutation will introduce one or more elements whose upper-index exceeds the lower, and that such are all zero. (vi. 6) SCHWEINS (1825). [Theorie der Differenzen und Differentiale, u.s.w. Von Ferd. Schweins. vi+666pp. Heidelberg. (Pp.317-431; Theorie der Producte mit Versetzungen.)] With much of the preceding literature Schweins, our next author, was thoroughly familiar. Cramer, Bezout, Hindenburg, Rothe, Laplace, Desnanot, and Wronski he refers to by name. The one notable investigator left out of his list is Cauchy, whose important memoir bearing date 1812 might have been known, one would think, to a writer who knew Desnanot's book of 1819 and Wronski's memoirs of 1810, 1811, &c. Still more curious is the omission of Vandermonde's name, whose memoir, as we have seen, is to be found in the very same volume as that of Laplace. 160 HISTORY OF THE THEORY OF DETERMINANTS Schweins' portly volume consists of seven separate treatises. It is the third, headed Theorie der Producte mit Versetzungen, which deals expressly and exclusively with the subject of determinants. The treatise is logically arranged and carefully written. It opens with an introduction of 4 pp., the main part of which serves as a table of contents and as a guide to the theorems which the author considered his own. It consists of four Sections (Abtheilungen), subdivided into portions which we may call chapters, the first section containing five chapters, the second also five, the third one, and the fourth four. Schweins' name for the function is Producte mit Versetzungen; (xv. 6) his notation is a modification of Laplace's, viz., he uses:I ) (vII. 6) where Laplace used simply ( ); and his definition is essentially the same as Vandermonde's; that is to say, he employs Bezout's law of recurring formation. His words at the outset are"Die Bildungsweise der Producte, welche hier untersucht werden sollen, geben folgende Zeichen an:AI As =I A1) 'A2 - | 'A. A', und allgemein a A A2 A = | I A2 A3 -A A2J A3 + A1 A2 AB, II~ a aa I 3 a... al 2 a3 a.I al a2 a A A A3 A = 2 | A1 A2 A3A A -| A1 A2 A3) A+ a aI I a3 a,3 \ aa a2 a3 a a4 a, a \ ai und allgemein - Al.............A._i An \/| Al. + ( A,,_, oder I a,...... a = (- / a x1..... a.n. nA...... A = (-)= Al......... An1. An x=0,l,)...,) n DETERMINANTS IN GENERAL (SCHWEINS, 1825) 161 The sequence of propositions as might be expected is not unlike that found in Vandermonde. The first six propositions are1. The under elements (Al, A2,...) being allowed to remain unchanged, the upper elements (a,, a2,.. ) are interchanged in every possible way to obtain the full development. 2. The sign preceding each term is dependent upon the number of interchanges of elements necessary to arrive at the term. 3. If two adjacent upper elements be interchanged, the sign of the dete-rmzinant is altered. 4. If an upper element be moved a number of places to the right or left, the sign of the determinant is changed or not according as the number of places is odd or even. 5. If several upper elements change places, the sign of the determinant is altered or not according as the number is odd or even which indicates how many cases there are of an element following one which in the original order it preceded. 6. If in any tern the said number of pairs of elements occurring in reversed order be even, the sign preceding the term must be positive; and if the number be odd, the sign must be negative. The proof of the 3rd of these, which gave trouble to Vandermonde, is easily effected in what after all is Vandermonde's way, viz., by showing that the case for n elements follows with the help of the definition from the case for ni-1 elements. (xi. 4) Schweins' 7th proposition is that there is an alternative recurring law of formation in which the under elements play the part of the upper elements in the original law, and vice versa. This amounts to saying, in modern phraseology, that if a determinant has been shown to be developable in terms of the elements of a row and their complementary minors, it is also developable in terms of the elements of a column and their complementary minors. The proof is affected by the so-called method of induction, and is interesting both on its own account and from the fact that Cauchy's development in terms of binary products of a row and column turns up in the course of it. The character of the proof will be understood by the following illustrative example in the modern notation:M.D. L 162 HISTORY OF THE THEORY OF DETERMINANTS By the original law of formation we have |a b2c3d4l = al b2c3d4 - a\ bc3d4 I+ a3\ b.c'2d4 I- a bc2d3; and, as the new law is supposed to have been proved for determinants of the 3rd order, it follows that lalb2c3cl4 = alb23d41 - a2{blc3d4l - clb3dl + dljb3c4 } + a3{b\lc9d4l - cljb2dj + dlb2c41} - Ca4{bljc2d3 - cllb2d3l + dllb2c31} Combining now by the original law the terms involving b, as a factor, the terms involving cl, and those involving dl, we obtain lalb2c3d4J = alb2C3d41 - blla2c3d4 + clja2b3d41 - d1 a2b3c41, and thus prove that the new law holds for determinants of the 4th order. (vi. 7) (IX. 3) Cauchy's development above referred to appears in the penultimate identity in the convenient form of one term cllb2c3d41 followed by a square array of 9 terms. The form in Schweins is{Ia A...... a,, || a....... a - -1.... a *,.1. =A 1 * * *.. An-1A (XXXVII. 2) + ( )y^-) -1 Ail... AXA A, A A,J Laplace's expansion-theorem is next taken up. To prepare the way a theorem in permutations is first given, the enunciation being as follows: If from n different elements every permutation of q elements be formed, and every permutation of n-q elements; and if each of the latter be appended to all such of the former as have no elements in common with it, all the permutations of the whole n elements will be obtained. Thus, if the permutations of 1 2 3 4 5, or say P(1 2 3 4 5), be wanted, we first take the permutations three at a time, viz., P(1 2 3), P(1 2 4), P(1 2 5),...., P(3 4 5) where 1 2 3, 1 2 4, 1 2 5,..., 3 4 5 are the orderly arranged combinations of three elements; secondly, we take the permutations two at a time, viz., P(1 2), P(1 3), P(1 4),..... P(4 5); DETERMINANTS IN GENERAL (SCllWEINS, 1825) 163 and, thirdly, we append each of the two permutations included in P(4 5) to each of the six included in P(1 2 3), each of the two in P(3 5) to each of the six in P(1 2 4), and so on. The identity here involved Schweins writes as follows, the only difference being that P is put instead of V (Vlerseftzngern): P(12 3 45)= P(1 23) xP(4 5) ~ P(1 2 4) xP(3 5) ~ P(1 2 5) xP(3 4) ~ P(1 3 4)xP(2 5) +P(I 3 5)xP(2 4) ~P(1 4 5)xP(2 3) + P(2 3 4) xP(1 5) ~P(2 3 5) xP(1 4) +P(2 4 5)xP(1 3) + P(3 4 5) xP(1 2). Another example isP(1 2 34 56) P(1 23). P(4 5) +P(1 2 4). P(3 5 6) +P(3 5 6). P(1 2 4) ~P(4 5 6). P(1 2 3). The proof consists in the assertion that no permutation can occur twice on the right-hand side, and in showing that the number of permutations which occur is the full number. From this lemma Laplace's expansion-theorem is given as an immediate deduction. The passage (p. 335) is interesting, as the mode of enunciating the theorem approximates closely to that of modern writers, and has a certain advantage over Canchy's, perfectly accurate, more general and more compact though the latter be. "Nach dieser Weise, alle Versetzungen zu bilden, weich e wir hier zuerst bekannt machen, k~innen auch die Summen der Produicte mit Versetzungen und mit veranderlichen Zeichen in niedrigere Summen zerlegt werden, wenn bei jeder Versetzung nach der oben gefundenen Vorschrift das zugeh,5rige Zeichen bestimmt wird; z. B. 164 HISTORY OF THE THEORY OF DETERMINANTS a a2 a3 a4 a, \ a a2 a3 a\ 4 as \ al a2 a3 \ a4 a5 \ A1A2A3A4A5 A1A2A)3 A4A5J A1A2A3) A4A5) a1 a2 a14 a3 a5 \ a1 a 2 a3 \ a4 a5 \ -!A1A2A3) A4A) A1AA) A3A5) 1 al a2 a,'5 a3 a4 \ a a 2 C3 \ | a4 a5 \ untern E e ve Z n + 1und - befolgen d a1 1 a3 a4 a2 a45 a1 a2 a3 a4 a5 \ + A1A2A3AA5 + AAA3AA.A2A) Giesctz in ' 140. Eben so ist 1 a 2 3 as \ || a2 a \ | 1 a2 a3 \ || a4 a5 \ a1 a4 a5 \r d a2 3 B\ 1 allgeme2 \ Z| 4 a5 l........ ni&) Cc- AI A,A........,A9AJ \\.AA, + A1A2A3) A A4A5) A+ AA4A5 A 2A3) a a2 a3 a \ I \ gla a a2 a3 \ | a4 a5 \ +jAlA2A3) |A4A5) | A2A3A4) 1A1A5) a2 Ca3.5,al a4 Al,A2 a... \ aO 2,5 Igefunden isA -lvAAA. AA 5) Gesetz in 140. Eben iso ist (Another example is given.) Wir wollen fur diese Bildungsweise folgende allgemeine Zeichen wahlen: I 01...... 2n 3 a2...... ** ^) 0 I a2... 2. ) \ Al * An ()|A; Aq) A+A. AA ) und, \ I a2.......... a ( g[ aq+l cjp+2........ an (n-q). = 2-) 1|(A1, A2,. A) *|(A A2,... An) ) wo * nach dem Gesetze bestimmt werden muss, welches in ~ 140 gefunden ist." (xiv. 5) The one imperfection in this is in regard to the question of sign. It is implied that the sign to precede any product, say the product A AA 2 A3) 3 A4 A ) DETERMINANTS IN GENERAL (SCHWEINS, 1825) 165 is fixed by making it the same as the sign of the term a2 a a al a5 A A A3 A A5; but nothing is said as to how this ensures that the 11 other terms of the product shall have their proper sign. Considerably less interest attaches to the next theorem dealt with,-Vandermonde's theorem regarding the effect of the equality of two upper or two lower elements. All that is fresh is the lengthy demonstration by the method of so-called induction. The identities immediately following from it by expansion Schweins expresses as follows:(-) | al......... Aa_ l a. aq+l......,, )-,) aq 0 Al... Ax-, Ax_-l................. al... ax-1 ax+l........ an" ax H(-) || Al..... Aqi A, Aq,... A) A, = ~ where x=l, 2,...., n. (xII. 10) This concludes the first chapter of the first section. The second chapter deals with a most notable generalisation and is worthy of being reproduced with little or no abridgment. The subject may be described as the transformation of an aggregate of products of pairs of determinants into another aggregate of similar kind. A special example of the transformation is taken to open the chapter with, the initial aggregate of products being in this case Iab2cdJ.le ef6g7l - Iacb2c3e4 l.d5f6g7 + I a1b2cJ3f41d5e6g7 - lalb2c3g'4l. d5e6.f71 Expanding the first factor of each product Schweins obtains { aj1b2C3 - d3l ab24cj1 + d2 ab3c4 - d1 a2b3C41 } e5f6g71 -{ ej albc3 - e3l ab2c4 + e2 1clb3cl - ela2b3c4).l }d5yf67 + {f4l abc - f3I ab2c ~ flabc44 + a - fi tCLb3c41)}.d5 e6g7] - tg4 ab2c3L - g31ahlb2C41 + g2a cb3C41- gi la2b3c41}.ld5e6 7 He then combines the terms which contain Ica1b2c31 as a factor, 166 HISTORY OF THE THEORY OF DETERMINANTS the terms which contain Ialb.2c4 as a factor, and so forth, the result being by the law of formation, alb2c3.Id-4e5f6g7 - alb2C4l- d3es5f6g + Iclb3c41. IdC2e5f6g - 9 a2b3c41.clde5f6g71 -The identity of this aggregate with the similar original aggregate constitutes the theorem. The only point left in want of explanation in connection with it is the construction of the aggregate of products presented at the outset, it being, of course, impossible that any aggregate taken at will can be so transformable. A moment's examination suffices to show that when once the first product of all I ab2c,d,4 lefg7 is chosen, the others are derivable from it in accordance with a simple law,-the requirements being (1) no change of suffixes, (2) the last letter of the first factor to be replaced by the letters of the second factor in succession, (3) the signs of the products to be + and - alternately. As for the first product of all, it is not difficult to see that the orders of the determinants composing it are quite immaterial. Instead of taking determinants of the 4th and 3rd orders, and producing by transformation an aggregate of products of determinants of the 3rd and 4th orders, we might have taken determinants of the (n + l)th and ntllh orders, applied the transformation, and obtained an aggregate of products of determinants of the nth and (m + )th orders. This is the essence of Schweins' first generalisation. His own statement and proof of it leave little to be desired, and are worthy of examination in order that his firm grasp of the subject and his command of the notation may be known. He says (p. 345)"Die Reihe, welche in eine andere iibertragen werden soil, sei V / a(- l,...l........ a....... b i= 2(S)^ | Al,........ B,_, B+... Ban) wo x=l, 2,....., m+1. Der erste Factor wird nach 515 in niedere Summen aufgel6st wI, l...... a-+l m +l a,... ay aiy+l..1. +lX al) A A B ) 2 A........ AA ) B, wo y=l, 2,..., n+1 DETERMINANTS IN GENERAL (SCHWEINS, 1825) 167 wodurch die vorgegebene Reihe in folgende uibergeht:.._ n+x-y\ a,... ay-..1..................... b y Q - l.......... Aa.|| B, Bx +l.. Bm+l Bz Es ist aber nach 522 qni +l-xli b].................... n ay...... n ay e(-) j| B,.. Bx-l Bx+l.. B.,J 'i Bx-iII B,...... Bm+l = (-) ||B,...... Bm+i); folglich,n-y a,.. a y+l.. a l.. - a+... l b2.... b Q = -) A........... A iB.... B,+1 oder es ist az-l||a............ n+l \ | &i.................. n \ E (-) | A. 1.... A Bx.)' B.) n-y+l a1... ay-1 ay+l.. a +l 1 y bl b2 *..* * 2 ) =^ (- A.|......... A B,......BJ+/ wo x=l, 2,..... m+l, y=1, 2...... n+l; oder es ist a........... n:+ 1 bl............ bin a.......... an+l bl........... b - A....n B3) B| 2 B4... B2+1),... an+1..... Al ||A1 A, +1 \ Bin ** 1 '. 1al......... an+l 1 bl.......... b a A....A.. - |l a BBB...... b, = A,............ BB+m I.... I n..... *, * \)II "2; A;.. I......... Bl| ) (hX A- 1|...... IB........ (XI.) 168 HISTORY OF THE THEORY OF DETERMINANTS The further generalisation of which this is possible, and which Schweins effects, depends on the fact that the law of formation twice used in proving the identity is but the simplest case of Laplace's expansion-theorem, and that the said theorem can be similarly used in all its generality. In other words, instead of taking only one of the B's at a time to go along with the A's to form the first factors of the left-hand aggregate, we may take any fixed number of them. For example, out of six B's we may take every set of three to go along with two A's, and we shall have the aggregate a 2 3 4 3 a4 a \ b b2 b3 \ a1 a2 a3 a4 a \ b b 2 b3 \ AI A2B B,2B ) BB B B A, A, B, B, B4 1 B B, B) Il a, a2 a3 a4 a5 \ || bb2 b3 \ 1 a a3 a4 as \ I|| b b2 3 \ + 1 A 1 A B2 B B B B A1A B1BB6) BB B B a )3 a2 a3 a41 5 \ b| b 2 3 a\ 2 3 a4 a5 b b23 2 3 + A1 AB AB B 2B5 B6 AA B B B a2 3 \ 2 \ 4 a b 2 b3 a, a1 2 1 3 a4 a5 \ b b2 b3 \ +I A1 A2 B, B3 B Bj B B) + A A, AB B, B,5 Bj B, BB) a 3 a 32 a 5 \ l b2 23 \ a, a, a3 a a \, bi 2 b3 \ - A A2B B B Bj B2 B B) + A A 2 A2 B1 I5 B6 j B, BB) a a21 a3 a4 a\ bN b2 b3 \ a 1 a32 a3 a4 a5 \ N bl 2 3 6 \ - A1 A2 B3 B4) Bs B6+ |j Al A B2 BB | B4 B,6 a Ct2 a0 a4 aS \ I 2 b3 \ 1 a a3 a4 a3 5 \ l 2 b3 \ a, a2 a03 a4 as \ N b, 22 b3 \ 1l a2 a3 a4 15 \ I bl 2 b3 \ A+ I A B, B, B5 l BI B, B6 A1 A2 B, B) 6 B1 B ) B A,1 As4 B BB 1 a2 bB3 + I1 A 2D3 AsBB, 6 B B BB - Al A2 B B, B, B3), -the sign of any term being determined by the number of inversions of order among the suffixes of all the B's of the term. In this particular case the first use of Laplace's expansiontheorem is to transform I a2 a3 a4 a A A2AB, B2,B3) DETERMINANTS IN GENERAL (SCHWEINS, 1825) 169 and the other similar determinants each into an aggregate of ten products, the two factors of any product in the expansion of 11 a 2 a3 a4 a5 \ A1 A2 B1 B2 B3) being, as we should nowadays say, a minor formed from the first two rows and the complementary minor. In this way would arise 20 rows of 10 terms each, and these being combined by a second use of Laplace's expansion-theorem in columns of 20 terms each, the outcome would be an aggregate of 10 products, viz., the aggregate a a2 \ a3 a4 a5 b b2 b3 \ a 3 \ a2 a4 a5 b b2 b3 \ A1 A ) B B2 B3 B B6 -| A1 A2j * B1 B2 B B B j || a a \ I a2 a3 a5 bi b b3 \ al a5 \ | a a3 a4 bl b2 b3 \ A)+ A A) B B B~ B~ B Bj -i Al A / B B, B, B B, B a2 a3 \ || 4 l 5 bl 3 \ 2 \.| aI.3 as b 2 b3 \ + A A2 * Bi B2 B3 B5 B - A1 A2I* B13 B BB B6 + | 2 a,5,1 a3 a4 1b b2 b3 + a3 a4 \ | a2 a5 b1 b2 b3 + A1Al2 B1 B B A B B B B B6 I a3 as \ a 1a 2 a4 b5l b2 b3 \ | I ~4 a 5\ a 2 a2 a3 b b2 b3 \ A, A2 B1B, B3 B B BJ + B A AJ B1B, B3 B B BB6. The following is Schweins' statement of this most general theorem:2(-)"/ A &B II B/...... Bij (XLIX. 2), v- \, ) atl............. a'n-q \ a'n-q+l...a'n b1.. bm-q\ A A1-........ *q B m. The only points about it requiring explanation are the exact effect to be given to the symbol 2, and the meaning of the dashes affixed to certain of the letters. The two symbols are connected with each other, the dashes not being permanently attached to the letters, but merely put in to assist in explaining the duty of the E. On the left-hand member of the identity, the two symbols indicate that the first term is got by dropping the dashes, and that from this first term another term is got, if we substitute for B1.... Bq, some other set of q B's chosen from 170 HISTORY OF THE THEORY OF DETERMINANTS B.... B., and take the remaining B's to form the B's of the second determinant,-the two sets of B's being in both cases first arranged in ascending order of their suffixes. On the other side of the identity, the use of the symbols is exactly similar, n -q of the n upper elements ac,...., a,, being taken for the first determinant of any term of the series, and the remainder for the second determinant. The number of terms in the series on the one side is evidently mz!/q!(m-q)! and on the other n!/q!(n —q)! In the demonstration of the theorem greater fulness is evidently necessary than in the case of the previous theorem, the rule of signs in particular requiring attention. This Schweins does not give. He merely tells the character of the first transformation, symbolising the expansion obtainable, and then says that a recombination is possible, giving the result. The succeeding five pages (pp. 350-355) are devoted to evolving and stating special cases. This is by no means unnecessary work, as in the case of a theorem of so great generality it is often a matter of some trouble to ascertain whether a particular given result be really included in it or not. To students of the history of the subject the special cases are doubly interesting, because it is in them we may expect to find links of connection with the work of previous investigators. The first descent from generality is made by putting some of the B's equal to A's, the theorem then being 1l...........a +,.............. bk+p Al [+~+~ |...A.......... z(-).A,... A,+8B1..... B.,. B+ei.. B' A;.....A (XLIX. 3) i all *.. p+a ) ' a' +s+)... a'+s+q B.. bx+p P+Sa+,+P........ ~+k.. 2 Al2-) I AP.8...~B. B.......... Bq+A-1.... Ap If in addition to this specialisation, some of the b's be put equal to the a's, the result is bl \ bh a'l ' -- | a+s-h+l.... a..'+s-h+q bl b~+k. (-)A,......... A.I B A........ A 1 (XLIX. 4) Z -)1...* A 7 1 *... * p +s+q-') B................. bA\) A2-1 A,.... Ap+8B1 B ]3q+l ~ ~ q.+h+k-P 1Ai Aj DETERMINANTS IN GENERAL (SCHWEINS, 1825) 171 -a notable theorem, which it would not be inappropriate to consider rather as a generalisation than as a special case of the theorem from which it is derived. Returning, however, to the preceding case, and putting = 0, we obtain a............. ap+s+q bl........ b,. E A,...A..B" A,......AJ a',l ***** a.f+,\ ||. S.l......... a,,+s+q b...... bp = -...A A+s B.... BqA............ A This may be viewed as an extension of Laplace's expansiontheorem to which it degenerates when p is put equal to 0. Though comparatively a very special identity it is considerably beyond anything attained by Schweins' predecessors. In fact, we only come upon something like known ground, when in descending further, we put in it q = 1. The result thus obtained is al'............ a +S............. bp s - ( 1A1....... Ap+.J, J1 A1.......A, (XLV. 2) which closely resembles a theorem of Desnanot's. The difference between them consists in the fact that here the second factor on the left-hand side is arny determinant of a lower order than the cofactor, whereas in Desnanot the second factor is a minor of the cofactor. A further specialisation, viz. putting B1=Ap+i, brings us to the result a'..... a' + b..a... A,..... Alj+s| Al...... Ap+ - 0 or (xxII. 8) (-)1..... AB B..1.. B ), b2 The form here is that of a vanishing aggregate of products of pairs of determinants, and identities of this form we have already had to consider in dealing with Bezout, Monge, Cauchy, and Desnanot. To the last of these only does Schweins refer. His words are (p. 352) 172 HISTORY OF THE THEORY OF DETERMINANTS "Wird in dieser Gleichung s = 2 gesetzt, so entsteht folgende:bl.....P+.... + A-l*Ai... A2;B' ) B'2BKs.*. B,+) = 0 wovon Desnanot einige ganz specielle Falle gefunden hat, oder vielmehr der ganze Inhalt seiner Untersuchung ist in folgenden dreien Gleichungen begriffen ( )* \i z B'~. || n,3 b, B' = 3 (- A B' 1) | B'2B'3B',B')=0, welche mit ermtidender Weitlaufigkeit bewiesen sind..... This statement is unfortunately not by any means accurate. As for the "ermidende Weitlaufigkeit," there can be no doubt about it, and to assert its existence is fair criticism; but to say that the whole of Desnanot's results are to be found in the three identities specified is a misrepresentation of the actual facts, and therefore quite unfair. The reader has only to turn back for a moment to our account of Desnanot's work, to verify the fact that the two most important general results attained by the latter (xxIII. 7 and XLV.) are ignored by Schweins altogether. The remaining paragraphs of the chapter are taken up with the very elementary case in which the products are three in number, and the theorem itself nothing more than one of the extensionals so lengthily dwelt upon by Desnanot, viz. the extensional of ClbC2I - b1la c2l c+ lab21 = 0. It is written in several forms, e.g.Ial.............*.. -n+, \ a a................. aZ+m+l \ Al........ An+m ' 11 Al... An-lA1+l '... A?+++, ) ai............. A+.n+ B +l A| A..n-lAn+l * An+m+l) || Al *A*** -f*nnm ~ Ia i I.............." ) ' +mA I....................l n + m + l O. The next chapter, the third, concerns the solution of a set of linear equations, although according to the title its subject is DETERMINANTS IN GENERAL (SCHWEINS, 1825) 173 the transformation of determinants into other determinants when the elements are connected by linear equations. It presents no new feature. The fourth chapter deals with a special form of determinant, the consideration of which must therefore be deferred. Suffice it for the present to say, as an evidence of Schweins' grasp of the subject, that the solution of the problem attempted is complete and the result very interesting. The fifth gives the solution of a problem on which the general Theory of Series is said to depend, the problem being the transformation of al a............... a BA... A 1A,2+.. A ) I a a.............. a \ A1A2.. A. ) into an unending series. The numerator, it will be observed, is of the order oo: the denominator is of the same order: and all the rows of the former except one occur in the latter. Indeed, if the first row of the numerator were deleted, and the nth row of the denominator, there would be nothing to distinguish the one from the other. The subject is best illustrated by a special example in more modern notation. Recurring to the extensional above referred to as the concluding theorem of the second chapter, and taking the case where the factors are of the 4th and 5th orders, we manifestly have a Ib2c3e4.lab2c314f5 - Iab2c3dj. c ab2c3e4f5 + catb2c3f4j. (ab2c3e4d5 =1 0 from which, on dividing by Iacbc3e4.cIab2c3e4d5, we obtain t1b2cd4f1 _ A Ilab2c4 l cb23e45 I 1b2c3f4 a 0 c1b2c3e4d5 ctLb2c3e4 | a l2c3e465 ctabc3e Similarly I alb2c3f4 1b2c3 I alb2e3f4 + alb2f3 - 0 acb2e3c4 Ia b2e3 b I caL1.2e3C4 I a1b2e3 1 I clb2f3 I alb2 I Cle2f3 I al I a1e2b3 a- ce2 a leb /3 ale, - and lalf2 al.lelf2 + fi = 0 I e1a2 I el I e12 I e 174 HISTORY OF THE THEORY OF DETERMINANTS the last fraction of each identity, be it observed, being the same as the first of the next with its sign changed. From the four by addition we have I cb2c3d45 = _ alb2c3(4 1. alb2ce4fs I alb2c3cde5 ab2c 3eI a1lb2c3d4e5 + I ab2c3 I xb2e3f/4 I I alb2e3 I | lb2ce3 I I cLb9 1 I ale2f3 I + a1 e1 el lale21 +A. e1 The general result, as stated by Schweins, is that a1.............. anz+m+l a..+ i. B Al... A._ A..... An + J+l (- I -l _ a an+m+l1 () V(-' L() ). V(1'+1) + L(n). V(.+l) -_....1 ()) V(2+-n+ l) al.....~............... a,,+m-1 a al.....................(-y'+ L^. V 1 1....................... an+n\ where L:' a, A A A,............. A..n+- and YV(+m) A I *I * * *. I * A1.+n-~ ). (L.) al....................... a,,+,, | Al...........,,,, A.Since the expression thus expanded is itself one of the L's, viz., L ) —as is readily seen by transferring the B from the beginning to the end, and denoting it by A,,+,+2,-and since L('")=l, the identity may equally appropriately be written with L("2) at the end of the right-hand member, and looked upon as the recurring law of formation of the L's in terms of the V's. This Schweins does, giving indeed the result of solving for L L(..1 DETERMINANTS IN GENERAL (SCHWEINS, 1825) 175 The Second Section (pp. 369-398), consisting of five chapters, and extending to 30 pp., is devoted to a special form of determinants, viz., those already partly investigated by Cauchy, and afterwards known as alternants. The Third Section (pp. 399-403), extending only to 4 pp., deals with another special form, whose elements are finite differences of a set of functions. The Fourth Section, (pp. 404-431), consisting of four chapters, and extending to 27 pp., has for its subject a third special form, foreshadowed by Wronski, the characteristic of which is that one of the indices denotes repetition of an operation involving differentiation. When these Sections come to be considered in their proper places, it will be seen that very great credit is due to Schweins for his labours, and that he has been most undeservedly neglected. The fact that he had ever written on determinants was only brought to light in 1884:' and, so far as can be gathered, his treatise had no influence whatever either on the work of succeeding investigators, or in diffusing a knowledge of the subject. *v. Philos. Magazine for November: An overlooked Discoverer in the Theory of Determinants. CHAPTER VII. DETERMINANTS IN GENERAL, FROM THE YEAR 1827 TO 1835. THE writers of this period are six in number, viz., Jacobi, Reiss, Minding, Cauchy, Drinkwater, Mainardi. Of these by far the most important both as regards quality and quantity is Jacobi; Cauchy contributes an investigation in which determinants are used; Minding makes some little use of the functions without knowing it; all the others are unimportant expositors. JACOBI (1827). [Ueber die Hauptaxen der Flachen der zweiten Ordnung. Crelle's Journal, ii. pp. 227-233; or Werke, iii. pp. 45-53.] [De singulari quadam duplicis Integralis transformatione. Crelle's Journal, ii. pp. 234-242; or Werke, iii. pp. 55-56.] [Ueber die Pfaffsche Methode, eine gewohnliche lineire Differential-gleichung zwischen 2n Variabeln durch ein System von n Gleichungen zu integriren. Crelle's Journal, ii. pp. 347-357; or Werke, iv. pp. 17-29.] We come here simultaneously on the names of a great mathematician and a great mathematical journal. Crelle's Journal fiur die reine und arngewandte lMathencatik, which began to appear at the end of the year 1825, and which without any of the symptoms of old age still survives, has rendered on more than one occasion important service towards the advancement of the theory of determinants. Its first contributor on the subject and one of its greatest was Jacobi. At a later date he published in the Journal an excellent monograph on Deter DETERMINANTS IN GENERAL (JACOBI,. 1827) 177 minants; but even his earliest papers show that he had begun to find it a useful weapon of research. In the first of the memoirs above noted, dealing with the subject of orthogonal substitution, constant use is, of course, made of the functions; but there is no special notation employed, nor indeed anything to indicate that the expressions used were members of a class having properties peculiar to themselves. In the second memoir, which likewise is taken up with a transformation, but in which the sets of equations involve four unknowns, any special notation is still avoided. Expressions, readily seen to be determinants of the third order, are even not set down, because, as the author expressly states, they would be too lengthy. The last clause of the passage in which this statement occurs is noteworthy. The words are (p. 236)"Dato systemate sequationum a t + P x + y y + 8 = Zn, a' U p' x + y' y + 6' z= n', a" U J+ V" + / y" yy + 8" Z = 11, a"'U + 3:"x + y"'y + 3"'z = i", ponamus earum resolutione erui: Am + A'1' + A"n"' + A"'"z"' = t, Bn+ B'Im' + B"m" + B"'i"' = x, Cm + C'' + C"'" + C"'1 = y, Dmi + DW')' + D"n'lZI + D"'1'" ' z. Valores sedecim quantitatum A, B, etc., supprimimus eorum prolixitatis causa; in libris algebraicis passim traduntur, et algorithmus, cuius ope formantur, hodie abunde notus est." On the next page, in eliminating D, D', D", D"' from the set of equations 0 = D(- x)+D'b' +D"b" +D"'b"', 0 = Db' + D'(a' +)+ D"c"' + D"'c", 0 = Db" +D'c"' +D"(" + x)+D"'c', 0 = Db" + D'c" +D"c' D (C"' + )) he arranges the resultant as one would now do who had expanded it from the determinant form according to products of the elements of the principal diagonal, viz., he says (p. 238)M.D. M 178 HISTORY OF THE THEORY OF DETERMINANTS "Fit illa, eliminationis negotio rite instituto 0= (a -x)(a' +x)(a" +X)(a"' X) -(a-x)(a' + x)c'c' - (a - ) (a" + ) cc" - (a-) (a"' + )c//'"' - (a" + x) (a"' + x) b'b' - (a"' + x)(a' + x) b"b" - (a' + x) (a" + x) b"'b"' + 2c'c"c"' (a - x) + 2c'b"b' (a' + x) (LI.) + 2c"b"'b' (a" + ) + 2c"'b'b"(a"' + x) +b'b'c'c'+ b"b"c"c" J+ b"'b"'c"'c"' -- 2b'b"c'c" - 2b"b"'c"c' - 2b'b'c"'c'." From the next paragraph we learn his sources of information, and infer that as yet Cauchy's memoir was unknown to him. The first sentence is (p. 239)"Inter sedecim quantitates a, /, etc. et sedecim, quse ex iis derivantur, A, A', etc. plurimse intercedunt relationes perelegantes, quse cum analystis ex iis, quse Laplace, Vandermonde, in commentariis academime Parisiensis A. 1772 p. ii., Gauss in disquis. arithm. sectio V., J. Binet in vol. ix. diariorum instituti polytechnici Parisiensis, aliique tradiderunt, satis notse sint, paucas tantum referam, quae casu nostro speciali ope sequationum (iv) facile ex iis derivantur." The third memoir is by far the most important to us. In the course of the investigation a new special form of determinants, afterwards so well known by the designation skew determinants, turns up; and three pages are devoted to an examination of the final expanded form of it. This examination, we cannot, of course, now enter upon; but it is of importance to note that in it Jacobi takes the step of adopting the name determinant,-a fact which doubtless was decisive of the fate of the word. The adoption thus made (although stated to be from Gauss), and the occurrence of the words " Horizontalreihen," "Verticalreihen," make it probable that Cauchy's memoir had now come to his notice. REISS (1829). [Memoire sur les fonctions semblables de plusieurs groupes d'un certain nombre de fonctions ou elemens. Correspondance math. et phys., v. pp. 201-215.] In Reiss we have an author who starts to his subject as if it were entirely new, the only preceding mathematician whom he DETERMINANTS IN GENERAL (REISS, 1829) 179 mentions being Lagrange. Like Cauchy he opens by explaining a mode of forming functions more general than those of which he afterwards treats, the essence of it being that an expression involving several of the n.y quantities, aa ad aY...a b. by be... b COL c3.c.. cp,r r, r3... p is taken, and each exponent ("exposant") changed successively with all the other exponents, a,,,, or each base changed with all the other bases, a, b,.... Only a line or two, however, is given to this, the special class known to us as determinants being taken up at once. His notation for alb2c3 - a1b3C2 _ a2bc3 + Ct2b 31 + a3blc2 _ a3b2C1 is __ (abc, 123), (vi. 7) a line being drawn above the exponents to indicate permutation. His rule of term-formation and rule of signs are combined after the manner of Hindenburg. Like Hindenburg he arranges the permutations as one arranges numbers in increasing order of magnitude; but, unlike Hindenburg, after the arrangement has been made he determines the sign of any particular term. On this point his words are (p. 202) "Cela fait, determinons generalement le signe du Mtme produit (soit M) de la maniere suivante. Le nombre M sera renferme entre les produits 1.2.3... I et 1.2.3... 1(1+ 1); soit M=m + Xx 1.2.3... 1, de sorte que X < 1 + 1, et m> 0 et < 1 + 1.2.3... 1. Cela etant, faisons _fi = m (-1 )X\" (III. 25) This apparently means that if the sign of the 23rd term in the expansion of (abed, 1234) * Or (abcde, 12345), or indeed (ala2.... a, 123... n). 180 HISTORY OF THE THEORY OF DETERMINANTS be wanted, we divide 23 by 1.2.3, getting the quotient 3 and the remainder 5, and thence conclude that the sign wanted is got from the sign of the 5th term by multiplying the latter by (- 1)3. Of course 5 has then to be dealt with after the manner of 23, the quotient and remainder this time being 2 and 1, so that we conclude that the sign of the 5th term is got from the sign of the 1st term by multiplying by (-1)2. And the sign of the lst term being +, the sign of the 23rd is thus seen to be (1)3+2 i.e. -. It would seem at first as if the case where M is itself a factorial were neglected. This however is not so, the condition m < 1 + 1.2.3... 1 being corrective of the opening statement that M must lie between 1.2.3... and 1.2.3... (1+1). For example, the term being the 24th, we put 24 in the form 3 x 1.2.3 + 6, and thus learn that the sign required is different from the sign of the 6th term: then we put 6 in the form 2 x 1.2 + 2, and thus learn that the sign of the 6th term is the same as the sign of the 2nd term; finally, we put 2 in the form 1 x 1+1, which shows that the sign of the 2nd term differs from the sign of the 1st: the conclusion of the whole being that the signs of the 24th and 1lt terms are the same, or that they are connected by the factor (-1)3+2+1 Though interesting in itself, a more troublesome form of the rule of signs for the purposes of demonstration it is scarcely possible to conceive, and, as might therefore be expected, it is on the score of logical development that Reiss's paper is weak. Through inability to use the rule later in the demonstration of the so-called Laplace's expansion-theorem, he is forced to supplement it by another convention. His words are (p. 203)"Avant d'aller plus loin, faisons encore la determination suivante. Soit o une fonction quelconque dans laquelle les k quantites A,B,C,... A2 entrent d'une maniere quelconque. Supposons que ces dernieres soient les k premieres de l'Nchelle A.. S 1 2 3 k... s' Qu'on fasse avec ces s elemens toutes les combinaisons sans repetition de la classe k, et qu'on les substitue successivment an lieu de A,B,... Ak dans la fonction. o; c'est-a-dire Ie premier element de .DETERMINANTS IN GENERAL (REISS, 1829) 18 181. chaque combinaison 'a A, le second h B, etc. Nous obtiendrons par la' autant de fonctions semblables 'a i qu'il y a de combinaisona de la classe k de s 616mens. Or, entre toutes. lea c ombinaisons qui en precedent une quelconque, il s'en trouvera une qui aura k- I elemens communs avec elie, tandis que lea deux 616mens qui restent isole's dans lune et l'autre se suivent imme'diaternent dans le'chelle. Donnons 'a la f onction qui contient la dernie're de ces combinaisona le signe oppose' 'a celui de lautre f onction ~ par consequent les signes de toutes les fonctions semblables 'a (0 seront parfaitenient determines, et de'pendront du signe de la premiere fonction,(f(A,B,C,. A`) Soit, par exemple, s =5, k~= 3; nous aurons successivement, en rempla~ant ABC, S par 1, 2, 3, 4, 5, et en donnant le sigue (+) 'a f(123), +f(123), -f(124), +f(125), +f(134), -f(135) +f(145), — f(234), +f(235), -f(245), + f(345). Voici comment on determinera le signe de chaque fonction scmblable 'a' w d'apre's celni d'une autre quelconque. Qu'on cherche les. noinbres qui se trouvent dana hi'chelle ABC...X,sous les e'lemens de 1'une et de l'autre de ces fonctions. Si l'on nomme h et h' leurs sommes respectives, on trouvera le signe de l'une des -fonctions ( ) i-h' x le signe de l'autre." Four theorems he considers fundamental, viz., those known to us as (1) Bezout's recnrrent law of formation, in all its generality; (2) Vandermonde's proposition that -permutation of bases leads to the same result as permutation of exponents; (3) Laplace's expansion-theorem; (4) Vandermonde's proposition regarding the effect of making two bases or two exponents equal. The two most important, viz. (1) and (3), he leaves without proof, and the 4 th he says he would at once deduce from the 3rd,doubtless by choosing the expansion in which the first factor of every term would be of the form (aa, a~3) and therefore equal to zero. The proof of the 2Qnd theorem, viz., (abe... r,aSy... p) = (abc....r,af3y... p), 182 HlISTORIY OF THE THEORY OF DETERiMINANTS is by the method of so-called induction, and may be illustrated in a later notation by considering the case al a2 a.3 al b, el b1 b2 b.3 a2 b2 02 Oi 02 03 (t3 b3 03 From theorem (1) we have 1 a2 3. b b b bb b b1 b2 b3= a1 2 3 a 1 3 + a b1 b2 02 001031O 02 el 02 03 a2 a. ~aa ai a2 a2 a3 b2 b I 2 03 01 03 Ci 02 a2 a3 a1 a3 + a, a2 = l b - 02 ~ 03 b b2 b3 b1 b3 b1 b2 But by hypothesis all the determinants on the right here may have their rows changed into columns; and this being done we have by addition and the use of theorem (1) a. a2 a3 al bi e1 3 bi b2 b3 =3 a2 b2 02 O 02 c3 a3 b3 03, and thence the identity required. (ix. 4) To this proof the following note is appended (p. 207): "Cette demonstration quoiqu.'assez simple semble reposer cependant sur un artifice de calcul: mais en cherchant une demonstration direte, j ai rencontre' nue diffilculte' d'nn genre partienlier. En effet, on tronve facilement que line terme de l'une des fonctions en question est aussi 6gal on an m~me terme de l'autre, ou. ge'neralement au Mmie, et que, dans le dernier cas, Imieterme de la premie~re est aussi egal anuin de la seconde, abstraction faite des signes. (ix. 5) Mais l'identite' de ces derniers (qui est de rigneur) exige des explications tre's-longnes et beancoup moins e'lementaires qne la demonstration que je viens de donner." DETERMINANTS IN GENERAL (REISS, 1829) 183 The remaining six or seven pages of the paper are more interesting, and concern the subject of vanishing aggregates of products of pairs of determinants. The theorems were suggested by taking, as we now say, a determinant of even order having its last n rows identical with its first n rows, e.g., the determinant (abab, 1234), and using theorem (3) to expand it in terms of minors formed from the first n rows and their complementary minors. When -n is even, a proof is thus obtained, as we have seen in the footnote to the account of Bezout's paper of 1779, that the first half of the expansion is equal to zero. When n is odd, the method fails, although the proposition is still true.* Reiss's enunciation is as follows (p. 209):"Theoreme V.-Soient les echelles /a b.. r, a, b a y... a9ll, a1 +l. p A 1 2.., I +l, l+2,. n 2n). 3..., +, 2, qu'on fasse avec les 6elmens py,y,..., p toutes les combinaisons de la classe (n -1), et qu'on les substitue successivement dans le premier facteur du produit (ab.,. A, a/3... r,). (abpy... ar, a..., p) au lieu de py... a"; qu'on remplace maintenant dans l'autre facteur les exposans an+... p par tous ceux qui ne se trouvent pas dans * It is worthy of note in passing, that a common method does exist for establishing the two cases, —a method quite analogous to Reiss's, but difficult of suggestion to one who used his notation, or indeed to any one who had no notation suitable for determinants whose elements had special numerical values. All the change necessary is to make the last n elements of the first column each equal to zero. This causes no difference in the result when n is even, e.g., from the identity a1 a2 a3 a4 b 2 2 b b4 - a2 a3 Ca4 b b 3 b4 we have, as before, I ab2\ 1. a3b4I -I a1b3 |.| a2b4 + I ab4 1. a2bI31 =0; and when n is odd, the second half of the terms which previously gave trouble do not occur. 184 HISTORY OF THE THEORY OF DETERMINANTS le premier, en ayant soin de les ecrire suivant l'ordre indique par les echelles. Si l'on donne au premier produit le signe (+), et qu'on determine les signes de tous les autres d'apres (II), la somme algAbrique en sera =0, que le nombre n soit pair ou impair." (xxII. 9) An example of it is (abc, 123)(abc, 456) - (abc, 124)(abc, 356) +(abc, 125)(abc, 346) - (abc, 126)(abc, 345) +(abc, 134)(abc, 256) - (abc, 135)(abc, 246) +(abc, 136)(abc, 245) + (abc, 145)(abc, 236) -(abc, 146)(abc, 235) + (abc, 156)(abc, 234) = 0, the left-hand side being nothing more than the first ten terms of one of the expansions of the vanishing determinant Ca1 2 CG a,4 aC5 s b1 b2 b3 b4 b5 b6 GC1 C2 C3 C4 C5 C6 c ca2 ca3 a4 a5 c b1 b2 b3 b4 b5 b6 C1 C2 C3 C4 C5 C6 or the other ten terms with their signs changed. Reiss's proof is lengthy and troublesome, the method being to expand each factor in terms of the a's and their complementary minors, perform the multiplications (e.g., in the special case just given the multiplication of aCL b2c3 - aC2lbc3l +% 3lbl1c by a41b5c6 -( a,,b4Cc6 + aGlb4c 5, &c.) and show that the terms of the final aggregate occur in pairs which annul themselves. The next theorem is of still greater interest, because it is that peculiar generalisation of the preceding which in later times came to be known as the 'extensional.' The way in which it is established is also noteworthy, viz., by deducing it as a special case from the theorem of which, as we have said, it may be viewed as a generalisation. The author's words are (p. 213): DETERMINANTS IN GENERAL (REISS, 1829) 18 185 "Ce the'ore'me nous conduit 'a une relation qui existe dans le eas, le plus general, savoir si v - n est un. nonmbre. quelconcjue ou positif on ne'gatif. Supposons v > n, et v - n N N; soient les ~e'helles, (b r a1 Ni'* ' N A BR et 1.NIN+1,... 2N, 2N~1, 2N~+2, v. Qu'on fasse avee, les e'lemens /3, y,...a~,a~ p toutes les combinaisons de la elasse N -I1; qu'on les substitue snecessivement an lieu de /3... aN danis le premier facteur du produit (ab..rAB.. RI a/3... a'A.B... P) x (ab..rAB.. R, aN~1. p AB... P); qn'on remplace dans l'antre facteur les exposans a N+l.. p par tous ceux qni ne se trouvent pas dans le premier: qu'on determine enfin le signe de chaqne prodnit d'apre's (II): la somme alge'brique en sera = 0. (XX111. 1 0) (XLIV. 5) "En effet, supposons les i'chelles 'b...r A, B,.., a, b, r A, B,... R \.12... N, N+1,I N+2,... v-N, v-N+1, v-N+-2,... v, v+1,I v+2,... 2v -2NJ t a P... aN, aN+1 I... p, AI B,... AV3NI Av-3N+1,I... P, A I....1 2... N, N+i,... 2N, 2N-)1 2N+2,... v-N, v-N+1,...v,v+1,... 2v-2N,} Formons avee ees e'limens la fonetion di'erite dans le dernier thi'oreinie: la somme totale en sera done = 0, et le premier terme aura la forme (ab....rAB... IR, a/3... pA....A V3N) Or, on voit facilement qne tons les termes qui ne contiennent pas dans ehaque facteur lous les exposans A, B,...P, s'evanoniront separiment, paree qu'il y anra -des exposans identiques dans inn on lantre des facteurs. II ne restera done que les termes qni, eontenant adans le premier facteur, y i'puisent snecessivement toutes les combinaisons de la elasse N - 1 des e'lemens /3, y/,... p. Mais les signes de ees termes sont e'videmment ditermine's comme uls devaient 186 HISTORY OF THE THEORY OF DETERMINANTS l'Ptre; partant la somme algdbrique de tous les termes est =0, ce qu'il fallait ddmontrer." This will be best understood by considering a special example. Going back to the previous theorem, and selecting its simplest case, we have laib,l a,%bj - la,b.1Hct2bjl + Ja1b4H-Ja2b31 = 0. Now what the new theorem asserts in regard to this is that we may with impunity extend each of the determinants occurring in it, provided the extension be the same throughout. For example, choosing the extension 16,,~71* we can, in virtue of the new theorem, assert the truth of the identity alb2b5Y67S7.a,3b4C5.i6S7 - 2a2b3$3,i6771. aLb4~5,6~71 + Jalb4b57Si67J. a2b3b37S761 = 0. That the two may be viewed as cases of the same theorem will be apparent when it is pointed out that just as the first is derivable from al a2 a. a4 b1 b2 b3 b4 = 0, a2 a3 a4 b2 b3 b4 so the second is derivable in exactly the same way from a perfectly similar identity,t viz. * In Reiss's notation the extension is AA BB... RP. t It is perhaps a little more readily seen to be derivable from a1 a2 a. a4 a. a6 a7 b, b2 b3 b4 b5 b6 b7 4`1 42 6 ~4 ~5 46 6 9 7i 72 973 974 975 776 977 a2 a3 a4 a5 a6 a7 b2 b3 b4. b5 b6 b7 97 973 94... 975 97 97 772 'q3 N 77 75 776 777 3~2 ~3 h 4 ~ 5 ~6 6 7 DETERMINANTS IN GENERAL (REISS, 1829) 187 a,1 O2 c3 a4 a5 a6 a7 a5 Ct6 C b1 b2 b3 b4 b5 b6 b7 b5 b6 b7 el $2 $3 &4 T 6 7 $ 5 & 7 n1 12 3 ' 4 '5 '16 '77 '5 '16 '7 *1 a2 ~3 a4 25 G6 ~ 7 75 26 27 _ 0. a2 a3 a4 a5 a6 a7 a5 a6 a7 b2 b3 b4 b5 b6 b7 b5 b6 b7 2 $3 $4 &5 6 $7 $5 6 $7 * '72 '13 '14 '5 '6 '7 '5 '6 '17 2 r 3 r4 r5 r6 r7 A5 r6 7 Many more products than three (126 in fact) arise in the latter case; but, for the reason stated by Reiss, only three of them do not vanish. CAUCHY (1829). [Sur l'equation a I'aide de laquelle on determine les inegalite's seculaires des mouvements des planetes. Exercices de Math., iv. pp. 140-160; or (Euvres (2) ix. pp. 172-195.] As the title would lead one to expect, the determinants which occur in this important memoir belong to the class afterwards distinguished by the name "axisymmetric," and thus fall to be considered along with others of that class. Since, however, the proof employed for one of the theorems therein enunciated is equally applicable to all kinds of determinants, it would be scarcely fair to omit here all mention of the said theorem. In modern phraseology its formal enunciation might stand as follows:S being any axisymmetric determinant, R the determinant got by deleting the first row and first column of S, Y the determinant got by deleting the first row and second column of S, and Q the determinant got from R as R from S, then, if R = 0, SQ = - Y2; 188 HISTORY OF THE THEORY OF DETERMINANTS and the theorem in general determinants whose validity is warranted by the proof given is in later notationIf b2c3clj = 0, then Ia2dc3c. b1c3d4l = - l ab2e3dJl c3d. (xx. 2) This, it is readily seen, is not a very obscure foreshadowing of Jacobi's identity A1B2 1 = I alb2ccd4 I \ C3 c d4 JACOBI (1829). [Exercitatio algebraica circa discerptionem singularem fractionum, quae plures variabiles involvunt. Crelle's Journ., v. pp. 344-364; or Werke, iii. pp. 67-90.] In the ordinary expansion of (ax + by + cz-t)- there are evidently only negative powers of x and positive powers of y and z; in the like expansion of (b'y+c'z+a'x-t')-l there are only negative powers of y and positive powers of z and x; and similarly for (c"z + a"x + b"y- t")-. It follows from this that the ordinary expansion of (ax + by + cz- t)-l (b'y + c'z + ax - t)- (c"z + a"x + b"y- t")-, looked at from the point of view of the powers of' x, y, z, contains a considerable variety of terms; for example, terms in which negative powers of x occur along with positive powers of y and z, terms in which x does not occur at all, and so forth. There is thus suggested the curious problem of partitioning the fraction 1 (ax + by + cz - t)(b'y + c'z + a'x - t')(c"z + a"x + b"y - t") into a number of fractions each of which is the equivalent of the series of terms of one of those types. This is the problem with which Jacobi is here concerned. In the case of two variables he counts three types of terms, viz., that in which the indices of both x and y are negative, that DETERMINANTS IN GENERAL (JACOBI, 1829) 189 in which the index of x only is negative, and that in which the index of y only is negative. n the case of three variables he counts seven types, viz., that in which the indices of x, y, z are all negative, the three in which the index of only one variable is negative, and the three in which the index of only one variable is not negative. These two cases are gone fully into, with the result that the expressions for the three aggregates in the former are all found to contain the factor (ab')-1, and the expressions for the seven aggregates in the latter the factor (acb'c")-1. The reciprocal of each of those factors is recognised as the common denominator of the values of the unknowns in a set of linear equations, a denominator " quam quibusdam determinantem nuncupamus et designemus per A." Its persistent appearance in the problem under discussion,-a persistency, in fact, sufficient to suggest the change of the numerator of the given fraction from 1 to (a b') in the case of two variables and from 1 to (a b'e") in the case of three,-is remarked upon:"Quam determinantem in hac quaestione magnas partes agere videbimus, videlicet omrbes illas series 'infiJnitas, quacs ut coeifcientes producti propositi evoluti invenimus, ex evolutione dcignitatum nefyativarum determinantis provenire." Then fixing the attention on a unique term of the expansion Jacobi ventures on the generalisation that the coefficient of l/(xxl 2. * Xn-l) in the expansion of I /( l l 2.... n -1), that is to say, of (ax+by+ez+...)- (b'y c'z+... ( )-1("+... )-1. is the determinant (ca "..... )-1. (LII.) No proof, however, is given, save for the cases where n 2 and n= 3. The proposition is most noteworthy, in that it supplies the generatizng frunction of the reciprocal of a determinant. To obtain a generalisation in a different direction, viz., from 190 HISTORY OF THE THEORY OF DETERMINANTS (ax+by)-l (bly +a1x)-l to (ax+by)-m(bly + ax)-, Jacobi proceeds in a very curious and interesting way. Denoting........ +/3-8a2 + -/2a+/3-l +a-l /+a+ -..+ -3B+. m= —0o or Eax-?nm-l m=+GO by * 1.......... /3-a a-, since it is the sum of the infinite series for (/3- a)-1 and (a -/3)-1, he proves after a fashion that its product by / -a or a -/ is 0, and that therefore its product by 1 1 or y+m(/3-a) y +r n(a -/3) is simply its product by. Turning then from this lemma to the product + 0I0 )A ( 1 1 + \ where o t-x by, = y + a e ss fr where u^0 = ac x + boy, ui = bly + acxlz, he substitutes for the first factor of it............................................................................................I.............................................................................................. a0tob x - b1t I + bo (o1 - ti) blto - aobl I - bo (a ~ -tl) his justification being the fact that bl(o - to) = ILobl I - blto + bo(l - t1); but, on account of the said lemma, he leaves the term bo (u1 - t1) out of both denominators. For the second factor there is thereupon substituted b {lob y - aCotll + { a0bll -x blto } +.......................................................................................................................................... + b1{l cttl - aob1 y} + al { bito - Iaobl l} *Jacobi writes it - + with the caution that the two parts are not P3-a a-3 to be taken as cancelling one another. Of course, also, lower down lie does not write | aob but a0b, - albO or later (aob1). DETERMINANTS IN GENERAL (JACOBI, 1829) 191 on the ground that we have the identity I a0b (1 - t) = bl { a0b, y - Iat 1} + a, { aob, 1 - Ibto 1} the term a1 { a0ob x - I b1tol} being subsequently left out of both denominators for the same reason as before. The result thus reached is consequently laoi4.( ~ 1 )- ( L + 1 /.............................................................L ) I o - tol t - U1 | tf tI - | tl x) I aOb, I a c0b1 = ob Ix - b1to I I b1to I- a"b-l'' + Il )., v0bl o y - aotll a0ot1 - laob y or, if we write I, r for the values of x, y which make 1u - t = 0, - tl = 0, Obl (UQ - to + o t- o U1 u - t t - U1) +(z-$2 - t = (xi$. +..(..... +.......) Since the general terms of the four doubly-infinite series here are tMon tn t V Ukm+l ' qn+l ' '+1 ) YV+1 ' we deduce /lb, ~I mztomt,4 tomt V b1 -^o^W4'1 ~ Zjx~y^ ~~i.e., Iab1 t mt1n z.e., | (ao x + b*o y)m+l(bi y + acx)a+l = lb1tol cl0tlIv 1 61b I A+Ib * x+lyv+l where m, n on the one side and,x, v on the other are to have all integral values from -oo to + cc. Since the coefficients 192 HISTORY OF THE THEORY OF DETERMINANTS of to0mtnl/xmyv on the two sides must be equal, we obtain the theorem:-The coefficient of -v in the expansion of X~xyv 1 (aox + boy)m+l(bly+aix)1+1 is the same as the coefficient of tomtln in the expansion of (b1to - b0t)g-'(aotl.- aito)v-1 a0ob,1 ^-1 it being rememtbered that m and n are of the same sign as a, and v respectively and that m + n = + - v - 2. (LII. 2) In similar fashion the author deals with the case of three functions uQ, Uz1, U2 of three variables x, y, z, proving laboriously and not very neatly the neat result (I 1 1 1 1............................................................ thence deriving na,~,c,/ ~~ t~n t1V t2 CV Y)7 1P 012 U 1 U2?-+u JI+y~z^ and ending with the theorem:The coefficient of 1 in the expansion of X/y yV Zp (aox + boy + Coz)"'+(b, y + cZ + z + ax)( a2x + b2y)r+l is the same as the coefficient of to"t1nt2r in the expansion of {iblc2tto + eibaot2 + b oll}-l{ + oailt2 + ela2lto}v-la0oblt2 + lab2to + la2bohtllaoblc21P+v+P-2 it being understood that m, n, r are of the same sign as a, r, p respectively and that m +-n + r = + v+ p- 3. (LII. 2) DETERMINANTS IN GENERAL (JACOBI, 1829) 193 The corresponding results for A functions of n variables are 'evident. They had already been enunciated in the introductory section of the paper, and Jacobi now merely adds "Omnino similia theoremata de numero quolibet variabilium, quae ~ 1 proposuimus, eruuntur." It has to be noted, however, that belief in the general fundamental theorem, viz., that which includes (a) and (,/) above, is more strongly induced by the elegance of the form of the theorem than by the mode of proof. In ~ 1 it stands approximately thusU(I t 0 to Uo Ui -~t i ti -1 K+A - n 1 1 (1o-po Po- Xi-Pi P -^l/" xn- - no P1n-1- n-1 and then follows the passage containing the two deductions, viz., "' quam aequationem etiam hune in modum repraesentare licet: oao lal... C 1 g o p1P1^./.. p - 1 0 oo+1,l+1..,_,L-1 A f o+l P1i+... +1,-1t+1 designantibus ao, al, etc. 0I,,, etc. numeros omnes et positives et negatives a - o ad + Go. E quo theoremate videmus, coefficientem termini 0fP+1 3311+...;i. +1 in expressione 0 1 ao"tl z+1- 1. al-1 %6bn — 1+1 aequalem fore coefficienti termini a t0la.... tan in expressione ~Po... nj/3^-i." (LII. 3) A 0p~~~ 1l - The use here of 0 - 1, 31+,.... rather than the change made in the two special cases to the less natural /0, P1,... is worth noting. The theorems of the remaining four pages of the paper have a less direct bearing on our subject. M.D. N 194 HISTORY OF THE THEORY OF DETERMINANTS MINDING (1829). [Auflisung einiger Aufgaben der analytischen Geometrie vermittelst des barycentrischen Calculs. Crelle's Journal, v. pp. 397-401.] Unlike Jacobi, Minding was unaware, apparently, of the existence of a theory of determinants. The functions occur at every step of his investigation, yet he makes no use of their known properties to obtain his results. He deals with four problems in his memoir, the second two being the analogues, in space, of the first two. Nothing noteworthy occurs in connection with the latter save that use is made of the identity, /s'y"- /I'~' a1' 7 = a(b'c"-b"c') + a'(b"c-b") + a"(bc'-bc), where /3'= ba'- b'a, /"= b'"- b', y '= c- ' C, y"= c- ca. This identity, it may be remembered, we have noted under Lagrange as an elementary case of the theorem afterwards well known regarding a minor of the adjugate determinant. Strange to say, it makes only its second appearance here fifty-six years afterwards. In the interim, too, no other special case of the theorem seems to have been established. The third is that if P, P' P", P"' be four points in space, given by the equations, q P =a A + b B + c C + d D, q'P' = a'A + b'B + c'C + d'D, q"P"= a"A + b"B + c"C + d"D, q"P"'= a"'A + b"'B + c"'C + d"'D; then for the bulk of the tetrahedron PP'P"P"', we have P P'P"P"' A+ A' +A" ABC D - qq'q"q' where A = ( "- ///y),y" A), A"= "'('y"-By DETERMINANTS IN GENERAL (MINDING, 1829) 195 /3=a' b-ab', y'=a'c- ac el =a' d-ad' =3 a" b' -a' b", y" a" e' - C'",8 a" d'- a' d" a"'- a"b"', y'Y a - a~~// 8" a"'d"- -'d' The transformation of A+~ A' + A" into the form a'"abV'cd"'l -a transformation all-important for Minding's purpose-is not made: but in the remark, "CMan kann den Ausdruck A + A'+~ A" leiclit entwickeln, und wird ihn dann durch a'a" theilbar finden," there is evidently a foreshadowing of the identity a'b la' c, la' d labj a"e', la" d' = a'a" lab/ecl"'dj The fourth theorem, concerning the tetrahedron enclosed by four given planes, A+xB+yC+(a +b xr+c y)C, A+xB+yC+(a' +b' xr+cl y)C, A+xB~yC+(a" +b1'x+c"y)C, A + xB + yC + (a"' + b"...x + c"4')C, is made dependent on the third. The intersections H, II', II", H"' of the four triads of planes are found to be given by q II = (b el )A + (c a' )B + (a b' )C + (a b c )D, q' Hr = (b' c"/)A ~ (c' a")B + (a' b"')C + (a' b' cl )D, q" H"~ = (b"c"')A ~ (c" a"')B + (a" b")C + (a" b"c&')D, q n"= (b"'c )A + (c'"a )B + (a"/'b )c + (a"'b"'"')D, where (be') =b(c'- c) + b'(c" -c) + b"(c- el), (ca') = e(a' -a"/) + e'(a" -a) + c"(a -a'), (ab') =a(b' -b") + a'(b" - b) + a"(b -b'), and (abc) =a(bc') + b(ca') + c(ab'), =a(b'c" -Vb') + a'(Veb~c b") + a"(bc' -b'c). 196 HISTORY OF THE THEORY OF DETERMINANTS Hence, by the third theorem, HIl'II"II"' _ A+A' +A" A BC D qq'q"q"'(b'c")(b"c/) ' where now A ("y - /yl) A = S(/("'- '), A = ("'- y"'), A" =."'(/'y" - /3By'), and ' = (b'c")(ca')- (b c')(c'ac"), 3"=.., y= (b'c")(ab') (b c')(ct'b"), 7". 8'= (b'c")(abc) - (bc')(cb'c'). 8"=... Minding then continues (pp. 399, 400):" Man setze,3=... =... or =... a"'(bc') - a(b'c") + a'(b""') - a"(b"'c) = M. Nach den nothigen Reductionen erhalt man: /3= -" -' )M, -' -"), = - (b' c" -b"c' )M, /" = + (c' - c" )M, Y7" + (b" - b") M, 8" = + (b" c"' - b"'c"l)M, 3I"'= - ( - c"') M, 7" = - (b' - b )M, 8"' = - (b"'c - b c'I)M. Hieraus erhalt man weiter: A = -M3(b"c' -b' c").(b""'), A' = - M3(b"'c" - b" c"'). {(b"c"') - (b")}, A" = - M (b c"' - "' ). (b'c"). Eine weitere Reduction ergiebt: (be"' - b"'c) (b'c") - (bc ("(b"'c" - b"c"') = (C"'b' - c'b"') (b"c"'). Hieraus folgt A + A' + A" = M3 (b'") (b"c"'), und als Resultat: IIL'Il"'"' M,, ABC D - q'q"'"' The first point to be noted here is, that since (bc'), (ct'), (ab'), are in later notation b b' b" c c' c" a a' a" c c' c" a a' " b b' b" 1 1 1 1 1 1 1 1 1 DETERMINANTS IN GENERAL (MINDING, 1829) 197 the identity a(bc') + b(ca') + c(ab') = a(b'c" - b"c') + a'(b"c - be") + a"(bc' - b'c) is the same as b b' bc" c c " a a' a" a ' a" a c e' c" +b c a" +c b b' b" = b b' b" 1 1 1 1 1 1 1 1 1 G C' ", -a disguised special case of Vandermonde's theorem (xnI.), the four elements of one row being each unity. (xnI. 11) The next point is, that since the expression denoted by M, viz., a"'(bc') - a(b'c") + a'(b"') - a"(b"'c) is in modern notation a a' a" a"' b b' b" b"' C C/ C// Cel c c' c" c"' 1 11 the identity ' = -(b'c" b"c') M is the same as b' b" b"' b b' b" ' c" c" C c' " a a, " a," 1 1 1 1 1 1 b b" b' b" b"' a' a" a a a " c c/ C c C" C/" b' b" b"' ' " 1 1 1 1 Ca Cth Ci / C Ci Cit and therefore is, like its eight companions, a fresh case of the theorem regarding a minor of the adjugate.* (xx. 3) *Instead of following Minding's lengthy process, a mathematician of the present time would of course observe that the coefficients of A, B, C, D on p. 195 are the principal minors of M, and using Cauchy's theorem would at once reach the desired conclusion, viz., that the determinant of them =M3. 198 HISTORY OF THE THEORY OF DETERMINANTS DRINKWATER, J. E. (1831). [On Simple Elimination. Philosophiccal Magazine, x. pp. 24-28.] Up to this date, almost 140 years after the publication of Leibnitz's letter to De L'Hopital, no English mathematician's name occurs in connection with the subject of determinants,-a fact most significant of the comparative neglect of mathematical studies in Britain during the 18th century. Apart from the contents, therefore, some little interest attaches to Drinkwater's short paper, as being the first sign to us of that revival which, as is well known otherwise, had taken place some few years before. Drinkwater knew of the investigations of Cramer, Bezout, and Laplace; and professed only to put the elements of the subject " in a more convenient form." His rule of signs is stated and illustrated as follows (p. 25):"Write down the series of natural numbers 1 2 3 4... n, and underneath it all the permutations of these n numbers, prefixing to each a positive or negative sign according to the following condition:"Any permutation may be derived from the first by considering a requisite number of figures to move from left to right by a certain number of single steps or descents of a single place. If the whole number of such single steps necessary to derive any permutation from the first be even, that permutation has a positive sign prefixed to it; the others are negative. For instance, 4 2 1 3... n may be derived from 1 2 3 4.... n, by first causing the 3 to descend below the 4, requiring one single step: then the 2 below the new place of the 4, another single step; lastly, the 1 below the new place of the 2, requiring two more steps, making in all 4. Therefore this permutation requires the positive sign." In this there is essentially nothing new: it at once recalls a theorem of Rothe's (III. 8). In the following paragraph, however, we find the discussion of a point not previously dealt with. The words are (p. 25):"The same permutation may be derived in various ways, and it is necessary, therefore, to show that this rule is not inconsistent with itself: thus the same permutation 4 2 1 3... n might have been obtained by first marching 1 through three places, then 2 through two; and, lastly, 3 through one [?], making six [?] in all, an even number DETERMINANTS IN GENERAL (DRINKWATER, 1831) 199 as before. Without accumulating instances, it is plain, if q be the smallest number of steps by which any number p reaches the place it is intended finally to occupy in that permutation, that if p should advance in the first instance mn places beyond this, it must subsequently return through m places: or, which is the same thing, it must at a later period of the march, allow m of those which it has passed to repass it, so that it will regain its proper place after the number of steps has been increased from q to q + 2m, which, by the rule, require the same sign as q. The same reasoning applies to every other figure; and hence the consistency of the rule is evident." (III. 26) He then establishes four properties of the functions, viz. (1) Vandermonde's theorem regarding the effect produced on the fu,nction by transposition of a pair of letters; (2) Bezout's recurrent law of formation; (3) Scherk's theorem regarding the partition of one of the functions into two; and (4) Scherk's theorem regarding the removal of a constant factor from one of the functions. The two latter theorems, which, as we have.seen, had been stated for the first time only six years before, are given by Drinkwater in the following form (p. 27):"(8) If any factor in f{XYZT... (n)}, as X, be divided into two parts, X = V + W, the function may be similarly divided, so that f{(V+W)YZT... (n)} =f{VYZT... (n)} +f(WYZT... (n)}, placing each part of X in the same relative position (which in this example is the first) which X itself occupied before the division. (XLVI. 2) (9) If any quantity which does not vary from one equation to the other, and which, therefore, is not liable to be affected with an index, is found under the symbol, it may be considered a constant coefficient of every term of the developed function; and written as such on the,outside of the symbol: of this nature are the unknown quantities themselves, so that for instance, f{XYzZT.... (n)} = xf{XYZT... (n)), and so of like quantities." (XLVII. 