The conditions of floatage and of stability of a body floating in a liquid are of great im portance. A coating body displaces a certain quantity of the liquid, and the weight of the solid body is equal to the weight of the liquid is that of the circular board, and whose height is the difference of the heights of the under surface of the top circular board and of the free surface of the water in the small upright tube. When shown in this form the principle here employed is often called the hydrostatic paradox, on account of the very great pressure that a very small quantity of water may be made to produce.
It is a well-known principle that liquids tend to find their own level. Thus, in the accom panying figure, showing a series of connected vessels, the liquid is seen to stand at the same height in the principal vessel and in the vari ously shaped tubes communicating with it, while from the short, narrow mouthed tube it spouts up to nearly the level of the water in the prin cipal vessel.
Archimedes is credited with discovering the principle that a body immersed in a liquid dis places its volume of the liquid; and that it is buoyed up by a pressure equal to the weight of liquid it displaces. This principle is illustrated in the hydrometer '(q.v.). When a solid is im mersed either partially or wholly in a liquid, a portion of the liquid is displaced. The solid is at the same time pressed at every point by the liquid, the pressure being always perpendicular that is displaced by it. To calculate how much of the body is submerged, and how much floats above the liquid, it is only necessary to consider what volume of the liquid would be equal in weight to the weight of the floating body. For example, the specific gravity of ice is about nine tenths of that of ordinary sea-water. Hence 9 cubic feet of sea-water weigh as much as 10 cubic feet of ice. Thus in an iceberg nine tenths of the ice is under water, and one-tenth is above the surface. In ships and other float ing bodies the stability depends on the form of the body. A sphere of wood floating in water is indifferent as to position. The slight est force is sufficient to overturn it from any given position or to set it rotating in the water. With a ship or other body that must float with one side upward, the stability is quite as impor tant as the floating power. The accompanying figure 4 illustrates the conditions of stability. When a solid body is slightly displaced from its ordinary position of equilibrium, the forces that act upon it are seen to be two-fold. First, there is the force of gravity on the solid acting verti cally downward, which, if c be the centre of gravity, may be considered to act downward through that point; and secondly, there is the resultant of the upward pressures of the vari ous portions of the liquid, which, if o be the centre of these upward parallel forces, or centre of buoyancy, may be considered as equivalent to a single force acting vertically upward through that point. In the figure these two
parallel forces are seen to form a mechanical couple whose tendency is to right the boat, and bring it back into its ordinary floating position. The metacentre M is a changing centre, representing the centre of the two forces of gravity and buoyancy. The higher the meta centre above the centre of gravity the more stable is the vessel.
Let A B in the figure be a line drawn through the point G, the centre of gravity of the floating body, and H the centre of the figure of liquid displaced (also the centre of buoyancy) when the body is floating with A a vertical. Let the body be then slightly displaced, and let o be the new position of the centre of the figure of the displaced liquid, and let rt be the point in which A B is cut by a vertical line through o: id is the metacentre which will be different or slightly different for every list. If the meta centre is above a then the equilibrium is stable; if it is below a the equilibrium is un stable, and the body being slightly displaced, it tends to fall farther and farther from its position of equilibrium. That a vessel may be stable and not overturn, it is apparent that the centre of gravity must be kept considerably be low the metacentre. This is why vessels load the keels at times with lead, and why vessels without freight in the bottom of the hold, take in ballast.
Among the instruments and machines founded on the hydrostatic principles here laid down are the barometer, the siphon, the hy drostatic press, and the hydrometer (qq.v.).
The engineering problems involved in the stability of dams, water gates and reservoir walls are those of hydrostatics. The pressure of water behind such structures exerts a con stant force toward overturning them, due to its tendency under the influence of gravitation to seek the lowest level. To counteract or coun terbalance this force is placed the weight of the dam or wall —in other words the larger at traction which the earth has upon the materials of which they are built than it has for the water. The higher the body of water behind a dam the greater the impulse to run down hill, and hence the greater push against the dam which stands in its way. The form of solid which is most difficult to overturn by pressing against one side of it is that of triangular cross section, and this 'is the one used by engineers for dams and reservoir walls. These trian gular bodies are built with broad bases, and one °toe of the triangle pushes up-stream under the water of the pond or lake, which helps in part to hold it down with its weight. Hydrostatic pressures exist also in all water supply pipes, tending to burst them apart from within. The greater the head of the water supply the greater the hydrostatic pressure, and the stronger the water-pipes must be made to hold it securely.