270 ON THE RECENT PROGRESS OF ANALYSIS. 



I may, however, mention a remarkable transcendental function 

 of the modulus k which is usually denoted by ^, and which 

 occurs perpetually in this part of the theory of elliptic func- 

 tions. If for the moment we denote this function by Fk, so 

 that q = Fk, then if for k we write 1e n , which we suppose to re- 

 present the modulus of the first real transformation of the nth 

 order, we find that f = Fk n9 so that if q n is the same function 

 of k n that q is of k, 



<ln = 2 n - 



This singular property, and others of an analogous character, 

 are of great use in establishing various formulas*. 



Before discussing the evolution of integrals of the third kind, 

 M. Jacobi has premised some important theorems. He proves 

 that the elliptic integral of the third kind, though it involves 

 three elements, viz. the amplitude, the modulus and the para- 

 meter, can yet be expressed in terms of other quantities severally 

 involving but two. In order to this we introduce either a new 

 transcendent t or a definite elliptic integral of the third kind, 

 whose amplitude is a certain function of its modulus and para- 

 meter. It is almost impossible to tabulate the values of a 

 function of three elements, on account of the enormous bulk 

 of a table with triple entry ; we therefore see the importance of 

 the step thus made. M. Jacobi announced this discovery as 

 generally true of elliptic integrals of the third kind, but his 

 demonstration applies to that subdivision already mentioned, 

 which was designated by Legendre * Fonctions du troisieme 

 ordre a parametre logarithmique,' and not to functions ' a para- 

 metre circulairej.' It is probable that this limitation was in 



* A method of calculating elliptic integrals by means of q was suggested by 

 Legendre. Vide Verhulst, p. 252, and M. Jacobi in Crelle. 



1* This transcendent is denoted by T, and is defined by the equation 



-/ 



where E (c0) is the elliptic integral of the second kind. If we introduce Ihe 

 inverse notation, and make = am u, we can readily establish the following result, 



sin 2 am udu 2 . 



T= * w 3 - c 2 If si 

 is the logarithm 



+ In the former species (r +n) f i +- ) is negative, and in the latter positive 



The function T, which is the logarithm of ft (vide infra, p. 288), has many re- 

 markable properties. 



