10. _Theorem._--If two liquids of different density are resting in vessels in communication, the height of the free surface of such liquid above the surface of separation is inversely as the density. For if the liquid of density [sigma] rises to the height h and of density [rho] to the height k, and p0 denotes the atmospheric pressure, the pressure in the liquid at the level of the surface of separation will be [sigma]h + p0 and [rho]k + p0, and these being equal we have [sigma]h = [rho]k. (1) The principle is illustrated in the article BAROMETER, where a column of mercury of density [sigma] and height h, rising in the tube to the Torricellian vacuum, is balanced by a column of air of density [rho], which may be supposed to rise as a homogeneous fluid to a height k, called the height of the homogeneous atmosphere. Thus water being about 800 times denser than air and mercury 13.6 times denser than water, k/h = [sigma]/[rho] = 800 × 13.6 = 10,880; (2) and with an average barometer height of 30 in. this makes k 27,200 ft., about 8300 metres. 11. _The Head of Water or a Liquid._--The pressure [sigma]h at a depth h ft. in liquid of density [sigma] is called the pressure due to a _head_ of h ft. of the liquid. The atmospheric pressure is thus due to an average head of 30 in. of mercury, or 30 × 13.6 ÷ 12 = 34 ft. of water, or 27,200 ft. of air. The pressure of the air is a convenient unit to employ in practical work, where it is called an "atmosphere"; it is made the equivalent of a pressure of one kg/cm²; and one ton/inch², employed as the unit with high pressure as in artillery, may be taken as 150 atmospheres. 12. _Theorem._--A body immersed in a fluid is buoyed up by a force equal to the weight of the liquid displaced, acting vertically upward through the centre of gravity of the displaced liquid. For if the body is removed, and replaced by the fluid as at first, this fluid is in equilibrium under its own weight and the thrust of the surrounding fluid, which must be equal and opposite, and the surrounding fluid acts in the same manner when the body replaces the displaced fluid again; so that the resultant thrust of the fluid acts vertically upward through the centre of gravity of the fluid displaced, and is equal to the weight. When the body is floating freely like a ship, the equilibrium of this liquid thrust with the weight of the ship requires that the weight of water displaced is equal to the weight of the ship and the two centres of gravity are in the same vertical line. So also a balloon begins to rise when the weight of air displaced is greater than the weight of the balloon, and it is in equilibrium when the weights are equal. This theorem is called generally the _principle of Archimedes_.