2) After these preliminaries the problem of the solution of n linear equations in n unknowns is taken up. The method followed is essentially the same as Scherk's. 200 HISTORY OF THE THEORY OF DETERMINANTS MAINARDI (1832). [Trasformazioni di alcune funzioni algebraiche, e loro uso nella geometria e nella meccanica. Memoria di Gaspare Mlainardio 44 pp. Pavia, 1832.] In his preface Mainardi explains that the algebraical functions referred to in the title are "funzioni rismltanti o determinanti." But although he thus speaks of them as if they were known to mathematicians by name, and mentions the researches of Monge, Lagrange, Cauchy, and Binet in regard to them, he does not take for granted that his reader has a knowledge of any of their properties. The one theorem on determinants,-the multiplication-theorem,-which forms the basis of the whole memoir, is consequently sought to be established without the use of any previously proved theorem. The attempt, as might be expected, is interesting. The first two sections (pp. 9-29) of the three into which the memoir is divided may be passed over without much comment. The first deals with the multiplication-theorem for two determinants of the 2nd order, and with those applications of it to geometry which arise on making the elements of each determinant the Cartesian co-ordinates of two points in a plane. No proof is considered necessary for this simple case, the opening paragraph of the memoir being;"Rappresentate con x,, x,,^, xb; y y,, ya, yb otto quantita qualsivogliano, ed indicati per brevita il binomio x,. x + y,.y, col simbolo (x,,x,), il binomio + con (xYb) e simili, si provera facilmente essere (a) (XmYn - z8Ym) (XaYb - Xba) = (Xma) (XnXb) - (X,,,b)(x.) All the seven other paragraphs are geometrical. The second section in like manner opens with an algebraical theorem, viz. (p. 13){Xm(yp yn)} {Xa(c 'Yb)} + {xn(Zp Zn)} {X a (-Zc-&b)} + {Ym(Zp-Z) } Ya (Zc - zb)} DETERMINANTS IN GENERAL (MAINARDI, 1832) 201 + + (Xm'Xa) (XpXc) (XCmXb)(XiVca) (XnXa)(XmXb) (XnXe) (XpXb) (XnlXb) (XmXce) - (XmX,) (XpXa) + + (XpXa) (XnXc) - - (XmXa)(XpXb) + + (XpXb)(XnXa) - - ~(XmXb)(XpvXc) ~ + (X~pXc) (Xnlb) - (XnXa) (XrnXe) (xpx0) (Xnra) (Xnflb) (XmXa) (XpXae)(XnXb) (XnXe) (XmXb) where {Xm(7/p - yj)} and (WniXa) stand for (Xm'Yp - XpYm) + (XnYm - Xjj2j) + (XpYn - Xnyjp) and Xrn~a + YmYa + Zrn~a respectively; and the remainder is occupied with the applications of the theorem to geometry and dynamics. Each factor of the, left-hand side of the identity is evidently a determinant of the third order, and the three pairs of lines on the right-hand side are each the expansion of a determinant of the same order;. so that in the notation of the present day the identity may be written IXm Xn xp Yrn Yn YP Yrn Yn yp 1 Xa Ya 1 Xb Yb 1 X0 Ye zM 1 Ya zn 1 Yb z~ 1 Ye 1 1 1 Za Zb ze Xm zrn 1 + Xn Zn 1 1 (~X.0) 1 = (XnXe) 1 (XePx0) Xa Xb xc (XmXa) (XnXa) (XpXa) ZaI1 ZbI1 Z~1 1 1 11 1 1. + ~ (x~nXa) (XmXb> (XnXa) (XnlXb) (xp1,Xa) (XpXb) (XmXb) (XmX,," There has been no previous instance of an identity perfectly similar to this; the nearest approach to such being, as the. numbering shows, a result obtained by Binet in 1811. The exact character of the affinity between the two, and the general, 202 HISTORY OF THE THEORY OF DETERMINANTS theorem which both foreshadow, will be most readily brought into evidence by a little additional transformation. Taking first the right-hand side of the identity, we observe that the three determinants have only twelve elements among them, being obtainable in fact from a single array of three rows and four columns. Their sum may consequently be put in the form 1 (cXmxa) (XmXb) (XmX,) 1 (XnXa) (X,,Xb) (X,X,) 1 (XjpXa) (X19Xb) (X~Xe) 0 1 1 1 Secondly, we observe that the first factors on the left-hand side are similarly obtainable from Xn Ym Zm 1 xn n On Z xi) yp zp and the second factors from X3n yn Zn 1 CXb Yb Zb 1 X Yc 1; and as the determinant which is the so-called product of these arrays is equa] to the said left-hand member diminished by Xm Ym Zm Xa Ya Za Xn Y7:n Zf- Xb Yb. Zb Xp Yp ZP Xe YC ZC Mainardi's theorem may be put in the much altered form1 (Xmxa) (XmXb) (XX,) 1 (XnXa) (xncb) (XnXe) X YM m Xn Yn On 1 Xb Yb Zb 1 I X~)(X,,Xb) (X~pXc) 0 1 1 1 X_ b') z 1 X0 Ye Z1 Xm YM ZM Xa Ya Zn Xn Yn Zn Xb Yb Zb XP Yp ZP Xc Ye Ze DETERMINANTS IN GENERAL (MAINARDI, 1832) 203 The constitution of the 3rd section is quite like that of the others, the first paragraph dealing with the multiplicationtheorem for the case of determinants of the 3rd order, the second paragraph with the same theorem for determinants of the 4th order, and the remaining eight paragraphs with geometrical applications. The mode of proof of the multiplication-theorem is partly indicated by saying that any particular case is made dependent on the case immediately preceding it; but its exact character can only be understood by a somewhat minute examination. The investigation for the case of determinants of the 3rd order stands as follows (p. 29):"Si considerino i due polinomj Xm (ynZp - y zn) + X, (Z^ - YmZp) + Xp (yz. - YnZm) (1) - {Xm, Yn. Z?}, Xa(Ybzc - YcZb) + Xba(ZYc - ZcYa) + c (YaZb - Ybza) = {Xa Yb -c,}. Se ne effettui il prodotto, il quale, mediante l'equazione (a) del primo articolo, si potrh disporre sotto la forma seguente xqnX (YnYb) (YpYc) + X.X. (ynyJ0) (YJb) + XPXa (yY)(9YA) + XmXb (YlYc) (,/Y.a) )+,,xb(y.ya) (yp c) + XpXb (Y,.Y) (YnY a) + X,.x(y,,JY) (IYYb) + TzXc (qmYlb (,YYJ(B) + XpXc (Yn.Yc) (YnYb) Esaminando ora la quantita - xnx (uJ m> (Yp/c) - XpXa (yYnc) (YYb) XfXb (YnY{) (YpYc) - XPXb (Y-nYc) (YpYa) - XXzb (Y.nVa) (YnYe) - XnXc (lY4) (YpY.) - Xnlc (ymyla) (YPYb) - xpx.c (Y.Yb) (YYa).) - XPc (y-Yb) (y.Y) +XmXa I X,.b (Y72c) + Xa { mXnc (YpYb) + %Xa {rmxi(ytY) + xpxc(YYlb) + XnXbcpXc - xnx (YpYb) - XpXb (YnYc) - XpXbc} + Xp(b mNYc) + XnXcCXpXb - XnXb (YYc) - XpXc(YmYb) - XmcbXy ) X } + X,,x (y,'Yb) + mXbX.Xc - XmXc (Y?a'b) - XnlXb (YmY) - ',,nxcXn}, 204 HISTORY OF THE THEORY OF DETERMINANTS e le due espressioni die si traggorio da questa, cambiando, prima a in b, b in c, c in a; poscia a in c, c in b, b in a; con facilith si scorge che la somma di questi polinomj e' nulla identicamente, per cui si potra' aggiungere al prodotto (h) senza punto alterarlo. Fatta quest' addizione, l'aggregato altro non sarat che lo stesso polinomio (h), ove si supponiga che i simboli (y~yb), (yy,&), eec. rappresentino rispettivamente, i trinornj seguenti XnXb + Ynb +ZnZb), %X0+,,Y +,.z ece. Se ora si ordineranno le espressioni (1) portando fuori dalle parentesi y o'vvero z in luogo di x, formeremo il prodotto delle medesime cosi' scritte, ed opereremo come sopra, il risultato sara' il polinomio die Si desume da (h) cambiando le x che sono fuori dalle parentesi in p ovvero, in z egualmente accentate. Se faremo per ultimo la somma di queste tre espressioni, tal somma si cavera' dal polinomio (h) scrivendo (xm,,xa) ovvero (y,npy,) invece di x,,,x,,; (ccX~a) in luogo di x-x,,X cc. cc. e sara eguale al triplo prodotto delle expressioni (1). Essendo poi quella somma divisibile per tre, effettuata la divisione per questo numero, avremo {X., Yn) Xjp.{X, Yb, Zc} (nXa.X) (Xfl1b) (zXX,) + (X,,a) (X~Xb) (XmXc) + (X~XA) (mXXb) (XX,,u) -(xmx,) (XpXb) (Xn2c) -(XnX.) (X,nXb) (XXc) -(XpXa) (XflXb) (XmXc)" (xvii. 6) That the essential points of this method of demonstration may be seen, let us apply it as it would be applied if adopted at the present day. The given determinants being Ia1b2c3I and laA/3731 we should say Ialb2c5, = a,1 b2C3 I- a2 Iblc3 I+ a3I bIe2, and Ia1/32y31 = al 32Y31 - a2 1/31y31 + a3 I 31y2j; hence, using the multiplication-theorem as established for determinants of the 2nd order, and (to save on the breadth of the page) denoting aa~~b/3~cy+... by a,...C).... a,3y,. we should have calb2c, C lIJ3273 DETERMINANTS IN GENERAL (MAINARDI, 1832) 205 b2, C2 b2, C2 a 272 33,73 - 3 2 b3, C3 b3 C3 2'72 /3,73 b2, C2 b2, c2 al"2 - 2,2/ 8 + a2C /3Pp)Yl P3373 1, 71 2, 72 60, C2 b2, C+ a3 a -, Y-2 02- - 62oC b C3 )b3, c3 O'YI 02, 72 That each line of this result a2, 2 C2 for b2, C2 a2,321,72 32'72 b3, c3 b3, ce I 02) 72 473 bl, Cl bl, cl b3, c3 b3, C3 321 72I /3373 bl Cl bl, c b1, c1 b1, c1 1 /3p'71,272 b3, C b3, C3 /I' Y1 32'7/2 + Ct1 bl, c1 b2, C2 /2)72 b2, c2 + a3a b1, e1 be, C2 P)2 02 3,' 73 b2, C2 /3'73 /)1, C1 /33 73 b2, c2 03,73 1), cl b2, C2 2'72 * is not altered in substance by writing a2, b2,2 for b2, 2 &c. C3,P3,573 P3' 73 would probably be shown by expressing the line in the form of a determinant of the 3rd order, e.g., the first line in the form bl, c bl, Ce 1 32'72 P3,Y3 b2, C2 b2, C2 q1 a2 2' 72 3, 73 b3, 3 b3, C3 3 /2 72 /P373 and increasing each element of the second column by a2 times the corresponding element of the first, and each element of the third column by a3 times the corresponding element of the first. The whole result would in this way be transformed into aa 1, b, C1,, bl, a2,02,^ 2 C7a3lp3373 Ca/ a, 25 c2 2, )2 C2 a2 L3P2, 72 a3', 33, a3, b3, C3 a3, b3, 3 a2, /32, 72 a3 3, 73 Cap, bl, C1 a61, Cl 1a aI,/l31y I a3,/33,y3 a2, b2, c2 a, b2, C aC,,l,1 a3,/33,y3 a,/3,y, a, 3, 3,,3 3 - a,, I 1,P,7i C13,33,73 206 HISTORY OF THE THEORY OF DETERMINANTS ap,b, c1 a, b, C1 a1a3 a1f,~3,y1 a2,/32,y72 + a a3 a25 b2, c2 a2, b2, c2 2 apo,yl a2,/32, 72 a~.a.,b3, C3 a3 b3, C3 (a,,' a2, a., a,, a2, as> (a,,a2, a., a,, a2, a.' 1', b2' b3' 01' 2' 513 Cl1, C2,) C3, Yi' 72'17 the first columns of this,-and the first columns only,-would be affected, the a's and a's becoming b's and /3's respectively in the one case, and e's and y/'s in the other; and as neither interchange could affect the left-hand side of our identity, we should consequently note that thus three different expressions would be at once obtained for la~b2c3l. a1 /32y3l. Adding these together, and combining the nine determinants of the sum. in sets of three by means of the addition-theorem (XLVI.), we should have finally apb., cl a,, b1, c1 a1, bl elc apop,/,y a21,32,y/2 a3 W3y 31a~~b -la ~2 ' 2 2 2' 2' 2 2' 2 C2 a.cbe a. b,, c. a. b., a 1,/31, y1 a2,/32,y/2 a3l,33,y3 from which it is only necessary to delete the common factor 3. JACOBI (1831-33). [De transformatione integralis duplicis indefiniti f~~~~~p JA + B cos + ~0 sin + (A' + B' cos q~+ C'sin q5) cos if+ (A' + B" cos P+ C" sin k)sii in formam simpliciorem I',a G- G' Cos 71 Cos 0 - G" sinq~ sin 0, Crelle's Journacl, viii. pp. 253-279, 321-357; or Werke, iii. pp. 91-158.] DETERMINANTS IN GENERAL (JACOBI, 1831-33) 207 [De transformatione et determinatione integralium duplicium commentatio tertia. Crelle's Journalt, x. pp. 101-428; or Werlee, iii. pp. 159-189.] [De binis quibuslibet functionibus homogeneis secundi ordinis. per substitutiones lineares in alias binas transformandis, quae solis quadratis variabilium constant; una cum variis theorematis de transformatione et determinatione integralium multiplicium. Crelle's Journal, xii. pp. 1-69; or TWerk-e, iii. pp. 191-268.] The first two of these memoirs may be viewed as continuations of a memoir with a similar title, which appeared in the second volume of Grelle's Journal, and to which we have already referred. They are noted here merely in order that the thread of investigation may be preserved unbroken, for the last memoir practically swallows up, by means of its splendid generalisations, all those that had gone before. So long as we confine ourselves, in problems of transformation, to three independent variables, the explicit employment of the theory of determinants may be dispensed with. When, however, a sufficient number of special cases have been investigated, and an alluring glimpse has thereby been got of a generalisation involving them all, he who attempts the establishment of the generalisation must have recourse to the new weapon. In this, latter position Jacobi now found himself. He wished to pass, from the problem of orthogonal substitution in the case of three variables to the analogous problem in which the number of variables is n, or in his own words (p. 7):"Investigare substitutiones lineares hniusmodi yj a, o, a2'"2 ++a.'x~,. 2 a," Tq a2" 2 +-. all"', XI' -n aIN 1 + a2 (n)X2 +. a.. + X,")x, quibus efficiatur Y1Y1 +y2Y2 +..+ynYfl= XIXI+ X2X2 +.. XnXfl simulque data functio homogenea secundi ordinis variahiliuni Xj.2 -.., x,, transformetur in aliam variabilium Y1, Iy2, 1 de qua binarum producta evanuerunt." 208 HISTORY OF THE THEORY OF DETERMINANTS This being the case he introduces determinants at the outset, fixing upon a notation which is practically Cauchy's, and immediately using properties of them without proof. Much that is contained in the memoir falls to be' considered later, as it concerns special forms of determinants,-those afterwards known as Jacobians, axisymmetric determinants, and, of course, determinants of an orthogonal substitution. Indeed, the halfpage of introduction is almost all that is of interest at present, but even in this a new and important theorem is enunciated. The first sentence of it stands as follows: "Supponamus, designantibus a,(")~ datas quantitates quaslibet, ex it equationibus linearibus propositis huiusinodi a, (m X + a2(m)X2 +...+ a,, MXI per notas regulas resolutionis algebraicle haberi oequationes formTe: Axk =P/'y1.+ /3 y2 +. + /3fY~. Ipsuin A supponimus denominatorem communem valorum incognitaruin, qui per algorithmos notos formatur: sive fit - 1 2i (n)9 signo summatorio amplecteate termninos omines, qui indicibus ant inferioribus ant super-ioribus omnimodis permutatis proveniunt;~ signis corum alternantibus secundum notam regulam, quam ita enunciare licet, ut terminio cuilibet per certain permutationem indicmm7 orto idem signum tribuatur, quo afficitur productum sequens confiaturn e differentiis numerorum 1, 2,..., m (2 -1)(3 -1).... (n -l). (3 -2) (4 -2).. n2). (4 -3) etc., cadem nwamer)orum permutatione facta." It will be at once observed here that Cauchy's italic letters S, a, h are simply changed into Greek 1, a, /3. The next sentence is: "Eadem notatione adhibita, sit -B = I 313.3 it(n) ubi ipsam B e quantitatibus 1 e~n odem modo compositam accipimus, quo A cx ipsis at~m') componitur. Quibus statutis observo ieri: B Aac generalius: II A 'I+ an+la r+2 a. (xx.4 DETERMINANTS IN GENERAL (JACOBI, 1831-33) 209 As for the first theorem thus formulated, the credit of it is, of course, due to Cauchy: the second, however, is new, being indeed the theorem referred to above under Minding as having been foreshadowed by Lagrange, and left for over fifty years undisturbed. Jacobi evidently knew it in all its generality, for he adds "De qua formula generali cum pro variis valoribus ipsius m, tum indicibus et superioribus et inferioribus omuimodis permutatis, permultae aliae similes formulae profluunt." Jacobi's mode of proving the two theorems occupies ~ 6 (pp. 9-11). Temporarily denoting by Xm the left-hand member of the Mth given equation a1 rn)x + a2x + )X2... + + a - 'irn, and by Ym the left-hand member of the M2th derived equation /Y, + 3r "yl3....~ lyn = Ax: and explaining that by he means the coefficient uOf X 1-1X2 —1.. Xjl in a certain specified expansion of U, he recalls his paper of the year 1829 on the "discerptio singularis," and affirms that he had there proved "fore X1X2., Xn, sive etiam, quod idem est, 1 1 1 LY1Y2..YJ1.Y Y. B ac generalius LX1'I+lX2r2+1... X nrnjl 1 X1X2... n, (LII. 4) I F Y"Y" Arl+1%2+ +r,+l Lsl+ly2S2+1.. y,Sn+1j 1 Y1.Y2 i/n designantibus rl, r2,..., r,. ac s1, s2,..., s, numeros quoslibet integros sive positivos sive negativos." M.D. 0 210 HISTORY OF THE THEORY OF DETERMINANTS A glance, however, suffices to convince one that the concluding general theorem here given differs considerably from the theorem which he had previously enunciated and possibly proved. As originally stated the theorem was lr I+~1. 1u Luyoa'+ltUa+'l..u. U+1] 1 X0o0+lXPll+l... XA 11-+1 0 1 qzl~- I which being altered into the the substitutions X,) xi,.. A ao, 1' ' ' - 6 1 * ~.. becomes [r i...X 1 [Xl'lq-lX2r2-l.. X~ t'nq-i] K Pon~pi~.... XI t l to tla l n. tam notation of his present paper by 1= XI) X2.... = xl, 2 _ X^, XX,.... -1 '2 Y Y 1 2 A' A'. - r 1 2.... -- 8 8 A.... 1 Asl+s2+... X s++ -sl+S2+. +_ [YlZ 1YYs22. Y n. Using on both sides of this the fact that if an expanded function be multiplied by the product of certain powers of the variables, any particular coefficient in the original expansion has now for facient its original facient multiplied by the said product, we obtain r xSlX2S2... X S X ri+lX 292+1... X r,+lj 1 X 1X2... Xn xx y y YY...yI ] AS+S2+. +Sl ylrl+ y22+1',.. y 'n+lj Y1Y2... Y DETERMINANTS IN GENERAL (JACOBI, 1831-33) 211 -a statement differing from Jacobi's in having r's and s's on the right-hand side where he has s's and r's respectively. The oversight was probably not noticed by reason of the fact that in the special instances considered by him the values of any r and the corresponding s are the same. In the first of these instances he puts ri = 2 =... = = - 1 81 = S2 -...= s= - 1, and obtains 1 - A=1 r_ 1,1 1 -An-i Y1Y2... Y] 1Yn B ' YIY2 'Yn thus arriving at Cauchy's theorem regarding the adjugate, viz., B = A9-1. (xxI. 3) In the second instance, he puts r = 2= r '' = r= - 1, rlm+al = m+2 =..= r = 0, S1 == 82 =. = 8 = - 1 8m+1 = 8rn+2. = n = 0, and obtains r I I _Xn,+X?+2... XnJ _ Xm+^lX+2 * * * Xn -2 2'. Yin Y lY2 Y112 Yins He then recalls the fact that by the conditions attaching to the,expansion of the expressions enclosed in rectangular brackets the powers of xI, x2,... x n contained in the one and the powers of yi,+1, Ym+2... yn contained in the other are all positive; and argues that as we are concerned only with terms that do not involve these variables, it is quite allowable to put them all 'equal to 0. This being done it is seen that 1r i 1 1 X~r,+ln~li+... Xz/j ___ _ (+rn-+i) (m+2) (n) XLMxt+ x,,,+2... Xn -' am+l ar +2.. a, Xin+llnm+2.. Xn and r 1 1 Ly1Y2... Yin I _2 m LY2,...Y,, Y1Y2 J r"m 212 HISTORY OF THE THEORY OF DETERMINANTS so that there is obtained -t (n A'2-1. r~ c(m+1)(m~2) as was expected. The oniy other point to be noted at present is contained in. the casnal remark that the O's may be expressed as differential coefficients of A. When dealing later (p. 20), with a special form of determinant, he says"Data occasione observo generaliter, si a,,x et ax, inter se diversi sunt, propositis n aequationibus linearibus hujusmodi:,jU1 + (aX,2ut2 + +.a... + aj VI a2jU, ~ a2,2?! ~... + atfla,, statuto a2,1flu + a,,222 + I... + an,nUn = Vn F = I~a a sequi vice versa 'au alp au ru, = V + V2 ~...+ V20 aal,,1 C3a2,1 CL,1 au au ar, Da1,2 V Dai,2 V2 + +,, = Dai,,,V1 + Da22 + ~ar aaa aa-T. aan,n (xiii. 5) (vi. 8 JACOBI (1834). [Dato systemate n aeqnationnm linearinm inter n incognitas, valores incognitarum per integralia definita (n -1)tnplicia exhibentnr. Crelle's Journ., xiv. pp. 51-55; or Werlee, vi. pp. 79-85.] This short paper is, as it were, a by-prodnct of the investigation which resulted in Jacobi's long memoir of the preceding year. Its only interest for us at present lies in the fact that values, which are ordinarily expressed by means of determinants are here given in the form of definite multiple integrals. Indeed,, instead of viewing the result obtained as being the solution of a set of simultaneous linear equations, it might be equally appro DETERMINANTS IN GENERAL (JACOBI, 1834) 213 priate to consider the investigation as belonging to the subject of definite integration. It will suffice, therefore, merely to give a statement of the theorem arrived at. In Jacobi's own words, it is, '"Sit propositum inter n incognitas z1, z2,..., Zn systema n aequationum linearium bllzl + b12z2 +.... + blzn = ml, b2ll + b222 +.... + b2n- =n 2, b +.... ~ *.. blAzl + bn2Z2 +.... + bnn,, - mnZ; istatuamus X = [blx, + b2l12 +. + bqXn]2 + [bl2x1 + b22X2... + b,,,, ]2 + [bl, + b2, +... + bnnX,]2, porro M m= nx + m2x2 +... +, ubi I,, = ^(1 - x -2 -.. - -_1) radicali positive accepto; porro ponamus V= ~ i b1lb22... bn, signo ancipiti, ante ipsum. posito, ita determinato, ut valor ipsius V positivus prodeat. Quibus omnibus positis, erit n z1 n-l M (blxv. + b21X2 +... + b,,,x1) 8&18x2* 8x.-. 2l -1 = V J + XX2 (n+2) n z2 I"- M(bl2xl + b22x +... + bn2x,)SxSx 2... 8x,_2-S' V J xJX (n~+2) 2n-lS 'o - - nX~X(n+2) fn Zn ('n-l M(blnxl + b2nx2 +... + bnx,,)x8x82... 6x_2n-S 'V J nX2X(i+2) integralibus (n- l)tuplicibus extensis ad omnes valores reales ipsorum x,, X2,..., x,,_ et positivos et negativos, pro quibus etiam xn realis sit sive pro quibus X,2 + 2 +.. + _ = 1. *et designante S aut 2.4... (n-2)(2 au 1.3.5... (-2)(2) prout n aut par aut impar." (LIII.) (xiII. 6) 214 HISTORY OF THE THEORY OF DETERMINANTS JACOBI (1835). [De eliminatione variabilis e duabus aequationibus algebraicis. Crelle's Jozrnacl, xv. pp. 101-124; or Nouv. Annales de LMath., vii. pp. 158-171, 287-294; or We7rce, iii. pp. 295-320.] In a memoir having for its subject Bezout's method of eliminating x from the equations aCnXn + -_1X "-1 +...+ 1X+60 = 0,n bx" + bn,_, -1 + + n-xbo = 0,. determinants are certain to occur explicitly or implicitly; and, the author being Jacobi, one is not surprised to find them introduced near the outset and employed thenceforward. It is of course only a special form of them which appears, viz., that afterwards distinguished by the term per'symmetric; consequently, for the present the main contents of the memoir do. not concern us. Note has to be made, however, of two points. -(1) that while Jacobi does not discard his former notation Ea ~ a, 0,.. a,,,,s,, he introduces and uses another, viz., J^~a; 2 ' ' i s (vII. 8),.So, S1,..., Sp 2 (2) that a page is devoted to a fuller statement of the abovementioned theorems regarding the adjugate determinant and a. minor of the adjugate. The final sentence of this statement is. all that need be reproduced. It is "Sint igitur r, r, '",..., r("-1 atque s, s",. s., s(- numeri omnes 0, 1, 2,..., n- 1, quocunque ordine scripti; erit () 8(m —+1)., Ir(n- 1).1 A,, *, }= LZ-(l —m). a (XX. 5) ), s (m). (+.. S(n-1) iS, sI., S (x 5 where L stands for ~ao,0ao1,.... a,-l,,-i and the adjugate of L. is:+A0,0Ai,..... Al,,_. No proofs of the theorems are given.. CHAPTER VIII. DETERMINANTS IN GENERAL, FROM THE YEAR 1836 TO 1840. THE writers of this period are nine in number, viz. Grunert, Lebesgue, Reiss, Catalan, Molins, Sylvester, Richelot, Cauchy, Craufurd. Of these the most prominent is Sylvester, who apparently in ignorance of all previous work discovers the functions for himself, gives a fresh investigation of some of their properties, and in a second paper makes an afterwards widely-known application of them to the theory of elimination; Richelot, Cauchy, Craufurd contribute papers dealing with the said' application; Lebesgue explains the results of another application previously made by Jacobi and Cauchy; and Grunert Reiss, Catalan, Molins give elementary expositions of the general theory. GRUNERT (1836). [Supplemente zu Georg Simon Kligel's WTrterbuch der reinen Mathematik. Art. Elimination (I. Gleichungen des ersten Grades), ii. pp. 52-60.] With Grunert it is necessary to take a long step backward. Although the memoirs of Bezout, Vandermonde, and Laplace were known to him, in addition to those of Hindenburg, Rothe, and Scherk, he advances only a short distance into the subject; his aim, indeed, is little more than the establishment of Cramer's rule for the solution of a set of simultaneous linear equations. His mode of presenting the subject, however, is fresh and interesting, the method of "undetermined multipliers" being taken to start with. 216 HISTORY OF THE THEORY OF DETERMINANTS Writing his equations in the form (1)jxj + (2)1X2 + PA1x + + ('n)x,X = [] (1)2X1 ~ (2 )2c2 + (3)2X3 ~ + (n)2mXn = [112 (1)3x1 + (2)3X2 + (3,x, + + ()c 1 (1),x, + (2nXx2+ (3)nX3 A-+ + (a~ [In. and takiing P1, P2' P3,.. p, as multipliers, he readily shows of course that if the multipliers can be got to satisfy the conditions (2)lpi + (2)2P + (2)3p,3 ~.. + (2)nP = 01 (3XP1 + (3)2P2 ~ (3)3pj +....+ (3)n~Pn = 0 (4)1p, ~ (4)2P2 + (4)3P3 +-. (4)nPn = (na)1P1 + (n)2P2 + (0)3PS +...+ (rt)nPn OJ the value of x will be [1]iPI + [112P2 + [l]3P3 ~ +. ~ n n (1)iPi + (1)2P2 + (l)3P3 +....+ (l),nPn' in other words, that xi can be determined at once if a function (l)1P1 + (1)2P2 + (l)3P3 ~... ()~ can be formed of such a character that it will vanish when instead of the coefficients (1)1, (1)2, (1)3,... I (1)n% we substitute the members of any one of the n -1I rows (2), (2)2 (2)3.... (2\n the said function itself being the denominator of the value.of X1 and the numerator being derivable f rom the denominator by inserting [11k1 [112, [1]3,..., [1n, in place of (1)1, (1)2, (1)3,..., (l),n. Further, as any one of the unknowns may be made the first, the complete solution is thus put in prospect. "Alles kommt demnach auf die Eutwickelung einer Function von der angegebenen Besehaffenheit an." (xiii. 7) DETERMINANTS IN GENERAL (GRUNERT, 1836) 217 Two rules, Grunert says, have been given for the construction of such a function, one by Cramer, the other by Bezout. The former he states, and illustrates by constructing the desired function for the case where n =4. The proof of it is then attempted, and is said at the outset to consist essentially in establishing the proposition that a permutation and any other derivable from it by the simple interchange of two indices must, according to Cramer's rule, differ in sign. This proposition is therefore attacked. The permutation........... (l)a..... A is taken in which the inferior indices are in their natural order 1, 2, 3,..., n, and k and 1 being interchanged, there arises the permutation.... (1)..... (..... B The part preceding (c), in A is called I., which thus of course also denotes the part preceding (1), in B; the part between (l)a and (1)a+~ in A or between (1)a and (lc)a+ in B is called II.; and the remaining part common to both A and B is called III. The number of inversions in both, when 1 and k are left out of account, is denoted by y; the number in both due to k and the division III. is denoted by X; the number in A due to k and the division II. by X'; and the number in both due to the division I. and k by X". The counting of the inversions then takes place for the two permutations. In the case of A there are the inversions due (1) to I. and I, which are X" in number. (2) to I. and II. (3) toI. and 1,...... a-..... (4) to I. and III. (5) to k and II.,...... '..... (6) to k and 1,..... 1..... (7) to 7 and III.,........... (8) to II. and 1,...... 3-1. (9) to II. and III. (10) to 1 and III.,.... 0..... 218 HISTORY OF THE THEORY OF DETERMINANTS and as those not counted here are y in number, the total is seen to be a + f + + X + ' + X"- 1. Similarly in the case of B the total is found to be a + f + y + X - X' + X" -2. But the former total exceeds the latter by 2X'+1, and this being an odd number, the proposition is proved. (iii. 27) Before proceeding further it is important to note that Grunert here establishes a more definite theorem than he proposed to himself, viz., the theorem of Rothe (III. 7). If he attains greater simplicity it is in part due to the fact that instead of taking any two indices for interchange, lc and r say, he takes k and 1. To prove now that the function constructed in accordance with Cramer's rule will satisfy the requisite conditions, it suffices to show by means of this theorem that on making any one of the n -1 specified sets of substitutions the function will be transformed into one consisting of pairs of terms which annul each other; in other words, to prove Vandermonde's theorem regarding the effect of making two indices alike. This is done;; and then it is shown how xK can be got by interchanging xK and x1 in all the given equations, the first step being of course to establish the fact that the denominator of x, and the denominator of xi only differ in sign. Bezout's rule of 1764 is next taken up, and shown to be identical in effect with Cramer's. The proof, by reason of the recurring character of the former, is inductive; that is to say,. it is demonstrated that, if the two rules agree in the case of n unknowns, they must also agree in the case of n+ 1. Paraphrasing the proof, but taking for shortness' sake the case where n = 4, we say that it is agreed that both rules give in this case. the signed permutations 1234, -1243, +1423, — 4123, -1324, +. Now for the case where n =5 Bezout's rule directs that to the end of each of these permutations, e.g., the permutation -4123,, a 5 is to be put, and asserts that the result -41235 will be one of the desired permutations with its proper sign. That it is a. permutation of the first five integers is manifest, and since the; number of inversions in 41235 is necessarily the same as the; DETERMINANTS IN GENERAL (GRUNERT, 1836) 219 number in 4123, its sign is correct according to Cramer's rule. In order to obtain four other permutations, Bezout's rule then proceeds to bid us shift the 5 one place and alter the sign, shift the 5 another place and alter the sign again, and so on. The result is +41253, - 41523, +45123, -54123. In regard to this, it is clear as before that permutations of the first five integers have been got, and that the altering of the sign simultaneously with the shifting of the 5 is in accordance with Cramer's rule, because every time that the 5 is moved one place to the left the number of inversions is increased by unity. The only question remaining is as to whether all the permutations are thus obtainable; and as it is seen that each of the 24 permutations of the first four integers gives rise to 5 permutations of the first five, we have at once grounds for a satisfactory answer. (III. 28) LEBESGUE (1837). [Theses de Mecanique et d'Astronomie. Premiere Partie: For — mules pour la transformation des fonctions homogenes du second degre a plusieurs inconnues. Journal (de Liouville); de Math., ii. pp. 337*-355.] This simply-worded and clear exposition is a natural outcome of a study of Jacobi's memoirs on the subject. Like these it mainly concerns determinants of the special form afterwards. individualised by the term axisymmetric; and, indeed, it is notable as being the first memoir in which a special name is given to a special form, the expression "determinants symetriques" being repeatedly used for the particular determinants referred to. His general definition is (p. 343):"Si l'on considere le systeme d'equations A1,11 + A1, +.....+ A1,+t, = mi, A2,1tl + A2,2t2 +...... A2,1,t, = i A, t, + A,2t2 +. +. A,t = m,, * N.B. —There are two pages numbered 337. 220 HISTORY OF THE THEORY OF DETERMINANTS le de'nomiuateur commun des iliconnues 11, t2,. 1,, est ee qiue lPon nommre le determinant du syst~me des nombres fAl,, A1,2... A, (17) A2,1 A2,2.....A Coname ce de'nominateur pent changer de signe, selon le mode de solution qu'on emploiera, on conviendra de le prendre de sorte que le No nse, however, is made of this for the purpose of establishing the properties of the functions, results being for the most part taken from previous investigators and merely restated. A notation for what are nowadays called the minors of a determinant is given in the following words (p. 344): (XL. 7) "1Ccci rappele', si l'on represente par D le determinant dn syste'me ~(17), par [g, i] le determninant du syste'me qui se tire du syste'me (17) par la suppression de la se'rie horizontale de rang g et de la se'rie verticale de rang i, et semblablement par la notation [~ ]le de'terminant dn syste'me cjni re'sulte de l'omission des series horizontales de, rangs g et i et des series verticales de rangs i et tk dans le syste'me (17), on pourra,..... Further, the determinants thus denoted are spoken of on page 346 as " determinants partiels." (XL. 8) IREISS (1838). [Essai analytique et ge'ome'triqne. Gorrespondctnce math. et phys., x. pp. 229-290.] 'Reiss's memoir, the first part of which appeared in 1829, was never completed. In the course of some remarks introductory to the present essay, he says by way of excuse: "Je m'aper~ns biento't, et pinsicurs savans me l'ont fait remarqner, que ces recherches, fussent-elles tre's-fe'condes en re'sultats e'legans, 'taient trop abstraites pour iuiteresserlepbi qui nap~i les theories que selon le degre' de leur utilit6. J'ai done ta'che' de montrer, par un exemple, de quelle manie're on pent se servir de ces fonctions dans la geometrie analytique: et j'ai choisi le te'ra~dre qui, DETERMINANTS IN GENERAL (REISS, 1838) 221 par le concours de plusieurs circonstances qu'on aura occasion de reconnaitre plus tard, permettait une application tres-facile et presque immediate des premieres consequences auxquelles j'etais parvenu." The analytical portion of the essay is to a considerable extent identical with the original memoir. In so far as there is a difference, the change is towards greater simplicity, less seemingly aimless plunging into widely extensive theorems, and in general a better and more attractive style of exposition. Less space too is given to it,-not even half what is occupied by the portion on the tetrahedron, the main aim now being to urge on mathematicians the capabilities of the analysis in its application to geometry. The matters falling to be noted as not having been given in the original memoir are few in number and of little importance. In restating the theorem (abc... r, a/3y... p) = (abc... r, a/3y... p) the remark is incidentally made that the order of the terms on the one side is never the same as that on the other except when the number of bases is 1, 2, or 3; for example, the number of bases being 4, we have (abed, 1234) = acb2c3d4 - a1b2c4d - alb3c2d + abc4d2 +.... whereas (abed, 1234) = alb2c3d, - acb2d3C4 - ac2b3d4 + ac2d3b4 +.... the difference first appearing at the fourth term. (IX. 6) Bezout's recurrent law of formation, formerly merely enunciated, is now accompanied by a demonstration. This is not without its weak point, the cause of which, as might be expected, is the awkwardness of Reiss's rule of signs. The first paragraph, which will suffice to show its character, is as follows (p. 233): — "Portons notre attention d'abord, seulement sur la fonction (abc... r, a/3y... p). Si l'on se represente la maniere dont on fait les permutations des n elemens a,3,'y,... p, on verra qu'a partir de la premiere, il y aura 1.2.3... (n- 1) complexions qui commencent. par a, et que, si l'on separe cet element par un trait vertical des autres, 222 HISTORY OF THE THEORY OF DETERMINANTS -on aura ' droite toutes les permutations des 'lemens 3,y,... p. Les 1.2.3.... (n - 1) premiers termes de (abc... r, ap/... p) commencent done tous par ac, et puisque les signes de ces termes sont:determines d'apres la maniereexposie plus haut, on trouvera leur somme = aa(be... r, /3y...). Vanderinonde's theorem regarding the effect, on the function,.of interchanging two bases is stated generally, and a demonstration is given. The mode of demonstration, which occupies one page and a half, will be readily understood by seeing it applied in later notation to the case where there are four bases, that is to say, where the theorem to be proved is I a,bpc,das k=a-bafLpC4Cds By repeated use of the recurrent law of formation we have aQbpc,d = ct, I bpc,cdl -a I b-c',da + ca, b cpds s - I bacpd, > - b c,da{ - b, cds + bajcpd,y - ct{b.- c,cds - b, cads + ba ccd,t} + cLa{bjc Cpd - bp ccdsa + b6 Cdp } - casb. CBgdyA - bp jiCady + by/ eCadp By collecting the terms which have ba for a common factor, hb for a common factor, and so on, this result becomes a,,bpeCrd = - b,(ap {Clc~daI - acpda + as Icpd, } + bpaC, I CYdIA - a, I CCdadI + aJCs ad, }I - b{Ca. cpda3/ - apcaC4,ds + aslcdp4 + b{C a, cpd,y - ct1 clct, + aT,1cadp }, = bac,,da + bp I aac,d8 - b, Caccpd8j + bs I aaCcpdy I, - bactpcyd8 K as was to be proved. (xi. 5) The suggestion readily arises that this process would be equally -applicable in proving Yandermonde's theorem regarding the vanishing of a function in which two bases are identical, and the process, it may be remembered, was actually so employed -by Desnanot. DETERMINANTS IN GENERAL (REISS, 1838) 223 One of the theorems given by Scherk, and later by Drinkwater, appears in the following form (p. 240), the peculiar notation adopted for a determinant with a row of unit elements being constantly employed throughout the remainder of the essay: "Si une des bases, par exemple a, est telle que la quantite6 qu'elle -repr~sernte soit la me'me quel que soit l'exposant dont elle est affecte'e,.C'est-h-dire, si a =0 a,' =...,oni aura (abc.... r, a/3y. p) = ~[b..r, /3y....p) - (bc....r, cy... p) + (be... r, 413... p) ~...]. La quantite' qui se trouve sous la parenthbse, peut doncehbre repre'sei-t~ee de la maniebre suivante: (Ibc e r, a1y 6. p); (XLVii. 3),enl admettant une fois pour toutes que le chiffre romain I soit tel que 1 = P = IP = IP =... Ii va sans dire que toutes les proprie'tes qui out lieu pour (abc... r, ap/Th... p) se rapportent 6galement 'a The character of the identities used in the treatment of the tetrahedron will be learned. from a glance at the following examples: -na(Ibc, 123) - b1(Iac, 123) + cl(Iab, 123) = (abc, 123).,(a, -a )lb, 2) - (b - b2)(Iac,123) ~ (el- c2)(Iab, 123) = 0. (ab, 12)(ac, 34) - (ab, 84)(ac, 12) =- al(abc, 234) + a2(abc, 134), = + a3(abc, 124) - a4(abc, 123). (lab, 123)(Jac, 124) - (lab, 124)(Jac, 123) =- (cL - a2)(Jabc, 12,34). (lab, 123) (abc, 124) - (lab, 124)(abc, 123) =+ (ab, 12)(Jabc, 1234). The first of these we have already seen u sed by Minding; the second is nothing more than the manifest identity, a. - a2 al a2 a3 a. ct1 a2 a3 or =0; b 2 b1 b2 b3 ~ b1 b1 b2 b3 cl- c2 Cl C2 C5 C1 C1 C2 C5 224 HISTORY OF THE THEORY OF DETERMINANTS the third is evidently the equatement of two expansions of al a2 a (3. a4 a, a2 (3 a.a4 a.3 a1 a2 a4 or bi b2 b b4 bN bi b2 b4 C1 C2 e3 c4 e3 c1 c2 C4 the fourth is a case of the fifth: and the fifth is itself a case of a theorem (C') of Desnanot's. CATALAN (1839). [Sur la transformation des variables dans les integrales multiples. Mgemoires couronne's par I'Acade'iie 'royale... de Brmxelles, xiv. 2m3 partie, 49 pp.] The first of the four parts into which Catalan's memoir is divided bears the title " Valeurs gene'rales des inconnuies dan& les 6qinatioms dua premier degre' et proprift s des deuominateurs commutns," and in the introduction it is said to contain several remarkable new properties of the functions called resultantsby Laplace "et connues aujourd'hui sons le nom de de'terminants." His method of dealing with the opening problem is to derive the solution of n, equations with n unknowns from the solution of n-1 equations with n-1 unknowns; or more definitely, to show that if the multipliers X 1 XX necessary for the solution of the set of equations, a1X1 ~ blX2 + c1x3 =I a2e1 + b2X2 + C2X3 a =C a0x1 ~ b3x2 + cmx =at3) be the determinants of the systems (4 b2 a3 b3 al b. a. b3, al b1, a2 b2, DETERMINANTS -IN GENERAL (CATALAN, 1839) 225 then the multipliers X., X2 X3, X4 necessary for the solution of the set a~x, ~ b1X2 + cAx + dAx = a, a2xl + b2X2 + CAx + dCx4 = a2 aAx + b3X2 ~ CAx + d3X4 = a3 ct4x1 + b4x2 + c4X35+ J4X4 = a4) are the determinants of the systems a2 b2 c2 a3 b3 c3 a,, b4 c4 a, b1 c1 a. b c3 a4 b4 c4 a. bi C1 ct2 b2 c2 a4 b4 c4 a, bi cl a2 b2 c2 a3 b5 c3 (X'iii. 8) 'This of course means that in the first case aX, + a'2X2 + ad3X = 0, b1Xj + b2X2 + b3X53 = 0, and X 3 X1a'1 + X2a2 + X3a5 3 Xicl + \2C2 + X3C3' and in the other a1X1 + at2X2 + aX, + a4\4 = 0 b1X1 ~ b2X2 + b3X53~ b4X4 = 0, c1X11 + C2X2 + C3X3 ~ C4X4 = 0, and X -X~a1 + X2Ca2 + Xka + X4at4 and X~x~d + A d2 Xd3 + 4d4 1 1 +X The proof is disappointingly weak and unsatisfactory, and, what -is still more surprising, rests at one point on a manifest inaccuracy. He says (p. 9)"ar un calcul. direct, on verifle la formule (6) et les relations (5) pour le cas de trois 6quations. En ni~me temps, l'on reconnait que "1 Le dd'nominateur de la valeur de x, par exemple, renferme toutes les combinaisons trois 'a trois des coefficients, chaque combi-naison ne contenant ni deux fois la me~me lettre, n i deux fois le m6me ~indice. 11 2o Deux termes qui, dans l'expression de ce de'nominateur, peuvent,se ddduire l'un de lFautre par une permutation tournante out m~me,signe. "13' Deux termes qui ne diff~rent que par le changement d'une lettre en une autre, et r~eciproquement, sont de signes contraires. * Note, however, the error in Sign of 'N2 and X4. M.D P 226 HISTORY OF THE THEORY OF DETERMINANTS "4' Par suite, le denominateur est le meme pour toutes les inconnues, pourvu que l'on preune convenablement le sigue du numerateur." He then proceeds"Supposons done que pareille verification ait 6t6 faite pour n - I equations entre n - 1 inconnues, je dis qu'elle se f era encore dans le cas de n equations." Now although the statement in 20 is true for the case of threeequations, it is not true generally, and therefore cannot be proved.* The theorems which follow this introductory matter concern a special determinant, viz., the determinant of the system, a1 b el..... I e2 b2 c2. k2 12 C,,, bi en.. k9 111, in which the elements are connected by the In(n - 1) relations alb, + o2b2 + a,b,, +... +a 3tnbn = 0 alc1+ ~ 2C2e + ae~. + ancn = 0 ll, + a212 + a315 +... + CLn = 0 b&ic, b2c2+ b3e3... bnen =O bid1+ b2d2 + b~d,~..+ bndn = 0 bll~ + b212 + b313..~+ bnlnO kill + ~k212 + k31~...+ ~knnl0= Such determinants are only a little less special than determinant, of an orthogonal substitution, and thus naturally fall to be considered later along with those of tie latter class. * In the proof he is fortunate (or unfortunate) enough to use another special case in which the statement is true. He says:-" Les deux termes ac7b6C1d3e,f2 et e7f~ajb,3cd2 qui entrent dans D4, et qui se deduisent P'un de l'autre par une. permutation tournante eutre les lettres ont m~me signe." DETERMINANTS IN GENERAL (SYLVESTER, 1839) 227 SYLVESTER (1839). [On Derivation of Coexistence: Part I.* Being the Theory of simultaneous simple homogeneous Equations. Philosophical Magazine, xvi. pp. 37-43; or Collected Math. Papers, i. pp. 47-53.] Sylvester was apparently first brought into contact with determinants while investigating the subject of the elimination of x between two equations of the mth and nth degrees. At the close of a paper on this subject (Phil. Mag., xv. p. 435) he says-" I trust to be able to present the readers of this magazine with a direct and symmetrical method of eliminating any number of unknown quantities between any number of equations of any degree, by a newly invented process of symbolical multiplication, and the use of compound symbols of notation." These last words, indicative of the method, exactly describe the matter dealt with in the paper we have now come to, and as will soon be seen, the functions which are the outcome of the said "compound symbol" of operations are determinants. It would also appear that Sylvester was unacquainted with any of the important memoirs of his predecessors regarding the functions: the twenty-seventh chapter of Garnier's Analyse Algebrique, to which he refers, may very probably indicate the extent of his knowledge. Premising that he is going to use such symbols as c a, a2,. he calls the letter a the "base," and the complete symbol "an argument of the base," a, being the first argument, a2 the second, and so on. Taking then a number of expressions, "each of which is made up of one or more terms, consisting solely of linear arguments of different bases, i.e., characters bearing indices below but none above," e.g., the expressions, a-1- b1, a1- c1; he alters them by writing the index-numbers above, e.g., a1- bl, a1-c1; takes the product of these resulting expressions in its expanded form a2 - alba - a1c1 + bcl; "Misprint for II., as an expression in the paper itself shows. 228 HISTORY OF THE THEORY OF DETERMINANTS and then reverses the operation on the index-numbers, thus finally obtaining a- 2 b - acC + b1c. The full series of these operations he indicates by the letter $, and denotes by the name of " zeta-ic multiplication." Thus, as results in zeta-ic multiplication, we have (a1 - bl)(C1 - C1) = a2- ab - ac, + b1c, and (a1 + b )2 = 2+2a+lb + b2.* Further ir is used to denote that, after the operations ~ have been performed, the indices are all to be increased by r, the result of so doing being called the zeta-ic product in its rth phase. He nexts recalls a notation previously introduced by him for the functions which came later to be known shortly as differenceproducts; denoting, for example, (b-a)(c-a)(c-b) by PD(abc), (b-a)(c-a)(c-b)(d-a)(d-b)(d-c) by PD(abcd), and abc(b-a)(c-a)(c-b) by PD(Oabc). Lastly, he combines the two notations; and any reader who remembers Cauchy's mode of solving a set of simultaneous linear equations can with certainty predict the result of the union to be determinants. A new notation and a new name for the functions thus come into being together, the determinant of the system a1 a2 a3 b, b, b3 C1 C2 C3 being represented by ~abcPD(abc) or ~PD(Oabc), (VII. 9) and being called a zeta-ic product of differences. (xv. 7) These special zeta-ic products being reached, the rest of the paper is taken up with an account of some of their properties, and the application of them to the discussion of simultaneous *He would not hesitate even to extend the use of the symbol, denoting, for example, 1 1 + 2a4 -. by cos(ax). 1.2 1.2.3.4 DETERMINANTS IN GENERAL (SYLVESTER, 1839) 229 linear equations. Some of the matter may be passed over as being already familiar to us, although its earlier appearances were certainly made in a less picturesque dress. The first really fresh theorem concerns the zeta-ic multiplication of a determinant PPD(Oabc... 1), by those symmetric functions of a, b, c,...,, which we should now denote by Ea, Sab, Yabc,.... but which Sylvester writes in the form S(abc... ), S(abc... ), S(abc....1), In his own words it stands as follows (p. 39):"Let a, b, c,... I denote any number of independent bases, say (n - 1); but let the argument of each base be periodic, and the number of terms in each period the same for every base, namely (n), so that a,. = a,., =a, = a a = a_br = b+, = b_ b = bo= b-, Cr +n= C = C,._ n Co = C C-n -r = 7rn= Ir-n _ r -= 1 0 == I -n r being any number whatever. Then C_1PD(Oabc... 1) = (S(abc... 1). 1PD(Oabc.... 1)) -2PD(Oabc... I) = (S2(abc... I). CPD(Oabc... I. )) r_,PD(Oabc... ) = C(S.(b.. ). ).PD(Oab... l))." The limitation made upon the arguments of the base would seem to imply that the theorem only concerned determinants of a very special kind. Such, however, is not the case. A special example in more modern notation will bring out its true character. Let the determinant chosen be ab2c3d4,, and let the symmetric function be ab + ac + ad + be + bd + cd. 230 HISTORY OF THE THEORY OF DETERMINANTS Multiplying the two together "zeta-ically," that is to say, in accordance with the law ar X aCs =: a+s) we find that 120 of the total 144 terms of the product cancel each other, and that the remaining 24 terms constitute the determinant lb2c4d5 1, the identity thus reached being (1 1ab2c3d4 1 lab)= I ab2cd I Now Sylvester's SPD notation being unequal to the representation of the determinant a1b2c4d5 in which the index-numbers do not proceed by the common difference 1, he would seem to have been compelled to give a periodic character to the arguments of the bases in order to remove the difficulty. At any rate the difficulty is removed; for the number of terms in the period being 5 the index-numbers 4 and 5 become changeable into -1 and 0, and thus we can have ClbA24d5 = Iab2c-1d0, =I a-,bcAd21, -a determinant in which the index-numbers proceed by the common difference 1, and which is obtainable from a 1b2cd4 by diminishing each index-number by 2. Sylvester's form of the result thus is {S2(abcd)..PD(Oabcd)} = _2PD(Oabcd).* *It is rather curious that Sylvester overlooks the fact that the legitimate equatement of two zeta-ic products implies an identity altogether independent of the existence of zeta-ic multiplication. Thus, the identity just discussed is essentially the same as the identity a a2 a3 a4 a a2 a4 a5 b b2 b3 b4 b b2 b4 b5 x (ab + ac + ad + be + bd + cd) = C C2 C3 C4 C C2 c4 C5 d d2 d3d4 d dd2 d4 d5 where the index-number denotes a power and the multiplication is performed in accordance with the ordinary algebraic laws. From this point of view the above quoted proposition of Sylvester's involves an important theorem regarding the special determinants afterwards known by the name of alternants. DETERMINANTS IN GENERAL (SYLVESTER, 1839) 231 Following this comes the application to simultaneous linear equations, or as they are called "equations of coexistence." The system is represented by the typical equation arx + bry c+ c.. + r.t = 0, in which r can take up all integer values from -oo to + o, there being really, however, only n equations, because of the periodicity imposed on the arguments of the bases. One socalled "leading theorem" is given in regard to the system, its subject being the derivation of an equation linear in x, y, z,..., t by a combination of the equations of the system. The theorem is enunciated as follows (p. 40):"Take f, g,...., k as the arbitrary bases of new and absolutely independent but periodic arguments, having the same index of periodicity (n) as a, b, c,.., 1, and being in number (n-l), i.e., one fewer than there are units in that index. "The number of differing arbitrary constants thus manufactured is n(n- 1). "Let Ax + By + C +... + L =0 be the general prime derivative:from the given equations, then we may make A= CPD(0af... k) B= CPD(Obfg... c) C = CPD(Ocfg... k) L = (PD(0fg... k)." (xIII. 9) As in the case of the other theorems, no demonstration is vouchsafed. In order, however, that the connection between it and previous work may be more readily manifest, it is desirable to indicate how it would most probably be established now. Taking the case where the number of unknowns is three and -the number of given equations four, viz.a x + b y + c 0z = 0 a2x + b2y + c2z = 0 ax b + b + z = 0 ax + b4y + c4z = 0, 232 HISTORY OF THE THEORY OF DETERMINANTS we should form an array of 4(4- 1), i.e. 12, arbitrary quantities, fi g1 h1 f3 93 h3 f4 g4 h4, from which we should select the multiplier If2g3h4 for the first. given equation, the multiplier Iflg3h4l for the second equation, and so on. The multiplication then being performed we should by addition obtain I alf2g3h41 + blf2g3A Y + I fCf2g3h4 z = 0, which is what Sylvester would call "the general prime derivative of the four given equations," the process being an instance of what he would similarly term the "derivation of coexistence." By proper choice of the arbitrary quantities it may be readily shown, as Sylvester proceeds to do, that the theorem gives (1) the result of the elimination of n unknowns from n equations; (2) the two equations of condition in the case of n +1 equations, connecting n unknowns; (3) the ratio of any two unknowns in the case of n - equations connecting n unknowns; and (4) the relation between any three unknowns in the case of n -2 equations connecting n unknowns. For example, the equations being axC + bly + cz = 0 a2x + b2y + c2z = 0 a3x + b3y + C3z =, the theorem gives the general derivative ai A g1 b1 J gi c1 fi g9 a2 f2 2g x+ b2 f2 g2 Y+ 02 A 2 2 = a3 f 93 b3f A 3 3 C 3 f 9.3 which is true whatever fl, f2, f3, g, g2, g3 may be. By putting fl f2, 9f, g2, g3 = b2, b3, C1, 2, c, this takes the form I alb2c3l x + blb2e3y + cbc3 Iz = 0, whence the equation of condition, or resultant of elimination, I ab2c3 I = 0. DETERMINANTS IN GENERAL (SYLVESTER, 1839) 233 As a corollary to one of the deductions from the leading theorem,-the deduction numbered (3) above,-the following proposition of a different character is given (p. 42):"If there be any number of bases (abc... 1), and any other, two. fewer in number, (fg... k), CPD(afg.., i) x PD(bc... )) + PPD(bfg... k) x PD(ac... 1) + PD(afg... k) x fPD(bc... I) + CPD(lfg... k) x CPD(abc... ) = 0, a formula that from its very nature suggests and proves a wide, extension of itself." (xxIII. 11) It belongs evidently to the class of vanishing aggregates of products of pairs of determinants, of which so many instances have presented themselves. There is a manifest misprint in the third product, which should surely be PPD(cfg... k) x iPD(ab... ); and there is an error in the signs connecting the products, which, instead of being all +, should be + and - alternately. When the determinants involved are of the third order, the theorem in the later notation is alf2gib31 bc2d I - Ibf2g3 l. lc2dl + Icjf2g3l ltb2d31 - Idjf2g3.jlab2c31 = ~y which is readily recognised as an identity given by Bezout. With this theorem the paper proper ends, but in a postscript an additional theorem of a curious character is given. As enunciated by the author-even his double mark of exclamation being reprinted-it is (p. 43):"Let there be (n - 1) bases a, b, c,...,, and let the arguments of each be "recurrents of the nth order," that is to say, let,/ 27ri\. / 27r\ / 2/i a= cos.S.-2, b co. 2,C O 7i = wrs/. 27ri.., t n= Coo 234 HISTORY OF THE THEORY OF DETERMINANTS Let R, denote that any symmetrical function of the rth degree is to be taken of the quantities in a parenthesis which come after it, and let - indicate any function whatever. Then the zeta-ic product, C(CR,(abc... 1) x CptPD(Oabc... I)) is equal to the product of the number 27- - 27\ 4w7 47\ Rt cos. -+ --. sin 1, cos.-+ f - 1. sin - cos.-+ -1. sin..... nn / co(2n- l)7r 1 2(n- )7r\ cos.(L_ _2+ J ---. sin- )) multiplied by the zeta-ic phase p-_tbPD(Oabc... )! " Unfortunately the meaning of the proposition is seriously obscured by misprints and inaccurate use of symbols. Instead of " rth" degree we should have tth degree; the ~ preceding Rt(cbc... 1) is meaningless, and should be deleted; P preceding nPD (Oabc... 1) in the first member of the identity is unnecessary when a i has already been printed at the commencement; and the subscript p, although giving an appearance of greater generality, serves no purpose whatever. Making the corrections thus suggested, and denoting 27 r. 27r 47r. 47r cos-+ si - 1- sn —, cos- C + -l sin,... n n n n' which are the roots of the equation xn-1+ xn-2 +. n-3 +2 + n-3 +... + X + 0, by a, 3, y,.., X, we are enabled to put the theorem in the more elegant form ~{R,(ac,b,c..., ).. PD(O,a,b,c,. O, ), = -t{Rt(a,3,y,..., X).. PD(O,a,b,c,.., )}. It is readily seen to be a generalisation of the first theorem of the paper, into which it degenerates when 5, instead of being any function of a,b,c,..., is a constant, and Rt, instead of being DETERMINANTS IN GENERAL (SYLVESTER, 1839) 235 any symmetric function, is one of the series Ea, cab, labc,.... As, however, the constant Rt(a,/3,y,... X) on the right-hand side will then be one of the series, Ea, Ea/y, a3,.... and will not therefore be +1 unless when t is even, there must be an inattention to sign in one or other theorem. The matter can be more appropriately inquired into when we come to the subject of alternants, because, as has been pointed out in a recent footnote, it is to this branch of the subject that identities between two zeta-ic multiplications of difference-products really belong. This early paper, one cannot but observe, has all the characteristics afterwards so familiar to readers of Sylvester's writings, -fervid imagination, vigorous originality, bold exuberance of diction, hasty if not contemptuous disregard of historical research, the outstripping of demonstration by enunciation, and an infective enthusiasm as to the vistas opened up by his work. MOLINS (1839). [Demonstration de la formule generale qui donne les valeurs des inconnues dans les equations du premier degre. Journ. (de Liouville) de Math., iv. pp. 509-515.] The real object of Molins was simply to give a rigorous demonstration of Cramer's rules. His literary progenitors, so far as determinants were concerned, were apparently Cramer, Bezout, Laplace, and Gergonne, the last of whom, it may be remembered, wrote a paper which might well have borne the same title as the above. The writer, however, whose work that of Molins most closely resembles is Scherk, and very probably the two were unknown to each other. Both had the same purpose in view, and both used the method of so-called "mathematical induction." The difference between them may be most easily explained by using a special example and modern notation. To make the solution of the set of three equations ax + a + a3z + = a4} bx + b2y + b3 = b4 C1X + cOy + caz = c4 236 HISTORY OF THE THEORY OF DETERMINANTS dependent upon the already obtained solution of two, Scherk put the first pair of equations in the form ax + a2y = a4 - Ca3 blx + b2y = 4- b3zf, solved for x and y, and substituted the values in the third equation. Molins, on the other hand, having used the multipliers mn, m2, 1, with the equations of the given set, performed addition, solved the pair of equations qma2 + m2b2 + c2 = 01 mla3 + rn2b3 + C3 = 0 for ml and im2, and substituted the obtained values in the result X - 1a4 -+ 2b4 + C4 (XIII 10) 1 ma, + M9b. + 10 His exposition is laboured and uninviting. SYLVESTER (1840). [A method of determining by mere inspection the derivatives from two equations of any degree. Philosophical Magazine, xvi. pp. 132-135; or Collected Math. Papers, i. pp. 54-57.] The two equations taken are anxn + anxn-l +... + aix + ao = 01 baxm + bmlX7m-l +... + blx + bo = OT, and rules are given for attaining three different objects, viz. (1) a rule for absolutely eliminating x; (2) a rule for finding the prime derivative of the first degree, that is to say of the form Ax-B=0; (3) a rule for finding the prime derivative of any degree. The first of these concerns the process afterwards so well known by the name " dialytic." Only part of it need be given (p. 132):"Form out of the a progression of coefficients m lines, and in like manner out of the b progression of coefficients form n lines in the. following manner: Attach m -1 zeros all to the right of the terms in DETERMINANTS IN GENERAL (SYLVESTER, 1840) 237 the a progression; next attach m - 2 zeros to the right and carry 1 over to the left; next attach m - 3 zeros to the right and carry 2 over to the left. Proceed in like manner until all the m - 1 zeros are carried over to the left, and none remain on the right. The m lines thus formed are to be written under one another. Proceed in like manner to form n lines out of the b progression by scattering n - zeros between the right and left. If we write these n lines under the mn lines last obtained, we shall have a solid square m + n terms deep and m + n terms broad." (LIV.) The rest of the rule deals of course with the formation of the terms from this square of elements, the old and familiar method being followed of taking all possible permutations and separating the permutations into positive and negative. As applied by Sylvester in the case of the elimination of x between the equations ax2 + bx + c = O lX2 + nx + n = O, that is to say, as applied to the development of the determinant of the system a b c 0 0 a b c I m n 0 0 m the method is lengthy. No hint at an explanation of this or either of the two other rules is given. The principle at the basis of them all, however, is essentially that of the preceding paper. A single example will make this plain, and will at the same time serve to give a better idea of the two remaining rules than could be got by mere quotation.* Let the two given equations be Cx3 + b + b2 + cx + 0 = axC4 + X3x + yx2 + dx + c = = and suppose that it is desired to obtain their " prime derivative" of the 2nd (rth) degree, that is to say, the derivative of the form Ax2 + Bx + C = 0. * The third rule is incorrectly stated. 238 HISTORY OF THE THEORY OF DETERMINANTS Taking the first equation followed by m -r -1 equations derived from it by repeated multiplication by x, and then the second equation followed by - r-1 equations derived from it in like manner, we have m + - 2r equations, ax3 +- bx2 + CX + cd = 0 ax4 + bx3 + CX2 + dx = 0 ax4 + f3x3 + yx2 + 8x + e = o, from which we have to deduce an equation involving no power of x higher than the 2nd. To do so we employ, as just stated,, exactly the same method as was used in obtaining the "leading theorem" of the preceding paper. That is to say, we form multipliers a b. a. a a /3 a a b effect the multiplications, and add, the result being.a b. a c. a d a b c x + a b d x+ a b c =+. (LIV. 2) a 3 y a 13 a e This is what Sylvester's third rule would give. His second rule is simply a case of the third, viz., where r= 1; and his first rule is another case, viz., where r=0. Had he followed the order of his former paper, he would have called the third rule his "leading theorem," and given the others as corollaries from it. RICHELOT (May 1840). [Nota ad theoriam eliminationis pertinens. Crelle's Journal,, xxi. pp. 226-234; or Nouv. Annvales de Math., ix. pp. 228 -232.] Just as Jacobi (1835) brought determinants to bear on Bezout's abridged method of eliminating x from two equations of the nth degree, so did his fellow-professor Richelot, in treating of the other method of elimination, Euler's and Bezout's, discovered in DETERMINANTS IN GENERAL (RICIIELOT, 1840) 23 9 the same year (1764). Euler's method, it will be remembered, consists in transforming the, problem into the simpler one of eliminating a set of unknowns from a sufficient number of linear equations; and Richelot in a few lines (p. 227) points out that this may, of course, be done by equating to zero the determinant of the system of equations. An inve stigation connected therewith occupies the main portion of the paper. Sylvester's method (1840) is described in passing, and the principle at the basis of it given. We have just seen that, when originally made known by the author, it was merely in the form of a rule without any explanation. Although no doubt exists as to the mode in which it was obtained, still this first published description of the mode by Richelot deserves to be put on record. The whole passage in regard to it is as follows (p. 226):"Quam vequationem* inveniendi methodi diversie a geometries adhibentur, ex quarum numnero emus, quoe a clarissimo Sylvester in diario The London and Ediniburgh Philosophical Mlagazine and Journal of Science nuper exposita est, mentionem faciendi hane occasionem hand prietermittere velim. Ibi illius eliminationis problema reducitur ad problema eliminationis mn + nt - 1 quantitatum cx systemate mn + it wcquationum linearium. Multiplicata enim oequatione f1 = 0 ex ordine per yn-', y"72.,2 f,0 nec non oequationc f2 =0 ex ordine per yn1, M21 0, adipiseimur sytmai + n cequationum linearium inter quantitates ynv1~, Y"" nn-2 I..., y0, quarum rn + n - 1 prioribus eliminatis, o-equatio inter coefficientes t a' et a" prodit. Quoc eliminatioe facillime ita instituitur, ut determinantem harum mn + n oequationum lincarium ponamus = 0. Determinans vero, cum quantitates a' et a" in cequationibus ipswe tantum lineariter involvantur, et quantitates a' in nt, nec non quantitates a" in in ceteris oequationibus solis reperiantur, respectu illarum dimensiones nta est, respectuque harum rntec. Unde concluditur, cam. positam = 0, esse quoesitam. illam. iquationem. finalem X = 0, quioe omni factore superfiua careat. Notissiina enim est proprictas ab Eulero, inventa ocquationis X = 0, quod emus dimensioe respectu quantitatum a' est = n, atquc respcctu quantitatum, a"= =n ita ut quoeque functio integra evanescens, inter quantitates a' et a", has dimensiones quadrans, pro genuina oequatiorne finali habenda, sit." (LIV. 3), * ILe., oequationem finalem. f The equations are taken in the form a/,ny -' + alm-iy"''I +....