82 Mr. Ivory on the Theory of the Figure of the Planets 
sure principles, by a process of reasoning not depending upon 
any dubious or intricate point of analysis. 
In treating of the figure of the planets there are three dif- 
ferent cases that principally engage attention. We may con- 
sider the equilibrium of a fluid mass that is homogeneous; or 
of one composed of strata varying in density according to any 
law; or we may suppose a solid nucleus wholly or partially 
covered with a fluid. Although the principles on which I 
proceed are equally applicable in every case, yet for the sake 
of brevity and simplicity I here confine myself to the first case 
only; namely, the equilibrium of a homogeneous mass of 
fluid. Such is the intimate connexion which binds together 
the different parts of the same theory, that if we can fairly 
overcome the difficulties which obstruct our progress in one 
case, every other case will readily be brought within our 
power. 
Now the principles by which Laplace has determined the 
equilibrium of a homogeneous fluid mass are these two: first, 
the direction of gravity must be every where perpendicular to 
the outer surface; secondly, the radius 7 of the spheroid must 
come under this formula, viz. 7 = a(1 + ay), y being a func- 
tion of the angles which determine the position of the radius, 
and « a small coefficient of which the square and higher 
powers are to be neglected. In the theory of Laplace, the 
equilibrium of a fluid of uniform density is a necessary conse- 
quence of the two conditions mentioned. Of these the first is 
entirely mathematical; the only purpose it can serve is to al- 
low the rejecting of certain quantities which would otherwise 
embarrass calculation; but it can in no respect contribute to 
make out the proof of the equilibrium, which must be deduced 
from hydrostatical principles alone. Whether there be an 
equilibrium. or not must depend entirely on the first condition. 
If that is sufficient, Laplace’s solution will be exact; otherwise 
we must conclude that it is defective, and we can consider it 
only as a method of calculation which accidentally leads to a 
result that we know to be true from other considerations. 
We have now then to inquire what are the conditions ne- 
cessary to the equilibrium of a homogeneous fluid. ‘The whole 
received doctrine on this head is contained in the single pro- 
position following. Conceive a homogeneous fluid contained 
within a continuous surface, and let x, y, z denote the rect- 
angular co-ordinates of a point in the surface drawn to three 
planes intersecting in the centre of gravity of the mass; then, 
$ denoting a function of x, y, z, if the equation of the outer 
surface be gia Cy; 
the fluid will be in equilibrio, if every molecule, whether si- 
tuated 
