Page:Scientific Papers of Josiah Willard Gibbs.djvu/168

We are not, however, absolutely certain that equation (200) will always be satisfied by a critical phase. For it is possible that the denominator in the fraction may vanish as well as the numerator for an infinitesimal change of phase in which the quantities indicated are constant. In such a case, we may suppose the subscript $n$ to refer to some different component substance, or use another differential coefficient of the same general form (such as are described on page 114 as characterizing the limits of stability in respect to continuous changes), making the corresponding changes in (201) and (202). We may be certain that some of the formulae thus formed will not fail. But for a perfectly rigorous method there is an advantage in the use of $\eta ,v,m_{1},m_{2},...m_{n}$ as independent variables. The condition that the phase may be varied without altering any of the quantities $t,p,\mu _{1},\mu _{2},...\mu _{n}$ will then be expressed by the equation

$R_{n+1}=0,$

(203)

in which $R_{n+1}$ denotes the same determinant as on page 111. To obtain the second equation characteristic of critical phases, we observe that as a phase which is critical cannot become unstable when varied so that $n$ of the quantities $t,p,\mu _{1},\mu _{2},...\mu _{n}$ remain constant, the differential of $R_{n+1}$ for constant volume, viz.,

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

(204)

cannot become negative when n of the equations (172) are satisfied. Neither can it have a positive value, for then its value might become negative by a change of sign of $d\eta ,dm_{1},$ etc. Therefore the expression (204) has the value zero, if n of the equations (172) are satisfied.

This may be expressed by an equation

$S=0,$

(205)

in which $S$ denotes a determinant in which the constituents are the same as in $R_{n+1},$ except in a single horizontal line, in which the differential coefficients in (204) are to be substituted. In whatever line this substitution is made, the equation (205), as well as (203), will hold true of every critical phase without exception.

If we choose $t,p,m_{1},m_{2},...m_{n}$ as independent variables, and write $U$ for the determinant

${\begin{vmatrix}{\frac {d^{2}\zeta }{dm_{1}^{2}}}&{\frac {d^{2}\zeta }{dm_{2}dm_{1}}}&...&{\frac {d^{2}\zeta }{dm_{n-1}dm_{1}}}\\{\frac {d^{2}\zeta }{dm_{1}dm_{2}}}&{\frac {d^{2}\zeta }{dm_{2}^{2}}}&...&{\frac {d^{2}\zeta }{dm_{n-1}dm_{2}}}\\...&...&...&...\\{\frac {d^{2}\zeta }{dm_{1}dm_{n-1}}}&{\frac {d^{2}\zeta }{dm_{2}dm_{n-1}}}&...&{\frac {d^{2}\zeta }{dm_{n-1}^{2}}}\\\end{vmatrix}}$

(206)