Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/77

INFINITESIMAL CALCULUS 67 then the transcendent function ty(x) will possess the general property expressed by the following equation ifX lL + ^,r. 2 + . . , ^ = ? + &! log ! + fc 2 log v.j + . . . +k n logv n . . . (6), u, v 1} v. 2 , . . . v,,, being rational functions of a, a , a", . . . and &_[, k. 2 , . . . k-n, constants. Damcmstration. To prove this theorem it is enough to express the first member of equation (6) as a function of a, a , a", &c. ; for thereby it will reduce to a rational differential, as we shall see. " First, the two equations (1) and (2) will give y as a rational func tion of x, a, a , a". . . Similarly the equation (3) will give for dx an expression of A he form dx = ada 4- a da + a"da" + &c. , where a, a , a" . . . are rational functions of x, a, a , a", &c. Thence it follows that the differential /To?, y}dx can be put under the form f(x, y]dx = <j>zda + ip^xda + <f>. 2 xda" + &c. , where <j>x, fav, . . . are rational functions of x, a, a, , a", &c. Integrating, we get $X =J (<pxda + <l>]Xda, + . . .); and from this wo conclude, since this equation holds when we put for x its /tt values, "In this equation the coefficients of the differentials da, da , &c. , are rational functions of a, a , a" . . . and of x lt x. 2 , . . . x/j. ; but they are besides symmetrical in x lt x. 2 , . . . ay ; therefore, by a well-known theorem, they can be expressed rationally in terms of a, a , a" . . . and the coefficients of the equation p = Q ; but these latter are themselves rational functions of the variables a, a , a" ... so that finally the coefficients of da, da , da", &c. , of equation (7) will be so too. Therefore, integrating, we have an equation of the form (6). " I propose on another occasion to develop numerous applications of this theorem, which will throw a great light on the nature of the transcendental functions it deals with. " 228. Abel died of consumption, April 6, 1829, having been con fined to bed nearly three months ; and of the applications promised nothing appeared or has since been found in his remains. More over, except the note that he had presented the memoir to the Academy, which appears in the paper On some general properties of a certain sort of transcendental functions " ( Works, i. p. 288), Abel hardly seems to have expressly referred to it, though he mentioned the theorem (November 25, 1828, Works, ii. p. 258) to Legendre, adding that on this general property he had in fact founded the whole theory of elliptic functions. 229. But notwithstanding, his contemporaries were not slow to estimate the value of his analysis. The statement in Crelles Journal revealed to Jacobi the entire significance of this "funda mental theorem of analysis," and his admiration breaks forth to Legendre on the 14th March 1829 : "What a discovery of Abel s is that generalization of Euler s integral ? But how docs it happen that this discovery, perhaps the most important our century has made in mathematics, though communicated two years since to the Academy, has been able to escape the attention both of you and your fellow members ?" To this question Legendre answers (April 8, 1829) : " I shall not close this letter without answering yours relative to M. Abel s beautiful paper, which was printed in the last number of Crdla s Journal, and which had been presented to the Academy by its author in the last months of 1826. M. Poisson was then president of the Academy. The committee named to examine the memoir were M. Cauchy and myself. We perceived that the memoir was almost illegible ; it was written with very faint ink, the characters badly formed ; it was agreed on that we should ask the author for a better copy, and one easier to be read. So things remained. II. Cauchy kept the manuscript up to this without doing anything further about it. The author, M. Abel, appears to have gone away without caring what became of his memoir. He furnished no copy, and no report was made. However, I have asked M. Cauchy to give me the manuscript, which never was in my hands, and I shall see what there is to be done, to make up, if possible, for the little attention he bestowed on a production which no doubt deserved a better fate." 230. The third and last supplement of Legendre to his great work is dated March 4, 1832, and concludes as follows : " Here we shall terminate the additions we proposed to make to our work, taking advantage of the recent discoveries of MM. Abel and Jacobi in the theory of elliptic functions. It will be remarked that the most important of these additions consists in the new branch of analysis we have deduced from the theorem of M. Abel, and which had hitherto remained quite unknown to geometers. This branch of analysis, to which we have given the name theory of ultra-ellip tic functions, is of infinitely greater extent than that of elliptic functions, with which it has very intimate relations ; it is composed of an indefinite number of classes, each of which divides into three kinds, as do elliptic functions, having besides a great number of