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MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
Problem 1st. — To determine the equilibrium of a homogeneous mass of fluid 
which is entirely at liberty, when the accelerating forces are known func- 
tions of the coordinates of their point of action. 
The equilibrium of a mass of fluid which is entirely at liberty, can depend 
only upon the action of such forces as tend to change the relative position of 
the particles with respect to one another. It is not affected by any motion 
common to all the particles, nor by any force which acts upon them all with 
the same intensity in the same direction ; the effect of such motion, or of such 
force, being to displace the centre of gravity of the whole mass without altering 
the relative situation of the particles. In estimating the accelerating forces 
upon which the figure of equilibrium will depend, we must therefore begin 
with reducing the centre of gravity, if it be in motion or urged by any force, 
to a state of relative rest ; which is accomplished by applying to every particle 
a force that would cause it to move with the same velocity as the centre of 
gravity, but in a contrary direction. In the investigation of this problem we 
may therefore suppose that the centre of gravity is at rest and undisturbed by 
the action of any accelerating force. 
Suppose now that a mass of homogeneous fluid entirely at liberty, is in equi- 
librium, and conceive three planes intersecting at right angles in the centre of 
gravity of the mass, to which planes the particles of the fluid are to be referred 
by rectangular coordinates. Let x, y, z , represent the coordinates of a particle, 
and having resolved the accelerating forces acting upon it into other forces 
that have their directions parallel to the coordinates, put X, Y, Z, for the sums 
of the resolved parts respectively parallel to x, y, z, and tending to shorten 
these lines. According to the hypothesis of this problem, the forces X, Y, Z, 
depend only upon the coordinates of their point of action ; and they are at 
every point the same functions of those coordinates. The equilibrium will 
therefore be impossible unless 
X d x + Y dy + Zc/z 
be an exact differential, this being necessary in order that the hydrostatic 
pressure be a function of three independent variables as the fundamental 
assumption of the theory demands. Let <p denote the integral, and p the 
