Theory of the Jacobian Elliptic Integrals. 279 



•"• ff p(X)sinXdX _ tt « (-)» 2 - P.fr) (5? 



) I* + k'- cos 2 X - W -o 2« + 1 r " W /tF^n + l) 2 ' l ' 



It follows that 



' n Jn & 2 + # 2 COS 2 X 1-1 4 



= |-1. . . . (58) 



Also 



J o n **& l)dt J PTF cos 2 X 2(2n + l) 2 (2« + 1) 3 ' ( 

 While squaring (57), and calling the integral on the left, I, 



r» « /""(—)" - V 2 



j_ i K°. l ' 2 rf M =2 o ^2^+r~(2« + l) 2 j 2« + l 



f P f** , # s FYX)F'(p) sin X sin /td*dXrf/» 



Jo Jo 3. -<* , + * B OOB»X)C*» + *r»OO^M) 



-i-V,+ ^-fcr(l-^+i...)- («» 



It may be noted that if 



" \/ C0SW 9 ""* C0S " 2 



then r^ °° 2 



^G 2 W S in^ = «i42 o(2n+1)5 



= 128 (l-^; 



| > *Q*(tf)sinAtf=>124a' ( ( 61 ) 



Jo 

 Similar results follow from the series 



cos.— |cos:3:+^,eos5:... = ^ (tt W) z between + J 



■9) 



