221 A. Mukhopaclliyay — Elliptic Functions and Mean Values. [No. 2, 

 so that the integrals become 



/ n su t dn j. 

 e^ + dtiH 



and 



/ fi sn j. on d ^, 

 i^ + dn^i 



respectively. 



It can easily be shown as follows that the result is expressible in 

 terms of circular and logarithmic functions only when the ellipsoid 

 degenerates into a prolate or an oblate spheroid. Thus, consider the 

 portion of the definite integral in (17) which is due to Rj and Eg ; then 



since 



this portion becomes 



Va («^-A2)^(X2-i2)^(X8-c2)2 I X2 J 



Assume 



d\. 



sin2 d ' 

 which gives 



6 cos ^ — c 



sin t 



b — c cos d 



sni t 



\c 



Substituting, we get 



Xd\ = - g-c) (6-e cos t 



siiiS^ 



'd9 



Bin 



e (-62 + 26c cos 61 -c2) , 



-a2cos2 6l + 26ccos6t+a2-62-c?'| ^ 



8 Jv/-a2c. 



a2cos2 6l + 26c coae+a^-h^-c^. d9 

 6d9 



?' cos^ 9 + 26c cos 9\a^-h'^- c^" 



