INFINITESIMAL CALCULUS when I learned with equal surprise and satisfaction that two young geometers, MM. Jacobi of Konigsberg and Abel of Christiania, had succeeded by their own studies in perfecting considerably the theory of elliptic functions in its highest parts." Abel and Jacobi have found not only points of connexion for their works with Legendre s investigations, but have been able to adopt a set of methods and points of view from his TraM, on the basis of which they have constructed the mighty edifice of the theory of elliptic transcendents. This Jacobi himself subsequently fully recognized. On May 27, 1832, he writes to Legendre : " In a note at the end of the eighth volume of M. Crelle, I have sought to extol the imperishable merits of the geometer who, besides the numerous and important discoveries with which he has enriched science, has effectually laid the foundations by the glorious labours of his life of two great and extended disciplines which shall henceforth form the a and the &> of every mathematical study. I have at the same time made use of this opportunity to speak of Abel and of his great theorem, which you again have the merit of having first penetrated, and of having shown to posterity that its development is the great task remaining for it to fulfil." 221. Before entering into the developments due to these two celebrated mathematicians we should make some mention of Gauss s labours in the same field. These were mostly found in incom plete sketches at the beginnings of different treatises, or as indi vidual formulae scattered among his other works. The editor of his collected works has brought them together in the latter part of the third volume, and states that there is evidence that Gauss was acquainted with the relations between the arithmetic geometric mean and the series proceeding by square powers in the year 1794. This arithmetic geometric mean is defined in the paper "Deter- minatio attractionis " published in 1818, where he speaks of it as a peculiar and most expeditious algorithm which he had for many years employed, and intended yet to treat of more fully. Let m, n be two positive quantities, andput m = (m + ri),n = Jrnn , so that m and n may be their arithmetic and geometric mean, taking the latter always positive. Now take m" = |(m + n ), n" = VmV > in " = |(w" + n"), n " == /m"n" > and so on. It may be seen that the series m, m , m", m ", and n, n , n", n ", &c., rapidly converge to a common limit, which we shall denote by/*, and simply call the arithmetic geometric mean between m and n. Now ve shall demonstrate that is the value of the integral f dr taken T = OtOT = 330. For suppose that the variable r is expressed by another r , so that 2m sin r it will easily be seen that as r increases from to 90, 180, 270, 360, r also, though not iiniformly, increases through the same range. But, effecting the substitution, dr _ dr Jrn 21 cos 2 r + n- sinV) ( m 2 cosV + n 2 sin V) accordingly the values of the integrals dr r dr /i r + n 2 sinV) 1/2 2 cosV + n" 2 sinV) . each taken from Oto 360 are equal ; and, since this maybe carried on as far as we please, plainly they are also equal to the value of the integral r d& 2 sin 2 ) from = to = 360, which is plainly . 222. N. H. Abel (1802-29) started in the summer of 1825 to pursue his studies abroad, chiefly at Paris. On his way he made the acquaintance in Berlin of A. L. Crelle, who had long conceived the project of founding a mathematical journal, but was decided to put this into execution by the importance of the numerous memoirs already prepared by Abel (as also by Steiner), who consented to their publication in it. The first number of the journal appeared during Abel s stay in Berlin, and each copy in the first four volumes contained papers by him. These and other published papers are reprinted in the first volume of his collected works (Christiania, 1839). The second volume contains mostly papers found after his death, nearly all in this volume having been written before Abel began his travels. These, therefore, first claim our attention after Legendre s independent investigations. 223. Elliptic integrals have hitherto occupied us. We have men tioned ( 207) the problem of inversion which leads to elliptic func tions, viz., that if = F(fc, c6), then < = amtt, and if r x dx_ =j we have x = sin am u. Paper viii. (vol. ii.), is headed "Remarkable properties of the func tion determined by the equation fy.dy-dx{(a-y}(a l -y)(a. 2 -y) . . . (a m - ?/)}* = 0, fy being any function of y which does not become zero or infinite when y = a, a lt a. 2 , , . . a m ." In it this problem of inversion of the more general (hyperelliptic) integral in which the square root contains a function of the degree m + 1 is attempted ; and, though it has since been shown by Jacobi (Crelle, xiii.) that the inversion of hyperelliptic integrals is a different problem from what is here proposed, Abel at any rate in this short paper had established the existence of two periods for elliptic functions. 224. We saw in 216 that Legendre, in comparing two elliptic integrals of the third kind, found a certain relation connecting with integrals of the first and second kinds two such integrals of the third kind, in which the argument and parameter are interchanged. This relation recurs to us in an extended form in the paper of Abel (ix., vol. ii. ) " On a remarkable property of a very extended class of transcendent functions." Defining y or tyx by the differen tial equation y . /e + . <f>x = 0, where (px = a + a^x + a.^x- + . . . ,
+ . . . , he gets x . dx (a - x}<pa . the integrals in x being taken from a value of x which makes tyx . <px vanish, and those in a from a value of a which destroys . When we put tt;c=* = in this, it becomes the theorem for the
- l<a V>a
interchange of argument and parameter for hyperslliptic integrals. 225. " The first works of Abel which attracted attention," writes his editor, "were his memoirs on the impossibility of the general resolution of algebraic equations higher than the fourth degree, and his researches on elliptic functions. Simultaneously with Abel, and without being acquainted with his works, M. Jacobi of Konigsberg began to treat the theory of elliptic functions. Thus a rivalry exists between these two men of exalted genius in their treatises on these functions. Abel told me that during his stay at Paris in 1826 he had already completed the essential part of the principles he sub sequently enunciated regarding these functions, and that he would have much wished to postpone the publication of his discoveries until he could compose a complete theory of them, had not in the meanwhile M. Jacobi entered the lists. " 226. On October 24, 1826, Abel wrote from Paris: "I have just finished a large treatise on a certain class of transcendent functions for presentation to the Institute, and that will take place next Monday. I dare without ostentation say it is a treatise which will give satisfaction. I am curious to hear the opinion of the Institute about it." He had not deceived himself in the significance and reach of this fundamental theorem ; yet in the Academy judgment upon the work was deferred, so that Abel two years later (Jan. 6, 1829), felt himself called upon to send to Crelle the following, which appeared in the fourth volume of the Journal ( Works, i.p. 324). 227. "Demonstration of a general properly of a certain class of transcendent functions. " Theorem. Let y be a function of a; which satisfies any irreducible equation of the form whereto, p lt . . . >-i are integer functions of the variable a". In like manner let g- , g lt . . . g, t -i be integer functions of a-, and = 2o + <?i2/ + &2/ 2 + . . . + ?-4/ n - 1 . . (2) a similar equation, and let us suppose the coefficients of the different powers of as in these functions variable. Let these be denoted by a, a , a" . . . . By reason of the two equations (1) and (2), x will be a function of a, a , a", &c. ; and we shall determine its values by eliminating y. Let us denote by P = ........ (3) the result of elimination, so that p will contain only the variables x, a, a , &c. Let /* be the degree of this equation in x, and let its p. roots be which will be so many functions of a, a , a", &c. " Now, if f(x, y} denote any rational function of x and y, and we make ^x=/f(x, y)dx ....... (5),
