440 HYDROMECHANICS [HYDROSTATICS. Consequently, if p, p be the pressures, v, v the specific volumes, p, p the densities, and t, t the absolute temperatures at two diffe rent states of a gas, when no heat has been allowed to escape, then p pv t since - r - i = -^ . pv t The value of 7 for air is about 1 4 ; for instance, the velocity of sound is A/ - = */ 7 ; and, for dry air at T C. , ^ = 7-8376x101 P and therefore the velocity of sound in centimetres per second = 33240 /fl + )= 33240 + 60r; nearly. V 273y We have defined the density of a substance to be the number of units of mass in the unit of volume ; for in stance, with English units the density is the number of pounds in a cubic foot, and the density of water would be 62 5, a cubic foot of water being 1000 oz. But the den sities of substances are generally tabulated relatively to water, and the old-fashioned name for the density relative to water was specific gravity. With French units of the centimetre and the gramme, a centimetre cube of water being a gramme, the density of water is unity. PART I. HYDROSTATICS. Hydrostatics is the science of the equilibrium of fluids. When a fluid has come to rest there can be no tangential stress, and consequently the stress across any surface is normal to the surface, and therefore the same in all direc tions about a point. We shall begin with a few elementary propositions about the equilibrium of liquids like water under gravity, and afterwards proceed to the consideration of the equilibrium of any fluid under any forces. Prop. I. The pressure is the same at all points in a horizontal plane of a liquid at rest under gravity. > For, taking any two points in the same horizontal plane, and joining them, and describing about the joining line as axis an inde finitely thin cylinder, then, since the weight and the pressures on the sides of the cylinder are normal to the axis of the cylinder, resolv ing parallel to the axis of the cylinder, the pressures on the ends must be equal for equilibrum, and must therefore be of equal intensity. Corollary. It follows that the free surface must be a horizontal plane, supposing the pressure uniform over it, and gravity to act in parallel vertical lines. That the free surface is a plane is verified experimentally by the fact that objects are seen by reflexion at the surface undisturbed as in a plane mirror ; and that it is a horizontal plane is verified by the fact that a plumb line and its image by reflexion at the surface always appear in the same straight line. Prop.- II. The pressure at any point of a liquid at rest under gravity is proportional to the depth. For, let P (iig. 1) be any point at a depth AP = z in the liquid, and about AP as axis describe a cylinder, of sectional area a suppose. Then, considering the equilibrium of the liquid in this cylinder, and resolving vertically, the pressure, p suppose, at P acting over the area must balance the weight of the liquid in the cylinder, neglecting the atmospheric pressure, and there fore pa. = p~a , p denoting the weight of the liquid per unit of volume. In practical hydrostatics the gravitation measure of forces is employed, and p the pressure is generally estimated in Ib per square inch, and then, an inch being the unit of length, s is given in inches, and p is the num ber of pounds to a cubic inch of the liquid con sidered. Thus with water, taking a cubic foot of water to be 62 -4 Ib, If, however, c be measured in feet and p in R> per square inch, Consequently the head of each foot of water produces a pressure of 433 lt> to the square inch. With the French metric system of units, a centimetre cube of water is one gramme, and therefore with the centimetre as the unit of length and gramme as unit of weight, using gravitation units of force, p = l, and p = z. Even if the point C considered (fig. 2) should not be vertically below the actual free surface, it still follows that the pressure at C is proportional to the depth below the free surface. For the pressure at C is equal to the pressure at D in the same horizontal plane, and this is proportional to the depth DE, that is, to the depth CF. If the atmospheric pressure & be taken into account, then p = w + pz , and generally, at different depths 2 and z , or the difference of the pressures at any two points of a liquid at rest under gravity is proportional to the difference of the depths. Fig. 2. Corollary. It follows from this that a liquid rises to the same level in a series of communicating vessels (fig. 3), since the fluid must have one horizontal plane as the free surface. A Fig. 3. If liquids of different densities p and p be poured into the two branches of a bent tube, the heights of the free surfaces above the plane of separation will be inversely as the densities. For the pres sure at the level of the common sur face being constant, and in one case due to a height h of liquid of density p and in the other case to a height h of liquid of density p , therefore ph p h , or h : h ***p : p . The barometer (fig. 4) is in reality an instrument of this character, for a column of mercury, of density a and height h suppose, supports a column of air, which is of density p suppose, and, if homogeneous, would reach to a height H. Hence the pressure at the level of the common surface p = ffh = pH , estimated in gravitation measure. If the tube AB should not be ex actly vertical, then h must be taken to denote the vertical distance between the level of the mercury at P and Q, which becomes difficult to estimate at sea when the ship is rolling. When his observed the temperature must also be observed; for at the same pressure the coefficient of linear expan sion of h is the coefficient of cubical expansion of mercury. For, if the density of mercury were halved,
the height h would be doubled, and so on in any proportion.
