Tlieory of the Jacobian Elliptic Integrals. ^81 



also ,# n 



- ^F^OsinXtan-M - cos X fdX 



Wdk]*- is-L*^ =£-l;(65) 



and squaring (62) and integrating to k from to 1, 

 , q r/ : V/-F\X)F'(/x) tan" 1 (^ cos x) tan" 1 (^ cos /^sin Xsin fid/cdXdfji 



r 



i/0 t * 



(£- + // 2 COS 2 X) (£ 2 + A' 2 COS* /*) 



_ 1 v « 2 /(-)V J 2(-)"-i \ 2 

 ~-4 w o 2w + l\ 2n + l (2/* + l; 3 / 



-j( 1 -i)*. +8 ( 1 -?h- fc? ( 1 -i)-* 



7 . 127 31 .„„, 



It may be remarked that formula* of these kinds can be 

 successively obtained, and the expansion of the integral 



J ( /^L tan U CQsX jJ ^ + ^cos 2 X 



in terms of Legendre's functions, when n is any positive 

 integer, by taking a succession of Fourier series of the kinds 

 already considered in the last two cases. 



We may write (b6) in a form involving only complete 

 elliptic integrals outside, for 



\/ c<^ - ; — cos-- 



V 2 2 



•"• by (56), 



2K< cos- *-J o ' K'(cos *)*=**, Jg= P. W -if ( £M, ; 

 .-. by (9), 



P.(*r-« l *) + iTK'(«.f)if-rf -^j,. (67) 



