AND THE FIGURE OF A HOMOGENEOUS PLANET. 123 
hydrostatic pressure at the point (x, y, z ,) : the equations that determine the 
equilibrium will be these two *, 
<p = f (X dx + Y dy + Z dz), 
p = C — <p. j 
If we make p — 0, we shall obtain the equation of the outer surface of the 
fluid, viz. 
<p = C. 
The differential equation. 
dp 
d<p 
dp 
cu dx + ry d y + Tz dz = o 
or which is the same, 
Y dy + Z d z = 0, 
is common to the outer surface and to all the interior level surfaces at every 
point of which there is the same intensity of pressure ; and it shows that the 
resultant of the accelerating forces is perpendicular to all such surfaces ■f*. 
The figure of the fluid being determined, it remains to inquire whether the 
equilibrium is secured. By varying the coordinates in the formula for p, we 
obtain 
which equation proves that, if a particle be moved from its place a very little 
in any direction, the variation of the intensity of pressure is equal and opposite 
to the action of the accelerating forces. A particle has therefore no tendency 
to move from inequality of pressure. But we must not from this hastily con- 
clude that there is no cause tending to change the figure of the fluid. For, as 
in the simple case of a fluid contained in a vessel, the equilibrium requires not 
only that the accelerating forces balance the inequality of pressure, but like- 
wise that the total pressures tending outward at the boundaries of the mass, be 
supported by the sides of the vessel ; so in the problem under consideration, 
there being no external support, the figure of the fluid will not be permanent 
* Equation (2) § 2. 
R 2 
f Equation (2) § 3. 
