214 A. Mukliopadliyay — Elliptic Functions and Mean Values. [No. 2, 



we have to determine the average value of the volume commou to this 

 ellipsoid and the ooncentrio sphere 



a;2 + 2/2 + 2;2 = r3 (2) 

 which always intersects it. We have obviously two distinct cases, 

 according as four or two vertices of the ellipsoid are exterior to the 

 sphere : in the first case, we have 



a -7 b -/ r y c, 



and in the second case 



a y r y b y 6, 



so that the limits of r are, in the two cases, 



r=b > r = a ■) 



r = c 5 'r = b 5 



respectively. In the following investigation, we shall consider the two 



cases separately. 



§§ 2—6. First Case. 

 § 2. Expression for tJie Common Volume. 

 Suppose four vertices of the ellipsoid to be exterior to the sphere, 

 and let V be the portion of the common volume lying in the positive 

 octant; then if v be the portion of the sphere outside the ellipsoid in the 

 same octant, we have 



V=i7rr8-v (3). 

 If z', z" be the ordinates of the spherical and the ellipsoidal surface 

 respectively, corresponding to the same system of values of x and y, 

 we have 



where 



Hence 



- = ff { (r'—'-y'y^ -C .Z:..?;/ (4) 



Eliminating z between (1) and (2), wo have for the equation of the 

 curve of projection on the coordinate plane of xy 



(l-^).H(l-g),/=,.^-c« (5). 



For integrating z' dx dy, put 



sc = pcosu, 7/ = psino), 



