22(5 A. Mukliopadliyay — Elliptic Functions and Mean Values. [No. 2, 

 If now, we assume 



Therefore 



a = a-a, /a = (r. — , y = crc, 

 wTiere cr is a constant to be suitably chosen presently, we have 



(ZS-27r coserf^ 



0 



= 7= —7 =rT + -i^^ Siu2 



As o- is an arbitrary quantity, we may, for the sake of symmetry, assume 

 1 



so that now 



and 



R 1 1 



f^S -. COS 0 d9 



a V^gg-c^ _ _ ^ 



277 ~ «, ■ 



6c 



Hence if d& be the superficial element of the ellipsoid whose axes are 

 reciprocal to those of the given ellipsoid, we have 



S+ — n = 7=^ sin B, 



which is the geometrical relation in question furnishing the meaning of 



n. 



§ 5. Canonical Form for Volume. 

 It may be observed that the expression for the common volume 

 furnished by (12) involves two definite integrals A, B, whose values as 

 given in (10) and (11) are not expressed in terms of known functions. 

 They may easily be reduced to the standard elliptic fo rms, but that would 

 only increase the difficulty of integration with respect to r. Thus, 

 from (10), we have 



