OF ROTATING COMPRESSIBLE MASSES. 
L61 
powers of the parameter e. The boundaries may accordingdy be regarded as distorted 
ellipsoids. 
In a previous paper* I showed how to obtain the potential of a homogeneous mass, 
expanded in powers of a parameter which measured its divergence from the ellipsoidal 
shape. As a preliminary to the present investigation, we must examine the 
corresponding problem when the mass is not homogeneous. 
Potential of a Non-homogeneous Distorted Ellipsoid. 
6. We shall assume, as being adequate for the present problem, that it is possible 
to expand the density in the form 
P = Po—p-2~pi—Pi~ •••>.(I) 
where p 0 is the density at the origin, which is taken to be the point of maximum 
density, and p 2 , p 3 , p 4 , ..., are functions of x, y, z of degrees 2, 3, 4 .... respectively. 
No terms of degree unity occur because, from our choice of origin, the first 
differential coefficients of p vanish at the origin. The value of p 2 is 
P^ 
= -flas 
0.T 2 
txy 
ex ey 
the differential coefficients all being evaluated at the origin. Our choice of origin 
has been such that p 2 is necessarily positive for all values of x, y and 2 , so that by a 
suitable choice of direction of axes, it must be possible to express p 2 in the form 
Let us further put 
P-S + P4+--- — (po — <r) ePo, 
where e is a numerical quantity which may for the present be left undefined, but will 
ultimately be taken to be the parameter defined by equation (3), and P 0 is a function 
of x, y and 2 . Then the general value of p, as given by equation (4), becomes 
and the boundary, defined by p = a-, has for its equation 
x 
2 
a 
V 2 T, T 
+ 7-5 + ■ 9 + ePri — 1 • 
l«) 
When e is not too great this is a distorted ellipsoid. In the paper already referred 
to, I showed how to write down the potential of a homogeneous mass whose 
* 1 Phil. Trans.,’ A, vol. 215, p. 27. 
