272 REPORT — 1897. 



systems of n quantities w,, w.,, . . . w„ ; 2/1, y-a . . . y,, ; ~i, ^2 • . . 2,, be 

 determined through the three systems of n transcendental equations 



^J N/y,RK) -^ J n/c«.R(xv) "^ J ,/aR(a)' 



(W=0, 1, . . . TC — 1) 



in which R(a;) is a given function of x of the 2/ith degree ; then there 

 exists among the integrals of the third kind the following equation : — 



(IV.) V I i = -v I '. 



/ XiX^ . . . X„ + i 



log 



When TC is made unity in the above formuL-e, they reduce to Legendre's 

 form of the addition-theorems for elliptic functions.' See Jacobi, 

 'Extrait d'unelettre addressee a M. Hermite' (Jacobi's ' Werke,' ii. p. 120). 



(33) Interchamie of parameter and argument of the integrals of the third 

 kind. — Legendre discovered this remarkable property of elliptic integrals, 

 and derived other formula in the same connection. ('Ex. de Calc. Int.' 

 t. i. p. 134 et seq.) The results of Legendre are implicitly contained in 

 the following formula, due to Abel (' Qlluvres,' t. ii. p. 43) : — 



r da 





:)v/^(a) 

 x"dx f a^da 



,^ , , ^ f x"dx f a^da 



Jn/^(x) Jv/'/)(a) 



where ^(.i') is any integral function of .r, o,„^„^.2 are constants, and m,n 

 integers. Jacobi (Crelle, bd. xxxii, p. 185; ' Werke,' bd. ii. p. 123) obtained 

 the analogous formula. Let /(.)•) be a rational integral function of .r, and 

 y,(x) andy"2(.x) any two rational integrals of a*, whose sum 



' In this connection see Heine (Crelle, bd. Ixi. p. 27G) ; Schumann (3fatJt. Ann. 

 bd. vii. p. 623) ; Scbeibner {Math. Ann. bd. xxxiv. p. 473.) We mention here a paper 

 by Serret, ' Memoire sur la representation guometrique des fonctions elliptiques et 

 ultra-elliptiques ' {Liouv. Journ. t. x. pp. 257, 28G, 351 and 421). Liouville (C'omjit. 

 Rend. xxi. p. \,2Z^,ox Liouv. Journ. t. x. p. 456) tjives a method of representing elliptic 

 and hyperelliptic curves. See Ellis' report, p. 72. 



