a faces must be proportional to the areas of the faces, and therefore the pressure is equal in any two planes which meet. Collecting our results, we see that the pressure is the same at all points, and in all directions, throughout the fluid mass. 688. Hence if a force be applied at the centre of inertia of each face of a polyhedron, with magnitude proportional to the area of the face, the polyhedron will be in equilibrium. For we may suppose the polyhedron to be a solidified portion of the fluid. The resultant pressure on each face will then be proportional to its area, and will act at its centre of inertia; which, in this case, is the Centre of Pressure. 689. Another proof of the equality of pressure throughout a mass of fluid, uninfluenced by other external force than the pressure of the containing vessel, is easily furnished by the energy criterion of equilibrium, $ 254; but, to avoid complication, we will consider the fluid to be incompressible. Suppose a number of pistons fitted into cylinders inserted in the sides of the closed vessel containing the fluid. Then, if A be the area of one of these pistons, p the average pressure on it, x the distance through which it is pressed, in or out; the energy criterion is that no work shall be done on the whole, i.e. that AAB,X A,B,X, +4,Pex, +...= (Apx) = 0, as much work being restored by the pistons which are forced out, as is done by those forced in. Also, since the fluid is incompressible, it must have gained as much space by forcing out some of the pistons as it lost by the intrusion of the others. This gives A,%, + A,x, + ... = (Ax)=0. The last the only condition to which x,, x,, etc., in the first equation, are subject; and therefore the first can only be satisfied if P= Pe=Pa = etc., that is, if the pressure be the same on each piston. Upon this property depends the action of Bramah's Hydrostatic Press. If the fluid be compressible, the work expended in compressing it from volume V to V-8V, at mean pressure p, is po V. If in this case we assume the pressure to be the same throughout, we obtain a result consistent with the energy criterion. The work done on the fluid is (Apx), that is, in consequence of the assumption, ps (Ax). But this is equal to pe V, for, evidently, (Ax) = 8 V. 690. When forces, such as gravity, act from external matter upon the substance of the fluid, either in proportion to the density of its own substance in its different parts, or in proportion to the density of electricity, or of magnetic polarity, or of any other conceivable accidental property of it, the pressure will still be the same in all directions at any one point, but will now vary continuously from point to point. For the preceding demonstration ( 687) may still S 2 2 2 be applied by simply taking the dimensions of the prism small enough; since the pressures are as the squares of its linear dimensions, and the effects of the applied forces such as gravity, as the cubes. 691. When forces act on the whole fluid, surfaces of equal pressure, if they exist, must be at every point perpendicular to the direction of the resultant force. For, any prism of the fluid so situated that the whole pressures on its ends are equal must experience from the applied forces no component in the direction of its length; and, therefore, if the prism be so small that from point to point of it the direction of the resultant of the applied forces does not vary sensibly, this direction must be perpendicular to the length of the prism. From this it follows that whatever be the physical origin, and the law, of the system of forces acting on the fluid, and whether it be conservative or non-conservative, the fluid cannot be in equilibrium unless the lines of force possess the geometrical property of being at right angles to a series of surfaces. 692. Again, considering two surfaces of equal pressure infinitely near one another, let the fluid between them be divided into columns of equal transverse section, and having their lengths perpendicular to the surfaces. The difference of pressure on the two ends being the same for each column, the resultant applied forces on the fluid masses composing them must be equal. Comparing this with $ 506, we see that if the applied forces constitute a conservative system, the density of matter, or electricity, or whatever property of the substance they depend on, must be equal throughout the layer under consideration. This is the celebrated hydrostatic proposition that in a fluid at rest, surfaces of equal pressure are also surfaces of equal density and of equal potential. 693. Hence when gravity is the only external force considered, surfaces of equal pressure and equal density are (when of moderate extent) horizontal planes. On this depends the action of levels, siphons, barometers, etc.; also the separation of liquids of different densities (which do not mix or combine chemically) into horizontal strata, etc., etc. The free surface of a liquid is exposed to the pressure of the atmosphere simply; and therefore, when in equilibrium, must be a surface of equal pressure, and consequently level. In extensive а sheets of water, such as the American lakes, differences of atmospheric pressure, even in moderately calm weather, often produce considerable deviations from a truly level surface. 694. The rate of increase of pressure per unit of length in the direction of the resultant force, is equal to the intensity of the force reckoned per unit of volume of the fluid. Let F be the resultant force per unit of volume in one of the columns of § 692; p and p the pressures at the ends of the column, l its length, S its section. We have, for the equilibrium of the column, (1-P) S=SIF. Hence the rate of increase of pressure per unit of length is F. or If the applied forces belong to a conservative system, for which V and V' are the values of the potential at the ends of the column, we have (8 504) V-V=-IFP, where p is the density of the fluid. This gives P-p=-P(V'- V), dp = - pd V. Hence in the case of gravity as the only impressed force the rate of increase of pressure per unit of depth in the fluid is p, in gravitation P measure (usually employed in hydrostatics). In kinetic or absolute measure (S 189) it is gp. If the fluid be a gas, such as air, and be kept at a constant temperature, we have p=cp, where c denotes a constant, the reciprocal of H, the height of the homogeneous atmosphere,' defined ($ 695) below. Hence, in a calm atmosphere of uniform temperature we have dp --adV; P and from this, by integration, P=PE-V; where p, is the pressure at any particular level (the sea-level, for instance) where we choose to reckon the potential as zero. When the differences of level considered are infinitely small in comparison with the earth's radius, as we may practically regard them, in measuring the heights of mountains, or of a balloon, by the barometer, the force of gravity is constant, and therefore differences of potential (force being reckoned in units of weight) are simply equal to differences of level. Hence if x denote height of the level of pressure p above that of po, we have, in the preceding formulae, V = x, and therefore Þ= C; that is, 695. If the air be at a constant temperature, the