112 
MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
and, if we substitute this value of Sjp in the formula (1), the independence of 
the variations will require these three separate equations, 
dp _ v 
dx ~ ~ A 
d JP _ v 
dy~~ Y 
iP_ __ r, 
dz~ ~ 
From this it appears that the algebraic expressions of the forces are not en- 
tirely arbitrary ; for they must be equal to the partial differential coefficients 
of a function of three independent variables. By differentiating we shall 
readily obtain the following equations which do not contain the function p, 
viz. 
dX _ dY dX_dZ dY _d_Z 
dy dx 3 dz dx’ dz dy ' 
Unless the forces possess these properties, which are the well-known conditions 
of integrability, the equation (1) will not hold in all parts of the mass of fluid, 
and the equilibrium will be impossible. But in the physical questions that 
actually occur, the forces of nature being either attractions or repulsions di- 
rected to fixt centres, and proportional to certain functions of the distances 
from those centres, they necessarily fulfil the conditions of integrability. 
The whole of what has been said is succinctly expressed by the two follow- 
ing equations, 
<p =y^(X dx + Y dy + Z cl z ), 
p == C - <p, 
Here <p represents a function of three independent variables x, y, z without any 
arbitrary quantity ; the constant C required by the integration is necessary 
only in the expression of p. 
3. The hydrostatic pressure at every point of the mass of fluid in equili- 
brium, is expressed by the second of the equations (2), viz. 
p = C-<p. 
But at all those parts of the outer surface of the fluid which are unconfined 
and entirely at liberty, there is no pressure ; wherefore we have, for the equa- 
tion of all such surfaces, 
<p = C. 
