202 A. Mnkhoyiadhyay—ElUplic Functions and Mean Values. [No. 2, 



Consider now 



P 



Let 



so that, wlien 



,.3 _ 1% 



r=b, 4=0, 

 and r« = a^ sin^ + cos«f, 



rrfr=(a3-62) sin ^. cos f. df, 



dr = (a2 - 62) smg-coBtc^g _ 



(a2 sin2 i+h^ cos2 



Hence, 



P=(ai>-6i>) 2 ^- sin |. cos j. d j 



J p (a» sin' 1+6® cos' f )^ 



= J^^ d J (a^ sin^ f + 6* cos^ f)^ j 

 parts we have 



P = ( f (a" sin« f + 6^ cos» i)i ] ^ ^ 



Integrating by parts we have 



~So^ sin5 i + cos= f)^ (?^. 

 = ™- (0,4 sin" f + 6^ cos= 



■ 2 



Next consider 



