144* Mr. Ivory on the figure requisite to maintain the equilibrium 
same density and the same velocity of rotation, because both 
the quantities increase as the radius of the sphere. Hence, in 
a planet of small oblateness, the value of q to the degree of 
approximation mentioned, is equal to the proportion of the 
centrifugal force to the gravitation at the equator ; and the 
proportion of 1 to 1 + i <7 is equal to that of the polar axis 
to the equatorial diameter. 
8. In the determination of the equilibrium of a homogene- 
ous fluid mass investigated in this Paper, two conditions are 
found necessary when the particles are endowed with attrac- 
tive powers ; whereas, in the usual solution of the problem 
one only is deemed sufficient, namely, that contained in equa- 
tion ( A ) , which expresses that the resultant of the accele- 
rating forces acting upon the particles in the outer surface 
shall be every where perpendicular to that surface. It is 
extremely remarkable that, of the two conditions, the one 
which is usually omitted, alone and without reference to the 
other, ascertains the kind of the figures of equilibrium. 
M’Laurin first proved synthetically that the ellipsoid, 
whatever be the degree of oblateness, fulfils all the conditions 
requisite for maintaining the equilibrium of a homogeneous 
fluid mass that revolves about an axis. If therefore the equa- 
tion (A) were alone sufficient for the equilibrium, the ellip- 
soid must be deducible from it, not in particular suppositions 
and approximately , but generally, and by an accurate process 
of reasoning. But this has not been accomplished, nor even 
attempted, by any geometer. No application has hitherto 
been made of the hydrostatical theory, except in the case of 
spheroids little different from spheres. 
If a homogeneous fluid of a spherical form at rest, and 
