228 A, Mukhopa.ahja.j—ElUpHc Functions and Mean Values. [No. 2, 



i_ r 2 de_ 



d-e^sin 



■^0 sin2 e)"^ 



But from a well-known transformation,* or from the identity 



sin ^ cos 5 1 1 1 _ 1 



— = (l-e2 sinM)' — 



we have at once 



^ (l-e8sin2 61)'(^e, 



and consequently 



/2 _j_ r 

 „ (l-e^sinM)*"^-'^' J„ 

 jquei 



X2 (76 

 o (l-j82eZ sin2 61) (l-e2 sin» ofi 



/32 ^2 d5 



(l-/32e2 sin2 0) (]-e2 sin^ ( 



-ir:^_T)jr' (l-e^sinM)V. 

 By making a similar transformation, we have, from (11), 



-r2) + (J2_yZ) tanS w > ^ , 

 ^ rfu) 



1 (i^-r^f r 



-f(i2_cZ) tan2 o) 



(l-/8'2e2 sin2 61) (l-e2 sin2 ( 



where 



* See my paper on " Poisson's Integral," J. A. S. B. 1888, vol. Ivi!, pt. ii, 

 pp. 100—106. 



