Page:Elementary Text-book of Physics (Anthony, 1897).djvu/142

128 In an incompressible fluid or liquid the pressure at any point is proportional to its depth below the surface. For, the weight of a column of the liquid contained in a vertical cylinder, terminated by the free surface and by a horizontal cross-section containing the point, is manifestly proportional to the height of the cylinder; and this weight is sustained by the pressure on the lower end cross-section, which must therefore be proportional to the height of the cylinder.

If the height of the cylinder be $$h$$ and the area of its cross-section $$s$$, and if the density of the liquid be $$D$$, the weight of the column is $$Dshg$$. If $$p$$ represent the pressure at the base, the upward force on the base is $$ps$$; so that we have From the foregoing principles it is evident that a liquid contained in two communicating vessels of any shape whatever, will stand at the same level in both. If, however, a liquid like mercury be contained in the vessels, and if another liquid, like water, which does not mix with it, be poured into one of the vessels, the surface of separation will sink, and the free surface in the other vessel will rise to a certain point. If a horizontal plane be passed through the surface of separation between the two liquids, the pressures at all points of it within the liquids, in both vessels, will be the same. These pressures, which are due to the superincumbent columns of liquid in the two vessels, are given by $$Dgh$$ and $$D'gh'$$, and since they are equal, we have $$Dh = D'h'$$; that is, the heights of the two columns above the horizontal plane passing through the surface of separation are inversely as the densities of the liquids.

There is nothing in this demonstration which requires us to consider both the columns as liquid: one of them may be of any fluid, and equilibrium will obtain when the pressure exerted by that fluid on the surface of separation is equal to the pressure exerted by the column of liquid in the other vessel on the horizontal plane containing the surface of separation; so that, if we know the