c 85 : 
V. On the figure requisite to maintain the equilibrium ofi a homo- 
geneous fluid mass that revolves upon an axis. By James 
Ivory, A. M. F. R. S. 
Read December 18, 1824. 
The theory of the figure of the earth, as delivered in the 
Philosophise Naturalis Principia Mathematical is liable to some 
objections. In determining the ratio of the axes, the illustri- 
ous author assumes that the terrestrial meridian is an ellipse, 
having the greatest diameter in the plane of the equator. 
M’Laurin afterwards proved, by a most elegant synthetic 
process of reasoning, that a homogeneous fluid body, pos- 
sessed of such a figure as Newton supposed, will fulfil all 
the conditions of equilibrium arising from the attraction of 
the particles, and a centrifugal force of rotation. In this 
manner the assumption of Newton was verified ; but the 
theory was still left imperfect, since it is necessary to deter- 
mine, by a direct investigation, all the figures of a fluid mass 
that are consistent with the laws of equilibrium, rather than 
to show that the same laws will be fulfilled in particular in- 
stances. We are indebted to Legendre for the first demon- 
stration that a homogeneous fluid body, revolving about an 
axis, cannot be in equilibrio by the attraction of its particles, 
unless it have the figure of an oblate elliptical spheroid. The 
researches of Legendre were rendered more general by 
Laplace, who gave a complete theory of the figure of rhe 
planets, distinguished by that depth and elegance which is so 
much admired in all his writings. It is assumed, however 
