126 
MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
at right angles, and the fluid presses perpendicularly and with the same inten- 
sity at every point of the lower surface which supports the stratum. What is 
here affirmed is true, however near the level surface be to the centre of gravity ; 
and as the accelerating forces urging the particles within the surface decrease 
without limit in approaching that centre, they may finally be regarded as eva- 
nescent when the internal body of fluid is no more than a drop occupying the 
centre of gravity. Wherefore, by taking the radius of the level surface small 
enough, the inclosed fluid may be considered free from any accelerating forces, 
and subject only to the external pressures ; and, these being perpendicular to 
the surface, and acting with the same intensity, the whole mass of fluid will 
be in equilibrium by the known laws of hydrostatics. 
It may be proper to add that the mass of fluid has no tendency to turn upon 
an axis. For no motion of this kind can be produced by the pressures propa- 
gated inward from the surface, the directions of which pass through the centre 
of gravity. Neither can the accelerating forces urging the particles, cause 
any such motion, these being wholly employed in counteracting the inequality 
of pressure. 
For the sake of illustrating the problem we have solved, we shall add one 
example, which is besides intimately connected with the principal subject of 
our research. 
Example . — To determine the figure of equilibrium of a homogeneous mass 
of fluid entirely at liberty, the particles being supposed to attract one another 
with a force directly proportional to the distance at the same time that they 
are urged by a centrifugal force caused by rotation about an axis. 
At first view the proposed problem may seem one in which the accelerating 
forces depend upon the figure of the fluid, since it is supposed that every par- 
ticle is attracted by every other. But, in the particular law of attraction 
assumed, the force which urges any particle is directed to the centre of gravity 
of the whole mass of matter, and is proportional to the distance from that 
point *. The hypothesis of the problem is therefore equivalent to the suppo- 
sition that the particles of the fluid are attracted to a fixt centre with a force 
proportional to the distance ; so that the accelerating forces are independent 
of the figure of the fluid. 
* Prin. Math. Lib. i. Prop. 88. 
