AND THE FIGURE OF A HOMOGENEOUS PLANET. 
115 
This equation must hold at every point of the mass of fluid without any rela- 
tion being supposed between the variations, wherefore p must be a function of 
three independent variables ; and in consequence the foregoing equation im- 
plies the three separate equations following, viz. 
d JP__ v _ 
dx @ J 5 dy 
fY, 
It now appears that the conditions of integrability must be fulfilled, viz. 
rf.gX d.g Y d.gX. d.g Z d.g Y d.gTj 
dy dx 5 dz dx ’ dz dy 
and unless the forces possess the properties expressed by these equations, the 
equilibrium will be impossible. 
Without pursuing the investigation in all its generality, we shall confine our 
attention to the case in which 
'K.dx + Y dy Zi dz, 
is an exact differential ; a supposition that comprehends all the applications of 
the theory. If we represent the integral of the differential by <p, so that 
d <p = X d x + Y d y + Z d % ; 
and convert the variations of equation (3) into differentials, we shall obtain 
dp + f d <p = 0 ; 
and hence 
p = C -f o d <p. (4) 
From this we deduce the equation of those parts of the outer surface which are 
at liberty, by making p = 0 ; and that of a level surface, by assigning to p 
some constant value. And if we differentiate the same equation (4) on the 
supposition that p is invariable, we shall get 
%d<p = g(Kdx-\-Ydy-\-Zdz) = 0, 
which differential equation is common to the outer surface at liberty, and to 
all the interior level surfaces ; and from which we deduce by the like reason- 
ing as before, that all such surfaces are perpendicular to the resultant of the 
accelerating forces urging the particles contained in them. 
The quantity under the sign of integration in the formula, 
P~C -f §d(p, 
