208 A. Mukliopadliyay — Elliptic Functions and Mean Values. [No. 

 Substituting, therefore, in the equation 



we have 



(a-6)n = -a6 I — — — j ^ dr. 



(a*-9-») = (r»-6=)tan='6l, 



so that, when 



r=a, 6-0 



r = h, e = l, 



and we have also the relations 



a» + - 2}-' = - a') cos 25, 

 a' + &2 _ ^2 _ ^2 sin' 0 + 6^ cos^^ 5, 

 a» -r' = (a^-&') sin=» (9, 

 (a^ - Z;'J cos^ 5, 

 dr — (^^ ~ s i^i 5 cos 5 

 (u^ cos^ 5+6* siii^ 5)^ 



Making these substitutions, we have 

 (a-6)n= ' ^ aS (a° -6") cos2gti 



cos' ^ + fc* sin^ 61)2 (a" sin" 6+6* cos' 



" _ (1-2 sin° ff) d6 



a6 (a + 6) J ^ (a' sin' 61 + 6= cos' 6) (a' cos' 9 + 6^ sin' 6)^ 



dO 



_2^i sin' g 



(a' sin' 5 + 62 cos' 5)(a2 cos' 5+6' sin' 



sin' 5 de 



(a' sin' 5 + 6' cos' 5) (a' cos' 5+6' sin' 



Putting 



cos^ 5=1- sin* 5, n = 7,2 = 



we get 