+ a', = 0, 12 e- c"Y + 0'-'1+....+ a",o = 0. 240 HISTORY OF THE THEORY OF DETERMINANTS Taking Sylvester's example, ax2 + bx +c= 01 ax2+ fx+y/= Oj, and doing as Richelot here directs, we should first multiply both members of the first equation by x2-' and by x-', then both members of the second by x2-1 and by x1-1, thus obtaining ax3 + bx2 + cx =0, ax2 + bx + c = 0, aX3 + fX2 + yx = 0, ax2 + O3x + y = 0, and finally eliminate from these four equations x3, x2, xl, by equating to zero the determinant of the system. The statement " Ibi illius..... linearium," which seems to contradict what we have above said in regard to the absence of explanation in Sylvester's paper, is not literally true. Richelot may have meant by it that Sylvester's result implied that the problem had been transformed as stated. CAUCHY (1840). [Memoire sur l'dlimination d'une variable entre deux equations algebriques. Exercices d'canalyse et de phys. math., i. pp. 385-422; or (Euvres compnletes, 2e Ser. xi.] After the appearance of the special papers on this subject by Jacobi, Sylvester, and Richelot, a review of the whole matter could not but be a desideratum. This was supplied by Cauchy in the singularly clear and able memoir which we have now reached. After an introduction of four pages there is an account (1) of Newton's method as expounded by Euler in 1748; (2) of Euler's and Bezout's method of 1764; (3) of Bezout's abridged method; and (4) of a method' by means of the differences of the roots of the equations. - Euler's, although not called so. DETERMINANTS IN GENERAL (CAUCHY, 1840) 241 Euler and Bezout's method is shown to lead to the same determinant as Sylvester's, and the cause is made apparent. Cauchy's says (p. 389): "Sup-posons, pour -fixer les ide'es, que les fonctions f(x), F(x) soient l'une du troisie'me degre', l'autre du second, en sorte qu. on ait fAx) =ax3 + bX2 + cx + d, F (x) = Ax2 + Bx + C. Alors u, v devront e'tre de la forme u= Px + Q, V= px2 + qX + q. et, si 1Pon 61imine x entre les deux 6cquations f(x) = 0, F (x) = 0, I'equation resultante sera priceise'ment celle qu'on obtiendra, lorsqu'on ehoisera les coefficients 19, q, r,) P, de manie're 'a faire disparaitre x de la formule (2)_ uf(x) + vF (x) =0, par consequent de la formule (Px +Q) f(x) + (_px2+qx + 2)F (x) = 0, que l'on peut encore 6crire comme il suit: (3) Pxf(x) + Qf(x) + px2F (x) + qxF (x) + rFP (x) = 0. Les valeurs de.A, 'r, P, Q qui remplissent cette condition sont celles qui ve'rifient les equations lmneaires, (aP Ap 0, bP +aQ +BP +Aq = 0, (4) cP +bQ~+Cp +B + Ar = 0 dP + cQ ~ Cq~+Br =0, + dQ + Cr =0. Done, 'pour obtenir la resultante cherche'e, il suffira d'e'liminer les coefficients P, Q, _p, q, r M.D.Q 242 HISTORY OF THE THEORY OF DETERMINANTS entre. les, equations (4), ou, ce qui revient au m~me, d'e'galer 'a zero la fonction alterne'e forme'e avec les quantite's que presente le tableau a 0, A, 0, 0, a,.(t B, A, 0, (5) ~ ~ ~ c, ~b, C, B, A, die 0, 0, B, 0 di0, 0, C. On arriverait encore aux me'mes conclusions en partant de la formule (3). En effet, choisir les coefficients P, Q, p, q, r, de mianie're, 'a faire disparaitre de cette formule les diverses puissances xi X2, X3,..., m n- de la variable x, c'est e'immner ces puissances des cinq equations, (6) xf(x)= 0, f(x) = 0, X2F (x) = 0, xF (x ) 0, F (x) 0, on cX 6 x3 + cx2 +dx =0, aX3 + bX2 + CX + d = 0 (7) Ax4 +Bx + CX2 =0, Ax' +BX2 +CX =0, AX2 ~ Bx +0=0O., C'est donc e6galer 'a zero la fonction alterne'e forme'e avec les quantite's que pre'sente le tableau, a, b, C, d, 0, B1, C, 0, 0, (8) 0,,) b, c, d, 0 A)B, C, 0, l 3A, B, C. Or cette fonction alterne'e ne diff~rera pas de celle que nous avons de'ja' mentionne'e, attendu que, pour passer du tableau (5) an tableau (8), il suffit de remplacer les lignes horizontales par les lignes verticales, et re'ciproquement." (LIV. 4) Bezout's abridged method for the equations aoxn + aXx'-1 +.. + a-1X + acci = box"t + bjxq'-' +....+ be1ol + bit = of DETERMINANTS IN GENERAL (CAUCHY, 1840) 243 is shown to lead to the final equation S = 0, where S is "une fonction alternee de lordre n formee avec les quantites que renferme le tableau, Ao0, Aol. A- A,n-1 Ao,1 AI,l.... Al,n-2 A1l,_-l Ao,_1-2 Al,n_-2... A_-s-2,an-2 An-2,n-l 0Ao,n-l A1,-1.... A -2,n-l An-l,n-l in which Ao,I = ctob+l - boCa+l, Al1, = albl+1 - blal+l- + Ao,l+, A2,1 = a2b1+1 - b2a++l + Al,1+1, In connection with this, however, no reference is made to Jacobi's paper of 1835. The fourth method, which occupies much the largest space (pp. 397-442), is not a determinant method. SYLVESTER (January 1841). [Examples of the dialytic method of elimination as applied to ternary systems of equations. Cambridge Math. Journ., ii. pp. 232-236; or Collected Math. Papers, i. pp. 61-65.] In returning to extend the method, here and generally afterwards called "dialytic," Sylvester takes occasion to say that "the principle of the rule will be found correctly stated by Professor Richelot of Kinigsberg in a late number of Crelle's Journal." It may be noted, too, that he now for the first time uses the word determiznant. Only the first and last of the four examples need be given, as the subject strictly belongs to the application rather than the theory of determinants. Even these, however, will suffice to show the masterly grip which Sylvester had of his own method. 244 HISTORY OF THE THEORY OF DETERMINANTS "To eliminate x, y, z between the three homogeneous equations Ay2 - 2C'xy + Bx2 = 0 (1), Bz2 - 2A'yz + Cy2 = 0 (2), Cx2 - 2B'zx + Az2 = 0 (3). Multiply the equations in order by - 2, x2, y2, add together, and divide out by 2xy; we obtain C'z2 + Cxy - A'xz - B'yz = 0 (4). By similar processes we obtain A'x2 + Ayz - B'yx - C'zx = 0 (5), B'y2 + Bzx - C'zy - A'xy = 0 (6). Between these six, treated as simple equations, the six functions of x, y, z, viz, x2, y2, z2, xy, xz, yz, treated as independent of each other, may be eliminated; the result may be seen, by mere inspection, to come out ABC(ABC - AB'2 - BC'2 - CA'2 + 2A'B'C') = 0, or rejecting the special (N.B. not irrelevant) factor ABC we obtain ABC - AB'2 - BC'2 - CA'2 + 2A'B'C' = O." (LIV. 5) The example, however satisfactory as illustrating the dialytic method, cannot be passed over without a note in regard to the unaccountable blunder made in developing the determinant involved. In later notation the determinant is C B -2A' C. A. -2B' B A... -2C' A'.. A -C' -B'. B'. -C' B -A'.. C - - -A' C Now neither of the factors given by Sylvester are re of this, the truth being that it = 2(ABC + 2A'B'C' - BB'2- CC'2- AA'2)2. ally factors DETERMINANTS IN GENERAL (SYLVESTER, 1841) 245 The fourth example concerns the elimination of x, y, z between the three equations Ax2 + By2 + Cz2 + 2A'yz + 2B'zx + 2C'xy = 0 Lx2 + My2 + Nz2 + 2L'yz + 2M'zx + 2N'xy = 0 px2 + Qy2 + Rz2 + 2P'yz + 2Q/'z + 2R'xy = 0 Using each of the three multipliers x, y, z with each of the three equations, we obtain nine equations linear in the ten quantities, x3, y3, Z3, x2y, X2z, y2x, y2z, z2x, z2y, xyz. Another such equation is thus necessary for success. Sylvester obtains it very ingeniously by writing the given equations in the form (Ax + B'z + C'y)x + (By + 'x + A'z)y + (Cz + A'y + B')z = 0 (Lx + M'z + Ny)x + (My + N'x + L'z)y + (Nz + L'y + M'x)z = 0 (Px + Q'z + R'y)x + (Qy + R'x + P'z)y + (Rz + P'y + Qx)z =, and then eliminating x, y, z. The work is not continued further. We may ourselves note, in conclusion, that the fourth example includes in a sense the three others, but that it does not follow therefrom that by giving the requisite special values to the coefficients in the result of the general example, we should obtain the results for the particular examples in the forms already reached. Indeed, it is on account of this apparent non-agreement that the dialytic method is valuable to the theory of determinants, some very remarkable identities being arrived at by its aid. An explanation is also thus afforded of the trouble we have taken to elucidate its history. CRAUFURD, A. Q. G.* (February 1841). [On a method of algebraic elimination. Cambridge Math. Journal, ii. pp. 276-278.] In Craufurd we have an independent discoverer of the dialytic method. A full account of his paper is quite unnecessary: the *Only the initials A. Q. G. C. are appended to the article. There can be little doubt, however, that they belong to Craufurd, whose name in full appears elsewhere in the Journal. 246 HISTORY OF THE THEORY OF DETERMINANTS few lines dealing with his introductory example will suffice to establish the fact. He says:"Let it be required to eliminate x from the equations x2 + x + q = 0, x2 + p'x + q' = 0. Multiply each of the proposed equations by x, and you obtain x3 + px2 + q = 0, x3 + p'x2 + q'x = 0. These two combined with the two given equations make a system of four equations containing three quantities to be eliminated, viz., x, x2, X3; and they are of the first degree with respect to each of these quantities. We may, therefore, eliminate x, x2, x3 by the rules for equations of the first degree. The result is.... " He enunciates a general rule, and then takes up the analogous subject in Differential Equations, where successive differentiation takes the place of successive multiplication by x. In a postscript he acknowledges Sylvester's priority which the editor had pointed out to him. He knew nothing of determinants. CHAPTER IX. DETERMINANTS IN GENERAL IN THE YEAR 1841. LIKE the year 1812 the year 1841 merits a chapter to itself; and in 1841 as in 1812 it is the work of only two authors that concerns us. Strange to say, however, the two notable years had an author in common, the writers of 1812 being Binet and Cauchy, and those of 1841 being Cauchy and Jacobi.* In 1841 Jacobi's contributions constituted a comprehensive monograph similar to that produced by Cauchy in 1812, and Cauchy's in 1841, as was to be expected, were more of the nature of an aftermath. CAUCHY (March 8, 1841). [Note sur la formation des fonctions alternees qui servent a resoudre le probleme de l'elimination. Comptes Rendus... Paris, xii. pp. 414-426; or ETuvres completes d'Augustin Cauchy, 1re Ser., vi. pp. 87-90.] Recalling the fact that the final equation, resulting from the elimination of several unknowns from a set of linear equations, has for its first member "une fonction alternee," and pointing out the further fact that the same holds good in regard to the elimination of one unknown from two equations of any degree, " puisque les methodes de Bezout et d'Euler reduisent ce dernier probleme au premier," Cauchy affirms the importance of being able easily to write out the full expansion of such functions. There can be little doubt, however, that it was the second fact alone,-in other words, the discoveries of Jacobi, Sylvester, and * Cauchy was born fifteen years before Jacobi and lived six years after him. 248 HISTORY OF THE THEORY OF DETERMINANTS Richelot,-which influenced the veteran Cauchy to return to a subject practically untouched by him for thirty years. The opening part of the paper is, of course, necessarily old matter. One thing to be noted is that Cauchy tacitly discards the term determinant, which he was the means of introducing, using uniformly the more general expression fonction alterne'e instead. Another is that he adopts the rule of signs which makes use of the number of interchanges. From this his own peculiar rule of signs is deduced, and made the starting point for the fresh investigation which forms the main portion of the paper. The exposition of his rule, which differs from that of 1812, is worthy of a little attention, both on its own account and because otherwise the matter following would be scarcely intelligible. In the case of any term (" terme" or "produit") of the determinant 4- a0,0a1,1a2,2a3,3a4,4a6,5a6,6 say the term a0,1a1,0a2,5(a3,3aC4,6a45,46,2 there is an underlying separation of the indices 0, 1,..., 6 into groups (" groupes"), by reason of the system of pairing; that is to say, since an index is found paired along with one index and not with another, there arises the possibility of looking upon those which happen to be paired with one another as belonging to the same family group. Thus, attending to the first a of the 'term, we see that 1 and 0 belong to the same group, and as on scanning the rest of the term, we find neither of them associated with any other index, we conclude that the group is binary ("un groupe binaire"). Again, we see that 2 is paired with 5, 5 with 4, 4 with 6, and 6 with 2; this gives us the quaternary group (2, 5, 4, 6). Lastly, 3 is seen to be paired with 3, and thus forms a group by itself. Now, if we wish to find how many interchanges of the second indices are necessary in order to obtain the given term a0,1a1,a2,6a3,3a4,6a5,4a6,2 from the typical term a0,a1,1a2,2a3,3a445a6,6a6,6, we may do the counting piecemeal, attending at one time to only that part of the term which corresponds to one of the groups of DETERMINANTS IN GENERAL (CAUCHY, 1841) 249 indices. In the case of the group (3), the number of interchanges is 0; in the case of the binary group (0, 1) it is 1; and in the case of the quaternary group it is 3-the number of interchanges being " evidemment" one less than the number of indices in the group. If, therefore, for a given term there be in all m groups, viz. f groups of one index each, g groups of two indices each, h of three, k of four, &c., the number of necessary interchanges will be 0.f +.g + 2.h 3.k+.. which - f+ 2.g + 3.h + 4.. -(f + g + h +. ), - n-rnm; and consequently the sign of the term will be + or -1 according as n-m is even or odd. (II. 29) The first step of the new investigation is to define "termes semblables ou de meme espece." Two terms are said to be alike or of the same species when the one may be obtained from the other by subjecting both sets of indices in the latter to one and the same substitution or permutation. Thus recurring to the term above used, aO,1a1,0a2,5a3,3a4,6a5,4a6,2 and substituting in both of its sets of indices 6, 0, 1, 4, 3, 2, 5, instead of 0, 1, 2, 3, 4, 5, 6 respectively-in order words, and with the notation of the memoir of 1812, performing the substitution O 1 2 3 4 5 6 6 0 1 4 3 2 5, we obtain the like term a6,o0ao,6a1,2a4,4a3,6a2,3as,1l (LV.) The groups in two like terms are evidently similar, the values of f, g, h,... for the one being the same as those for the other. Indeed, since it is in this matter of groups or cycles that the terms have any likeness at all, the expression " cyclically alike" would have been a better term for Cauchy to use. From the definition there arises the self-evident propositionTerms which have similar index-cycles or are cyclically alike in their indices have the same sign.. 30) 250 HISTORY OF THE THEORY OF DETERMINANTS Also, the full expansion of a determinant may be represented by writing a term of each cyclical species, and prefixing to each such typical term the symbol z with its proper sign, + or -. (Lv. 2) To obtain a term of any given cyclical species, that is to say, corresponding to given values of f, g, h,..., all the preparation that is necessary is to write the indices 0, 1, 2, 3,..., (n-l), enclose each of the first f of them in brackets, enclose in brackets each of the next g pairs, then each of the next h triads, and so on. This gives the groups of the term, and the term itself readily follows. For example, if we desire in the case of the determinant ~_ a,,oaa1a22a33a44a5ac65 a term corresponding to f= 2, g = 1, h = 1* we take the indices 0, 1, 2, 3, 4, 5, 6; bracket them thus (0), (1), (2, 3), (4, 5, 6): and with the help of this, write finally a0,0 a1,1 a2,3 a3,2 a4,5 a6,6 a6 4 (II. 7) The number of different cyclical species of terms in a determinant of the nth order is evidently equal to the number of positive integral solutions of the equation f+ 2g+ 3h+... + nl=n. (LV. 3) Cauchy's illustration of this is clearness itself. He says (p. 419):"Si, pour fixer les idees, on suppose n=5, alors, la valeur de n pouvant ftre pr6sentee sous lune quelconque des formes, 1+ + 1 + 1 +1, 1 + + 2, 1 + 2 + 2, 1 + 1 + 3, 2 + 3, 1 +4, 5, * It would be convenient to say, a term whose index-cycle scheme is 2(1)+1(2)+1(3). DETERMINANTS IN GENERAL (CAUCHY, 1841) 251 les systemes de valeurs de /, g), b, i, se reduiront a Fun des sept systemes f=5, g=0, h=0, k=0, 0, = 3, g =, = 0, = =0, = f=, g=2, h=0, k=0, = 0, f= 2, g=, h= 1, k=0, = 0, f=0, g= 1, =, k=0, I =0, f=, g=0, h=0, k=1, 1=0, f=, g= 0, h= 0, k=0, =1; et par suite, une fonction alternee du einquieme ordre renfermera sept especes de termes." The next question considered is as to the number of terms of a given cyclical species which exist in any determinant of the nth order. The species being characterised by f groups of one index each, g groups of two indices each, h groups of three indices each, &c., the required number of terms is denoted by Nf, g, g,.., i ~ Now all the terms of the species will certainly be got if we write in succession the various permutations of the n indices 0, 1, 2, 3,...., n-1, and then in the usual way mark off each permutation into the specified groups, viz., first f groups of one index each, then g groups of two indices each, and so on. As a rule, however, each term of the species will, in this way, be obtained more than once. For, if we examine in its grouped form the particular permutation which was the first to give rise to a certain term, we shall find that changes are possible upon it without entailing any change in the term. For example, the set of groups (0), (1), (2, 3), (4, 5, 6), instanced above as corresponding to the term a,0 a,1 a2,, a,2 a4,s c5,0 a6,4, might be changed into (1), (0), (2, 3), (4, 5, 6) or (1), (0), (3, 2), (6, 4, 5) or......... 252 HISTORY OF THE THEORY OF DETERMINANTS which, while still corresponding to the term a0,Oa 1,1 2,3 a3,2 a4,5 a6,6 a6,4 are derivable from different permutations of the seven indices 0, 1, 2, 3, 4, 5, 6. In fact, the f groups of one index each may be permuted among themselves in every possible way, so may the g binary groups, the h ternary groups, &c. Further, with like immunity to the term, each separate group may be written in as many ways as there are indices in it,-the group (4, 5, 6), for example, being safely changeable into (5, 6, 4) or (6, 4, 5). The number, therefore, of different permutations of 0, 1, 2, 3, 4, 5, 6, which will give rise to any particular term, is (1.2.3...fx 1.2.3...g x 1.2.3...h x... x 1.2.3.../) x (lf/23h.3...1), or say, (f!g!h!... l!)(lfY3Sh... ). There thus results the equation (f!g!h!... 1!)(1237lf... )Nfg,,,, =!, whence Nf,,h, = (!...! )(lf23'... l) (LY. 4) Following this interesting result a few deductions and verifications are given. First of all it is pointed out that since the total number of terms of all species is n! we must conclude that n I n! ~ g(/(f!g'h!...!)(1f2931... A1) where f + 2g + 3h +... + l = n. Cauchy says (p. 423):"Cette derniere formule parait digne d'etre remarquee. Si, pour fixer les idees, on prend n = 5 l'quation donnera 1. 2.3.4.5 = N5,0,0,0,0 + N3,1,0,0,0 + Nl,2,0,0,0 + N2,0,1,0,0 + N0,l1,,0,0 + Nl,0,0,1,0 + N0,0,0,0,1, et par suite 1.2.3.4,5 = 1 + 10 + 15 + 20 + 20 + 30 + 24 = 120, ce qui est exact." Again, since the number of positive terms in a determinant is equal to the number of negative terms, and since the terms, DETERMINANTS IN GENERAL (CAUCHY, 1841) 253 whose number Nhf,,h,..., has just been found, have all the sign-factor (-1)n-(f+S+h7+... +) we have on leaving out the common factor (-I)n the identity O = ( _ )f+ +++h... +l! (/f!g!h!...!)(lf237S... n,)' which like its companion may be illustrated by the case of n= 5, viz., 0 = 1 - 10 + 15 + 20 - 20 - 30 + 24.* Lastly, attention is directed to the fact that when n is a prime, and therefore not exactly divisible by any integer less than itself, the number (f!g!h!...!)(1f23.... o) must be exactly divisible by n, except in the case f= n, g =0, h =0,..., =0, when it has the value 1, and in the case f=0, g=0, h=0,..., I=1, when it has the value (n -1)! It, therefore, follows from either of the two preceding identities, that the sum of these two values must be divisible by n,-which is Wilson's theorem. The remaining two pages are occupied with the expansion of a determinant of special form, viz., that afterwards known by the name axisymmetric. JACOBI (1841). [De formatione et proprietatibus Determinantium. Crelle's Journal, xxii. pp. 285-318; or Werke, iii. pp. 355-392.] The value which Jacobi attached to determinants as an instrument of research has already become well known to us: we have * In connection with this and in illustration of a previous remark regarding a mode of expressing the full expansion of a determinant, we have Z a0ooa22a33a44 = aooalla22a33a4 - 1aooalla22a34a43 + Zaooa12a21a34aa4 + Zaooalla23a34r942 - aola010a23a34a42 - n00aoo2a23a34a41 + Za1a12a23a34aa40. (LV. 2) 254 HISTORY OF THE THEORY OF DETERMINANTS found him, indeed, in almost constant employment of the functions. In the memoir now reached, however, we have still stronger evidence of his interest in the subject, and of his opinion as to its importance. Knowing of no succinct and logically arranged exposition of their properties readily accessible to mathematicians, he deliberately set himself the task of preparing a memoir to supply the want. In his few words of preface he says:"Sunt quidem notissimi Algorithmi, qui aequationum linearium litteralium resolutioni inserviunt. Neque tamen video eorum proprietates praecipuas, ita breviter enarratas atque in conspectum positas esse, quantum optare debemus propter earum in gravissimis quaestionibus Analyticis usum. Scilicet illae proprietates quamvis elementares non omnes ita tritae sunt, ut quas indemonstratas relinquere deceat, et valde molestum est earum demonstrationibus altiorum ratiociniorum decursum interrumpere. Cui defectui hic supplere volo quo commodius in aliis commentationibus ad hanc recurrere possim; neutiquam vero mihi propono totam illam materiam absolvere." While Jacobi was aware, as we have already partly seen, of the labours of Cramer, Bezout, Vandermonde, Laplace, Gauss, and Binet, his main source of inspiration is Cauchy. Of all the writers since Cauchy's time, indeed, he is the first who gives evidence of having read and mastered the famous memoir of 1812. It scarcely needs be said, however, that his own individuality and powerful grasp are manifest throughout the whole exposition. At the outset there is a reversal of former orders of things; Cramer's rule of signs for a permutation and Cauchy's rule being led up to by a series of propositions instead of one of them being made an initial convention or definition. This implies, of course, that a new definition of a signed permutation is adopted, and that conversely this definition must have appeared as a deduced theorem in any exposition having either of these rules as its starting point. The new definition has its source in Cauchy, and rests on the well-known agreement as to a definite mode of forming the product P of the differences of an ordered series of quantities. This being settled to be DETERMINANTS IN GENERAL (JACOBI, 1841) 255 (a - a0)(a2- ao)(a3 - a.). (a. - a0) ( (a2 a)-)(a, - a)... ( - a,) (a.- )..... (an -2) (an -,-1) for the quantities a0, a,,.... a,, while in the order here written, the definition stands as follows (pp. 285-286):"Vocemus eas indicum 0, 1,...., n permutationes, pro quibus P valorem eundem servat, positivas; eas pro quibus P valorem oppositum induit, negativas; sive priores dicamus pertinere ad classem positivam perrmutationmm, posteriores ad classera negativam." This implies of course that the original permutation 0, 1, 2,...., n is to be considered positive; and, such being the case, there seems to be a certain appropriateness in applying the term ~negative to a permutation whose corresponding differenceproduct is of the opposite sign from the difference-product corresponding to 0, 1, 2,...., n. The propositions which lead from the definition to Cramer's rule may be enunciated as follows:(a) One permutation performed upon another gives rise to a third, and the combined effect produced by performing the second and first in succession is the same as the effect of performing the third. (b) Two given permutations belong to the same class or to opposite classes according as the permutation by means of which the one is obtained from the other belongs to the positive or negative class. (c) If the same permutation be performed on a number of permutations which all belong to one class, the resulting permutations will still all belong to one class, viz., the same or the opposite according as the operating permutation is positive or negative. (d) The order of compounding a set of permutations is, as a rule, not immaterial. 256 HISTORY OF THE THEORY OF DETERMINANTS (e) The permutations which arise by compounding a set of permutations in every possible order belong all to the same class. (III. 32) (f) The interchange of two indices is equivalent to the performance of a negative permutation. (g) The interchange of two indices causes all the positive permutations to become negative, and all the negative to become positive. Definition.-Two permutations may be called reciprocal which being performed in succession do not alter the order existing before the operations. (xxiv. 2) (h) Reciprocal permutations belong to the same class. In the original, it must be borne in mind, these are not separated and numbered, but appear merely as consecutive sentences in a paragraph. The words "classem negativam" of the definition above given are followed in the same line by "Binis propositis permutationibus quibuscunque, certa exstabit permutatio, qua post alteram adhibita altera prodit. Pertinebunt duse permutationes propositse ad classem eundem aut ad classes oppositas, prout permutatio, qua altera ex altera obtinetur, ad classem positivam aut negativam pertinet," &c. -that is to say, by the propositions which have been paraphrased into (a), (b), &c. The most essential point to be considered in connection with them is the probable meaning of the expression "permutationem adhibere," or the free English translation of it, "to perform a permutation." An example will make it clear. To perform the permutation 35412 would seem to be the operation of removing the 3rd member of a series of five things to the first place, the 5th member to the second place, the 4th member to the third place, and so on. With this explanation the proposition (a) is self-evident, an example of it being (if we may improvise a symbolism) (35412)(41352) = (32541), where 35412 is the operating permutation. Cauchy's usage, it DETERMINANTS IN GENERAL (JACOBI, 1841) 257 may be remembered, was to speak of " applying a substitution to a permutation." * Of the proposition (b) a proof is given, which may be paraphrased as follows:-Let the three permutations referred to change P, the original product of differences, into elP, e2P, e3P, respectively, the e's of course being either +1 or -1. Then as the performance of the first two permutations in succession will result in the change of P into e1.e2P, we must have e1 e2 = e3, so that e, and e3 have the same or opposite signs according as e2 is +1 or -1; and this is virtually the proposition to be proved. (III. 31) A demonstration of (d) is also given. The two permutations being A and B, I the first index of A, and m the first index of B, the performance of A on B implies that the Ith index in B is to take the first place, and the performance of B on A that the fmth index of A is to take the first place. The resulting permutations will consequently not agree in the first index, unless the Ith index of B is the same as the rnth index of A, which manifestly need not be the case. t To prove (f) is of course the same as to prove that the interchange of two indices ir and s, r being the greater, alters the sign of the product of differences; and this is done by separating the product into three portions, viz., (1) the portion which contains neither ar nor a,, (2) the single factor which contains both, a- a., and (3) the product of all the factors having either one or the other for a term. It is then asserted that the interchange of r and s cannot alter the last of these, because it is symmetrical with respect to a. and a,; also, that no alteration is possible in *He says, for example (Journ. de 1VEc. Polyt., x. p. 10), "Si en appliquant successivement a la permutation Al les deux substitutions (2 et (A), on obtient pour resultat la permutation A6; la substitution (A6) sera equivalente au produit des deux autres et j'indiquerai cette equivalence comme il suit A= (A (As)A t This also is a paraphrase of Jacobi's proof. M.D. R 258 HISTORY OF THE THEORY OF DETERMINANTS the first, and consequently that the change in the second accounts for the validity of the proposition. (III. 33) As for the permutations which are called reciprocal they are, exactly those whose existence we have seen noted by Rothe, and called by him "verwandte Permutationen." Jacobi's definition, however, presents them in a slightly different light, the property involved in it being readily deducible from Rothe's. The latter's illustrative example was, as may be seen on looking back, 3, 8, 5, 10, 9, 4, 6, 1, 7, 2 A 8, 10, 1, 6, 3, 7, 9, 2, 5, 4 B. Now the performance of either A on B or B on A* gives rise to 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, the original arrangement: consequently A and B satisfy Jacobi's definition. The proposition (h) is also Rothe's. After these propositions, as already intimated, the subject of other rules of signs is taken up, the first rule considered being Cramer's. Since in the product of differences corresponding to any permutation every factor in which an index is preceded by a smaller index would require the sign-factor -1 to be annexed to it in order that the said product might be transformed into the original product of differences, it is clear that the determination of the class to which the permutation belongs is reduced to counting the number of such inversions. But the pairs of indices in the product of differences corresponding to the given permutation are exactly the pairs of indices to be examined in applying Cramer's rule. The identity of the two rules is thus apparent. (III. 34) To the demonstration Jacobi adds "quam regulam olim cel. Cramer dedit ill. Laplace demonstravit." The last assertion is notable for two reasons: first, because the rule like Jacobi's own is incapable of proof being a definition, postulate, or convention according to the mode in which it is expressed: secondly, because an examination of Laplace's memoir shows that there is no ground for the statement. The fitness of the rule for the deter-In the compounding of reciprocal permutations the order is immaterial. This is the exception hinted at in (d). DETERMINANTS IN GENERAL (JACOBI, 1841) 259 mination of the signs of the numerators and denominators of the unknowns in a set of simultaneous linear equations may of course be demonstrated, and perhaps this was in Jacobi's mind, but prior to the statement the abstract subject of permutations had alone been discussed. The other rule of signs dealt with is Cauchy's, in which permutation-cycles are counted instead of inversions. The existence of such cycles is the first point to be established, that is to say, it has to be shown that any permutation of 1 2 3... n may be obtained from any other by the performance of one or more cyclical permutations. Let 3271654 be the permutation sought,* and 2647513 the permutation from which it is to be derived. Placing the former under the latter, thus 2647513 3271654,' we see that 2 has to be changed into 3, then seeking 3 in the upper line we see that it has to be changed into 4, similarly that 4 has to be changed into 7, 7 into 1, 1 into 5, 5 into 6, and 6 into 2, the element with which we started. Now the proof turns upon the simple fact that the elements in the two lines being exactly the same, by following a string of changes like this we are bound sooner or later to reach in the second line the element we started with in the first. It may be that as here one cycle suffices for the second transformation; but if not, as in the case of the two permutations 264751 3 4157236, where the short cycle 245 is obtained, we turn to the remaining *This is a paraphrase of Jacobi's demonstration, which is not so simple as it might have been. The notation of substitutions, which Jacobi did not follow Cauchy in using, is here a great help toward clearness. 260 HISTORY OF THE THEORY OF DETERMINANTS elements, and knowing that those in the first line are of necessity the same as those in the second, we see that the application of the same process to them must, for the same reason as before, lead to a cycle. The possibility of arriving at any permutation by means of cyclical permutations alone is thus made manifest. The next point to be established is that a cyclical permutation of r elements can be accomplished by r-1 interchanges of pairs of elements. Little more than the statement of this is necessary. For if the elements of the cycle be a, a2, a.,..., ar, it is clear that to change a, into a2, ca into a., &c., has the same effect as to interchange a1 and a2, then a1 and a,, then a1 and a, and so on, the final interchange being that of a, and ar; and there are in all r-1 interchanges. This being proved, the final step is taken as in Cauchy's Note of 8th March. (II. 35) This rule of Cauchy's Jacobi deservedly characterises as beautiful. It is important, however, to take note that it possesses the other quality of usefulness in as marked a degree; and such being the case one is surprised to find that it has not received the attention which was its due. Any reader who will make a comparison of it and Cramer's by actual application of them to a number of examples will soon find that Cramer's is more lengthy and requires more care to be given to it to avoid errors.* The preliminary subject of permutations having been thus dealt with, determinants are taken up. In the first section regarding them there is little noteworthy. Cauchy's word "terme" is supplanted by the fitter word element, and term (" terminus ") is put to a more appropriate use; that is to say, ca) is called an element of the determinant 2 ~+ aa'a2... a. and aka',a",..... aC a term. Further, the word degree is employed in place of Cauchy's more suitable word order, " ipsum R dicam determinans n + lti gradus." A section of two pages is given to considering the effect *The best way perhaps of applying Cauchy's rule is to write the primitive permutation, 123456789 say, above the given permutation, 683192457 say, draw the pen through 1 and the figure below it, seek 6 in the upper line and draw the pen through it and the figure below it, and so on, marking down 1 on the completion of every cycle. DETERMINANTS IN GENERAL (JACOBI, 1841) 261 produced upon the aggregate of terms by the vanishing of certain of the elements. The propositions enunciated, with the exception of one made use of at an earlier date by Scherk, are as follows (pp. 291, 292):"I. Quoties pro indicis k valoribus 0, 1, 2,..., m -1 evanescant elementa a(), m+l).... determinans ~aai Ca2."... a.( abire in productum a duobus deterIminantibus:k aa (-.... ) a ~.... a(. (xIv. 6) II. Evanescentibus elementis omnibus, m (m+1) a(n) 7G,..., ^ f in quibus respective index inferior k indicibus superioribus m, m + 1,..., n minor est, fieri aa aCa....a -, ()a(m,+l). n) ~ aal'.... a m-) (VI. 9) IV. Evanescentibus elementis omnibus, a(m) a(n+l) a(n) in quibus indices inferiores superioribus minores sunt, si insuper habetur, a(m) - ( a m+l fit ~ aa1'a2".... a (n = aa '.... a(:. (VI. 9) As immediate deductions from the definition these are somewhat out of place, the trouble of demonstrating the first of them being virtually thrown away. The trouble taken by Jacobi, too, was less than required, the question of sign, for example, being inadequately discussed. In the course of the next section which deals with what we have called the recurrent law of formation, and with the vanishing aggregate connected with this law, Jacobi gives an expression for the complete differential of a determinant, the elements being viewed as independent variables. The passage is (p. 293):"Determinans R est singularum quantitatum a() respectu expressio linearis, atque ipsius a() coefficientem, qua in determinante R afficitur, 262 HISTORY OF THE THEORY OF DETERMINANTS vocavimus Al~; unde adhibita differentialium notatione ipsum Al~ exhibere licet per formulam, 3. A(' R Hinc si quantitatibus a(~ incre-menta infinite parva tribuimus, da~i) simuique R incrementum dR capit, fit 4. dR 7p, A('da~i. (LVI.) siquidem sub signo summatorio utrique indici i et k valores 0, 1, 2,.. n conferuntur." The recurrent law of formation and its dependent neighbour formula he is enabled, by means of (3), to view as the partial differential equations which the determinant must satisfy. His words are (p. 295): "Substituendo formulas (3), inventas formulas sic quoque exhibere licet: 9. a~)DR D~I)R DaR DR,DR DR a,, + - +a Tj Da~ + ~ Da', + + Dat'7 ( 10 0 i) DR DRa 0DR 0. 0 a, 5-+ +a + 1n DR,DR D1)R Quae sunt aequationes differentiales partiales quibus determinans R satisfacit." Passing over a section (7) on simultaneous linear eqnations, and a short section (8) in which Laplace's expansion-theorem is enunciated, we come to two sections dealing with what at a later time would have been called the secondary minors. No name is given to them by Jacobi; they only appear as co-factors of the product of a pair of elements, the aggregate of the terms containing <ag ag as a factor being denoted by a17 af).i Af,f7 (XL. 9) DETERMINANTS IN GENERAL (JACOBI, 1841) 23 263 From observing that the interchange of f and f' or of g and g' alters R into - R and cannot alter A~,. it is concluded that -and that the full co-factor of Af,<f is a. a6f1 - a.,) a~f' in accordance with the expansion-theorem of the previous section. The remark that A~f can be expressed in terms of n of the quantities Aff'~ leads up to a curious set of equations the determinant of which belongs to the special class of determinants known afterwards as zero-axial skew determinants. The passage is,(pp. 300, 301): "Designemus br. causa per (k, ic) expressionem 10. A7,,f = (k, k'), ita ut sit (,I')=-(kic Ic). Fit e (8) ipsi g substituendo numeros 0, 1, 2,. n: Af)= * + a (')(0, 1) + a~f')(0, 2)..+ a,(f'`(0, n) AV)= a~'(l,) I 0) + * + a~f')(l, 2)..+ a (1l(, n) IL. APf = a~f')(2, 0) + a~f')(2, 1) + *..+ a~f U (2, a) A~)= a~~'`(n, 0) + a`f'(n, 1) + a2"'\n, 2)..+ *,Similes formulae e (9) derivari possunt. In aequationibus (11) ipsorum a~' aP etc. co~fflcientes in diagonali positi evanescunt, bini quilibet,codfficientes diagonalis respectu. symmetrice positi valoribus oppositis gadn.Quae est species aequationum linearium memorabilis in -variis quaestionibus analyticis obveniens." (LVIL.) The simple step from. the expression of A~f) as a differential 'Coefficient to the similar expression for Af' f' is next made p.301):"Ex ipsa enim aggregati definitione eruimus formulas 1. A~ = D2R 'a2R =a. Af ='a. A~~ =~ a. DA~f_-'a. A (gf) Da~f)-Da) 'a) a2' g1 g g 264 HISTORY OF THE THEORY OF DETERMINANTS By taking the identities 0 = aAk, + a'A' +. +. ( )A(), 0 = alA-k + 1A - + * + lA(A), R = aAk + a',A', +...+. a()A 0 = aA+k + aA +.... + a)A; using the multipliers Au, i'At..i,,,,'. AOk" A171k k, '' An,' * and adding, there is obtained 4. R. A', =-A()A("') A( )A(') ' k' r ' ' ~"' '""'k — '-"k I'"7 k -a result at once recognisable as a case of the theorem regarding a minor of the adjugate. Next by starting with Bezout's identity connecting any eight quantities, the particular eight taken being A(i) ) A(i ) A (i) k c ' "c'' s" IGc"" A(") A("') A(kl//) ^A(") and making six substitutions of the kind A(7)A h2 - Ak IAk R. A<,, I^c ^k' - -c'k,~ ' just seen to be valid, there arises the identity Ah:" A';. + A' A7', + A A'," = 0. (XXIII. 12) Aki,iA i,P1, k A k,k' A kk' ' This clearly belongs to the class of vanishing aggregates of products of pairs of determinants; but in order that its true character may be seen, and comparison made possible between it and others of the same class already obtained, a more lengthy notation is necessary. Taking for shortness the case where the primitive determinant is of the 8th order, but writing it in the form abc23d4e5gf 7h I and making i, '= 3,6 and k, k',k", '= 5, 6, 7, 8, DETERMINANTS IN GENERAL (JACOBI, 1841) 265 we find the identity to be alb2,d3e4g7hs I. ab2d,3e4g5h6 I - I a1b2d3e4g6h8. I alb2d3e49g6h7 + I albd3e4g6h7. ab2d3e4gshs I = 0, a glance at which suffices to show that it is nothing more than the extensional of 1 g7h8 1. 6hG - I 6h8 1 1g6h7 1 + g 1. Igh8 = the very identity of Bezout which was taken as a basis for it. As the same extensional has already been found among those of Desnanot, any new interest in it is due to the peculiar way in which Jacobi obtained it. By the same method, viz., by substituting for secondary minors an expression (4) involving primary minors and the primitive determinant, he shows that A()A,, + AA(giA'A = 0. (xxIII. 13) This being translated in the same manner as the preceding, becomes I abd3e4,fg7hs I. l alb2d3e4g6h8 - ai b,da3e4fg7,h I. aLb2d3e4g6hS + I ab2de4fg6h8s I. ab2d3e4g7h8 | = 0, and is thus seen to be another of Desnanot's results, viz., the extensional of tfo671 5 - /f567 I96 + f6g 1.g9 = 0. (xxIII. 13) The deduction A(') A(i) k ' 'k OA(t \ (i)A ' i, AWi AA(i") ii *^c _ _ *f C^k'k"" k, k'_ _ c k' (il) A('A) <(ill)- A(-)A(') ' is made from it by substituting appropriate differential coefficients for the primary and secondary minors involved in it. (LVIII.) The eleventh section is devoted to the establishment of the general theorem which includes the theorem R. A 'i, A(i)A(i) -A(i)A(i ) kc, - fc k k k' k' kc of the preceding section, and which, as we have seen, Jacobi had first enunciated in 1833. To start with it is repeated that the system of equations 266 HISTORY OF THE THEORY OF DETERMINANTS a t + al tl +...+ c tk + ak+ltk+l +..+ + an tn = u, a' t + at +...+ a tc + ak+ltk+l+...+ a tn =, a ( )t + a kt1 + + k+ a 7tk + a t(+i +... + k == ( )t + q + at1 +.. q t + a+t + t *~ +... a t = an, gives rise to the system A u + A' u, +... + A(c) + c + A(+ k+l... + A(f + n) = R.t A'ue + A', a1 +... + A k),Z + A(+l)n+ +... + A(n) = R.t| Akm + A', +... + A7 k)Z. + A+)... + A n, = R.t Ab + AG1 +... + A() + A+l, +... A(") = R.t, in which R = +~aa.... ( A( = aa'.... ac.(-1 Then taking only the first 7c+1 equations of the first system and eliminating t, t,,... t,, there is obtained CktA + C+lt,+, +.... + C,,( = D + D11 +... + DkUk, (X) where the multipliers D, DI,..., Dk, by which the elimination is effected, are (-) +a1 a' ''... a cand consequently by C, C,,,...., C are denoted z +M1 @2 ''''ak7c-1 k7c.. +C.... v +.. 2 R, (z-1h +_ aa /.... a(7_1 b DETERMINANTS IN GENERAL (JACOBI, 1841) 267 Similarly, taking only the last n-k + 1 equations of the second system and eliminating z, +,2,. +2, U n there is obtained Eu + Ell +... + Ektk = RFktk + RFk+lt+l +... + RFntn, (Y) where the multipliers F, F+l,.. F,, by which the elimination is effected are I 4 A (k+1)A (k 2).... An) - A(k+l)A(k+2) A(n) 7+ k+2., (-1 )f-k - - A(k+l)A(k+2).... A () and consequently by E, E1,.., E, are denoted - +-A A A(+l) A() + AA' k+l).... A), + A(Ak)A(k+1)....A These two derived equations (X), (Y), however, must be identical, because they may be both viewed as giving to in terms of tk~+l,tk+2 *..., tn, U1),, ~. ~, and, as the first system of equations shows, this can only be done in one way. We thus have the deduction D,_ Ek C - R. FJ' e 2 + a/n'a2" ff,(kl-) +4 A(k+)A+l) A(n) i.e...... k —1 -k kd-1...,L,',".(k) 1~ +A (k+1)A1k+2) (n) ~ + ~ aaI a2 a.... R - A Ak+A+2.... A,(, This is the keystone of the demonstration. The simple continuation of it may for sake of historical colour be given in Jacobi's own words (p. 304):t"In hac formula generali ipsi k tribuendo valores n- 1, n- 2, - 3,..., 1, prodit: * The demonstration in the original is considerably disfigured by misprints. 268 HISTORY OF THE THEORY OF DETERMINANTS ~aa' /.... 0`2 A( 1 ~1 ~aa~l.... a(z' R n ~aa'...n~-2 R Y ~~-') n a - ~1l"2.. A? "Harum aequationum prima suppeditat, ~A~n- A")= RY, ~ aa 1..a:) -RA quae cum formula (4) ~ pr. convenit. Delude aequationum. (10) duas, tres, quatuor etc. primas inter se multiplicando, prodit formularum. systema hoe: 991 A 99-' A - R I ~aa /. a(n9-2) ~A~n-2 k" A_ - R2 I ~ aa1....a n3 Quas formulas ampleetitur f ormula generalis, I ~ Aij (k )A k.... A(z = Rnk1~ ~ aa'1 a..k). (xx. 6) Cauchy's theorem (n) which-may be viewed as the ultimate case of this, Jacobi arrives at by expressing I ~+ AA,'... A(") in terms of A, A19...,) A~ and their cofactors, substituting for the said cofactors their equivalents as jnst obtained, viz. aRn-l, a1Rn - I, a2Rn 1.,~nand then using the identity Aa +Ala, +.. +Anan=R. Passing over the twelfth section, which relates to certain special systems of equations, we come to two sections devoted DETERMINANTS IN GENERAL (JACOBI, 1841) 269 to the multiplication-theorem. Of the five formally enunciated propositions which they contain, two, the second and fourth, need not be more than referred to, as their substance comes from Binet and Cauchy, and as the mode in which they are established will be sufficiently understood from the treatment of one of the others. The general problem of the two sections is the investigation of the determinant E + cc1 c2..c.. c where ) a()c ( )) + ct(). (i)C(k) Taking a single term of the determinant, we have of course CC1 C2.... C) - (a a + a, al, +.... + apalc) X (a a'+al/a/+.... + ap'/a/) X (a(f)a() + a(2)(2) + + a(n),) and we see that if the multiplications indicated on the right be performed there must arise a series of (p+l)81+l terms of the type a,f. as as ' ata...... a() or by alteration of the order of the factors aras at //.. a(). aCsat. () where each of the inferior indices r, s, t,.., w may be any member of the series 0, 1, 2,.., p. If we bear in mind the meaning which we thereby assign to the summatory symbol S we may write this in the form,cc, S (a,a S /,,a...(),. aa'a,"... ~01 02.. = t IV. ) ~ The next point to consider is the transition from the single term cc,'c2"... () to the full aggregate + cc1'c2.... c). A glance at the sum of terms denoted by c(i) shows that by permuting the superior indices of cc'c2"... c() the superior indices of the a's are subjected to the same permutation, and that, on the other hand, when we permute the inferior indices of cc1'c2"... c() it is the a's that are affected, the like permutation being given to 270 HISTORY OF THE THEORY OF DETERMINANTS the superior indices. Making the choice of the superior indices of the c's, let us permute them in every possible way, and to each term thus derived from ccl'c2"...c(n) prefix the sign + or - according as its superior indices constitute a positive or negative permutation. By so doing the left-hand side of our identity becomes i ~ ccl'c"... c(); and, owing to the consequent permutation of the superior indices of the a's, each term on the right-hand side gives rise to 1.2.3...(n +1) terms whose signs are the same as the signs of the terms corresponding to them on the left-hand side;-in other words, each term ral at... a(). ar,'a... a) gives rise to the compound term ars at "... ). Z aaat'..... We thus reach the result Jr CZ0/ * 6(n) = S((}rus (I * (9b) * aCC.las. at.a) ~ C1c2"... = S(a,,'a,"... -- a a.... ). Although the number of terms on the right is the same as before, viz. (p+l) 1)+, arising from giving to each of the n+1 indices r, s, t,..., w any one of the p+1 values 0, 1, 2,...,p, it has now to be noticed that a goodly proportion of them must vanish because of the fact that 2 -+ a.a 'a"/... a() = 0 when any two of its inferior indices are alike. The right-hand side will thus not be altered in substance if the summatory symbol be now taken to mean that i, s, t,..., are to be any n+ 1 of the p+1 indices 0, 1, 2,..., p. If p be less than n it will be impossible to have r, s, t,..., w all different, so that in that case the right-hand side must be 0. This is Jacobi's first proposition, and it constitutes his addition to the multiplicationtheorem. His formal.enunciation of it is (p. 309):"Sit (5 ) ()W (i) (k (i) (7) c( = aa + aa, + ~ a +. C, quoties p n< evanescit determinans i CC1 'C2...... C(." ' (xvIIm. 6) The consideration of the case when p=n leads to his second proposition. The natural addendum is then made regarding DETERMINANTS IN GENERAL (JACOBI, 1841) 27 271 the multiplication of more than two determinants of the samedegree (p. 310):"'Datis quoteunque eiusdem gradus determinantibus, eorum productum ut eiusdem gradus exhiberi posse determinans, cuius elementa expressiones sint rationales integrae elementorum determinantium propositorum." (xvii. 7). The equally natural transition to the subject of the multiplication of two determinants of different degrees results in the~ proposition (p. 311): — "Sit pro indicis i valoribus 0, 1, 2.,M Ck =a a + al a +.. a. pro indicis i valoribus maioribus quam in, k =-. a i+ cuj~laj+1 + ai~+2a.+g +. 4.. a(%)aA~ erit aa~.... a. aal.... ~ct1 a,, - cI~.... Cit. (xvii. 8), Proposition IV. concerns the case where p > n,. Proposition V. is but a corollary to the combined propositions I., II., IV., its~ subject being the effect of the specialisation - WThe enunciation is as follows (p. 312): — "Posito = k C7 == a ja~ ) + a('a.17. a,,a+ ) sit determinans ubi p < n fit P =0; ubip = n fit P { ~aa1'.... aW}2 ubi p> n fit P = 2 {~ a a Mt.... a(~} siquidem pro indicibus inferioribus m, in' &c. sumuntur quilibet n + 1 diversi e numeris 0, 1) 2. p." (xviii. 7y~ 272 HISTORY OF THE THEORY OF DETERMINANTS The two remaining sections (15 and 16) deal with a special system of simultaneous linear equations, interesting application being made to the theory of the Method of Least Squares-an application probably due to a suggestion of Binet's in his note of November, 1811. It is important to note, in conclusion, that from one point of view Jacobi's memoir was but the introduction to two others of really greater importance, both treating of a special class of determinants. The first concerns determinants of the kind afterwards deservedly associated with his name, and bears the title "De determinantibus functionalibis." It occupies the forty-one pages (pp. 319-359) immediately following the general memoir. The other, with the title "De functionibus alternantibus earumque divisione per productum e differentiis elementorzm conflatum," treats of those determinants, first considered by Cauchy, in which the members of one set of indices represent powers, and to which the name alternants afterwards came to be assigned. It extends to twelve pages (pp. 360-371). The three memoirs together constitute an excellent treatise on the subject, and are known to have been markedly influential in spreading a knowledge of it among mathematicians. The second and third memoirs, from the nature of their subject-matter, fall to be considered later. On the last page of the third memoir, however, where a possible simplification of a special determinant has to be effected, the general theorem on which the simplifying operation rests is enunciated; and as this theorem does not appear in the first memoir, it calls for attention now. The wording is:"Constat enim non mutari Determinans si singulis seriei horizontali terminis addantur earundem serierum verticalium termini multiplicati per quantitates quascunque, quae tamen pro omnibus eiusdem seriei horizontali terminis eaedem esse debent." (LTX.) One cannot but wonder why the afterwards familiar fact regarding the effect of " increasing a row by a multiple of another row" was not formulated long before this date. DETERMINANTS IN GENERAL (CAUCHY, 1841) 27 273 CAUCHY -(1841). [Note Sur les diverses suites que l'on pent former avec des termes donne's. Exercices (['analyse et de phys. math., ii. pp. 145-4l50; or Wuvres completes, 2e, Si'r. xii.] [Me'moire sur les fonctions alterne'es et sur les sommes'alterne'es. Exereice~s ct'arnayse et de Ahy&. math., ii. pp. 151-159; or (Euvres completes, 2e Se'r. xii.] [Me'moire sur les sommes alterne'es, connues soils le nom de re~sultantes. Exercice's d'artalyse et de phys. math., ii. pp. 160-176; or (iEtvres completes, 2~ Se.xi [Me'moire sur les fonctions diff~rentie11es alterne'es. Exercice's d'analyse et de phys. math., ii. pp. 176-187; or f7Emvres compl)tes8, 2e Si'r. xii.] From internal evidence, there can be little doubt that this series of papers, containing the fundamental conceptions and salient propositions of the theory of determinants, was prompted by the appearance of Jacobi's memoirs, and by the consequent conviction that the work of 1812 had begun to bear fruit. The first paper, called a "note," is introductory, on the subject of signed permutations; the three others, called" memoirs," correspond to Jacobi's,-the first of them to Jacob's third, the second to Jacobi's first, and the third to Jacobi's second. The note, although on so trite a subject as the division of permutations into positive and negative, is most interesting. Cauchy's original stand-point with regard to the subject is so far unaltered that the rule of signs specially known by his name is made fundamental, and all others deduced from it. The explanations preparatory for the rule are, however, on the lines of his paper of 1840, that is to say, it is groalps and not circular substitutions that are spoken of. The preference is a little difficult to justify; for notwithstanding Cauchy's assertion that groups come naturally into evidence, the idea is far-fetched as compared with that of circular substitutions. He says (p. 145):"Si 1'on compare uue quelcouque des nouvelles suites* A la premi~re, on se trouvera naturellement conduit par cette comparaison 'a distribuer les divers termes a, b, c, d. I Le., permutations of a, b,,ci, d)... 5 M.D. 274 HISTORY OF THE THEORY OF DETERMINANTS en plusieurs groupes, en faisant entrer deux termes dans un meme groupe, toutes les fois qu'ils occuperont le meme rang dans la premiere suite et dans la nouvelle, et en formant un groupe isole de chaque terme: qui n'aura pas change de rang dans le passage d'une suite a l'autre." The question of the natural order of ideas and the best mode of presentment is really, however, of small importance, for in application a grotup and a circular substitution are essentially the same. The difference is entirely one of stand-point, nomenclature, and notation. The permutation e, a, b, d, c, g, f, being in question, and comparison between it and the primitive permutation c, b, c, d, e, f, g, having been instituted, we are directed to form the members (" termes") of the permutation into groups, commencing to form a group with e and a, because they occupy like positions in the two permutations, putting b in the same group because it occupies the same position in the second permutation as one already in the group occupies in the first permutation, putting c in for the same reason, making d constitute a group by itself, and finally putting f and g together to form a third group. We are directed further, to write the members of each group in such an order that any member and the one following it may be found to occupy like positions in the primitive and derived permutations respectively. The result thus is (a e, c, b), (d), (f, g), or (e, c, b, a), (d), (g,f), or it being possible to write the first group in four ways, and the last in two. Now all this is nothing more than an unreasoning way of arriving at the circular substitutions which are necessary for the derivation of the given permutation from the primitive one. Cauchy himself, indeed, in pointing out that there would only be one way of writing a group if the members were disposed in a circumference instead of in a straight line, says:"C'est par ce motif que dans le tome x du Journal de l'Vcole DETERMINANTS IN GENERAL (CAUCHY, 1841) 275 Polytechnique j'ai designe sous le nom de substitution circulaire l'operation qui embrasse le systeme entier des remplacements indiques par un meme groupe." It must be borne in mind, however, that not only the operation, but the symbol of the operation, was so denoted, and such being the case, we may then very pertinently ask, What is a group in Cauchy's usage but the symbol of a circular substitution? The peculiarity of using the number of groups to separate the various permutations of a, b, c, d,.... into two classes makes its appearance in the following sentence (p. 147):"De plus, ces memes suites ou arrangements se partageront en deux classes bien distinctes, la comparaison de chaque nouvel arrangement au premier a, b, c, d,.... pouvant donner naissance a un nombre pair ou a un nombre impair de groupes." Of course, the primitive permutation is looked upon as having its groups also, viz., one for every letter in the permutation. Then comes the important proposition-The interchange of two letters increases or diminishes the number of groups (substitution-cycles) by unity. In proving it the two letters are first taken in different groups, (a,b,c,..., h,), (I,m,n,..., r,); and since any member of a group may occupy the first place, the letters a and I are fixed upon. Now what the groups imply is that the letters a, bc,.... h, k, I, m,,.... r, s in the primitive permutation are changed into b, c,....., a, m, n....s, I respectively to form the given permutation. If therefore in the given permutation the letters a and t be interchanged, the new permutation so obtained will be got from the primitive by changing ac, b,c, h,c,l, I,,...,,,s into b,c,..... k,, n...., s, a; 276 HISTORY OF THE THEORY OF DETERMINANTS that is to say, by the changes indicated by the single group (ca,b,c,..., h,lc,m,n,,...,,s). The interchange of two letters belonging to different groups is thus seen to reduce the number of groups by one. On the other hand, it is clear that had this single group belonged to the given permutation, the interchange of two letters, a and I say, would have had the effect of breaking up the group into two, (a,b,c,..., h,k) and (l,m,n,..., r,s). The theorem is thus established. (II. 36) It is next pointed out that the transformation of the primitive permutation into any other may be accomplished by interchanges only, because by this means any given letter may be made to occupy the first place, then any other given letter to occupy the second place, and so on. From this also it follows that any system of circular substitutions may be replaced by a system of interchanges. Should the transformation of one permutation into another be effected by interchanges, the number of these will be even or odd according as the two permutations belong to the same or different classes; for, by the above theorem, every interchange makes only one group more or one group less, and consequently the total number of interchanges, and the net increase or diminution of the number of groups, must be both even or both odd. The counting of interchanges may thus be substituted for the counting of cycles. (III. 37) Finally, Cramer's rule is introduced, in which, as we know, it is neither cycles nor interchanges that are counted, but inverted-pairs, or, as Cauchy, like Gergonne, calls them, inversions. To establish the rule, it is clear that two courses were open, viz., to connect inversions directly with cycles or to connect them with interchanges. The latter course is taken, the requisite connecting theorem being that the interchange of two elements of a permutation increases or diminishes the number of inversions by an odd number, an odd number of interchanges thus corresponding to an odd number of inversions, and an even to an even. The proof is not direct, like Rothe's, being effected with the help of a fourth related entity, the difference-product. The order of thought in it is as follows: DETERMINANTS IN GENERAL (CAUCHY, 1841) 277 If we define the difference-product of the primitive permutation a, b, c, d,... to be (a-b)(-c)..... (b-c)..... then it is clear that in the difference-product of any derived permutation there will be found exactly as many factors with changed sign as there are inversions of order in the permutation. A change of sign in the difference-product thus becomes a test for the existence of an odd number of inversions, and consequently, instead of the theorem just enunciated, it will suffice to show that the interchange of two elements of a permutation alters the sign of the difference-product. This Cauchy says must be true, for, the elements being h and k, it is manifest that the factor which involves them both, h-Ik or k-h, must change sign, but that the factors which involve them and any third element s constitute a partial product (h-s)(Jc-s) or (h-s)(s-A), the sign of which cannot change. (II. 38) Of the three memoirs, the first and third, like Jacobi's third and second, do not at present require attention. A slight reference to the first-on alternating functions-is, however, necessary, because Cauchy, unlike Jacobi, makes determinants a special class of alternating functions, and it is therefore of importance to see the exact position he assigns to them. It will be remembered that in 1812 he partitioned symmetric functions into permanent and alternating, and made determinants a class of the latter; that is to say, his scheme of logical relationship was ((a) Determinants. (A) Symmetri(a) Alternating [(A) Symmetric [ Functions [ (b) Permanent [(B) The memoirs we have now come to indicate a departure from this, both verbal and substantial. The change is made too without any reason being assigned; indeed, there is not even 278 HISTORY OF THE THEORY OF DETERMINANTS a word to imply that any change had taken place. Alternating functions are, as in his Cours d'analyse, put on the same level as symmetric functions; the term pernanent is dispensed with; a new entity, alternating aggregates, is introduced; what were formerly called determinants are made a class of these alternating aggregates; and for the name determinant resultant is substituted. The scheme of relationship is thus transformed into f (a) Resultants. '(a) Alternating Aggregate ) Resultants ((A) Alternating l(,8) 1(b) Functions (B) Symmetric (C) Neither scheme, we must at the same time remember, is really as simple as here indicated, being complicated by the fact that a function may be alternating in more than one way. This is brought out much more explicitly and clearly in the present memoirs than in that of 1812, as the following quotations will show. We have first of all (p. 151), an alternating function of several variables. "Une fonction alternee de plusieurs variables x, y, z,..., est celle qui change de signe, en conservant, au signe pres, la meme valeur lorsqu'on 6change deux de ces variables entre elles." Next we have an alternating function with respect to several indices (p. 155):"Quelquefois on represente ces m6mes variables par une seule lettre affectee de divers indices 0, 1, 2, 3,..., n, et l'on peut dire alors que la fonction ou la somme dont il s'agit est altern6e par rapport a ces indices. Ainsi, par exemple, le produit (XO - ) (X- X2) (Z - X2) est une fonction alternee par rapport aux variables xo, X1, x2, ou, ce qui revient au meme, par rapport aux indices 0, 1, 2." DETERMINANTS IN GENERAL (CAUCHY, 1841) 27 279 This example being an, alternating function according to the first definition, it would. Sfem that here we have a mere abbreviation or variation of language. There are, however, it must be borne in mind, functions which are alternating with respect to indices, and are not alternating according to the first definition. For example, any determinant, like a~b2c3 + at.3 bic2 + oi2b3c1 -a 3b2c1 - t2b~c3 -ab C2) is alternating with respect to all the indices involved, but is not* alternating with respect to all or any other number of the variables a,, ca2, a., b1, b2 b3, e 1, c2, c3. Strange to say, Cauchy makes no mention of this, but goes on to a third definition, by means of which alternating functions are made in another way to include determinants. He says (p. 156): "On pourrait obtenir aussi des f onctions qui seraient atterne'es par raport " diverses suites, c'est ' dire, des fonctions qui auraient la proprie'te de changer de signe, en conservant, au signe prt's, la me'me valeur quand on e'chaugerait entre eux les termes correspondants,de. ces m~mes suites. Conside'rons, par exemple, m suites diff6rentes composees chacune de n termes qui se trouvent represente's, pour la premiere suite, par Xe, xi,....,X pour la seconde suite, par YO, y1,.., pour la troisie'me suite, par zo, Si., -1 etc..; et soit f(x0, x1,...., ~ n- Y05 Yv.,y Yn1; z'O -, z.. z..;..n. une fonction donne'e de ces divers termes. Si A cette fonction l'on ajoute toutes celles que l'on peut en de'duire, th laide d'un ou de plusieurs 6'changes ope'res entre les lettres X, y, 2 prises deux 'a deux, chacune des nouvelles fonctions e'tant prise avec le signe + ou avec le signe -, suivant qu'elle se de'duit de la premie're par un nombre pair, ou par un nombre impair d'6cehanges; le re'sultat de cette addition sera une somme alterne'e par rapport aux suites dont il s'agit. " It is a little unfortunate that this definition proceeds on different lines from the others, being rather indeed a rule for the formca 280 HISTORY OF THE THEORY OF DETERMINANTS tion of an alternating function with respect to several sets of variables than a definition of such a function. It would have been much more appropriate and instructive to have said that a function was called alternating with respect to two or more sets of the same number of variables when the interchange of each member of a set with the corresponding member of another set altered the function in sign merely. Examples like the following could then have been given to make the two usages of the term perfectly clear, and to show the exact relation between them. To illustrate the first usage, the expressions ac- be, (a - b)(c-cl), (a - b) (a - c)(b-c), might be taken, where ac-be is an alternating function with respect to the variables a, b; (a-b)(c-d) an alternating function with respect to a, b, and also with respect to c, d; and (a-b)(a-c)(b-c) an alternating function with respect to a, b, with respect to a, c, and with respect to b, c, or shortly, an alternating function of all its variables. On the other hand, the expressions a2b - c2d, a b-cd, would illustrate the second usage; a2b - c2d being an alternating function with respect to the sets of variables ab, cd; and ab - cd an alternating function with respect to the sets ab, cd, and also with respect to the sets ac, bd. In a word, the alteration which produces change of sign is, in the case of the first usage, interchange of two individual elements; in the case of the second usage it is interchange of two ranks or sets of elements. The entity to which the new name somme alternee is given is explained as follows (p. 160):"Soit f(x, y,,... ) une fonction quelconque de n variables X, y, z.... DETERMINANTS IN GENERAL (CAUCHY, 1841) 28 281 et ajoutons 'a cette f onction toutes celles qu'on pent en de'duire par la transposition des variables, on, ce clii revient au na~me, par un ou plusieurs e6changes ope'res chacun entre deux variables seulement, chaque nouvelle fonction e'tant prise avec le signe + ou le signe -, suivant qu'elle se de'duit de la premie're 'a laide d'un nombre pair ou impair de semblables 6'changes. La somme s ainsi obtenue scra la soinme alte'in que nous repre'sentons par la notation S [~f (x, y Z,z. ] On trouvera, par exemple, en supposant it = 2, S -- f (X, y) - f (y, s); en supposant it = 3, s = f(x,y,z) - f(x,z,y) + f(y,z,m) -f(y~x~z) + f(z,x,y) -f(z,y,x), etc." The only matter now remaining for explanation is the mode of transition from sommies alterne'es to re'smltantes,th difcl point being, as in the memoir of 1812, to include all kinds of the latter as special cases of the former. The two pages which Cauchy devotes to the subject are curious to read, and deserve a little attention. He says (p. 161): "Concevons maintenant que la fonction f (x,y,z,... ) se reduise an produit de divers facteurs dont chacun renferme unc suite des variables xi, y, z, en sorte que l'on ait, par exemple, f(X, y, z,... ) = ~ (X) X W(z) (.... alors, pour obtenir la somme altern~e' S = s+W (Kz. il suflira. and having shown the mode of formation, and given the examples $ O (X)X(Y) - 0,)() = k(PWX WbVfr(Z) - o X(x)x Ver W +. he adds " Les sommes de cette espe'ce sont celles que M. Laplace a de'signles. sons le nom de re'ultantes." 