AND THE FIGURE OF A HOMOGENEOUS PLANET. 
113 
It may be proper to remark, that although this equation is universally true, 
yet it is no new or independent condition of the equilibrium ; it is merely an 
inference from the general expression of the hydrostatic pressure. 
If we assume two points (a?, y, z) and (a + dx, y + dy, z + dz) indefi- 
nitely near one another in a part of the outer surface at liberty, we shall have, 
in consequence of the foregoing equation. 
d 
d x 
d <f> 
dy 
izdx + ^7 dy + Y~ z dz 
or, which is the same thing, 
XrfcT + Yrfy-f- Zrf* = 0 ; 
and if ds represent the distance of the two points, we obtain 
x£+y£+z£=o. 
ds 
d s 
dx dy dz 
ds ’ ds ’ ds 
Now 3-r, are the cosines of the angles which the directions of the 
forces make with the line d s ; wherefore the expression on the left side of the 
foregoing formula is the sum of the partial forces which act in the direction of 
ds ; and as this sum is equal to zero in all positions of the line ds round the 
point (a, y, z), the resultant of the forces produces no effect in the plane touch- 
ing the surface, and consequently its whole action is perpendicular to that 
plane. The nature of the case requires further, that the same resultant be 
directed towards the surface of the fluid. 
What has been deduced from the algebraic expressions is evident in another 
view. For, could we suppose that the resultant of the forces is not at every 
point perpendicular to the surface at liberty, it might be resolved into two 
partial forces, one acting in the tangent plane, and the other perpendicular to 
that plane ; and as the first force is opposed by no obstacle, it would cause the 
particles to move, which is contrary to the equilibrium. 
If we suppose that p is constant in the general formula of the hydrostatic 
pressure, we shall have an equation, 
<p = C—p, 
which is exactly similar to that of the surface at liberty, and which will deter- 
mine an interior surface at every point of which there is the same intensity of 
MDCCCXXXI. 
Q 
