118 
MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
initial and final pressures being the same, the impulses of the molecules in the 
whole circuit will balance one another. But in this case, the reasoning we 
have employed will not be exact, unless p, the algebraic expression of the 
pressure, be such a function as admits of only one value for any three given 
co-ordinates ; a restriction however, which, in every point of view, seems in- 
dispensable. 
6. The whole theory, it will readily appear from the foregoing investigations, 
is built on the assumption, That the hydrostatic pressure at every point of the 
fluid is the same function of the co-ordinates of the point. The accelerating 
forces are represented by the partial differential coefficients of the pressure ; 
and therefore they are likewise the same functions of the co-ordinates of their 
point of action in every part of the mass. The whole reasoning rests on these 
fundamental points ; and if the state of a fluid were such that they are not 
verified, the equations for determining the required figure could not be formed, 
and the equilibrium would be impossible. As the hydrostatic pressure is known 
only by means of the given accelerating forces, it seems most suitable to em- 
ploy the properties of the latter in laying down what is required for the equi- 
librium of a mass of fluid. It is necessary, and it is sufficient for the equili- 
brium of a homogeneous fluid, first, that the accelerating forces acting in the 
directions of the co-ordinates be, in every part of the mass, the same functions 
of the co-ordinates ; and, secondly, that these functions possess the conditions 
of integrability. When these two conditions are both fulfilled, the determina- 
tion of the figure of equilibrium is reduced to a question purely mathematical. 
For we can form the equation (1) which makes the accelerating forces balance 
the variation of pressure ; and, by integrating this equation, we obtain the hy- 
drostatic pressure, from which is deduced the equation of all those points at 
which there is no pressure, or in other words, the equation of all those parts 
of the outer surface which are at liberty. Nothing more is required for se- 
curing the permanence of the figure of the fluid, except that the pressures pro- 
pagated through the mass be either supported or mutually balance one another. 
The conditions for the equilibrium of a homogeneous fluid, as they are here 
laid down, do not enable us in all cases to form immediately the equation of 
the figure of equilibrium. If the particles attract or repel one another, the 
accelerating forces will, for the most part, vary as the fluid changes its form, 
