128 
MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
more difficult part of this research, when the accelerating forces urging the 
particles of the fluid, depend upon the very figure of equilibrium which is to 
be investigated. This must happen in fluids consisting of particles that 
mutually attract one another, if the attractive force acting upon a particle vary 
with the figure of the attracting matter. In this division of our subject, the 
law of attraction that prevails in nature being in reality the only one which it 
is of much importance to consider, will chiefly engage attention. 
Problem 2nd. — To determine the equilibrium of a homogeneous fluid entirely 
at liberty, the particles attracting one another with a force inversely 
proportional to the square of the distance, at the same time that they are 
urged by a centrifugal force caused by rotation about an axis. 
The fluid being supposed in equilibrium, the axis of rotation must pass 
through the centre of gravity of the mass. For, abstracting from any motion 
or force common to all the particles, that centre may be considered at rest 
and free from the action of any accelerating force ; and, as the attractive forces 
balance one another at that point, the centrifugal force must likewise vanish 
at the same point. 
Conceive three planes intersecting at right angles in the centre of gravity of 
the mass, one of them being perpendicular to the axis of rotation : let x, y, z 
represent the coordinates of a particle in the surface of the fluid, x being 
parallel to the same axis ; and put V for the sum of the quotients of all the 
molecules of the mass divided by their respective distances from the particle : 
then the attractive forces urging the particle inward in the directions of x, y, z, 
will be respectively equal to 
dV dV dV 
dx 3 dy ’ dz’ 
Further, if /’denote the centrifugal force at the distance unit from the axis of 
rotation, the action of the same force at the distance >Jy 2 -f- z 2 from the same 
axis will be f *Jy 2 + z 2 ; and the resolved parts of this force urging the par- 
ticle to move in the prolongations of y and z, will be fy and f z. Wherefore 
the total forces parallel to x, y, z, and tending to shorten these lines, are 
respectively, 
