HYDROMECHANICS 439 If an area a be pushed by P units of force, then the mean p pressure p over the area a is ; or, if /; be variable, then a p at any point p = It , a being any small area enclosing the a point at which/* is required, and P the number of units of force with which a is pushed. The pressure across any surface being normal to the surface, it follows from the general equations of internal stress (see ELASTICITY) that at a point the pressure is the same in all directions about the point. This may be proved independently by considering the equilibrium of a tetra hedron ABCD. Let p, // be the pressures on the faces ABC, BCD, and resolve the forces parallel to the edge AD ; the face ABC will be pushed by a force p x area ABC, and the face BCD will be pushed by a force p x area BCD ; and the projections of the areas ABC, BCD on a plane per pendicular to AD being equal, it follows that p = p. If forces act throughout the fluid so that the pressure varies from one point to another, we must suppose the tetrahedron taken indefinitely small, and then the impressed forces, depending on the volume, may be neglected in com parison with the forces acting on the faces, which depend on the surface of the tetrahedron. When fluids such as exist in nature are in motion, the stresses across any surfaces are no longer normal, but tangential stresses are called into play, the intensities of which depend upon the relative motion of the parts of the fluid ; those tangential stresses are said to be due to the viscosity of the fluid. The difference between a solid and a fluid is that in a solid the tangential stress must exceed a certain amount before permanent shearing takes place, otherwise the stress being removed the solid regains its shape ; in a fluid the slightest tangential stress produces a permanent deformation, and if continued long enough will cause a complete change of form, however great the viscosity of the fluid may be. 1 But little progress has been made in the theory of the motion of viscous fluids, those cases which have been worked out mostly falling under the head of the practical subject of hydraulics. In the theoretical subject of hydro dynamics the motion of the so-called perfect fluids only is considered, fluids in which no viscosity is supposed to exist, and in which therefore the pressure at a point is the same in all directions. Fluids are divided into two classes, incompressible fluids called liquids, and compressible fluids called gases. The so-called incompressible fluids are in reality compressible, but the compressibility being small is neglected in ordinary problems. The compression of a substance is defined to be the ratio of the diminution of the volume to the original volume ; and the compressibility is defined to be the ratio of the compression to the pressure producing it. The elasticity or resilience of volume is the ratio of the pressure to the compression produced, and is therefore the reciprocal of the compressibility. Fluids, from the definition, possess only elasticity of volume ; an elastic solid possesses in addition an elasticity of figure, called also the rigidity. Arable of the elasticities and compressibilities of liquids is given in Everett s Units and Physical Constants, ex pressed in C. G. S. units; for instance, at 8 C. the elasticity of water is 2 08 x 10 10 , and the compressibility per atmo sphere is 4-81 x 10~ 5 . The elasticity of water is also proved to exist, and can be determined from the velocity of sound in water ; for instance, in water of density unity, the velocity is the square root of the elasticity, and there- 1 For examples of the difference between a soft solid and a very viscous fluid see Maxwell s Heat, chap. xxi. fore 144,000 centimetres per second, which agrees closely with the velocity determined by experiment. Compressible fluids or gases are assumed to obey the two gaseous laws. The first gaseous law, discovered by Boyle, and gene rally called Boyle s law, asserts that the pressure of a given quantity of a gas at a given temperature varies inversely as the volume, or directly as the density. The density is defined to be the number of units of mass in the unit of volume, and the specific volume is the volume of the unit of mass. Hence Boyle s law asserts that the pressure of a gas at a given temperature varies as the density, or inversely as the specific volume. Dalton generalized Boyle s law for a mixture of gases by enunciating the law that the pressure of a mixture of gases is the sum of the separate pressures each gas would have if it existed alone in the containing vessel. If we suppose all the gases the same, we are led to Boyle s law, since, if a gramme of air in a centimetre cube produces a certain pressure, then two grammes will produce double the pressure, three grammes treble the pressure, and so on ; hence the pressure varies as the density. This method of exhibiting Boyle s law is due to Piankine (Max- , well, Heat, p. 27). The second gaseous law was discovered by Charles, but j it generally goes by Gay-Lussac s name. It asserts that | under constant pressure the volume of a given quantity of 1 every gas increases by about -jig- of its volume at C. for a rise of temperature of 1 C. Therefore, if v be the volume at T C. and v the volume at C., then v 273 273 If we reckon temperature from -273 C., and put 273 + r = t, then t is called the absolute temperature, and - 273 C. is called the absolute zero ; and the second gaseous law asserts that at a constant pressure the volume of a given quantity of gas is proportional to the absolute temperature. Combined with Boyle s law, this leads to the result that pv<xt, = R< suppose, where p is the pressure, v the specific volume, and t the absolute temperature ; also, if p denote the density, 1 To determine the numerical value of ^ = R, in C.G.S. units, sup- . pose at 80 C. a centimetre cube of air is 001 of a gramme, the height of the barometer being 76 centimetres, and the numerical value of g to be 981 ; then jw = 981 x 1342 x 76 = 1 6 very nearly, = 273 + 80 = 353, If the temperature is kept constant, then, an increment of pressure dp producing a compression -, the elasticity of volume v dp dp = -v -~- = p~f =p , dv dp - and therefore at a constant temperature the elasticity is equal to the pressure. Professor Maxwell (Heat, p. 171) has proved from first principle? that the ratio of the elasticity of volume when no heat escapes is to the elasticity at constant temperature as the specific heat at constant pressure to the specific heat at constant volume. Consequently if 7 denotes the ratio of the specific heat at constant pressure to the specific heat at constant volume, and if the gas be compressed and no heat allowed to escape, then dp dv --7P dp do . + y , p V
pi^ = constant.