pressure diminishes in geometrical progression as the height increases - in arithmetical progression. This theorem is due to Halley. Without formal mathematics we see the truth of it by remarking that differences of pressure are ( 694) equal to differences of level multiplied by the density of the fluid, or by the proper mean density when the density differs sensibly between the two stations. But the density, when the temperature is constant, varies in simple proportion to the pressure, according to Boyle's law. Hence differences of pressure between pairs of stations differing equally in level are proportional to the proper mean values of the whole pressure, which is the well-known compound interest law. The rate of diminution of pressure per unit of length upwards in proportion to the whole pressure at any point, is of course equal to the reciprocal of the height above that point that the atmosphere must have, if of constant density, to give that pressure by its weight. The height thus defined is commonly called the height of the homogeneous atmosphere,' a very convenient conventional expression. It is equal to the product . of the volume occupied by the unit mass of the gas at any pressure into the value of that pressure reckoned per unit of area, in terms of the weight of the unit of mass. If we denote it by H, the exponential expression of the law is pepocă, which agrees with the final formula of $ 694. The value of H for dry atmospheric air, at the freezing temperature, according to Regnault, is, in the latitude of Paris, 799,020 centimetres, or 26,215 feet. Being inversely as the force of gravity in different latitudes (S 187), it is 798 533 centimetres, or 26,199 feet, in the latitude of Edinburgh and Glasgow. 696. It is both necessary and sufficient for the equilibrium of an incompressible fluid completely filling a rigid closed vessel, and influenced only by a conservative system of forces, that its density be uniform over every equipotential surface, that is to say, every surface cutting the lines of force at right angles. If, however, the boundary, or any part of the boundary, of the fluid mass considered, be not rigid; whether it be of flexible solid matter (as a membrane, or a thin sheet of elastic solid), or whether it be a mere geometrical boundary, on the other side of which there is another fluid, or nothing [a case which, without believing in vacuum as a reality, we may admit in abstract dynamics (S 391)], a farther condition is necessary to secure that the pressure from without shall fulfil the hydrostatic equation at every point of the boundary. In the case of a bounding membrane, this condition must be fulfilled either through pressure artificially applied from without, or through the interior elastic forces of the matter of the membrane. In the case of another fluid of different density touching it on the other side of the boundary, all round or over some part of it, with no separating membrane, the condition of equilibrium of a heterogeneous fluid is to be fulfiled relatively to the whole fluid mass made up of the two; which shows that at the boundary the pressure must be constant and equal to that of the fluid on the other side. Thus water, oil, mercury, or any other liquid, in an open vessel, with its free surface exposed to the air, requires for equilibrium simply that this surface be level. 697. Recurring to the consideration of a finite mass of fluid completely filling a rigid closed vessel, we see, from what precedes, that, if homogeneous and incompressible, it cannot be disturbed from equilibrium by any conservative system of forces; but we do not require the analytical investigation to prove this, as we should have 'the perpetual motion' if it were denied, which would violate the hypothesis that the system of forces is conservative. On the other hand, a non-conservative system of forces cannot, under any circumstances, equilibrate a fluid which is either uniform in density throughout, or of homogeneous substance, rendered heterogeneous in density only through difference of pressure. But if the forces, though not a conservative, be such that through every point of the space occupied by the fluid a surface may be drawn which shall cut at right angles all the lines of force it meets, a heterogeneous fuid will rest in equilibrium under their influence, provided ($ 692) its density, from point to point of every one of these orthogonal surfaces, varies inversely as the product of the resultant force into the thickness of the infinitely thin layer of space between that surface and another of the orthogonal surfaces infinitely near it on either side. (Compare $ 506). 698. If we imagine all the fluid to become rigid except an infinitely thin closed tubular portion lying in a surface of equal density, and if the fluid in this tubular circuit be moved any length along the tube and left at rest, it will remain in equilibrium in the new position, all positions of it in the tube being indifferent because of its homogeneousness. Hence the work (positive or negative) done by the force (X, Y, Z) on any portion of the fluid in any displacement along the tube is balanced by the work (negative or positive) done on the remainder of the fluid in the tube. Hence a single particle, acted on only by X, Y, Z, while moving round the circuit, that is moving along any closed curve on a surface of equal density, has, at the end of one complete circuit, done just as much work against the force in some parts of its course, as the forces have done on it in the remainder of the circuit. 699. The following imaginary example, and its realization in a subsequent section (S 701), show a curiously interesting practical application of the theory of Auid equilibrium under extraordinary circumstances, generally regarded as a merely abstract analytical theory, practically useless and quite unnatural, 'because forces in nature follow the conservative law.' 700. Let the lines of force be circles, with their centres all in one line, and their planes perpendicular to it. They are cut at right angles by planes through this axis; and therefore a fluid may be in equilibrium under such a system of forces. The system will not be conservative if the intensity of the force be according to any other law than inverse proportionality to distance from this axial line; and the fluid, to be in equilibrium, must be heterogeneous, and be so distributed as to vary in density from point to point of every plane through the axis, inversely as the product of the force into the distance from the axis. But from one such plane to another it may be either uniform in density, or may vary arbitrarily. To particularize farther, we may suppose the force to be in direct simple proportion to the distance from the axis. Then the fluid will be in equilibrium if its density varies from point to point of every plane through the axis, inversely as the square of that distance. If we still farther particularize by making the force uniform all round each circular line of force, the distribution of force becomes precisely that of the kinetic reactions of the parts of a rigid body against accelerated rotation. The fluid pressure will ( 691) be equal over each plane through the |