Theory of the Jacobian Elliptic Integrals. 275 



and 



Jo 2 (2/1 + 1)- (2/t— l;(2n + 3) 



Now in (31). write (tt-0) for 6. 



E(cos^ = -f „ r f«W ' . . (43) 



V 2s o (2/i + l)(2/i— l)(2/i + 3)' v ; 



whence 



£ E (~ D p " (cos " } siu <^ = 7 i W -i) ( 2,+3)(^ +i)' • (44 > 



Combining (7) and (43) 



El cos^)sinx^=— 22 7 ^ 1wo , , ( t — — r^ 



V 2/ 2 o (2n— l)(2n + 3)(2w + l) 2 



9. 9 



t 3 -»' + * 



}■ 



•'• ^E(i)«= jl-A+I. ..}+!. . (45) 



We note that 



i l Edk=l)Kdk + l, .... (46) 



Jo -vo * 



while /m 1 /-«i 



^ E7tt==;1 K'dk (17) 



Jo - 1 o 



These might be obtained by integration by parts. 

 By (43) 



\_V (cos ,) dp = 22 ^ + i)^_ 1)2(2n + 3 )J 



.-. by previous result 



\ [EWPAa-sJgff, (48) 



Again 



J^BE / ^ = i 2^+] • (2n-l) 2 (2n+l) 2 (2w 



+ar 



T2 



