230 A. Mukhopadliyay — Elliptic Functions and Mean Values. [No. 2, 



Hence these two surfaces have the same relation as the sphere and the 

 ellipsoid in the present case. The volume common to these two surfaces 

 may, therefore, be obtained from (12) by putting 



a=a', h = b', c=:c', r=l. 

 Hence, remembering that 



dx dy dz — ahc dx' dy' dz', 

 we infer that the volume common to the surfaces 

 a;2 + 2/2 + 22 = 5-2 



~.S »2 



when a -7 r 7 hV c 



is obtained from (12) by writing 



r r r , 



a-~, 0 = -, c = - , r = l, 



c 0 a 



and theia multiplying the result by ahc. 



Making these transformations in (10) and (11), we find, calling the 



now values of A, B, ^ B', ^gA', respectively, 



TT 



08 ,_ r2 gS ^ r^-(l^ sin2 o> + c^ co&i ^ ? 



9-8 ~J rS ( a2-(i2 eos2o)+c2 sin2 «)) 3 ^' ^ 



oS r 2 ^ r2(^2 cosg (o+c^ sin2 (o)-^)M ^ - 

 r3 ~J rS |a2(,ia cos2 iu+c2 siu2 a>)-t2c2 5 '^'^ — ^ 



Making those substitutions in (12) and multiplying the result by ahc, wo 

 get for the required volume 



V = i7rr3-io3A'+|a&c B'. 

 Hence, if M bo the average value requiredj we have 



M '^l 



8 



dr 



so that 



Ydr 



b 



