326 Mr. Ivory on the figure requisite to maintain the equilibrium 
geneous fluid mass in equilibrio, namely, that all the level 
surfaces are similar to the outer one. 
Again, since R and R' ; are always in the same proportion 
to one another, O — O, will be independent of a, b, c, if we 
make r disappear in Q. Now, by expanding the expression 
of O, and equating the co-efficients of the several powers of 
r to zero, we get, 
Q=fJ 
o =// R'. C^d^dw' 
. «=//— V '" 1 
0 =//- c< . V^ 
and generally. 
(?) 
- /■ f c dy! d', 
J R ,? — 2 
In the first place, all these equations are satisfied if we suppose 
R' constant ; that is, if the figure of the fluid be a sphere. But 
the supposition of a sphere is inconsistent with the equation 
(A), unless « be evanescent. Wherefore, a homogeneous fluid 
body of a spherical figure cannot be in equilibrio by the 
attraction of its particles, unless it have no rotatory motion. 
Again, it follows from what has been shown, that R' is a 
function of y,Vi — y n . Sin -ar, V i — y n . Cos ; y being 
the cosine of an arc 6' reckoned from the pole of a great cir- 
cle on the sphere, and t*' the angle between 0' and a given 
great circle passing through the same pole. Now if we sup- 
pose that R ; is an even function, or that it contains only the 
squares and the combinations of the squares of y n V i — y 2 . 
Sin -ar', V ! — y n . Cos sr' ; the values of R', which are always 