282 HISTORY OF THE THEORY OF DETERMINANTS In regard to this the first comment clearly must be that it is not a little misleading. The sums referred to are only a very special class of those functions which Laplace called resultants; they belong, in fact, to that peculiar type for which in later times the name alterrant was coined. In the second place, Cauchy's virtual renunciation of his own word "determinant" must be noted,-a renunciation all the more curious when we consider that the word had now been adopted by Jacobi, and had thereby become the recognised term in Germany. It may be that Laplace's word "resultant" had proved more acceptable in France, and that Cauchy merely bowed to the fact; but there is little or no evidence to support this.* In the paragraph following the above Cauchy proceeds, as it were, to rectify matters. He says (p. 162):"Les formes des fonctions designees par (x), X (), (), etc. etant arbitraires, aussi bien que les variables x, ), y z,. permettent aux divers termes qui composent le tableau (2) d'acquerir des valeurs quelconques, et representons ces variables A l'aide de lettres diverses x, y, z...., t affectes d'indices differents 0, 1, 2,., n-1, dans les diverses lignes verticales. Alors, au lieu du tableau (2), on obtiendra le suivant XO, x1, X21.,.. XYo, YiV Y2..., yn-l (5) Z, Z, 2. - tO, t1, 12..... tnl * Liouville, in a paper published in the same year as Cauchy's memoirs, uses resultant, but adds in a footnote, "Au lieu du mot rdsultante, les geometres emploient souvent le mot determinant" (Liouville's Journ., vi. p. 348). DETERMINANTS IN GENERAL (CAUCHY, 1841) 283 et la resultante s des termes dans ce dernier tableau sera s = S[~ x OY1'2.. t._]-. The general determinant is doubtless here reached, but the transition requisite for the attainment of it, viz., from ((x), X(x), /(x),.... to the perfectly independent x0, xl, x2,.... is not made without considerable strain. This is all the more surprising, too, when we consider, that a much less troublesome and less objectionable mode of bringing determinants under alternating aggregates lay ready to Cauchy's hand. Bearing in mind the definition given above, of fonctions alternees par rapport a diverses suites, we see that a determinant of the nth order could have been made to appear as an alternating function with respect to n ranks of n variables each. For example, the determinant alb2C3 + 3+ a3b,+ 2b3,c - a3b2c - a2blc - alb32, could have been introduced as a function alternating with respect to any two of the three ranks, a1 a2 a3, bl b2 b3, C1 e2 C3; and indeed, as we know, it is alternating also with respect to any two of the ranks a b61 C a2 b2 C2 a b3 C3, that is to say, according to another phrase of Cauchy's, used above, it is alternating with respect to the indices, 1, 2, 3. The fourteen pages (pp. 163-176) which follow, are taken up with the properties of determinants as thus defined and with the application of them to the solution of simultaneous linear equations. Most of the matter is already familiar to us, and may be altogether passed over. One of the theorems it is necessary to give verbatim, not because of its importance, but 284 HISTORY OF THE THEORY OF DETERMINANTS because it serves to make evident the untenable position Cauchy had taken up in so peculiarly bringing determinants under the head of alternating aggregates. The theorem is (p. 164):"Si, avec les variables comprises dans le tableau (5), on forme une fonction entiere, du degre n, qui offre, dans chaque terme, n facteurs dont un seul appartienne a chacune des suites horizontales de ce tableau, et qui soit alternee par rapport a ces memes suites, la fonction enti6re dont il s'agit devra se reduire, au signe pres, A la resultante s." This not only justifies the definition proposed above to be substituted for Cauchy's, but it also entitles us to say that Cauchy having started by including determinants among alternating functions of one kind, viz., functions alternating with respect to every pair of n variables, soon succeeds in showing that they are alternating functions of an entirely different kind, viz., functions alternating with respect to every pair of n ranks of variables. The only other noteworthy matter is a theorem in regard to the solution of a set of simultaneous equations. Viewing the equations aix + bly + c z = } a2x + b2y + c2z = ax + bsy + cs3 = as giving each of the three variables,,,, in terms of the other three x, y, z, we see that on solving for x, y, z, we obtain a converse system, that is to say, a system giving each of the three x, y, z, in terms of g, ], a. The latter system is, as we know, A A A2 A = A + Aq + A, B1 B2 +B3 = gs+ ~-+ P, C C2 C3 where A is the determinant of the original system. and Al, B1, C1, A2,.., are the cofactors in A of b. bl, cr, a2,...., respectively. Multi DETERMINANTS IN GENERAL (CAUCHY, 1841) 285 plying the determinants of the two systems, we obtain the determinant of the qnantities 0 0 1. Hence (p. 176): "Si, n variables e'tant lie'es 'a n autres variables par n equations line'aires, on suppose les unes, exprim~es en fonctions lin'aires des autres, et re'ciproquement; les deux re'sultantes forme'es avec les coefficients que renfeimeront ces fonctions line'aires dans les deux hypotheses, offriront un produit equivalent 'a l'unite'." (xxi. 4) CHAPTER X. DETERMINANTS IN GENERAL, FROM 1813 TO 1841: A RETROSPECT. THE characteristics of this period are best brought out by comparison with those of the preceding period, it being carefully borne in mind, in making the comparison, that the two are markedly unequal in length, the period of pioneering, as we may term it, extending to 120 years, and the next to only about 30. In the first place, then, the evidence shows that as time went on there was considerable increase of interest in the subject, and a more widely spread knowledge of it; for, whereas to the longer period there belong 21 papers by 16 writers, for the shorter period the corresponding numbers are 38 and 19. Among the 19 writers, too, are represented nationalities which had previously not put in an appearance, viz., English, Italian, and Polish. In both periods the French language greatly predominates in the writings, even although in the second period the number of German contributors is about equal to the number of French. The details on this point are:(1693-1812) (1813-1841) French, - - - - 16 17 Latin, - - - - 3 9 German, - - - - 2 6 English, - - - 5 Italian, - - - - - 1 The Latin papers are mainly those of Germans, Jacobi alone being responsible for 8 in the later period. DETERMINANTS IN GENERAL 287 In the second place, we have proof that the early period was by far the more fruitful in original results. The pioneers had mapped out most of the prominent features of the new country; their successors had consequently to concern themselves in a considerable degree with filling in the details. During the second period one finds the fundamental propositions of the first period reproduced in new varieties of form; also, there are not awanting new proofs, extensions, and specialisations of old theorems; but of absolutely fresh departures there are comparatively few. An examination of the results numbered XLIV.LIX. will show the character of these departures. It will be seen that they are due to Desnanot, Scherk, Schweins, Jacobi, Sylvester, and Cauchy. The most notable name of the period is Jacobi's, and next to it perhaps that of Schweins. There is no one name, however, which stands out in this period so conspicuously as Cauchy's does in the first period. Sylvester, unlike the others, it must be remembered, was only beginning his career, and we have yet to see him,in the fulness of his power. It is worthy of note, too, that the striking figure of the first period is not by any means dwarfed in the second, his name occurring five times in the chronological list, and his papers at the close of the period showing much of his old insight and vigour. In the next place, the second period contrasts with the first in that during it important work was done on the subject of special forms of determinants. This will become more apparent after consideration of the chapters which follow. It will then be seen that of the five most important forms there dealt with, viz., those subsequently known as Axisymmetric Determinants, Alternants, Jacobians, Skew Determinants, Orthogonants, three had their origin during the second period; and, further, that although the two others originated during the first period the greater bulk of the work done on them belongs to the second. Here, again, the noteworthy names are those of Jacobi, Cauchy, and Schweins. Lastly, it having been noted in the retrospect of the first period that the subject of determinants was almost entirely a creation of the French intellect, we must not fail to take 288 HISTORY OF THE THEORY OF DETERMINANTS cognisance now of the fact that in the second period the preeminence belongs to Germany, France however taking still a fairly good second place. To aid in bringing all these facts more clearly home to the reader a table similar to that supplied for the elucidation of the first period (see page 132) is annexed. The cautions formerly given as regards the imperfections of such a table and the care consequently necessary in using it are expected to be again borne in mind. TABLE- SHOWING THE ADVANCE OF THE THEORY OF DETERMINANT ) FROM 1813 TO 1841. EFacizng pcage 288. 0) 0, a) Co IL II. II. VII. Vill. IX. VIII. XI. XII. XIIL. XSIH. XIV. XV. XVII. XVIII. XX. XXIC. XXIII. XXXVII. XL. 1-34-5 135 re )fs I._ 1 1-1 136 oi CTo 0t 03 1-1 St (U 2 a) wa C) i 1o 1:9 GO r-1 I u c) v )6 ^~~~ cq r-1q a-, a 9 X *cC o " ~R = C.) C) U2 2 2 I 145 143 148 140, 142, 145 139, 140, 142, 145 145 I 156, 158 159 151 162 160 162 161 -165 t164 160 1813. Gergonne, 1814. Garnier, 1815. WNronski, 1819. Desnanot, 171 162 171 154 154 158 167, 169, 170(2), 171 174 cza 1-z Id 178.Z pti a)s 9a 179 179 182(2) 184, 185 185 "I o c3 0 188 -C( 0. - To To ca ys a; - 199 199 CO co 1-I 0 i" h-z 212 212 208 211 1-1 Q c) 213 213 To c) i-s 214 214 ore; L0 00 00 I cO 00 r —q I 218, 219 216 IL-~ I CO i i l o CO I Co 00 I GO I r- TrI i a) I a O Ode d;9 197 197 204 201 220 220(2) 221 222 225 231 228 233 236 z -4 — I k pa a)S 0 i w 9)l 122 6 rr o, GO I _-q -)w V 02 XLIV. XLV. 1821. CauchTy, 1825. Scherk, XLVI. XLVII. XLVIII. 199 199 223 0 Q C.) 239 261 271(2) 270, 271 268 264, 265 2.56 262 I p) cd v ot T-l 00 l — 1) 0) 02 4 z ICS F-l U = t-So 250 249(2, 256, 257, 258(2), 260 276(2), 277 261.0 0 Co 285 v0 C-a Schweins, 1827. Jacobi, 1829. Reiss, Cauchy, Jacobi, XLIX. L. I L. 209I 2091, LII. s189, 192, 193 MBinding, 1831. Drinkwater, 1832. ainarcdi, I 183i1. Jac 0i, Ua~- I, ),i. t.,''. i LIII. 1835. Tacobi, -- 1.836. /:rniert, 1 837. Lehesgue,.1838. Reiss, 1839. Oatalan, Sylvester, 1840. Sylvester, LIV. ___J l 213 237, 238 242 244 Richllelot, Cauchy, 184 -.Sylvester. 'raulf3rdii, I I I ~1 _~12 '-auceh. L II. Tacobi, LVI. LVII. LVIII. LIX. 249, 250(3), 2521 262 263 265 272 -auchy, I CHIAPTER XI. AXISYMMETRIC DETERMINANTS, FROM 1773 TO 1841. ATTENTION has already been drawn to certain identities of Lagrange's which might possibly be viewed as contributions to the theory of determinants. Among these were the following published in 1773:(xy'z" + yz'x" + zx'y" - xz'y yIz" _ zy"I/)2 = (x2 + y2 + 72) (x2 + y,2 + z'2) (x"2 + y"2 + z12) + 2(xx'+ yy'+ zz')(xx + yy" +zz")(x'x"+ y'y"+ '") - (x2 + y2 + z2)(XIX'" + yy + z/z-)2 - (x'2 + y2 + /2) (xx" + yy" + zz')2 - (x2 y2 + 1 + 2) (xx' + yy' + zz')2; (yX'/- y"')2 + (z'IXII _ z")2 + (x'y"-I X")2 = (X12 + y'2 '+z'2) (x2 +y"2 +"2) - ('x" + Yy'y + z'z")2; and (pr- q2)(Mn - NN)2 = (pM2 + 2qM~n + rm2)(pN2 + 2qNn + 2) - (MN + qMn + qNqn + rmn)2. Four of the expressions here occurring would doubtless at a later date have been viewed as axisymmetric determinants, and in Cayley's notation of 1841 would have been written x2 + X + y2 + z x' + ' + ' " + yy" + ' + yy + + zz' x'2 + y2 + z'2 x'x"+ y'y" + z'z" xx" + yy"+ zz" x'x"+ y'y"+ z'z" x"2 + y12 + z"2, etc.; but a reference to the original papers, already described, will M.D. T 290 HISTORY OF THE THEORY OF DETERMINANTS make it almost perfectly certain that Lagrange did not view them in this light. The like is true of Gauss (1801) who discovered the next case of the third of the preceding identities. ROTHE (1800). [Ueber Permutationen, in Beziehung auf die Stellen ihrer Elemente. Anwendung der daraus abgeleiteten Sitzen auf das Eliminations-problem. Sammlung combinatorischnacaytischer Abhanrdllngen, herausg. v. C. F. Hindenburg, ii. pp. 263-305.] The position of Rothe was quite different from that of Lagrange and Gauss, as his paper dealt explicitly with determinants (or, rather, with the functions afterwards known as determinants), and the case of axisymmetry is definitely referred to, although not by name. His one theorem may be illustrated by the case where the number of given equations is 4, and is then to the effect that if we have ax1 + bx2 + cx3 + tdx4 = s bx1 + ex2 + fx3 + gx4 = s2 cx1 + fs2 + hx + iX4 = 83= dxl g2 + s + i3 + jx4 = 84, where the array of coefficients on the left is axisymmetric, then the same peculiarity of axisymmetry must make its appearance in the derived set which gives each of the x's in terms of the four s's. Starting with the more general set of n equations 11.x, + 12.x2 +... + l.x,= s 21. x + 22.xs +... + 2n.x, = s2 nl.x 1. -2.x 2+.. -.. s, and denoting the determinant formed from the coefficients on the left by N, and the cofactor in N of any coefficient pq by fpq, he proves in Laplace's method that AXISYMMETRIC DETERMINANTS (ROTHE, 1800) 291 fll.s1 + f21.s2 +....+ fnl.s, = N. f12.s1 + f22.s2 +....+ f2.s, = N. 'l1.s2 + f2.s +....+ fnn.s = N.xnJ where, be it observed, the coefficients of s1 are not the cofactors of the coefficients xi in the original set of equations but the cofactors of the coefficients of xl, x2,..., xn in the first equation of that set: in other words, the first column of coefficients in the derived set of equations corresponds to the first row of coefficients in the original set. Then taking another set of n equations having the same coefficients 11, 12,.... differently disposed, viz., 11.y1 + 21.y2 +.... + 2l.yn = v 12.y + 2n.y2 +.... + nn.y= vn ln.yF + 2n.y9 +.... nn.yF = v, but where of course the determinant of the coefficients is in substance the same as before, and therefore denotable by N, and where consequently the cofactors of the elements of which the determinant is composed are also the same as before, he proves, rather unnecessarily, that fll.vl + f12.v2 +.... + fln.vn = N.y f21.vl + f22.v2 +....+ f2n.v, = N.y2 fnl.v1 + fn2.v2..+. fnn.vn = N.yJ In this way it is made to appear that the coefficients of the one set of derived equations are the same as the coefficients of the other set of derived equations, the difference in the arrangement of them being exactly the difference observable in regard to the primitive sets. From this he passes to the case where the array of coefficients of the primitive set of equations possesses the property of axisymmetry, his words being (p. 301) 292 HISTORY OF THE THEORY OF DETERMINANTS II1st endlich f dr j edes _p und q, pq = qpq, oder ist bey den gegebenen 0-leichungen, ffur jedes rn, die mte ilorizontaireihe der Coeflicienten mit der mten Verticairefihe derselben einerley; die Horizontairefihen nelimlich von oben herab, und die Verticairefihen, von der Linken nach der Rechten zu gerechnet, so ist auch aligemein fpq =fqjp, oder die mte Horizontalreihe der Coefficienten, mit der mnten Verticairefihe derselben, auch bey den Aufltisungsgleichungen einerley." It may be noticed in passing that as the determiinant of the coefficients in the derived set of equations is the conjugate of the. adjugate of 'the determinant of the original set, there is involved in Rothe's proposition the well-known proposition of later times, viz., that the acljugctte, of an axis ymmetric determinant is also axis ymmetriw. BINET (1 81 1). [Me'moire sur la the'orie des axes conjugue's.... Jomrn. de I Ecole Polytechniq~ue, ix. (pp. 41-67), pp. 45, 46.] [Sur quelques formules d'alge'bre, et sur leur application 'a des expressions qui ont rapport aux axes conjugues des corps. Nouv. Bnul. des Sciences par la Soci~e`t Philomatique, ii. pp. 389-392.1 With Binet we have a recurrence to those axisymmetric determinants which appear as equivalents to second powers of determinants or to sums of second powers. His theorems mx2 +,nmX12 + m x 2 +..MXy + M~xy + mqnyx+. m + m~xz + mZ~ z mxy + 9)~xIYI + MAxY2 +.. m2 + 9nly1 + 71~A2 +..MyZ +?1,yl-1 + rn2y2Z2 + mmZ + m1x1Z1 + m~2X2Z2 +..my.+ MOyIZ1 + MAy~z2 +.. 12f + 91i 1Z12 + nfl2Z2 2+ x 1 x2 x 1 x3 =mmIm2 Y 1 Y,12 + mmim3 Y1 I/iIf +.; Z Z1 Z2 0 1 03 g ~~~~hi g mIX2 + MJX12 +..mxy + MAx yi +..MXZ + mAxz1 ~ h mxy +m mx Jy1 + M. my2l+ m12 +..myz_~ Mlylz1 + imxz ~ mixiz1I +..My_ -Pm1yJz1 +..mz2 ~ mlz12 + AXISYMMETRIC DETERMINANTS (BINET, 1811) 293 g x xi 2 g X X2 2 =MM h yy1V + mm2 hy Y2 + 0 1 0 2 g X1X 2 M02 0j 02; m2 +'t1 2 +. 4u+Th1Ql +. %UZ +Th10I +.. r 2 + xi2 +. xzy+xZy1+.. cay +xly1+.. y2 ~ y12 ~ i/O-, + Ylzl + xmo + calo1 +. Ylf0~lf-Oi + o2 ~zi 2 ~. Y Y I Y2 2/3 0 zi 02 03 1% U1 U' I 2 + c x1 x2 4 2/ Yi 12 1/4 0 01 02 04 ~ These all indicate most important advances, and, be it noted, the last of them is given by its author as the third of a series the law of which he considered " facile 'a saisir." In view of this, and the fact that in the following year he published his great memoir containing the multiplication-theorem in all its generality, we are bound to conclude that whatever credit in the latter he must share with Cauchy, the axisymmetric case of it is entirely his own. Further details need not be given, as this has already been done when dealing with determinants in general. JACOBI (1827). [Ueber die Hauptaxen der Flachen der zweiten Ordnunug. Orelle's Jov~rnal, ii. pp. 227-233; or Werke, iii. pp. 45-53.] [De singulari quadam duplicis integralis transformatione. Cretle's Journal, ii. pp. 234-242; or TWerke, iii. pp. 55-66.] In these two papers, which owe their inspiration to the famous memoir * of Gauss on the " Determinatio Attractionis... Jacobi concerns himself with two problems of transformation, * Commentationes societcatis regice scientiarum Gottingensis recentiores, iv. (1818): or Gauss, Werke, iii. pp. 331-355. For abstract see Gtitingische gelehrte, Anzeigeni (1818, Feb.), pp. 233-237: or Werke, iii. pp. 357-360. 294 HISTORY OF THE THEORY OF DETERMINANTS the first of which explicitly deals with the transformation of the ternary quadric AX2 + By2 + CZ2 + 2ayz + 2bzx + 2cxy into the form LC2 + M;7 + NT2, and the other implicitly with the corresponding change in the case of a quateruary quadric. The papers will be fully discussed when we come to deal with "determinants of an orthogonal substitution." It suffices for the present to note that in the first Jacobi virtually gives as an equivalent for the axisymmetric determinant which we should now write in the form x-A X Cos V -c x cosu - b Xcos p-C x-B x cos X-a x cos -b x cosX-a x-C the expansion (x —A)(x —B) (x —C) - (x —A) (xcosX-a I)2 - (X- B) (x cos j - b)2 - (X - C)(X COS v - C)2 +2(xcosX-a)(xcosu-b)(xcosv- and in the second paper for the axisymmetric determinant a-x b' b" b"' bl ca'/+x C// C/ b" c'" a/"+ X C' b/// CII a/"' + x the expansion (a - x) (a+ x) (a" + x)(a"' + x) (a - x) (a'+X +X)c2. - (a" + iX)(a"'/ + v)b'2 -(a - x)(a"+ X)C1"2 - (a"' + x)(d' +x)b"2 - (a-x)(a "'+x)c"'2 - (a' +x)(a" +X) b"'2 + 2c'c"c"' (a - x) + 2c'b"b"'(a' a+ x) + 2c"b"'b' (a" + x) + 2c"'b'b"(a1"' + ) + b'/2'2 + b"/2c"2 + b"'2c'/12 - 2blblCC - 2b Wb"'CIC"I - 2b"ble -that is to say, the expansion arranged according to products of elements of the principal diagonal. A clause of the paper refers to the writings of Laplace, Yandermonde, Gauss, and Binet. AXISYMMETRIC DETERMINANTS (CAUCHY, 1829) 295 CAUCHY (1829). [Sur l'equation ' laide de laquelle on determine les inegalites seculaires des mouvements des plane'tes. Exercices de Math., iv. pp. 140-160; or (~Emvres completes, 20 s6r. ix. pp. 172-195.] The equation which Cauchy refers to in his title is exactly the equation with which we have just seen Jacobi occupied. Cauchy, however, comes upon it from a different direction, and it is no longer with him a cubic or quartic, but an nthic. The problem he sets out to, solve is the finding of the maxima and minima of what we should nowadays call an n-ary quadric, viz., A,,X 2 ~ A~yy2 + A zZ2 +... ~ 2Axyxy + 2Azxz +. subject to the condition that the sum of the squares of the n variables x, y, z,... equals 1. In a few lines it is ascertained that the equation in s, S=O say, whose roots are the extreme values in question, is obtainable on eliminating x, y, z,... from -the set of n equations (A,,-s)$X + Ayy + + =0 AyXx + (Ay-s)y + Ay+.... =Z0 A1,,x + Azyy + (A,,-s)z + 0 = 0 -where =Axy, R.Remembering Cauchy's great paper of 1812, we are quite prepared to find him at this stage proceeding -to say:"S sera une fonetion alternee des quantites comprises dans le Tableau A - A1 - S A... A,, A1, A,, -s;savoir celle dont les diff~rents termes sont represent6es, aux signes pres, par les produits qu'on obtient, lorsqu'on multiplie ces quantites, n 'a n, de toutes les manieres possibles, en ayant soin de faire entrer dans eichaque produit un facteur pris dans ehacune des lignes horizontales du:Tableau et un facteur pris dans chacune des lignes verticales." 296 HISTORY OF THE THEORY OF DETERMINANTS The lengthy discussion of the character of the roots of S=0 which thereupon follows, and in which the properties of "fonctions alternees" are freely used, belongs almost entirely to a different portion of our subject: for the present there concerns us only one theorem subsidiary to the said discussion. In modern phraseology this lemma is-S being any axisymmetric determinant, R the determinant got by deleting the first row and first column of S, Y the determinant got by deleting the first row and second column of S, and Q the determinant got from R as R from S, then if R=0, SQ= - Y2. The mode adopted for testing the truth of this is applicable to any determinant S, whether axisymmetric or not; and when the second condition, viz., the vanishing of R, is also removed, there emerges the simplest case of Jacobi's theorem of 1833 regarding a minor of the adjugate. JACOBI (1831 Dec.). [De transformatione integralis duplicis indefiniti A + B cos + C sin ~ + (A'+ B' cos, + C' sin ~p) cos ~ + (A" + B" cos ~ + C" sin b) sin in formam simpliciorem aJ-ainsimli G - G' cos 1 cos 0 - G" sin V sin ' Crelle's Journal, viii. pp. 253-279, 321-357; or Werke, iii. pp. 91-158.] As the algebraical transformation effected in this paper is an extension of that dealt with in Jacobi's second paper of 1827, it is only what might have been expected to find expressions contained in it which may be viewed as axisymmetric determinants. Such expressions are two forms of the square of A(B'C"/- BC') + B(C'A" - C"A') + C(A'B"- A'B'), or A, and the non-zero side of the cubic equation therewith connected. upon which the whole investigation depends, viz., X3 - X2{A2 + B2 + C2 + A'2 + B'2 + C'2 + A"2 + B"2 + C/2} + x {(B'C"-B"C')2 +.. } - {A(B'C/ - B"C') + B(C'A" - C"A') + C(A'B" - AB')}2 AXISYMMETRIC DETERMINANTS (JACOBI, 1831) 297 No hint, however, is given of these expressions being determinants,-a fact which is all the more noteworthy in view of the reference made in the second paper of 1827 to the writings of Laplace, Vandermonde,..., and in view of the reference made on p. 350 of his present paper to Cauchy's of 1829, where, as we have just seen, "fonctions alternees" are explicitly used throughout. As a mere aid to the memory it would appear to have been worth while to note that if one of the said squares of A be the determinant formed from ~ 9/' m' n' 'n in' r/' I' n the non-zero side of the fundamental cubic is the determinant formed from x-I X,' m' n' x - I' ' x - n-, and that the coefficient of -x~ in the cubic is the square of A, the coefficient of x1 the sum of the squares of what came afterwards to be called the "primary minors" of A, and the coefficient of x2 the sum of the squares of the secondary minors. JACOBI (1832). [De transformatione et determinatione integralium duplicium commentatio tertia. Crelle's Journal, x. pp. 101-128; or Werke, iii. pp. 159-189.] This last paper of the three dealing with the transformation of integrals contains less regarding our present subject than either of the others. The only thing worth noting is the curious cubic equation 3{ abc - ad2 - be2 - cf + 2def } _x2 a'(bc-d2) + b'(ca-e2) + c' (ab-f2) + 2d'(ef-cad) + 2e'(fd-be) + 2f(de-cf) a(b'c'-d"2) + b(c'a'-ced') + ' c(a'b'_f2) t+ + 2d(e'f - a'd') + 2e(fd'- b'e') + 2f(d'e' -c'f') - { a'b'c' - ad' - b'e'2 - c'f'2 + 2d'e'f } = 0, 298 HISTORY OF THE THEORY OF DETERMINANTS where the first and last coefficients are in modern notation a f e a' f/ e f b d f b' d e d c, e' ' c' the second coefficient from the beginning is a' f' e' a f e a f e f b d + f' d' + f b d e d c e d c e' d' c' or Aa' + Bb' + Cc' + 2Dd' + 2Ee' + 2Ff'; and the second from the end a f e a' f' e' a' f' e' f b' d' + f b d + f' d' e' d' c' e' d' c' e d c or A'a + B'b + C'c + 2D'd + 2E'e + 2F'f, or A'a F + + E'e + F'f + B'b + D'd + E'e + D'd + C'c JACOBI (1833). [De binis quibuslibet functionibus homogeneis secundi ordinis per substitutiones lineares in alias binas transformandis, quae solis quadratis variabilium constant; una cum..... Crelle's Jourral, xii. pp. 1-69; or Werke, iii. pp. 191-268.] As in this great memoir Jacobi sums up and generalises the results of his papers of 1827, 1831, 1832, in which, as we have seen, axisymmetric determinants were implicitly made use of, it is at first somewhat surprising to find very little reference to properties of determinants of this special form. The reason, however, doubtless is that when he came to extend his theorems from the third or fourth order to the nth, he also withdrew the restriction as to axisymmetry and gave the results in quite general form. In support of this the fifth and sixth sections AXISYMMETRIC DETERMINANTS (JACOBI, 1833) 299 (pp. 8-1 1) may be ref erred to,-sections which on account of being concerned with determinants in general have already been dealt with in the proper place. Even when he comes, as before in the particular cases, to his equation for determining the coefficients of the squares of the new variables, that is, the equation F = O where F is described as the expression got from -4~ a11a22...an by changing all, 22... into a1 - x, a22 - x,.. he gives an expansion of F according to ascending powers of x~, which holds whether aKx = ax, or not. The passage is" Quod attinet ipsam ipsius F formationem, observo, si signo summatorio S amplectamur expressiones inter se, diversas, quae permutatis indicibus 1, 2, 3,..., n proveniunt, fieri: xI' = +a11a22... an-, n, + X2Sj ~ all a22.. afl-2, n-2 ~ ~S~11 a22:F xn"1SE~ al1 ~ cc-. Qua in formula, expressio SE a1122. a,2,, designat suimmam n (n -1)..(n - m+1) pesinm.1. 2... m exrsoum quie e 1~ala2.. am proveniunt, Si -in a11 a22...am. loco indicum priorum simul ac posteriorum 1, 2,..,m scribimus,omnibus modis, quibus fleri potest, m alios e numeris 1, 2, 3,... n" There are, however, two minor instances in which it is the special determinant that is alone concerned. The first occurs -after proving (p. 13, footnote) the theorem (see Rothe's paper of 1800) that if the solution of aixxl + akX2 +... + an Xn UX (X=l, 2,.. n) be 300 HISTORY OF THE THEORY OF DETERMINANTS then the solution of axlYi + cX2~Y2 +. + axn'Yn (NVx1, 2,...,I must be yK 1 4~a,,a22... ann, = blKVl+b2KV2~ *. + bjjKv (K l, 2,..,n when he adds the corollary that if atKX = a~k, then also I)X bXK. The second occurs quite similarly when, having poiuted out, (p. 20) that The coefficients bx in either solution are expressible as differential-quotients of I ~i a1la22... ann,, he adds the sentence, "Quoties aKX1 aX K differentialis semisse tantuin surni debet Si K et X diversi sunt." JACOBI (1834). L Dato systemate n aequationuln linearinni inter n incognitas,, valores incognitarum per integralia definita (n - )tuplicia exhibentur. Crelle's Jourual, xiv. pp. 51-55; or Wer-ke, vi.. pp. 79-85.] Jacobi having already pointed out in his long memoir of thepreceding year that the cofactor of a~ in al~a a22.. ann, or N say, is and having now to deal with the case where aX = aAK, draws attention again to the fact that in solving the equations a,, yj + a12 Y2 ~ + + a1n y, = inl a21 Y, 1+22Y2 +. +a2n yn= i21 an 1+ '9 '2y2 +....+ O'n =n ma,J we no longer obtain N (Na (DN 'a(DNT\ NYi \aMl +1 + + M\D n but Nyl = (~Ml + MD~ ~ D \Da11/1 kaa 2 + +2 anMn AXISYMMETRIC DETERMINANTS (JACOBI, 1834) 301 -his explanation being that the differential-quotient of N with respect to a,x, where K and X are unequal, is obtained by first viewing acy and,a^c as being different, adding together the differential-quotient with respect to ac, and the differentialquotient with respect to a,, and then putting aK, = aK. His own words are" Si vero aKx = aCX differentiale partiale secundum aKX sumtum, quoties non K ==, obtinetur, si primunm cta et ax^ diverse statuuntur, atque differentialia partialia secundum aKx et secundum aXK sumta itnguntur, ac deinde aK^ =a c statuitur: quo facto cur utraque differentialia sequalia fiant, casu quo aKX = a^K valor duplus emergit eius qui in formulis (3) locum habere debet." LEBESGUE (1837). [Theses de Mecanique et d'Astronomie. Premiere Partie: Formules pour la transformation des fonctions homogenes du second degre a plusieurs inconnues. Jokur'. (de LioLville) de Math., ii. pp. 337-355.] Lebesgue's subject is exactly that dealt with in the first part of Jacobi's memoir of 1833, viz., the transformation of a general homogeneous function of the second degree into one containing only squares of the variables. Indebtedness to Jacobi, Cauchy, and Sturm is indirectly intimated at the outset, and the paper is modestly offered as being new in manner rather than in matter. Like Cauchy and Jacobi, the author of course is led to the set of equations from which by elimination there is deduced the equation for the determination of the coefficients of the new variables; and recognising that "le premier membre de cette equation n'est qu'une de ces fonctions nommees determinants," he devotes his second section of five pages to the properties of these functions. Throughout this section prominence is notably given to determinants having the elements Aap, Ago equal; and such determinants are spoken of as "symetriques,"-a noteworthy fact, since up to this time no separate name had been applied to any specific form. " On pent dire alors," Lebesgue says, "que le systeme est symetrique, puisque les nombres qui le forment sont places symetriquement par rapport aux nombres a indices egaux All, A22,...., A,,,, qui forme la diagonale du systeme." 302 HISTORY OF THE THEORY OF DETERMINANTS The first proposition is that in a symmetric determinant [g, i] = [i, g], where [g, i] is used to denote the determinant got from the original determinant D by suppressing the gth row and i" column. The second is that"Pour tout determinant nul on a [g, g]. [i, i] = [, g]. [g, i] et par consequent pour un determinant a la fois nul et symetrique [g, gA. [i, i] = [i, g2 = [g, i]2. This is proved independently, but, of course, it is nowadays best viewed as a special case of Jacobi's theorem (1833) regarding a minor of the adjugate. The third and fourth propositions combined are to the effect that in every perfectly general determinant dD d7A i,= while in a symmetric determinant dD dD d g g], = (-1 )* 2 [i, g]. A proof of the last of these is given,* the starting-point being the identity D = An,, [nn,n] - A+,,_ [,'n - 1] A,_2 [n,n - 2] -.. where D is expressed in terms of the elements of the last row and their cofactors. By differentiating both sides of this with respect to the particular non-diagonal element A,_x there is obtained dD [= - 0 [rcl [n,n - l } + An2 d[,n- 2] dA~0. 10c[ n,n-1 ]An dA,_2 A An,n -l n' ' nn-I The differentiands on the right of this, viz. [n,n- 1], [n,- 2],... although not involving A,,,_ do involve A,_,,, which is the same as Ani_: consequently their differential coefficients are other than zero and have to be found,-that is, we have to find * There are several misprints in the original, and the paging of the volume is hereabouts all wrong. AXISYMMETRIC DETERMINANTS (LEBESGUE, 1837) 303 d[n, i] d -- ni where i < n. dAn-,n Expanding [n, i] after the manner of D above, but now in terms of the elements of the last column, we obtain,An1,n,2fJ 3- * -1, n - -2, ' 2 -3,n - "'" and therefore, since the second factors on the right do not contain A,_,, or A,,_i (both the nth row and fath column being gone in all of them), there results d[,i] _ r n,i dAd_1, nL- l, ' Substituting this above we see that dD A rn,1n-l+ A r2n,n-2~ dA_ -=[n^n -1] - A..1 Lv' - l ]_ + Ab,- 2 [, n = - [n,n-] - - [n,n-1], =- -2[n,n-]. The theorem having thus been proved for the case of the suffixes (n -1, n), the passage to the case of any unequal suffixes is made by saying "Par un deplacement de series horizontales et de series verticales, on trouvera dD gdjq= (- 1)+^ 2[i, g] comme il est dit dans 1'enonce." Save for a page in which the development of a symmetric determinant for the cases z= 2, 3, 4 is given, the rest of the paper is taken up with the concluding portion of the solution of the problem of transformation. It may be well to note, however,, that on the page referred to (p. 347) the determinant of the system A-11 A12.. A1, A12 A22 -. ~ ~ A21, Ai, A2n... Ann -- is denoted by det. [All-u, A.22-Z,..., A - ~. :304 HISTORY OF THE THEORY OF DETERMINANTS JACOBI (1841). [De formatione et proprietatibus Determinantium. CreUle's Jo crnal, xxii. pp, 285-318; or Werke, iii. pp. 355-392; or Stackel's translation 'Die Bildung und die Eigenschaften der Determinanten,' 73 pp., Leipzig, 1896.] As already noted (see above p. 271) Jacobi formally enunciated in this his great memoir Binet's case of the multiplicationtheorem when the product-determinant is axisymmetric. CAUCHY (1841). [Note sur la formation des fonctions alternees qui servent a resoudre le probleme de l'dlimination. Comptes Rendus... Paris, xii. pp. 414-426; or twEovres completes, ire ser. vi. pp. 87-99.] The early part of this paper, in which the finding of the terms of a general determinant (" fonction alternee") is made dependent on a study of the properties of " groups," or index-cycles as they would more appropriately be called, has already been described. The nature of it will be readily recalled from the mode of writing the expansion of the determinant of the 4th order, viz., alCa22a33a44 - z a611c22a34a43 + - cl'.a23a34a42 + a12221a34~a43 - a12a23aC34a41, where under the last E are included all terms (6 in number) whose indices form one quaternary cycle, under the preceding E all terms (3 in number) whose indices form two binary cycles, and so on. On coming to consider a determinant in which ai=a ji, Cauchy points out that because of this peculiarity every term will be found repeated unless those whose index-cycles are all lower than ternary: for example, in the case of the determinant of the 4th order, the six terms having a quaternary index-cycle are condensed into three with the coefficient 2 prefixed, and the eight terms having a ternary index-cycle into four with the same coefficient, the whole result being AXISYMMETRIC DETERMINANTS (CAUCHY, 1841) 305 a1a22aC33a44 - a11a22ct34 + 2 - ~ L1a23a34(124 l+ 2 a13a24 - 2 2 a12a23a34a14. The definite theorem reached by him on this point may be formulated in later phraseology as follows:If v3, '4,...be the number of ternary, quaternary, and higher index-cycles in any term of an axisymmetric determinant, the coefficient of the term when condensation takes place is 2v83+4+ ~ ~. By way of proof it is stated that when we have got a term with index-cycles higher than binary, we may, by reversing the order of the indices in one of the said cycles, obtain another term of the development, and that this will be equal to the former. For,example, if a term have the quaternary cycle (1, 2, 3, 4), another term is obtainable by simply changing this into (4, 3, 2, 1), the,effect on the original term being to change it from.... Cts2aC23aC34a1... into.... a~aaai~a^.... *. * 43a32ct21 14 which, in the circumstances, is equivalent to no substantial change at all. "Pour fixer les iddes" he takes the case of the 6th order, giving the following as the development of what we should nowadays denote by Iacla22a33a44a655a66 Is=sr, viz., a22a3344a566 - 22a33a44a2 +- 2 222 2 ^\1a2aa a 6656 2 22a'a - 56 2a34 56 2al~a a~a~a~a^ -- 2 I a 4~a 2a c4 q- 4 2] a 4,a6a6a4a~ — 2 a11ac22a3aa4aa63a + 2 N a22aCC34C4,C66a63 + 2 Ca11C23a34a45a66a62 - 2 Z a12a23a34aC45Ca56ac 6 where it will be seen that the first four types of terms correspond to the following partitions of 6, viz., 1,1,1,1,1,1 1,1,1,1,2 11,2,2 2, 22,2 and the remaining types to the remaining partitions, 1,1,1,3 1,2,3 3,3 1,1,4 2,4 1,5 6. M.D. U CHAPTER XII. ALTERNANTS FROM THE YEAR 1771 TO 1841. THE first traces of the special functions now known as alternating functions are said by Cauchy to be discernible in certain work of Vandermonde's; and if we view the functions as originating in the study of the number of values which a, function can assume through permutation of its variables,* such an early date may in a certain sense be justifiable. To all intents and purposes, however, the theory is a creation of Cauchy's, and it is almost absolutely certain that its connection with determinants was never thought of until his time. PRONY (1795). [Leqons d'analyse. Considerations sur les principes de la; methode inverse des differences. Journ. de l'Ec. Polyt., i. (pp. 211-273) pp. 264, 265.] In the course of his investigations Prony comes upon a set of equations i(1+ P + 2 ++ n =- O pl\1 + p2M2 + + PnMn = }l P IIU+ p22 +. + p/2'w = Z_ _ Pn1 il + Pn 12 + -,+ pJ_ = / — *The history of this subject is referred to in Serret, M. J.-A.: "Sur le nombre de valeurs qui peut prendre une fonction quand on y permute les lettres qu'elle renferme," Journ. (de Liouville) de Math., xv. pp. 1-70 (1849). ALTERNANTS (PRONY, 1795) 307 where the coefficients of each unknown are the 0th, 1st, 2nd, &c., powers of one and the same quantity, and where, therefore, the determinant of the set is that special form long afterwards known as the simplest form of alternant. The full solution is given for the first four cases, but without any indication of the method employed. Thus for four variables the results appear in the form =- P2P3P4z0 + (P2P3+ P.2P4+ PP4)z - (P2+P3+P4)z2 + Z3 1(P1- P2)(P1- P3)(P- P4) - PiP3P4Z + (P1P3+P1P4+P3P4)z1 - (PI+P3+P4)z + Z3 rCA~2 (P2- P1)(P2- P3)(P2- P4) M4 =...... and the writer then adds:"En general, quelque soit le nombre n, pour avoir le numerateur de la fraction qui donne la constante s/, il faut prendre toutes les racines, excepte la racine p,, et des n - 1 racines restantes, en trouver le produit total, la somme des produits n -2 a n- 2, n-3 a n- 3, n-4 ia n-4,...., 2 a 2, 1 a 1, multiplier, respectivement, le produit total et chacune des sommes par z0, z1 2, z2-,.. ajouter z,_, et donner a tous les termes des signes alternatifs, en commiengant par - ou +, selon que i, est pair ou impair. "Pour avoir le denominateur, on soustraira, successivement, de p, chacune des autres racines, et on fera un produit de toutes les differences donnees par ces soustractions." It is, of course, quite possible that Prony was not acquainted with Vandermonde's memoir of 1771, or Laplace's of 1772, or Bezout's of 1779; and, further, that in seeking for the solution of his equations he was lucky enough to hit upon the set of multipliers which, being used, would, on the performance of addition, eliminate all the unknowns except one; e.g., in the case of four variables the multipliers - P2PP4, + ( P2P3 + P2P4 + P3P4) - p p3+P4), 1. If, however, he was familiar with the method of any one of 308 HISTORY OF THE THEORY OF DETERMINANTS these memoirs, and applied it to the set of equations und er discussion, it would scarcely be possible for himi not to anticipate Canchy and Schweins in the discovery of the elementa ry properties of alternants. Thus, to take again the case, of four variables, say the equations x + y + z+ w=p ax + by ~ cz + dw = q a2x + b 2y + eC2Z + d 2wV= a3x + bly + clz + dlgw = sJ Laplace's process would have given the value of x in the formn Ibtmc2d31 p - b0c2d' q ~ b0c'md3 Iri - WYe 2 b 1e2d' a -. be2 d3j a + Ib0ec1d3 (62 - b0emd2 a3' and Prony obtaining it in the form bc. p - (bc +bd +cd) q + (b +c +d),r - s bed. a0 - (be+bd+ed)a ~ (b+e~d)a(2 - a could not have failed to know in their general forms the theorems 1V d1c l + b0cd2 -=bed, bOe d3 b0emd2 be + bd + ed, j1b0eld 31 I eld 2~ -= b+e+d, and 1a~b ed3 b~e1d2 =(d - a)(e - a)(b - a), and.a~ble2 d3 1= (d -a)(e - a)(b - a)(e - b)(e - a) (b - a). CAUCHY (1812). [Me'moire sur les fonctions qui ne peuvent obtenir que deux valeurs e'gales et de signes contraires par suite des transpositions ope'rees entre les variables qu'elles renferment. Journ. de l'e. Polyt., x. pp. 29-51, 51-112; or (Euvres completes, 2" se'r. i.] By reason of- the fact that Cauchy viewed determinants as a class of alternating functions, it has already been necessary to give an account of a -considerable portion of the first -part ALTERNANTS (CAUCHY, 1812) 309 (pp. 29-51) of this memoir: in fact, only five pages (pp. 45-51) remain to be dealt with if the portion referred to be borne in mind. From observing the substitutions which result in the vanishing of the function, he derives the following theorem:"Soit S(~K) une fonction symetrique alternee quelconque. Designons par a, f3, y, &c., les indices qu'elle renferme, et par -a, a3, a.... ba, b3, by,.... Ca, C13, Cy, les quantites qui dans cette fonction se trouvent affectees des indices a,, 7,..... Si l'on remplace ba, a...., bC, c,...., by, Cy...., par des fonctions semblables des quantites a,, a1, a,....; la fonction sym6trique alternee deviendra divisible par chacune des quantites aa - a,, aa - ay, a, From this he passes to alternating functions "which contain only one kind of quantities," and deduces the result that S ( ~a+C... a4) is divisible by (a2- a)(a3- a,)..... (an- Cl1)(a3 -2).... (a - a2)... (an- a-l). The question as to the remaining factor is then dealt with in the three simplest cases:(1) In the case of S(+ a0c.... aC-1) it is found as follows to be 1. "La somme des exposans des lettres a,, a,..,, a,, dans chaque terme de la fonction symetrique alternee 0 1 2 n —2 n —l S ( ~ aa2a3..... <l-ln ) sera 0+1+2+.... +(?-2)+(n-1) ( = (n ).... o~~~~~~ 810 HISTOIRY.OF THE THEORY OF DETERMINANTS Mais les facteurs du produit A [i.e., (a2 - a,) (a5 - a,).... (an, -an0 4'tant aussi en nombre e'gal 'a In (n - 1), la somme des exposans des lettres al, a2.,an, dans chaque terme du d~veloppement de ce produit sera encore 4gale 'a ce nombre; par suite, le quotient qu'on obtiendra, en divisant la fonction syme'trique altern~e par le produit, sera une quautit6i constante. Soit c, la quantitl6 dont il s'agit, on aura S -i(~a'ca'... n-'~) = cA. Pour determiner c on observera que le terme 01 2 3? a pour coefficient l'unite6 dans la fonction donn~e et dans le produit A; on doit done avoir c = 1." Before proceeding to the next case he calls to mind the fact that the produet or quotient of two alternating functions- of order n is a symmetric function of the same order, and is thus, enabled to amplify one of the preceding propositions by affirming that the result of dividing S 8 cel',aq. a') by S (~a~al..a1) is a symmetric function of a,, a2,., a,,., (2) In the case of S (~a'al n. a n~~ the quotient is found to be al+a2 + +. +an, For the quotient "sera n~cessairement du premier degre' par rapport aux quantite's a2, a2,...- a.,: et comme elle doit e6tre syme'trique et permanente par rapport 'a ces quantite's, on sera oblige' de supposer 6'gale 'a..l. 2 +an,) =cna,) 6 tant nne constante uin pent diff~rer ici de l'unit'." (3) In the case of S ( ~i a'a,2. a"') the quotient is, of course, found to be aja2..a,,. The memoir closes with the conditions for the identity of two alternating functions, these being stated to be (1) that all the terms of the first functions be contained in the second; (2) that the term-s have the same numerical coefficients in both; (8) that one of the terms of the first has the same sign as the corresponding terra of the second. ALTERNANTS (SCHWEINS, 1825)31 311 SCHWEINS (1825). [Theorie der Differeuzen und Differentiale, u. s. w. Von Ferd. Schweins. vi +666 pp. Heidelberg, 11825. (Pp. 317-431: TLheorie der Produete mit Versetzmngen.)] It may be remembered that Schweins' large volume contains seven separate treatises, that the third treatise deals with determinants (Prodcimte mit Verseftzngen), and is divided into four sections (Abtheilungen). The first of the four almost entirely concerns general determinants, and consequently an account of -it has already been given. The second section (pp. 369-398) 110w falls to be undertaken, its heading being " Determinants in which the upper index denotes a power " (Producte mit Versetzungen, wenn die oberen Elemente das Potentiiren angeben). His first theorem is hhh h Ial a2 at3 ap Ih+alih+ 8 h+a3 h+an\ A1A2A3... A ItAlA2A3.... A3 IAl A2 A3.... An, which is seen to be an extension of one of Cauchy's; but, besides this, in the first chapter there is practically nothing worth -noting. The remaining four chapters, however, are full,of interest, and deserve every attention, as until the present day they have been utterly lost sight of and contain a theorem or -two which are still quite new. The second chapter concerns the multiplication of an alternant,of the nth order by the sum of the p-ary combinations of the variables in their hth power. In Schweins' notatio-n'this product is represented by Al, A2 )....I-n A A2.. 'An/ -in later notation, the case where n =3, p =2, h =5 would be 7written (aVb + alc' + blc5). a,' a,, at' or l alb5. lar'bbct b'l b' bt Cr c' ct 'The case where p =1I is first dealt with, and the proof is written out at length without specialising n; but as this does 312 HISTORY OF THE THEORY OF DETERMINANTS not add to clearness or conviction, n may here, for convenience in writing, be taken = 4. Let, then, the alternant be larbsctdlu so that the multiplier is ha ~ bb + ch + dh. Expanding the multiplicand first according to powers of a, we perform the multiplication by a"; expanding next according toi powers of b, we perform the multiplication by bh; and so on, the sum of the products being naturally arrangeable as a square, array of sixteen terms, viz., ar+h bsctdu - as+hI bM'ctdu + at+h b-csdu - au+A bc'dCs - br+lh aS Ct IJ + bs+b aS'ctdu - bt+h arCsdlu + bu~l ahWVdt& + 0r+h asbtduLJ - cS"blar-btdul + ct+7b a1'bsd"u - Cu+,lbawbsct - d c+h sbtc6 ~ dsh IaWVbtcI - dt+A I allbbScu + du+AL I Co:',I. Rlecombination of these, however, is possible by taking them in vertical sets of four, and the result of doing this is lab+hbsctd'u - jaS+hb1bctdul + lat+hbb CSdb I- Ia&bCbrebsdt so that we have Iaabsctdul. 7a b - Iab'+lbsctddu + Jabbs+hctdbI + larbsectfduj + Ijarbsctdu+hbI and generally larbsctcdbevu.... alb = a+h bsetduel.... + a'bs+Ct&uev.... + ab'bsct+7bduel'... The special case where r~, s, t, U.. proceed by a common difference, h, is drawn attention to, as then all the alternants on the. right vanish except the last: that is to say, we have 7r+h r+2b ar+(n7 -I1) h 2,1b - b' b+h 2'+27 i (-2)hr+b a a2 a.. a b aa a lb a~b a 12 3 n 2 3 n 1 i a result which may be looked upon as an immediate generalisa-_ tion of one of Cauchy's. When p>1, the mode of proof is totally different, being an., attempt at so-called " mathematical induction." It is not by any means readily convincing, and is much less so than it might have, ALTERNANTS (SCHWEINS, 1825) 313 been, as, although there are two general integers involved, viz., p and n, Schweins attends only to the second of them. He begins with the case of n-= 4, p = 2,-that is to say, the multiplication of' a,'bsctdl by ahb7, the result being (A, A 3, A3 A 1A. 2A A A3 A34 J 2A A A3 A4 h +a a h 76+a3c- a + |A A2 A,4 +1 h7+al, a2 a3 h+a4 +IAl A, A2 A,3A + a- h+a2 h+a3 a4\ + IA1 2 A3 A4 ) a, I (3 a2 a3 h+a41. al a 2 hf +a 76-+a4\ + [AA A, A,. +'A a, 1 c2 ha3 h"a4 To indicate the mode of formation of the alternants on theright from the given alternant on the left, he says:"Hier entstehen alle Vertheilungen von h, h zu zweien in vier Abtheilungen namlich h+a h+a2 a3 a4 h+ a a2 h+a 3 a4 h+ a a a3 h+a4 al h+a2 ht+a3 a4 a1 h+a2 a3 h+a 4 1 + a h+3 h+ 4 aI h q- a~. h + a4 He next takes the case where n =5 and p =3: that is to say, the case of |a1bsctdeev. abc, 314 HISTORY OF THE THEORY OF DETERMINANTS and gives as his result / h h h h h (3) | a 2 a3 g a5 \ A1, A2, A, A,4 A | Al A2 A3 A4 A Aj I_ h+l h+ A 1 A2 3 A 4 A] \ = Au A, A A A, + 1 h+a h+al h+a a ha4 as\ = A1 A A3 A4 A +................. | AaIA aA h!a3 h2{-a4 h'5s + A1A2 A3 A4 A3 ), wo h, h, h in funf Abtheilungen zu dreien vertheilt werden, namlich h+a1 h+a2 ha +3 a4 a5 in ach o e e a as h it o a i a +a1 h+a2 a3 h+a4 a5 h +al h+a2 a3 a4 hea h + a, a2 h + a3 h + a, a5 h + a a2 h + a3 a4 h + a5 h+a1, a2 a3 h+a4 h+a5 a, h + a2 h +a3 h + a4 a, a1 ha2 h+a3 a4 h+a5 a h +a2 a3 h +a4 h+ a5 a, a2 h+ a3 h+a4 h+5,, the table being intended to make clear the fact that the five indices of each of the ten alternants on the right of the identity are got from the five a1, a2, a3, a4, a5 of the given alternant on the left by adding h to three of them. The mode of formation, seen to hold in these two cases, being then supposed to hold for h h h \(p) a, a2 a,, Al, A2,...., A_.|. Al A2 An-..., ALTERNANTS (SCHWEINS, 1825) 315 is attempted to be shown to hold for / h h 7h \(p) ai, fa an-1 n\ A1, A2,,.A.., A_.. A] A... An A; that is to say, the case for n variables, A,..., An, is sought to be made dependent on the case for n-1 variables, A,..., An_1, p remaining the same in both. The process followed is to change the first factor into / h h h \(p) Al A2,, Ax_, A +..., A,... / A h h 7 & h (p -1) 7l + Al, A A2,..., AZ _, Ax+,, A~}. A express the second factor-the alternant-in terms of n alternants of the (n-l )th order, and then perform the required multiplication and condense the result. This being satisfactorily accomplished, it would not of course follow from the two special cases previously dealt with that the theorem had been established in all its generality, but merely that it held for any number of variables A1, A2,... so long as p was not greater than 3. The passage from one value of p to the next higher-which is left unattempted by Schweins-is not free from difficulty, as will be seen on trying a particular instance,-say the passage from Iabsctdcu. (a7bb" +a c1' + aldh + b &Ch + b"d" + cddh) to a arbsctduI. (Ca"b7h + a +C J (ahCh + b hChd ). Several special cases of the general theorem are noted, where a number of the alternants on the right vanish and where consequently a comparatively simple result is attained. The first of these is where the indices of the alternant to be multiplied proceed by a common difference h: the identity then is / h7 h ^7? \(p) 11 a+7 a+2 ca+nh\ A, A2..., An).| A A2.... An I+i C+h8 +27 a+(n-p)h a+(n-p+2)h a+(n+l)h) - A1 A2....A-P A-p+l....n A The second is where h=-h, and the indices proceed by a common difference h, the result then being -h -h - h\(p) a+h a+2h a+nh A1 ) A2. An )1Al A2... A.n a a+h a+(p - 1)h a+(p+l)h a+nh\ Al A 2 Ap Ap+i....A, t 316 HISTORY OF THE THEORY OF DETERMINANTS The third is where the series of indices consists of two progressions proceeding by the common difference h, and where, of course, there are fewer vanishing terms in the product. In the next chapter the subject matter is quite similar: in fact, the only difference is in the constitution of the multiplier, which is more extensive than before by reason of the fact that in forming the p-ary combinations there is now no restriction as to non-repetition of an element. Thus, instead of the example lCarbSct. (c1bh +a ~"c7 + bhch) we should now have abset I. (ab1 + a + bhC + b2h + c2+). The method followed is exactly the same as before. Three simple cases are carefully worked out, viz., I aC'b. (a2" + b2" + ahb7), arbst. (,a2 + b27, + C2h +~ ahb + a c ' + a c + ), a 'bsct. (ac37 + b3h + c3 + a27bh + at277c" + b2hah + b27bCh + c 2ah + cb2bh~ + albhc)), the results in Schweins' notation-where the change to rectangular brackets should be noted-being r ha 7(2) a| a2 2h+aI 2\ a I a, 2h/a2\ [A1, A A A2 2= alA+) +i2 Al ) + AA,2 7r h h 71,(2) a, a1 al a j I 2A+a1 a2 a3 I1 a1 2h+a2 a3 A, A2,A A. 1 1 2 A = 1 A1 A2A3 + 1A1 A2 A3 + A A2 A3 ) + A A l A2 A2A) + |A A,2A ) + IAl A A, ) r h Ah h(3) a1 a A2 ac \ |1 37+a1 a2 a3 \ i ai1 3h+a2 a1A + AA A2A3) + 1A1A2 A3) + | IA1 A A+) + A1 A +2 a3) | 21~ a+al 2 a3 \ I | a l 2h+a2 a 3 + A1 A2A3 + 1lA2 A 3) | A1 f2 2A6t3) + 76+al h+a2 h+a3 + |AA, )+ A A, A ALTERNANTS (SCHWEINS, 1825) 317 Each result is seen, as in the preceding case, to be a sum of alternants differing only in the indices from the alternant which is the subject of multiplication. Further, it is observed that this difference is a difference in excess, the indices of the multiplicandappearing in all the terms of the product, so that the only difficulty is to ascertain what addendum is to be made to each. The next observation is that the addendum is a multiple of h, and that in the three cases the multiples are the following: 2h, Oh 2h, Oh, Oh 3h, Oh, Oh lh, Ih Oh, 2h, Oh Oh, 3h, Oh Oh, 2h Oh, Oh, 2h Oh, Oh, 3h Ih, lh, Oh 2h, Ih, Oh lh, Oh, Ih 2h, Oh, Ih Oh, lh, lh Oh, 2h, Ih lh, 2h, Oh Ih, Oh, 2h Oh, Ih, 2h I/,, Ih, Ih. The law of formation seen by Schweins in these coefficients of h is to be gathered from the sentence, "Hier werden alle mogliche Zerfallungen einer Zahl in mehrere Abtheilungen gebracht," and is nothing more nor less than the solution of the problem of putting p things in every possible way into n compartments. Thus, to take another example, if p were 2 and n were 4, the coefficients would be 2, 0, 0, 0 0, 2, 0, 0 0, 0, 2, 0 0, 0, 0, 2 1, 1, 0, 0 1, 0, 1, 0 1, 0, 0, 1 0, 1, 1, 0 0, 1, 0, 1 0, 0, 1, 1. 318 HISTORY OF THE THEORY OF DETERMINANTS Assuming this law to hold in the case of n 1 variables A,.... An_P, his mode of writing it being r h h h (p) ] aa, a, 1 ph+a a2 an-1\ _A1, A2 1....... A.1) - ZA.-L|Ai AA2l.. A_1), he tries to show that it will hold in the case of one additional variable An, the possible variation of p being ignored as before. To do this he changes the first factor rh h7 7h(P) Ai, A2., An, into 7r h^ h h, -|(p)_ r / A, (np - 1) h Al, A2, A... _ + L.Ai, A2,.. A) n-. A?,, and the second factor exactly as it was changed in the preceding chapter, performs the required multiplication, and condenses the result. The rest of the chapter is occupied with the consideration of special cases, the lines of specialisation being exactly those followed in the case of the previous general theorem. Only the first need be noted: it is r 1 h 7 l(p) ( a+h7 a+2j a+nh\ A, A2,...,A An.... A ) 1 a+7h a+2h a+-(n - l)h a -+(n+p)h) ~ A A,.... A A } ={A1 A2 n_-1 The fourth chapter does not impress one favourably, although the author speaks of its importance in connection with later investigations. It is almost entirely dependent on a very special case of the theorem of the second chapter, viz., the case where all the indices, except the last, of the multiplicand proceed by a common difference h, and where consequently all the alternants in the result vanish except two. In the original notation it is / 76 h7\(n-p) |I ca+7 Ca+2h a+(n-I)h s (A, A. A,.... An- Al) 11 a+h7 a +p76 a+(p+2)h a+nh s+h\ I= A Al.... An _ Ai ) a +h a +-(p-i)h a+(p+1l) a+nh s + A.... Ap- A... A n- An), but for convenience in what follows it may be shortly written Np. A, = M +,+ + Mp,.?Z~~p ~ PS-lp+, s+A ALTERNANTS (SCHWEINS, 1825) 319 Using it n -p +1 times in succession, we have Nnp. A = Ms+l,s+ + Mps, - Nn-p-_ * As+7 = - M+2,s+2~ - Mp+l+,s+h Nn-p-2. A+2h = M)+3,s+31 + MP+2.s+2h, - N_ p3. As+3h = - MlVi+4,s+4-,- -VI+3,s+3, (-)-PNo. As+(_-) 0 + (-)?MPMns+(_p)l, and therefore by addition Mp,s = n-p_. A, - N__. As+7 + Nn-_-2. As+2 -... (-)-No. As+()p) or a+h a+2h ca+(p-1)h a+(p+l)h ac+nh s AI 1 A2 A_, A.....A A_1 A AlA h Ai \(n-P) 31 a+h a+1 (n -)h s \ A1, A2.... An) 1..A. 1- An) h h 7A(n-p-l) a+t a+(n - 1)7 s+h) A, A2.A i.... An ' 1- - / h 7 h7\(n-p-2) 1 a+h7 a+(n-l)h s+27\ iI-pt h h h\(0) a ~+76 a+(n-l)h s+(n -p)h +(-1) A1, A2,.... A).A.. A..A_1 A, a theorem which may be described as giving an expression for an alternant having two breaks in its series of indices in terms of alternants which have only one such break and that at the very last index. On account of the fact, however, that alternants of the latter kind are multiples of the alternant which has no break at all-that is to say on account of the theorem r h hh h(p. I a+I c+27 a+czh Al A, A2. A 2, AJ.... A, ah a,+-2 a+27 +( - )h a+(n+-p)h7 A1 A2 A A A= A A2.... A_ A ) above given as an important special case of the general theorem of the third chapter-substitutions may be made which will result in the appearance of the last mentioned simple alternant :320 HISTORY OF THE THEORY OF DETERMINANTS in every term. Consequently, if we divide by this alternant and put s=a+(n+m)h, we have the theorem I a+h a+2h a+(p-l)h a+(p+l)h a+nh a+(n+m)h\ IA1 A A2....Ap Ap -1A,_ A, I +h a+2h a+nhh (I AAA1 Al A 2... )(.. ).A1, A2..., An / h h h\ (n -p) r h h h ( nm) - \1 A2, A2,., A. L.A1, A2,... AnJ / h h 7h (n-p-1) r 7 h A l("1+l) + (A1 A2, A..., A. A1, A2,..., A n-.p/ h h h\O ] ~r h h h(~z 2+n-p) (-) A 2, A2,..., A). A1, A2,., A.An Again starting from the same initial identity we obtain the analogous series Ip, s + Mp_l,s-h = N -p+l. As-h, - Mp-l, s-h - MP-2,s-2h = - Nn-p+2 As-2h, + Mp-2,s-217 + Mp-3,s-3h = + N7-.p+3. A-,-Q31 (-)P-lM,ls.