166 Compressible and N on- Homogeneous Masses [CH. vn 



then the general value of the density given by equation (426) becomes 



+ jJ + J+eP ) ............... (427), 



and the boundary, which is defined by the condition p o- has for its 

 equation 



,+f| + J-l+P. = ..................... (428). 



If eP is small, this represents a distorted ellipsoid. Now for a perfectly 

 incompressible mass, the boundaries of all stable configurations have been 

 seen to be spheroids and ellipsoids, and so are all included in equation (428) 

 with P = 0. Moreover the general argument of 142 144 has shewn that 

 the stable configurations of compressible masses can be derived from these 

 spheroidal or ellipsoidal configurations by continuous distortion. Thus it 

 appears that the boundaries of compressible rotating masses may be supposed 

 given by an equation of the form of (428) ; in this equation eP will be 

 small if the matter is only slightly compressible, but may become com- 

 parable with the other terms of the equation for highly compressible matter. 



A preliminary problem must accordingly be the determination of the 

 potential of a mass whose boundary is determined by equation (428), while 

 the density at any point a?, y, z in its interior is given by equation (427). 



The potential of a non-homogeneous distorted ellipsoid 

 167. Let q be a function of the density p, defined by 



<2 2 = ^^ .............................. (429). 



po~<r 



As we pass from the centre to the boundary, p will vary continuously 

 from p to G-, so that q will vary continuously from to 1. The surface 

 of constant density p has for its equation 



+ + + ^. = 9 2 ........................ (430), 



and this may be regarded as arrived at by distortion from an ellipsoid of 

 semi-axes qa, qb, qc. Equation (428) is a special case of (430), arrived at by 

 taking q = 1. 



In Chapter IV we found how to write down the potential of a dis- 

 torted ellipsoid such as that determined by equation (430), the density 

 being supposed uniform. Let the potential of a homogeneous mass of unit 

 density bounded by the surface (430) be denoted by F (q) when evaluated 

 at a point outside the surface, and by Vi(q) when evaluated at a point 

 inside the surface. 



