Page:Scientific Papers of Josiah Willard Gibbs.djvu/353

expressions we may suppose to be derived from the fluid mass. These expressions, therefore, with a change of sign, will represent the increments of entropy and of the quantities of the components in the whole space occupied by the fluid except that which is immediately contiguous to the solid. Since this space may be regarded as constant, the increment of energy in this space may be obtained (according to equation (12)) by multiplying the above expression relating to entropy by $-t$, and those relating to the components by $-\mu _{1}'',-\mu _{2}$, etc.,^[1] and taking the sum. If to this we add the above expression for the increment of energy near the surface, we obtain the increment of energy for the whole system. Now by (93) we have

$p''=-\epsilon _{V}''+t\eta _{V}''+\mu _{1}''\gamma _{1}''+\mu _{2}\gamma _{2}''+{\text{etc.}}$

By this equation and (659), our expression for the total increment of energy in the system may be reduced to the form

$\int [\epsilon _{V}'-t\eta _{V}'-\mu _{1}''\gamma _{1}'+p''+(c_{1}+c_{2})\sigma ]\delta NDs.$

(660)

In order that this shall vanish for any values of $\delta N$, it is necessary that the coefficient of $\delta NDs$ shall vanish. This gives for the condition of equilibrium

$\mu _{1}''={\frac {\epsilon _{V}'-t\eta _{V}'+p''+(c_{1}+c_{2})\sigma }{\gamma _{1}'}}\cdot$

(661)

This equation is identical with (387), with the exception of the term containing $\sigma$, which vanishes when the surface is plane.^[2]

We may also observe that when the solid has no stresses except an isotropic pressure, if the quantity represented by $\sigma$ is equal to the true tension of the surface, $p''+(c_{1}+c_{2})\sigma$ will represent the pressure in the interior of the solid, and the second member of the equation will represent (see equation (93)) the value of the potential in the solid for the substance of which it consists. In this case, therefore, the equation reduces to

$\mu _{1}''=\mu _{1}',$

that is, it expresses the equality of the potentials for the substance of

1. The potential $\mu _{1}''$ is marked by double accents in order to indicate that its value is to be determined in the fluid mass, and to distinguish it from the potential $\mu _{1}'$ relating to the solid mass (when this is in a state of isotropic stress), which, as we shall see, may not always have the same value. The other potentials $\mu _{2}$, etc., have the same values as in (659), and consist of two classes, one of which relates to substances which are components of the fluid mass (these might be marked by the double accents), and the other relates to substances found only at the surface of discontinuity. The expressions to be multiplied by the potentials of this latter class all have the value zero.
2. In equation (387), the density of the solid is denoted by $\Gamma$, which is therefore equivalent to $\gamma _{1}'$ in (661).