This machine consists essentially of two communicating cylinders (fig. 1a), filled with liquid and closed by pistons. If a thrust P lb. is applied to one piston of area A ft.², it will be balanced by a thrust W lb. applied to the other piston of area B ft.², where p = P/A = W/B, (1) the pressure p of the liquid being supposed uniform; and, by making the ratio B/A sufficiently large, the mechanical advantage can be increased to any desired amount, and in the simplest manner possible, without the intervention of levers and machinery. Fig. 1b shows also a modern form of the hydraulic press, applied to the operation of covering an electric cable with a lead coating. 8. _Theorem._--In a fluid at rest under gravity the pressure is the same at any two points in the same horizontal plane; in other words, a surface of equal pressure is a horizontal plane. This is proved by taking any two points A and B at the same level, and considering the equilibrium of a thin prism of liquid AB, bounded by planes at A and B perpendicular to AB. As gravity and the fluid pressure on the sides of the prism act at right angles to AB, the equilibrium requires the equality of thrust on the ends A and B; and as the areas are equal, the pressure must be equal at A and B; and so the pressure is the same at all points in the same horizontal plane. If the fluid is a liquid, it can have a free surface without diffusing itself, as a gas would; and this free surface, being a surface of zero pressure, or more generally of uniform atmospheric pressure, will also be a surface of equal pressure, and therefore a horizontal plane. [Illustration: FIG. 1b.] 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 [alpha] ft.², the pressure at A and By by p0 and p lb./ft.², and by w the density of the liquid estimated in lb./ft.³, p[alpha] - p0[alpha] = w[alpha]·AB, (1) p = w·AB + p0. (2) Thus in water, where w = 62.4lb./ft.³, the pressure increases 62.4 lb./ft.², or 62.4 ÷ 144 = 0.433 lb./in.² for every additional foot of depth.