86 Mr. Ivory on the figure requisite to maintain the equilibrium 
by the eminent geometers we have mentioned, that the figure 
of the fluid mass is but little different from a sphere which 
is a restriction not essential to the problem, but introduced 
for the sake of overcoming some of the difficulties of the in- 
vestigation. In the following Paper, the figure of a homo- 
geneous fluid body, that revolves about an axis, and is in 
equilibrio by the attraction of its particles, is deduced by a 
direct analysis in which no arbitrary supposition is admitted. 
1 . It is necessary to begin this research, with laying down 
some general properties of the attractions of bodies ; and we 
cannot better accomplish this end, than by considering the 
function, which is the sum of all the molecules of a body 
divided by their respective distances from the attracted point. 
Conceive any material body to be divided into an indefinitely 
great number of molecules, one of which is represented by 
dm; and having drawn three planes intersecting at right 
angles within the body, let x, y, z, denote the co-ordinates 
that determine the position of d m, and a , b, c , those that de- 
termine the attracted point : then, if we put 
r = Vd+b 2 + c* 
f=V\a — xy+ (6 — y) a + (c — z) % ; 
r will be the distance of the attracted point from the origin of 
the co-ordinates, and f that of dm from the attracted point. 
Now let V ( r) = J' 
the sign of integration, extending to all the molecules of the 
body : and V (r) will be the function alluded to, and which 
we have to consider. 
It need not be mentioned that V(r) is not a function of r, 
but of the three co-ordinates a, 6, c ; or it is an abridged 
