226 The Configurations of Rotating Liquid Masses [ch. yiii 
forces acting on the body of mass M, which we shall call the primary, may be 
derived from a potential 
discussed later, and will be found to be so small that the results now to be 
obtained are hardly affected. 
Omitting these terms, and combining the two potentials (205*1) and 
(206*1), we find that, apart from its own gravitation, the primary may be 
supposed to be acted on by a total field of force derived from a potential 
(206*2). 
Denoting this potential by V', the equations of equilibrium of an element 
of the primary are, as in § 188, 
where V is the potential of the primary, and these have the common integral 
Thus the primary will be in equilibrium if V + V' is constant over its 
surface. 
Since V' is quadratic in x, y and z, it is at once clear that ellipsoidal 
configurations of equilibrium are possible, as in the former problem. The 
surface 
will be a possible configuration of equilibrium if (cf. equation 189*7) 
— irypabc (J A x 2 + J B y 2 + J c z 2 — J) 
is constant over its surface. 
To make this expression constant over the surface, we must first remove 
the term in x by assigning to &> 2 the appropriate value 
special problem dealt with by Roche. In this simple case the gravitational 
We shall for the present be content to omit all terms beyond those 
written down. The correction required by the neglect of these terms will be 
dp dV dV' j 
7 T- = p ~—I - p -x— etc. 
dx P dx P dx 
- = V + V + cons. 
P 
+ W> x ( y - wfu) + IT ■- **’> + i '“* (*■ + y*) ( 206 ' 4 ) 
, v(M+M') 
0 ) = — 
R 3 
We next equate the expression, as in § 190, to 
(206*5).