150 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



104. When the ellipsoid is of the oblate or "planetary" 

 form, the appropriate coordinates are given by 



x k cos 6 sinh 77 = fc/zf, \ 



y = CT cos o>, = sr sin o>, V (1). 



where w = k sin 6 cosh 77 = k (1 - /* 2 )* ( 2 + 1)*. j 



Here f may range from to oo (or, in some applications from 

 oo through to + oo ), whilst fi lies between + 1. The quadrics 

 f= const., fi= const, are planetary ellipsoids, and hyperboloids of 

 revolution of one sheet, all having the common focal circle x = 0, 

 OT = &. As limiting forms we have the ellipsoid f = 0, which 

 coincides with the portion of the plane x = for which w < k, and 

 the hyperboloid //, = coinciding with the remaining portion of 

 this plane. 



With the same notation as before we find 



l)*^ (2), 



so that the equation of continuity becomes, by an investigation 

 similar to that of Art. 100, 



or 



i)* 



} d? 



This is of the same form as Art. 100 (4), with if in place of f, 

 and the same correspondence will of course run through the 

 subsequent formulae. 



In the case of symmetry about the axis we have the solutions 



and $ = P n (p) <?n (f ) (5), 



where 



L fn-z 



+ "^ K - 2 ^- 3) r-' + .4...(6). 



