250 REPORT— 1897. 



limit a; as a function of u, and with Legendre we write a;=sin <f>, so that 

 the integral above becomes 



Jos/l-/c2a 



We consider as a function of m, and write 0:=amplitude of M^am(M), 

 so that x= sin am (M)=sn u. 



The function x^snu enjoys all the properties of a rational fractional 

 function, and, as is shown later in connection with the ^-function, the 

 numerator and denominator of this fraction may be developed in rapidly 

 convergent series for all real or imaginary values of m. Hence the elliptic 

 function x=snu has one, and only one, definite value, corresponding to a 

 given value of u. 



(10) Periods of the inverse /unctions. — Abel and Jacobi recognised 

 that the elliptic functions have at the same time the nature of circular 

 functions and of exponential functions in that they are periodic for both 

 real and imaginary values of the arguments. They saw that the function 

 x=snu, for example, remained unaltered when u is changed into m4-4K 

 or into M + 2K' V — 1, where K and K' are definite constants. 



Jacobi often repeated that the introduction of the imaginary was a 

 complete solution of all the enigmas that had previously beset this 

 subject.* 



The introduction of the imaginary and the necessity of treating the 

 limit as a function of the integral were two great advances made by 

 Jacobi and Abel. 



(11) Abel's investigations took different directions from those of Jacobi. 

 Abel devoted himself to problems that have to do with the multiplication 

 and division of elliptic integrals, their double periodicity, and their defini- 

 tion by infinite products. By the help of the principle of double periodi- 

 city he penetrated deeply into the nature of the roots of the equation 

 upon which the division depends, and made the unexpected discovery that 

 the general division of the elliptic integral with arbitrary limit may be 

 performed algebraically (i.e. through the extraction of roots) as soon as 

 the special division of the so-called complete integrals is presupposed 

 performed. 



The simplest case of this special division is for the modulus to which 

 the lemniscate corresponds ; and Abel shows that the division of the 

 entire lemniscate is completely analogous to that of the circle, and may 

 be performed by geometric construction in the same cases as the circle 

 admits. The solution of the circle had been solved some twenty-five years 

 before by Gauss. The admirers of Gauss with Dirichlet, from whom the 

 above extracts have been made (loc. cit. p. 11), contend, from certain 

 remarks (among others) made by Gauss in connection with the division of 

 the circle and the lemniscate, that the principle of double periodicity was 

 also known to Gauss. Some persons might, however, insist that Gauss 

 too was beset by some of the enigmas above referred to, and that it was 

 more likely that Gauss omitted to mention these dilemmas than to keep 

 silent about the remarkable doubly periodic property of the functions 

 that are connected with the lemniscate. In this connection Enneper 

 (• Elliptische Functionen,' p. 7) says that it is to be regretted that Gauss 

 did not communicate his remarkable discoveries to his contemporaries and 

 invite their co-operation. 



■ See Dirichlet, Qeddchtnissrede auf Jacobi (Jacobi's Werke, i. 10). 



