Page:Scientific Papers of Josiah Willard Gibbs.djvu/397

existence in virtue of properties which prevent the commencement of discontinuous changes. But a phase which is unstable with respect to continuous changes is evidently incapable of permanent existence on a large scale except in consequence of passive resistances to change. To obtain the conditions of stability with respect to continuous changes, we have only to limit the application of the variables in (14) to phases adjacent to the given phase. We obtain results of the following nature.

The stability of any phase with respect to continuous changes depends upon the same conditions with respect to the second and higher differential coefficients of the density of energy regarded as a function of the density of entropy and the densities of the several components, which would make the density of energy a minimum, if the necessary conditions with respect to the first differential coefficients were fulfilled.

Again, it is necessary and sufficient for the stability with respect to continuous changes of all the phases within any given limits, that within those limits the same conditions should be fulfilled with respect to the second and higher differential coefficients of the pressure regarded as a function of the temperature and the several potentials, which would make the pressure a minimum, if the necessary conditions with respect to the first differential coefficients were fulfilled.

The equation of the limits of stability with respect to continuous changes may be written

$\left({\frac {d\mu _{n}}{d\gamma _{n}}}\right)_{t,\mu _{1},...\mu _{n-1}}=0,$or$\left({\frac {d^{2}p}{d\mu _{n}^{2}}}\right)_{t,\mu _{1},...\mu _{n-1}}=\infty ,$

(15)

where $\gamma _{n}$ denotes the density of the component specified or $m_{n}\div v$. It is in general immaterial to what component the suffix $_{n}$ is regarded as relating.

Critical phases.—The variations of two coexistent phases are sometimes limited by the vanishing of the difference between them. Phases at which this occurs are called critical phases. A critical phase, like any other, is capable of $n+1$ independent variations, $n$ denoting the number of independently variable components. But when subject to the condition of remaining a critical phase, it is capable of only $n-1$ independent variations. There are therefore two independent equations which characterize critical phases. These may be written

$\left({\frac {d\mu _{n}}{d\gamma _{n}}}\right)_{t,\mu _{1},...\mu _{n-1}}=0,$$\left({\frac {d^{2}\mu _{n}}{d\gamma _{n}^{2}}}\right)_{t,\mu _{1},...\mu _{n-1}}=0.$

(16)

It will be observed that the first of these equations is identical with the equation of the limit of stability with respect to continuous