Page:Scientific Papers of Josiah Willard Gibbs.djvu/319

where the coefficient of $z$ is to be regarded as constant. Now the value of $z$ cannot be very large, in any surface of sensible dimensions, unless $\gamma ''-\gamma '$ is very small. We may therefore consider this equation as practically exact, unless the densities of the contiguous masses are very nearly equal. If we substitute for the sum of the curvatures its value in terms of the differential coefficients of $z$ with respect to the horizontal rectangular co-ordinates, $x$ and $y$, we have

${\frac {\left(1+{\frac {dz^{2}}{dy^{2}}}\right){\frac {d^{2}z}{dx^{2}}}-2{\frac {dz}{dx}}{\frac {dz}{dy}}{\frac {d^{2}z}{dxdy}}+\left(1+{\frac {dz^{2}}{dx^{2}}}\right){\frac {d^{2}z}{dy^{2}}}}{\left(1+{\frac {dz^{2}}{dx^{2}}}+{\frac {dz^{2}}{dy^{2}}}\right)^{\tfrac {3}{2}}}}={\frac {g(\gamma ''-\gamma ')}{\sigma }}z.$

(620)

With regard to the sign of the root in the denominator of the fraction, it is to be observed that, if we always take the positive value of the root, the value of the whole fraction will be positive or negative according as the greater concavity is turned upward or downward. But we wish the value of the fraction to be positive when the greater concavity is turned toward the mass specified by a single accent. We should therefore take the positive or negative value of the root according as this mass is above or below the surface.

The particular conditions of equilibrium which are given in the last paragraph but one may be regarded in general as the conditions of chemical equilibrium between the different parts of the system, since they relate to the separate components.^[1] But such a designation is not entirely appropriate unless the number of components is greater than one. In no case are the conditions of mechanical equilibrium entirely independent of those which relate to temperature and the potentials. For the conditions (612) and (614) may be regarded as consequences of (605) and (617) in virtue of the necessary relations (98) and (508).^[2]

The mechanical conditions of equilibrium, however, have an especial importance, since we may always regard them as satisfied in any liquid (and not decidedly viscous) mass in which no sensible motions are observable. In such a mass, when isolated, the attainment of mechanical equilibrium will take place very soon; thermal and chemical equilibrium will follow more slowly. The thermal equilibrium will generally require less time for its approximate attainment than the chemical; but the processes by which the latter is produced will generally cause certain inequalities of temperature until a state of complete equilibrium is reached.

1. Concerning another kind of conditions of chemical equilibrium, which relate to the molecular arrangement of the components, and not to their sensible distribution in space, see pages 138–144.
2. Compare page 146, where a similar problem is treated without regard to the influence of the surfaces of discontinuity.