1889.] A. Uu.'k'ho-pa.Ahjfiy— Elliptic Functions and Mean Values. 203 

 Let 



so that, when 



and 



h /a^ -r^Y 



r = a, '?> = 0, 

 r—l), <p 



2' 



an^ 



Hence, 



_ — g^) sin f cos <^ (^<^ 



(a'' sin^ <l> + cos^ ^)'' 

 g& (b' — a^) sin 1^ cos0 d<l> 



(g^ sin'' <?> + 6'» cos^ <p)^ 



"3 if sin <l> cos ^> f^^) 



(a^ sin^ </> + 6^ cos^ (p)i 



= _ i^s J^^ j sin^ I?. + cos» ~* j 

 Integrating by parts, we get 



} = - igSJB 



L (g^ sin" if + &2 cogS J ^ - 0 

 J„ (g» 



(g" sin^ i/> + &2 cos"^>)^ 



(g^ sin" 4- 6" cos" 

 Therefore, we have 



-2a6l (g 



" sin* ^ + &" cos" I) 



