Page:Scientific Papers of Josiah Willard Gibbs.djvu/260

as the masses $M'$ and $M''$ respectively. We shall then have, by equation (12), if we regard the volumes as constant,

$\delta \epsilon '''=t'\delta \eta '''+\mu _{1}'\delta m_{1}'''+\mu _{2}'\delta m_{2}'''+{\text{etc.,}}$

(485)

$\delta \epsilon ''''=t''\delta \eta ''''+\mu _{1}''\delta m_{1}''''+\mu _{2}''\delta m_{2}''''+{\text{etc.;}}$

(486)

whence, by (482)–(484), we have for reversible variations

$\delta \epsilon '''=t\delta \eta '''+\mu _{1}\delta m_{1}'''+\mu _{2}\delta m_{2}'''+{\text{etc.,}}$

(487)

$\delta \epsilon ''''=t\delta \eta ''''+\mu _{1}\delta m_{1}''''+\mu _{2}\delta m_{2}''''+{\text{etc.}}$

(488)

From these equations and (477), we have for reversible variations

$\delta (\epsilon -\epsilon '''-\epsilon '''')=t\delta (\eta -\eta '''-\eta '''')+\mu _{1}\delta (m_{1}-m_{1}'''-m_{1}'''')+\mu _{2}\delta (m_{2}-m_{2}'''-m_{2}'''')+{\text{etc.}}$

(489)

Or, if we set^[1]

$\epsilon ^{S}=\epsilon -\epsilon '''-\epsilon '''',$$\eta ^{S}=\eta -\eta '''-\eta ''''$

(490)

$m_{1}^{S}=m_{1}-m_{1}'''-m_{1}'''',$$m_{2}^{S}=m_{2}-m_{2}'''-m_{2}'''',$ etc.,

(491)

we may write

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

(492)

This is true of reversible variations in which the surfaces which have been considered are fixed. It will be observed that $\epsilon ^{S}$ denotes the excess of the energy of the actual mass which occupies the total volume which we have considered over that energy which it would have, if on each side of the surface $S$ the density of energy had the same uniform value quite up to that surface which it has at a sensible distance from it; and that $\eta ^{S},m_{1}^{S},m_{2}^{S}$ etc., have analogous significations. It will be convenient, and need not be a source of any misconception, to call $\epsilon ^{S}$ and $\eta ^{S}$ the energy and entropy of the surface (or the superficial energy and entropy), ${\frac {\epsilon ^{S}}{s}}$ and ${\frac {\eta ^{S}}{s}}$ the superficial densities of energy and entropy, ${\frac {m_{1}^{S}}{s}},{\frac {m_{2}^{S}}{s}}$, etc., the superficial densities of the several components.

Now these quantities ($\epsilon ^{S},\eta ^{S},m_{1}^{S}$, etc.) are determined partly by the state of the physical system which we are considering, and partly by the various imaginary surfaces by means of which these quantities have been defined. The position of these surfaces, it will be remembered, has been regarded as fixed in the variation of the system. It is evident, however, that the form of that portion of these surfaces which lies in the region of homogeneity on either side of the surface of discontinuity cannot affect the values of these quantities. To obtain the complete value of $\delta \epsilon ^{S}$ for reversible variations, we have

1. It will be understood that the $^{S}$ here used is not an algebraic exponent, but is only intended as a distinguishing mark. The Roman letter $S$ has not been used to denote any quantity.