Page:Scientific Papers of Josiah Willard Gibbs.djvu/266

which may be applied to any portion of any surface of discontinuity (in equilibrium) which is of the same nature throughout, or throughout which the values of $t,\sigma ,\mu _{1},\mu _{2}$, etc., are constant.

If we differentiate this equation, regarding all the quantities as variable, and compare the result with (501), we obtain

$\eta ^{S}dt+sd\sigma +m_{1}^{S}d\mu _{1}+m_{2}^{S}d\mu _{2}+{\text{etc.}}=0.$

(503)

If we denote the superficial densities of energy, of entropy, and of the several component substances (see page 224) by $\epsilon _{S},\eta _{S},\Gamma _{1},\Gamma _{2}$, etc., we have

$\epsilon _{S}={\frac {\epsilon ^{S}}{s}},$⁠$\eta _{s}={\frac {\eta ^{S}}{s}},$

(504)

$\Gamma _{1}={\frac {m_{1}^{S}}{s}},$⁠$\Gamma _{2}={\frac {m_{2}^{S}}{s}},$⁠ etc.,

(505)

and the preceding equations may be reduced to the form

$d\epsilon _{S}=td\eta _{S}+\mu _{1}d\Gamma _{1}+\mu _{2}d\Gamma _{2}+{\text{etc.,}}$

(506)

$\epsilon _{S}=t\eta _{S}+\mu _{1}\Gamma _{1}+\mu _{2}\Gamma _{2}+{\text{etc.,}}$

(507)

$d\sigma =-\eta _{S}dt-\Gamma _{1}d\mu _{1}-\Gamma _{2}d\mu _{2}-{\text{etc.}}$

(508)

Now the contact of the two homogeneous masses does not impose any restriction upon the variations of phase of either, except that the temperature and the potentials for actual components shall have the same value in both. {See (482)–(484) and (500).} For however the values of the pressures in the homogeneous masses may vary (on account of arbitrary variations of the temperature and potentials), and however the superficial tension may vary, equation (500) may always be satisfied by giving the proper curvature to the surface of tension, so long, at least, as the difference of pressures is not great. Moreover, if any of the potentials $\mu _{1},\mu _{2},$ etc., relate to substances which are found only at the surface of discontinuity, their values may be varied by varying the superficial densities of those substances. The values of $t,\mu _{1},\mu _{2},$ etc., are therefore independently variable, and it appears from equation (508) that $\sigma$ is a function of these quantities. If the form of this function is known, we may derive from it by differentiation $n+1$ equations ($n$ denoting the total number of component substances) giving the values of $\eta _{S},\Gamma _{1},\Gamma _{2}$, etc., in terms of the variables just mentioned. This will give us, with (507), $n+3$ independent equations between the $2n+4$ quantities which occur in that equation. These are all that exist, since $n+1$ of these quantities are independently variable. Or, we may consider that we have $n+3$ independent equations between the $2n+5$ quantities occurring in equation (502), of which $n+2$ are independently variable.