130 MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
forces which urge the molecule inward in the respective directions of x, y, z, 
viz. 
d\{r) 
d x ’ 
and if these forces be multiplied, each by the variation of its direction, the sum 
of the products will be the variation of the intensity of pressure, which is equal 
and opposite to their action, according to equation (1) of the general theory; 
thus, we have, 
V d . \ (r) v d . V (r) ^ d.V(r)^ s, * t s \ ^ /n\ 
*P- -d7~ lx ~~d^~ l y ^Sz-/(;?/^ + 3 5z) = 0; (2) 
and, as this equation is true at every point of the mass, we further obtain 
p =V (r) + y ( y 2 + z 2 ) — C, 
( 3 ) 
the constant being the same as in the equation (1) of the upper surface, be- 
cause the two equations must coincide when the interior molecule ascends to 
the surface. It must be observed that p represents the intensity of pressure 
caused by the principal forces alone, and not the whole pressure upon the 
molecule, if besides these forces there exist other causes of pressure in the in- 
terior parts. 
From the nature of the function V or V (r), it has its maximum at the centre 
of gravity of the mass, or when r = 0 ; for at that point we have the equations 
d . V (r) _ 
d x 
0 , 
d.V(r) 
dy 
= 0 , 
d.\ (r) 
d z 
= 0 , 
because the attractive forces balance one another. While r, without any 
change in its direction, increases to be equal to R, V (r) continually decreases. 
In whatever direction the radius R be drawn to the surface, there is always a 
point in it, the coordinates of which will satisfy equation (3), supposing that 
p has any assigned value less than the maximum which takes place at the 
centre of gravity. All the points in which p has the same given value will 
form an interior surface, returning into itself and pressed with equal intensity 
by the action of the principal forces upon the exterior fluid. Such interior 
surfaces are likewise perpendicular to the resultant of the principal forces 
