only exter

Gravity the of liquids of different densities (which do not mix or combine nal force. 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 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.

Rate of increase of pressure.

752. 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 § 750; and p' the pressures at the ends of the column, 7 its length, S its section. We have, for the equilibrium of the column,

Ρ

Hence the rate of increase of pressure per unit of length is F.

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 (§ 486)

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 measure (usually employed in hydrostatics). In kinetic or absolute measure (§ 224) 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 (753) below. Hence, in a calm atmosphere of uniform temperature we have dp/p-cd V; and from this, by integration, pp where p. is the pressure at any particular level (the sea-level, for instance) where we choose to reckon the potential as zero.

increase of

When the differences of level considered are infinitely small in Rate of comparison with the earth's radius, as we may practically regard pressure. them, in measuring the height of a mountain, 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 p1, we have, in the preceding formulæ, V=x, and therefore p = p. That is to say

in a calm

of uniform

ture.

753. If the air be at a constant temperature, the pressure Pressure diminishes in geometrical progression as the height increases atmosphere in arithmetical progression. This theorem is due to Halley. temperaWithout formal mathematics we see the truth of it by remarking that differences of pressure are (§ 752) 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 and Mariotte'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 Height of atmosphere," a very convenient conventional expression. It geneous atis 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 p=pe-x/H, which agrees with the final formula of § 752.

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 (§ 222), it is 798,533 centimetres, or 26,199 feet, in the latitude of Edinburgh and Glasgow.

the homo

mosphere.

Analytical investigation of the preceding theorems.

Let X, Y, Z be the components, parallel to three rectangular axes, of the force acting on the fluid at (x, y, z), reckoned per unit of its mass. Then, inasmuch as the difference of pressures on the two faces Sydz of a rectangular parallelepiped of the fluid dp

is Sydz 8x, the equilibrium of this portion of the fluid, regarded dx

for a moment (§ 564) as rigid, requires that

Conditions

which are the conditions necessary and sufficient for the equi

This shows that the expression Xdx + Ydy + Zdz must be the complete differential of a function of three independent variables, or capable of being made so by a factor; that is to say, that a series of surfaces exists which cuts the lines of force at right angles; a conclusion also proved above (§ 749).

When the forces belong to a conservative system no factor is required to make the complete differential; and we have

if V denote (§ 485) their potential at (x, y, z): so that (2) be

comes

dp = pd V.....

..(3).

This shows that Ρ is constant over equipotential surfaces (or is a function of V); and it gives

showing that P also is a function of V; conclusions of which we have had a more elementary proof in § 752. As (4) is an analytical expression equivalent to the three equations (1), for the case of a conservative system of forces, we conclude that

754.

It is both necessary and sufficient for the equilibrium librium of of an incompressible fluid completely filling a rigid closed

of equi

pletely fill

vessel.

vessel, and influenced only by a conservative system of forces, fluid comthat its density be uniform over every equipotential surface, ing a closed 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 (§ 438)], a farther condition is necessary to secure that the pressure from without shall fulfil (4) 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 fulfilled 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 Free surface liquid, in an open vessel, with its free surface exposed to the vessel is air, requires for equilibrium simply that this surface be level.

in open

level.

closed

vessel,

755. Recurring to the consideration of a finite mass of fluid Fluid, in completely filling a rigid closed vessel, and supposing that, if the potential of the force-system (as in the case referred to in the sixth and seventh lines of § 758) be a cyclic* func- of forces.

* We here introduce term "cyclic function" to designate a function of more than one variable which experiences a constant addition to its value every time the variables are made to vary continuously from a given set of values through some cycle of values back to the same primitive set of values.

Examples (1) tan-1 (y/x). This is the potential of the conservative system referred to in the first clause of the third sentence of § 758.

(2) ƒ (x2 + y2) tan-1 (y/x). This expresses the fluid pressure in the case of hydrostatic example described in the next to the last sentence of § 758.

(3) The apparent area of a closed curve (plane or not plane) as seen from any point (x, y, z).

under a nonconserva

tive system

Fluid, in closed vessel,

under a non

conserva

tive system of forces.

Fluid under any system of forces.

tion, the enclosure containing the liquid is singly-continuous, we see, from what precedes, that, if homogeneous and incompressible, the fluid 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 conservative, be such that through every point of the space occupied by the fluid a surface can be drawn which shall cut at right angles all the lines of force it meets, a heterogeneous fluid will rest in equilibrium under their influence, provided (§ 750) 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 § 488.)

The same conclusion is proved as a matter of course from (1) since that equation is merely the analytical expression that the force at every point (x, y, z) is along the normal to that surface of the series given by different values of C in p=C, which passes through (x, y, z); and that the magnitude of the resultant force is

T

of which the numerator is equal to 8C/T, if τ be the thickness at (x, y, z) of the shell of space between two surfaces p= C and p = C + SC, infinitely near one another on two sides of (x, y, z).

(4) Functions of any number of variables invented by suggestion from (2).

The designation "many-valued function" which has hitherto been applied to such functions is not satisfactory, if only because it is also applicable to functions of roots of algebraic or transcendental equations.