212 A Mukliopadhyay — Elliptic Functions and Mean Values. [No. 2, 



to find the mean value of 4r\|/ ; so that, if o- be the average value re- 

 quired, we have 



4ir dr 



J -.a 



Dved in § 1, 



Now, from the formulas proved in § 1, we have 

 h (a^'-r^ 



sm i(/ = 



Therefore, 



Assume 



so that, when 



and we have 



r = a, 1^ = 0, 

 J- = 6, = 



a« sin2 <?) + 6^ cos^ 

 _ _ a^jg - a%) sin cos 0 ^fift 

 ~ (a2 sina 0 + cos^ 



Hence, by substitution, 



_ I 2 a2;,2 (a2 -?>g) ift sin <|> cos d^ 

 (a - 6) (7 - 4 J ^ ^.^g ^ _^ ^2 ^^^g 



= - 2a%^J'^ ^ </> d J (aZ sinS Z;^ cos^ 1>)~'^] 

 Integrating by parts, we have 



