Page:Scientific Papers of Josiah Willard Gibbs.djvu/169

and $V$ for the determinant formed from this by substituting for the constituents in any horizontal line the expressions

${\frac {dU}{dm_{1}}},\,\,\,{\frac {dU}{dm_{2}}},\,\,\,...\,\,\,{\frac {dU}{dm_{n-1}}},$

(207)

the equations of critical phases will be

$U=0,\,\,V=0.$

(208)

It results immediately from the definition of a critical phase, that an infinitesimal change in the condition of a mass in such a phase may cause the mass, if it remains in a state of dissipated energy (i.e., in a state in which the dissipation of energy, by internal processes is complete), to cease to be homogeneous. In this respect a critical phase resembles any phase which has a coexistent phase, but differs from such phases in that the two parts into which the mass divides when it ceases to be homogeneous differ infinitely little from each other and from the original phase, and that neither of these parts is in general infinitely small. If we consider a change in the mass to be determined by the values of $d\eta ,dv,dm_{1},dm_{2},...dm_{n}$ it is evident that the change in question will cause the mass to cease to be homogeneous whenever the expression

${\frac {dR_{n+1}}{d\eta }}d\eta +{\frac {dR_{n+1}}{dv}}dv+{\frac {dR_{n+1}}{dm_{1}}}dm_{1},...+{\frac {dR_{n+1}}{dm_{n}}}dm_{n}$

(209)

has a negative value. For if the mass should remain homogeneous, it would become unstable, as $R_{n+1}$ would become negative. Hence, in general, any change thus determined, or its reverse (determined by giving to $d\eta ,dv,dm_{1},dm_{2},...dm_{n}$ the same values taken negatively), will cause the mass to cease to be homogeneous. The condition which must be satisfied with reference to $d\eta ,dv,dm_{1},dm_{2},...dm_{n},$ in order that neither the change indicated, nor the reverse, shall destroy the homogeneity of the mass, is expressed by equating the above expression to zero.

But if we consider the change in the state of the mass (supposed to remain in a state of dissipated energy) to be determined by arbitrary values of $n+1$ of the differentials $dt,dp,d\mu _{1},d\mu _{2},...d\mu _{n}$ the case will be entirely different. For, if the mass ceases to be homogeneous, it will consist of two coexistent phases, and as applied to these, only $n$ of the quantities $dt,dp,d\mu _{1},d\mu _{2},...d\mu _{n}$ will be independent. Therefore, for arbitrary variations of $n+1$ of these quantities, the mass must in general remain homogeneous.

But if, instead of supposing the mass to remain in a state of dissipated energy, we suppose that it remains homogeneous, it may easily be shown that to certain values of $n+1$ of the above differentials