Page:Elementary Text-book of Physics (Anthony, 1897).djvu/141

§ 113] forces which are everywhere in the opposite direction to the gravitational force and make equilibrium with it. The cylinder being in equilibrium, by hypothesis, the forces on the end surfaces, which alone can produce movement in the direction of the axis, must also be equal, and the pressures on those surfaces are therefore equal. Surfaces of equal pressure are equipotential surfaces; in small masses of liquid they are horizontal planes; in larger masses, such as the oceans, they are curved so as to be always at right angles to the divergent lines of force.

The surface of separation between two fluids of different densities in a field in which the lines of gravitational force may be supposed parallel is a horizontal plane. For, take two points, $$a$$ and $$c$$, in the same horizontal plane in the lower fluid, and from them draw equal vertical lines terminated at the points $$b$$ and $$d$$, respectively, in the upper fluid. The horizontal planes containing $$a$$ and $$c$$, $$b$$ and $$d$$, respectively, are surfaces of equal pressure. Now with these lines as axes construct right cylinders with the same small radius and terminated by equal cross-sections in the upper and lower horizontal planes. The pressures on the cylindrical surfaces, being everywhere normal to them, will have no effect in sustaining the weights of these cylinders. Their weights are sustained by the difference in pressure between the upper and lower cross-sections, and, since these cross-sections are in surfaces of equal pressure, the difference of pressure is the same for both cylinders, and the weights of the cylinders are therefore equal. By the construction the cylinders contain portions of both the fluids, and since these fluids are of different densities the weights in the cylinders can only be the same when each cylinder contains the same quantity of each fluid, that is, when the surface of separation between the fluids is parallel with the planes which contain the end cross-sections. The surface of separation is therefore also a horizontal plane. This theorem may be extended so as to prove that the surface of separation between two fluids in any gravitational field is at right angles to the lines of gravitational force, or is an equipotential surface.