_Analytical Equations of Equilibrium of a Fluid at rest under any System of Force._ 14. Referred to three fixed coordinate axes, a fluid, in which the pressure is p, the density [rho], and X, Y, Z the components of impressed force per unit mass, requires for the equilibrium of the part filling a fixed surface S, on resolving parallel to Ox, _ _ _ _ _ / / / / / | | lp dS = | | | [rho]X dx dy dz, (1) _/ _/ _/ _/ _/ where l, m, n denote the direction cosines of the normal drawn outward of the surface S. But by Green's transformation _ _ _ _ _ / / / / / dp | | lp dS = | | | -- dx dy dz, (2) _/ _/ _/ _/ _/ dx thus leading to the differential relation at every point dp dp dp -- = [rho]X, -- = [rho]Y, -- = [rho]Z. (3) dx dy dz The three equations of equilibrium obtained by taking moments round the axes are then found to be satisfied identically. Hence the space variation of the pressure in any direction, or the _pressure-gradient_, is the resolved force per unit volume in that direction. The resultant force is therefore in the direction of the steepest pressure-gradient, and this is normal to the surface of equal pressure; for equilibrium to exist in a fluid the lines of force must therefore be capable of being cut orthogonally by a system of surfaces, which will be surfaces of equal pressure. Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force; 1 dp 1 dp 1 dp ----- --, ----- --, ----- --, or X, Y, Z (4) [rho] dx [rho] dy [rho] dz are the partial differential coefficients of some function P, = [int]dp/[rho], of x, y, z; so that X, Y, Z must be the partial differential coefficients of a potential -V, such that the force in any direction is the downward gradient of V; and then dP dV -- + -- = 0, or P + V = constant, (5) dx dx in which P may be called the hydrostatic head and V the head of potential. With variation of temperature, the surfaces of equal pressure and density need not coincide; but, taking the pressure, density and temperature as connected by some relation, such as the gas-equation, the surfaces of equal density and temperature must intersect in lines lying on a surface of equal pressure. 15. As an example of the general equations, take the simplest case of a uniform field of gravity, with Oz directed vertically downward; employing the gravitation unit of force, 1 dp 1 dp 1 dp ----- -- = 0, ----- -- = 0, ----- -- = 1, (1) [rho] dx [rho] dy [rho] dz _ / P = | dp/[rho] = z + a constant. (2) _/