Theory of the Jaeohian Elliptic Integrals. 273 



If cos t =l\ this becomes 



\' (l-^sin 2 *.)!"" SF 



= 2(E / -£ 2 K / ) by (32 6). 



" s oW_r ! (2/i-l)(2«+l)(2rc+3) 



-^.{,VScr- W :o + ^ fl -.«'} 



22 \/2 



= E'. 



•" E f COS - = 1 -r, ; , iw.» ,qx if At = cos ^ . (34) 



V 2/ (2/<-l)(2n + l)(2w + 3) ^ y 



The series being absolutely convergent. 

 It follows that 



£*W'M=( u-i^w*+q ■ ■ ■ < 35) 



■"" f E< ( C0S a ) F " (eOS * } Sin ^ g = (2 n-l) ( (7n+"3K2n + l)- ( 36) 



Combining (7) and (34) 



I E' (cos- ) sin -d6= — 2 2 tt; rfTTs -— T-f, 77* 



Jo \ V 2 o (2n-l)(2» + 3)(2n + l) 2 



= 2 S 777- 7^9 on reduction 



o (2?i + l) 2 



_ wf 



~ 4 " 



■■' |V(*>» = ^ (37) 



Squaring (34) and integrating, 



J Vw*. = 2 2 (2n _ 1)8(2n 1 +3)8(58m+x)8 - 



P/«7. .1%. 8. 'I. Vol. 9. No. 50. Feb 1905. T 