(p-)76 + 0 = (-,)P-1Nn. A,.-p and therefore by addition have Mp,s = Nn-p+l. As- - N^n-p+2. As-2h +.... (-)P-lN. As-ph, or I a+h a+2h a+(p-l)h a+(p+l)h a+nh s\ A1 Ag.... Al A......AA_1 A_ / h h h\(n-p+l) I c+h a+2h a+(n-l)h s-h / h h h\(n - p+2) cc+h a+h a+2 a+(n-l)h s-2h\ - 1 A2..., A,.A,. A1 A2....A A,- A / h h h\(n-p+3) 1 a+h a+2h a+(n-l)h s-3h\ +A A A A A....A, A A.. A A.. IA A.... A*. A... p-l/ h h 7\(n) f1 +h a+2h a+(n-l)h s-ph\ (-) \A1, Az,., A/. |A A. A _-. A; ALTERNANTS (SCHWEINS, 1825) 321 so that by substituting as above for each of the alternants on the ca+h a+2h a+n7t right and dividing both sides by A A... A ) there results the alternative theorem a+h a+2h a+(p-1)h a+(p+l)h a+nh a+(nt+m)h 1 A A2....Ap A P,....A 1 A, ) a+7-h o+2- a+h\ -Al A2,.......... A, / It h(n-p+l) r 6mI= A,....,. LA1,....,A / hw h\(n -p+2) A h h/(.- 2) h h6 (n - p+S) r h h l(w-3) + (A...., AnJ LA1,.... A, ~ (Al., A l, A..,., A p-1/ h h\(n) r h (mrp) (- A,...,A. LA,....,A. (a) A, A). LA1., AXn] Lastly, attention is drawn to the case where a=0, h =, s=1, and to a case where the order of the alternants is infinite, viz., to the fraction b1 a) a+h a+2h a+(- 2)h6 a+nh A1 A2 A 1~.....A-) Ia +h a+2h Ac ) AA A,................A - The fifth and last chapter (pp. 395-398) concerns the simplest form of alternant above met with, viz., that in which the indices proceed throughout by a common difference, the main proposition being in regard to the resolvability of the alternant into binomial factors. The property with which Cauchy and almost all later writers start is thus that with which Schweins ends. The mode of proof is interesting from its farfetchedness and ingenuity, but need not be given in full generality or in the original notation; the case of laoblc2d31 will suffice. The first step, then, is to select a row, say the last, and express the alternant in terms of the elements of this row and their complementary minors. In this way we obtain aCCbl2d31 = d3lctblc21 - d2 o0blc3l + dlaob21 - lalb2C3. Now each of the alternants on the right is expressible as a multiple of Ic~blc21 by means of the theorem above given regarding M.D. X 322 HISTORY OF THE THEORY OF DETERMINANTS alternants with one break in the continuity of the equidifferent progression of their indices. Using this we obtain a0,b1c2d31 = {d3-d 2(a,b, c)l + d(a, b, c)2 - (a, b, c)3}. ablc21, = {d3-d2(Ca+ b+c)+d(ab+ac+be)-c)abc}. a0bblc21, = (d-a)(d-b)(d-c). I blc2, when it only remains to continue the selfsame process upon the alternant of lower order now reached. It may be remarked in passing that the identity la~bc2d3 = Ad3bc2 - d21 a0blc3l + d lab2c3 - |alb2c31, which expresses the alternant in descending powers of d, when taken along with the identity known to Cauchy [ca0b12d3 = (d-c)(d-b)(d-a)(c-b)(c-a)(b-a), the right side of which may likewise be arranged in descending powers of d, viz., {d3-d2(a + b+c) + d(ab+ac+ be)- abc} (c-b)(c-a)(b -a), may have been the means of suggesting to Schweins his theorem regarding alternants like la~b2c3, ca~b1c31 which have one break in their series of indices. In other words, the order in which he gives his theorems was very probably not the order of discovery. The remaining portion of the chapter is an investigation of the quotient of two alternants of infinite order, viz., Ia a+h a+27i a+(- 1)7V a+nh B Al A2.... A_ An+1.... A. ) a a+h a+2h7 C \ ' A1A2 A3.............. A ) SYLVESTER (1839). [On Derivation of Coexistence: Part I. Being the theory of simultaneous simple homogeneous equations. Philos. Magazine, xvi. pp. 37-43; or Collected Math. Papers, i. pp. 47-53.] As has been already shown, Sylvester's first approach to the subject of determinants was similar to Cauchy's, the basis of both being the outward resemblance of the two expressions ALTERNANTS (SYLVESTER, 1839) 323 be2 + a2c + tab2 - C2b - ac2 - b2c, blc2+ aCl+ a2 -c t 2b- ac2 - b2c1. As the former is equal to (c-b)(c-a)(b-a) or PD(abc), i.e., product of the differences of a, b, c, Sylvester denoted the other, viz., the determinant 1a a2 1 b b2 1 c C2, by ~PD(abc), ~ being the sign for multiplication according to the law a,. a, =a,.+. Using this notation he rediscovered, as has also already been seen, Schweins' theorem regarding the multiplication of the alternant I alb2Cc.... I by such symmetric functions as (a+b+c+... ), (ab+ac+... +bc+...),..... his form of statement being {S,.(abc... 1). gPD(Oabc... 1)} = _.PD(Oabc... 1), where _,. implies that after 'zeta-ic' multiplication the subscripts are all to be diminished by r. His attempted generalisation of this theorem has likewise been spoken of, its validity, however, being left undecided upon. Instead of the multiplier S,.(abc... I) he proposed to take any symmetric function whatever of a, b, c,..., l1,-or, rather, any function whatever followed by any symmetric function. This would have been a most noteworthy extension which Schweins had not foreseen, but unfortunately there are grave doubts as to the truth of it,-indeed, one may go so far as to say that there would be no doubt whatever about the author's inaccuracy, were it not that there are doubts also as to his meaning. By way of test let us take the case where the multiplier of aCb2c3d41i is the symmetric function 2a2bc. From later work * it is known that aI l2cdc4. a2b1c1d0 = - alb3C4d61 - 3 a2b34d See Muir, "Theory of Determinants," p. 176 (1882). 324 HISTORY OF THE THEORY OF DETERMINANTS whereas, according to Sylvester, there ought to be on the left only one alternant. Now although we know that Sylvester was in the habit of making guesses, and that these guesses though often brilliant were not always so, it would be next to impossible to find a generalisation of his which had no individual instances in support of it. There thus remains the curious and interesting question as to what amount of truth there is in the theorem as enunciated, and whether an amendment of the enunciation would not give something not merely unexceptionable but of important value. In trying to pass from symmetric functions like la, lab, Eabc,... which are linear in regard to each of the variables, and to extend the theorem to any symmetric function, Sylvester probably thought-at least it would be quite natural for him to do so-of expressing the latter in terms of the former and then applying the theorem already obtained. It is desirable, therefore, to see what such a process may lead to. Taking the case of the multiplier Ea2bc we have |b2c3c4 1. 2 a2bc = ab2c3d4 1. { a a. Z abc - 4 Z abccl, = {] clb2c34 1. c a. Z abc - c b2c3d4.4 abcctd, = alb2c3d5. c abc - 4 a 2b3C4t5 1 At this point we encounter a difficulty, for the previous theorem, although it teaches us to multiply I ctb2c3d4] by cab, does not help us in the case where the multiplicand is C alb2ccd51. Proceeding, however, with other assistance we find the desired product = al2b3c4d5 + Calb3c4d6 - 4 2b3c4d5 1, - alb3c4d6 1- 3 a2b3C4d5, agreeing of course with what has already been found. Now the difficulty referred to would present itself to Sylvester also, but in a slightly different form by reason of the periodicity which he assumed in the elements. Thus, instead of writing {I a bcZ3d4!. aC} ' abc =- a'b2c3 d5. p abc, - a2bIC4d5 + |a'b3C4d6, * See Crelle's Journal, lxxxix. pp. 82-85. ALTERNANTS (SYLVESTER, 1839) 325 he would write {< PD(Oabed). S (abed) ). S3 (abcd) = I{ 1PD (Oabcd). S3(abcd)} and there pause for a little, not having specifically provided for the 'zeta-ic' multiplication of such an expression as _-PD(Oabcd) by S3(abcd). The result forced upon him, however, would be the single term 4_PD(Oabcd), which in modern notation is oa2b3c4d5 In the course of the work, therefore, the term alb34d6 would be dropped altogether out of sight. The cause of this is undoubtedly the imposition of the condition just mentioned;-indeed, if we take the result of the work as above performed in the modern notation, viz.:I aclb3c4 1-3 a2b3Cc4d, and make the elements periodic, i.e., make a6, b6, C6, d6 = al, b, c,1 d, the first alternant will vanish by reason of having two indices alike, and we shall be left with a result agreeing with Sylvester's. The conclusion, therefore, which we are tempted to draw is that if Sylvester's general theorem be correct it is only when the elements are subjected to periodicity. JACOBI (1841). [De functionibus alternantibus earumque divisione per productum e differentiis elementorum conflatum. Crelle's Journal, xxii. pp. 360-371; or Werke, iii. pp. 439-452; or Stackel's translation in 'Ueber die Bildung und die Eigenschaften der Determinanten,' 73 pp., Leipzig (1896).] After having treated of determinants in general (pp. 285-318), and of the special form which afterwards came to bear his own name (pp. 319-359), Jacobi turned to another special form which he had learned about from his great predecessor Cauchy. As, however, he differed from Cauchy in his mode of defining a 326 HISTORY OF THE THEORY OF DETERMINANTS determinant, Cauchy's definition, which, it will be remembered, made use of the difference-product, now appears as a theorem, and with it Jacobi makes his start; that is to say, he proves that If in the determinant - a0bc2d... 11,_ the suffixes be changed into exponents of powers, the result obtained is equal to the product of the ~n(n-1) differences of a, b, c,...,, viz., the product (b-a)(c-a)(d-a).... (1-a) (c-b)(d-b).... (1-b) (d-c).... (1- c) With the help of Sylvester's notation, which symbolizes the opposite change, viz., from exponents of powers to suffixes, this may be expressed in the compact form ~PD(abc... 1) = + a0obl2... l,.In proving it he takes for granted (1) that the product in question merely changes sign on the interchange of any two of the elements, and (2) that in the developnment of any function of this character there can be no term in which two or more exponents are equal, for the reason that, if there were one such, there must be another exactly like it but of the opposite sign. Combining with this latter-which includes of course the case where the index 0 is repeated-the fact that, for the particular function under consideration, the indices must all be + and the sum of them equal to ~n(n-1), he concludes that no term can have any other indices than 0, 1, 2,..., n-1. Next, as there is only one way of getting an element, 7k say, in the (n-l)th power, viz., by multiplying all the n-l binomial factors k-a, 7 k-b,... in which k occurs, and after that only one way of getting an element, h say, in the (n - 2)th power, viz., by taking from out the remaining binomial factors all the n -2 factors in which h occurs, and so on, it is inferred that no term can have any other coefficient than +1 or -1. Summing up ALTERNANTS (JACOBI, 1841) 327 rather hurriedly, he consequently finds that the development of the product may be got by permuting in every possible way the indices of the term cA0bl2 I. n-1 and determining the signs in accordance with the law that the interchange of any pair causes the aggregate of all the terms to pass into the opposite value. This being exactly the mode of formation of the determinant ~+caoblc2... n- with the difference that suffixes take the place of exponents of powers, the theorem is held to be established (.... "signis insuper ea lege definitis ut binorum indicum commutatione Aggregatum omnium terminorum in valorem oppositum abeat. Quse ipsa est Determinantis formatio, siquidem exponentes pro indicibus habentur"). In passing, he remarks on the large number of vanishing terms in the development of the product, viz., 2n(n~-l)-n!, and the consequent desirability of obtaining this development from that of the determinant and not vice versa. The fundamental relation between the determinant 2 ~t a0ob1...** _nand the product of the differences of a, b, c,..., having been established, it is then sought to find properties of the latter from the known properties of the former. What properties of the determinant are used Jacobi does not mention, all that is given being a bare enunciation of the results. It may be as well, however, to point out at once that all of them flow from one general theorem, viz., that of Laplace regarding the expansion of a determinant in terms of products of its minors. The first is indicated by using as examples the case of three,elements, a1, a2, a3, and the case of four elements, al, aC 3, a4, viz., (a2 - a,)(a3 - a1)(3 - a2) = c3(ca3- c2) + ac3a (1 - a3) + a12(ca- aL), C2 - a1)(3 - 1) *.. (4 - ) = a2a34(a3- a2)(a4 - Ca2)(a4- a3) - Ca3C4aC1 (a - C3)(C - a3)(a - a4) + a4a1a2(a- - a4) (a2 - a4)(a2 - a1) - a,42a3(a2 - al) (a3 - a1) (a3 - ca2 328 HISTORY OF THE THEORY OF DETERMINANTS it being pointed out that any term of the expression is got from the preceding by cyclical permutation of the suffixes, and that the signs are all + when the number of elements is odd, and alternately + and - when the number of elements is even. The case of Laplace's expansion-theorem, which is here used, is easily seen to be that where the orders of the minors are. n-1 and 1. Thus using later notation, we have 1 a a2 a3 1 b b2 b3 P%(abcd)= 1 c C2 C3 1 d d2 d3 = bl2d31 l a'd3l + ab2d3- I ab2c = bed b0cld2 - acdla0ld2 + abdla0bld2 - abc l0blc2, which is the desired result. In connection with this, it is perhaps worth noting that the result being, by the same case of Laplace's theorem, also equal to 1 c a2 bed 1 b b2 cdct 1 c c2 dab 1 d d2 abc we may view Jacobi's first theorem as being equivalent to one of later date, viz.2 n-2 (lac*... ca) = (-)-1 ~ 1 4 (.. Con,W., 1 a a2... a2 1a3a4... an 2 - 2 1 an n... a1a2a3... aC -1 When the determinant is of even order, it is possible to use that case of Laplace's expansion-theorem in which all the minors are of the 2nd order. Thus ALTERNANTS (JACOBI, 1841) 3~ 329 ~'abcd) 1= 2 a 1 b b2V b3 1 cc2 c3 1 d d 2 dl 1a c2 c3 1a b2 b3 1a b2 b3 1 b d2 d3 c d1 d3 1 d 0 1 b a2 a3 b a2 a3 1 c a2 a3 +1I c cd2 d3 1d C2 c3 d b2 b3, = (b -a)(d- C)C2d2 - (c-a)(cl-b)b 2d2 + (d -a)(c -b)b 2c2 + (c -b) (d -a)a 242 - (d - b)(c - a)a 2c2 + (d-c)(b-a)a~b, = (b-a)(d-c){a 2b2 +c2d2} + (c -a)(b - d){la 2C2+d2b2} + (d -a)(c -b){a 2d2~ b2C2} By Jacobi, however, the result here established is given merely as an example of an improved general theorem, which is. enunciated in the form of a 'rule,' as follows: "Fingatur expressio (a~ -ao) (a3 - a)... (ca,, - a,, 2) 2a4a 4aa"a? quam quo clarius lex appareat sic seribam (a, -. a,) (a,, - a2)... (a,, - an-0~ I (a0aj1) (a2a3)2 (a4a5)4... (.-a, sub signo 1: omuimiodis permutatis exponen'tibus 0, 2, 4.., n -I. Iin expressione illa cyclum percurraut prirno elementa tria secundo elemeuta quinque a,,.4, a,, 3, a,,-2, a,,-.1,a,, et sic deineeps ita ut postremo cyclum percurrant elementa aD a2, a3,.., a,, Omnium expressionium proveniienitiun aggregatum oequabitur ipsi P."' 330 HISTORY OF THE THEORY OF DETERMINANTS The meaning will be made quite apparent by taking a case other than Jacobi's above referred to, say the case where there are six elements, c, a 2, a2,...., c5. According to the rule, what we have got to do at the outset is to form the term (a1 - aO) (a3 - a2) (5 - a4) Z(a0a1)~(a3)(a)4a5)4; then derive from it two others by the cyclical substitution (CLa3 C4 as a4 a5 a; and finally, from each of these three derive four others by the cyclical substitution a/ a2 aC3 a4 a5 CLa a C~4 05. l This being done, the sum of the fifteen terms so obtained can be taken as an expansion of the difference-product of a0, aX, (a2....2 5 -Although, as has been said, the theorem is given without proof, it has to be noted that Jacobi draws attention to the fact that the number of ultimate terms in the expansion of the compound term (a1 - )(c3 - aC)... (Ca, - C,,_)(i% (o1)0(2a3)2(a94a5) (-)4an)2 is 2"1.2.8 2 that the number of ultimate terms obtainable from all the compound terms of this form is 2. (1.2.3.....(35.... ): and finally that this is equal to 1.2.3.... (n+ 1), a result which agrees with what we know of the differenceproduct from its determinant form. From this general theorem regarding the difference-product of an even number of elements, an advance is made to a theorem of still greater generality, the means employed in obtaining it ALTERNANTS (JACOBI, 1841) 331 being in all probability the same as before, viz., Laplace's expansion-theorem. The most general form of the latter theorem, it will be remembered, gives an expansion in terms,of products of more than two minors. Jacobi was familiar with this, for in his famous fundamental memoir regarding general determinants a whole page (pp. 298, 299) is devoted to an illustration of it. Now, if we take the case where the number of minors is three, and apply it to the determinant which is the equivalent of the difference-product, we obtain a result which is transformable without difficulty into I(Co, *a,..., a,,) = ~ '{ (aia+lai+2... k)i+l(ak+l +2... a,, )k+1 x II(ao,,,,..., ai)II(a+1a+2 * * a)(aT(c+a+2.. ca,) and this is the theorem "of still greater generality" above referred to. Jacobi then proceeds to the consideration of alternating functions in general. The definition which he gives, and to which he attaches Cauchy's name, is somewhat different from Cauchy's, being to the effect that an alternating function is one which, by permuttation of its variables, is either not changed at all, or is changed only in sign. In the matter of notation he also introduces a variation, but this time with more success. It will be remembered that, when Cauchy denoted a determinant by prefixing S+~ to the typical term, he was simply following his practice in regard to alternating functions in general, which he denoted by S +- 0(a, b,c,..., ), the rule for determining the sign of any term of the aggregate being left unexpressed. Instead of this, Jacobi uses p f(p(a,b,cI.., I l\ where P stands for the product of the differences of a, b, c,...,; and as the P which is inside the brackets is subject to permutation of its variables, and therefore automatically, as it were, changes sign with every interchange of a pair of variables, 332 HISTORY OF THE THEORY OF DETERMINANTS while the P which is outside the brackets remains unaltered, it is clear that the rule of signs is here fully expressed. Thus if q(4a,b,c,..., 1) were ab2c4, we should have v Clb2C4\ caclb2c4 a 1C2b4 Z P (b- a)(c - a)(c -b) + (c-a)(b-ca)(b-c) bla2c4 blc 2a4 (a-b)(c-b)(c-a) (c-b)(a-b)(a-c) cla2b4 clb2c4 (a - c)(b-c)(b-a) (b-c)(a-c)(a -b)' CCb2C4 - alc2b4 - bla2c4 + blc24 + cla2b4 - cb2a4 (b —a)(c-a)(c —b) and therefore P (cb) = a lb c4- aoc24 -blc24 a + blc24+C1l2b4 - clb2a which is an alternating function written by Cauchy in the form S( alb2c4), and which, being a determinant, was; written by Jacobi himself also in the form 2 ~ cab2c4. It is pointed out that any term of p which remains unchanged by the interchange of two of the variables may be left out of account; but the question raised by Cauchy regarding possible and impossible forms of cp is not touched upon. As a corollary, it is stated that if (aO, L1...., aa) = aOct1....an the indices o, ap..., a, must be all different if the alternating function is not to vanish. He then recalls the known fact that, when the indices a, ap..., a, are integral, the alternating function aao aal an 0 4 - -1 n or Za^1C'.... ia" or Pa.P.. is divisible by P, the difference-product of ca, a,..., a,, and puts to himself the problem of finding the generating function of the quotient a0Ca1 a), IaCn c....a o Ld P In the course of this quest his first proposition is ALTERNANTS (JACOBI, 1841) 333 If p be any rational integral fmnction of m + 1 variables, II their difference-product, and f be a function of the (n+ 1)th degree in one variable and be of the formn (x-ao)(x-a1).... (x-a), then when m > n no single term of the expansion of 11(to, t... ~, tll)..0(to, t,...., t1n) f(to)f (t).... f(tm) according to descending powers of t0, t1..., t,,, can contain negative powers of all these variables. To prove it he of course uses the identity 1. 1 f(x) (x- O0)(X - a().... ( - aC,) 1 + 1 + f'(ac). (X- O) f'(ac) (a -1). +'(a).(-a,.) and thus changes the expression into the form I. (' O) f(ao) (t - a o) + f'(a,). (to- a) + a(to - a,) >< tf'(a0).(1 a0 + + 1 1 1 (x f(o) (t, - a) + f(a).(t, - () + an'(). (t-am) ~................... o X + - f '''''(an).(t,,,-c,,) ' o).(t, ao) +f'(a). (t~,-i + +f(..(t,,) He then says that the result of performing the multiplication of these bracketed factors is to produce terms of the form f'(a)f'().... f(p). (to - a) (t -b).... (t -p) where each of the m +1 quantities a, b,..., p is necessarily one of the n+1 quantities a0, a,..., a, and where, therefore, on account of mz being greater than n, the quantities a, b,..., p cannot be all different. But terms of this form can be changed into f _____ in 1 '(c)f(b)....f (p) t,-b-to+a to-a t-bJ ' (t-c)(t-1).... (t,-p) 334 HISTORY OF THE THEORY OF DETERMINANTS which shows that in the case of two of the quantities a, b,..., p being alike, say a and b, the second factor would become II 1 - t0 and therefore could be simplified by having t -t0 struck out of both numerator and denominator. This means that when m > n the second factor, like the first, can have only positive integral powers of the variables. As for the third and fourth factors, their product is the difference of the two fractions (-)( C)(-d).. ) and (t-a)(t2- c)(t3- d)... (tm -p) (t1- a)(t2- C)(t3- d)... (t-in p)' the former of which yields no negative powers of t1, and the latter no negative powers of to. The proposition is thus established. To prove the next proposition he utilizes the theorem that If F be any rational integral function of a number of variables, the coefficient of x-ly-lz-1.... in the expansion of F(x, y, z,....) (x-a)(y-b)(z-c).... according to descending powers of x, y, z,.... is F(a, b, c,....) This is spoken of as being well-known, and no proof of it is given. It is readily seen, however, that as the expansion referred to is got by performing the multiplications indicated in F(x,y,z,....). {x-l+ax-2+a s+....} {y-l+by-2+b2y-3+....} { -1+ c-2+ 2Z-3+....} any term in F, say the term Axoayzv...., would require to be multiplied by x-'Z-l, y —, z-v-l.... in order to produce a term in x- y- z-1...., and that these multipliers being only found associated with the coefficients a, b3e, c.... the term so produced would have for its coefficient AabWcT.... The full coefficient of x-ly-l-1.... would thus be F(a,b,c,....). ALTERNANTS (JACOBI, 1841)33,935 He also uses an identity regarding difference-products which it may be as well to state separately, viz., that (a$a~... a.). ll (anm,an-m~i.., an) =(lI)PA(?T~1)H(a0, a,).,. an-m-1). f'(an-m)f'(n-m+0...i) a where f'(a,.) stands f or the product of the n factors got by subtracting from a,. each of the quantities a0, a, c... ya except a,. This he holds to be true,* because the product Pa._,jf~n-'w+1).... f'(cn) *The factors of a difference-product may always be, and usually are, arranged in the form of a right-angled isosceles triangle; for example, 2(abcdefg) (b -a)(c -a)(d -a) (e -a) (f -a) (g- a) (-b) (d- b) (e - b) (f - b) (g - b) (d -c) (e- c) (f -c) (g- c) (e -d) (f -d) (g-d) (f- e) (gY- e) (g-f). Consequently there must be an algebraic identity corresponding to the geometrical proposition-Iffrom a point in the, hypotenuse of an isosceles right-angled triangle straight lines be drawn par-allel to the other sides, the, triangle, is thereby divided into two triangles of the same kind and a rectangle. This identity it is which is at the basis of Jacobi's, for drawing the lines thus(b -a) (c -a) (d -a)(e -a)(f -a) (g -a) (c - b) (d - b):(e -b) (f -b)(g - b) (d - Cy(e- c)(f -c) (g- c) (e - d) (f - d,)(g - d) (f-e6)(g - e) we obtain ~1(abedqfg) - (abcd). ~(efq). (e, - a) (f - a) (g - a) (w (e- b)(f- b)(g- b) (e - c)(f - c) (g - c) (e - d) (f - d) (g - d). But the expression here which corresponds to the rectangle in~"the geometrical. proposition =(e-a)(f-a)(g-a) (e -b) (f — b) (g - b) (e - c)(f - c) (g- c) (e - d) (f - d)(g - d) + (eq -O(gfe) (f- e) (g - e) (e -f). (g -f) (e -g) - g) f'(e).f'(f).f'(g) (-)'~I(efg). - (efg) - 336 HISTORY OF THE THEORY OF DETERMINANTS contains as factors the differences of all the elements aoo, a1,..., a, except those which go to make I(a,, a,,..., a^-m_, ) and contains a second time but with opposite signs the ~nm(m+ ) factors which go to make II(a,_,, a-,,,+1,..., a.). These preliminaries having been given, the second proposition may now be proceeded with. It isIf ( be any rational integral function of m + 1 variables, II their difference-product, and f be a function of the (n + I)th degree in Consequently ](abcdefg). I == H(-)3f(e)' f'(f) f'(g) ( ) f(abcd) which is Jacobi's identity. It is easily seen that there is a corresponding theorem to (o) obtained when the point through which the parallelsare drawn is taken inside the triangle: thus, corresponding to the diagram.................................................. we have the identity _t (abcde). j2(cc defg) ( f]~c(abdef) f(ae)'.(cdefg). n) (/ - -a)(f- b) (g - b). (') 1(abcdefg) (c d__ ]2(cde) Here, however, it is no longer possible to treat the final group of factors as was done in the case of (X). To Jacobi's identity (0) the absolutely perfect geometrical analogue is got by taking a rectilineal figure of the form ABCDE, where AB=BC, CD=DE, B=C=D=90~, and then equating the sum of the two parts got by drawing CE to the sum of the two parts got by producing DE to meet AB in F. Further, the exact analogue to his proof would be to say that the rectangle BCDF contains all of the triangle ABC except the triangle AEF, and contains the triangle CDE in addition. A B D X C ALTERNANTS (JACOBI, 1841) 337 one variable and be of the form (x-ao)(x-aa).... (x-a1), then when m j> n the coefficient of to-t — 1.... - t,- in the expansion of HI(to, t,... t,l). 0(to, ti,..., tin) f(to)f(ti).... f (t) according to descending powers of to, t,...., t, is +aOala2 a an 111-1n l 0 a * a 0 1 2. n-m — i-nll/a, a a,) "n-m-l(v (~an-m' an-m+l ' O n) (._)n(amo, alp...., al) effect being given to the sign of summnation by permuting in every possible way the quantities a0, a,..., a,. As has already been seen the expression to be expanded is equal to an aggregate of terms of the form 0(to0 tl,.... tM). I (to0, tl. *, tin) f'(a)f'(b).... f'(p). (to-Ca)(t -b).... (tin-p) where each of the n +l quantities a, b,...., p is one of the n+1 quantities a0, al,...., (. Since, however, we are now in search of the coefficient of tt'l.... t~1 we may leave out of account all terms of this aggregate which have two or more of the nm+1 quantities a, b,...., p alike, for it has been shown that the expansion of such a term cannot contain tolt.... t,1. We are thus left with an aggregate which may be represented by C____ +_(to0 t,...., tm) * (to,...., t __,) J (a,, - )/(aX,, - in+1).. Jf'(a). (to- Ct-,(?,)(tl- a1-,,-,+)... (t - an) it being understood that for a,_,, a,,-+,i * *., a,, is to be taken any permutation of n +1 quantities of the group ao, al,...., an. But, if the coefficient of to-tlt.... t-1 in this be denoted by H, we have by the first of our auxiliary theorems f ( a,,) Hn _ 8 ~(an_,an_ m+l (?, 1 a. * I -~ _,,a) * Ia-n-'-m+l7, '. * * n) S ^f (a?'t-m,)f'(aX,-m+l) * + * f"(Ctan) and using the second to substitute (- l),(+l)H(II (ao,...., a,,, _,,)IIo, (al, c1...., 7,,) for II(a,, _,a,,_+i,,...., an)/f'(an, - M)f(a,, - II/-).... /(a,), M.D. Y 338 HISTORY OF THE THEORY OF DETERMINANTS we have T _ (_ 1 )2a \~+l) l 'c I I *(a 'alv * Cnan —a. can.l) 1) (an,,an7)..., n a ) where, be it remembered, the n+1 elements ao, ac,...., an are to be separated in every possible way into two classes containing 9n-m and m+1 elements respectively, and all permutations of the elements of the second class are to be taken. In this expression, however, another substitution can be made by reason of the identity Hn(aoal,..... - __ V a0al I.. 2w_-l P -Z P where under the sign E all possible permutations of the indices 0, 1,...., n-m-1 are to be taken. When this substitution has been made, we shall consequently have to take every possible permutation of both classes of elements. But to take every possible separation into two classes and permute the elements of each of the classes in every possible way is the same as to take every possible permutation of all the elements. Our result will therefore be a -_,,al. 4.(c.._m.a,~_,,~+l,., aan) fi = (l)l?fl(?T2+I) aacl a.. anz7...ma,. a,) H- = (-'l>t),+') a a o.. I_ *n-m. -+l p* n) if it be understood that under the sign of summation all possible permutations of aal,,...., are to be taken: and this is what we set out to prove. The case where m= n is then considered, because of its special interest. The first expression obtained above for H becomes in this case S P. (a0a.o,... aa) f /(a0)f'(a>).... f(an) where under E all permutations of a0, a,...., a. are to be taken. Making in this the substitution which is possible by reason of the identity f'(a)f'(al).... /'(a = (-1)(1()+=)p2, we have H = (- 1)4+ 1(' 0 "1) ~.....'.) ALTERNANTS (JACOBI, 1841) 339 The formal enunciation of the result thus obtained is:If pq be any rational integral function of n +1 variables, II their difference-product, and f be a function of the (n + 1)th degree in one variable and be of the form (x-ao)(x-a).... (x-an); then the coefficient of to1lt1.... t-' in the expansion of )n(ntl) (to, tan). (to., tt. tn) f(to)f(t).... f(tn) is 4(a0,ao., an) H(aoa, a,...., an) effect being given to the sign of summation by permuting in every possible way the elements a0, a,..., a,. As we have seen above that Y (f((a0,a 1 '., an) n(ao,0C,...., a) is the quotient of any rational integral alternating function by the difference-product of its elements, and that this quotient is often in request, it is important for practical purposes to note that what this last theorem of Jacobi's gives is the generating function of the said quotient. After giving a line or two to the case where m = n-1, Jacobi returns to the general theorem and specializes in another direction, viz., by putting (t0, t...., tim) = tt*l... t. Division of both sides by 0 is in this case possible, and the resulting theorem is one of considerable importance:The expression a~a1.... an M 1 aY1 fl^ l * n-m-1 n-nM n-m+l a n a (1- an(a- ara).... an-an-i) which is the quotient of an alternating function by the difference-product of its elements, is equal to the coefficient of t- (+l)t- (Yi+l) t -(Ym+l) 0 1 m 340 HISTORY OF THE THEORY OF DETERMINANTS in the expansion of (to - tl)(to - t2).... (tn-l -t) f(to)f(t).... f(tm) according to descending powers of to, t,...., t,, where f(x) = (x-ao)(x-aa)....(x-an). This is followed up by actually working out the expansion in question, the numerator being of course changed into -+ tt0 - 1 I - and its cofactor 1 1 1 f(to)' f(t j(tm) into ( ) I C, L C2 + Cs + + I + + t....+ 2+ +3 n+.+s+ VO '0 '0 X(tn+-1 t"-t+2T t^n+3 n+l+ss+.... 1 1 1 1) (tj.+.... s X it.-+"l I t+2 + t27+3 + ' ' ' ' n+ +6 + 't where C, is the sum of all the products of s elements, different or equal, taken from a0, a1...., an. Multiplication of these m + 1 partial factors has next to be performed, the general term of the result being seen to be CSoCS.... Csn tn+l+sotn+l+sl.tn+l+s, 0 1 rM All that remains, then, is the multiplication of this result by the corresponding expression for the original numerator, i.e., by 2 t_ t - l... t l,_ which, be it noted, consists of (rz +1)2 terms, the Z referring to permutation of the indices, rn, min-,.... 1, 0. Without further delay, Jacobi merely adds that the general term will therefore become CSoCS,....C M to. Cs/i.... T tn - m+l+so ttnm+1+s1 t - III mtl+ S o 1 m ALTERNANTS (JACOBI, 1841) 341 and that consequently the proposition last formulated will tc suggest" the identity 1 2 nz-rn-i y C/Y cta a ani~.... a a a/n 1 2 l -M + i (a- a) - a).... (al -a,,, I- 0) ~C( m n '12Cn where the I in the first case refers to permutation of a,, a1)., an, and in the second case to permutation of yl)...., y. in a couple of lines it is next pointed out that the putting of in =0, m =1. in this suggested identity gives 1 2 it1 - I yy a,1 2 2. y n-lan P T-1 1-2 2i-in a a... C`C Ct a 1 2 n- -2 n 1 n Cy+lnCyn - +incyn, P VI- 1-n yfln~n then, rather unexpectedly, there is given a mere restatement of the identity itself, viz. "Generaliter cequatur quotiens proposituts aCa2. e_._. a aV. a... a 2 n-m-n-n n-m + P determinanti quod per-tinet ad systemc quantitatum Gy+m - n CY +1iiin.... Cm - n Gyfm-n-l G/1+71 —1 ' YCm - -t Gy-11 G/ It 91 This is the last result of the memoir, the few additional linesused being merely for the purpose of showing how the determinant just mentioned may be simplified. The simplification consists in leaving out the element an2 in forming the C's of the, second row from the end, the elements an, an, in forming the C's of the third row from the end, and so on. The reason in the first case is that this will have the same effect as subtracting 342 HISTORY OF THE THEORY OF DETERMINANTS from each element of the row an times the corresponding element of the last row, and the reason in other cases is similar. If C' be used to stand for the same as C, but to concern one element less, viz., an, and C" be used in similar manner, the identities at the bottom of the simplification areCs+ - aC, = C s+1, Cs+2 (a + a,-1) 08+ + anan-C = C s+2 the truth of which is apparent when we remember that C,, C2.... are practically defined by the equation 1 1 C1 C2 (x -a) (x-a1).... (x-an) =x+1 X+ 2 X1,+3 It is noted also that in the determinant a C with the suffix 0 is to be taken as 1, and a C with a negative suffix as O. CAUCHY (1841). [Memoire sur les fonctions alternees et sur les sommes alternees. Exercices d'analyse et de phys. math., ii. pp. 151-159; or (Euvres completes, 2e ser. xii.] As has before been pointed out, the preceding paper of Jacobi's was the last of a triad which was followed up by a similar triad from the pen of Cauchy. Cauchy's first paper, which corresponds in subject to Jacobi's third, comes up therefore quite appropriately for discussion now. What is really new in the first part of it concerns the finding of the symmetric function which is the quotient of an alternating function by the difference-product of the elements; that is to say, in Cauchy's notation, the finding of s[+_/(X,Y,,....)] (x-y)(x-z).... (y-).... or, in Jacobi's notation, the finding of E f(x,y,z....) H(x, y,z,....) ALTERNANTS (CAUCHY, 1841) 343 It therefore opens with the reminder:"Une fraction rationnelle qui a pour denominateur une fonction symetrique et pour numerateur une fonction alternee des variables x, y, z,.... est evidemment elle-meme une fonction alternee de ces variables. Reciproquement, si une fonction alternee de x, y, z.... se trouve representee par une fraction rationnelle dont le denominateur se reduise a une fonction symetrique, le numerateur de la meme fraction rationnelle sera n6cessairement une autre fonction alternee de x, y, z..... This prepares us for the consideration of the alternating aggregate S[ ~f(, z..)] where f is fractional and rational, and where, although Cauchy does not explicitly say so, the numerator and denominator are integral. In regard to this he asserts that the various fractions which compose the aggregate may be combined into one fraction U/V, where V is an integral symmetric function divisible by all the denominators, and where, therefore, U will necessarily be an integral alternating function and, as such, be divisible by the difference-product of its variables. We are thus led to the proposition that the given alternating function of x, y, z,.... can be resolved into two factors, one of which is the difference-product (P) of x, y, z,...., and the other of the form W/V, where W and V are integral symmetric functions of the same variables. As an illustration of this, full consideration is given to the,case where f(xyz,)....) = C)(y - b)(z-) the number of variables being n. The appropriate symmetric function V, which is divisible by all the denominators of the aggregate E [ +f(x,, z....)] is evidently in this case,(x-a)(x-b)(x - c)... (y-a)(y-b)(y-c).. (z-a)(z-b)(z-c)...,or say, F(x). F(y). F(z)....; and the corresponding numerater U, always divisible by the *differenqe-product of x, y, z,.... is in this case, because of the peculiar form* of the denominator of the function f, also * The form is such that the result of any interchange among x, y, z,.... is.attainable by a corresponding interchange among a, b, c,.... 344 HISTORY OF THE THEORY OF DETERMINANTS divisible by the difference-product of a, b, c,.... It is thus seen that the given alternating aggregate ____________ PP1 [ ( - a)( -b)( - -c)... = V ' where P, P', V are known, and lc has still to be found. An easy step further is made by inquiring as to the degree of k, it being noted in this connection that the degree on the one side is -n, and that on the other side the degree of P= n(n( -1), the degree of P' likewise =-n(n -l), and the degree of V= n2 The resultant degree of PP'/V on the right is therefore inferred to be = 1n(n-1) + -n(n-l) - n2, = -n; and as a consequence the degree of k must be zero. In other words, 7k must be constant in regard to x, y, z,...., a, b, c,..... so that for its full determination the best thing to do is to select as easy a special case as possible. Cauchy's choice falls on the case where x = a, y = b, = c,....; and preparatory for this substitution he transforms the above result, +(i i -)(Y-b)( 5PP -- (x- a)(y- b)(z-c)....i = J V ' into k.PP' =. (x- a)( - b)(z-c).... A f eL ( -a)(-b )(z-c).... As for the right side of this, it has to be noted that, since V contains each of the binomials x-a, y-b, z-c,.... once and once only, any one of the 1.2.3.... n terms under Z will vanish when the substitution X y,,.... = a, b, c,.... is made, unless the denominator of the term also contains all the said binomials. But by reason of the interchanges which produce the other denominators, the first term is the only one ALTERNANTS (CAUCHY, 1841) 345 of thlis kind: and the value of it after the substitution has been made is (a -b)(a — c).... (b -a)(b -c)...(c -a)(c -b).. an expression which, as we have already seen in the preceding paper of Jacobi's,* is equal to As the left-hand side, UPP, becomes under the same circumstances k. P2, we have as our last desideratumn and are thus enabled to formulate the proposition 4- 1 (x - ac)(y - b)(z - c)...I = (-1)Pn(n —') P(x,q,Z,...). P(a,b,c,...) (x-a)(x-b)(x-c)... (y-.a)(y-b)(y-c)... (z-a)(z-b)(z-c)... a noteworthy result which in later notation takes the form (x-a)'" (x-b)'- (xr-c)'... (l'n~1)Q(x,y,z,...). ~i(a,b,c,... (x-a)-' (x-b) —l (x-c)-l.. (y-a)-I (y-b)-l (y-c)-I lx.y..z where n is the number of variables, and F(x) = (x-a)(x-b)(x-c).... * Since V=F(x). F(y). F(z)...., the first term of the alternating aggregate may be written F(x) Fly) F'(z)_ x - a y-b z-c which, on the substitution being made, becomes F'(a). F'(b). F'(c).... and it is this form which in Jacobi is replaced by (-l)1fl(f - 1)p2* CHAPTER XIII. JACOBIANS FROM THE YEAR 1815 TO 1841. IT is not improbable that determinants in which the number of a row is distinguished by differentiation with respect to a definite variable, and in which the number of a column is distinguished by a particular function set for differentiation, may have appeared long before the time of Cauchy and Jacobi, the likelihood probably being the greater the fewer the number of functions and variables involved. There can be little doubt, for example, that expressions like 'au av Atu?v Dx ay ' y Dx may be found repeatedly in the writings of mathematicians belonging to the eighteenth century. It would appear, however, that the first who got beyond the second order, and clearly associated the expressions with determinants, was Cauchy. CAUCHY (1815). [Theorie de la propagation des ondes a la surface d'un fluide pesant d'une profondeur indefinie. Me'm. presentgs par divers savants a t'Acad. roy. des Sci. de l'Inst. de France.... i. pp. 1-312 (1827); or (Euvres completes, Ie ser. i. pp. 5-318.] Cauchy was a competitor for the prize for mathematical analysis in the 'concours' of 1815, and gained the prize. His work, however, like others belonging to that interesting political JACOBIANS (CAUCHY, 1815)3& 347 period, was not printed until long afterwards. In the form which it takes in the collected works the essay proper extends to only 108 pages, the remaining 210 being occupied with notes: this was probably due to the circumstances unuder which the -paper was first written. In the same way is explained the writer's action in referring in it to himself by name, the ob~ject being to preserve his anonymity. There is only one passage in it which directly concerns the,student of determinants, but it is interesting from more than one point of view. The exact wording of the passage (pp. 11, 12) is as follows:"Cela pose' concevons que le sommet de ia molecule rn, auquel.appartenaicut, dans le premier instant, les trois coordonne'es a, b, c, se trouve, au bout de temps I, transport6' en un point dont les,coordonn~es soient x, y, z. Les trois arbtes de in molAcule qui -aboutissnient an sommet dout ii s'agit, et qui, dans lorigine, se trouvaient paralleles aux trois axes des coordonne'es, auront alors,cesse' de l'bre, et les projections de ces m~mes arbtes sur les axes dont il s'agit, projections qui dans 1'origine e'taient respectivement egales, pour la premibre arefte 'a (Ia, 0, 0, pour la seconde, 'a.. 0,1 db, 0,3 pour la troisib'me, aN. 0, 0, de, seront alors devernues pour la premiere arebte dda, d'y d a,dz da, da 'da (lda pour in seconde dr (lb) dy Ad z (l p (l~~~~b db (lbd pour in troisib c me -X d de, - dIC. de d I Ii est aise' d'en conclure (voir in Note I.) que le volume de ia mnol~cuie, qui e'tait primitivement 6'gal 'a da dA dc, sera devenu, nut bout du teraps I, xds dy dz dx dy dz dx dy dz dx dy clx (da dA do dad db -c d b cia dehdb d eda dx dy dx dx dy dA da db dc - dc d b d a +de da dA 348 HISTORY OF THE THEORY OF DETERMINANTS et, comme ces deux volumes doiven &tre 6quivalents, on aura, par suite, dx dy dz dx dy dz dx dy dz dx dy dz la db de dadedcbdb a'dc +db deda dx dy dz ~dx dy dx d cd b da d,cdFadbA Si, pour plus de simplicite', on fait usage de la notation adopte'e par M. Cauchy dans son Me'moire sur les fonctions syrne'triques,* e'6quation prendra la form-e suivante: dx dy dx\ ci(a (lb cie - le sigue S e'tant relatif 'a la permutation des trois ]-ettres a, b, c." Here we have clearly the Jacobian of x, y, z with respect to a, b, c: and we have it expressed also in the determinant notation then in use. The second point of interest is centred in the note to which the anthor directs his reader. This note, which consists merely of the formal statement of a theorem, and extends to only ten lines, is as f ollows "1Si l'on rapporte la position des sommets d'un parallelepipede 'a trois plans rectangulaires des x, y, et x; que lPon de'signe par A, B, C, les longueurs des trois arbtes de cc para~llelpipbde qui aboutissent 'a un meme sommet, et par A1, B1, el) A2, B 2, 02, A3, B3, 03, les projections respectives des mebmes areftes sur les axes des x, y, et z, le volume du parallelepipede aura pour mesure A1BC0 - A1BC + A9B3 - A9B, + A3B, - AB2 = S(~- AIB2C3)." *There is a curious oversight here. In a footnote, Canchy says " Le Me'moire dont ii est ici question a 6te imprimi' en partie dans le XVije. Cahier du Journal de l1cole Polytechnique. " Now, as a matter of fact, there is no memoir bearing this title. The well-known memoirs contained in Cahier xvii. are beaded IIMe'moire sur le nombre des valeurs....and " Memoire sur les fonctions qui.... The second part of the latter bears the approximate designation, "IDes fonctions syt-ietriques alterndes.... "; hut the notation in qnestion occnrs in both parts. It is also not clear what was intended by the words I imprime' en parties' in Cauchy's footnote. Both in the original and in the reprint fonr signs are twice printed incorrectly, and in the reprint D's have been substituted for d's of the original. JACOBIANS (CAUCHY, 1822)34 349 Here we have one, of those so-called "Capplications of determinlants to geometry " which are often snpposed to belong to a much later date. CAUd-"lY (1822). [Me'moire sur 1'inte'gration des equations line'aires aux diffireuces partielles et 'a coefficiens constans. Journ. de 1'JPc. Polyt., xii. 19'e Cahier pp. 511-592; or (Euvres completes, 2e s er. i. ] Here again bnt under quite different circumstances the same form of determinant presents itself to'Cauchy. Having found the value of a certain multiple integral to be wbere D = ~ab'c"..., he adds "II est essentiel d'observer que, si lPon de'signe par L le de'nominateur 'commun des fractions qui represente les valeurs de u, V, W) tire'es des equatioiis dM d1M dIM U~ F- V dv- d W,: (IN dN dN itU + V + W dw+.=1 (IP dv LIP dP dP ad~ d ~Vv 7T+zv dw+ =1 et par Lo ce qui devient L quand on y pose JPL,=Po V V0, WW5.,on aura identiquement D =L0. JACOBI (1829). [Exercitatio algebraica circa discerptionem singularem fractionum, quae plures variabiles involvunt. Grelle's Journal, v. pp. 344-364; or Wer-le, iii. pp. 67-90: also abstract in Nomtv. Anrnales de M'ath., iv. pp. 533-535.] To the great mathematician whose name was ultimately associated with determinants of this special form, they first 350 HISTORY OF THE THEORY OF DETERMINANTS appeared in a totally different connection. He was considering a problem of the partition of a fraction with composite denominator into others whose denominators are factors of the original, and the paper to which we have come concerns given fractions of the form (ax+by-t)- (b'y+a'x-t')-1, (ax +by + cz - t)-(b'y+c'z + a'x - t')(c"z + x+b"y -t")-, The expansions of these clearly contain a variety of terms, the reciprocal of each linear expression contributing negative powers of its first term and positive powers of the others; and the 'discerptio singularis' consists in obtaining fractions which produce, each of them, the aggregate of the terms of a particular type found in the expansion. Thus, to take the simplest example, viz., (ax + by) - (b'y + a'x)- 1, it is seen that the expansion of (ax +by)-1 will contain one term with negative power of x and others with a negative power of x and a positive power of y, that the reverse will be the case in the expansion of (b'y+a'x)-1, and that the product of the two expansions will therefore contain a term in x-ly-1, a series of terms with negative powers of x and positive powers of y, and a series of terms with positive powers of x and negative powers of y. Now Jacobi establishes the identity (ax+by)1(b'y+a'x)- = 1ab'l-1. b {1 a }__ x y x ax+ by y by+axj' where on the right there are three parts; and as the first is a term in x-ly-1, the second equivalent to a series of terms consisting of negative powers of x and positive powers of y, and the third equivalent to a series of terms consisting of negative powers of y and positive powers of x, it is clear that the three portions of the expansion of (ax+by )-l(b'y +a'x) - have been isolated and summed. Now it will be noticed that a common factor of the three parts is the reciprocal of the determinant acb'l, or as Jacobi, following Cauchy, writes it (ab'). The corresponding factor in the next case, where there are three linear expressions and three variables, JACOBIANS (JACOBI, 1829)35 351 is found to be (ab'c") -1I; an d Jacobi then makes a generalisation regarding the first of the partial fractions in each case, viz., to the effect that the coefficient of -1 -1I - 1 -1 X XI X2. n. 1 in the expansion of U, 1 2 n. i. e. of is ((ab'". -the resnit being, so to speak, the discovery of the generating function of the reciprocal of a determinant. Shortly after this follows the passage which is interesting in the history of Jacobians. It stands as follows: "1At theorematis, de quibus in hac commentatione agimus et quorum modo mentionem injei' s latissimam conciliare licet extensionem. Ponamus enim., u - 1,,u m1 -t,. mam series esse quaslibet, sive -finitas sive infinitas, ad dignitates integras positivas elementorum x,1, x1j... procedentes, quaruni serierum 1, 1'....sint termini constantes. Sint porro in seriebus illis it, u1, U2...termini, qui primus ipsorum x, x~,.2 ''dignitates continent, respective ax, /x1, c"x2., ac ponamus, uti in easu lineari, fractiones (it - 1)-i, (it1 - t')', (u2 -t")-1,.. evolvi respective ad dignitates descendentes ternminorum ax, b'xj, C"X2. -' Vocemus porro A determinantem differentialium partialium. sequentium: Du Du D- Dau Di1 D1 D21Dn Dx' Dx1' Dx2. Du l N2 Din2 ZU2 Die2 Dx' DX' DxD_' Dx 'ax ' x Dax Erit e.g. pro tribus functionibus un, "In, in2, tribusque variabilibus x, y, z.A Du Du, D?12 Die Du1 Din2 Din1 Die2 Dan Dx - ay az Dx DX- a 'y a y ax Dz _ i2 D De1 D Di1 Di2 D D1 Di2 Dz- Dy D x + Dy Dz' Dxr Dz D Z,-x D y' 352 HISTORY OF THE THEORY OF DETERMINANTS quam patet expressionem casu, quo 'a, it 15 1a2, sunt expressiones lineares, in expressionem ipsius A~ supra exhibitam redire. Quibus positis dico, siquidem x =p, x1 = PI) 2 =-P2. Xn-1 =An-1 satisfaciant aeqnationibus ua= t, 'a1 = t', 'a12 = t",'a,= -',producti dictum in modum evoluti, partem earn, quae omnium simul elementorum...p- dignitates nlegativas neque ullius positivas coutinet, ut supra in casu multo simpliciore, fleri (X P (I P) X2 P) X-1 -1 It will be observed that Jacobi looks temporarily upon the ordinary determinant (ab'c"...) as the particular case of the Jacobian in which the involved functions are linear in all the variables concerned. JACOBI (1830). [ De resolutione aequationum per series infinitas. Crelle's Jozo-nal, vi. pp. 257-286; or Werloe, vi. pp. 26-61.] Although the general subject here is new, there is a certain link of connection with the preceding paper, in that one of the results of that paper is employed, and also that Jacobi is using once miore the method of 'generating functions.' Passing over the first two cases, let us note how he proceeds with a set of three equations and three variables. As a preliminary he introduces after the manner of the preceding cases the determinant of the partial differential coefficients, the sentence in regard to it being [page 263]"Ut similia eruamus de tr-ibus functionibus, tres variabiles x, y, z involventibus f(x, y, z), ~h (x, y, z), ~b (x, y, z) adnotetur aequatio ideijtica: D[4/,(y)Wb(z) - jV(z)W1(y)] D+(Z) t/(x) - 4'(x) ~b'(z)] + D quam differentiationibus exactis facile probas. E qua, posito brevitatis causa V =~ WW + f'(y [q', (Z) V '(x)W+ f'(z)[ '(x) (xy) - JACOBIANS (JACOBI, 1830) 353 fluit sequens: Df[qS(y)/ (Z) - 4'(z) ~t(y)] P fMj4/ (Z) b'(~) - l/(X) I/ ()] ax +ay + ~ (Y - 0, (y)~b() Here the concluding identity has to be noted. He then establishes certain results concerning the coefficient of x-zy-l1-l in the expansion of V, or, as he writes this coefficient, Lv] Y 1 -11 Thus prepared he attacks the given equations (p. 284)=ax + by + cz + dx2 + exy + r= a x + bVy +c'z + d'X2 + e'xy + v a" x + b"y + c"z+d"X2+e"xy + obtains first the derived set s = Ax + ax2 + fxy + yy2 + t = Ay + a X2 + J3'Xy + _ y'y2 +... U = Az + a"X2 + /3"Xy + yly2 + where the values of s, A, a,..., t, A, a',..., mA, a",... are sufficiently suggested* by giving one of them, viz., A = (ab'c"); and then seeks to find any function of the roots, F(x, y, z) say, in the form of a series proceeding according to powers of the constants s, t, U, the result being that the coefficient of sPtq9r in the said series is shown to be in general F(x, y, z).V] Xr+lyq+l~r+lX1 Iwhere X, Y, Z are the variable members of the derived set of equations, and V is the determinant of their partial differential coefficients with respect to x, y, z. * In later notation the derived equations would of course be writtena b c - a b c x+ d b C r b' c' a' b' c ' d' b' cf v b"N a~1 cc" e d" V c" M.D. Z 354 HISTORY OF THE THEORY OF DETERMINANTS No other case is dealt with, but the paper closes with the sentence"Quae autem hactenus de duabus, tribus aequationibus inter duas, tres variabiles propositis protulimus, eadem facilitate ad numerum quemlibet aequationum et variabilium extenduntur." JACOBI (1832, 1833). [De transformatione et determinatione integralium duplicium commentatio tertia. Crelle's Journal, x. pp. 101-128; or Werke, iii. pp. 159-189.] [De binis quibuslibet functionibus homogeneis secundi ordinis per substitutiones lineares in alias binas transformandis, quae solis quadratis variabilium constant; una cum variis theorematis de transformatione et determinatione integralium multiplicium. Crelle's Journal, xii. pp. 1-69; or Werke, iii. pp. 191-268.] The latter of these memoirs is by far the more important; in fact, it may be fully described as the summation and development of a series of memoirs of which the former is the last. As is natural, therefore, six pages of it at the outset are occupied with an introduction, in which the main points of the said series of papers are recapitulated. Seven-and-twenty pages are then devoted to " Problema I," which may be roughly characterised in later phraseology as the problem of the linear transformation of an n-ary quadric. Then follows "Problema II," the solution of which occupies pages 34-50. Its subject is the transformation of a very general multiple integral, and is closely connected with the subject of the preceding problem by reason of the fact that the integrand involves a power of an n-ary quadric. It is in this portion of the memoir that the special form of determinant which we are now considering makes its appearance. From two particular results in the previous papers referred to, Jacobi infers the existence of a general theorem which he states, adding that the demonstration is, however, not so easy, and that as a contribution towards it he will enunciate certain general theorems, the proofs of which for the sake of brevity are suppressed. His starting-point is the following: JACOBIANS (JACOBI, 1832, 1833) 355 If, 2' *..., &n-2 be any given functions of v, v2,..., n_-2, then CID2... V^-n = a (S^ ~ _ 2.,, DI —2 vl Z... vn-_.. VaU1 DV-2nFrom this he proceeds to cases where there are additional $ functions not independent of the others, enunciating and proving the result where there is one connecting equation, contenting himself with the mere enunciation for the case where there are two connecting equations, and leaving these to suggest the general theorem. The two enunciations areTHEOREMA 1. Datis $l, 2'..., n- ut functionibus ipsarum v1, v,..., v,_,- si inter variabiles illas datur aequatio F(1, $,..., -)= 0, erit 3g'342 * *. *$n-2 - ~ _ __ D 3 vA 'at2..* vn-2 3En-1 a n-1 THEOREMA 2. Datis I, $2,...,, ut functionibus ipsarum v, v2,..., vn, si inter variabiles illas proponuntur duae aequationes: F(,.... 2, _ ) = ~,_ 1(,,....,,) = DEl'ae2 P DE-2 D4 ____ _ __.... ol-2 3F 0b -F 3< V D v av..... * 9IJ,, F 3 aFF 3 ' 3n-1 D3en Dn Dn-1 aU-1 Van Dvn vn-1 His third theorem is a special case of his first, and may therefore be passed over. Then we have THEOREMA 4. Supponamus l, f2,.... In-1 datas esse sub forma fractionum _U _ U2 _Un-1 1- U e2 - u'.n- - U fit V 3$- a2 aen- 1 aul aU un-1 I1 ~ 2..1..1_-1 U-_1 u- 2 _V n-V_ ubi in altera summa inter indices permutandos etiam referri debet index 0 seu index deficiens. From this is deduced THEOREMA 5. Si loco functionum u, up, u,....u _ ponitur u UI U2 Un,_ t t t t 356 HISTORY OF THE THEORY OF DETERMINANTS designante t aliam functionem quamlibet, expressio Z nl aU _1 2?un-1 V~ Iu2 D' u u2 abit in I Uc~- ~aU IU2 aun-1 1 ~ Uwu1 U2 2 3Un_ sive in differentiationibus instituendis denominatorem communem t ut constantem considerare licet. The last of the series is THEOREMA 6. Sint u, u,, u2,..., Un_ expressiones lineares aliarium functionum w, w1, w,..., w,_d, datae per aequationes huius modi Uk = akW + 0kW1 + akW2 +,...+ akn- )Wn_ fit aulu, a u -,a2 3/U, awl (wn-n-1 V~U... - - = 2E + aa^... a W ~W-1.. a n )1 'a'IDvuu 2 v. i =- kdaa'a2 a1 n \O\n-ii to which is added the remark that if there were one additional independent variable v we should similarly have aV a l= (Z,' () DV1 Vn-l CATALAN (1839). [Sur la transformation des variables dans les integrales multiples. M6moires couronne's par l'Academie... de Bruxelles, xiv. 26 partie, 47 pp.] Having devoted the first part (pp. 7-18) of his memoir to the properties of determinants (see above pp. 224-226) Catalan is ready to use them in the second part in dealing with his main problem, viz., that specified in the title. The integral as given being IF(sx, x2..., xn dx. dX2.... dxn, and the equations connecting x, x2,..., x with the new variables ul,,2)..., un being 1 =, 52 =0...,I = 0, he first seeks to remove the variable xi. In doing this JACOBIANS (CATALAN, 1839) 357 x2, x3,..., x are constants, and the connecting equations involve the n +1 variables x1, ul, ui2,..., un, so that he obtain dx du2 du 2 dun d2 dl + ddu, + dou, 2d2 +.... d = 0, dx + dul l+ du2 du, = 1 d1 2i2 ^n dx + d, + d+. + -d% + d-du +. and thence N di = D dul. By dealing similarly with x2, xs.... in succession the result reached is Nd NN Nd dx1. dx2.... dxz = (-1)N 2 Dn d d.... d,.. Di D2 Dn and as N2=D1, N3=D2,...., N,=D,_, the ultimate form is dx1. d2.... dn = ()n 1. dd. d2.... dcn, Dn where " N. est le denominateur de la valeur des inconnues dans les equations d^ l+d +* dqbl dpl, Z + doJ1 2 _....+ du Zn 02 Z1 + d0oZ + *.* * + d Zn = 1| dp2 dx2 dx__ du Z + du 2 d. + tandis que D, est celui qui correspond aux equations l Y1 + dz Y2 +. Yn dx1 dx2 dXn dx: Y1+ dx Y2+.... + =1" 358 HISTORY OF THE THEORY OF DETERMINANTS -a lengthy and (because of its y's and z's and l's) an awkward way of saying that N= E f..... don 1 E ( duI du2 dun and Dn= (dp~dx2 clq5 ~ — s( —.Xcl~ dX2 dx. Catalan refers to Lagrange and 'others' for the cases n =2, n=3, but does not mention Jacobi's paper of 1833.* JACOBI (1841). [De determinantibus functionalibus. Crelle's Journal, xxii. pp. 319-359; or Werke, iii. pp. 393-438; or Stackel's translation 'Ueber die Functionaldeterminanten,' 72 pp. Leipzig (1896).] Up to this point, as will have been evident, the special determinants which we are considering have turned up merely incidentally in the course of other work. Now, however, we come upon a separate and direct investigation of their properties, the memoir under consideration being the second of the three portions into which Jacobi divided his formal exposition of the theory of determinants. From the mere fact that separate treatment is bestowed by him on only one other special form, it is clear that the subject of the memoir had come to be considered of particular importance. The same is rendered still more strikingly apparent when it is recalled that of the 87 pages occupied by the whole exposition, as many as 41 are devoted to this second portion concerning a subordinate form, while only 34 are assigned to what we are bound to consider the main portion, viz., that dealing with determinants in general. At the outset the preceding memoir 'De formatione et proprietatibus determinantium' is referred to, and intimation made that there is now about to be considered the special case where the elements are partial differential-quotients of a set of n functions each of the same n independent variables, and that in this case the special name functional determinants may with * On the question of the authorship of the theorem of transformation see MANSION, P.: Discours sur les travaux mathematiques de M. Eugene-Charles Catalan. M6m. de la Soc. R. des Sci. (Liege), 2e ser. xii. pp. (1-38) 10-12. JACOBIANS (JACOBI, 1841) 359 convenience be used. Jacobi takes pains, however, to explain that this relation of general to particular may appropriately be taken in reverse order, going, in fact, so far as to say that from the properties of functional determinants the properties of what he calls algebraic determinants may be deduced. He is careful to note also another relationship of the same kind, his statement being that in various questions relating to a system of functions the functional determinant is the analogue of the single differential-quotient in the case of a function of one variable. The subject of the notation of partial differential-quotients is then entered on at some length (pp. 320-323), and the decision made to use D in the manner which soon afterwards came to be familiar. The insufficiency of this notation is not forgotten, however, although its advantages over the different devices of Euler and Lagrange are recognised, his illustrative example being the case of Z where z is a function of x and u, and u is ax a function of x and y. He puts the whole matter in a nutshell when he says that it is not enough to specify the function to be operated on and the particular independent variable with respect to which the differentiation is to be performed, but that it is equally necessary to indicate the involved quantities which are to be viewed as constants during the operation.