318 Some Results in Elliptic Functions. [Apr. 30, 



4 f q . nit . a* . 2ttu . o 3 . 3^ . "1 



— < — sm- — sm + — - — sin + ...)■. 



Kll- 2 a k l-V K 1— K J 



(/3.) Putting ^=2wK, we deduce a simple identity, viz. : — 

 If n be an integer, then — 



(l + 2n)r»+ (3 + 2n)r* + * n + (5 + 2n)r &+5n + (7 + 2n)r™ + 1 n + . . ad infin. 

 = (1 — 2n)r~ n + (3— 2rc)r 2 - 3 *+(5-2rc)r 6 - 5 *+(7 — 2n)r 12 "7«+ ..ad infin. 



(7.) From the formula dmt= v^VV"^ < ^( 2t ~^-^-) ? 

 we have ^?=4^J 



r i (r i + r- 4 )+rl(rt + r-f)+r¥(rf + r-f)+ . . . 



— 2 ^ + ^+ y^+^r* 9 + . • • 

 ~ l + 2r + 2r 4 +2r 9 + 2r 16 + . . . 



This result is, in fact, Jacobi's 

 \/fc=2 1 ^ 2 g + 2g 4 + ' ^ WG cliail g in g * 



into and consequently r into gr. 



(a.) From OW = / y // ^^e^ 2U<WM wededi. 



ce 



A /___ =$(0) 



= 2{4/r+>9+^r 25 + yr 49 + . . .} 

 a/— = l + 2r + 2r 4 +2r 9 + 2r 16 + . . . 



V 7T 



5. Extension of the above Method to a £i(u) Function connected with 

 Elliptic Integrals of the Third Kind. 



In a former note by the present writer mention was made of a £(u)> 

 function of the form £(u) = — ^^ (p- |? M )> 

 where 



dG 



(o de n=[r 



V J o (l + rcsin 2 0) ✓ l-fcWfl' )\ (1 + 



wsin 2 6»)v/i_^ ^25, 



*=-V5-', P' = K' — — ^— II', 



* n K' l+n 



