Page:Scientific Papers of Josiah Willard Gibbs.djvu/338

surfaces, by $\sigma$ and $s$ the tension and area of one of these surfaces, and by $E$ the elasticity of the film when extended under the supposition that the total quantities of $S_{1}$ and $S_{2}$ in the part of the film extended are invariable, as also the temperature and the potentials of the other components. From the definition of $E$ we have

$2d\sigma =E{\frac {ds}{s}},$

(643)

and from the conditions of the extension of the film

${\frac {ds}{s}}=-{\frac {d(\lambda \gamma _{1})}{\lambda \gamma _{1}}}=-{\frac {d(\lambda \gamma _{2}+2\Gamma _{2(1)})}{\lambda \gamma _{2}+2\Gamma _{2(1)}}}\cdot$

(644)

Hence we obtain

$\lambda \gamma _{1}{\frac {ds}{s}}=-\gamma _{1}d\lambda -\lambda d\gamma _{1},$ $(\lambda \gamma _{2}+2\Gamma _{2(1)}){\frac {ds}{s}}=-\gamma _{2}d\lambda -\lambda d\gamma _{2}-2d\Gamma _{2(1)},$

and eliminating $d\lambda$,

$2\gamma _{1}\Gamma _{2(1)}{\frac {ds}{s}}=-\lambda \gamma _{1}d\gamma _{2}+\lambda \gamma _{2}d\gamma _{1}-2\gamma _{1}d\Gamma _{2(1)}.$

(645)

With this equation we may eliminate $d\sigma$ from (643). We may also eliminate do- by the necessary relation (see (514))

$d\sigma =-\Gamma _{2(1)}d\mu _{2}.$

This will give

$4\Gamma _{2(1)}^{2}d\mu _{2}=E(\lambda \gamma _{1}dr+2d\Gamma _{2(1)}),$

(649)

or

${\frac {4\Gamma _{2(1)}^{2}}{E}}=\lambda \gamma _{1}{\frac {dr}{d\mu _{2}}}+2{\frac {d\Gamma _{2(1)}}{d\mu _{2}}},$

(650)

where the differential coefficients are to be determined on the conditions that the temperature and all the potentials except $\mu _{1}$ and $\mu _{2}$ are constant, and that the pressure in the interior of the film shall remain equal to that in the contiguous gas-masses. The latter condition may be expressed by the equation

$(\gamma _{1}-\gamma _{1}')d\mu _{1}+(\gamma _{2}-\gamma _{2}')d\mu _{2}=0,$

(651)

in which $\gamma _{1}'$ and $\gamma _{2}'$ denote the densities of $S_{2}$ and $S_{2}$ in the contiguous gas-masses. (See (98).) When the tension of the surfaces of the film and the pressures in its interior and in the contiguous