Page:Scientific Papers of Josiah Willard Gibbs.djvu/265

Rh when the bounding surface is fixed, and the total entropy and total quantities of the various components are constant. We may suppose $$\eta^S, \eta ', \eta , m_{1}^S, m_{1}', m_{1}, m_{2}^S, m_{2}', m_{2}'',$$ etc., to be all constant. Then by (497) and (12) the condition reduces to (We may set $$=$$ for $$\geqq$$, since changes in the position of the dividing surface can evidently take place in either of two opposite directions.) This equation has evidently the same form as if a membrane without rigidity and having a tension $$\sigma$$, uniform in all directions, existed at the dividing surface. Hence the particular position which we have chosen for this surface may be called the surface of tension, and $$\sigma$$ the superficial tension. If all parts of the dividing surface move a uniform normal distance $$\delta N$$, we shall have whence the curvatures being positive when their centers lie on the side to which $$p'$$ relates. This is the condition which takes the place of that of equality of pressure (see pp. 65, 74) for heterogeneous fluid masses in contact, when we take account of the influence of the surfaces of discontinuity. We have already seen that the conditions relating to temperature and the potentials are not affected by these surfaces.

In equation (497) the initial state of the system is supposed to be one of equilibrium. The only limitation with respect to the varied state is that the variation shall be reversible, i.e., that an opposite variation shall be possible. Let us now confine our attention to variations in which the system remains in equilibrium. To distinguish this case, we may use the character $$d$$ instead of $$\delta$$, and write Both the states considered being states of equilibrium, the limitation with respect to the reversibility of the variations may be neglected, since the variations will always be reversible in at least one of the states considered.

If we integrate this equation, supposing the area s to increase from zero to any finite value s, while the material system to a part of which the equation relates remains without change, we obtain