1889.] A. Mukliopadhyay — Elliptic Functions and Mean Values. 225 



In order that these may be reducible to logarithmic and trigonometric 

 functions, the expression under the radical must be a perfect square, the 

 condition for which is 



or (c^-a^){b^-a^) = 0 



therefore, cither 



(•2 = a2 



or = 



§ 4. Geometric Interpretation. 

 It is interesting to remark that the geometry of the ellipsoid fi- 

 nishes an interpretation of the quantity called in (22). First consider 

 the ellipsoid 



a:^ 



^ + |8^ + /-^' 



then if S be its superficial area, we have, by Dr. Jellett's theorem,* 

 S = 2./ + 2.a/J a-e^eW^ 



Jo i^-e 



where 



Hence, if tZ S be an element of the superficial area, we have 



M " 1 1 



Assume 

 Therefore 



ex = sin 9, cdx = cos 9. d9 

 rZS COB Ode 



L l-e'2sinZ5 ,^ 

 r a 0 



(l-f,sin..) 



d9 



which is an expression of the same form as dQ. 



* Hermathena, vol. iv, 18S3, p. 477. 



