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

Rh 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., 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 By this equation and (659), our expression for the total increment of energy in the system may be reduced to the form  In order that this shall vanish for any values of $$\delta N$$, it is necessary that the coefficient of $$\delta N Ds$$ shall vanish. This gives for the condition of equilibrium This equation is identical with (387), with the exception of the term containing $$\sigma$$, which vanishes when the surface is plane. 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 that is, it expresses the equality of the potentials for the substance of