Page:Scientific Papers of Josiah Willard Gibbs.djvu/274

remain uniform, and on account of its infinitely greater size to be infinitely less altered in its nature than the first part. Let $\Delta \epsilon ^{S}$ denote the increment of the superficial energy of this first part, $\Delta \eta ^{S},\Delta m_{a}^{S},\Delta m_{b}^{S}$, etc., $\Delta m_{g}^{S},\Delta m_{h}^{S}$, etc., the increments of its superficial entropy and of the quantities of the components which we regard as belonging to the surface. The increments of entropy and of the various components which the rest of the system receive will be expressed by

$-\Delta \eta ^{S},$⁠$-\Delta m_{a}^{S},$⁠$-\Delta m_{b}^{S},$⁠etc.,⁠$-\Delta m_{g}^{S},$⁠$-\Delta m_{h}^{S},$⁠etc.,

and the consequent increment of energy will be by (12) and (501)

$-t\Delta \eta ^{S}-\mu _{a}\Delta m_{a}^{S}-\mu _{b}\Delta m_{b}^{S}-{\text{etc.}}-\mu _{g}\Delta m_{g}^{S}-\mu _{h}m_{h}^{S}-{\text{etc.}}$

Hence the total increment of energy in the whole system will be

$\Delta \epsilon ^{S}-t\Delta \eta ^{S}$	$-\mu _{a}\Delta m_{a}^{S}-\mu _{b}\Delta m_{b}^{S}-{\text{etc.}}$	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$ (516)
	$-\mu _{g}\Delta m_{g}^{S}-\mu _{h}\Delta m_{h}^{S}-{\text{etc.}}$

If the value of this expression is necessarily positive, for finite changes as well as infinitesimal in the nature of the part of the film to which $\Delta \epsilon ^{S}$, etc. relate,^[1] the increment of energy of the whole system will be positive for any possible changes in the nature of the film, and the film will be stable, at least with respect to changes in its nature, as distinguished from its position. For, if we write

$D\epsilon ^{S},$⁠$D\eta ^{S},$⁠$Dm_{a}^{S},$⁠$Dm_{b}^{S},$⁠etc.,$Dm_{g}^{S},$⁠$Dm_{h}^{S},$⁠etc.,

for the energy, etc. of any element of the surface of discontinuity, we have from the supposition just made

$\Delta D\epsilon ^{S}-t\Delta D\eta ^{S}$	$-\mu _{a}\Delta Dm_{a}^{S}-\mu _{b}\Delta Dm_{b}^{S}-{\text{etc.}}$
	$-\mu _{g}\Delta Dm_{g}^{S}-\mu _{h}\Delta Dm_{h}^{S}-{\text{etc.}}>0;$

and integrating for the whole surface, since

$\Delta \int Dm_{g}^{S}=0,$⁠$\Delta \int Dm_{h}^{S}=0,$⁠etc.,

we have

$\Delta \int D\epsilon ^{S}-t\Delta \int D\eta ^{S}-\mu _{a}\Delta \int Dm_{a}^{S}-\mu _{b}\Delta \int Dm_{b}^{S}-{\text{etc.}}\geqq 0.$

(519)

Now $\Delta \int D\eta ^{S}$ is the increment of the entropy of the whole surface, and $-\Delta \int D\eta ^{S}$ is therefore the increment of the entropy of the two homogeneous masses. In like manner, $-\Delta \int Dm_{a}^{S},-\Delta \int Dm_{b}^{S}$, etc., are the increments of the, quantities of the components in these masses. The expression

$-t\Delta \int D\eta ^{S}-\mu _{a}\Delta \int Dm_{a}^{S}-\mu _{b}\Delta \int Dm_{b}^{S}-{\text{etc.}}$

denotes therefore, according to equation (12), the increment of energy of the two homogeneous masses, and since $\Delta \int D\epsilon ^{S}$ denotes the

1. In the case of infinitesimal changes in the nature of the film, the sign $\Delta$ must be interpreted, as elsewhere in this paper, without neglect of infinitesimals of the higher orders. Otherwise, by equation (501), the above expression would have the value zero.