Hence the theorem.—The free surface of a liquid at rest under gravity is a horizontal plane. This is the characteristic distinguishing between a solid and a liquid; as, for instance, between land and water. The land has hills and valleys, but the surface of water at rest is a horizontal plane; and if disturbed the surface moves in waves.
9. Theorem.—In a homogeneous liquid at rest under gravity the pressure increases uniformly with the depth.
This is proved by taking the two points A and B in the same vertical line, and considering the equilibrium of the prism by resolving vertically. In this case the thrust at the lower end B must exceed the thrust at A, the upper end, by the weight of the prism of liquid; so that, denoting the cross section of the prism by α ft.2, the pressure at A and By by p0 and p ℔/ft.2, and by w the density of the liquid estimated in ℔/ft.3,
pα − p0α = wα·AB,
(1)
p = w·AB + p0.
(2)
Thus in water, where w = 62.4℔/ft.3, the pressure increases 62.4 ℔/ft.2, or 62.4 ÷ 144 = 0.433 ℔/in.2 for every additional foot of depth.
10. Theorem.—If two liquids of different density are resting in vessels in communication, the height of the free surface of such liquid above the surface of separation is inversely as the density.
For if the liquid of density σ rises to the height h and of density ρ to the height k, and p0 denotes the atmospheric pressure, the pressure in the liquid at the level of the surface of separation will be σh + p0 and ρk + p0, and these being equal we have