properties. We have only been able to glance at this matter, but no doubt it will be gradually enriched by the labours of geometers, and at length will form one of the finest parts of the analysis of transcendents. " At nearly the same time Legeudre wrote to Crelle " The work, so far as I am concerned, has given me the profound satisfaction of rendering conspicuous homage to the genius of M . Abel, in making felt all the merit of the beautiful theorem which was his discovery, and which maybe characterized as Monument um fere pcrcnnius." In his remarks on this third supplement of Legendre (Grellc, yiii.) occur the notable words of Jacobi : " We hold it (the Abelian theorem) to be the greatest mathematical dis covery of our time, although it remains for a future, perhaps long distant, work to manifest its whole significance." 231. The name which Jacobi thus applied, the Abelian theorem, has since adhered, and the functions to which it refers have been called Abelian functions, the term hyper- or ultra-elliptic having been restricted to that particular class in which the square root of a polynomial is the only irrational quantity introduced ; while Abelian functions may depend on any irrationality. The neglected paper of Abel appeared in the Memoirs of the Institute in 1841. 232. But, though the Abelian theorem was thus published during its author s stay in Paris, his labours in other departments of the theory of transcendents suffered no intermission. In December 1826 he writes "I have written a large memoir on elliptic functions which contains much that is curious, and which I flatter myself will not fail to fix the attention of the literary world. Amongst other things it treats of the division of the arc of the lemniscate. I have found that with rule and compass the lemniscate can be divided into 2" + l equal parts, when this number 2" + l is prime. The division depends on an equation of the degree (2" + l) 2 -l, but I have found its complete solution by means of square roots. This has revealed to me at the same time the mystery in which Mr Gauss s theory of the division of the circumference of the circle has been wrapped up. I see clearly how he arrived at it," referring to the last section of the Disyuisitioncs Arithmetics of Gauss, published in 1801. 233. C. G. J. Jacobi (born Dec. 10, 1S04, died Feb. 18, 1851) appeared first as a discoverer in connexion with our subject in the "Extracts of two letters to the editor," published in September 1827 in the Astronomisehc Nachrichten of Schumacher, and re printed at the beginning of the collected Works, 1881. We have seen that Legendre discerned the vast importance of the relations which he called Landen s transformation, and discovered with increased wonder the further transformation of the third order, which became public in January of this same year 1827 in the Traite. But in his first letter Jacobi states : " The integrals of the form /- -. for different moduli c, belong to different J Vl - c 2 sin 2 <j> transcendents. Only one system of moduli is known for which they reduce into one another, and M. Legendre in his Excrcices says even that there is only this one. But in fact there are as many of these systems as there are prime numbers, that is to say, there is an infinite number of these systems, all independent, each answering to a prime number ; the system heretofore known answers to the prime number 2. " This is already, without proof, a statement of the general theorem of transformation of elliptic integrals of the first kind. If U be a certain odd function of sin ty of any degree , and V a certain even function of sin ^ of degree n-1, then, putting sin = - , the co efficients of these functions may be determined so as to satisfy /d<j> f rfy /, s=^== = m / "TT 7 ", ,~T; 5 and each of these substitu- / 1 - <r sm 2 < J VI - k* sm-ty tions gives a new system of moduli. Further, Jacobi notices that sin ty can be in an almost analogous manner expressed by sin 6, so as by composition of the two integral equations to satisfy the relation r d<p r do I . -L = nl ; J V 1 - c 2 siiiV J V 1 - 1 c 2 sin-0 Thus the substitution which serves to give n times the tran scendent can be divided into two of a simpler nature, and this substitution gives sin <p expressed by a fraction whose numerator contains the odd powers of sin up to n 2 , and its denominator the even powers of it up to n 2 - 1. Without giving the general proof, the transformations of the third and of the fifth degrees are here actually effected, and connected with multiplication and division for the numbers 3 and 5 ; and thus for the first time the algebraic solution of the equation of the ninth degree which trisects the transcendent is given. 234. Legendre could not at first believe in the existence of an alge braic transformation belonging to any arbitrary degree, and thought Jacobi trusted to mere induction. But he soon admitted the pro fundity and rigour of Jacobi s analysis on receiving from him a letter, dated Aug. 5, 1827, in which it is stated that, if p be any odd num ber, we can by a rational substitution,