Page:Scientific Papers of Josiah Willard Gibbs.djvu/287

The total quantity of the substance indicated by the suffix $_{1}$ is

$\gamma _{1}'v'+\gamma _{1}''v''+\Gamma _{1}s.$

Making this constant, we have

$\left(\gamma _{1}'-\gamma _{1}''+\Gamma _{1}{\frac {ds}{dv'}}\right)dv'+\left(v'{\frac {d\gamma _{1}'}{d\mu _{1}}}+v''{\frac {d\gamma _{1}''}{d\mu _{1}}}+s{\frac {d\Gamma _{1}}{d\mu _{1}}}\right)d\mu _{1}=0.$

(546)

The condition of equilibrium is thus reduced to the form

$\sigma {\frac {d^{2}s}{dv^{'2}}}>-{\frac {\left(\gamma _{1}'-\gamma _{1}''-\Gamma _{1}{\frac {ds}{dv'}}\right)^{2}}{v'{\frac {d\gamma _{1}'}{d\mu _{1}}}+v''{\frac {d\gamma _{1}''}{d\mu _{1}}}+s{\frac {d\Gamma _{1}}{d\mu _{1}}}}},$

(547)

where ${\frac {ds}{dv'}}$ and ${\frac {d^{2}s}{dv'}}$ are to be determined from the form of the surface of tension by purely geometrical considerations, and the other differential coefficients are to be determined from the fundamental equations of the homogeneous masses and the surface of discontinuity. Condition (540) may be easily deduced from this as a particular case.

The condition of stability with reference to motion of surfaces of discontinuity admits of a very simple expression when we can treat the temperature and potentials as constant. This will be the case when one or more of the homogeneous masses, containing together all the component substances, may be considered as indefinitely large, the surfaces of discontinuity being finite. For if we write $\textstyle \sum \displaystyle \Delta \epsilon$ for the sum of the variations of the energies of the several homogeneous masses, and $\textstyle \sum \displaystyle \Delta \epsilon ^{S}$ for the sum of the variations of the energies of the several surfaces of discontinuity, the condition of stability may be written

$\textstyle \sum \displaystyle \Delta \epsilon +\textstyle \sum \displaystyle \Delta \epsilon ^{S}>0,$

(548)

the total entropy and the total quantities of the several components being constant. The variations to be considered are infinitesimal, but the character A signifies, as elsewhere in this paper, that the expression is to be interpreted without neglect of infinitesimals of the higher orders. Since the temperature and potentials are sensibly constant, the same will be true of the pressures and surface-tensions, and by integration of (86) and (501) we may obtain for any homogeneous mass

$\Delta \epsilon =t\Delta \eta -p\Delta v+\mu _{1}\Delta m_{1}+\mu _{2}\Delta m_{2}+{\text{etc.,}}$

and for any surface of discontinuity

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

These equations will hold true of finite differences, when $t,p,\sigma ,\mu _{1},\mu _{2}$, etc. are constant, and will therefore hold true of infinitesimal differences, under the same limitations, without neglect of the