Theory of the Jacobian Elliptic Integrals. 267 



Put 4> e 



tan — = tan . cosec A, 



2 tan ^ . cosec A. cot A <^A 



... # = ■i— e 



] + tan 2 - cosec 2 A, 



q 

 n , sin 2 ^ cot 2 X 



cos- - — cos- - = 



1 + tan 2 cosec 2 A, 



6 

 sec g cosec A, d\ 



■■ J v^i-~ 1 W i+ 



tan 2 ^ cosec 2 A 



— 2K/ cos » j, 



where K(£) is Jacobi's complete integ] 



1 £ 2 sin 2 A,' 



^K) is tiacoors complete integral I "— 



Jn VI- 

 Hence 



>ral H^J 

 Jo VI- 



V 2/ 2n + l v } 



where fi = cos 6. 

 The formula 7r 



= cos c — ^-cos 3^ + icos oz, 



holds from 2= — - to z= + - , both exclusive. 



tJo 4 ' v'2(cos^-costf) o^n+l " wv ' 



J /I 



To transform this integral, put tan ^ = tan- cos\. 





 — 2 tan - - . sinA d\ 



••• d * = ~ 



l + tan 2 ^cos 2 A 



Q * ,0 sin 2 2 sin 2 X 



cos- 1 - cos- ^ = - ^ 



1 + tan 2 = cos 2 A 



