Hydrostatics explains the pressure and equilibrium of liquids, or of what have been generally termed inelastic fluids. A fluid is a body whose parts are put into motion among one another by the slightest force, and which return to their former state as soon as the impressed force is removed. Fluids have been divided into elastic and inelastic; but recent experiments have proved that there is no fluid that may not be compressed by a sufficient force, and that will not return to its former state when the compressing force is withdrawn. The terms may, however, without much inconvenience, be retained, as the difference in compressibility is so great, that a sufficient distinction obtains. Thus the same power that would reduce air to one half of its former dimensions would effect a compression not exceeding one twenty-thousandth of its bulk in water. Hydrostatics, then, is concerned with those fluids, the compressibility of which may, for most practical purposes, be considered as inappreciable. The laws of this science are generally proved by experiments on water, as a fluid that is most plentiful, and most easily adapted to the purpose. The principal and most important propositions with respect to pressure are, - 1. That fluids press equally in all directions. 2.
That in fluids of equal density the pressure is proportional to the perpendicular distance from the surface. That fluids press equally in all directions is seen in a variety of cases, and may therefore be easily made evident. If a cylindrical tin vessel have two holes of equal size, one in the bottom and the other in the side as near the bottom as may be, it will be found that each of these holes will permit the contents of the vessel to run out in equal times. Now as the velocity of the running water is dependent upon the pressure, it is manifest that the pressure upon the side, or lateral pressure, is equal to that on the bottom, or downward pressure. That the upward pressure is also equal is inferred from the fact, that water poured into one of the legs of a bent tube will rise to an equal height in the other leg. This upward pressure is a feature that distinguishes fluids from solids, and is due to the extreme mobility of its particles, owing to the repulsive energy which is exerted when the least compressing force is applied. We have next to show that the pressure upon any particle in a fluid is proportional to its perpendicular depth.
In the annexed diagram, if x be a particle of water, the pressure exercised upon it will be proportional to the depth B x; for if a hole were made at x, the fluid (making allowance for the resistance of the air) would spout up to A e. In like manner the pressure at C is equal to the perpendicular depth B C; and as every particle on the bottom is at an equal depth, the whole pressure will be equal to that produced by a column filling the whole space BCDe. From this it appears that the pressure on the bottom of a vessel is equal to the area of the base multiplied by its perpendicular height. In vessels having equal bases the pressure will be proportional to the height, and in vessels of equal depths the pressure will be as the area of their bases, and this without any regard to the quantity of fluid employed. This has given rise to the hydrostatic paradox - "that any given quantity of water, however small, may be made to balance any other quantity, however large." Also the hydrostatic bellows depend on the same principle. In the cut, A and B are two circular boards connected by leather after the manner of a pair of bellows, so as to be water-tight. A tube, C D, is made to communicate with the interior.
If a small quantity of water be now poured in so as to separate the boards, and a number of heavy weights be placed upon the upper board, the water in the tube C D will be seen to rise till it balance the weights placed upon A. If the quantity of water in the tube, above the level A g, be noticed, it will be found to be so much less than the weights upon A, as the area of the bore of the tube falls short of the area of the board A. To make the subject more evident, let us suppose the sectional area of the tube C D to be half a square inch, and that of the lower board of the bellows to be one square foot, or 288 times greater; it will then be found that one pound of water in the tube C D will support a weight of 288 pounds on the board A g. In a similar way, a long narrow tube may be insulted perpendicularly into a cask or other vessel; after the vessel has been filled, a few ounces of water poured into the tube will burst it. If a fissure in a rock should communicate with an internal cavity of considerable magnitude, situated at some depth below the top of the fissure, and filled with water, the pressure may be so enormous as to burst the rock.
The same effect may be produced by rain falling into, and filling a long slender chink that may have been left in the walls of a building; whether the chink is of equal diameter throughout, or vary in its size, and whether it be straight or crooked, provided it be watertight, so as to get full of rain, the effect will be the same, the pressure being always proportionate to the perpendicular height. This principle has been ingeniously applied by the late Mr. Bramah, in what is termed the Hydrostatic Press, a machine by which an almost incredible force may be obtained in a very small compass. (See Bramah's Press). From the equal pressure in all directions arises the tendency in fluids to find their own level, so that the surface of every fluid at rest is horizontal; and for a similar reason, if two fluids be in the same vessel, and do not mix, their common surface will be parallel to the horizon. From what has been stated on the nature of fluid pressure, it will be easy to calculate the pressure on any horizontal surface. Thus, if a cubical vessel be filled with water, the pressure on the base will be equal to the weight of the fluid. The same pressure will obtain if the vessel be of a conical form, provided the area of the base and the height be the same.