Page:Scientific Papers of Josiah Willard Gibbs.djvu/313

masses, and the other as belonging to these masses. The elements of intrinsic energy, entropy, etc., relating to an element of surface $Ds$ will be denoted by $D\epsilon ^{S},D\eta ^{S},Dm_{1}^{S},Dm_{2}^{S}$, etc., and those relating to an element of volume $Dv$, by $D\epsilon ^{V},D\eta ^{V},Dm_{1}^{V},Dm_{2}^{V}$, etc. We shall also use $Dm^{S}$ or $\Gamma Ds$ and $Dm^{V}$ or $\gamma Dv$ to denote the total quantities of matter relating to the elements $Ds$ and $Dv$ respectively. That is,

$Dm^{S}=\Gamma Ds=Dm_{1}^{S}+Dm_{2}^{S}+{\text{etc.,}}$

(597)

$Dm^{V}=\gamma Ds=Dm_{1}^{V}+Dm_{2}^{V}+{\text{etc.}}$

(598)

The part of the energy which is due to gravity must also be divided into two parts, one of which relates to the elements $Dm^{S}$, and the other to the elements $Dm^{V}$. The complete value of the variation of the energy of the system will be represented by the expression

$\delta \int D\epsilon ^{V}+\delta \int \epsilon ^{S}+\delta \int gzDm^{V}+\delta \int gzDm^{S},$

(599)

in which $g$ denotes the force of gravity, and $z$ the height of the element above a fixed horizontal plane.

It will be convenient to limit ourselves at first to the consideration of reversible variations. This will exclude the formation of new masses or surfaces. We may therefore regard any infinitesimal variation in the state of the system as consisting of infinitesimal variations of the quantities relating to its several elements, and bring the sign of variation in the preceding formula after the sign of integration. If we then substitute for $\delta D\epsilon ^{V},\delta D\epsilon ^{S},\delta Dm^{V},\delta Dm^{S}$, the values given by equations (13), (497), (597), (598), we shall have for the condition of equilibrium with respect to reversible variations of the internal state of the system

${\begin{aligned}\int t\delta D\eta ^{V}&-\int p\delta Dv+\int \mu _{1}\delta Dm_{1}^{V}+\int \mu _{2}\delta Dm_{2}^{V}+{\text{etc.}}\\+\int t\delta D\eta ^{S}&+\int \sigma \delta Ds+\int \mu _{1}\delta Dm_{1}^{S}+\int \mu _{2}\delta Dm_{2}^{S}+{\text{etc.}}\\&+\int g\delta zDm^{V}+\int gz\delta Dm_{1}^{V}+\int gz\delta Dm_{2}^{V}+{\text{etc.}}\\&+\int g\delta zDm^{S}+\int gz\delta Dm_{1}^{S}+\int gz\delta Dm_{2}^{S}+{\text{etc.}}&=0.\\\end{aligned}}$

(600)

Since equation (497) relates to surfaces of discontinuity which are initially in equilibrium, it might seem that this condition, although always necessary for equilibrium, may not always be sufficient. It is evident, however, from the form of the condition, that it includes the particular conditions of equilibrium relating to every possible deformation of the system, or reversible variation in the distribution of entropy or of the several components. It therefore includes all the relations between the different parts of the system which are necessary for equilibrium, so far as reversible variations are concerned. (The necessary relations between the various quantities relating to each element of the masses and surfaces are expressed