Page:Scientific Papers of Josiah Willard Gibbs.djvu/167

determined by a pair of coexistent phases, and to ${\frac {m_{n}}{v}}$ a series of values increasing from the less to the greater of the values which it has in these coexistent phases, we determine a linear series of phases connecting the coexistent phases, in some part of which $\mu _{n}$—since it has the same value in the two coexistent phases, but not a uniform value throughout the series (for if it had, which is theoretically improbable, all these phases would be coexistent) must be a decreasing function of ${\frac {m_{n}}{v}},$ or of $m_{n},$ if $v$ also is supposed constant. Therefore, the series must contain phases which are unstable in respect to continuous changes. (See page 111.) And as such a pair of coexistent phases may be taken indefinitely near to any critical phase, the unstable phases (with respect to continuous changes) must approach indefinitely near to this phase.

Critical phases have similar properties with reference to stability as determined with regard to discontinuous changes. For as every stable phase which has a coexistent phase lies upon the limit which separates stable from unstable phases, the same must be true of any stable critical phase. (The same may be said of critical phases which are unstable in regard to discontinuous changes, if we leave out of account the liability to the particular kind of discontinuous change in respect to which the critical phase is unstable.)

The linear series of phases determined by giving to $n$ of the quantities $t,p,\mu _{1},\mu _{2},...\mu _{n}$ the constant values which they have in any pair of coexistent phases consists of unstable phases in the part between the coexistent phases, but in the part beyond these phases in either direction it consists of stable phases. Hence, if a critical phase is varied in such a manner that $n$ of the quantities $t,p,\mu _{1},\mu _{2},...\mu _{n}$ remain constant, it will remain stable in respect both to continuous and to discontinuous changes. Therefore $\mu _{n}$ is an increasing function of $m_{n}$ when $t,p,\mu _{1},\mu _{2},...\mu _{n-1}$ have constant values determined by any critical phase. But as equation (200) holds true at the critical phase, the following conditions must also hold true at that phase:

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

(201)

$\left({\frac {d^{3}\mu _{n}}{dm_{n}^{3}}}\right)_{t,p,\mu _{1},\mu _{2},...\mu _{n-1}}\geqq 0.$

(202)

If the sign of equality holds in the last condition, additional conditions, concerning the differential coefficients of higher orders, must be satisfied. Equations (200) and (201) may in general be called the equations of critical phases. It is evident that there are only two independent equations of this character, as a critical phase is capable of $n-1$ independent variations.