162 MR. J. H. JEANS ON THE CONFIGURATIONS 
boundary was given by an equation of the type of (6), the solution occurring as a 
series of powers of e. It is now necessary to extend the method so as to be able to 
write down the potential when the internal density is variable and given by 
equation (5). 
As a matter of convenience, of which the advantage will appear later, we shall 
permit P 0 to include terms of degree 2 as well as terms of degree 3 and higher. 
This is equivalent to regarding the mass under discussion as being arrived at by 
distortion from an ellipsoid of density, 
P = —( a —+ ^ + , ' .(7) 
the only restriction on the distortion being that the point of maximum density must 
remain the origin, and that the density at this point must remain equal to p 0 . But 
distortions which are themselves ellipsoidal are not prohibited, as they would be if 
second-degree terms were excluded from P 0 . 
7. Let q be a function of the density p , defined by 
r / = /AZfi..( 8 ) 
Po-v 
so that, as we pass from the centre to the boundary, q varies continuously from 
0 to 1. The surface of constant density p has for its equation 
which may be regarded as a distorted ellipsoid of semi-axes qa , qb, qc. The boundary 
now figures as the special case of q — 1. 
Let the potential of a homogeneous mass of unit density bounded by the surface (9) 
be denoted by V 0 (q) when evaluated at a point outside the surface, and by V, (q) 
when evaluated at a point inside the surface. 
The whole heterogeneous mass of density given by equation (5) may be regarded 
as split up into shells of constant density, and the potential of the whole will be 
found to be given by the formulae : 
Vo-^(l) + r v 0 (q)dp . 
J d 
. . . (10) 
v. = <rV.(l)+r V i (q)dp+\'\(q)dp, . . . 
J a J p 
• • • (ID 
the first formula giving the potential at a point entirely outside the mass, and the 
second formula giving the potential at a point inside the mass at which the 
density is p. 
