Hydraulics is the term applied to that collection of facts which describe the phenomena of fluids in motion. In treating of hydrostatics, we have shown that fluids press in all directions, and that the pressure is as the depth: hence if a small hole be made in the bottom or side of a vessel, the velocity with which the water will issue will depend on the pressure above it, that is on the depth of the hole. If a vessel be kept quite full of water, so that the height may remain fixed, and holes be made at various depths beneath the surface, it will be found that the velocity of the issuing fluid will be as the square root of the depth of the hole. Hence to produce the velocities 1, 2, 3, 4, the corresponding depths must be 1,4, 9, 16, which are the squares of the velocities. To produce a double velocity we see a fourfold pressure is required. To explain this, we must bear in mind that when the velocity is doubled, a double quantity of fluid must issue, and as this issues with a double velocity, the power requisite to produce this effect must be four times greater. In like manner three times the quantity of fluid issuing with three times the velocity, will require a power nine times greater, and so on.

This law, it will be seen, is coincident with that of bodies falling in free space by the power of gravity. To produce a double velocity, a fourfold height is required; to produce a threefold velocity, a ninefold height is required. Thus it appears if a particle of water were to fall unresisted from the surface of the fluid to the orifice, it would acquire the same velocity as the issuing water actually exhibits. If a small tube were attached to the orifice and turned upwards, the water would ascend to a height equal to its source, were it not for the resistance of the air, and the friction against the sides of the tube. The absolute velocity of the fluid cannot be ascertained by calculation alone, and must therefore depend partly upon experiment. In the annexed cut it will be seen that the particles of water, on account of the equality of pressure on all sides, will so far interfere with each other's motion, that the stream will be contracted a little below the orifice. This contracted part was called by Sir l. Newton, who first observed it, the vena contractu, or contracted vein; and its magnitude compared with that of the orifice, was found to be as one to the square root of two nearly, or as 1000 to 1414.

Now the velocity found by the rule we have before stated is the velocity at the vena contracta. But as the velocity of fluids in a channel varying in diameter is inversely as its sectional area, the velocity at the orifice is less than at the vena contracta in the proportion of 1 to 2, or as before stated, as 1000 to 1414. Hence it follows, that the velocity at the orifice is that which a body would acquire in falling through half the altitude of the fluid above the orifice. From this theorem we may easily calculate the quantity of water that would escape through a given orifice in any time. Let a cistern or other vessel be six feet high, and kept full of water, and it be required to ascertain what quantity of water would run out through a hole a quarter of an inch area, near the bottom, in ten minutes. By the laws of mechanics we find that the velocity acquired by a body falling through half the height, viz. three feet, is fourteen feet in a second; if, then, we multiply the area of the orifice, namely a quarter of an inch by 14, and then by 600, the number of seconds in ten minutes, we shall obtain 2100 cubic inches, or seven gallons and a half as the issuing quantity.

The quantity of water that issues through a hole in the bottom of a vessel may be varied by inserting small tubes in the holes. Thus it was found by Venturi that when a small tube was applied whose length was equal to twice the diameter of the hole, it discharged eighty-two quarts of water in one hundred seconds, while the hole without the pipe discharged but sixty-two quarts in the same time. If the pipe, instead of being level at its top with the bottom of the vessel, projects some distance within it, it diminishes the discharge to less than would occur with the simple hole. In the preceding view of the velocity of discharge we have considered the surface of the water as maintaining the same level; but in ascertaining the time in which a vessel would empty itself through a given hole, it must be evident that attention must be given to the varying depth of the fluid. If the fluid flow with a velocity of sixteen feet per second, when the vessel is full, it will flow with a velocity of eight only when the vessel has discharged three-fourths of its contents, that is, when the height is reduced to a quarter of its original altitude.

If it be considered that the pressure is continually diminishing with the flow of water, it will easily be conceived that the quantity of water that will flow out will be only one half of that which would be discharged if the vessel were kept full. In this case the surface would sink with a gradually retarded motion; and if the sides of the vessel were marked with a series of numbers representing the hours of the day, it would form the clepsydra, or water-clock of the ancients, the surface of the water being the index. The same law applies to the conveyance of water through valleys in pipes, as we have shown to exist in the issue of water from holes in the bottoms of vessels. If it were required to convey water across a valley of considerable depth, the pipe employed must have great strength to withstand the pressure arising from the height of its source. If a small hole were made in the pipe the height to which the water would ascend, would indicate the great pressure existing. In this way various fountains may be constructed. In conveying water from a higher to a lower level, when it is inconvenient to form a channel, the syphon may be employed. This consists of a bent tube, A, B, C, having a shorter and a longer leg.

The shorter leg is placed in the water to be removed, and the air being then drawn out through a tube communicating with the longer leg, or by means of a stop-cock, the water will rise in the shorter leg by the pressure of the external air, and going over the bend of the tube, will run in a continuous stream as long as the level of the water at A is lower than that at C. In the flowing of water through holes in the sides of vessels, the same proportions obtain as in the discharge through holes in the bottom. If A B represent a vessel kept full of water, and holes be made in the sides at a b c, the water will be found to spout to different distances, and yield different quantities according to the depth of the orifice. Thus the quantities of water that would flow from a and c would be as 1 to 2, because their depth are as 1 to 4; and from any other hole the quantity would still be as the square root of its depth. If a semicircle be imagined to stand with its diameter on one side of the vessel, and lines be drawn perpendicular to the diameter, as a d, b e, and of, these lines will show the proportionate distances to which the fluid will spout.

The vena contracta, in this case, being very near the vessel, the velocity of projection at this point must be considered as the true velocity; and this is equal to that acquired by a body falling through the whole height of the fluid above the hole. The curve c C described by the water, is a parabola, whose vertex is at c: and by a property of the parabola, B C, the distance to which the fluid spouts from B is equal to twice the square root of (Bc multiplied by c A) which is equal to twice c f. In a vacuum, therefore, double the lines ad, e b, e f, represent the distances to which the fluid would spout. If the water, instead of flowing through a very small hole, had to flow through a long slit, the velocity would differ at the top and bottom, and in this case, the point which may be taken as that of the mean velocity, is two-thirds of that at the lowest point. In the action of liquids, as in solids, a considerable quantity of power is constantly consumed in friction; hence the velocity of water through pipes, or jets is considerably diminished, as is also the motion of rivers, a circumstance which is essentially beneficial to navigation; for otherwise, the velocity of the water, continually increasing in its fall, would become so great as to be unmanageable.

For further information in this science, consult the articles, Pumps, Water-Wheels , and the various hydraulic machines