216 A Mukhopadhyay — Elliptic Functions and Mean Values. [No. 2, 



Therefore, finally, from (3) we get 



y = ~abo+ic^A- labc B (12) 



§ 3. Expression for the Mean Value. 

 We now proceed to calculate the mean value of the volume common 

 to the sphere and the ellipsoid. If M be the average value sought, 

 we have 



Ydr 



M 



f. 



b 



dr 



which gives 



^b 



^M.= I V dr 



— \Trabc 



whence 



3 (6-c) 



c 



To calculate 



j: 



Xb r'b /^h 



dr I ^dr-labc I B dr, 



B dr. (13) 



substitute in (10) 



b 



Adr, 



"t2cos«co + a2 sin^w' 

 so that 



, , . o,»/ o ION sin o) cos (D tZft) 



(62cOs2a) + a2sin2a))8 



d<i3= —ab — ■ 



A(a2-X2j2(A2-i2)a 



