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 | |||
| ρ | dx | ρ | dy | ρ | dz |
(4)
are the partial differential coefficients of some function P, = ∫ dp/ρ, 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, |
| dx | dx |
(5)
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,