Page:Scientific Papers of Josiah Willard Gibbs.djvu/253

 all the conditions which we have obtained for ordinary solids, and which are expressed by the formulæ (364), (374), (380), (382)–(384). The quantities $$\Gamma_{1}', \Gamma_{2}^{\prime}$$, etc., in the last two formulæ include of course those which have just been represented by $$\Gamma_{a}', \Gamma_{b}'$$, etc., and which relate to the fluid components of the body, as well as the corresponding quantities relating to its solid components. Again, if we suppose the solid matter of the body to remain without variation in quantity or position, it will easily appear that the potentials for the substances which form the fluid components of the solid body must satisfy the same conditions in the solid body and in the fluids in contact with it, as in the case of entirely fluid masses. See eqs. (22).

The above conditions must however be slightly modified in order to make them sufficient for equilibrium. It is evident that if the solid is dissolved at its surface, the fluid components which are set free may be absorbed by the solid as well as by the fluid mass, and in like manner if the quantity of the solid is increased, the fluid components of the new portion may be taken from the previously existing solid mass. Hence, whenever the solid components of the solid body are actual components of the fluid mass, (whether the case is the same with the fluid components of the solid body or not,) an equation of the form (383) must be satisfied, in which the potentials $$\mu_{a}, \mu_{b}$$, etc., contained implicitly in the second member of the equation are determined from the solid body. Also if the solid components of the solid body are all possible but not all actual components of the fluid mass, a condition of the form (384) must be satisfied, the values of the potentials in the second member being determined as in the preceding case.

The quantities being differential coefficients of $$\epsilon_{V'}$$ with respect to the variables  will of course satisfy the necessary relations  This result may be generalized as follows. Not only is the second member of equation (468) a complete differential in its present form, but it will remain such if we transfer the sign of differentiation (d) from one factor to the other of any term (the sum indicated by the