68 INFINITESIMAL CALCULUS arrive at the equation dx dz Moreover, this substitution can be replaced by two in succession, _y(a + V + . . . +aV~ 1 ) - the first substitution transforming the elliptic function into another of different modulus, so that dx M.dy and the second returning to the original dz Now, giving^; different values, we see that each given modulus is one in an infinite scale of moduli into which it may be transformed by an algebraic and even rational substitution. This letter, moreover, contained the two theorems giving the general formula in a trigono metric form. 235. Subsequently Jacobi, on January 12, 1828, brought under the notice of Legendre the works of Abel on elliptic functions, which Crelle had published, but in his own notation. Abel, he says, begins with the analytic expression of all the roots of the equations of higher degrees on which the division of elliptic func tions depends. Taking sin <p = i tan J/, where i /- 1, and J Vl-^sm^ we have, if K be the complement of K, d<t> whence sin am (if, /c) = i tan am (f, K }, which is a "fundamental theorem of M. Abel." Further, we have in general sin am (f + 4mK + im iSL 1 ) = sin am f , where m and mf are positive or negative integers, and K is the com plete function and K the complete function corresponding to K . "We see then that the roots of this higher equation for dividing the elliptic function f into n parts will be of the form a formula which involves as many as ?i 2 roots, if we make m, m successively take the values 0, 1, 2, ... n 1. Abel next reduces the division of any elliptic function f to the division of the complete function K. In fact, if a, /3 be any roots of x n =l, the expression ing m, m all the values 0, 1, 2 . . . n-1, will not change if we put instead of sin am -- any other root, sin am Ll_ + ^ l it will n n thus be symmetrical in these roots, and may therefore be expressed by sin am f , and by constant but irrational quantities of the form Now giving a, jS all possible values produces n 2 combinations, and thereby the values of all the roots. The division of the com plete function, which depends in general on an equation of the J(?i 2 -l) degree, is reduced to one of the n + l degree, n being a prime number. For if w = - V l - , and g be a primitive root of the congruence x"- 1 ! (mod ?i), also <f> (u) any trigonometric function of the amplitude of <a, and a a root of a; n - 1 = l, we attain this by considering the expression O(o>) + a(f>(gca) + a?(j)((j-<c} + . . . + a. n -^(g n --w}] n - 1 which is symmetrical in 0(), 0(r/), rty-u) . . . <t>(g"--u). But symmetric functions of these quantities can only have + l different values, answering to /j. = 0, /u = l; /u=l, fjf=0; /u=l, n = 1, 2, 3, . . . n-1. So they will be determined by an algebraic equation of the degree n + l. In conclusion, Jacobi mentions researches of his own, which led him to the conclusion that if a modulus K can be transformed into another A, they are connected by an algebraic equation of the degree n + 1, if the transformation be of the order of the number n, supposed prime. These symmetric equations are, for n = 3 and 5, M 4 _ fl42i>(l - wV) = 0, M 6 -v* + 5u z v-(u- - v-)4uv(l - u^) = , putting u = %/K, v = %J . These equations he names modular equations, and notices as remarkable that they have their simplest forms when expressed in the fourth roots of the moduli. He also gives the differential equation of the third degree which all these algebraic equations satisfy, viz., Moreover, in some cases the same modulus reappears, and the trans formation becomes multiplication. This takes place in_all cases when n is the sum of two squares, w = 2 + 4& 2 , K being Vi, and the elliptic function becomes multiplied by a2W. Similarly with all moduli which are connected by any scale with K = Vi> a kind of multiplication not having an analogue in circular arcs. 236. In answer to a request of Legendre that he would furnish him with the clue to his discoveries, Jacobi wrote, April 12, 1828: " Having found (March 1827) the equation T Vf-Uf , M dx dx I recognized that for any number, n, transformation was ^determinate problem of algebraic analysis, the number of arbitrary constants being always equal to that of conditions. By indeterminate coefficients I formed the transformations for the numbers 3 and 5. The biquadratic equation to which the former led me having nearly the same form as that which serves for trisection, I began to sus pect some relation. Fortunately I happened to remark in these two cases the other transformation complementary to multiplication. At this stage I wrote my first letter to M. Schumacher, the method being general and verified by examples. Subsequently, examining T i ,, , ... ,. att + by 3 a x+b x A more closely the two substitutions z = -^ %- , ?/ = ~ "
1 + c a: 2 under the form presented in my first letter, I saw that when we put =sin am , z must vanish, and, as in the said form was 3 a positive, I thence concluded that y must vanish also. In this manner I found by induction the resolution into factors, which being confirmed by examples, I gave the general theorem in my second letter. Having remarked the equation sin am (if, K) = i tan am (f , K), I next drew from it the transformation from K to A . I had then two different transformations, one from K to a smaller modu lus , the other from K to a larger one A . Thence I conjectured that exchanging inter so K and A, K and A , the analytic expression of the complementary transformation would be got. The demon strations were found only subsequently." 237. Equally interesting is Legendre s reply (June 16, 1828): " As to what you told me of the train of ideas which led you to your beautiful discoveries on elliptic functions, I see that we have both run some risks, you in announcing discoveries not yet in vested with the seal of a rigorous demonstration, and I in publicly and unrestrictedly giving them my full and entire approbation. We have neither of us to repent of what we have done. ... I saw very clearly that results such as those you had obtained could be no effect of chance or of a faulty induction, but only of a profound theory based upon the nature of things." 238. Of Gauss s investigations in this branch of mathematics Jacobi makes mention in his first letter to Legendre (August 5, 1827). These researches "are not the only ones which have been undertaken in Germany in the same subject. M. Gauss having heard of them let me know that he had developed as far back as 1808 the cases of division into 3, 5, and 7 parts, and found at the same time the new scales of moduli referring to them." Again, April 12, 1828: "As to M. Gauss, he has not yet published anything in elliptic functions, but it is certain he has made beautiful discoveries. If he has been anticipated and perhaps surpassed, it is a penalty due to the veil of mystery he has spread over his works. I am not personally acquainted with him, as I studied philology at Berlin, where there are no distinguished geometers." Legendre, however, cannot believe that discoveries of such reach can be left unpublished, as was actually the case with Gauss. " If M. Gauss, " Legendre writes to Jacobi, April 14, 1828, "had fallen upon such discoveries, which in my eyes surpass all hitherto done in analysis, most assuredly he would have lost no time in publishing them." 239. Simultaneously with the announcements of Jacobi just men tioned there appeared in September 1827, in Crelle, the first part of Abel s " Eecherches sur les fonctions elliptiques," and accom- janyingthe second part (Feb. 12, 1828) a statement that, "having finished the preceding memoir on elliptic functions, a note on the same functions by Mr C. G. J. Jacobi, inserted in No. 123 of M. Schumacher s NachricMen, has reached me. M. Jacobi gives the
