1889.] A. Mukhopadhyay — Elliptic Functions and Mean Values. 221 

 Again, 



X8 d\ 



X 



JL 



^ Jl (a2y-l)2(l_&2i,)i L'' 



-~8^- (21) 

 In order to evaluate the parts involving Rj, Rg, let us first assume 



and substitute 



wliicli give 



(o2-X2)2(X2-62)2(X2-c2)^ 



X2 = a2cos2,(, + 52 sin^^ 

 X X= — (a^ — 6^) cos ^ sin i(/ tiliii 

 a^-X^ = {a^-l^) sm^^ 

 X2-&2=(a2-i2) cos2.(,, 



a ^ cos^ ij/ + J2 sin2 

 ctn= — — — 



(a^ cos2 ■(, + 62 sin2 - c^) 2 

 c2 + J (a2-c2) -(a2-i2) sinS ^ j 



J (a2 - c2) - (o2 - i2) sinZ J 2 



= ^~ — T- (a^-c2)^(l-/t2sin3^)l 



(a2_cS)2 (1-A:2 sin2^)2 



where 



* -a^-c^- 



Hence we have 



n= ^_F(^, +)-(aZ-cZ)iE(/f, I), (22) 



