498 
MR. HENNESSY’S RESEARCHES IN TERRESTRIAL PHYSICS. 
figure which the forces acting on its different parts would ultimately produce with- 
out any knowledge of the peculiar physical nature of the fluid. It appears to be 
sufficiently demonstrated, that the figure which such a fluid mass would, assume in 
virtue of the mutual attractions of its particles and of centrifugal force would be an 
ellipsoid of revolution. It is superfluous therefore to enter upon that portion of the 
subject. Assuming then the figure of the mass to be an ellipsoid of revolution, 
I shall endeavour to obtain general expressions for its ellipticity, and for the variation 
of gravity at its surface. 
From what has been stated in the last paragraph of art. 2, it appears to be unne- 
cessary to transform equation (1.) into the equation of a similar form which the con- 
ditions of the present problem would require, and from this cause the whole investi- 
gation may be considerably abridged. 
Let the origin of the coordinates be at the centre of gravity of the fluid mass, let 
r represent the radius vector drawn from this origin to any point in any surface of 
equilibrium of the fluid, let 6 represent the angle formed between this radius and the 
axis of or axis of rotation, let co represent the angle formed by the plane which 
passes through the radius r and the axis of rotation with the plane of zx, then 
2 = rcos^, ^=r sin ^ cos iy, ;^=r sin ^sina;. 
Let J represent in this case the fourth term at the left side of the integral of (1.), and 
let f represent the centrifugal force of a point situated at distance unity from the axis, 
hence 
If it be assumed that the polar and equatorial axes of the spheroid always differ 
but very little, we shall have for the determination at any period the centrifugal 
force at the equator. 
a'P’ 
( 3 .) 
where represents the semipolar axis at that period, and a' the same semiaxis at 
present,/'' being the centrifugal force at the equator at present. Let g represent the 
density at any point of a surface of equilibrium, the polar radius of which is a, and 
the ellipticity e, then as the figures of the mass and of its surfaces of equilibrium differ 
so little from the spherical form, r may be supposed nearly equal to a, and hence for 
the determination of e we shall have the equation* 
where U® is a function of the coordinates, (Ba constant, and [jb— cos 6. If E represent 
the ellipticity, and ^ the density at the exterior surface, it follows that 
E= 
AE 
{5lEifga^da + fa^^dg — Eg') 
( 4 .) 
* Mecanique Celeste, tom. ii. p. 88. 
