INFINITESIMAL CALCULUS 69 following theorem. . . . This elegant theorem, which M. Jacobi gives without demonstration, is contained as a particular case in formula 227 of the foregoing memoir (which is the formula of of 246 infra), and is fundamentally the same as that of formula 270. " This he proceeds to show. 240. The " Recherches " present a great and complete theory of elliptic transcendents. Starting with the inverse function <f>(a) as /"" clQ that detennined by a=/ r= and sin = rf>a = ;r, which by J VI -&SUY-Q / f dx dd V 1 - sin - e = d<pa = dx gives a / / , - , Abel noticed Jo V(i-x 2 )(i-cV) that the formuke become simpler by supposing c 2 negative = - e 2 , and for symmetry writes 1 c-x^ instead of 1 -a; 2 , so that the func tion <pa = x will be given by the equation r a=/ J
- c 2 x")(l + C 2 x 2 ) or by tp a = V(l - c 2 2 a)(l + e-<f> 2 a), and for brevity two other functions/a = Vl - c - 2 a, Fa = Vl + e 2 2 a are introduced. After establishing the double periodicity, and determining the zero and infinite values of these functions, Abel proceeds to the development of the formulae of multiplication to determine <p(na), f(na), F(na) in rational functions of <p(a), /(a), F(a). He next enters ou the solution of the more difficult problem of the division of elliptic functions, which is the principal object of the memoir, Abel proves the algebraic expressibility of the func- tions <p /(a), F(a) in the form 1 in which 2+iJ as functions of <f>(a}, and the quantities C, D are rational functions of t 08), while the quantities A, B are similar functions of <p(2n + 1)/3. Thus these equations give 0(0) algebraically expressed by ^(yS), and then 0^)3) algebraically by </>(2?i + l)j3. So, replacing ft by weget / a ^ (j) I - - I as an algebraic function of <a ; and similarly for /
ll> ~r 1 y
and F. 241. The priority of this beautiful discovery Jacobi ascribes un conditionally to Abel. To Legendre he writes (March 14, 1829) : "You suppose I have found means of expressing algebraically trigo nometric functions of the amplitudes you denote by a m , adding that without that my formula would contain coefficients I could not determine. But that is quite impossible in the general case, and is done only for special values of the modulus. My formula, which gives the algebraic expression for sin am u by means of sin am nu, supposes known the section of the complete function. In this manner, for more than a century, the division of an: arc of a circle could be solved algebraically, supposing known that of the complete circumference, this latter having been given generally only in these later times by the works of M. Gauss.... You see then that M. Abel has proved this important theorem, as you call it, in his first memoir on elliptic functions, although he has not dealt in it with transfor mation, and does not appear even to have thought when he wrote that his formulas and theorems would find such an application. The transformed modulus, or, which amounts to the same thing, the regulator, being supposed known, it is still necessary to resolve an equation of degree ()i-l) to arrive at the quantities sin 2 am(2^co), or at the section of the complete function. Thus you had only to solve a quadratic in the case of w = 5. M. Abel proved that M. Gauss s method applies nearly word for word to the solution of these equations, so that it is only the modular equations that we are unable to solve algebraically. " 242. Starting from the solution of the problem of multiplication and division, 0(2)1+1)3 is exhibited by Abel as the quotient of two double products, the factors of which depend linearly on $(&). Thence, putting /3 = , and % = co, the development of the in verse function <(<x) is derived in double products and double sums, the factors of which are linear in a ; accordingly a unique analytic expression is found for the function heretofore defined only by its properties. The reduction of the double products and double sums to simple products and simple sums, the product development and breaking up into partial fractions of the elliptic functions, follow then without further difficulties. 243. With the publication of the "Recherches " Abel clears at one bound the limits of the investigations of Jacobi hitherto published, though the first part devotes no attention to the problem of trans formation of elliptic integrals. Moreover, this work drew from Gauss the remark: "M. Abel has anticipated me at least in a third part. He has just trodden precisely the same path I went along in 1798. And so I am not at all astonished at his arriving, for the most part, at the same results. Besides, as in his deduction he has displayed so much sagacity, penetration, and elegance, I feel myself by it relieved from the publication of my own researches." 244. The same volume of Crelle contains, besides the first part of of the "Recherches," indications in the paper "Problems and Theorems " that Abel was at the time in possession, not only of the theory of rational transformation which Jacobi treated, but of the general algebraic transformation, as has been made manifest subse quently in his collected works. 245. Before Jacobi had read the " Recherches " he published a proof of the general theorem of rational transformation in No. 127 of Schumacher s Nachrichtcn, December 1827. It is based on enumeration of the constants available, and fixing the conditions in order that the substitution y^- may satisfy the differential equation V dy dx MV(1 - He introduces the unique inverse function which he calls sine of the amplitude, sin am, and gives the value of "*"( *- ^Y. -A- 1-,= ^ Mn 2?i + ] ( 1 - * 2 a 2 siu 2 am -M- V . . ( 1 - K *x 2 sin 2 am 2nK
2n+lJ 27i+ 1;
as satisfying the differential equation (M being constant) dx , r dy The value of y is derived from this, and Jacobi remarks that this theorem holds generally, but does not embrace all the solutions of the problem. 246. The second part of the " Recherches " was finished by Abel February 12, 1828, and appeared immediately in Crelle. The first problem treated is the algebraic expressibility of the function </>( ) V n / when certain relations, as for the lemniscate, hold between e and c. The principal application of this is the expression of the function by square roots whenever n is of the form 2" or 1 + 2 n , the latter being prime. He then proceeds to deal with the general treatment of rational transformation, which he presents in the following form. If -, (m + u)&> + (7)1 uWl T , , f ,, . , Jt a be -! i _ , where at least one of the integers ~~ ^liit ~T~ I m and ^ is prime to 2;i + l, we shall have f__ dy _ ~ J [(1 -c t " _ ^ [(1 -cV where yf.se. +- )] /being an indeterminate, so that there only exists a single relation between the quantities c 1 , c lt c, c. The section concludes with the words "To have a complete theory of the transformation of elliptic functions, it would be necessary to know all the transformations possible ; now I have succeeded in demonstrating that they are all got by combining that of M. Legendre with those contained in the above formula, even iclien we are looking for the most general relation betiveen any number of elliptic functions. This theorem, the con sequences of which embrace nearly the whole theory of elliptic functions, has led me to a very great number of fine properties of them." 247. The same number of Crelle contained, in an extract from a letter by Jacobi, " Note sur les fonctions elliptiques," the exhibi tion of sin am as the quotient of two series & and H, or as they
