AND THE FIGURE OF A HOMOGENEOUS PLANET. 
133 
which are no other than the partial differentials of the equation, 
V (r) — V' (r) = constant. (4) 
This equation must hold at every point of every interior surface, such as abc-, 
and, as its differentials are separately equal to zero, it must not contain the 
coordinates of the surface. If such a figure can be induced on the mass of 
fluid as will possess the property expressed by equation (4), every particle of 
the mass will be urged by the principal forces alone, the equilibrium will be 
possible, and it will be determined in the very same manner as in the first 
problem. 
We have now obtained a mathematical property that distinguishes the figures 
with which the equilibrium is possible from all others. We have also, in an- 
other place*, investigated the figures that alone possess this property ; and it 
appears from what is there shown, that ABC can be no other but an ellipsoid, 
and that every interior surface, as abc, is similar to the outer surface, and simi- 
larly posited about G. 
Having demonstrated that the fluid in equilibrium must be an ellipsoid, it 
readily follows that the axis of rotation must be one of the three axes of the 
geometrical figure. For, as the axis of rotation passes through G, the centre 
of gravity, it is a diameter of the ellipsoid ; and the centrifugal force being 
evanescent at the extremities of this diameter in the surface of the fluid, the 
only force in action at those points is the attraction of the mass of matter. 
But the whole force urging every particle in the outer surface of the mass in 
equilibrium, is perpendicular to that surface ; wherefore, the attractive force 
of the ellipsoid is perpendicular to its surface at the extremities of the diameter 
about which the fluid revolves ; and as there are no points on the surface of 
that geometrical figure at which the attraction of its mass is perpendicular to 
its surface, except the extremities of its three axes, it follows that with one or 
other of these, the axis of rotation of the fluid in equilibrium must coincide. 
Let us now determine the relations between the axes of the ellipsoid and the 
centrifugal force. Of the three planes of the coordinates, one, which is per- 
pendicular to the axis of rotation, is a principal section of the ellipsoid ; and 
we may suppose that the other two coincide with the two remaining principal 
sections. We may therefore compute V (R) for a point in the surface ; and 
by substituting this value in the equation. 
* Phil. Trans, for 1824. 
