Page:Elementary Text-book of Physics (Anthony, 1897).djvu/140

126 If a vessel filled with a fluid be fitted with a number of pistons of equal area $$A$$, and a force $$Ap$$ be applied to one of them, acting inwards, a pressure $$Ap$$ will act outwards upon the face of each of the pistons. These pressures may be balanced by a force applied to each piston. If $$n+1$$ be the number of the pistons, the outward pressure on $$n$$ of them, caused by the force applied to one, is $$npA$$.

The fluid will be in equilibrium when a pressure $$p$$ is acting on unit area of each piston. It is plain that the same reasoning will hold if the area of one of the pistons be $$A$$ and of another be $$nA$$. A pressure $$Ap$$ on the one will balance a pressure of $$nAp$$ on the other. This principle governs the action of the hydrostatic press.

113. Relations of Fluid Pressures due to Outside Forces.—If forces, such as gravitation, act on the mass of a fluid from without, Pascal's law no longer holds true. For, suppose the fluid to be acted on by gravity, and consider a cylinder of the fluid, the axis of which is vertical, and which is terminated by two normal cross-sections. The pressure on the cylindrical surface, being everywhere normal to it, has no effect in sustaining the weight of the cylinder. The weight is sustained wholly by the pressure on the lower cross-section, and must be equal to the difEerence between that pressure and the pressure on the upper cross-section. As the height of the cylinder may be made as small as we please, it appears that, in the limit, the pressure on the two cross-sections only differs by an inflnitesimal; that is, the pressure in a fluid acted on by outside forces is the same at one point for all directions, but varies continuously for different points.

If, in a fluid acted on by gravity, a surface be considered which is everywhere perpendicular to the lines of gravitational force, the pressure at every point in this surface is the same. To show this we draw a line in the surface between any two points of it, and construct around it as axis a cylinder terminated at the chosen points by end-surfaces drawn normal to the axis. The pressures on the cylindrical surface, being normal to it, occasion resultant