* * I may state in passing that in 1869 when lecturing on the subject I found it very useful to write X, y,,.9, t,, v,.... in place of f, (x,, z), f (s, t,, ).... and then indicate the number of times the function had to be differentiated with respect to any one of the variables by writing that number on the opposite side of the vinculum from the said variable; thus 1 3 2 132 <x,y,z meant the result of differentiating once with respect to x, thrice with respect to y, and twice with respect to z. Using this notation to illustrate Jacobi's example, we see that if it were given that Z =- 4X,oM we should have ax- 1 but that if it were given that z = x,u and u= 'xy 360 HISTORY OF THE THEORY OF DETERMINANTS The dependence or independence of equations is the next preliminary subject (pp. 323-325), the starting-point being the definition of an identical equation as one in which every term is destructive of another, and from which, therefore, it is impossible to express one of the involved quantities in terms of the rest. On this the definition of mutually independent equations is made to hang, such equations being defined as those of which no one at the outset is an identical equation nor can be transformed into an identical equation by aid of the others. Then taking m +1 equations, U = 0, U1 = 0,.... ',, = 0 involving n+ 1 quantities x, x,..., xn he contemplates the possibility of solving u =0 for x in terms of x, x2,..., xn, and the substitution of the expression in place of x in the remaining equations. The latter equations as altered he supposes to be dealt with in the same way, and the process continued until k+1 quantities have been eliminated and m-k equations left involving x1+,, xk +2, X., x. Reasoning from this, he concludes that a number of given equations are mutually independent or not according as by their help the same number of involved quantities can or can not be expressed in terms of the remaining quantities. In this connection he does not omit to draw attention to the existence of exceptional cases, such as that in which two of the quantities, x,, xk, say, occur indeed in all the equations, but always in the form x, +xk; and this leads him, for the sake of greater definiteness, to introduce the qualifying phrase, 'with respect to certain quantities' in using the expression 'mutually independent.' His words are"Aequationes u=0, u1=0,...,. %,=0 quibus totidem quantitates x,, -..., x,, quas involvunt, determinantur, harum quantitatum respectu dico a se independentes." From the independence of equations he naturally passes (~ 4) to the independence of functions, with the remark that exactly then we should not be certain as to the meaning of -, as it would stand for ax' 1 1 1 1 X,llt or qx,- + x,,u. x,,y according as u or y was to be considered constant. JACOBIANS (JACOBI, 1841) 361 similar propositions are found to hold in regard to the latter,-a statement which it is not hard to believe when we recall that any function, x2+ y2-4xy say, may be denoted by a functional symbol, f say, the equation f=x2 + y2-4xy thus resulting; and that any non-identical equation connecting two or more quantities implies that any one of the latter is a function of the others. Functions of several variables are said to be mutually independent when no one of them is constant or can be expressed in terms of the rest. This is extended and made more definite by saying that if functions of x, x,...., x,, involve also the quantities a, ac, a2,..., the functions are said to be mutually independent with respect to the quantities x, x,..., x,, if no equation subsists between the functions and the quantities a, a, a2,... These definitions will suffice to indicate the analogy above referred to, and the deduced propositions (pp. 325-327) need not be entered on. All this introductory matter having been disposed of, Jacobi proceeds (~ 5, p. 327) to deal with the subject proper, his starting-point being the fact that if there be n+1 functions f, f 2, f.,., f, of the same number of variables x, x1,..., x, there arise in connection with these the (n +1)2 quantities Dfc The determinant formed therefrom, viz., Df atf Dfif he calls the "determinant pertaining to the functions f, f,...,, of the variables x, x1,..,,," or the "determinant of the functions f, fi,..., f,,, with respect to the variables x, xc,..., xv,." The case where n=0 is then referred to in a line, after which cases are taken up where it is the functions that are specialised. The first of these is that in which f=n+l = c,,+1, fm+2 = Cm+2),... f. n = X,, it being pointed out of course that the order of the determinant is then lowered, being equal to - Df Dfi af, 362 HISTORY OF THE THEORY OF DETERMINANTS Another is that in which the functions f/m+, fm+2,..., f, do not involve the variables x, x,..., xm, the peculiarity then being that the determinant breaks up into two factors similar to itself, being equal to af.... afm afin+i. fm +2... fn X 3ZD y l DXm^ ~m~ + D X)m+2 Dx The important proposition regarding a vanishing functional determinant is then dealt with (~ 6), viz., the proposition "functionum a se non independentium evanescere Determinans, functiones quarum Determinans evanescat non esse a se independentes." The proof of the first part of it opens with the assertion that since the functions are not mutually independent, there must exist an equation n(/,A f.., / n) - such that on substituting for f, fi.. fg their expressions in terms of x, x,...., xn we shall obtain an identity. From this by differentiating separately with respect to x, x,..., x, there is obtained the set of equations o = L II~ fi II fn x ax f D' a Df +' ax a 'fn a an + 3/i an + + 3fn all 0 xf a+ a. + l af+ an oX/n af a Xrn aDi 3 Xn Ifn Then it is recalled that in a set of linear equations a11x + a12X2 +.. + 1ax, = 0O an1X1 + an2X2 +.. + cnnn = the determinant of the coefficients must vanish unless all the unknowns vanish. And as the vanishing of an: an an 3f' af' ' a/ would imply that the expression II(f, f.,..., f) was free of JACOBIANS (JACOBI, 1841) 363 f,f A...... fi the conclusion is reached that the determinant of the coefficients of these differential-quotients must vanish, i.e., that the functional determinant axcr ax, ax" The proof of the converse proposition, Jacobi owns, is 'paullo prolixior.' It is of the kind improperly known as 'inductive,' and the first part (Q 7) of it goes to show that if the proposition holds in the case of 'afj f. a a'a it will also hold in the ax1 Ox2 axn case ofE ~f af1 D~f* As a preliminary, there is estabDax ax1 ax lished the lemma thatIf f, f, fI,..., f, are mutually independent, then af af 3 Z a l f 3 -4-D-.f n f)-f1 = 2 fn axi~ ax, ax Ex ax, ax Oxx DxD1 nx 1 2 nx where the brackets enclosing a differential-quiotient are meant to indicate that f there is to be taken as a function of X, fl, f2,. ) L Denoting the cofactors of Df Df af ax) ax, afn in the determinant ~ Df af,.. by -ax ax, Ox", A, Al1An we have of course f _f _af a Df Df1 afn A+ AA + + A,= A ~ ax ax1 x x ax O1 n f A+afl l+ afl n ax axi axn ~fn abaf fA+2fn A, ~ + An ax ax1 a1 But since fl, f2'..., f, are mutually independent it is possible by solution to obtain x1, x2,..., xn each in terms of the 364 HISTORY OF THE THEORY OF DETERMINANTS remaining variable x and fl, f2,..., f; and as a consequence it is possible by substituting for x1, x2,..., x. to obtain f in terms also of x, f/, f2,.., f. Differentiating f with respect to x, xl..., Xn, and using brackets to indicate that the f within them is to be viewed as a function not of x, x,..., but of x, fl,.... f, n we have = (f (t) DX De2) + *** n DeX' Df1 _ (Dff D fl + (Lf D%2 (D Of D.f -3^ W \3/./... \2/ t2) * \2fJ Baf Dx -x Dx f + + D+af Df, (D) Df1 + (L f2 + + a.... Dfn D3 \Df/i xC, \af2/ 3\Dx "" J x / Dox afn ^ M JL l\ af DA j- \ fafn ax \af/ +^ +' Dx a " 1\/ 3^' From these, on multiplying both sides of the first by A, both sides of the second by Al, and so on, and~then adding in columns, there is obtained, with the help of the n+1 equations immediately preceding, + / fDfi,, n -- = A + o +..+ + o -,x Dx1 Dxn x.. which is what was to be proved.* Now suppose that in any way it has been established that if +-f.... Dfn = 0, the functions involved are not mutually Dx1 Dxn * Using this theorem upon itself we have x Dia', D Dx6~ -l*4 x2 ax.. provided that on the right is expressed as a function of,, fA, f,...., fn, and.f as a function of x, xl, f,..... f,; and ultimately Df 'af i (D I (Dfi 4 (Df- ) oD (x1 Dx x \D, - \X \Dx '/' provided that in every instance on the right fj is expressed as a function of x, x1, X2,.'.*.. X X, f+l...*.,' A theorem like this ultimate case Jacobi enunciates and proves quite independently at the end of his memoir (v. ~ 18). The one, however, is seen to include the other if we note the simple fact that DxfXDf1 EDf3 - DfD~ 1 DYf Dx'x1....) = V D' -Dn1.... DX Dx, ax, 3x-i " Z"1?axn^ ^'^-i " JACOBIANS (JACOBI, 1841) 365 independent, and that the next higher case is to be investigated, viz., where +ff...V fn = 0. Of the n+1 functions - ax x ax" involved in the latter determinant the last n of them must either be independent or not. If they are not independent, there is nothing more to be proved. And if they be mutually independent, then the lemma gives (Df) A 0, and therefore (f) = 0 for it is impossible that the functional determinant A could be 0, as the functions involved in it are by hypothesis mutually independent. The vanishing of j(a) however implies that x is not involved in f; and this, if we bear in mind the meaning of the brackets, further implies that f is a function of only f1, f2,..., fn -that is to say, that f f1, f2,..., f are not mutually independent. There now only remains to establish the proposition for the case where the determinant is of the second order,-that is to say, where f f _ af = 0. Dx ax1 Dx1 ax Here the function f must involve both variables, or only one of them, x say, or be a constant, If it be not a constant, x1 is expressible in terms of f, or at most in terms of fA and x, and thus, by substitution, f is expressible in terms of the same. In this way we have af = f + af i f ax \wx -' \ 'aS/ ax af= () ~ a \ f and therefore, by elimination of (-), (f)Afi _ DfD, af f afi \x ax x ax 3Dx1 Dx = 0. 366 HISTORY OF THE THEORY OF DETERMINANTS Now it cannot be that the second factor on the left vanishes, because fl is known to involve x,; consequently \ax/ From this it follows that of the two variables f, and x, supposed to be involved in f, the second is awanting, and that therefore f is expressible in terms of f1 alone. Our result thus is that either f, is a constant or that f and f, are dependent. The general theorem being thus established, Jacobi refers in a line or two to the corresponding contrapositive proposition, viz., that if the functional determinant do not vanish, the functions are mutually independent, and to the contrapositive of the converse, viz., that if the functions be mutually independent, the functional determinant cannot vanish. Finally, in recalling the ultimate case where the determinant is of order 1, he notes that, as in that case, so generally the four propositions may be combined in a single enunciation, viz., According as the functions f, f,.., fn of x, xl,..., x,, are not or are mutually independent, the functional determinant does or does not vanish. The next subject taken up (~ 8) is the solution of a set of linear equations af + Af f ~-r r+... + r = s, Dx ax1 axn ZA r r + 'af i = 1)I ax........ ~fn r rl fn.. + n rnSn, in which the determinant of the coefficients of the unknowns is the functional determinant _,af iff fn a~x ax ax, 1 Dfn Of course a condition of solution is that this determinant does not vanish, and therefore that the functions f, f,..., f, are JACOBIANS (JACOBI, 1841) 367 mutually independent. But this being the case it follows that x, x1,..., xn are expressible in terms of f, f,..., fn and, as a consequence of substitution, that any other function k of x, x,..., xn is expressible in terms of the same, thus giving us a3 = D~ x + i'A ai3 +.. + '.fax -af; Dx Df axk Dfk Ixi, afj (a). Now if we turn to our set of equations, and multiply by Df' f a' ''' af respectively, with the result f axr + af ax r + a.X _f Df1 f+ 3a ix'fr +, 3 + x afl V Z xf I DA Df xj. k 3xk. + a;,., r f - s,... +f x,~', af-, V, aDfn 3a + f aX + a x fn -'aX1 f,,n + f n -xk x _ K + DXfc =an Dafn nJ it is evident from (a) that on adding column-wise we shall find the coefficients of all the unknowns equal to zero, except the coefficient of Xk which is unity, and therefore that r = +a81 + + *+ n. Jacobi is thus led to formulate the proposition:"Sint variabilium x, xz,.., x functiones f, f,..., f, a se invicem independentes, si proponetur hoc aequationum linearium systema, Df f r + r + /x 3ax fir + ai I + Dx DX1i af... r = s, ax/... + r'n = s,, ax" -.r +f r + Ds 3;x1 f +- r,=s. Zx." = 368 HISTORY OF THE THEORY OF DETERMINANTS earum resolutio semper est possibilis et determinata eruntque incognitarum valores: Ax ax x '=f S + fi + + Sw, Dx1 e1 Dxl.ax, x.., I x, af?fS + f,, =f af s +... +,,. Put into the form of a 'rule' this amounts to saying that any one of the r's is got by taking its 'reversed' coefficients and multiplying each of them by the corresponding s, and then adding. Of course, having obtained a solution by using the peculiar eliminating multipliers a- Dx1, 3x1, af?fe ' afn and being aware of the existence of a general set of such multipliers, Jacobi had already to hand the means, which he did not fail to use, of finding an important identity. For, if the functional determinant be denoted by R, it had long been known that 1 (?R +R R A rj~ =R - 'vS }** "^ ^H a O ~ aafi l+ a a a and a comparison of this with the value of i. above obtained gives at once 1 DR Dxk -R,af, afi a particular case being I 3Z~Dll af, ax This result may be viewed as giving any one of the differentialquotients -a in terms of the differential-quotients; or, if we afi ox, i w multiply both sides by R, it may be viewed as giving an JACOBIANS (JACOBI, 1841) 369 expression for the cofactor of any element of the functional determinant. Before leaving this part of the subject it may be noted that Jacobi discusses with equal fulness a set of linear equations in which the determinant of the coefficients is the conjugate (as it afterwards came to be called) of the functional determinant, with the result that the practical 'rule' above given is shown to hold here also. The expression obtained for the cofactor of any element of the functional determinant is utilised (~ 9) to find an equally interesting expression for the differential-quotient of the functional determinant with respect to a quantity a which may be x, x,,... or any other involved in the functions. As R can be considered a function of its (n+1)2 elements i, it is DR clear that OR is expressible as the sum of (n+ 1)2 terms of the form D aft BR X7/ a afi?a This, however, by substitution of the expression just referred to, becomes DDff RXk aXe R Dfi Da or a afi axk; so that we have i=ni 7a=% 4/fn DR - R. Da Xk Da =~~ l Xk=, 'Dxk Df' i= 0 k=O. Dfi i=O M.D. 2A 370 HISTORY OF THE THEORY OF DETERMINANTS or, as Jacobi puts it, log R _ a a a. = V +3+, f. + + af With this theorem is associated another having no connection with it save the fact that the proof of it is dependent on the use of the same expression for the cofactor of an element of the functional determinant. Recalling the general theorem proved in his memoir De fornmatione... viz.,,(m+l) (n) _ AA'... A(m) = (2 ~aa... )), a- he applies it to the functional determinant, viz., to the case where w(i,_ ~ft as afi and where the cofactor A(t= RDax The immediate result, on dividing both sides by Rm, is + af Xf1. n,__.E ax3Z., Da xm, _ Dfrn+l I.. fn -ax ax aXn af afi afm DXm+l Dax, The theorem R Eax - ifi a a already obtained, is the special ase of this where 0. Another already obtained, is the special case of this where in = 0. Another special case is at the other extreme, viz., where m= n, when we have Dx x ax 1 -- -- ^1 n _- 1________ 7x, = 1?xax axx the generalisation-or the analogue, as Jacobi would seem to prefer to view it-of the theorem dy _ 1 dx dx' dy The next part (~10) of the subject relates to the case where the functions, whose functional determinant is sought, are not JACOBIANS (JACOBI, 1841) 371 given explicitly in terms of the variables-where, in fact, we have n+1 functions of x, x1,..., xn, f, f...., f,, viz., F = 0, F1 = 0...., F 0 and where we are asked to find OE +f., Af. - Ax D'O x, Differentiating any one of the F's with respect to any one of the xs we have a0=, aFD.. aDF,af, F a f__. 0 F F fFf +..y + =x, ~ +f aDx', D / x,, 'a / x,' DF. which may be viewed as giving an expression for - Using OXk this expression (n +1)2 times we obtain for.. an equivalent determinant each of whose elements is the sum of n+ 1 products, and which from the multiplication-theorem we know to be equal to aF aF aF Dn aff afn Df DOf1 Dfn X nxc Dxi, It thus follows that the result sought is DF aF1 DFn +3.3 I.... rn E -' fi, = (_l)n+l vL DX aDx ax1 a OF FD}1 DFn Oaf Daft afn a 'theorem which Jacobi again takes pains to have noted as the analogue of the theorem which holds when F(f, x) =0, viz., df. BF 'F dx Of 'x By way of corollary it is remarked that as the equations F=0, F1=0,...., F,=0 cannot be more appropriately viewed as giving the f's in terms 372 HISTORY OF THE THEORY OF DETERMINANTS of the x's than as giving the x's in terms of the f's, we therefore have the twin result F F1 _ F J 'W D1 ~f1 Df + zSG/id Df~ = (-1)n+1' 'F DE — D-f' af fi aF FF -ax ax ax", and consequently from the two a theorem already obtained by a different method, viz.,,ax ax f, _ = 1 -4-* *] +.. f'.... af af, aBfn L f, af, Saf n afn Steadily pursuing his analogy, Jacobi next takes up (~ 11) the case where the f's are not given immediately in terms of the x's, but are given in terms of functions 5, <1,.., op of the x's. Here, of course, we have aft = Af.?o + 'afi eI + + afi.af 3xk 30 3Dx - 3+ I ax3 0a, 3Xk and therefore by (n+ 1)2 substitutions there is obtained for af 3fi,,?f f Vf.....n — x 3ax aXn an equivalent determinant, each of whose elements is the sum of p + 1 products. This latter determinant, however, we know from Binet's multiplication-theorem is equal to 0 when p < n; is equal to the product of two determinants L —.af.... y -- D 1 30 a3n ax Dax1 3 x" when p = n; and when p > n, is equal to a sum of such products, viz., S -E'... Z' V-+ a 'x1.x v where the different terms included under the S are got by taking all the different sets of n +1 I's from the p +1 available. This tripartite result Jacobi carefully enunciates at length in the form of three propositions. He notes, too, that the first is practically JACOBIANS (JACOBI, 1841) 373 a result already obtained, because the functions in that case are not independent; that the second has for its analogue or ultimate case the theorem df _ df dy. dax dy d ' and similarly that the third is the extension of a result actually used in the proof, viz., Of -.f a- +f 1 + + f >P.?x 3a ox 3<, Ox a (p ax He even enunciates formally a variant of the second proposition, calling the variant "Proposition iv," viz., If f, f1,..., fl, be functions of y, y1,..., yn, and it be possible to express both the f's and the y's in terms of n+1 other~ quantities x, x1,..., Xn, then Df af, afn, f f1 af -aDx Dxn +....... _ =~x'3xi Dyx E+ 4 ---. x1 ^DY y Dyi ~ i^y Dyl, xyn ax ax1 3Xii The analogue also is again referred to in the form df df dx _ = _, dy - dy' dx and the special case, already twice obtained, where f=x, f,=xi,.. f* n=xnStill further importance is given to the second proposition by assigning the next section (~ 12, pp. 341-343) to the consideration of certain deductions therefrom. First there is taken the special case where =x, 0 1,.., n = =Xm,, and where therefore LO._1._... = 1,n_ - m+i 0+2 + 0 x O3x ax", x,,,,+ 3++, 2 3a,'2 the result clearly being that If f, fl,..., be functions of X, X1,... Xm, qm+i...,, aC nd qm+i... 5, be functions 374 HISTORY OF THE THEORY OF DETERMINANTS of x, x1,..., x, then the functional determinant of f, f,..., n with respect to x, x1,..., xn f=. af afm m+....l afn m +1.... an Dx Dx' aDX - aq m+a x~, DX~+, " Xn' Here the first factor would reduce to 4, f 'A....m -ax ax1 ax, if it were possible to put m+l == fm+l, Om+2 = fm+2), * * O il =-nf; hence there follows the further proposition, which Jacobi speaks of as "prae ceteris memorabilis," that the functional determinant of f, f,..., f, with respect to x, x,..., xn is equal to \ Dx x/ \xm -axxI3l xm+2 xn if the brackets in the first factor be taken to mean that the functions therein occurring, viz., f, f,..., f,, are considered to be expressed in terms of x, x,.., x, fm-i,m fm+2-, ', fn An extreme case of this, viz., where the first determinant factor is of the order 1, has already been given. In order that we may be able to substitute ax+1 a o E-t —1 ~ Vm+2 ~ a x, Dx- 4_ D., for C 4- + Dm+1 a m+2 _n.m, _ _m2+1 +2 n in the former of these two propositions it is necessary that from the equations which give mPl+1, m~+2,., 0n in terms of x, Xi,..., x, we obtain xm+l, Xm+2,..., x, in terms of the other x's and km+l, m+2.., On. Consequently, we have the proposition, If f, f1,..., fn and Xm+1, xm+2,..., X be expressed in terms of x, x,..., Xm, x m+1, 1 m+2, 0m+, qn the functional determinant of f, fl,..., f with respect to x, X1,..., Xn is equal to JACOBIANS (JACOBI, 81 375 of af, Dfm Df Df D ~~4Df~~f1 in1+i m+2... Ox ax, Dxrn Drn,1i Dkm+2 Dp11 Similarly, in order to be able to substitute I ~~~for ~~D fn2 D, afM+1 aDx m+2 fX in the other proposition it is necessary that from the equations which give fm+i, fm,+2, *..,I fn, in terms of X, X1,...,I Xn We obtain xmr1 I Xm~2,., IX,, in terms of the other x's and fm~1' fm+2, *.f,.We therefore have the result-If he expressed in terms of the, functional determinant of f, fl,.., f with respect to x, XI, I.,Xn~, is equal to?m 1 D~+ Xm _am+1 Dfm+2 D Leaving this, Jacobi barks back (.~ 1 3) to an earlier proposition with'a- view to a generalisation now possible, viz., the proposition where the functions are given implicitly in terms of the independent variables by means of n +1 equations 'The extension arises from the number of equations now given being n +m, viz., F =0, F1 = 0. 1F+M = 0, and each of the F's being a function not only of x, x,,..., x f~f1..,,,but also of If11+1, f1~2).. fn+m. 376 HISTORY OF THE THEORY OF DETERMINANTS If the last m of the equations were solved for the last m of the f's, and these fs thus eliminated from the other n +1 equations, the functional determinant desired would, by the proposition sought to be generalised, be equal to (_)n+ \axi. v+()F 31 (DF aFn} where the brackets are used to indicate that the enclosed F's are in the altered forms resulting from the substitution referred to. Multiplying numerator and denominator by DF+l DFn+2 Fn+_m ~ fn +1 Dfn+2 fn+m we obtain a new numerator which by a later proposition is equal to F _T_ __n Fn+l DFn+2: F n+ and a new denominator which for a similar reason is equal to DF 3F1 DFn+F 73f 'fi t27 nrm The functional determinant desired is thus found to be equal to F DF1 DFn aFn+1 DFn+2 DFn+m ax Dx D, f~1f2 "t f ' D1. af f n+m where, it will be observed, all the F's are in their original form. As usual, the extreme case is noted, viz., the case where a single function f of one variable x is given by means of m+ equations connecting x, f, f,.., f, the result then being oF DFD1 Fm dx =X /fi fm. DY D- D 1 MDf' v- ~ *Df-i-* Df m JACOBIANS (JACOBI, 1841) 377 and also, as usual, the occasion is utilised to draw attention to the analogy between the differential-quotient and the functionaldeterminant. "Quam formulam si cum generali comparas, et hic vides perfectam locum habere analogiam inter differentiale primum functionis unius variabilis atque Determinans systematis functionum plurium variabilium." The theorem next brought forward (~ 14) is said to be useful in connection with the preceding general theorem for finding the functional determinant when the functions 'quocumque modo implicito dantur,' and is also spoken of as being in itself 'prae ceteris memorabilis.' It is-If f, fl,..., f, be functions of x, x,..., xn and there be given the equctions f = a, fl = al,., fn= an in which a, a,..., an are constants, then the functional determinant -ax ax1 xn wzill not be altered by any transformation made upon the f's by the use of the given equations, it being understood, of course, that the equation fi = ai is not used in the transformation of fi. Taking first the case where only one of the functions, say f, is. transformed, this becoming q by the use of the equations f =, f2 a2, ' = n.,, we see that p in addition to x, x1...,.,, may involve a, a2,..., a, and that therefore we have /f = v + v +fl.' 2 + + af/, ax ax Da, ax aa2 )x an a af =_ ~+ a( afl + 0f 2 +P afn ax3 3XI Oa, Ox, a2 3i a an 3ax, af _ 0+ a0 afi + ) 42 __ + 3 fn vx =. +-v + v..'v -'v... DXn uXn a axn a fa2 ax 3 d n an tn By substituting in the functional determinant df Dfi ', a `ax xDx1 ax,, 378 HISTORY OF THE THEORY OF DETERMINANTS the equivalents here given for af af Df' sx ax1 Dxn there is obtained a determinant which is expressible as the sum of n +1 determinants, all of which vanish except the first. We thus arrive at Dx ax ax, -ax' ax ax" as was to be proved. Passing to the case where two of the functions are changed, he says, first, that if, by the use of the equations = a, a, /f = 3,.;.,n = an the function fl is changed into 1,, then exactly as before it can be shown that 'a Df' afn D 4 )g9 i =)1 Df2 _ fn -X D31 i Xn ax ax, 2 ax More questionable is the logic of his second step, which is to the effect that from this and the previous result it follows that +af 2-)/ a* ^n i_ V^ 0 2 e*n ax~ff ax ~ afx,_ \~~ kf a2f -~_x D/ DVxn -faDx V 1 Dx2 Dx' His third step is simply the assertion that by proceeding in this way we may prove generally that if by use of the equations f = a, f= a1,...., il-==a, /fii l = ai-,.... = an fi becomes changed into qSi, then v / fl af/n = _+.-....; Dn - aDX 1 Bxn - x ax, ax and the matter is concluded with the further assertion that if in the elements of the second determinant there be substituted for.a, al,..., an the functions which they represent, that determinant will be identically equal to the other. Considerable space is next given (~ 15) to the discussion of the case where the number of variables x, xI,..., x+,m which JACOBIANS (JACOBI, 1841) 379 the functions involve is rn greater than the number of functions. First it is noted that if the functions be not mutually independent, they are not independent with respect to any n +1 of the variables, and therefore each functional determinant formed with respect to n +1 of the n+-m+l variables must vanish. Then the converse proposition is taken up, viz., that if all these determinants vanish, the functions are not independent. The method of proof is that known as mathematical induction, that is to say, the assumption being made that the proposition holds for n functions f, fl,.., f,it is shown to hold for n +1. Clearly we may start by viewing f, fi,..., fnl as being independent, for if they be not, there is nothing to prove; and this being the case, the various determinants of these functions with respect to n of the variables cannot vanish. Denoting the first of the said determinants, viz., Lf af af b B,_l..f. 1 A by B, - x ax, axnand choosing from the given vanishing determinants of the (n+?1)th order those having n of their variables the same as those of B, viz., the m+1 determinants, f. f, f-,. fn Oaf afs, fwt afn -x D xf 1 Dfn-1 fn+1, y4..... -ax ax, ax,_, ax+n ' x a Dx9 Dzn-1 ^n+m we see that from a previous proposition these are respectively equal to ax, ax +l/ (n+2) ax +m) where the operation indicated within brackets is meant to be performed on fn as expressed in terms of f? fin * * *? fn-1l ) X Sn- l * * * X Sen+nb 380 HISTORY OF THE THEORY OF DETERMINANTS As B does not vanish, it follows from this that (fn ) ( = 0,. ( ( ) -0, ax = ' X ' ' ' ',n+, and consequently that f/ involves only f, fi,...., f,-; that is to say, that f, fi,..., fn are not independent. This result being obtained, it only needs to be noted that the proposition being manifestly true in the case where the number of functions is one, must be true generally. As an addendum, it is pointed out that since the vanishing of the m+ 1 determinants af Sf, Dfn-l f,, L — ~x ~l ~x~a_ ~l Cx,' -3x ax, axn-l aDx+l +3f af'i af-1 Dfn -dx Dax, ax,_, axn,,m when the determinant, B, and the n2 elements common to all these do not vanish, implies that fn is a function of f, /,...,,: and since this mutual dependence of f/, f,..., f,-, f, implies the vanishing of all the functional determinants formed with respect to any tn+1 of the n + m+l independent variables, we are led to the conclusion that, provided B does not vanish, the vanishing of all these functional determinants of which the number is (n+m+l)(n+m)... (m+1) (n+rs + )(n+i)... (n ) 1.2.3... (n+1) o 1.2.3... (m+1) is a consequence of the vanishing of a certain m+1 of them. In order that the connection between the members of this group of functional determinants formed from the differentialquotients of f, f,..., fn with respect to any n +1 of the variables x, x,,..., n+m may be better looked into, several identities regarding square arrays of functional determinants are next given (~ 16). Taking in addition to f, f,..., f, the m arbitrary functions fJn+, fn+2,). ) f n+m JACOBIANS (JACOBI, 1841) 381 of the same n +mn+1 variables, and denoting the determinant E]+ f;, Dfl Dfn- 1 afn+i by bV<) - x ax, 3-xn_ 1 x+k k where i, k may each have the values 0, 1, 2,..., m, we see that from a previous result we have ) = B. ), if f,,+ within the brackets involves f, f,,..., f,/_ in place of x, xl,..., xa,_I. From this it immediately follows that 46 bb'.... bm) = B E-( f)... (') L'VV 1? \Ax,/ \CWn i/ \0}axn-m But the theorem already obtained regarding the factorisation of any functional determinant gives yEf.Dfi.. afnm -A-Df?f 9af Dx a +1 D +rn+ fax x, aDn - ( fn)(fusil ( Dfnn) = DX DX)1 - DI/ DXn/+m 1 = B - -\x \x+.. '.( \Xn+2nm' Consequently, by substitution we have finally -7- }bb1 Db(m) n,.-af.f; aSfn+m ~b....... -B' 4 - a result which Jacobi says is of frequent use in dealing with questions regarding determinants.* * A curious interest attaches to this result. On the right-hand side are two determinants whose elements are differential-quotients; but the first, B, being a minor of the second, the total number of different elements is simply the number in the second determinant, viz., (? +m+ 1)2. On the left-hand side is a compound determinant of the (m +l)th order, each of whose elements is a determinant of the (n + l)th order; nevertheless the number of different elements is again (n +m+ 1)2 and not (m+ 1)2(n + 1)2, because all the (m+ 1)2 elements of the compound determinant have the n2 elements of B in common, and of the 2n+ 1 which border these n2 elements, only one, viz., the cofactor of B, is different throughout, each of the 2n others being repeated m + times, so 382 HISTORY OF THE THEORY OF DETERMINANTS Next, -,() being used to stand for A, + Of 'fi laf-1 af_ _ S f+l _ afn z-.x ax, aXi-,1 x,,+k 3Dx+1?x21 it is sought-to find an equivalent for X ~f0(i).... (m). Clearly -f/i) is the determinant formed from B by using x,,, as an independent variable instead of xi; but for our purpose it is of more importance to note that it is the cofactor of afv+ in b(1, ax, for then the determinant under consideration, viz., being thus the cofactor of afn afn+l aDfn+m in the left-hand member of the identity just found, it only remains to seek the cofactor of the same expression in the that the total number of different elements is 2 + (m+ 1)2 + 2n(m + 12) (m + ). Further, on both sides the degree in these (n+m+1)2 differential-quotients is the same, being clearly (n +l) (m + I) on the left, and mn + (n + m+1) on the right. It is thus at once suggested to us that the identity is not necessarily an identity connecting differential-quotients only, but is true of any (n + + 1)2 elements whatever; and the suggestion is readily verified when the ubiquitous presence of B as a coaxial minor raises the suspicion that the identity must be an 'extensional.' The case where n=2 and m=3 is given on p. 215 of my text-book (Treatise on the Theory of Determinants) in the form5a,1 e5 Al 2CS e5 C I% e5 1 2 l lla eA fl Idb e5f6\ 1I12 e5f6l Ic13 e5fAl Id 4 e5fl 1 I C1 e5 % l I I2 %- Il I e Af 1 C4 e5 f6l alb2C e5. 6 where it is iewed an extensional of the ifest identity where it is viewed as an extensional of the manifest identity a, a2 a03 a4 b, b2 b1 b4 la~b2csd4l. C1 c2 c3 C4 d, dC d3 d4 The theorem in its general form may be enunciated as follows:-If from the determinant I al,n+m+l I there be formed all minors of the (n+ l)th order which have I a,, I for the cofactor of their final element, and these be orderly arranged in square array, the determinant of this square array of the (m + l)th order is equal to, al In". I ain+m7+1. JACOBTANS (JACOBI, 1841) 383 right-hand member. Now, in the first factor, Bi, of this righthand member the expression does not occur at all, and in the other factor, which is transformable into ( 1)n(n+1)'E ' f, fi fn - r afn af... — Xm+1 axm+2 DXm+, ax D ' it occurs multiplied by (_i)n(m+ln') E + _f _mn+I r'-in+2 rn+n The desired result thus is +i (l)../ () = (1lny+l)gm B f 1)l.Bf n-_ Dxm+l DXm+2 xm+ which Jacobi formally enunciates as follows: "E Determinante - - f af, 'f vl fx f fax,_l deducantur (m+ 1)2 alia Determinantia, uni cuilibet differentialium ipsarum x, xz,..., x respectu sumtorum substituendo successive differentialia ipsarum n,,x+1i,..., x,,+ respectu sumta: illarum (m + 1)2 quantitatum Determinans aequatur expressioni )(-){m+l)('~+l)Bm. 3,f.J f1.+ ' _ v-/" ax )? '12 7}ax From equating cofactors of?fn afi b + 'af' 3a3l Dx ax,... Jacobi proceeds to equate cofactors of afn-+l 'afm+n aX ' '3mi-1 in the same fundamental identity, the resulting theorem now being af Xf, Df. -l) = (-l)m(n+)gm. ~ a*.... af and then he adds, "Eodem modo obtinetur generaliter z abb.... b^:/3.I+ 3.... /p-l =~B"m. ' af,fi _fn+i-i _ X:-+B.,-:+2 f E~ oX7rn-ii1 '?Xmfi+2,_~n 384 HISTORY OF THE THEORY OF DETERMINANTS qua in formula signo ~ substituendum est aut (-1)n(m+1) aut (-l)m("+' prout i par aut impar est."* The second derived identity,-that is to say, the case of the final general identity where i= 1,-Jacobi proceeds to utilise for the purpose of proving his proposition regarding the effect of the vanishing of certain m +1 functional determinants. The path which he follows to reach his result is not a little surprising. Instead of saying that the vanishing of b, b, b2,.., bm,-for these are the m+1 determinants in question,-entails the vanishing of the left-hand side of the identity, viz., IzA bo~ o(.),.. (nt-1)' * The fact that these identities can be derived in the way here indicated from another which the preceding footnote has shown to be true, not merely of functional determinants but of determinants in general, is convincing proof that they also (i.e., the derived identities) are not restricted to any special form of determinant. Using the fundamental identity as enunciated in the footnote, and taking the special case of it where n=4 and m=2, and where therefore the given determinant may be written acb.2c3d4e5f6g7 1, we have la1b2 c3d4ej1 alb2c3d4e61 la\lb2c3d4e7 a1b24c3d451 laCb2c3d4f61 lalbc,2fC3\/ = laIBac, l2C. Ia b2 c3 d4e55 671. I b2 C3 d4 1 a, b2 C3 d4g6 I a b C3 d4 Now in each of the determinants forming the first row on the left here, el occurs as an element, in the second row f2 similarly occurs, and in the third row g3, while on the right these only occur in alb2c3d4efSq7 I. Consequently, equating cofactors of ef2g3 we have I a2 b3 C4 d5J I a2 b3 C4 do I I a2 b3 C4 d7 - ab3 c4 d - I a b3 C4d -I a b3C4d7 = [ a, b c d412. I a4 b5 C6 d7I I a b c d5 I a b2 c4 d6J I al b c4 d7 which when put in the form I I5b2C3 d41 | a6b2 C3 d41 I a7b2c3 d4 a I bb5c3d41 Cab6c3d4 lalb7c3d41 = -lalab2e3d412. ja4b5sc47 a, b2 c5 d41 aI, b2 c, d4l I a1 b2 c7d4 is a case of the first derived theorem. The original theorem, it should be noted, is true for all values of n and m; the derived holds only when m < n,-in fact, if we do not, in seeking to obtain the latter, take m<n in the former, we shall fail in our aim. Thus, taking n=2=m in the former, the given determinant being I ajb2c3d4e l, we have quite correctly lab2 C31 I a b2 c41 al b2 c5l \alb2d3l \ab2d41 Iab2d5 = I alb2 12. al a C3 d4 e5l; lalb2 e31 lab2 e4l la lb2 e5 JACOBIANS (JACOBI, 1841) 385 and that therefore, if the factor B on the right-hand side do not vanish, the other factor Df ffi..afn 3XAM 3Xm+l axm+n must vanish, he takes some pains to obtain a new identity and then applies this very reasoning to it. Denoting the cofactors of b, b,...., b in Z -b,1 ).... (m -l) by X,,...., m, he points out that as none of the 3's involves f, the same is true of the X's. On the other hand, the b's do involve f, but only one of them, viz., b,, involves the differential-quotients of fn with respect to x~+, this differential-quotient being in fact the last element of all and having B for its cofactor. In this way it appears that on the left-hand side of the identity the cofactor of f+ is XB. axn+k If in the same manner we take +- 4f - f n and -dXm Dax~ +l axm+n denote* the cofactor of xnk in it by,, it must follow that on but while cl occurs in each element of the first row on the left, and d2 similarly in the second row, e, does not so occur in the third, and consequently the cofactor. of cd2e3 on the left takes a different form from that given by Jacobi. The first derived theorem in its general form may be enunciated as follows:If there be two determinants D and A of the nth order such that the last n- m columns of D are the same as the first n- m columns of A, and if there be formed a square array of new determinants by supplanting each of the first m columns of D by each of the last m columns of A, the determinant of this square array of the mtlh order is equal to ( - 1)m(n+i) Dn-1 A. To illustrate the second derived theorem we may equate cofactors offig2 where we formerly equated cofactors of elf2g3, the result clearly being alb2c3d4e51( cab2c3d4e6[ abb2,acd4e71l a2 b c4 d5 a2 b3 c4 d6 I a2 b3 c4 d7 = - a b2 c d4 2. I a3 b4 c d e7 ' a 1b3 c4d5 a b3 C4d6 I a b3 c4d7 The next of the series would be got by equating cofactors of g,. * This is not the same as putting, with Jacobi, V^-f.f. af~ ~f~ Vf + Dfn E " ) afm+l..+.l = ^ + wf+ 'X+1 + am-+n for the determinant on the left being of the (n+ )th order there should be n+ 1 terms on the right instead of m + 1. M.D. 2B 386 HISTORY OF THE THEORY OF DETERMINANTS the right-hand side 'the cofactor of aaf is (-1)Ta(n+l)BP.k axn~ The connection between the X's and the Ax's is thus Xk = (-1)m(n+1)Bn-1/Ak,, so that Xb + x1b1 +... + Xmbm = (-1)m(nf+l)Bm-l(,Lb+,lb,+... ~ixb,nb). The left-hand member here, however, being equal to the lefthand member of the identity with which we started, it follows that the two right-hand members must also be equal, and therefore that af afi ~~~afn, ub + Alb, +... + imbm = B. ~ Df f1f oax ax ax?n rn-F m~n Of course this shows, exactly as the original identity did, that if b.= =... b. = 0b,= and B~~0 then af?fl 3f, 4- -.... =_ =0 -?xM axm,, axmf~lb -that is to say, the functional determinant of f, f,...,f, with respect to another set of n +1 variables vanishes also. Jacobi, however, does not at once say this, but drawing his reader's attention to the fact that the new set of variables contains n -a taken from x, x1,. x,_ and m+1I others, viz., Xn,,, Xn,~ Xnm' he affirms that the identity reached shows how the functional determinant of f, f,,..., f,, with respect to any set of n +l variables is expressible in terms of the m +1 functional determinants whose variables are cc, Xc1. I X,,1, cc,,, cc, X1. cc,,1, ccXn+15 cc, X1)., x.,..., cc-,~Xn,. His words are"Unde formula docet quomodo e functionum f, f,..., f, Determinantibus bk per idoneos factores multiplicatis et additis proveniat carundem functionum Determinans quarumcunque variabilium respectu JACOBIANS (JACOBI, 1841)38 387 formatum atciue per ipsum B multiplicatum. lHine bene patet, quod ~ pr. demonstravi, quomodo omnibus b, evanescentibus neque ipso B evanescente, simul cuncta illa Determinantia evanescant." * Continuing the work of deduction, Jacobi lastly equates the cof actors of L'in the two members of the identity ax j~b + Mlbj + + ttmbm = BZ~f DJ ff_?x~n?xM+i?xm+n noting that this differential-quotient does not occur at all on the right-hand side, nor in the IA's on the left-hand side, but in boccurs with the cofactor N Z~~~1 The result is the proposition t "Sit 1k functionum f, fl,.., fi,, Determinans quod in Determinante ~~f af1 __ Di per afmultiplicatur, ubi m n, erit axi ax I x2 an-1 aX,,1 aX1 Dr2 DXni1 + l~rn~+D f A Df9 __A Dfll 0. + [tin 4 D Dr Dr n 'm X1;2 nx-i The case where mi n is specially noted. * Of course this theorem also is not limited to determinants having differentialquotients for their elements. The general enunciation may be put as follows: If in determinants of the ntl1 order all have the same n - 1 columns in common, and vanish independently, then every determinunt of the nth order whose n columns are chosen from the m + n - I different columns must vaniish likewise. (See Proc. Roy. Soc. Edin., xviii. pp. 73-82.) t This proposition, and that from which it is derived, are again propositions which hold regarding determinants in general, the class to which they belong being that which concerns aggregates of products of pairs of determinants,-a class, the first instances of which have been seen to occur in Bezout (1779). In connection with Jacobi's remark regarding the case where m = n, it is worth while to note Sylvester's enunciation in Philos. Magazine (1839), xvi. p. 42. 388 HISTORY OF THE THEORY OF DETERMINANTS Leaving now these general theorems which involve two suffixes m and n, and which concern groups of functional determinants, Jacobi returns (~ 17) to the consideration of the properties of a single functional determinant, the specialisation being made, not by giving a particular value to m the second suffix introduced, but by leaving it unrestricted, and putting the original n =1. In the theorem..+.bb.), b -m) = - B"'m 2 + f fA _fm+i I~ f " m.a- ^1 X m+i B then stands for -f and Wb( for ~,.- f~+.l Making in this ax -- x x7k+~ the further specialisation, f=O, so that x has to be considered as a function of x1, x2,...., x a- we have ' +if ax =0 Xk+1i ax 3ax+1 and consequently b()' i.e. f tf+l _,/,f+I k 'ax Xk+1 aXk++1 Dx = f af~i+ -af1l ax Dx [C~,+ a~x ax+ ' af f (3/+1>> 3 \3k+/' if the brackets in the last line be taken to indicate that the fi+ enclosed by them does not involve x but its equivalent in terms of x1, x2,...,. By use of this substitute for b) there results xf -./ (/l)(D.f m,,), =37 _ f =fl _ afv +,d axiO31 2 am+lx at 3Z1 axm+1 and if we denote the cofactor of f in the determinant on the axr right by Ak we have of course the said determinant - a + A A/ +....+ A f Iand.. = A m-+A -. -A 1 and.~.A - A af ax -A f c ax 1ax DX - ax m +i. xi -- ~-~ -- Ai~-~. ~x~....fi JACOBIANS (JACOBI, 1841) 389 af by reason of the deduction from f= 0. Division by -L thus gives us the 'formula memorabilis' 4xi) f /("/m+i A-A -- A 2.. - Am+ where A =- af f2 Dfax+li 2- 1. 3t2 Dm+. and Ak is derivable from A by putting differentiation with respect to x in place of differentiation with respect to xk, and prefixing the proper sign. Jacobi adds "Formula inter egregia inventa illustrissimi Lagrange censetur," and asserts that for the purposes of proof it is not necessary to put f= 0.* This is immediately followed by the theorem-If u, ul, u2,.... Un be functions of x1, x2,..., x,, then E;uiu D1 3unu- 1 u 1 E _ _u,2 un Dx2 Dn unl Dx1 32u DXn the connection being somewhat distant, and probably lying in the fact that in both the b's are of the 2nd order. The mode of proof is curious. In the first place, by putting x - x * No indication is given of where Lagrange published the theorem attributed to him. As for the particular way in which x is given as a function of xl, x2...,,.xml, whether by the equation /=0 or not, it is clear that this cannot affect the truth of the result, because the latter contains nof at all, the requisites for validity being (1) that ff,..., fm+l are functions of x, xI, x2,.... Xm+l; (2) that x is a function of x xI, X.., Xm+i; (3) that on the left thef's have by substitution been freed of x before differentiation; (4) that on the right this has not been done, but they are there differentiated as if x were a constant. The subsidiary result f.fi+1 f (.fi+i ~x?ak+l - Dx \ k+l/ is certainly true whether f=O or not, all that is-requisite (see p. 374 above) being that fi+ on the right shall be what fi+i becomes by substituting for x its value in terms of f, x,, 2.., x m-li and that in the differentiation of itf shall be. viewed as constant. 390 HISTORY OF THE THEORY OF DETERMINANTS and therefore b(i) = aa(i+1) (+ = aa- k a - a there is obtained the identity + bb'l.. (' m +- aa ~.... a(m+l) Then since the a's here may denote any quantities whatever (" qua in formula cum ipsa a) quantitates quascunque designare possint ") a further substitution * bringing us back to differentialquotients is made, viz., a(i+l) =, +1) _ ai+1 w -- k4-1 ~ "fc+l — Xk._l where u, u, uV^..., Um+1 are functions of x1, 2,..., Xm+. In this way 7 (i) ^^z-)-l ^ ^au U.= -> ij + 1- == UU.~i+ k - ',Sk+1 aXk+l aXk+l and the aforesaid identity gives us 'W1 au2 qm+l,2m+2q + _ - 2U 1. 22 - m+X w / i -2- /-^ * m q-..... — ''' — ) - X1 O X2 D.m+l1 ax1 2 'max+1 as was to be proved. As a corollary to this it follows that if in the determinant ~. l axU2 _Un 'Xi 'a32 eaXrn tu, twu,..., be put instead of u, z1,..., t being any function whatever, the effect is the same as if the determinant were simply multiplied by tn+1. "Quod iam olim alia occasione adnotavi," the reference being to the 5th theorem of his paper of the year 1833, already dealt with. * A combination of the successive substitutions is impossible, by reason of the fact that in the second case the equation a(i+l)_ zui+ 7+1 -xk+1 is not meant, as in the first case, to include the definition of a(i+1), which has thus to be defined by a supplementary equation. The result of the first substitution is very noteworthy, in view of previous footnotes. JACOBIANS (JACOBI, 1841) 391 The next section, the 18th and penultimate, shows how by modifications made upon the functions, the functional determinant may reduce to one term. The first function f being expressed in terms of cc cc 2,.. xn) it follows that theoretically x is expressible in terms of f, X,, X2,..., c,,, and that therefore by substitution so also are fA,....A f.. Similarly A, being now a function of f, XI X2,..., IX, we conclude that theoretically x, is expressible in terms of f, fl, cc, x..., X., and that therefore by substitution so also are f2, f,..., fI. Supposing this process to be completed, Jacobi denotes the new forms of fl, f2, ff by fi(f, X1, X2..., Ic), fA(f, Af cc,., XC), JUV A) fit 1) i xn) )n> and the difference between any one of the old forms and the corresponding new by F with the appropriate suffix. He thus has an+l equations 0 =F =f - f (X, XI, X21 ~~ X01) o = F, =f - fj (, X17 X2X,..., xc), O = F2 =f2 - f2(f, f, X2,..c Xn), o F, fn fnf~,f, A I... I fn-,, Xnz), connecting 2n +2 variables X, c,)..., XI, f, fA,...f, n now viewed as independent. By a previous theorem there is thus obtained DF DF1, aF f _ _________ ax __ a_ a- - -F DF, _ aF E f1 q afn Now the numerator here reduces to one term, viz., DF aF, aFn ax ax, DDc' 392 HISTORY OF THE THEORY OF DETERMINANTS and the denominator in similar fashion to 3F DF1 DFn f fafl ' afn as Jacobi might briefly have justified by a reference to Prop. III. of ~ 5 of his "De formatione et proprietatibus Determinantium" -this proposition being that which concerns a determinant whose elements on one side of the 'diagonal,' as it afterwards came to be called, all vanish. Further, the factors of the reduced numerator are equal to (of\ _( (.f\ \x' \Dxa ' \x ' ' ' 'ax and those of the reduced denominator to 1, 1, 1,.... Our final result thus is -af _a afn = / (a na D~x ax, axn \ \3..j \ax where the brackets on the right are meant as a reminder that fi is there expressed in terms of f, f,,..., fi-l, Xi, Xi+,.., Xn. The last section (~ 19) is occupied with a theorem of the Integral Calculus, already twice enunciated (see pp. 357-358), namely, |U3/.3/,... ^ = | (2>|i **4 |n -.. |f 'x D1 a ' -f. -- / D.... a-ax ax,..", it being noted that the cases where the number of variables to be changed are 2 and 3 had been already dealt with by Euler and Lagrange.* Catalan's memoir of 1839 is not referred to. Then come the final words, "Et haec formula egregie analogiam, differentialis et Determinantis functionalis declarat,"-a not inappropriate ending in view of the author's attitude throughout the memoir. * The memoirs referred to here and by Catalan seem to beEULER, L.-De formulis integralibus duplicatis... N.ov. Comm. Acad. Petrop. (1769), xiv. i. pp. 72-103. LAGRANGE, J. L.-Sur lattraction des spheroides elliptiques. Nouv. Mdrm. Acad.... Berlin (1773), pp. 121-148; or (Euvres, iii. pp. 619-658. JACOBIANS (CAUCHY, 1841) 33 8931, CAUCHY (1841). [MWmoire sur. les fonctions diff~rentie'lles alternees. Exercices d'analyse et. de phy. mat.,ii pp. 177-187; or (IEmvrescomple'tes,:21 se'r. x-ii-.] This w-as evidently sugested by Jacobis memoir justdel with, but it belongs to ~a quite different class, being merely a. simply-written exposition containing two of the -fundamental properties of the functions and a few illustrations. in the first' part ('Considerations ge'nerales') he explains the, meaning of. on the understanding that the n variables x,- y,,.. t are connected with the n others x, ly, z, ) t by-n equations: and then establishes the theorems ~~~~~~~~~~~S —xp y... 1,t (a) S[~D~x. Dyy.... Dtt] = )n TI.X.De] () -4-DxP.Dy...DtWl making reference in -both cases to Jacobi's memoir, and pointingout in connection, with' the latt-er theorem that -it leads to " la formule donne'e par- M. Catalan pour la transformation d'une integrale multiple." In the second part ('Exemples') he instances first the case where x, y, z...t are linear functions of x, y, z and shows how a result obtained in a previous -memoir (see above, p. 285) affords 'a verification of: the theorem (a). The. nex t example is of' greater interest, being the case where x= A(x-a)(y-a)(z-a)...(t-a), y = B(x-b)(y - b) (z -b). (fb) z = C(X. -c)(y-(z-c).. (t-) t =H(X -hA)(y-.h)(,z-'h)...(t -h). 394 HISTORY OF THE THEORY OF DETERMINANTS As is readily seen we have here S[ + Dx. Dsy. Dz.... Dtt] = xyz...t.[ ~ 1 1 1 x-a y-b z-c t-h' which, according to a previously obtained result (see above, p. 345), = (-1)(- 1). xyz... t S [+ ct0blc2... h-1. S1] I+ xgy2 t... — 1] ( - )( - b)... (y - a)(y-b)....(2 - )(z-b)... (t - -a)(t-b)... so that if we substitute the given expressions for x, y, z,....t there is obtained finally S[ + Dx. Dyy. Dz....Dtt] (-1)(n-') )ABC... H S[+a0blc2... h-1]. S[+~ 0yZ2... t-1]. The third example is equally worth attention, the connecting equations being A (x- Ca)(y-ca)(z-a).... (t-a) (x- l)(y-k )(z-k ).... (t-/) ' y = (x-c)(y-c)(z-c).... (t-) Yz (x- _k)(y- k)(z- k).... (t- k) t = H(x-h)(y-h)(z-h).... (t-h) H(X - )(y -)( -* ).... (t -) ' (x-h)(y-h-)(z-ho)... (t-h) so that at the outset it is found that:S[ + Dx. Dyy. D,z.... Dtt] (a - k)....h [ 1 1 "I =xyz....t. - c [-..... -.( ) (_7C (tC) ' (Gx-aC y-b z-c t-h] and ultimately after substituting as before S[Dx D yy. Dy zz.... Dtt] (a- 7o). A(h- lo) (- )n(A->. ABC...... _ ) (-)n+ 1) ~['on2,,. A-)]. S[{-_0). 2... t1]. S [~a0bc2... hn-l]. S[4X 0ylz2.. tn-1.. CHAPTER XIV. SKEW DETERMINANTS FROM 1827 TO 1845. SETS of equations of the form a12x2 + a13x3 + a14x4 +... + a.. aJ =, - 612X1 + a23X3 + a24X4 +.... + C2ln =x- 2| - C'13X1 - a23x2 + a34X4 + '. - aX3nX7n = 3X - Ct14X- - a24XC - a34x3 + + a4nxn = C4 - ClX1 - a2X - asns3 - a4nx4 -.... =., where the coefficient of x, in the sth equation differs only in sign from the coefficient of x8 in the rth equation, had often made their appearance in analytical investigations before the determinant of such a set came to be considered. An instance is to be found in a memoir of Poisson's, read before the Institute in October 1809, and printed in the Journal de V'Ecole Polytechnique, viii. pp. 266-344; and similar instances of an earlier date in writings of Lagrange and Laplace are therein referred to. A curious example occurs in one of Monge's papers already dealt with (see above, pp. 67, 68), there being additional interest attaching to it by reason of the fact that in it the a's and x's are themselves determinants. It is to be found in an earlier portion (pp. 107-109) of the same volume as Poisson's. Denoting by a, /3, y,; M, N, P, Q, R, S the six-termed expressions which at a later date would have been written bc2d, alc2d3, | alb2d, lab2c 1; ab2e3 I Ib1ce,, cd2e3, - a1d2e3, a1c2e3, bd2e3, * See especially p. 288. 396 HISTORY OF THE THEORY OF DETERMINANTS Monge established the set of equations Qa + S3 - Py = 0 Ra - N3 + P = 0 Ma -1Ny- + SS = 0 - M/3 + Ry + Q6 = 0 from which he eliminated a, 3, y,, and obtained RS+ QN- PM = 0. On altering the signs of the last two" equations the result of the elimination would a generation afterwards have been put at once in the form Q S -P R N-N. P - M. N -S =, M -R -Q and the left-hand side would have been recognised as a 'skew determinant and altered into (RS+QN-PM)2 = 0. No prophetic glimpse of this, however, occurred to Monge. The mathematician who first referred definitely to the determinant appears to have been Jacobi. PFAFF (1815). [Methodus generalis, aequationes differentiarum partialium, nec non aequationes differentiales vulgares, utrasque primi ordinis, inter quotcunque variabiles, completi integrandi. Abhandl..... Acad. der Wiss. (math. Kloasse),. Berlin, 1814-1815, pp. 76-136; or Kowalewski's German Translation, 84 pp., Leipzig, 1902.] After seven pages of historical introduction and preliminary explanation Pfaff arranges the subject of his memoir in the form of a series of fourteen problems with their solutions. Problem i. is to integrate completely a partial differential equation in three variables; problem ii. is to transform any differential equation of the first order in four variables into, SKEW DETERMINANTS (PFAFF, 1815) 397 an equation in three variables, and to integrate the latter by means of a system of two equations; and problem iii. is to integrate an ordinary differential equation of the first order in five variables by means of a system of three equations. Problems iv., v., vi. correspond respectively to i., ii., iii., the number of variables being one more in any problem of the second triad than in the corresponding problem of the first triad. A similar step onward is taken in problems vii., viii., ix. which form a third triad, and again in problems x., xi., which are the first two members of a fourth triad. At this stage the 'methodus generalis' is supposed to be sufficiently foreshadowed, and in the remaining three problems the restriction to a definite number of variables is withdrawn. Of these three it is the thirteenth (xiii.) which concerns us here, namely, the reduction of an ordinary equation of the first order between 2m variables to a similar equation between 2mn-1 variables, and the performance of the integration by means of a system of n equations. It (xiii.) is the generalization of problems ii., v., viii., xi., these being the cases of it where m=2, 3, 4, 5. The solution consists in expressing 2m-1 of the given variables as functions of the 2jnth and 2m-1 new quantities, and introducing the latter in place of the former. By considering the new quantities as constants 2nz-1 auxiliary differential-equations arise, the integration of which supplies the desired functions; and for the formation of the auxiliary equations 2m-1 quantities are needed whose values are determinable from the same number of conditional equations. It is in the solution of this set of conditional equations that our interest centres. As Cramer and Bezout had dealt with a more general set, Pfaff naturally made trial of their methods; but they were found, he says, of little service. He therefore sought for and discovered two laws of formation which sufficed for his wants. His words are (p. 119)"Haec determinatio, si consueta eliminandi methodo tractetur, calculos nimium complicatos et operosos postulat: ipsaque precepta generalia, quae Bezout et Cramer de eliminatione tradiderunt, in casu substrato parum commodi afferre videntur. Accuratius vero 398 HISTORY OF THE THEORY OF DETERMINANTS considerando praedictas aequationes conditionales et formulas ex earum solutione actu. evolutas, ad duas leges satis simplices easque generales perveni, quas hec breviter exponere sufficiat." In exposition of the first law he begins by repeating the results for the cases m =2, 3, 4, using for brevity's sake the symbol *(AE to stand f or AdC -CdA CdE -EdC EdA -AdE de ~ da + dc That is to say, he recalls (1) that when the given equation is Adct + Bdb + Cde + Ede,= 0 the auxiliary equations are da db O>- BCE) +(ACEJ) (2) that when the given equation is Ada + Bdb + Cde + Ede + Fdf + Gdg = 0, the auxiliary equations are cla 0 (CBE)(CFG) - (CBF)(CEG) + (CBG)(CEF) db + (CAE)(CFG) - (C/AF)(CEG) + (CAG)(CEF)' and (3) that when the given equation is Ada + Bdb ~ (Cde + Ede + Fdf + Gdg + lldh + Idi =0, the auxiliary equations are da dc *This may be nothing more than a coincidence; but if so, it is a curious one, the expression replaced by (ACE) being the determinant A dA da C dC E dEk de, and Pfaff even drawing attention to the fact that (AEC):- (ACE). SKEW DETER~MINANTS (PFAFF, 1815)39 399 where = (BCE)(BFG)(BHI) - (BCE)(BFH)(BGI) + (BCE)(BF1)(BGH) -(BCF)(BEG)(BHI) + (BCF)(BEH)(BGI) - (BCF)(BEI)(BGH) + (BCG)(BEF)(Bf11) - (BCG)(BEH)(BFI) + (BCG)(BEI)(BFH) - (BCH)(BEF)(BGI) + (BCH)(BEG)(B1F1) - (BCH)(BEI)(BFG) + (BCI)(BEF)(BGH) - (BCI)(lBEG)(BFH) + (BCI)(BEH)(BFG). Now the denominators here are functions of the kind afterwards known as Pfaffians, and such as would now be written (BCE), (CBE) (CBF) (CBG) (CEF) (CEG) (CFG) (BCE) (BCF) (BCG) (BCJI) (BCI) (BEF) (BEG) (BEll) (BEI) (BFG) (BFH) (BFI) (BGHl) (BGI) (BHI) Their law of formation as given by Pfaff is therefore of the greatest interest. His words as regards WX are (p. 124): "Separando litteram, B, termini hujus expressionis complectuntur permutationes litterarum. reliquarum. C, E, F, 0, H, I (exciusa prima A), quae sub hac restrictione fleri possunt, ut litterae in quavis complexione (cx. gr. C, G, E, H, F, I in termino octavo i-of- W) prima, tertia, quinta (ex. gr. C, E, F) in genere imparem. locum, obtinentes inter se rite sint ordinatac, et litterarum, quaevis pai loco constituta (0, H, I) sit ordine aiphabetico posterior littera in loco impari proxime praecedente (C, E, F). His formis rite inter se ordinatis i.e. secundum. ordinem. lexicographicum (e.g. C, G, E, H, F, I ante C, G, E, I, F, H) terminorum. signa alternant. ilac lex restrietiva permutationum, etiam, sic enuntiari potest, ut singtilas complexiones dispertiendo in dyades, sive classes binorum. elementorum, ipsac dyades tam quoad sua elementa, quam. inter se invicem, rite debeant esse ordinatae." This practically means that the terms of 21 are got (1) by taking every permutation of C, E, F, 0, H, I which is such that each odd-placed letter in it and the letter immediately following are not an inverted-pair, and that the full group of oddplaced letters is also free of inversions; (2) by placing a B before, .400 HISTORY OF THE- THEORY OF 'DETERMINANTS each odd-placed letter and marking off with brackets the triads so formed; and v(3) by making the signs alternately + and - when the terms have been arranged in dictionary order. Not content, however, with this rule Pfaff immediately gives another of equal interest, the passage being (p. 125): "Processus autem combinatorius, quo permutationes praedictae exhibentur, satis, commodus hie est: Sint litterac, quarum perm'utationes sub restrictione supra commemorata quacruntur a, b, el e,. k, 1, M rn,: supponamus inventas esse permutationes litterarum c, e, m, n, exelusas duabus a, b: tum 1) singulis his permutationibus vel complexionibus praeponatur binio ab; 2) cx hac prima serie complexiones totidem aliae formentur, permutando b et c; 3) cx his -porro aliae, permutando c et d, sieque progrediendo ex quavis serie complexionum ~nova formet ur, litteram aliquam cum proxime sequente permutando, donce postremo m et a invicem permutentur. Qu a ratione obtinentur omnes perinutationes litterarum a, b, c,..., in, a quas restrictio praedicta admittit." Here the case for 2m letters is made dependent on the case for 2m - 2 letters, so that if six letters a, b, c, d, e, f were given, we should begin by forming the permutation for the last two letters, namely ef to -this we should prefix ct,_ and by performing the interchanges dt L e, e 2-7f obtain the, permntations for the four letters c, d, e, f, namely ccte. ef -ce. df +cf.cde; and lastly we, should prefix ab to each of these and perform the interchanges b -1- c, c 2-7d, d e, e 27 4 the resnlt reached being ab.ed. ef - ac.bd.ef + ad. be.ef - ae. be.df + af. be.de -ab. ce. df + ac. be. df - ad. be. cf + ae. bd. cf - af. bd. ce 4~ ab. cf. de - ac. bf. de + ctd. bf. ce - ae. bf.ecd + af. be.cdA As a manifest deduction from this rule* Pfaff states that for 2m letters the number of such restricted permutations is ____________ 18.3.5..;(2mn- 1). -* The other rule, however, would have been equally useful towards this end. For if we remove the restriction each one of the N restricted permutations would give rise to 2-. (1I. 2. 3... mn) unrestricted permutations; so that we should have and therefore N ~~ 2.3.42rn) 1 3.5... (2m -1).2-. (i.2.3... m) SKEW DETERMINANTS (PFAFF, 1815) 401 Later he points out that since each factor of each of the fifteen terms of A is itself six-termed, the total number of terms in the final expansion of W ought to be 63.15, i.e. 3240; but that 2400 of them cancel each other. To obtain the 840 really needed is the reason for his propounding a second law, which he does in ~ 18 (pp. 126-129). JACOBI (1827). [Ueber die Pfaffsche Methode, eine gewohnliche lineare Differential-gleichung zwischen 2n Variabeln durch ein System von n Gleichungen zu integriren. Crelle's Jornal, ii. pp. 347-357; or Werke, iv. pp. 17-29.] An essential part of Pfaff's method is the solution of a set of equations which Jacobi writes in the form NXax = + (0,1)ax, + (0,2)x2.... + (0,p)3xP NXx = (1,0)ax + * + (1, 2)D2x ~... + (l,p)3ax NXx = (2,0)ax + (2,1)ax, + * + + (2,p)3x, NXpox = (p,0)3x + (p,1)6x1 + (p,2)Dx2 +....+ where (0,0)= -(1,0) and generally (a,/3)+(/3,a)=0. This form of his own he frankly characterises as "elegant and completely symmetrical"; but the same description would apply equally appropriately to the solution which he gives. Unfortunately, the method by which the latter was obtained is not indicated, and we can only hazard a guess in regard to it. The balance of probability would seem to be in favour of the method of devising a set of multipliers which, when applied to the given equations, would after the performance of addition bring about the elimination of all the unknowns except one. In the case of four equations this would not be at all difficult. For M.D. 2c 402 HISTORY OF THE THEORY OF DETERMINANTS example, if we wish to eliminate x2, x3, x4 from the equations ax2 + bx3 + ex, = 6 - ax,. + dx, + ex4 = 4 - bx, - dx2. +fx4 = C - cx, - ex2 - fxA = 43 the multipliers are readily seen to be 0, f, -e, d, so that after multiplication and addition there results (-af+be-cd)x, = fe - ee3+ d4. Similarly by using the multipliers -f, 0, c, - b, we find (- aff+ be - cd)X2 =-fe + ce - bC4 and the 'other two are (-af+be-cd)x3 = ee - ce2+ aC4 (- caf+ be - ec)x4 = -de, + be2- aC3 Jacobi's corresponding result is to the effect that the numerators of the values of the four unknowns are Nax{ * + (2,3)X, + (3,1)X2 + (1,2)X}, Nx((3,2)X+ ~ (0,3)X2 + (2,0)X}, NaIx{(l3)X ~ (, X + * + (0,1)X},I N-x{(2,1)X + (0,2)Xl + (1,0)X2 + * and the common denominator (0,1)(3,2) + (0,3)(2,1) + (0,2)(1,3), or, as he thereafter writes it (0,1,3,2). When the similar set of six equations had to be dealt with, the devising of the multipliers requisite for elimination would SKEW DETERMINANTS (JACOBI, 1827) 403 necessarily be harder; but keeping in view the analogous mode of arriving at the solution of a11 + a2X2 = 1 bllx + b2x2 -= 2 and then proceeding to the solution of aCx, + a22 + a3X3 = 1 } bjxj + b2X2 + b33 = 2 X cl1x + c2X2 + C3AX =3 J, where, it will be remembered, the multipliers requisite for elimination are of the same form as the common denominator of the values of the unknowns in the preceding case, Jacobi would have little real difficulty in finding that corresponding to the four multipliers requisite for eliminating Dx,, aX2, ax, in his first case, viz.,0, (2,3), (3,1), (1,2) he would now require to have the six multipliers 0, (2345), (3451), (4512), (5123), (1234). As a matter of fact, he gives for the numerator of the first unknown N3Z{ * +(2345) (351)X(3451)+(4512)X3+(5123)X,+(1234)X5}, the others being NDZ{(3245)X + * +(4350)X2 +(5402)X3+(0523)X4+(2034)X5} The common denominator is not mentioned; we should have expected him to say that it was (10)(2345) + (20)(3451) + (30)(4512) + (40)(5123) + (50)(1234) or - (012345). It is then pointed out that when the first coefficient has been got in one of the numerators, the others are arrived at by 404 HISTORY OF THE THEORY OF DETERMINANTS. circular permutation, the elements permuted being 12345 in the case of the first numerator, 02345 in the case of the second, 01345 in the case of the third, and so on; also that the first coefficient in one line is got from the last in the preceding line by changing 012345 into 123450 and then transposing the first two elements; and that these laws hold generally. A general mode of finding the ordinary expression for the new functions here introduced and symbolized by (1234), (123456),.... is next explained. It is first stated that the number of terms represented by (2,3,4,...., ) where p is necessarily an odd integer is 1.3.5.....