Page:Scientific Papers of Josiah Willard Gibbs.djvu/143

we see that the stability of any phase in regard to continuous changes depends upon the same conditions in regard 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 in regard to the first differential coefficients were fulfilled. When $n=1$, it may be more convenient to regard $m$ as constant in (145) than $v$. Regarding $m$ a constant, it appears that the stability of a phase depends upon the same conditions in regard to the second and higher differential coefficients of the energy of a unit of mass regarded as a function of its entropy and volume, which would make the energy a minimum, if the necessary conditions in regard to the first differential coefficients were fulfilled.

The formula (144) expresses the condition of stability for the phase to which $t',p',$ etc., relate. But it is evidently the necessary and sufficient condition of the stability of all phases of certain kinds of matter, or of all phases within given limits, that (144) shall hold true of any two infinitesimally differing phases within the same limits, or, as the case may be, in general. For the purpose, therefore, of such collective determinations of stability, we may neglect the distinction between the two states compared, and write the condition in the form

$-\eta \Delta t+v\Delta p-m_{1}\Delta \mu _{1}...-m_{n}\Delta \mu _{n}>0,$

(148)

or $\Delta p>{\frac {\eta }{v}}\Delta t+{\frac {m_{1}}{v}}\Delta \mu _{1}...+{\frac {m_{n}}{v}}\Delta \mu _{n}.$

(149)

Comparing (98), we see that it is necessary and sufficient for the stability in regard to continuous changes of all the phases within any given limits, that within those limits the same conditions should be fulfilled in 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.

By equations (87) and (94), the condition (142) may be brought to the form

$\psi ''+t''\eta ''+p'v''-\mu _{1}'m''_{1}...-\mu _{n}'m''_{n}-\psi '-t'\eta ''-p'v'+\mu _{1}'m'_{1}...+\mu _{n}'m'_{n}>0.$

(150)

For the stability of all phases within any given limits it is necessary and sufficient that within the same limits this condition shall hold true of any two phases which differ infinitely little. This evidently requires that when $v'=v'',m_{1}'=m_{1}'',...m_{n}'=m''_{n},$

$\psi ''-\psi '+(t''-t')\eta ''>0;$

(151)