Page:Scientific Papers of Josiah Willard Gibbs.djvu/264

is evidently consistent with the suppositions made on page 219 with regard to this surface.

We may therefore cancel the term

${\tfrac {1}{2}}(C_{1}+C_{2})\delta (c_{1}+c_{2})$

in (494). In regard to the following term, it will be observed that $C_{1}$ must necessarily be equal to $C_{2}$, when $c_{1}=c_{2}$, which is the case when the surface of discontinuity is plane. Now on account of the thinness of the non-homogeneous film, we may always regard it as composed of parts which are approximately plane. Therefore, without danger of sensible error, we may also cancel the term

${\tfrac {1}{2}}(C_{1}-C_{2})\delta (c_{1}-c_{2}).$

Equation (494) is thus reduced to the form

$\delta \epsilon ^{S}=t\delta \eta ^{S}+\sigma \delta s+\mu _{1}\delta m_{1}^{S}+\mu _{2}\delta m_{2}^{S}+{\text{etc.}}$

(497)

We may regard this as the complete value of $\delta \epsilon ^{S}$, for all reversible variations in the state of the system supposed initially in equilibrium, when the dividing surface has its initial position determined in the manner described.

The above equation is of fundamental importance in the theory of capillarity. It expresses a relation with regard to surfaces of discontinuity analogous to that expressed by equation (12) with regard to homogeneous masses. From the two equations may be directly deduced the conditions of equilibrium of heterogeneous masses in contact, subject or not to the action of gravity, without disregard of the influence of the surfaces of discontinuity. The general problem, including the action of gravity, we shall take up hereafter; at present we shall only consider, as hitherto, a small part of a surface of discontinuity with a part of the homogeneous mass on either side, in order to deduce the additional condition which may be found when we take account of the motion of the dividing surface.

We suppose as before that the mass especially considered is bounded by a surface of which all that lies in the region of non-homogeneity is such as may be traced by a moving normal to the dividing surface. But instead of dividing the mass as before into four parts, it will be sufficient to regard it as divided into two parts by the dividing surface. The energy, entropy, etc., of these parts, estimated on the supposition that its nature (including density of energy, etc.) is uniform quite up to the dividing surface, will be denoted by $\epsilon ',\eta '$, etc., $\epsilon '',\eta ''$, etc. Then the total energy will be $\epsilon ^{S}+\epsilon '+\epsilon ''$, and the general condition of internal equilibrium will be that

$\delta \epsilon ^{S}+\delta \epsilon '+\delta \epsilon ''\geqq 0,$

(498)