(p-2), and that one of them is (23).(45).(67).... ( - 1, p). We are then told to permute cyclically the last p-2 elements 3,4, 5..., p, and we shall obtain from this p-2 terms in all; thereafter to permute cyclically the last p-4 elements 5,6, 7,... p in each of the p -2 terms just obtained, and so on. For example, (234567) = (23)(45)(67) + (23)(46)(75) + (23)(47)(56) + (24)(56)(73) + (24)(57)(36) + (24)(53)(67) + (25)(67)(34) + (25)(63)(47) + (25)(64)(73) + (26)(73)(45) + (26)(74)(53) + (26)(75)(34) + (27)(34)(56) + (27)(35)(64) + (27)(36)(45). It is important to note in conclusion, that the case of an odd number of equations is not neglected by Jacobi, a proof being given by him that in that case the determinant of the system vanishes. In his own words-which are interesting in view of what has been said elsewhere regarding the evidence which the SKEW DETERMINANTS (JACOBI, 1845) 405 paper affords of the progress made by him in the study of determinants"Nun bleibt nach dem bekannten Algorithmus, nach welchem die Determinante gebildet wird, diese unverandert, wenn man die Horizontalreihen und Verticalreihen der Coefficienten mit einander vertauscht. Fur unsern besondern Fall nun wird, wenn wir die Determinante mit A bezeichnen, hieraus folgen: A=(-i)P+lA, da jedes Glied der Determinante ein Product aus pp+1 Coefficienten ist, von denen jeder durch Vertauschung der Horizontal- und Verticalreihen sich in sein Negatives verwandelt. Diese Gleichung A=(-l)P+A aber kann nur bestehen, wenn p+1 eine gerade Zahl ist, wofern nicht = 0 sein soll." Thus, though only Pfaff's expositor as regards the functions which came long afterwards to be known and are still known as 'Pfaffians,' Jacobi was the first to discover and prove the now familiar-worded theorem " A zero-axial skew determinant of odd order vanishes." JACOBI (1845). [Theoria novi multiplicatoris systemati aequationum differentialium vulgarium applicandi. Crelle's Journal, xxvii. pp. 199-268, xxix. pp. 213-279, 333-376; or Math. Werke (1846), i. pp. 47-226; or Gesammelte Werke,* iv. pp. 317-509.] As is well known, this long and exhaustive memoir of Jacobi's is broken up into three chapters,-the first giving the definition and various properties of the new multiplier, the second explaining the application of it to the integration of differential equations, and the third illustrating this application by means of particular differential equations of historical interest. One of the latter is the equation associated then, and still more since, with the name of Pfaff, the discussion of it occupying ~~ 20, 21 on pp. 236-253 of vol. xxix. We are thus prepared to find the function, defined by Jacobi eighteen years before, again referred to. The old definition of the function, which he here denotes by R, is practically repeated, the initial and originating term being * In all preceding references of this kind Werke has been used for Gesanmelte Werke: here the longer but more correct name is necessary for distinction's sake. 406 HISTORY OF THE THEORY OF DETERMINANTS now of the form ac12a.4.. a,2m-1,2': and then he makes the pregnant general remark that the properties of R are analogous to those of determinants. Prominence is given to the theorem regarding the effect of interchanging two indices. This is followed by the twin pair of identities DR aR + aR a1l,s + a2,s8D +***+ 2m, s DR DR DR 0 = al,,- + a2, +... + a2ms?, aR in the latter of which s differs from r, and the term a,.-,a is awanting; and finally, it is pointed out that the differentialquotients of R with respect to one or more elements are functions of the same kind as the original, and, probably as a consequence, that certain second differential-quotients are identical. No proofs are given; indeed, the statements themselves are in the most concise form possible, the whole passage being as follows:" Designantibus i, i', i", etc., indices inter se diversos, si sumuntur differentialia partialia aR 'a2R Dai, jt Daaj, i Zp, "'. ea erunt aggregata ad instar aggregati R formata, respective reiectis Coefficientium binis, quatuor,... seriebus cum horizontalibus tum verticalibus, eritque D2R D2R 32R,, aZaiD, Za.i/ aa, l ai'// ll,, aa,,, iaai,, ll,, It should be carefully noted that both in this paper and in the preceding, Jacobi views the new functions as separate from and independent of determinants, and not at all in the light in which, at a later time, they came to be looked upon-viz., as a subsidiary function arising out of the study of a particular kind of determinant with which it had a definite quantitative relation. CHAPTER XV. ORTHOGONANTS FROM THE YEAR 1827 TO 1841. THE special form of determinant to which we have now come is connected with a problem in coordinate geometry-the problem of transformation from one set of rectangular axes to another set having the same origin. The actual appearance of determinants in any of the attempts to solve the geometrical problem did not take place until comparatively late in its history, probably because the connection between the two subjects was less patent than in other cases, the problem when transformed into algebraical language being not a mere matter of elimination of unknowns from a set of linear equations. The earlier portion of the history of orthogonal substitution, although of considerable interest, is thus not sufficiently germane to our subject to warrant detailed treatment of it. For those interested in this earlier portion it will suffice to give the following chronologically arranged list of papers from 1770 to 1840:1748. EULER. Introductio in Analysin Infinitorum. Tomi duo. Lausannae et Genevae (v. ii. Appendix de Superficiebus *). 1770. EULER. Problema algebraicum ob affectiones prorsus singulares memorabile. Novi Commentarii Acad. Petrop., xv. pp. 75-106; or Commentationes Arith. Collectae, i. pp. 427-443. 1772. LAPLACE. Recherches sur le calcul integral et sur le systeme du monde. Hist. de l'acad. roy. des sciences (Paris), 2e partie, pp. 267-376. * Or in Labey's French Translation, ii. pp. 370-378. 408 HISTORY OF THE THEORY OF DETERMINANTS 1773. LAGRANGE. Nouvelle solution du probleme du mouvement de rotation d'un corps de figure quelconque qui n'est anime' par' aucune force acce'liratrice. Nowv. Wem. de l'acad. roy. (Berlin), pp. 85-120. 1775. EULER. Formulae generales pro translatione quacunque corporum rigidorum. Novi Commendarii Acad. Petrop. xx. pp. 189-207. 1776. EULERp. Nova methodus motum corporum, rigidorum determ minandi. Novi Commentarii A cad. Petrop., xx. pp. 208-238. 1776. LEXELL. Theoremata nonnulla generalia de translatione cor — poruim rigidorum. Novi Commentarii Acad. Petrop., xx.. pp. 239-270. 1784. MONGE. Sur l'expression analytique de la generation des, surfaces courbes. Mum. de l'acad. roy. des sciences (Paris), [pp. 85-117], p. 114. 1802. HACHETTE et POISSON. Addition au me'moire preicedent... Journ. de l'e'. polyt., cahier xi. pp. 170-172. 1806. CARNOT, L. N. M. Sur la relation qui existe entre les distances, respectives de cinq points quelconques pris dans l'espace,;, suivi d'un... Paris, 1806. 1810. LACROIX, S. F. Traite' du calcul diff6rentiel et du calcul int'gral. 2e edition, i, p. 53 3 1811. LAGRANGE. Meicanique analytique. 2~ e dit.,~i. p. 267. 18 18. GAuss. Determinatio attractionis.... Commentationes Soc. Gottingensis, (Classis math.) iv. pp. 21-48; or Werke, iii. pp. 331-355. 1827. JACOBI. Euleri formulae de transformatione coordinataruni.Crelle's Journal, ii. pp. 188-189; or Gesammelte Werke, viiL pp. 3-5. 1827. JACOBI. Ueber die ilauptaxen der Flitchen der zweiten Ordnung. Crelle's Journal, ii. pp. 227-233; or Gesammelte Werke,. iii. pp. 45-53. 1827. JACOBI. De singulari quadam dupli~cis integralis transforma — tione, Crelle's Journal, ii. pp. 234-242; or Gesammelte Werke,, iii. pp. 55-66. 1828. CAucHY. Sur les centres, les plans principaux et les axes. principaux des surfaces du second degr4. Exercices de Math.,, iii. pp. 1-22; or GEuvres compUgtes, 2e s6r. viii. pp. 8-35. ORITHOGONANTS4O 409, 1828. CAUCHY. Discussion des lignes et des surfaces du second degre' PExercices de Math., iii. pp. 65-120; or UEuvres completes, 2T se'r. viii. pp. 83-149. 1829. CHA SLES. Sur les proprie'tes des diame'tres conjugues des hyperboloides. Cori-esp. miath. et phys., v. pp. [137-157] 139-141. 1829. CLAUJSEN. Ueber die Bestimmung der Lage des Haupt-Umdrehungs-Axen eines Kdrpers Crelle's Journal, v. pp. 383-385; or J\ouv. Annales de Math., v. pp. 81-83. 1829. CAUCHY. Sur e'~quation 'a 1Faide de laquelle on determine les. ine'galites se'culaires des mouvements des plane'tes. Exercices de Math., iv. pp. 140-160; or ~E'uvres eomnl s,2 er x pp. 172-195. 1831. JACOBI. De transformatione integralis duplicis indefiniti. in formam simpliciorem....Crelle's Journal, viii. pp. 253-279, 321-357; or Gesammnelte Werlee, iii. pp. 91-158, 1832. GRtUNERT. Ueber die Verwandlung der Coordinaten im Raume. Crelle's Journal, viii. pp. 153-159; or Nouv. Annales de Math., v. pp. 414-419. 1832. ENCKE. Ableitung der Formein. von Monge fuir die Transformation der Coordinaten in Raume. Berliner Astron. Jahr — buch (1832), pp. 305-310; or Corresp. math. et phys., vii. pp. 273-277. 1832. JACOBI. De transformatione et determinatione integralium duplicium commentatio tertia. Crelle's Journal, x. pp. 101-.128; or Gesammielte Werke, iii. pp. 159-189. 1833. JACOBI. De finis quibuislibet functionibus homogeneis secundi ordinis.... Grelle's Journal, xii. pp. 1-69; or Gesammelte~ Werke, iii. pp. 191-268. 1833. GRUNERT. Supplemente zu Kliigel's Wbrterbuch: Art. "G oordinaten." 1835. JACOBI. Observationes geometricae. Crelie's Journal, xv. pp. 309-312; or Gesamnmelte Werice, vii. pp. 20-23. 1839. GATALAN. Sur la transformation des variables dans les integrales. multiples. M~m. couronne's par I'Acad. de Bruxelles, xiv. ii.. pp. 1-47. 1839. IREISS. Sur les neuf angles que forment re'ciproquement deux syste'mes d'axes rectangulaires. Correspond. m~ath. et phoys., xi.. pp. 119-173. 410 HISTORY OF THE THEORY OF DETERMINANTS 1840. RODRIGUES. Des lois g6ometriques qui regissent les deplacements d'un systeme solide dans lespace,... Journ. (de Liouville) de Math., v. pp. [380-440] 404-405. Of these only seven need be taken account of because of their connection with determinants, viz., two by Jacobi in 1827, one by Cauchy in 1829, three by Jacobi in 1831-3, and one by Catalan in 1839. JACOBI (1827). [Ueber die Hauptaxen der Flachen der zweiten Ordnung. Crelle's Journal, ii. pp. 227 —233; or Gesammnelte Werlee, iii. pp. 45-53.] Without unnecessary preliminaries Jacobi enunciates the problem which he wishes to solve, viz., the transformation of an expression of the form Ax2 + By2 + Cz2 + 2ayz + 2bzx + 2cxy, where x, y, z are the coordinates of a point referred to an oblique coordinate-system, into an expression of the form L2 + M + + N2, where,?, are the coordinates of the same point referred to a rectangular system having the same origin. This implies that the things directly sought are the nine coefficients which give each of the original coordinates in terms of the new. Jacobi, however, prefers to begin with a related set of unknowns, taking the equations which give the new coordinates in terms of the old. These being assumed to be = ax + 3Y + yZ = a"x + /"y + y" the equivalent set giving the old in terms of the new is of course A.x = (3'" - ( - 'y"') > + ( 3y' - 3'y)~ A.y (y'a' - y"a') + ('y"a -ya"), + (ya' - y'a) l.z = (a'/ - a/3') + (a"/ - c/3") + (ca/3 - U'3)S where A= a/3'y" + /y/'a" + ya'/" - - a/y"' a - ya"yS'. ORTHOGONANTS (JACOBI, 1827) 411 Denoting the known angles between the original axes by X, /x, v, there is obtained at once the set of six equations a2 + a'2 + a"2 = 1, /32 + /'2 + /"2 =1, 72 + 72 +y"2 =1, Iy + 0I3' + //3V = COSX; ya + 'a' + I"a" = cos N, a/3 + a'3' + a"/" = cos v; and, since the expression L(ax+ y + yz)2 + M(a'x + f'y+ ) + z)2+ N(a"x + "y y")2 has to be identical with Ax2 + By2 + Cz2 + 2ayz +- 2bzx + 2cxy, we have thus by implication another set of six equations, viz.: La2 + Ma'2 + Na"2 =A, L32 + Mf'2 + N3"2 =B, Ly2 + My'2 + Ny"2 =C, L/y + Mr3'' + N /"y" = a, Lya + My'a' + Nya" = b La/s + Mai/' + Na"/3" = c. What, therefore, remains to be done is the solution of these twelve equations in the twelve unknowns a, 3,: a',3',y': a", ", y": L, M, N. Jacobi's mode of accomplishing this is very interesting. He notes first that A may be looked upon as known, by reason of the fact that it is expressible in terms of A, A,, the connection in modern notation being a2 +a 2 +a"2 a3 +a'/3' +a"f3" ay + a'y' a"y" A2 = a/ + a'!3'+ a"/3" )32 + 3'2 + f"/2 y3 + y'3' + y"/I3 +ay + a'y' + ay" /3y + /3I' + 3"y/ Y2 + y'2 + y2 1 cos v cos P = cos v 1 cos cos P cos X 1 412 HISTORY OF THE THEORY OF DETERMINANTS In the next place he draws attention to the resemblance between the two sets of six equations, and points out that as a consequence any equation legitimately obtainable, from the, second set is matched by an equation which might in like manner be obtained from the first set, but which is much more readily got by using the substitution L =M=N =A = B= 0= a, b, c =Cos X, cosMU, Cosv f He then from the second set of six equations forms three groupsLa. a + lMIa'. a' + INa".-a" =A La./3~+Ma'.13'~+Nal " =3"c La-y + Ma'.y' +Na".y" = bJ L3. a +MO'a/'+ Nol"-a" =0 L13.13 + M13'. 3' + N/3". 3" = B L13.y + Mo3. y' +N N"y" = a) Ly -a + My'.a'a/ + Ny"-a"l= b Ly. 03 + My'. O/' + Ny"/. p3" = a Ly. y + My' y' + N7" y" C. and solves the first group for La, Ma', Na"; the second for Lf3, Mo/' N/3"; and the third for L7, My', Ny"; the results being A. Lal = (/3y"- - o3y') A + (y'a" - ya') c A. Na"l = (/3y' - /3'y/)A + (ya'/ - y'a)c + (a'/3" - a'43') b + (a"/3 - a/") b + (a/3 - a'/3)bJ A. Lo3 = (0-3/ 3"7)C A. Mo/ = (T31y -/3y",)c A.Noll"= (/3y' -f3'y/)c A. Ly = (""- /y) A. My' = (/3"y -0/3y") b A. Ny"/ = (/3y' -$3'y)b + (y'a"l - y"a') B + (a'O/" - a1/3')a + (y //a - ya") B + (a"/3 - af3") a + (ya' -y-'a)B + (a/3' -a'/)a + (y'a" - y"a') a + (a'!3" - a"/O') C + (y //a - ya") a + -(a"/3 - a/3") C + (ya' - y'a) a + (a!3' -a'3) C ). Making the substitution above referred to he derives the ORTHOGONANTS (JACOBI, 1827) 413 corresponding results which are obtainable of six, viz.: A. a = (/ - ) + ('a" -y"a')cos v. a' = (/3"y - /3 ) (y'a - ya")Cos A. a" = (3y' -/3'Y) + (ya' - 'a)Cos v A. / = (/'y" - /3"') cos v + (y'a" - y"') A. / = (/3y / -ly")cos v + (y'a - ya") A. /" = (3y' - 'y) cos v + (ya' -y'a) A. y = (/3'y"- i"y')cOS U + (y'a"- y"a')cos X A. y' = (/"y/ -ly")cos Ju + (y"a -ya")cos X A. y" = (f3y' - /'y)Cos /x + (ya' -y'a)Cos X from the first set + (a'/"- a"f3')cos j + (a"l3 -a/'")Cos IU + (a3' -a',)cosu + (a'3" -a "/3) cosx + (a"/3 - a")cos X + (a/' -a'S)cos xj + (al'/" - a"/') + (a"/ - a/3") +(a/3' - a/l) I He then takes each of these nine equations along with the one of which it is a special case, and by subtraction obtains nine new equations, which he groups as follows:0 = (L - A) (P'y" - /3y') + (L cos v - c) (y'a" - y"a') + (L cos ~ - b) (a'" -a"P') 0 = (L cos v - c)'( (/3" - P/"') + (L - B)' (y'a" - Vy"a') + (L cos X - a) (a'P" - a"/') 0 = (L cosx - b).(3'y" - i"y') + (L cos X - a) (y'a" -y "a') + (L - C) (aI" - a"j') O = (M - A) '(f"y - /y") + (Mcos v - c) (y"a - ya") + (Lcos - b)'(a"p - af") 0 = (M cos v - c) ("y - ^y") + (M-B)(^y"a - yTa") + (McosX-a)'(a"3 - ap") 'O = (Mcosu -b) (/"y - -y") + (McosX-a)'(y"a - -ya") + (M- C)'(a" - a") 0 = (N-A) (py' - '/y) + (Ncosv-c)'(ya' - y'a) + (Ncos/-b)'(a3' - a'p) = (Ncosv-c)' (y' - ^'y) + (N-B)'(ya' - y'a) + (NcosX-a) (ac' - a') = (Ncos -b) (f-y' - S',y) + (NcosX-a)'(ya' - y'a) + (N-C)-(a/' - a'/). Now from the first of these groups of three it is possible to eliminate f'y"- i"y', y'a" —y"a', a'3"- a"/'; from the second, i^"y- y'", y"a-ya", a"/l-afl'"; and from the third,.y'-/3'y, ya'-y'a, afi'-a'f3; and this being done there is obtained the set of three equations 0 = (L-A)(L-B)(L-C) + 2(Lcos X-a)(Lcosu -b)(L cos v-c) -(L- A)(L cos - a)2 - (L- B)(L cos u- b)2 - (L - C)(L cos - )2, 0 = (M-A)(M-B)(M- C) + 2(M cos X-)(M cos M- b)(M cos -c) - (M- A)(M cos X - )2 _ (M- B)(M cos - b)2 - (M- C)(M cos v -c)2, 0 = (N-A)(N-B)(N-C) + 2(N cosX-a)(N cos,- b)(N cos v-c) -(N - A)(N cos X - a)2 - (N - B)(N cos - b)2 - (N - C)(N cos. -c)2; 414 HISTORY OF THE THEORY OF DETERMINANTS from which it is clear that the unknowns L, M, N are the three roots of the equation in x, 0 = (x - A)(x - B)(x - C) + 2 (x cos X - a)(x cosa - b)(x cos v - C) -(x - A)(x cos X - a)2 - (x - B)(x cosIA - b)2 - (x - C)(X COS V - C)2' and therefore may be considered as expressible in terms of the nine knowns, A, B, C, a, b, c, X, c, v. To obtain the remaining unknowns-which, be it noted, are not a,~ a"1, pi, >1 but /y../3/, 'a" - "a', " -y", ya a, a',' - a"/', a",3 - a!31, aO'- -do -recourse is had to the two original sets of six equations. In the first equation of each set a2 occurs, in the second P2, and in the sixth a,3. Eliminating these in succession we have (L-M)a '2 +(L-N)a"2 -L-A, (L-M)ft12 + (L-N)o "2 L-B, (L - M) a'/' + (L - N) a'3" = Lcos v - c; and thence (L - NM)(L - N)(a3" -a"/3')2 = (L -A)(L - B) - (Leos v-c)2; so that one of the nine unknowns (L A)E -B) - (L cos v _ C)2 4- M( -RI)(L - N) the others being like it, and indeed derivable from it, although Jacobi does not say so, by cyclical permutation of triads of letters. The solution thus reached we may formulate as follows:The Cartesian equation AX2 + By2 + CZ2 + 2ayz + 2bzx + 2cxy = 0, where the axes ar~e inclined to one another- at angles X, M v, mcay be transformed into L$2 + Mil2 + N 2 = 0, ORTHOGONANTS (JACOBI, 1827) 4155 where the axes are rectangular, by means of the suhbstitution {(L-B)(L-C) -(LcosX-a)2}C + (M - M- -(AosX-a)2 f (N-B)(N - C) - (N cosX{(L -C)(L -A) - (L cos p b)2 I A(-L)N-M A2(L-M)(L-N) $.... '= i~~~Bc ~ ~.Ni~~i where L, M, N are the roots of the equation x - A x cos v - c x cos tk - b X COS v - C x - B x cosX -a =a0, xcosm - b xcosX - a x - C and 1 COS v COs M A2 - cosv 1 cos X CosfX cosX 1 The paper closes with a reference to the case where cosX = cos 1A = cos v = 0, and to the case where a = b = c = 0; the equation for the determination of L, M, N being in the former case X3 - (A~B+ C)X2 + (AB+BC+CA-a 2 - b2 - C2)x - ABC + Aa2 ~ Bb2 + Ce2 - 2abc = 0, and in the latter case A2x3 - (A sin2X ~ B sinu21 + C sin2v)X2 + (AB+BC+CA)x - ABC = 0, "weiche beide Gleichungen schon sonst gegeben sind." JACOBI (1827). [IDe singulari quadam duplicis integralis transformatione. Orelle's Journal, ii. pp. 234-242; or Gesammelte Werke, iii. pp. 55-66.] Although the title of this paper is qnite unlike that of the preceding, it will be seen that the two are in essence most closely related. .416 HISTORY OF THE THEORY OF DETERMINANTS The double integral referred to is Jjsin where p = a+ a' cos2 \b3+ a" sin2 ifr COs2 2+ a'..sin2if sin 2 ~ 2b'cos \r - 2b"sin 4r cos p + 2b"'sin \lt sin ~ + 2e'sin2VZ cos p sin 9 ~ 2c"cos V, sin Vf sin p + 2c"'cos Vz sin V/, cos -that is to say, where p is a quadratic function of cos IVz, sin Vt cos p, sin 4t sin 0; and the purpose of the paper is to show that the integral can be transformed into F F ~~~sin P. DP.D0o JiG + G' cos2 P + GI"sin2P cos2 0 + G/"' sin2 P sin2 0' where the denominator is a quadratic function of cosP, sin P cos 0, sin P sin 0, but contains only the squares of these quantities. The transformation is avowedly suggested by Gauss',solution of a simpler problem of the same kind, viz., the transformation of J,,/[(A - a cos E)2 + (B - b sin E)2 ~ C2] into the form I V(G ~ G'cos2 P + GI" sin2P)' As in the preceding paper, Jacobi does not begin with the substitution which is really sought, but with the reverse substitntion,-that is to say, the substitution necessary for the transformation of fF ~~~sin Rp.DP ___ino _sn _____~,a JJG + G' Cos2 P + GI' "sin2P cos2 0 + Gsiu2 P sin20 into p -knowing that from the latter substitution, when found, the former will be obtainable. This substitution he takes in the form cos P = a ~ a' cos / + a" sin V, cos ( + a."' sin 4' sin p + 3 ' cosqz + 6" sin V cos g + 6+." sin Vz sin ' sin P cos 0 = + ~3'cos V + /3"sin Vz cos 4 + 8/...siqZ usin9 3+ 8'cos Vz + 6' sin x/cos7$ + 6"' sin fr sinp' sin Psin 0 = y + Y'cos V + y" sinu., cos p + y"..sin uz sin7$ 6 + 6' cosxfr + 6" sin4.-cos (p + 6"' sin V sinp' ORTHOGONANTS (JACOBI, 1827). 41 the three new facients, cos P, sin P cos 0, sin P sin 0 being expressible as fractions whose numerators and common denominator are linear fnnctions of the original facients. It rests with him therefore to prove that the sixteen quantities a, a, a, a 7, 7 V,, 7 a, a', a", a'",and the four G P, (', (/ 1, I //,are so determinable that the performance of the substitution may bring back the original integral. By reason of the fact that cos2P ~ sin2P Cos22 0 + sin2Psin20 = 1 for all values of P and 0, it follows that the expression (a + a' cos V/ + a" sin 4r cos /5 + a"' sin ~fr sin ()2 + (3 + O' Cos +f A- /3"sin V~r cos /5+...'sin sfr sin (/)2 + (Y +y' Cos +fry / sin if cos 5 + y"'sin xfr sin /)2 - (a-I 'cosp-~a" / sinqlrcosp + 6."' siunfr sin 0)2 must vanish for all valnes of 4r and p, and that therefore a number of relations must exist between products of pairs of the coefficients. These relations Jacobi might have obtained by giving special values to Vt and qS: for example, by putting Vfr = 0 and V. = r he might have obtained (a2+/32+y2-a2)+ 2(aa'+/3/3a+yy' a') + ('2+/'2+7y12-'2) = 0 and (a2 + 2 + / 2+y62) -2(aa'~/3/3'+ /y'a- 6'6) +(a'2 + /'2 + v'2-a'2) = 0 and thence aa ~ 3/'+77'-aa' = 0 -and a2~ /2+ Y2- a2 = -(a'2+0/2+2y'2-a'2). As a matter of fact, however, taking a hint from Gauss, he concludes that since cos21, + sin2V/ coS2p + sin2Vt sin2, -1, the expression must be of the form lc(cos24r + sin2 V cos2/ p+ sin2V/ sin2(p - 1) M.D. 2D 418 HISTORY OF THE THEORY OF DETERMINANTS and that therefore by equalisation of coefficients a2 +/32 + y2 -_2 = -, a2 + P/2 + y'2 -_ 82, a"/2 + 1"2 + "/2 _ 8"2 = - i a//2 + 13///2 + y//2 _ 8"'2 = 7, aa' + 33' -+ y7' - 8' = 0, aa" + 13 + yy" -88" = 0, aa"' + 133'"' + yy"' - 8' = 0, a //a + I1/1/// + yQy'" - "8"/'/ = 0 'a' + 13"'1' + 7/y' - 8"'8' = 0, a'a" + 3'13" + 'y7" - '8" = 0, where k is arbitrary.* Again, since by making the substitution in the denominator G + G' cos2P + G" sin2P cos20 + G"' sin2P sin20 a multiple of the original denominator p must be obtained, it follows that the expression G' (a + a' cos J + a" sin V cos q + a"' sin q sin q)2 + G" (3 + /' cos 1, + +3" sin ir cos q + I"' sin V sin q)2 + G"'(y + 7' cos Vz + y" sin r cos q + y"' sin f sin q)2 + G (8 + 8' cos V + 8" sin I cos 0 + 8"' sin i sin 0)2 must also be a multiple of p. Putting it equal to kp, and equalising the coefficients, we obtain another set of ten equations G'a2 + + /G"/y2 + G + G82 = ak, G'a'2 + G"13'2 + G'"'y'2 + G8'2 = a'lk, G'a"2 + G"/3"2 + G"'y"2 + G8"'2 = "k, G'a"'2 + G"13"'2 + G"'y"'2 + G8"'2 = a'c," G'aa' + G"WO' + G"'7yy + G8.' = b'k, G'aa" + G" " G" + G'yy"I + G66" = bllk G'aa"' + G'"/313/'" + G"'yy"' + G88"' = b"'ll G'a"a"' + G"i3"pP3" + G"'y"y"'l + G8"6 = c'k, G''a' + G""a + G"yy/"'' + G "' + G8"' = c"k, G'a'a" + G"/3'1" + G"'y'y" + G'"16 = c"'k. *The fact that these equations imply I apy'y" "' = k2 is not alluded to. ORTHOGONANTS (JACOBI, 1827) 419 We have thus a score of equations from which to determine the score of unknowns, a,/3, y,, a ',..., G, G', G", G"'. From this point onward the procedure closely follows that of the preceding paper. Noting that the specialising substitution - G = G =G= "/ = G' = -a =a' =ca" = a' =1.. b = b" = b"' = C = C/ = C = O changes the second set of ten equations into the first, he confines himself at the outset to the second set. From this four sets of four equations are selected, e.g., the set a.G'a + /3.G"/3 + y.G~"y + 6.G8 = akCe a'.G'a + 3'.G"3.G' + y'.Gy +.G = b'k a".G'a + /3".G"/3 + y".G"'y + 8".G~ = b"l a"'.G'a + /3"'.G"/3 + y"'.G"'y + ~"'.G8 = b"'kl and solved as sets of linear equations, the results being put in the form kc(Aa + A'b' + A"b" + A"'b") = G'a, k(Bc + B'b' + B"b" + B"'b"') = G"3, le(Cc + C'b' + C"b" + C"'b"') = G"'y, k(Dct + D'b' + D"b" + D"'b"') = G6; k(Ab' + A'a' + A"c'" + A'"c") = G'a', k(Bb' + B'' + B"c"' + B"'c") = G"/3', c(Cb' C' + C + C"c"' + C"'c") = G"'y', k(Db' + D'a' + D"c"' + D"'c") = G6'; k(Ab" + A'c"' A"a" + A"'c') = G", k(Bb" + B'c"' + B"ct" + B"'c') = G"/3", k(Cb" + C'c"' + C"a" + C"'c') = G"'y", k(Db" + D'c"' + D"ca" + D"'c') = G8"; kc(Ab'" + A'c" + A"c' + A"'c"')= 'a"', k(Bb"' + B'c" + B"c' + B"'"') = G"fi'", k(Cb"' + C'c" + C"c' + C"'a"') = G"''y" k(Db"' + D'c" + D"c' + D"'a"')= GO"'; 420 HISTORY OF THE THEORY OF DETERMINANTS where it is readily seen what is denoted by A, B, C, D, A', B',... The corresponding results from the other set of ten equations are a = -kA a' = kA' a" = /cA" a" = kA'" a/// = kA/// /3= /3' "_ - cB leB' kB" k~B"' 7, = = =: kC' kC" kcC"' = ' = 8" = 6' = -D, - cD', - kD" -HD"', these being most quickly obtainable by means of the specialising substitution just referred to. By taking each result of the former collection along with the corresponding one of the latter, four new sets of four are deduced, which on rearrangement stand thus: A(a+G') + A'b' + A"b" Ab' + A'(c'-G ') + A"c"' Ab" + A'c'" + A"(a"- G') Ab'" + A'c" + A"c' + A"'b"' + A"'c' + A"'(a"'-G') + BS"'b"' +3 B"'c" + B"'c' + B"'(a"'- G") = 0, - 0, =0; = 0O = 0, - 0, =- 0 = 0; B(a+G") Bb' Bb" Bb"' + B'b' + B'(a'-G") + B'c'" + B'c" + B"b" + B"c"' + B"(Ca"- G") + B"c' C(a+ G"') + C'b' + C"b" Cb' + C'(a '-G"') + C"c"' Cb" +C'c" + C"(ca" - G"') Cb'" + C'c" + C"c' + CG"'b"' = 0, + CI"c" = 0, + C'"c' = 0, + C"'((a'"- G"') = 0; D(a-G) Db' Db" + D'b' + D'(a' +G) + D'c" + D"b" + D/"c"' + D"(ca"+ G) + D"c' + D"'b"' + D"'c" + D"'c' = 0, = 0, -- 0. Db"' + D'c" + D"'(a"'+G) = 0. The elimination of A, A', A", A"' from the first set of four; B, B', B" B, "' from the second set of four; and so on; gives * Observe A is not the cofactor of a, viz., j /'y""' I, but i /'y""'l + a+p'y"'" i. Attention has been drawn elsewhere to the fact that at this point a passage occurs which contains Jacobi's first printed reference to determinants. The words are "Valores sedecim quantitatum A, B,... supprimimus eorum prolixitatis causa; in libris algebraicis passim traduntur, et algorithmus, cuius ope formantur, hodie abunde notus est." ORTHOGONANTS (JACOBI, 1827) 421 rise to four equations, the first of which is a quartic in G', and the second, third, and fourth differ from the first merely in having G", G"', -G in place of G'. This, of course, is the same as saying that G', G", G"', -G are the roots of a certain quartic in x, which would nowadays be written a-x b' b" b" b' a'~+x c"' c"a b" c"' e/ a+x c' = b"' c" ' a" '+- x but which Jacobi writes in the form (a - x)(a' + )a + x) + (a" + x) -(a-x)(a'+ x) -(a- )( + 2-a- )((a +) - ( x)-)(a"' + x)c2 -(a" + x)(a'+ )2-("' + ) - )(' + X)b"2 - (a' + x)(" + x) b"'2 + 2c'c"c"'(a - x) + 2c'b"b"'(a' + x) + 2c"b"'b'(a" + ) + 2c"'b'b"(a"' + X) + b'2c'2 + b/"20 C2 + b"'2C'2 - 2b'b"c'c" - 2b"b"'/c"//c/ '- 2b"'b ', just as if he had expanded the determinant according to products of the elements of the principal diagonal. Interrupting the process of solution for a moment Jacobi draws attention to the fact that elegant relations between the sixteen quantities a, a', a, a".. and the sixteen A, A', A", A"',.... have been handed down by Laplace, Vandermonde, Gauss, and Binet,-an interesting remark as showing what writings on determinants were then known to him. Upon the subject of these relations, however, he does not enter, contenting himself with giving two sets of equations derivable from them with the help of the sixteen results a = - kA, /3 = -EB, The first set resembles the half-score of equations obtained near the outset, being -a2 + a2 + a"2 + a"'2 = - a/3 a + a/'/' + " / + "/ = 0, - y, + y"" + '//d// + y"I'"' = 0. 422 HISTORY OF THE THEORY OF DETERMINANTS The other set consists of sixteen of the type a/3 - a'P = - (y"6"' -y 6 )e, where e= +1, and is in effect a prolix way of stating the fact, nowadays familiar, that any two-line minor of laf'y'""' differs from its complementary minor only in sign, if it differ at all. Further he inserts at this stage the reverse substitution of that with which he started, viz., - 6' + a'cos P + 3'sinPcos 0 + 'sin P sin ~ - a cosP - 3 sinPcos - y sinPsinO' - 8" + a" cos P + /3" sin P cos 0 + " sin P sin 0 sin cosp= - a cosP - sin P cos 0- y sin P sin ' - 8"' + a"' cos P + A"' sin P cos 0 + y"' sin P sin 0 sins sin; = 8 - a cosP - / sinPcos 0- y sin Psin0' to which is added the fact that the common denominator here is the quotient of k by the common denominator in the original substitution. These results, he states, are easily proved,doubtless by solving the three equations of the original substitution for cos r, sin 4 cos, sin f sin p, or by taking the results as already found, and verifying them by substituting the values of cos P, sin P cos 0, sin P sin 0. On returning to the main line of investigation, viz., the solution of the set of twenty equations, Jacobi unfortunately does not proceed with the same fulness of explanation as before the interruption. In fact, the values of the remaining sixteen unknowns are merely put on record without any indication of the mode in which they have been obtained, " brevitati ut consulatur," the first four of the sixteen being a2 (a'- G')(a"-G')(a"'- G'-) - '2(a'- G') - c"2(a"- G') - "'2("'- G') + 2c'c"c"' k - (G'+ G) (G'- G")(G'- "') a'2 (a"- G') (a- G') (a + G') - b"'2(6"- G') - b"2(a"'- G') - c' (a + G-') + 2b"b"'c' k = (G' + G) (G' - G")(G' - G"') "2_ (al"-G')(a+G')'-G')- '2(a"'- G')- "/2(C+-G')- b""2 (a'-G') + 2b"'b'c" k (G' + G) (G' - G") (G' - Gr"') '2 _ ( + G')(a'- G') ("- G') - c"'2(a + G' -) - b"2(a'-G') - 2(a"- G') + 2b'b"c"' k (G' + G)(G'- (")(G'- G"') ORTTHOGONANTS (JACOBI, 1827) 423 and the others obtainable therefrom by the change of a2, a2,, a a,,a/2 G G', G", G, into 32, /2, // 2, // 2, G, G", G', G"', 2 7'2 "2 y,"'2 G"', G', -2 _22, _-2, _, /2, _G/, -G, G", G"'. The difficulty of the double sign which appears in every case is got over by merely fixing at will the sign of a, /3, y, 8, —the reason being that there are rational expressions for aau, aa, aa,11 3',.. ~.,yy~... 6.,6 and indeed also for a a, a, a.. similar to those just given for a2,.... For example a' b' (a" - G') (a"' - G') - c"b"' (a" - G') - c"'b" (a"' - ') - b''2 + b"c'c" + b"''c"' T (G' G) (G' - ") (G' - G"') There is nothing to suggest that the numerators of all these expressions are determinants, and still less that in the case of 2 / x/ /// a2 aa aa aa k' k' l' k /2 / // / /// a aa aa Ic ' ic ' "2 // /" a at a 7J' k a"'2 the numerators are* the ten principal minors of * For the modern reader the following substitute for the missing demonstration will suffice:If the cofactors of the elements in the four-line determinant given at the top of next page be denoted by [11], [12],..., then from the equations - (a+G')a + b'a' + b" a" + ba"' 0 -b' a + (a' - G')a' + c'" a" + c" a"' = - b a + c"'a' + (a"-G')a" + c' a"' = O - b'" a + e" a' + c' a" + (a"' - G')a"' = 424 HISTORY OF THE THEORY OF DETERMINANTS a+G' b' b" b' b' a'- G' c"' c" b" c"' a"- G' c' b"' c" c' a/"- G' The next and concluding paragraph of the paper is of course occupied in showing that by making the substitution whose coefficients have just been obtained, the given integral can be transformed as desired. It is worth noting here that although this paper and the previous one are contiguous in the original volume of publication, and the problem solved in the second is in essence quite similar to that solved in the first, there is not a word to indicate that the author viewed them in this common light. we have a a' a/ a/// [11] - [12] - [13] - [14] a a' --:.... [21] [22] a [31] a [41] Multiplying in these lines by a, a', a", a"' respectively we see that a 2 aa 2 /2 a//2 [11] -[22]- [33] [44] and therefore that each of them is equal to t2 _ a'2'- a/2 _ - a'2 [11] - [22] - [33] - [44]' and thus equal to -k [11] - [22] - [33] - [44] But by the rule for differentiating a determinant the denominator here is the differential-quotient of the determinant with respect to G'; and this because of the theorem d-{ (x - rl) (x - r2) (x - r)... (r - r2) (r - r3.. L Jrd a=ri is equal to -(G' -G") (G' - G"')(G'+G): consequently IGG kOcG G+" a2 (G/'-G")(G'-G"')(G' +G) - [11] ORTHOGONANTS (CAUCHY, 1829) 425 CAUCHY (1829). [Sur l'equation a l'aide de laquelle on determine les inegalites seculaires des mouvements des planetes. Exercices de Math., iv. pp. 140-160; or (Euvres completes, 2e ser. ix. pp. 172-195.] The equation as it arises with Cauchy would be more fitly described as the equation whose roots are the maxima and minima of a homogeneous function of the second degree with real coefficients, and with variables subject to the condition that the sum of the squares equals unity. Denoting the function by Ax+2 + Ayyy2 + Azzz2 +.. + 2Ayxy + 2Azxz +... or for shortness' sake by s, he of course begins with the known equations for determining the extreme values in question, viz., the equations 3s Ds as ax_ =y _ _a ~ y z An elementary algebraical theorem gives each of these ratios 's as as x + y- + z- +.... _ x ay 3z X2 + y2 + z2 +. and, therefore, by the fundamental theorem regarding the differentiation of homogeneous functions and by the abovementioned condition, = 2s. He thus obtains the set of equations 1 3s, as s -x" 1 =...... 2 = SX, -ax SY 2 = - or (Ax -s)x + Axyy + Azz.. = 0 j Aysx + (Ayy-s)y + Ayzz +. = 0 Azxx + Azyy + (Azz-s)z +... = 0 and therefore concludes that, on eliminating x, y, z,... from the set, the resulting equations in s, S = 0 426 HISTORY OF THE THEORY OF DETERMINANTS say, has for its roots the maxima and minima values of s. The third chapter of the Couers d'Analyse is then referred to and taken as warrant that "S sera une fonetion alternae des quantites comprises dans le Tableau AX - s A, AX, Ay Ayy -s Ay. Axz AL Az. - s and the developments of the function are given for the cases n = 2, n 3, )n = 4 exactly in the form adopted by Jacobi. The question of the particular values of the variables xv, y, z, which correspond and give rise to each of the n extreme values of s is next taken up, the equations for the determination of them being clearly the set from which the equation S = 0 was obtained (a set, be it remarked, which of itself can only give the ratio of any twAo) and the additional equation X 2 + y2 + Z2 +.... = 1. A series of identities connecting these n2 values is however first obtained. Denoting by xr, Yr, Zr,... the values of x, y, z,... which corresponds to the extreme value 9,. of s, he has, by a double use of each equation of the set, the n, pairs of equations (A,-s 1)cc + Axyy 4 A,,z1 +... = 0 (AXX - s, 2 + AyY2 + A,Xz +... = 0~ A,,xj + (Ayy-sj)yj + Ay,z1 +.+.. 0 Axx 2 + (Ayy -S2)Y2 + Ayz2 +... = oJ AxZij + Ayzy1 ~ (Azz - s1)z, + 0+ AXx2 + A,y2 + (Azz-S2)z2 +... From the first pair Axx can be eliminated, from the second pair A,,, and so on. Consequently there is in this way obtained -the n equations (I2 - S1)X1X2 + A,,x(yyX1 -xly2)+ Az,(X2z1l - xiz2) +. = Ax5(y(x1-YI,1x2) + (82 81)YlY2 + A,(y~z1- yz'2) +... = 0 A Xz 7-2I z X2) ~ Az(zy2 - ZOy,) + (s2- s1zl ~.+.. = 0 ORTIIOGONANTS (CAUCHY, 1829)42 427 and from these by addition @xI9~YlY2+zlzS.. )(S2-'S) =0, the, conclusion being " Done, toutes les fois que les racines sl, S2' serout ine'gales entre elles, on aura xIx2 YII+y-y1~z2z + 0; et, si le'quation S = 0 n'offre pas de racines 6'gales, les valeurs de x, y, Z,.. correspondantes 'a ces racines verifieront toutes les formules comprises dans le Tableau suivant: xi 2 + Y2 + =1, rxI2~+yy2 +....=0.. XIX,,~ +?1In +.. 0 x2x1 ~ Y2y1 +..=0, x2 2+ y2 2 +... =1,X2Xn + Yd/n +..=0 This interlude over, the fundamental set of equations is returned to, and, the first of them being deleted, there is got from the remainder Pxx P where the denominators are seen to be what we now call certain 'principal minors' of S; or, as Cauchy says, where P,,, is "cc que devient S, lorsqu'on supprime dans le Tableau les termes qui appartiennent 'a la me'me colonne horizontale que le bino~me A., - s avec ceux qui appartiennent 'a la me'me colonne verticale que A, - s on bien encore les termes compris dans la m~me colonue verticale que A,,, - s, et ceux qui sont renfermn's dans la m~me colonne honizontale que A,,, - s. " The ratios x: y:z.... having thns been got, there only remains, for the determination of x,, Z.,to nse the,equation x2+ Y 2+ Z1. IBut before doing so it is temporarily convenient to introduce an,alternative notation, viz., denoting the signed minors PXX, -Pyy, -Pzz.... -by x, Y Z. so that the values of these corresponding to Xr., Y11, Zr... and 428 HISTORY OF THE THEORY OF DETERMINANTS therefore to 8r, may be denoted by X,., Y,, Zr,. We thus have from the additional equation X Il Z 1 X Y Z VX2 +Y2+Z2~.. and therefore XI Yi ZI - /X122+Y 12~ Z12~2 X2 Y2 2 1 -4 -X2 2Z"/X2SFY2+Y Z22+..2 2 2 2 2 2 2 Xn Yn 2n 4- 1 Xn y, x, I~~Xn+, 2 X,7 Yn Z, - IX2,Y/ 2~Z 2+... Of course this supposes that the special values of X1, Y1, Z1,... occurring in the denominators do not vanish; and Cauchy's conclusion therefore is "les expressions zi, y1, z2. X2, 1/12 -'2. seront, aux signes pre's, completement determinees...., a moins que des racines de Phiquation S =0 ne verifient en m-nme temps la formule Px 0 ' The next step is to prove that the roots of the equation S =0 are all real so long as the coefficients of the quadratic s are real. If the contrary be supposed, viz., that one of the roots s,is of the form X + f\/l -1, this will of course entail the existence of another sq of the form X - l-1. Also, X, being the same function of s,, that Xq is of 8sq it will follow that X., and Xq will be of the form M+NN/-1 M- N-1 and therefore that XX = M2~N2. This means that XXq will be positive or zero, and similar reasoning would prove the same regarding Ypyq, ZpZq,.. None of them, however, can be positive; for since XpXq +Yp~.. =q0, ORTHOGONANTS (CAUCHY, 1829) 429 it follows from the values obtained for xp, Xq,..., that XpXq + YpYq +.. = 0. And since they are all zero, and each the sum of two squares, we are forced to the conclusion that Xp = Xq = 0, YP = Yq = 0, Zp = Zq =0, which is the same as to say that the roots xp, Xq satisfy the 'equations 0=X =Y Z =.... *i.e., 0 - Pxx = Pxy Pxz. The supposition therefore that the equation of the nth degree,S = 0 can have a pair of imaginary roots leads us to assert that a perfectly similar equation, PX =0, of the (n_-l)th degree, will have the same pair of roots. It is thus seen that the supposition and reasoning, if persevered in, will ultimately land us in an absurdity, when we reach, as we are bound to do, one of the equations of the first degree Axx - s = O, Ayy- =,.... " Donc l'equation S =0 n'a pas de racines imaginaires." The next object being to fix the limits between which the roots of the equation S=O are comprised, a theorem necessary for the accomplishment of this is first attended to. Formally *enunciated in modern phraseology it is:S being any axisymmetric determinnant, R the determinant got by deleting the first row and first column of S, Y the deter-,minant got by deleting the first row and second column of S, and Q the determinant got from R as R from S, then if R= 0, SQ = - 2. As the mode of proof employed by Cauchy applies equally well when S is not axisymmetric, let us take ] alb2cd for the given determinant, and write the proof as it would nowadays be given. To begin with, if A, A,... be the complementary minors of the elements al a2,... in aclbc3 I we have 430 HISTORY OF THE THEORY OF DETERMINANTS a1A1 - a2A2 + a3A3 - a4A4 = | 1b2c3d4 I bA1l - b2A2 + bA3 - b4A4 = lA -- c2A2 + cA3 - cA4 = 0 dlAl - d2A2 + d3A3 - d4A4 = 0 Putting A =0, and leaving out one of the last three equations, we obtain - 2A2 + C3A3 - a4A4 = I alb2C3d4 j - c2A2 + c3A3 - 4A4- =0 - d2A2 + d3A - dA =0 solving for A2 there results A a,b2cs3d4. | I I | 2 -. a2c~dA I from which by that is, a62C3d4 ' I b6C13d4 =- I ab2C3d4 i 03. C3d4 and this, when the original determinant is axisymmetric, becomes b1c34 2 = - I acb2 4.I,1 or, as Cauchy writes it, - Y2 = SQ. The first four cases of S =0 are then considered, viz., the series of equations S =0, S=0, S=0, S4=0,.... or, as at a later date they would have been written, AX~ - Azy Axz Axu >: We know from a identity is = 0, AZZ - s _ 0, AZQ AA - Ay - Ay Ayu Ayz AZZ-s AZ = O, Ap AZA, - s AZ Ayu AZU AU - - 8 Axy AxZ AwU A yy 8 Ayz Ay lny AZZ - AZo t AyB AZU A U- S later theorem (Jacobi, 1833) that when A1 is not 0 the I AB2 |I=| alb2cd4 I C3d4 |I ORTHOGONANTS (CAUCHY, 1829) 431 where each determinant is the complementary minor of the element in the place (1, 1) of the next determinant. The root in the first case is evidently A,. In the second case the solution is s = {A + AX (A z-A)2 + (2A,)2}, where the reality of the roots s8, s2 is manifest; and as their sum is Az + AR, it follows that s2 —AU may be substituted for A. - s in Azz- sl Azu |AZu AuA- sl with the result that we have (AuuI'-s2)(ALuu -s) = - Azu and are able to conclude that the roots sp, s2 of the equation S2 = 0 lie on opposite sides of the root A, of the equation S, = 0. Coming now to the case of 3 =0 we proceed differently, the three roots being localised by observing the changes of sign in S3 as we pass from one value of the variable s to another. Four values of s which suffice for the purpose are -oo, s8, S2, +00. No reasoning is necessary to show that, when s is =-oo, S3 is positive, and when s= +00, S3 is negative. When s = s we have S2 = 0, and therefore know from our auxiliary theorem that 81 and S3 must have different signs,-a fact from which we deduce that S3 is then negative. Similarly, when s=2', it is seen that S3 is positive. We thus have the set of values S = -GO, S1, 82, +GO, and S3 = +, - +, -, which shows that one value of s which makes 3 =0 lies between -o0 and sl, another between s, and s2, and the third between 82 and +oo. In other words, the roots s', s", s"' of S3=0 are such that between each consecutive two of them there lies a root of S2=0. The case of S4= 0 is treated similarly, the five values given to s in S4 being - o ',s 8, s, s, +oo. 432 HISTORY OF THE THEORY OF DETERMINANTS As before, there is no difficulty about the first and last of these, the value of 84 being seen to be positive for both. When s is put = s' we know that S3 vanishes, and that therefore Sand S4 must have different signs. The sign of S2 is settled from recalling that s' lies between - o and s1, and that for these values of the variable S2 is equal + o and 0 respectively: consequently the putting of s= s' makes S4 negative. Similar reasoning enables us to complete the set = — o, S/, 8S, S8, + 00 S4= +,, +, -, + from which we learn that one value of s which makes S4=0 lies between - oc and s', a second between s' and s", a third between s" and s"', and the fourth between s"' and - o. Having reached this point Cauchy adds'Les memes raisonnements, successivement etendus au cas ou la fonction s renfermerait cinq, six,..., variables, fourniront evidemment la proposition suivante: THEOREME I.-Quel que soit le nombre n de variables x, y, z,... I'quation S =0 et les equations de meme forme R =0, = 0,.... auront toutes leurs racines reelles. De plus, si l'on nomme S' S/ " Sf 'tf... S(11-l) les racines de l'quation R=0 rangdes par ordre de grandeur, les racines reelles de l'equation S=O seront respectivement comprises entre les limites - CO, St, S S..,, s(n- oo." Considerable space (pp. 188-192) is next given to extending this theorem to the case where several values of s satisfy at the same time two consecutive equations of the series S=0, R=0, P=0,.... Then follows a series of noteworthy deductions, which bring us round to the solution of a general problem of a quite different character, viz., the problem of transformation which ORTHOGONANTS (CATJCHY, 1829) 433 we have seen Jacobi attacking in detail. Denoting, as before, the extreme values, all different, of the quadratic function AXzx2 + A'yy2 +... + 2Axyxy +. by s, s2,..., sn, and by x,., y,,,.., the values of the independent variables which give rise to S,., we know that we have (Azx-sl)x1 + Azyy1 + Azzl +...= 0 Azyx, + (Ayy-sl)y1 + Ayz +...= o0 Azzl + Ayzy1 + (AzZ-s1)z1 +... = xl2 + y2 + l2 +.. = (Ax- s2)x2 + A.yy2 + Axzz2 +... = Ayxy2 + (Ay,- 82)y2 + Ayz2 + *. = Axx2 + AYy2 + (Azz-s82)z +... x2 + y2 + z2 + _ 1 (Axx-S,s)x,, + Axyyn + Axzz + Axyxn + (Ayy-Sn)yn + Ayzzn + AzXn + Ayz,, + (Azz- s)z + Xn2 + Yn2 + 2 Z2 + and that, further, when r and s are unequal xsx s yrys + ZrZs + s. = 0. Recalling this, Cauchy says that if a new set of taken.. = 0. = 0 v= O. = 1 n variables be i, V,,.. related to the old by the equations x = X1+ + X21 + X3 +.. Y Y1 + Y27 + Y3+ + Z = yZ1 + yZX + Z3 + z =.... +. +...* it is at once verifiable that X2 +2 + +.y2.. +22 +. 2. M.D. 2E 434 HISTORY OF THE THEORY OF DETERMINANTS In the second place, if we take any one, say the first, of the set of equations connecting s,, x1, Y z,.., the corresponding equation of the set connecting 82' x, Xy2, z2,..., and so forth, writing them in the form A,,x1 + Axyl + Axzo +... = I A,,x2 + AXyY + Az-2 +. 2 = multiplication by ~ y, r,... respectively, followed by addition, gives Axxx + Ax y + Axz +... = IC + s2X21' + 3X3 + Ax x + Ay y + Ayz + + Axx + Ayzy + Azz +... = s+z1S + sz9, s~ ~ +. [ In the third place, if the equations giving x, y, z,... in terms of %, j, 5,. be taken, multiplication by xi, Yr, Zr,... respectively, followed by addition, gives = xlx + yly + zlz +... =X2X + NY2~ + -7'- + = X3X + YOY + Z3Z +.. In the fourth place, if we take the second of these derived sets of equations, multiplication by x, y, z,... respectively, followed by addition, gives Axc2 Ay2 +... + 2Axy ~XY + = 12 + SY2 f 32 +. le 82+s22+3 +.3 With these results before him Cauchy is led to formulate the following proposition previously given "dans le dernier volume des Me'moires de I'Academnie des Sciences" "THE'ORLME II. Etant donnee une fonction homoge'ne et du second degre' de plusieurs variables x, y, z,..., on pent toujours leur substituer d'autres variables $, -, ~,... liees 'a x, y,... par des, equations lineaires tellement choisies que la somme des carr6s de XI y, z, s.. soit equivalente 'a la somme des carres de $, rj, 9,..., et que la fonction donnee de x, y, z,... se transformie en une fonction ~, r, 4,... hornogene et du second degre, mais qui renferme seulement les carr's de $, r, 9,. ORTHOGONANTS (JACOBI, 1831) 435 The validity of this rests on the supposition that the equation R= 0 has all its roots unequal; but Cauchy is careful to point out that even if this were not the case, the requisite inequality could be brought about by giving an infinitely small increment e to one of the coefficients Axx, Axy..; and as e could be made to approach indefinitely near to zero without the theorem ceasing to be valid, the validity would remain even at the limit. After a reference to the special case of three variables, the paper closes with the announcement that Sturm had arrived independently at the theorems marked I. and II., and had offered his paper on the subject to the Academy on the same day as Cauchy's.* JACOBI (Deer. 1831). [De transformatione integralis duplicis indefiniti A + B cos q + C sin q + (A' + B cos - + C' sin ) cos ^ + (A" + B" cos + C" sin q) sin in formam simpliciorem _ in formam impliciorem G - G' cos r cos 0 - G" sin -V sin 0 Crelle's Journal, viii. pp. 253-279, 321-357; or Gesammelte Werke, iii. pp. 91-158.] In his previous paper with a similar title to this Jacobi confined himself strictly to the consideration of his double integral, without saying a word as to the purely algebraical problem of transformation which lay at the root of it. Had he acted otherwise he would have been forced to note that this *A short account of Cauchy's memoir is given in the Bulletin des Sciences Math., xii. (1829), pp. 301-303, by C. S(turm), who says, "M. Cauchy a bien voulu observer, en terminant son article, que j'etais parvenu, de mon c6te, a des theor6mes semblables aux siens, sans avoir connaissance de ses recherches. Le Memoire de M. Cauchy, et le mien, dont je donne plus loin un extrait, ont ete offerts le meme jour a l'Academie des Sciences." A few pages further on in the same volume we come to an article entitled "Extrait d'un M6moire sur l'integration d'un systeme d'dquations diff6rentielles lineaires, pr6sente a l'Academie des Sciences le 27 Juillet 1829, par M. Sturm." The abstract occupies nine pages (pp. 313-322), and though it does not contain explicit statement of the two theorems referred to by Cauchy, the theorems themselves are evidently implied. There can be little doubt, therefore, that the memoir here condensed is that which was presented on the same day as Cauchy's. 436 HISTORY OF THE THEORY OF DETERMINANTS algebraical problem differed from that dealt with in the earlier paper of the same year merely in having four independent variables instead of three. Using modern phraseology, we may say that the one paper dealt explicitly with the transformation of a ternary quadric into the form Le2 + M2 + N2, and the other implicitly with the transformation of a quaternary quadric into the form Ge02+ G'l2 + G/"e22 + G/"'32; and such being the case, it is a matter for some surprise that the consideration of the corresponding problem for an n-ary quadric was left to Cauchy. In the lengthy paper we have now come to, the algebraical problem is no longer kept in the background, but forms one of the three parts into which the subject-matter naturally divides itself. The first is the "Introductio," occupying ~~1-9, pp. 253-264, and containing a brief account of previous related work, followed by an indication of the new results reached. The second is headed "Problema I." and occupies ~~10-15, pp. 264-279, its subject being an algebraical transformation pure and simple. The third and longest is headed "Problema II." and concerns the closely related, not to say dependent, problem of the transformation of a double integral. With this clear-cut subdivision there is no need for any process of sifting: we turn at once to Problema I. It is stated by Jacobi as follows:-Proponitur, per substitutiones lineares x = as + a's + a"s" w = at + bu + cv y = f3s + /'s' + fs" w' = at + b'u + c'v z = ys + y's' + y"s" w" = a"t + b"u + c"v quae identice efficiant x2 + y2 + s 2 + s'2, W2 + w'2 + w"2 = t2 + U2 + 2, transformare expressionem (Ax + By + Cz)w + (A'x + B'y + C'z)w' + (A"x + B"y + C"z) w" in hanc simpliciorem Gst + G's'u + G"s"v. ORTHOGONANTS (JACOBI, 1831) 437 Among the problems of the previous papers its closest relative is the first of all, the relation being that of general to particular. In modern symbolism the expression now given for transformation is x y z A B C w A' B' C' w A/ B" C" w", whereas in the first paper of 1827 it is x y A -B C x B B' C y C C' C" z. Cauchy's extension was from one set of three or four variables to one set of n variables; Jacobi's from one set of three variables to two sets of three variables. The preparation for solution begins with the reminder that the condition x2 + y2 + z2 = 82 + 8/2.+ /12 associated with the substitution x = as + a's' + a"s" y= 3s + 3's' + f"3 z = ys + y'S' + y"s" entails the six relations a2 + /2 + y2 =1, a'a" + 3'" + " = 0, a'2 + 3'2 + y'2 = 1, a"a + 3+ " y"y -= 0, a"2 + /"2 + y72 = 1, aa' + ' yy = 0; that from these and the given substitution we obtain the reverse substitution s ax + /3y + yz s' =a'x + 'y + ~ YZ s" = a"x + 3"y + y"z; 438 HISTORY OF THE THEORY OF DETERMINANTS and that, this latter substitution when taken along with the original condition gives the second set of six relations a2~+a'12 +a"2 = 1, /3y+ 31y' + 31'y"O, 032~/3/2 +/"21= ya +y'/a'+y7#a" =O, y2 + y'2 + y"2 = 1, a/ + a'f3' + a"f3" = 0. Further, it is pointed out that if we 'put for a (/3'y" - /3"y') + 03 (y'a" - y"a') + y (a'/3" - a"/'O) the ordinary solution of the given substitution results in 68 =x(~/1y// - 3",y') + y(,y'a"' - y/la') + Z(a'/13" - al/ es' =0 x(3"y - /3y") + y(ylca - ya") + z(a"/3 - af311) ES x8 y - f3'y) + y (y/a' - y'a) + Z(a/3' -a/' and that a comparison.of this with the reverse substitution as already -obtained produces ea = 3y" - /3y' ea' = /3 "y - 3y" ea" = / y' - ef3 = y'a"/ - ylla', e/3 = y//a - ya/", e0/3 = ya' - y'a, E7 = a/3/ - a"/3', cy7 = a"!~3 - a!31t, ey" = a/3' - a/f3. In the next place it is noted that with the help of these the left side of the identity (y a -ya")(a/3' -a'/3) - (,yea-y'a)(a"/3 - a"") ae becomes first and then 2 E.a and that consequently 2 - Lastly, attention is very pointedly drawn to the fact that if the nine quantities a, a', a"l, 3, /3', /3", y, y', y" be such as the foregoing results imply, and any three quantities X, Y, Z be connected with other three P, Q, R by the equations X = aP + a'Q + a"R)j Y = 3P + 3'Q + /"R Z = yP + y'/Q + y"/R ORTHOGONANTS (JACOBI, 1831) then it follows that P = aX + 3Y + Z Z Q = aX + 3Y + 'Z R = a"X + 13Y + y"Z and X2 + 2 + Z2 = P2 + Q2 + R2.* 439 (0) The next preliminary step is to formulate the equations which result from the identity of (Ax+ By +Cz)w.... with Gst + G's'u + G"s"v. These aret A =Gaa +G'a'b +G"a"c B = Gaa +G'p'b +G"i"c C = Gya +G'y'b +G"3y"c A' = Gaa' +G 'a'b' + G"a"c' B' = Gaa' +G'/'b' + G""c' C' = G^a' + G'y'b' + G"y"c' A" = Gaa" + G'ab" +G"a"c" B" = Gala" 'b + " + G""c" C" = Gya" + G'y'b" + G"y"c". Along with the twelve relations previously obtained, they give in all twenty-one equations for the determination of the three G's and the eighteen coefficients of the substitutions. The actual process of solution consists in a long series of deductions from the last-obtained set of nine equations, the repeated use of the twelve other equations being disguised by employing the theorem above called (0). Thus from the first column of equations this theorem gives Ga G'a' o/a/ = aA + a'A' + a"A" A" = bA + b'A' + b"A" = cA + c'A' + c"A"; the second column gives similar expressions for Gf3, G'3', G"/3"; and the third column for Gy, G'y', G"y". The whole set is in later notation ^( G Ad ae, ( a' a )b ' b" c c' c" ) Ga G'a/ GWa" Ga G'a Ga A A' A" A A' A" A A' A" ca a' a/" b b' b" c c' c" GP GP'3' G"" B B B B B' B " B" B B B",Gy G,y '/, ct Q a a c" b b b c c' c" Gy Gy bGC C' C" C C C' " C c' C" In leaving these preliminary deductions, it may be worth remarking that the like results which flow from the second given substitution and its associated condition are not taken entirely for granted by Jacobi, but are given with equal fulness, the two series indeed appearing in parallel columns. t v. next page. 440 HISTORY OF THE THEORY OF DETERMINANTS where h k where g h i is used to denote gp + he + kcr. Similarly by taking the same set of nine equations in rows there is obtained G'b G"c ( a 3 y a' /' y' a" /3 " ") Ga G'b We c A BC A B C A B C pa P Y 1 p yl a// pl/ yll Ga' G'b' G"c' = C Ga c A' B' C' A' B' C' A' B" C' Ga" G'b" G"c" a y a ' y' a" "y" A" B" C" A" B" C" A"// B" C" From these two sets of equations it is clear how the coefficients of one of the substitutions may be obtained when the G's and the coefficients of the other substitution have become known. Separating the latter of these new sets of nine in a similar fashion into column-sets of three, but solving this time in the ordinary way, Jacobi obtains a further set, which, if only to save space, we may write in the form ( a a' aa ) ( ) (Gv G - j aB'C"l IbB'C" IoB'C"I G G' Gp"77= (c aC'A"! | bC'A" cC'A" G ' G" G, a G' a'A'B" bA'B" iA cA'B" where A = ] AB'C", or, as Jacobi of course writes it, A = A(B'C"-B"C') + B(C'A"-C"A') + C(A'B"-A"B'). From a set giving the Italic coefficients in terms of the Greek coefficients we have thus got a reverse set. The other reverse set obtainable in the same way need not be given; but it is easily seen that the two have the same practical value as the two from which they are derived. tJacobi writes the nine equations in one column: they are better arranged in three, however. Cayley at a later date would have preferred to write more luminously A B C (G, Gi, G"a, a', a"a,, c) (G, G', G"I3, A', '"'a, b, c). A' B' C' = (, G', aG"a, a, aa', b', c') (G, G', W"/O', ', "'a, b', c') A" B" l " ORTHOGONANTS (JACOBI, 1831) 441 To make another advance, either of our latest sets of nine is taken and separated into row-sets of three, and theorem (0) applied. The result which Jacobi gives in nine separate equations of the type B'C"-B"C' aa aab a"c A ~ G +G' + G" may be written more compactly and more instructively in the form |B'C" | | GA" | | A'B ) ( act a'b a"c pca /3'b l"c ya y'b Y'c ) -+ +c - - +V - + A A A G~-G' (r" G GJ+G'-+ G +G' +G7 A A a A I G G~G ^ G" G G' G" G ' G'" BC' | CA' I A"B' aa" a'b" a"c" a3" 'b" p"c" ya" y'b" V"c" A A A G+ G' G" G+ G' + G" G+ G' +..+ Any one of the nine here, however, may be matched by one deduced directly from the set of nine which we obtained at the very outset. Thus* B'- B"C' = (G/3' + G'/'b' + G"/"c' )(Gya" + G'y'b" + G"y"c") - (Go3a" + G'3'b" + G"/"'c")(Gya' + G('y'b' + G"y"c'), = G'G"(/3'^y - 3"y')(b'c" - b"c') + G"G (8"1y - /3" )(c'c" - c"ca') + GG' (3y' -f3'y )(act'b"-"), = G'G"aa + G"Ga'b + GG'a"c. With this we have to compare A A A, aa + a'b + a"c, the result being that we obtain GG'G" = A, and thus reach the first resting-stage on our journey. * Nowadays we should rather put IB'C" I Goa'+ G'P'b' +G"/c' Gca" + G''b" + G"3"c" Gya' + G'y'b' + G"y"c' Gya + CG'y'b"+ G'yC" " G_ 3 G',p' G"P" a' b' c' Gy G'-y' G 'y" a" all c" -= I.GG'. | l I. I a'b" I + GG". I ". I a'c" i + G'G". I P'y"/. I| 'c/" I 442 HISTORY OF THE THEORY OF DETERMINANTS At the outset of the next stage it is found desirable, for brevity's sake, to. introduce six additional letters to denote certain functions of the known quantities A, A', A")...viz. p for A 2 ~ B2 + C2, q for A'A" A- B'B" + C'C", p' for A'12 + B'2 + C'2, q' for A"A + B"B + C"C, p " for A"12~ B"12 + C"12, q" for AA' + BB' + CC'. These are said to entail the six identities jpj"- 2 (B'C" - B"/C')2 + (C'A"1 - C"/A')2 ~ (A'B"/ - A"B')2, P "P q = (B"/C - BC"/ )2 + (C"/A - CA" )2 + (A"/B - AB"/ )2, jPP' - q"/2 =(BC' - B'C )2 + (CA'" C'A )2+ (AB' - A'B )2, qq" - pq =(B"/C - BC" ) (BC' - B'C ) ~ (C"A - CA" ) (CA' - C'A) + (A"/B -AB"/)(AB' -A'B) q"q - p'q' (BC' B'C )(B'C"/ - B"/C') + (CA' -CA )(C'A"/ - C"/A') + (AB' -A'B ) (A'B"/- A"B'), qq' - p"q" =(B'C"/ - B"/C')(B"/C - BC/1) + (C'A"/ - C"/A')(C"/A - CA"/) + (A'B"/ - A"/B')(A"/B -A/) and A2 = pp'p" - pq2 - p'q'2 p q"/2 + 2qq'" The original set of nine equations, giin AAAI.i em of the three G's and the coefficients of the substitutions, is then returned to, and the following equations derived,Q G2a2 ~ G'12b2 + G" 2c2 = G 2a'12 + G'12b'2 ~ G"2c'2, Q 2a"2 + G' 2b"/2 + G" /2 "2, =G G2a'a" + G'2b'b" ~ G"/2/c'" = G 2a"ca + G' 12bb + G" /2c//el q"= G 2aC' ~ G'2bb' + G"2ec'; the first three being& got by use of the second part of theorem (0), but all of them readily verifiable by merely substituting the said values of A, A',I A"l,*.. In exactly the same way from another set of nine equations, viz., those beginning = (B'C" - B"/C')a + (B"1C - BC"/)a' ~ (BC'- B'C)a' ORTHOGONANTS (JACOBI, 1831) 443 there is obtained pp" - q2 a2 b2 c2 a'2 b'2 c'2 A2 G + + A2 - G2 GQ'2 G"2' pp' q/- 2 _ 2 b " C/2 A2 = G2 +Go2+G G2' q'q"-pq _ ab" b'b" ' c"t + + A2 - G2 G'2 G"2' "q -p'q' a_ a b"b c"c A2 - G '2 +G G"2 qq-p" q" _ aa bb' + cc' -A2 - G2 G- 2 G+ 2 Then, by mere addition, half of the first derived set gives G2 + G'2 + G"2 = p +p' +p"; and the corresponding half of the second set 1 I + 1 _ 'p'+p"p+pp'-q2-_qq /2 G2 G2 G"2 - A2 which on putting GG'G" for A becomes G'2G"2 + G"2G2 + G2G'2 = pp" + p"p + pp - q2 _ q'2 _ "2 Lastly, by taking all of the first derived set and using the first part of theorem (0), there is obtained a reverse set of nine,G2a = pa + q"a' + q'a", G2' = q"a + p'a' + qa", G2" = q'a + qa' + p"a", G'2b = pb + q"b' + q'b", G2b' = q"b +p'b' + qb", '2b" = q'+ qb' + p"b", '2C = pe + q"c' + q'c", G"2' = q"c + p'c' + qc", G"2c" = q'c + qc' + p"c", 444 HISTORY OF THE THEORY OF DETERMINANTS and by the second part of the same theorem p2 ~ q"/2 + q'2 = G4a2 + G'4b2 + G"/4c2, jp2 + 2 + q2 -=G4a'2 + G'4b'2 ~ 4 G"e2 p"/2 + q'2 + q2 -G4a"12 + G'4 b"2 + G"/4c"/2 The existence of similar results obtainable from the second. derived set is pointed out, but separate investigation of the two sets is shown to be clearly unnecessary in view of' the following theorem: ccE qualibet formularum propositarum derivari posse alteram, si in locum quantitaturn A B C A' B' C' Al" B"/ C5f CG, G', C-" substituantur respective selquentes.. BC1- BWC' C'A"f - C"A' A'B"1 -A/B A A ' A BWC - BC" C",,A - CA" A"B - AB" 1 A ' A ' A ' BC' - B'C CA' - C'A AB' - A'B A A ' A unde, e.g., etiam pro A ponendum ~.Quod patet reciprc ubi illa in haec abeant, sirnul etiarn haec in illa miutari." 