141. ELLIPSOID WITH UNEQUAL AXES 



For the general ellipsoid with equation 



2 2 2 



x^ y^ z^ 



— + + — =1, [141a] 



a^ b^ c^ 



the appropriate ellipsoidal coordinates A^, A , A, are defined in terms of x, y, z as the three 

 roots of a cubic in A, which can be written 



2 2 2 



x^ y^ z^ 



+ ^— + = 1. [141b] 



a^ + k S^ + A c^+A 



The use of these orthogonal coordinates in potential problems involves special functions 

 known as Lame' functions, and no further details will be given here. 



For translation of the ellipsoid through fluid at rest at infinity, parallel to one of its 

 axes, which will be taken as the a-axis without regard to its relative magnitude, the kinetic 

 energy of the fluid is found to be 



2-«„ 3 



nabcpV'^, ■ ■ [141c] 



r dx 



!«„ = abc I 



1 (a2 + A)3/2(j2^;^)l 



[141d] 



{a^ + xy"- {b^ + Xy'^ (c^ + A) 



/2 .^2^^x1/2 



The definite integral can be expressed in terms of elliptic integrals, which are tabulated; 

 see Reference 3 or Reference 235, as listed later, where a>b>c. 



For rotation at angular velocity w about an axis, here taken as the a-axis, 



T = ;; 77 abc p w^, [141e] 



2(62-c2) + (62 + ^2) (^Q-y^) 15 



^„ = abc f — , [I4in 



° ^ (a2 + A)i/2(62 + A)3/2(c2 + A)l/2 



373 



