1889.] A. MnkhoTpadhjaj— UlHptic Functions and Mean Values. 219 

 Substituting from (14) and (16) in (13), we have 

 3(6-0) f M ^a&c-) , 



Bdr 



Jo 



/(X).X3(X2-c2)dA 



^ (a2-XZ)^(X2-i2)2(X3-c8)* 



\ =ah f 



Therefore, finally, 



3(?^jM r" imj^ 



where / (X) has the value defined in equation (5) ; this equation gives 

 the required mean value. 



To evaluate this definite integral, we notice that there are six 

 I terms in / (X), say 



r /(X) = P,+P, + Q, + Q,+R,+R,, 



!' where 



8 X? 



J3 



Now, 



4 X* 

 _c3 (X2-c2)^ 

 ^^-4 ~X~ 



R,=-sm 1^ 

 R.= -|sin-^. 



Pi X3 (?X 



(a2 - X2) * (X2 - 62) a (X2 - c2) * 

 ■6 



_56 XdX 



^ J" (a3_X2)^(X2-c2)i' 



= I T 1. [« = XZ. 



