Page:Scientific Papers of Josiah Willard Gibbs.djvu/148

112 minors successively obtained, and $$R_{1}$$ for the last remaining constituent. Then if $$dt, d \mu_{1},... d \mu_{n-1}$$, and $$dv$$ all have the value zero, we have by (172) that is, - =i. In like manner we obtain

Therefore, the conditions obtained by writing $$d$$ for $$\Delta$$ in (166)–(169) are equivalent to this, that the determinant given above with the $$n$$ minors obtained from it as above mentioned and the last remaining $$\frac{d^2 \epsilon}{d \eta^2}$$ constituent shall all be positive. Any phase for which this condition is satisfied will be stable, and no phase will be stable for which any of these quantities has a negative value. But the conditions (166)-(169) will remain valid, if we interchange in any way $$\eta, m_{1},...m_{n}$$ (with corresponding interchange of $$t, \mu_{1},...\mu_{n}$$). Hence the order in which we erase successive columns with the corresponding rows in the determinant is immaterial. Therefore none of the minors of the determinant (173) which are formed by erasing corresponding rows and columns, and none of the constituents of the principal diagonal, can be negative for a stable phase.

We will now consider the conditions which characterize the limits of stability (i.e., the limits which divide stable from unstable phases) with respect to continuous changes. Here, evidently, one of the conditions (166)–(169) must cease to hold true. Therefore, one of the differential coefficients formed by changing $$\Delta$$ into $$d$$ in the first members of these conditions must have the value zero. (That it is the numerator and not the denominator in the differential coefficient which vanishes at the limit appears from the consideration that the denominator is in each case the differential of a quantity which is necessarily capable of progressive variation, so long at least as the phase is capable of variation at all under the conditions expressed by the subscript letters.) The same will hold true of the set of differential coefficients obtained from these by interchanging in any way $$\eta, m_{1},...m_{n}$$, and simultaneously interchanging $$t, \mu_{1},...\mu_{n}$$ in the same way. But we may obtain a more definite result than this.