01 1 )CUM esse, id est, The reason for this dualism is at once perceived on noting that the original set of nine equations is matched by a derived set,, perfectly similar in form, but having (B'C"` —B"C`)/A,.. in place of A,...Of course, as Jacobi notes, the dualism extends. to the transformation which is the object of the whole memoir; that is to say, the equation (Ax +By +Cz)w + (A'x +B'y + C'z)w' + (A"x +B"y +C" Z)w"1 = Gst + G',s'u + G"s'v is necessarily accompanied by [(B`C" -B"C')x + (C'A" - C"/A)y + (A'B" - A"B')z] w + [(B"/C - BC")x + (C"/A - CA"/)y + (A"/B - AB"/)z]W' + [(BC' - B'C)x + (CA,' - C'A)y + (AB' - ABzw =G'GWvt + G"Gs'u + GG's"v. ORTHOGONANTS (JACOBI, 1831)45 445 This means, in modern phraseology and nomenclature, that the linear orthogonal suhbstituhtwon which change x y z A' B' C'w' A"/ B"/ C"/ w" will at the saine time change x y z B'C"/ C'A"/ A'B"/ w B"/C C"A A"/B w, into G'G"st + G"Gs'u ~ GG's"v. BC' CA' AB' w/ In parallel columins with these results regarding the p's and q's Jacobi places a series of others perfectly similar to them, the twin series originating in the fact that in squaring IAB'C"/ j, as we should nowadays put it, the multiplication may be performed either row-wise or columin-wise. The chief points in the second series we may state rapidly in modern compact form as follows By way of defining the new letters introduced we start with n' m' A2 + A'12 ~ A"12 AB +A'B'~A"/B" AC~A'C'A"C" +A// n' M~ 1' BA+B'A'+B"/A" B2 + B'2 + B"/2 BC +B'C'-hB"/C" m' l' n CA + C'A' + C"/A" CB + C'B' ~ C"/B" C2 + C'2 + C"/2 whence it follows that the, determinant of either matrix is. equal to IA B'C" 2, and the, secondary minors equal to I C'A"12~+ C"A12 +I ICA' 12 jC'A1jA"IBIA'" AIAB" 2 + Then from the original set of nine equations we have n' m' Ga ~ G' a' ~ a" "2G~q ~ G'2a'3' + G"2a&'f3 G2aty + G'2a',y'+ G"/2a-y" m 1' V G2~32 + G'2f3'2 ~ G"120/32 G203y + G/201'y' ~ G"/2013y" m' 91' G2y2 + G/2 y'/2 + G"/2y12 and from this, in passing, by the addition of diagonal elements, 446 HISTORY OF THE THEORY OF DETERMINANTS Next, as the matrix on the right ( G2a G'2a' G'2a ) = G G2/2' G/ "2/3" G2y '2y' G'y27"/ there follows a /3 y ) a' /3' y' a" 13",y/ (G2a G/2a' G23 G'2/' G27 G'2y' G"2a" ) ( G"2/3" = n' G,.,, //, O' m' I' n ) (a a' a" /,B//, / /3' 3" 777, ' i la + n'/3 + m'y = n'a + m/3 + l'y m'a + 1'/3 + ny la' + n'3' + m'y' n'a' + m/3' + I'y/ m'a' + ['/3' + ny' la" + n'/3" + m'y" ) n'a" + n/3i" + ['y" m'a" + ['/3" + ny", whence, by summing the squares of the elements of each row separately, we have G"a2 + G'4a'2 + G"4a"2 = [2 + r'2 + m'2, G"'32 G42 + G'42 "42 m + 'm 2 + '2 + '2, G 72 + Gy47'2 + G 47y"2 = 2 2 + 12 +2 J Among the results obtained up to this point, there are sufficient to determine the twenty-one unknowns, and to this Jacobi now definitely devotes a section (~ 14). First the G's are dealt with. There having been obtained G2 + G2 + G"[2 = +m++ =p+p +p ", G'2G"2 + G"2G2 + G2G'2 = (mn-1'2) +... = ("- q2) +. G2G'2G"2 = A, it is perceived at once that G2, G'2, G"2 are the roots of the equation X3 -x2((l+m+n) + x(mn+nl+lm-' 1-m'2-_2_n2) - (lnnf + 21'm'n- 1'2- mm'2 _- M 2) =nn 0, or X - X2(p +p'+p") + X(p'p" +pp +pp' -q2 - q2 q2 - (pp'2 + 2qq'q" -2 _-pq'2 -p"q"/) = 0; ORTIOGONANTS (JACOBI, 1831) 447 which respectively are the same as ( - l)(x- M)(x - n) - '2(x -) _ m'2(X - _) - ' n2( - )- 21'mn' = 0, (x-p)(x-p')(x -p") -q2(x-p)-q/'2(x- p') -q"'2(x-p") - 2qq'q" = 0; and either of which is X3 -x2(A2 + B2 + C2 + A'2 + B'2 + C'2 + A"2 + B"2 + C"2) (B'C" - B"C')2 + (C'A - C"A')2 + (A'B" - AB')2 +4x +(B"C -BC" )2 + (C"A -CA" )2+(A"B -AB"//)2 + (BC' -B'C )2 +(CA' -C'A ) + (AB' -A'B )2J - {A(B'C"/-BC') + B(C/'A"-C/"A')+C(A'B" -A"B')}2 0. As an alternative to this, however, it is pointed out that we might, by putting the equations G2a = la +n'/3+m'y' (O = (1-G2)a+ n'3+ m'y G23 = n'a +m/3+l'y in the form 0 = n/'a+( n-G2),/+ 'y G2y = m'a + ' +ny J oI = Mn'a+ 1 + (n -G2)y, eliminate a, 3, y and obtain a cubic in G2; then by similar action obtain the same cubic in G'2 and the same cubic in G"2. In this way the left-hand side of the equation, whose roots are G2, G'2, G"2, would naturally recall determinants, although Jacobi does not say so; and after Cayley (1841) it might have been written l-x n' m' p-x q" q' n' 'm-x I' or q" p'-x q m' 1' n-x q' q p"-x In the next place, four equations having been found in a2, a'2 a12, viz., a2 + a'2 a"2 = 1, a a + a G2a2 + G'2a'2 + G"2a/2 = 1 2 1 a2 + "2 mn-' 2 G-2 G'G'2 G"'2 a = A2 G4a2 + G'4a'2 + G"4Ya"12 = 2~+mn2.2+ ' 2, 448 HISTORY OF THE THEORY OF DETERMINANTS if the first three be taken there is obtained for a2 the value (GI - m)(GI2- 1)-1'2 (G2_ G'2)(G2- G /2), and, if the 1st, 2nd and 4th, the alternative form (I- G'2)(- G" 2) + m'+ 2n'2 (G2 G12)(G2_ G /2) where the identity of the two numerators is readily verifiable. In the same way the expressions for the squares of the six other coefficients of the first substitution may be obtained. The difficulty of the double sign resulting from the extraction of the square root is readily got over, because rational expressions similar to those for a2, a'2,... are given for the nine binary products a/3, a'/3', aOc3" y,..., from which, when the sign of one of the coefficients is fixed, the signs of the others at once follow. It is not noticed, however, that the numerators of these eighteen values are the principal minors of the three eliminants, t- G 2 ' ' I - G'2 n' 'I - G"2 n mn',' -G2 V ' m-G'2 I t' n- G"2 G" n' V' tn -G2, ' i - G'2, mI' 1' n-G' 2 above referred to, the corresponding unknowns being ( ) ( ) ( ) 9 /~ / i7 /9 ' /s'7' /9 /s" 7" a ap3 ay a2 a' / a'yI a/2 a"/3" ay// 32 13y I 2 3Y1 ^2 I / Y2 2 '2 y2 and the corresponding denominators, (G2 G'2)(G2- _ "2) (G/2_ G"2)(G'9G-G2), (G"2 _G2)(G,/2 0/2). As an alternative to this process for finding a2, a',... there is given another, which in some respects is the more interesting of the two. Beginning with a different set of equations, viz., the set (I-G2)a + n's + My = a'a + (n-G2)/3 + o Ut'a + 'o/3 + (n-G2)y =, ORTHOGONANTS (JACOBI, 1831) 4499 Jacohi drops out the first and finds a: 0: y, drops out the second and finds 3: y: a, drops ont the third and finds y: a i3. Then since these three sets of ratios are the same as the three sets a2: a/3: ay, 12: 3y: Pa, y2: ya: y1; and as the expressions found proportional to a,3, ay in the first set are respectively equal to the expressions found proportional to the same unknowns in the other sets; it follows that a2, a1, ay P32, 18Y are proportional to (m - G2)(n - r2)' -['2, ['m'- n'(mn - G ql' -2), (n' - G2)([- G2) -_ M'2, m"n' q - ['(1- G2), (1- G2)(M - G2) - -12 and therefore that a2 or or...... (m - G2)( - G2)- 1[2 or ' - n'(n - Q 2) o a2 +,32 + y2 - (rn - G2)(n - Cr2) + (na - Gr2)(1 - G2) + (1 - G2)('nl - G2)- 11'2 _ rn'2 - n12 Here, however, the numerator is equal to 1: and the denominator, being obtainable by differentiating (x - l)(x - Mrn)(x - fn) - ['2(X _ 1)-_ n'2(x -'m) - 'nI2(x - 'n) - 21"'V'n' with respect to x, and substituting G2 for xc in the result, must be what is obtainable in the same way from (x - G2) (X - G'2)(X - Cr"2) and therefore must be equal to (GC2 - G'2)(G2 - 0//2). 'There thus result the same values for a2, a3,.., 'as before. The values of a2, 2a',a.. are throughout given side by side with those for a2, a/3,..; thus(Cr2 - Mn)(Cr2 _ n) [2 2 _ (Cr2 2.p')(G2 -.p") -q C(G2 - G' 2) (Cr2 - 0/12) a (Cr2 -G'12)(Cr2 -//2) M.D. 2F 450 HISTORY OF THE THEORY OF DETERMINANTS At this point " Problema I." stands fully solved: one or two interesting addenda, however, are given in a concluding section (~ 15). From the equations Ga =Aa + B/3 + Cy, s = ax + /y +yz, G'b = Aa + B/' + Cy', and s' = a'x + 'y + y'z, G"c = A + B" C", 8" a" + y + y"z Ga' =....... by multiplication and addition* there are obtained Ax + By + Cz = Gas + G'bs' + G"cs", A'x + B'y + C'z = Ga's + G'b's' + G"c's", A"x + B"y + C"z = Ga"s + G'b"s' + G"c's"; and then from these by the second part of theorem (0) (Ax + By + Cz)2 + (A'x + B'y + C'z)2 + (A"x + B"y + C")2 = G2S2 + G'2s'2 + G"2s"2, which may also be written in the'form lx2 + my2 + zz2 + 21'yZ + 2mn'zx + 2n'xy = G2s2 s' + G"2 + Gs"2. To this of course may be appended the derivative from it by BICG-_ -B//C the substitution of,.... for A,.... viz., {(B'C"- B"C') + (C'A"- C"A')y + (AB"- A'"3)z}2 + {(B"C -BC")x + (C"A -CA")y + (A"B-AB")z}2 + {(BC' -B'C )x + (CA' -C'A )y + (AB' -A'B)z}2 = G2G"/282 + G"2G2s'2 + G2G/2,,2. "' We may formulate for use here the following theorem in modern dress:If I ao'y" f be an orthogonant, then A, B, C a, 3, A, B, C a', ', y' A, B, C a", O y"_' A, B, C a, x, y x, y, z a ',', ' x, y, z a, ",y" x y, z x, y, z ORTHOGONANTS (JACOBI, 1831)45 451 Further, it will be observed that oniy one substitution is here involved, and that consequently in connection with the other substitution there must be analogous results, beginning with pw2 ~2p'W'2 +]"W"2 ~ 2qw'w" ~ 2q'w'w + 2q'ww'=- G2t2 + G'2U2 + G"12v2 All of them, manifestly, may be described as transformations,simultaneous with the main transformnation, and, like one which appeared earlier in the paper, may be usefully enunciated in modern form as follows: The linear orthogonal sqnbstitations which change x y z A B C W it A' B' C' w' 1?t ` B" El" w"/ will at the s.ame Mtie change G'St + G's'n ~ G"'S"v 1. n' 'm'I x rn 1' yn into G 's2 + G'128'2 + G"2s"2, x n- nn' I! l'' nn' M n'n /-11' z n'l' -mm' M n'n-ll' lm,-nI x ft z.into GI'2G"2,32 + G"//2G2s'12 + G2G' 2s"/2, i29 q q' Iw in to q" }'I q W ' q, p" W" G2t2 + G'2'ut2 + G"I2v2, The second result, however, is seen to follow from the first, and a fourth fromi the third by the previously euunciated theorem of this kind. 452 HISTORY OF THE THEORY OF DETERMINANTS JACOBI (1832). [De trausformatione et determinatioue iutegralium duplicium commentatio tertia. Crelle's Jomr~nal, x. pp. 101-128; or Gesammielte Werlke, iii. pp. 159-189.] This memoir, although classed by its author with the two others of which we have giveu au account, is of much less interest on the purely algebraical side. In fact it consists almost entirely of the transformation of integrals like,.JJ/R sinp p dfr, do cy,.... by means of substitutions like rnlcosn. insiu9cosf sinsisiu n nAl COS ~i sin q cos O= where R - nj2cos2( + n2sin2 Cos2r + 2 sin2q5 sin2\fr. When, however, an advance is made from R to U, i.e. to a Cos2 -1- b2sin2 &cos2fr + e2sin2 sin2 / + 2sin Si, cosfr sin Vr + 2e cos p sin q sin lr + 2fcos p sin 0 cos ifr, the underlying algebraical problem becomes of more importance; for example, such a problem (p. 122) as the finding of the coefficients of the substitution 'i~ =yx +hIy ~-iz, v = g'x + h'y + i'z4 w = y"x + h"y ~ i"zj which transforms aCx'2 +_ by2 + Ce2 + 2dyz ~ 2ezc 2f Jxyz, a/X2 + b'y2 + c'Z2 + 2d'yz + 2e'zx + 2f'xy, into U2 + v2 + W2 n2 v2 W2 M2 17,2P 2 ~n 'p. p22 respectively. Still there is nothing calling for more than this passing mention. ORTHIIOGONANTS (JACOBI, 1833)43 453 JACOBI (1833). [De binis quibuslibet, funetionibus homogeneis secundi ordinis per substitutiones lineares in alias binas transformandis, quae solis quadratis variabilium constant: una cum. Crelle's Journal, xii. pp. 1-69; or Gesoanmelte Werke, iii. pp. 191-268.]I This memoir, the general plan of which has already been indicated (see above, p. 354), naturally divides into two main portions in accordance with the title, these being prefaced by an introdnction referring to both. The first portion, now to be dealt with, is the natural outcome of a thorongh re-examination of the author's own previous work viewed in the strong light of Cauchy's memoir of 1829. In the 'Introduction' (pp. 1-7) the general problem is at the outset concisely stated and shown to be determinate. Th e opening words are (p. 1): "Propositis inter variabiles XI, x2., - )X9, et Y I21, y..., 1 n aequationibus linearibus huiusmodi i/n cjx2 + n. + facile patet, co~fficientes a() quo rum est numerus nm, ita deuerminari posse, ut data functio quaclibet homogeuca secundi ordinis variabilium. xp, X2,.. - xn transformetur in aliam. variabilium Y1, Y2,..., y,, quae solis earum. quadratis constet, simuique summa quadratorum. variabilium non mutet valorem, sive fiat XIX1 + X2X2 +... + X"-X- =7- /Y1/ + Y/2/2 +..+ Y/flffl Nam hac altera conditio sibi poscit aequatioucs conditionales numero, 2~ porro cumn de functione transformata supponatur abiisse producta e binis variabilibus conflata, accedunt aequationes 2 ) ita ut habeas aequationes conditionales numero nn, qui est numerus co~fficientium substitutionis adhibitac. Unde problemia, determinatum. est." Referring shortly to Cauchy he next intimates the chief of his own new results, and illustrates it in recounting the contents of his previous papers. 454 HISTORY OF THE THEORY OF DETERMINANTS Problem ii. is then attacked, the direct consequences of the condition 2 + 2 + '+ XI y2+ 1 + 2 + +1 2 C +...+ = ++Y n Y... + Y, being first noted, namely a a+ + aax + + aKa,, == a'a+a aJ+.. + a)a() = 1 from which comes a(n) x = aKYl+a 2 + aY +.. + aK n and thence a(')a( ) + a(K)a() +. + a(X) = I 1 2 2 ' r it a(")a(K) + a(K)a(K +. + a()a (K), 1 1 2 2 it n In the second place it is recalled that if the determinant of the coefficients of the substitution, Z ~ aa (.../ a( be denoted by A, the cofactor of a(8) in A by /(), and the determinant _ ~q'/".,. BT) by B, there are at our disposal three results independent of the conditioning equation, namely, AxmK = fKY1 A + NY2 +. + Ynn B = An-l -4I1... B(m) = Am_-l. 1 -4- a(m+l)a(m+2) a(n). 2 ~ 2 1... -m Am-. t m+ an ' and it is then pointed out that a comparison of the first of these with one already obtained gives in our special case (- n)- Aat(), from which follows by substitution B i.e., ~ /3,/3. - A. - ) t a... a = A+1 that a comparison of this with the second general result gives A2 = 1; and that a like substitution changes the third general result into -A. _ a1a2__.a(m) = +a(+l)a(m+2) (n) +a'a" aam) a 2 a.....12 tm -- m+1 mn+2... In later language these are the propositions: The square of an orthogonant is unity and The product of an orthogonant ORTHOGONANTS (JACOBI, 1833) 455 and one of its coaxial minors is equal to the complementary minor. The other conditioning equation is next utilized, namely, that the homogeneous function of the second order aE aKXKX or V, K, X where aKX = ax, shall be transformable into and from Gly12 + G2y22 +.... + G.Yi2*2 The latter transformation at once gives a = GaIa -+ Gaaa +.... ++ GI (t.)aX Taking from this set of n12 equations the sub-set in which = 1, 2,....,, namely, ab, = GiaA. a, + G2a. a +.... + Gl *a a, I ',,r) ",1 (~. ) aC2A = GlaX. a 2 +... ++ ax G a,' " a ~^ / / i (T) (n) Ca, = Gla1 a% + G2a, +.... + G a (2)a and using the result (0), so strongly insisted on in his paper of the year 1831 (see above, pp. 438-439) and here again spoken of as something "quod maxime tenendum est," he deduces the n (or n2) equations (G ) (m)a,, (+ ) +. a (mt 01m) - a1 a1 + a2 a2 +.... + an Ct" and the equation C'Ix a + a2 + a.+ = (Gl,)2 + + (.... +, (G,x)2, pointing out however that from the former a more general result than the latter is obtainable,* namely, acIl A + Ca2 +a. + a 2c = (GCa' )(GKA) + (G2,)(G2) +.. (Ge< (G))(GanTLa). i The mode of deduction is not given, but evidently (Ga)(G A) + (G2a)(GK) -....+ )(Gn ) 456 HISTORY OF THE THEORY OF DETERMINANTS From the same source and in the same way he derives GCa',. y1 + G2a(. Y12 +.... + Gna<. = Cal?? +a CtSXX +..- ~ + CaLjexf wx say, exactly as Cauchy did (see above, p. 434): bnt taking X =1, 21., n and using the second part of the theorem (0) he steps ahead of Cauchy with the result (G1y1)2 + (G2Y2)2 +.. + (Gnyn)2 =: [aixx1+ +a2X2+ +... +a aCxxr]2 and, what is more important, he notes that the n equations on which (0) has just been used, viewed as connecting Glyi' GOyD,.., O/n and w1, w2,...I W1, are exactly the equations originally connecting /1, I 12,., 7/n1 and XI, '2'. I Xn; and thus draws the important conclusion that any relation between the y's and the x's will still hold when y, is ch-anged into Gmy. and x, into arx1 ~ arxS +... a,,x,. For example,, corresponding to and deduced from the relation z2 + y2.. 2 = XI+ X +... +XI Y1 2 ~ IY 2 we have the result just obtained by means of the theorem (0). Further, any new relation derived in this way may be treated in by 2n substitutions becomes a1, Ct2. "'a: a1~"I, a2., alK, 2)...., I an ap a2.. a al, 2.... an al, a2,.... an a1K, a2,..... a1lK a,., a2x...... an, (n) ( n) (it) (i) (it) al), a2j. a l, a2,. a1K, a2Kv afK a,,, a2,...,...an which by the proposition already formulated by me (see above, p. 450) a1,K a2K)...... a, ai,, a2,.. a, In the case of the next deduction n of the 2n substitutions would be for y's. ORTHOGONANTS (JACOBI, 1833) 4557 the same fashion as that from which it was obtained; con — secuently it is seen that for any positive integer p Gijy + GI)J +...+ Giy2 may be expressed in terms of the x's. Returning now to the set of n equations alxx~ it 2XX2 + ~ a~~ X, Ct XX where the w's are known as linear fnnctions of the y's, and solving for the x's we have, on putting brs for the cofactor of ar,, in ~ ~i- a11a 22 ' 'a1W XI. ~ -ta11Ca22... an = bKiW1 + bK2W2 + + bcilwn, Should we now snbstitute for the w's in this the appropriate expressions in terms of the y's we should have a set of n equations corresponding and in a sense equivalent to the original set XC = UKY1 + aN,2 +... C+ UY7? By doing this and comparing the two sets there is obtained C(~ ~ a11a22... G bj b = Cl~baiCj ~ bK2r( +. + bKK2(Th) a( KK2 b b,, (.. or G al)+bI2a + l~~ a result distinguished as a "formula meiorabilis" because of' the fact that on comparing it with the previously obtained equation Gm I a( a ) (in) a.+a K aKIKa ~ a2.cq + tc we are led to the result that all equations involving the a's, the G's and the a's will still hold if a,\ be repclaced by bK +'A, Gin be replaced by 1 +Q, c and the a's be left unchanged. For' example, having already found that aKX = GK,a/ a + G 2a x + G, U1nCK& and GjI + G~ + + Ga~ - ax, + ax + + a 458 HISTORY OF THE THEORY OF DETERMINANTS we conclude at once that bK aKaX a a,, an(n) + -~ +.... A G, + G2 + Gl, and ayl aY2 a( )y blAlI + b2x2 +.... + bce X. _+ +_ _....+. Gl 1 G, - A and similarly, from another previous result, that 2 2 2 Gfi G GP can be expressed in terms of the x's. From the n equations embraced in the second of these we obtain by multiplying by z, 2,.... x,, respectively and by adding G+ +..G +~ 2 - K A ' A, 1 2 n G1 Y+.... Y a result which may also be viewed as a fourth example of the efficacy of the general theorem. Lastly it is noted that the same second example teaches that all relations between the x's and y's will still hold if for yYm we substitute y,. Gm and for x^ put bixl + b2X.2 +.... + b,,x, A The next section (~ 8) concerns the equation F=0 for determining the G's, and need not detain us because the set of n equations from which the said equation is derived by elimination of a(, a ),..., a() has already been more than once referred to, namely, the set G = a2 %. + +...+a Ci a?)l, K IE * + 2K * 2 +IK rn or, as it may also be written 0 = (a,, - Gm) a"1 2 + C.21a2 +... a1a 0 = aa() + (a. -G a(i) +,,a(m) () 0 = a^c) ~ Ct2612U61 + a, +.. (a1,,, - 2- G 2,) ) J............. G1 ORTHOGONANTS (JACOBI, 1833) 459 The only addition to be made to what has been said above (see p. 229) is that in r the term involving the highest power of x evidently comes from the determinant-term (a,, - x)(ta x).... (Cn - x), and therefore is (- 1)nx". Consequently we must have identically r = (G-x)(G2-x)....(G, -x), which on putting x =0 gives Z ~a11Ca22.... a,, or A = G1G2.... G,. The ninth section (~ 9, pp. 15-19) deals after the manner of Cauchy with the finding of the values of the coefficients aex, and the character of the roots of the equation F=0. Denoting by B() what bx becomes when a1 - GC, 22 - G, G... are substituted for a11, a2,..., a consequence of which is that BB( ) RB() Jacobi leaves out the Xth equation from the set used to determine the G's, and derives from the rest al: a( 2) ' n = A ': B(. B. 1 2 I - X 2,\. nX' and thence a ( B) 2_ KA C {(B ) +2 (B2+2X )+... + (B(")2} exactly as Cauchy did. He also however supplies an alternative procedure and result. Writing the above chain of equal ratios in the form 1 a 2 aV h lk 2X nX' (qu.)) ("Oil ) ("O)('") (^)_(" - B1): B: O) (in whence it is evident that BKa () a(a) is independent of K, he puts P a() a()) - B(K thence derives of course p("). a()a(2) =B0() and consequently p(?) p(m). K --- 460 HISTORY OF THE THEORY OF DETERMINANTS In other words, he proves that the ratio is independent of X also, and may therefore be denoted by p(nt). Knowing this, it only remains to use the equation (a(?n))2 +(n2 " ( + + (am))2 1 as before, and we obtain p(rn) = B(n) + B +... B() L'11 22- VA2 '' 1~2 whence immediately we have ^U^(}))) _ MA(p ), K A pB(?) I.(..) + +B(91) B11 B22 + + I nn and finally a1(m1a) = _KK _ K /(p("' i p "12) R i _R( \) (B11 +B22 +... +B, n)' Although the solution is thus complete, Jacobi takes the opportunity to add that from the value P,, found for the ratio it follows that B](m) B(mA) = B(m). B() K,\ K' -- KX' K'/) and therefore also that B(m16) W(n) = (("90\2 KK KIKI KK' - ' Not only so, but he gives another mode of investigating the value of P(0) itself. This consists in noting that if n of the coefficients of the original function V, namely, the coefficients al1, a22,..., ca, all receive the same increment e, the corresponding increment of V will be ( x2 + X2 +... + ); and that consequently when V by substitution alters its form into G2 + G2 + + Gy, this increment may, by reason of the relation 2+ + +J + 2 = y2 +y+ 2 + y2, l 2 ' ' ' ' be written (y? + Y2 +... + y2), -a result which shows that G1, G2,..., GG all receive simultaneously the same increment e. Now knowing this, and ORTHOGONANTS (JACOBI, 1833) 461 applying it with the increment - G, to the previously established result,,, GS ' " (n) (n)} bKX = G1G2... G- {K + K ~ + a X, we see that the left-hand member becomes BW), and that the KAX terms of the right-hand member all vanish except one, namely, the term a(n)a (r) G1G2.. G K A G711 which becomes (G1 -G) (G2- ) ( - (G-G j - G,)... (G,-Gi). G a). ) The new value obtained for the ratio B(na). x(')c() is thus the product of the differences got by subtracting G, from all the other G's in succession. Nor is this all, for since r = (G-X)(G2 - )... (G - ), it follows that if we differentiate r with respect to x and subsequently put x=Gm, we shall obtain this very new value changed only in sign; so that the equation with which Jacobi legitimately closes ~ 9 is B('") (in) (m) __ ___ _ afc aA = - w We now come to the section (~ 10, pp. 19-21) containing the notable result to which in his introductory pages Jacobi, as we have indicated, specially directs attention. Using the fact that EcKxXKAx is transformed into KX Gly2 + G2y~ +... + Gjy by the substitution x = a'y + a +. +.. (+ )Y K K/A 2 A K YT he deduces the results xYa a(5na4)A = 0, E-J KX K A X KX the latter of which is more clearly comprehended and easily 462 HISTORY OF THE THEORY OF DETERMINANTS remembered by writing the original function V in the modern notation x1 X2 X3 a,, a12 C13.... X ac2l a22 a23,.... X (31 Ca32 Ct33.... X3 and noting that Gfl% is what this becomes when a<(n) o(m) a(m) are substituted for the x's. Then, preparatory to differentiating the second of the two results, he points ont that the variation of the a's on the left-hand side of this result may be neglected, becanse, if it be not, the snm of all the terms involving diff'erentials of the a's will be and that this 2>31 a Ua(i2a(01), 2, a(").E E[ ().Gam K = Gm. D {a((>)2 + (jni))2 ~.. + (a(t))2}2 0. Thns prepared and differentiating with respect to aKx, we obtain at once h, ~(K \ X), and a(mt)ot(n) DG K K DaKK -resnlts which Jacobi deservedly styles "formnlae perelegantes." DG, As, however, we have another expression for an, namely,?aKX *In the original X is incorrectly placed below the second I. ORTHOGONANTS (JACOBI, 1833) 463 where the numerator is an abbreviation for ( D G a/ JQ= Gj,z it follows that DFm DFm K ~ r,, K K IF 2a(Koa(Qt) =- -^A and a(;'2)a0')=- raK; and thence with the help of the last result of ~ 9, 2B(m) _ I'm, and B(~) a_ r KKk a-Ik K~ -aa~ K -a verification of a case of the general theorem regarding the differentiation of a determinant (see above, p. 212). The section closes with an extension of the result of ~ 7 regarding GIy+ G1l y +G2 + G2.. In ~11, the last which concerns our present subject, Jacobi brings himself into touch with Cauchy's starting-point, namely, the problem of finding the extreme values of 2ccKxxK,. The other sections (~~ 12-16) deal with special forms of V. Passing over these and the 17 pages devoted to Problem ii. concerning the related subject of the transformation of multiple integrals, we find a return made to the original purely-algebraical subject, the new problem (iii.) being more general than the first in that for the condition x2 + x2 +.... + = y2 + y2 +....+ y2 there is substituted ScACK X - Hlyl + HB2y +. K..+ Hyl,. KA The investigation (pp. 51-57) does not, however, so far as determinants are concerned, contain any new departure. LEBESGUE (1837). [Theses de Mecanique et d'Astronomie. Jomrn. (de Liouville) de Math. ii. pp. 337-355.] Of the two parts into which Lebesgue's memoir is divided it is the first which concerns us, the sub-title being 'Formules 464 HISTORY OF THE THEORY OF DETERMINANTS pour la transformation des fonctions homogenes du second degre a plusieurs inconnues.' The authorities cited are Cauchy, Sturm and Jacobi, and little credit is taken for novelty. All the same the exposition is singularly clear and elegant. The given function 1 A,.,x,8x is written in the form x1(A 1x, +A... + A 2 +.+ Ax) + x2(A21x +A22s2+....+A2z) +.......... + x,(Anx ll A,-2X2+ t...+ * * * An) where An = Ap; and the substitution Xr = alyil + ar2Y22+ *. + a (r= 1, 2,..., n) being made, the result is necessarily taken to be of the form + y2(B21yl + B22Y2+.... + B2,y,) +........, + y~(Bqlyl + B?2y, +.... + B ) As for the values of the B's, if for shortness' sake there be put Ci, for Ailal + Ai2atC +-.... Aina,, it is found that Baa = C laC la 22 + +.... + aC, Ba a= (lC,,a + a2C2. +....+ t3 aC,,, Ba = alaC1 + a2aC2p +....+ a6naCpn, and, that, because of A, and Aa being identical, Ba = B~. Should it be desired to have the result of transformation in the form Ulyl2 + U2y2... + Uy2 it is necessary to put B1= U1, B12= 0, B,3= 0,... B = 0, B21=0, B22 = 2, B23= 0,..., B2n=0, B = 0, B2 = 0, B, = O,.., B,,, U,, ORTHOGONANTS (LEBESGUE, 1837)46 465 aset of equations of which only I (n2 ~ n) are distinct, but in which are involved double that number of unknowns, namely the %2 a's and the n U's. "On doit donc," says Lebesgue, "encore se donner l (n 2 +n) equations entre les inconnues, afin d'6~ter an proble'me son indetermination. II est. bien de manie'res,d'obtenir ces nouvelles relations." Taking, first, for this purpose the condition xi12 + X22 +. + Xn2 = y12 + y2 2 +... + Yn2, and deducing the said relations, he then shows how by partition-.ing the thus completed set of n2 ~ n equations into n sets of n +1I equations each, it is possible to find the values of the unknowns Q I+ at a time. The passage I's (p. 341):"Par exemple, si l'on veut obtenir -le syste'me qui donnera la valeur.des. n ~+1 inconnues Ua, ala, a2a,... a on prendra les equations BOL=O,) B2U = 0,. Baa=Uay.. Ba., auxquelles on joindra e'6quation a2 +a2 +a2 la 2a n~~~~fa Les n premieres equations reviennent 'a a12Cja + a22C2a +..+ aniCna = 0, al2Cla + a2.C2a +.. an2CnL 0, a1 lCla + a2C2. + +. anaCna = Ua d'oii l'on tire tre's facilernient Cla I daqa C2' a2.U,., C n aU; la premie're, Cla = atiaUa, s'~obtient enl multipliant les equation's pr6'ce'dentes par a11l, a12,..., a1,, (coefficients de C1,,) respectivement, et en faisant la somme des re'sultats.' Les autres Is'obtiennent d'une M.D. 2 466 HISTORY OF THE THEORY OF DETERMINANTS maniere toute semblable. Rempla~ant Ca, C.,)..., C,, par leurs valeurs, on aura done definitivement le syste'me (An - U.) a,, + Al ai +a.... + n = 0, A21a. + (A22- U)a2a +. + A2nan. 0, A.3ali + A32aIa ~. + A3~%nan 0, Anlal. + An2a2a. +... + (A,n - U)a,,) = 0, a2 + 2 + 2 la 2 nL~a 3. '' l - dont la solution n'offre. d'autre difflcult6 que la simplification des resultats auxquels conduit Ie'limination." Here a digression (Q ii.) is naturally made into the subject of determinants (see above, pp. 301-303), after which (in ~ iii.) the reality of the roots of the equation in Ua or mt (namely, U 0) is considered, this being done in three steps: (1) when n =-,; (2) when all the A's vanish except All, A221.. A nn AAln) A * A0,_-,; (3) when n = rn, the previous case n m -1I having been already established.' The differential expressions for the coefficients of the substitution are then found, the starting point being the equation a,,a [mn] from ~ ii. By putting ['n,n].[i,i] for [n,i]2 and using the equation a2 + C2 ~ + =1a2 there is obtained 2~ [n,n/] 'ta [1,1] + [2,2] +... + L1,2]' and generally a2 [K, K] aK [1,1] + [2,2] + +[Tr]* Since, however, ~ ii. gives us [n, K] as a differential-quotient, and since by reason of the special form of U we know that dU dU dU dU - u dWA11~dA 2 dA,, * Lebesgue says this proof is essentially the same as Poisson's for the case =3, reference being made to the latter's memoir of the year 1834 in 3lMm. de lacad. roy. des. sci.... (Paris), xiv. pp. 275-432. ORTHOGONANTS (LEBESGIJE, 1837)46 467 it follows that aK. 1 dU dU aa 2dAKa dAfl?1 and a~2 dU dU and aK U. dudA (t5 in the latter of which qb is to be changed into U,, after differentiation. The remaining section (~ iv.) concerns a second mode of obtaining I (n2 + nt) equations to make the original problem determinate, namely, from- laying down the condition It has no present interest. CATALAN (1839). [Sur la transformation des variables dans, les integrales multiples. Me'rnoires couron-as par: l'Acad....de Bruxelles, xiv. 2 e partie, 47 pp.], After his -introduction on the solution of a set of linear equations (see above, pp. 224-226). which he writes in the form a~xl + bIX2 + cIX3 +.. + h~x,-1 + lix,. = a,1 a~ce1 + b2x2 + c2x3 ~.. + h 2Xn- 1 ~ =2 a2 anx1+ +bne2 +cOX3 +...~ n- +tx,=a he, sets himself to consider the special case where the n2 Coefficients are connected. by the In (n - 1) relations alb1 + a2b2 ~ a3 b3 +..+ anbn = 0, a~c1 + a2e2 + a3e3 +..+ ancn = 0, a111 + a212 + a313 +. + a,11 = 0,) bic1 + b2c02 + b3c3 +..+ bocd = 0, bid, +. b2d2 + b3d3 + + bnd+, = 0, bill + b212- + b313 +.,+ bnl4,3 0, kill1 + 1'2 12 + 10313- +. + k~t = 0, 468 HISTORY OF THE THEORY OF DETERMINANTS with the object "de trouver d'autres relations entre ces coefficients." In the first place, following, as he says, Poisson and Lacroix, he squares both sides of each of the given equations, and adds, thus obtaining 2 Ax2 + Bx2 + + Lx[; 1 if for shortness' sake there be put A- a2, B=b2,..., L= lv 1 1 1 In the second place, multiplying both sides of, each equation by the coefficient of x, in it, and adding, he obtains in succession the equations Ax1 = a.Cla1 + 2a2.. + C+ a2al, Bx2 = bla1 + b2a2 +... + bnaq, Lx,, = 1lc + 12,2 +.. + izacs, which constitute of course the solution of the original set. In the third place he treats the derived set as the original set was first treated, save that he divides by A, B,..., L respectively before performing the addition. This enables him to put his result in the form In the fourth place, taking advantage of the fact that the first and third results have a mnember in common, he equates the other members, and thence concludes that' a 2 b2 c_ 2 2 a2 b 2 e 2 [29 *A B L A + B-+-+' " +L + = J ORTHOGONANTS (CATALAN, 1839) 469 and a a b b 1 i A2 + b2 +...~ 2 0 a a b b I I 1 C3 + 1 3 1 3 3\j ~j...~L3- =0 A B L a (t. b bn ~ 1 1, 1 +~ ~ 1...~-I A B L ataa b2 1b 1213 A +B + + L n0 aLZa,, + +,,, A + B L - ~+ + A B L These are additional relations of the kind sought, the number of them being n-1n(n-1), that is, 'n(n+1). At this point opportunity is taken to effect contact with the work of previous writers by means of the sentence "Ordinairement, dans les proble'mes de mecanique, on suppose les quantite's A, B,..., L egales a l'nnite': et alors les formules ci-dessus se simpliflent conside'rablement." In the fifth place he takes the ordinary solution of the original set of equations, that is to say, denoting the determinant of the coefficients by A and the cofactor of a, in A by D,., he obtains = a1D1 + a2D2 +..+ CnDn xi = 'A 2nn This he compares with the first line of his second result, and deduces Di D2 _ D3 __ _ n _ a al a2 a3 a,, A' and thence D2+D 2...+D~= (2V+-a 2. ~) 2 2a 1 2?b(a, a2+.. +a n) A2 470 HISTORY OF THE'THEORY OF DETERMINANTS Lastly he devotes four pages to establishing in an uncon-_ vincing manner* an immediate result of Binet's multiplicationtheorem. Thus, according to Binet but in a later notation, d d2 3 d4 d5 2 de11f 61 62 3 e4 e,, or:d~el el ~elf, =d~e2}%12 +I cle2 f412 + 3. d~~52 Should it be given that ~d e1 i.e. d ej1+ d~ +..+ d~ ~ 0 d1f1+d02f9...~d5 f5 = 0 e1f1~ e2f2 +.+ e5f5 = 0, the product-determinant reduces to its diagonal term, and we have an instance of' Catalan6's result, namely, the sum of the squares of ten (, )three-line determinants equal to the product of' three sums of five squared elements. A. special case of the, result is of course A2-A.BC... L -which permits a previous theorem to. be changed into D2 D + D2= B. C... L. POSTSCRIPT. Le 'dernier volume' referred to on p. 434,turns out to be the 'ninath; and the title of the memoir, which does not extend to three pages (pp. 111-113), is "L'6/quation qui a pour racines les moments d'inertie principaux d'un corps solide, et sur diverses c'quations du me'me genre." As it was read to the Academy on 20th November 1826, and therefore preceded all Jacobi's papers * The so-called property of general determiniants which Catalan uses as his foundation is, when accurately stated, a truism. The I'demonstration' is vitiated by an oversight, in regard to signs, similar to that made in connection with a permutation tournante' 'in the opening portion of the memoir, n~amely, he takes dlje1f2g31 + e41f192d31 + f4jf/jd2e31 + g4 1dle2f'31 as the equivalent of I C4ePfg3 I' OIRTHOGONANTS 471 on the subject, it deserves very special attention. The second theorem enunciated in it is: "Si l'on nomme s la somme des carre's de n variables indeipendantes X, y, z, U,... et r une fonction homogene du second degr6', compos~e avec ces meimes variables, et si l'on cherche les valeurs maximum on minimum du rapport -, la determination de ces valeurs de'pendra d'une S equation du rn-e degre' dont toutes les racines sont re'elles." To this Cauchy adds the remark that the method followed in proving it had led him to other' propositions, and he quotes one, namely, that given above on p. 434, adding: "Le dernier the'ore'me entraine e6videmment plusieurs relations,entre les. coefficients des equations line'aires par lesquelles les variables' $, ~,...~sont li~e's aux variables x, y, z. Ces relations sont smlbes 'a celles qui existent entre- les cosinus de agles que forment -trois axes rectangulaire's donn~s avec les axes des coordonn~es suppose's,eux-m~mes rectangulaire-s." CHAPTER XVI. MISCELLANEOUS SPECIAL FORMS FROM 1811 TO 1841. THERE now only remains to deal with those special forms which prior to 1841 had not excited much interest, and whose properties had consequently been little investigated. The most fertile originator of such forms was Wronski: unfortunately he had only one follower, and still more unfortunately the work of this follower, Schweins, was almost immediately lost sight of and remained of none effect until 1884 (see above, p. 175). Others, whose similar contributions fall to be noted, are Scherk, Jacobi and Sylvester. The writings will not be grouped according to subjects, but will be taken in order of date. WRONSKI (1812). [Refutation de la Theorie des Fonctions Analytiques de Lagrange. Dediee a l'Institut imperial de France. 136 pp. Paris.] As has already been pointed out (see above, pp. 78-79) Wronski's first mention of 'sommes combinatoires' was in connection with a special form of them. The form was not the product of fancy: it made its appearance, like so many others, when a set of linear equations called for solution in the course of an attack on a seemingly unconnected problem. This is important to have noted, and it is made quite clear by a note appended to the highly controversial ' Refutation' and bearing the title "Sur la demonstration de la loi generale des series, servant de principe a cet ouvrage." The law itself is stated as follows (p. 15): "Or, si Fx est la fonction qu'il s'agit de developper en serie, kx la fonction arbitraire qu'on prend, dans la serie, pour la mesure algorith MISCELLANEOUS SPECIAL PORMS (WRONSKI, 1812) 473 mique de la fonction propose'e Fx, c'est-a'-dire, pour la fonction generatrice-* du de'veloppement; et si, de plus, on considere les series. dans leur plus grande ge'n~ra'lit6, d'apr~s la formie donnee plus haut, savoir, Fx = AO + A1l. kx + A2. / 2/4 + A2. ~r31 +.. qui proce'de suivant les facult6s progressives (~, c/JX2I4, 4x3/, etc., de la fonction g~'neratrice, 1 aecroissement ~ d'tant arbitraire, onl aura, pour la determination des coefficients A,, A2, A 3, etc,, exprime's ge'neralement par Am~ ji etant un indice' quelconque de'puis was jusqu'a' linfini, la. loi A~ [zaXa~. Ab~X2/4.- AkkrX/.. Alr(1xi) - 1)/4 Al)Fx] jLk =Aa/X.Ab~~X2/4 Ae~X3I4.... AIX~ - 1)lt. mX, / en observant de faire e'gal 'a $ l'accroissement dont dependent lesdiff~rences, les valeurs a = 1 b =2, C = 3, d = 4.. =.,m. et 'a la variable x, une valeur telle que 4~x = 0. Quant 'a la quantit6 A0, il faut savoir que, suivant la loi de continuite' de ces fonctions, on a A0 = Fx, en donnant toujours 'a la variable x la valeur qui results: de la relation ~x = 0." No demonstration of the law was given in the original conmuunication to the Institute. That supplied in the I'Refutation' (pp. 131-133) proceeds as follows: "1Prenant done, des deixx membres... les difflirences des ordres. successifs 1, 2, 3, 4, etc., et donnant ensuite 'a x la valeur qui re'duit a zero le facteur (k, nous aurons, en vertu de l'expression (91~ [i.e. A4r,~xI = 0], la suite inde'flnie d'e'quations AFx = A1..~ A2Fx A1. A2~x + A2. 2X/,,,A3Fx = Al. A3~ ~ A2. ' AB2/4 + A,.A3. x La premie're de ces equations donne imme'diatement En second lieu, puisqu'en vertu de l'expression (91) on a _A4X2/f 0, les deux premieres des equations pr6ce'dentes sont identiques avec: celles-ci,A Fx = A1. A~x + A2. A~X 2/4,A2Fx = A1. A2.rX+ A2. A2(~X2/4 *In Wronski's own use of the word. 474 HISTORY OF THE THEORY OF ]DETERMINANTS equations qui donnent'de me'me imme'diatement, A =z[,A'kX. A2Fx] In similar manner A. is found from the equations:,A Fx = Al "A obx + AA.A o5x2 + A3. kXe A2Fx = Al. A2(pX + A2.Ak20X2/ + A,3.A25x 3/t,A3Fx A A.395x ~ A.A3q0X2/t + A 3.Aq3/4, and the proof is considered complete when it is pointed out that "Ia somme combinatoire formant le d'nointeur se r~duit 'a son premie r terme." On leaving the matter it may be as well. to note that this first special determinant-form of Wronski's is not only -specialized, in having differences for- its elements, but in having zeros in all the places. included between -the diagonal and the last column. Also, attention may be called to the fact that on page 33 he has an instance in which 'differentials' take the place of 'differences' as elements. WRONSKI (1815). [Philosophic de la Technic. Algorithmique. Premiecre Section, contenant la loi supre'me et universelle. de mathe'matiques. xii + 286 p. Paris.] Instead of the "loi generale des series " we have now the. much more extensive "loi supre'me," which Wronski writes in the form Fx =A0.Q0 ~ Al.Q~1 ~ A2.Q2 + and which appears inscribed on the pedestal of the androsphinx adopted later by him as an authenticating sta mp- for his works. Notwithstanding the increased generality, the Q~'s being now any functions of xr whatever,* the law of formation of the A's is,expressed by means of thc same kind of 'schin' functions as 'The interestingly guarded report of Lagrange and Lacroix on Wronski's first -memoir to the Institute (see Gazette, Nationale for 15th June, 1810) shows that the statement F (x) =A09, ~ A10, +A2Q2 + ihowever vague and undefined occurred in that memoir. MISCELLANEOUS SPECIAL FORMS- (WRONSKI, 181,5) 475 before. In the short exposition (pp. -1705-182.)-given of the latter functions preparatory to the treatment of the "loi supre'me" there is therefore little new to be expected; and as a matter of fact it is only the last sentence that is worth transcribing, the reason being, that~ it brings ont a conspicuous width of view in connection with the functions. The words are:,,II faut enfin observer que si la caracte'ristique'-A, au lieu de de'siguer les diff~rences prises sur les fonctionsX X X 2,x,.. denotait -tout autre syste'me de fonctions algoritlimiques prises sur les me'mes fonctions X1, X, X,. tuccque. nous veuons de dire concernant les f onctions schins, se trouverait 6'galement vrai." It may be added that in a laterj part of the work specimens of such 'schin' functions actually appear, e.g. [Qiai.1 Q.2C2. Q30t3 QJ-ak 131) j~k + +)32) "'(Ak+J33)1.1+ The demonstration of the law occupies fifty pages (pp. 188-238), and is preceded by the intimation that it depends on a hige -theory of the 'schin' functions and that consequently this theory, -reduced to a lemma and a theorem and three corollaries, must first -be dealt with. This is done at considerable length, thirty-four,of the fifty pages being so occupied. The, most important portion to be reproduced is the enunciation of the said theorem, which,stands as follows (p. 193):' "Soient YO0, Y1, Y0 Y. Y". des fonctions d'une variable x, et soit -pour cette fois, A la caract~ristique des difrecspis oot, 'snivant la voie progressive ou la voie regressive, par rapport a~ un,aecroissement quelconque de la variable x. Si, avec ces fonctions,,on construit, d'une part, les quantite's X = = ['A1oY0. Ally1d, X2= =~~[A~oY0. A~iY 2], 3= I[A'OYO 'A"Y3] X.= tt[A~oY0. A1i,j.;,,et, d'une autre part, les quantite's =i' AWOY + i. AW'0, T= A80oY + 2i. A8iY0 + j2. A2 T= AMoYo + 3i. A81Yo ~ 3j2. A82Y0 + i3. A3 476 HISTORY OF THE THEORY OF DETERMINANTS et g6neralement, pour un indice quelconque p, Tp = AoYo + l. A o p ( - 1) i2. A2 + (p - )( - 2) i3. 3yo +.. 1.2.3 en faisant i = + 1 lorsque les differences A sont prises suivant la voie progressive, et i = -1 lorsque ces differences sont prises suivant la voie regressive; on aura la relation d'egalit6 [AiX, iX. AA2X3.... AAXo] =(T1. T2. T... T,_1). Y. [AOyo. A sYl. AY... Atw]; en donnant aux exposans /, P 1 2, i,..., et 8, 8, 8,., 8 de la permutation desquels dependent ici les fonctions schins, les valeurs suivantes P1 - ~X /2 = 1 2 + A,2 P 3 = + 2)p, = V -, 68 = 8, 1 = 1 +, 32 = 1 +6,..., 68 = 1 + 8 = 6 + 1; 8 etant un nombre entier quelconque, et v un indice arbitraire." WRONSKI (1816-1817). [Philosophic de la Technie Algorithmique. Seconde Section, contenant les lois des series comme preparation a la reforme des mathematiques. xx+646 pp. Paris.] The contents of this larger volume are more or less methodically concerned with specializations from the "loi supreme," the word 'series' being used by Wronski in a way of his own which enables him to view all series as being derivable from AoQO + All + 2.... by taking S0 = (/x, a, b, c,..)1/, a, P, v... = (x, a, b, c,...), Q2 = ((x, a, b, c,,)2, a,', = p(x, a, b,,...) *. (x+e, a+a, b+/3, c+y,...), Q=g = $(x, a,,, c,...)3/a 3,,... = (X(, a, b, c,...).p~ <(C+ a+, b+/3, c+y,...) (x + 2 a + 2a, b + 2/, c+ 2y...) MISCELLANEOUS SPECIAL FORMS (WRONSKI, 1816-1817) 477 This being the ease it is natural that the ground covered by his earlier writings, namely, by his first paper to the Institute and by the 'Refutation' should be partly retraversed, and that specialized forms of the 'schin' functions which form the numerators and denominators of the A's should have to be considered. Thus we at once come again upon the case where I~ = OXI 2 - Ix'43, ~23 cI~0,. and the sub-case where $ is indefinitely small, and where therefore 2. = (x)". In the latter case it is shown that A w d0px 0. d19x1. d~pX2.... d~L-U~ 2C2rF X] (10/1. 11/1. 12/1.... pI'). (dpx)1-(tk~k)L 'and further that the A's are then so related that A - dFx, 1.. dox A2 - 1 {d2Fx - dA2. p" A2 12/1. OX2 I Fx Al A, = 1 {3. Fx - Ail d3'px _ A2. d 3pX21 in all of which, be it noted., x is ultimately to be given the value a which makes Pa = 0.* A little fusrther on (p. GO) but in the same connection, and while considering Burmaun's series, the result is reached. * In a foot-note (p. 13) it is curious to find the identity 1)(fk - >1)/1,, (a)]( 3.0 X Yp-2 p-i pi d-1 p-2 1 + [.0(0) ).. - 2) D(p - 1) 'F(P)] p240) 41() Q.(P - 2.(P - 1).. p-2 p 0 - 2p-1 namely, the 'extensional' of anjb~c.2j -- b,j[a,c f + cl cIb- I =0O reached two years later by Desnanot. This should have been noted in its proper place. 478 HISTORY OF THE THEORY OF DETERMINANTS.Still later (p. 399), when another form of k is being dealt with, a quite different form of determinant makes its appearance, namely (p. 42~1) the, determinant where (o)=0 when -> p. But although Wronski notes that it and evaluates it in a number of particular instances, he does not appear to have gone further. 'WRONSKI (1 819). [Critique de la The'orie de Fonctions Ge'neratrices de M. Laplace. iv+ 136 pp. Paris.] In this his last work on pure mathematics Wronski comes across (p. 67) still another special form of 'schin'' function, viz., ~, VIg)2~43.... where vj, v2 v3)...are indices of powers. Again, however, in working with them he does not get beyond the use of the identities la b C d...I= ab2C3d,4 I.- a9lb~c d 4... ~+ a3,Ib~c2d4 I..- a4l blc2d 3..I 0=b 1 3d -. belbd..+ b lb c d b lbe2d bed4 I b 3 4..I 124.. 4 123 -SCHERK (1825). [MathematischeAbha-ndlungen. iv +116 pp~ Berlin. (pp.31-66).] The second'-portion. of the appendix ( 8) to the second of Scherk's m;emo6irs" (see above, pp. 150-159) concerns the solution of the set of equations s ax, 1 2 s = ax + ax, 2' 2 1 221 1 2 3 s = ax + ax ~ ax, 3 21 32 3 3 1 2 -3 s ax +ax +ax~.. + ax n n n 2 n 3 n n MISCELLANEOUS SPECIAL FORMS (SCHERK, 1825) 479 This, it will be observed, is exactly the kind of set which Wronski obtained for the determination of his coefficients in the Third. Note- of the 'Refutation.' As a -consequence Scherk is led to seek for the final expansion of a special determinant, which we should nowadays write in the form a.8 1 2 a a s9 2 2 2 1 2 3 a a a.... s 3 3 3 3 1 2 3 a a a.... ic k k k His result is (p. 60)".,Foiglich ist der Zaihier von.x, das Aggregat aller der Glieder, die entstehen, wenn man alle Permiutationsformen aus den Gr,5ssen 1, 2,...,) h -1, h +1 1...,9 k, die so gebildet werden, dass nur 1 in der ersten, 2. nur in den beiden ersten, 3 nur in den 3 ersten....Stellen stelt, ildtdese nach einander unter a' a2...a'-' setzt, mit Si multiplicirt, und h nach einander die Werthe k, k - 1,..,2, 1 giebt." To provide a check on the calculation, he draws attention to the fact that the nnmber of terms in the development is 2 k'; and to establish this, he consider's (p.: 61) in succession the cases where h =k,) k-i,7 lc2, k-:3, k -4. As an.illustration both of the law of formation of the expression for X~adof the check upon it, he takes lo = 4, giving ala2a.s4 - aa2a4s3 +a~a3a492- aa4a382- a2a3a4sl+ a2a4a.s, + c3a2a4.s1 -a4a2a3s, X4- 1 23 4 ala 2a3a4 and then specializes by finding the 4th Bernoulli number and the 4th coefficient of the secant-seres SORWEINS (1825). [Theorie der Differeuzen: und Differentiale vi+666 pp. Heidelberg.] There can be little doubt that Schweins was one of the few men who were -not daunted. by the egotistic- and exhaustingly 480 HISTORY OF THE THEORY OF DETERMINANTS wearisome style of Wronski's works. This appears not only from the fact that Wronski is repeatedly referred to by Schweins, but from a striking coincidence which occurs in connection with their study of determinants. As we have seen there are four special forms of those functions which are to be found in Wronski's writings, three of them having appeared there and nowhere else previously: and these four are exactly those which receive attention at the hands of Schweins. Therefore, while refusing to accept the estimate of the 'loi supreme' which its author in season and out of season insisted upon, let us not forget that some of the concepts which sprang from the hot brain of the poor Polish enthusiast provided material for exercising the industry of an exemplary German professor. Taking the forms in the order in which Schweins deals with them, we have, then, first of all, Wronski's =:[(1, 0)(2, 1)(3, 2).... (w, -l)], (p, o) >p =0 or, in modern notation, a1 Ca2 a3 a4.... b1 b2 b3 b4. * 2 C3 c4..... d3 d4... e4.... This Schweins treats of in a portion of his treatise which we have called the fourth 'chapter' of the first Abtheilung (see above, p. 173), a chapter headed "Auflosung der Producte mit Versetzungen, in welchen einige Factoren verschwinden, in Producte bestehend in gedoppelten Verbindungen." His solution of the problem set for himself, namely, the finding of the final expansion of the determinant, is complete though clumsy and unpleasing in form, being stated as follows:11 () (ni+1) (n+2) An+lAn+2.... A _+m+1 I1n) (tn+,++) - D['n, (+1, +2,....., +2, +m)"n, +in+l] A, A,_+i An+m - D[), (n+-t, n +2,...., n+wm)(m"-'), n+ -n+l] (-)mD['n, (n+ 1, + 2,..., n,+ mn )~, 'n+m+ 1], MISCELLANEOUS SPECIAL FORMS (SCIHWEINS, 1825) 481 n) (IL) where D, =An48 The notation on the right may be underA, stood from the example D [lo, (1, 2, 3, 4, 5)3, 61, which stands for the ten-termed sum 0 12 3 0 12 4 0 12 5 0 34 5 1D1D2D3D6 + DlD2D4D6 + D1D2D5D6 +.,. + D3D,4D6D, the first part of the symbol, '0, indicating that the upper index of the first D is always to -be 0, the second part, (1, 2, 3, 4, 5)3, that the last three upper indices and the first three lower indices are to be those of a set chosen fromt 1, 2, 3, 4, 5, and the third part that 6 is to be always the lower index of the last D. Thus, taking the determinant of the 4th order, we have a01 12~ 1(53 (t54 all a12 ((33 a14 a22 a23 a~24 Ct3((3 a34 =dl~d2,1,3d4 - ~Cl(A C1dC3C14+ kc2d,4) + (d>d4 + d2d4 + c13d4) 0 - c_14, a_ lalC(5C23(t34 (a02(,(23Cu34+c(la33c64+ a,3al2a24\ c(00a11a22('33 \a00a22c(33 aooaia3133 a00a31c(22/ +(a53a34 + (a24 0112l14) a200 and therefore the determinant =a0la12a23a34 - tl. aC(51253Ct34 -. The forced introduction of a,) should be noted, and the object gained in doing so. The form is really the same as Scherk's. The next special form taken up by Schweins is that to which the name alt ernant has since been assigned, and his contributions M'D. 2n 482 HISTORY OF THE THEORY OF DETERMINANTS to the theory of it, occupying all the five chapters of the second Abtheilung, have been already recounted (see above, pp. 311-322). The third form occupies similarly the whole of the third Abtheilnng, but much less space is given to it, there being only one chapter of four pages. The title at once recalls the 'loi supreme'; it is "Producte mit Versetzungen, wenn die oberen Elemente h~ihere Untersehiede angeben." Beginning with the expression for the A of a product in termls of the A's of the factors, he deduces from it the A of the special determinant form in question, his contracted mode of writing the result being A AaA,. AbA2 NACA A"A4) - aI aA1.aA,.A A3. AdA4) at b c+l d a b+l c ci a+l b c. d +~I AaAj. AbA2.Ac+1A3.Ad+1A4) a b - 1 c d +1 a b+I1 c+1 d a -Ii b c c1 + I at+ I b c+1 cd a+1 b~1 c cd + a aAblA AClA,. Ab+,1 A(-F1A4) a+1 b c~1 d+1 ca+1 b+1 c cd+1 a+1 b~1 c+1 d ~ Aa1+'A,. Ab+lA2. 'ACA3 -A11+1 where there has of course to be noted the large number of determinants which vanish when b=a +l1, c=b+l 1. The only other matter- is an investigation of the A of the Wronskian quotient A"aA.,Aa+lA.... Aa~+n-2A,1. _Aa+n-lAi, ) A'AaA. Aa+lA,...................Aa+A MISCELLANEOUS SPECIAL FORMS (SCHWEINS, 1825) 483 The mode of procedure is perfectly straightforward, the results used being P QA &P - PAQ Q Q(Qz &8,AQ)' the expression just obtained for the A of a determinant, and a theorem giving a product of two determinants as a sum of like products (see above, p. 171, result XLV. 2). Putting A,,+, in place of A,,+, there is obtained by division the corollary AaA,. Aa~1Aa..A. Aa+n-lAn Aa~nA~2) AaAl. Aa+lA2... Aa~n-lA,. na+fA ) A AaA,. Aa+1A2..A. Aa+n- 2A. Aa~+n-lAn+2) AaA,. Aa+lA2.................AatAn-'An AatA;. Aa~1A2.. Aa+n-2 An-. Aa+nl'An+i) Aa, aIA.........Aa~n-lAn To the fourth and last special form is devoted the whole of the fourth and last Abtheilung, which consists of four chapters and occupies pp. 404-431. The title under which it appears is " Producte mit Versetzungen, wenn die Elemente das Differentiiren mit abwechselndem Vervielfachen angeben"; that is to say, the subject is the determinant derivable from that of the 'loi supreme' by changing A into Zd, where by Zd is meant the double operation of differentiating and subsequently multiplying by Z. The first chapter, which closely corresponds to the first and only chapter of the third Abtheilung, opens with the expression for the Zd of a product in terms of the Zd's of the factors: and thence there is obtained the Zd of the special determinant-form under consideration, namely, Zd II(Zd)al N,.(ZCJ)a2A...(Zdl)anAn) - (Zd)a'+lA,. (Zd)'nA2. (Zd)asA..... (Zd)anAn) ~ Z (Zd)a1+'A. (Zd)a3A,.... (Zd)anA,) ~ (Zd)a'Al. (Zd)a2A2. (Zd)as+1A3.... (Zd)aA,2) + j (Zd)aA. (Zd)a2A2.... (Zd)an,1+lAn.1.(Zd)anAV) M. D, ~~2 H2 484 HISTORY OF THE THEORY OF DETERMINANTS the rth determinant on the right being derivable from the original by increasing the rth upper index by 1. When a, = a + r all the determinants on the right except the last evidently vanish, and we have Zcl (Zd)aA,. (Zd)a+lA... (Zd)a+n-'An) = (Zd)aA,. (Zd)a+lA,... (Zd)a+n- 2An,. (Zd)a+nAn). The Zd of a generalized Wronskian quotient is next investigated, with the result Zd Il (Zcl)aA. (Zd)a+1A2. (Zd)n-lA ) I (Zd)a'+l,. (Z)a2A2... (Zd)a+n -A - -| (Zd)A.. (Zd)a+t-2Anl) I(ZJ)a+lAi... (Zd)F+n A, ) (Zc)a+lAl... (Zdc)a+n - lAn ) (Z)a+A... (Zctl)+ -'A_) Similarly there is obtained Zcgo J(Zd)aA,. (Zd )a+iA2... (Zd)a+-2 An-1 (ZZd)a+- 1A ) d (Zd)A,. (Zd),... (Z(d))a+n2A,,-. (Zd)a+n -1A ) -_ (Zd)aAl. (Zd)alA2... (Zd)a+n-2A-1) I (Z. (Zd)cA1.(Zd)a)a+ — A (Zdc)a A1(Z). (Z)a+A2.. (Zd)a-lAn )' (Zd)A. (Z)A (Zd))a+n-lA by a double use of which and by division it follows that Zd (Zd)aA,. (Zd)a+lA2.... (Zd)a+n-2An _. (Zc)a+n+-A,n+2)t {L (Zd)aA. (Zd)al+A2.... (Zd)a+i - 2An1. (Zd)a+n - A ) Zd f I(Zd)aAl. (Zd)a+lA,.... (Zd)a+n-2A-1. (Zd)a+nA-lA )| t (Zd)aA,. (Zd)a-A,....(Zd)a+n-2 A 1. (Zd)a+n-lA )J = (Zd)aA1. (Zd)alA....(Zd)a+n-A. (Zd)a+ An+2) -'(Zd)aA. (Zd)a+lA2.... (Zd)a+n-An. (Zd)a+n An+1) The second chapter, which is much longer, is devoted to the consideration of the quotient (Zd)1Aal. (Zd)2Aa2.... (Zd)n-lAn-i. (Zd)nB) (Zc)A. (Zd)A.(...Zd)-, or Qn say, ( cl)Al(Zd)2A a (ZA......... (Zd)nAan) MISCELLANEOUS SPECIAL FORMS (SCHWEINS, 1825) 485 in other words, to those special eases in which the functions to be operated on are all of them powers of one function A. Several interesting results are obtained, such as OG, 1 c tai,-a i c tai,-a i cc___ _ _ _ _ ___ __ __ _ Qn-at- n1-la- a n-,-2 QniL-1, a., 1a - a, a( - a2 a n flan2 d(Aan- a,-b-1) the ]ast of them being specialized down until (Zd)OAO. (Zd)1A1.... (Zd)nAn) = li/i. 12/1.... IIiA.(ZdA)l+2+ *.;+n and to this is appended the note "Setzen wir noch in dieser Gleichung Z=1, so erha]lten wir endlich jene particulare Gleichung, weiche Wronski Seite 110 findet." The third chapter of three pages begins with the consideration of 2(fX)" ~ " aJ)".... dn-l(i~fx"a~n-2.dn) and ends with the result dn - +1 (fx)n-q. dii-q+I2(X 7 +....(fx)n2l - ()q Ia/I. In-i/i *.... ) iq/1 x2)..+n dqy~xa n ] q/i la-i-i/i (dfx d(al)(a ) daf) and the sentence "Diesen ganz speciellen Fall findet zuerst Wrouski Phil. d. 1. T Seite 60 auf einem ganz verschiedenen Wege." The fourth chapter of like extent specializes in a different. direction, but ends in quite the same manner. JACOBI (1835). [De elimiinatione variabilis e duabus aequationibus aigebraicis. Crelle's Jotbrnal, xv. pp. 101-124; or Nomv. Annales de Math. vii. pp. 158-171, 287-294; or lWerke, iii. pp. 295-320.] This memoir which we have already referred to (see above, p. 214) contains an investigation of questions arising out of Bezout's method of eliminating the unknown from two equations of the nth degree in x. If the given equations be f(x) = anXX + a-,_1Xn-... + a,, 95(x) = bqI3a ~ b X,_lX+n-i.. + b0, Bezont, as is well known, reached the desired end by deriving from them a set of n equations of the (n -1 )th degree, and eliminating. x, XI, x2,..., )3i-1 from the said set. This process 486 HISTORY OF THE THEORY OF DETERMINANTS Jacobi explains at the outset, and, writing for shortness' sake the set in the form 0= aX0 + a~oxl +... + aq,,ox?` 0 a01X0 + ajx1 ~...~c40 + 0ao,jasx0 + ao, n-X' + + ni i that the determinant of the coefficients is axisynimetric.* Then he first shows that ar, asr,,, or, as would have been said later, denoting the cofactor of a,., in this determinant by A,., he next proves not only that A,,= A,,. but that A,., = ALlr, s in every case where v ~s=v'+s'. A,,, thus depeuding only on the sum of the suffixes he suggests that in what follows it would be better to write A A,, 8 and he thereupon formulates his result as follows (p. 105): "Quo adhibito notationis modo, videinus, earn esse naturam coeflicientium a,., quac aequationes lineares afficiunt, e quibus eliminatione incognitarum facta aequatio finalis quaesita petitur, ut posito: a00r0 + a,,2i +.... f -,,,,~x,,,- = a10X0 + an1Xi - + i,?z-in-I aU80%;0 + a21X +. + 2, nlXni - M a,,,-iOIOO + a8-I, 11 X+.+ a8,,-l, n-Iot-1I 4 nn * Taking for shortness' sake the case where 88 3, it is immediately evident that if f(x) =0 and 95 (x = 0 we must also have ao a, +- a2x + a,X2 = 0 b, b, mu bst as b3X2 ao + a+ a2 + a3x2 bo + bbx b3x2 b 0, bo+b a,x b8+baX2 a = 0, 0o + bix + bax2 a because the first column increased by a multiple of the second column gives in each case a column whose elements are f(x), q5(x). These are the 8 derived equations in question. Further, by transforming them into Iaab,1 + Ja0b8x + I a0b31 X2 = 0 asbI f -poD, { lasb3 I + 1aSb2!}bx + I1 ab31X2 I Oj Ic Ibb3l+ cvalx b.1 X2 - the axisymmetry not only comes into evidence, but the reason for it is apparent. MISCELLANEOUS SPECIAL FORMS (JACOBI, 1835) 487 aequationes inversae, quibus quantitates x,. per quantitates m,. exhibentur, formam sequentem induant L. o = Aomo + A + A21+ +.... A,^_ L. x = A1mo + A2mn + A'3m2 +.... + A~mi_ L.x2 = A2mo + A3mn + A4m2 +.... + A1+m,,1 L. x,_1 = Amo0 + A,,rn + A,,+11z2 +.... + A2,_n 2n. Tlhe determinant of the coefficients of the latter set of equations here is of the type which Sylvester afterwards distinguished by the name 'persymmetric.' Of course L in the same set stands for ~ a,00a1... c-1, n-. SYLVESTER (1840). [A method of determining by mere inspection the derivatives from two equations of any degree. Philos. Magazine, xvi. pp. 132-135; or Collected Math. Papers, i. pp. 54-57.] As we have already seen (see above, pp. 236-238) the eliminant of ao + ax + acx2 -+... + a,x = 01 bo + bx + b2x2 +.. + Ct,, = 0o arising from Sylvester's dialytic process is a determinant of the (met+n)th order, in which the coefficients of the first equation appear as elements of each of n successive rows, and those of the second equation as elements of each of the remaining rows, each coefficient appearing in any row one place in advance of its position in the immediately preceding row. Determinants of this bi-gradient form, e.g. a( a. a2 3. a, a, a(2 a3 bo b1 b2 bo b, b2 bo b1 b2, were not long in attracting attention. Up to the year with which we close, however, no property of them had been noted, although any one able to compare the new process of elimination with Bezout's process had a result ready to his hand. CHAPTER XVII. RETROSPECT ON SPECIAL FORMS FROM 1772 TO 1841. A GLANCE over the preceding six chapters shows the extent to which the study of special forms of determinants had been carried prior to 1841, an extent probably hitherto unsuspected. In all, ten different forms had made their appearance, and about half of them had more or less engaged the attention of several investigators and had had a number of their properties brought to light. Of the dozen writers to whom one or more special forms had become familiar, by far the most conspicuous was Jacobi, to whom six forms were known and by whom five of them at least were carefully studied. After him came Cauchy, Wronski and Schweins. The only form to which a distinguishing name had been assigned was that called symlmetric by Lebesgue in 1837. The fruitful era of nomenclature, but not of that alone, was ushered in by Sylvester shortly after the date with which we close. INDEX TO THE NUMBERED RESULTS IN PART I. I. p9. I.pp. 9, 17, 52, 55, 75, 101, 250. i.pp. 9, 13, 17, 52, 55, 56, 56, 57, 57, 57, 58, 59, 59, 61, 75, 77, 78, 99, 101, 102, 134, 135, 156, 158, 179, 199, 2 18,5 219, 249, 249, 257, 256, 258, 258, 260, 276, 276, 277. iv. pp. 13, 52. v. p. 1 3. vi. pp. 17, 61, 103, 103, 145, 159, 162, 212, 261. vii. pp. 24, 33, 54, 78, 99, 160, 179, 214, 228. VTILi pp. 24, 99, 220. ix. pp. 24, 103, 162, 182, 182, 221. x. pp. 24, 96. xi. pp. 24, 33, 103, 161, 222. xii. pp. 24, 33, 52, 61-2, 97, 103, 117, 135, 143, 165, 197. xiii. pp. 24, 33, 148, 151, 212, 213, 216, 225, 231, 236. xiv. pp. 24, 33, 52, 116, 164, 261. xv. pp. 33, 64-5, 78, 98, 136, 160, 228. xvi. p. 33. xvii. pp. 41, 41, 66, 81, 109, 204, 271, 271. xviii. pp. 41, 70, 72, 81, 119, 270, 271. xix. p. 4 1. xx. pp. 41, 188, 197, 208, 214, 268. xxi. pp. 41, 110, 211, 285. xxii. pp. 41, 66. XXIII. pp 11, 52, 68, 117, 140, 142, 145, 171,~ 184, 185, 233, 264, 265. XXIV. pp. 59, 256. XXV. p. 60. XXVI. pp. 63, 63. XXVII. pp;- 65, 104. XXViii. p. 68. XXiX. pp 71, 201. XXX. pp. 86, 121. XXXI. pp. 86, 92. XXXII. pp. 87, 92. XXXIII. p. 88. XXXIV. p. 88. XXXV. pp. 91, 92. XXXVI. P. 91. XXXVII. pp. 105, 162. XXXVIII. pp. 106, 110. XXXIX. P. 1 1. XL. pp. 112, 114,114,114,114, 120, 220, 220, 262. XLI. pp. 11 8, 118. XLII. pp. 120, 121. XLI pp. 121,~ 122. XLIV. pp. 139, 140,142,145, 185. XLV. pp. 145, 171. XLVI. pp. 154, 199. XLVII. pp. 154, 199, 223. XLVIII. p. 158. XLIX. pp. 167, 169, 170,170,171. L. p. 174. LI. p. 178. LII. pp. 189, 192, 193, 209. LIII. p. 213. LIV. pp. 237, 238, 239, 242, 244. LV. pp. 249, 250-253, 250, 252. LVI. p. 262. LVII. p. 263. LVIII. p. 265. LIX. p. 272. LIST OF AUTHORS WHOSE WRITINQS ARE REPORTED ON. PAGE Be'zout (1 764), - 14-17 (1779), 1 41-52 Binet (1811), - - 69-71. (1811), - - - 71-72 (1812), - - - 80-92 (1 81 1) Axisymnietric, 292-293 Catalan (1 839), - - 224-226 (1839) Jacobians, 356-358 (1839) Orthogonants, 467-470 Cauchy (1812), - - 92-131 (1821), - - 148-150 (1829), - - 187-188 (1840), - - 240-243 (1.841), - - 247-253 (1841), - - 273-285 (1829) Axisymmetric, 295-296 (1841),, 304-305 (1812) Alternants, 308-310 (1841),, 342-345 (1815)- Jacobians, 346-349 (1822),,349 (1841),, 393-394 (1829) Orthogonants, 425-435 (1826) (Postscript), 470-471 Cramer (1750), - - 11-14 Craufurd (1841), - - 245-246 iDesnanot (1819), - - 136-148 PAGE Drinkwater (1 83 1), - 198-199 Fontaine (1748), - - 10-11 Giarnier (1814), - -135-136 Gauss (1801), - - - 63-66 Gergonne (1.813),. 3-3 Grunert (1836), - -215-219 Hindenburg (17 84), - 53-55 Hirsch (1809), - - - 69 Jacobi (1827), - - -176-178,, (1829), - 188Q-193,, (1831-33), 1 206-212,, (1834), 212-213 (1835),- - - 214 (1841), - -- 253-272 (1827) Axisyrmnetric, 293-294 (1831),, 296-297 (1832),, 297-298 (1833),, 298-300 (1834),, 300-301 (1841),,304 (1841) Alternants, 325-342 (1829) Jacobians, 349-352,, (1830),, 352-354 ~j (1832, 33),, 354-356 (1841),, 358-392 (1827) Skew, 401-405 (1845),, 405-406 (1827) Orthogonants, 410-415 Jacobi (1 827) Orthogonani 1 ~(1831) 1 (1832) ) (1833) 1 (1835) Miscell. Sp( LIST OF Al PAGE Is, 15-424 435-451 452 453-463 -cial, 485-487 JTHORS. 491 Lagrange (1773), - - 33-37 (1773), - - 37-40 (1773), - - 40-41 Laplace (1772), - - 24-33 Lebesgue (1837), - - 219-220 5) (1837) Axisymmetric, 301 -303 (I1837) Orthogonants PADRichelot (1840), - -238-240 iRothe (1800), - - - 55-63 (1 800) A xisymmetric, 290-292 Scherk (1825), - -150-159 55(1825) Miscell. Special, 478-479 Schweins (1825), - -159-175 (1825) Alternants, 311-322 (1825) Miscell. Special, 479-485 Sylvester (1839), - - 227-235 (1 840), - - 236-238 (1841), - - 243-245 (1 839) AlIternants, 322-325 (1840) Miscell. Special, 487 Vandermonde (1771), - 17-24 Wronski (1812), - - 78-79 (1815), - 136 (1812) Miscell. Special, 472-474 (1815),, 474-476 (1816-17),, 476-478 (1819),, 478 Leibnitz (1693), Mainardi (1832), Minding (1829), Molins (1839), Monge (1809), Pfaff (1815), - Prasse (1 81 1), - Prony (1795), iReiss (1829), - 'I,(1838), - 463-467 - - 6-10 - - 200-206 - - 194-197 - - 235-236 - - 67-68 - - 396-401 - - 72-78 - - 306-308 - - 178-187 - - 220-224 GLASGOW: PRINTED AT THE UNIVERSITY PRESS BY ROBERT MNACLEHOSE AND C0. LTD.