Page:Scientific Papers of Josiah Willard Gibbs.djvu/166

of these relations, although it may prevent an experimental realization of the phases considered. For the sake of brevity, in the following discussion, phases in the vicinity of the critical phase will generally be called stable, if they are unstable only in respect to the formation of phases entirely different from any in the vicinity of the critical phase. Let us first consider the number of independent variations of which a critical phase (while remaining such) is capable. If we denote by $n$ the number of independently variable components, a pair of coexistent phases will be capable of n independent variations, which may be expressed by the variations of $n$ of the quantities $t,p,\mu _{1},\mu _{2},...\mu _{n}$ If we limit these variations by giving to $n-1$ of the quantities the constant values which they have for a certain critical phase, we obtain a linear^[1] series of pairs of coexistent phases terminated by the critical phase. If we now vary infinitesimally the values of these $n-1$ quantities, we shall have for the new set of values considered constant a new linear series of pairs of coexistent phases. Now for every pair of phases in the first series, there must be pairs of phases in the second series differing infinitely little from the pair in the first, and vice versa, therefore the second series of coexistent phases must be terminated by a critical phase which differs, but differs infinitely little, from the first. We see, therefore, that if we vary arbitrarily the values of any $n-1$ of the quantities, $t,p,\mu _{1},\mu _{2},...\mu _{n}$ as determined by a critical phase, we obtain one and only one critical phase for each set of varied values; i.e., a critical phase is capable of $n-1$ independent variations.

The quantities $t,p,\mu _{1},\mu _{2},...\mu _{n}$ have the same values in two coexistent phases, but the ratios of the quantities $\eta ,m_{1},m_{2},...m_{n}$ are in general different in the two phases. Or, if for convenience we compare equal volumes of the two phases (which involves no loss of generality), the quantities $\eta ,m_{1},m_{2},...m_{n}$ will in general have different values in two coexistent phases. Applying this to coexistent phases indefinitely near to a critical phase, we see that in the immediate vicinity of a critical phase, if the values of $n$ of the quantities $t,p,\mu _{1},\mu _{2},...\mu _{n}$ are regarded as constant (as well as $v$), the variations of either of the others will be infinitely small compared with the variations of the quantities $\eta ,m_{1},m_{2},...m_{n}.$ This condition, which we may write in the form

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

characterizes, as we have seen on page 114, the limits which divide stable from unstable phases in respect to continuous changes.

In fact, if we give to the quantities $t,\mu _{1},\mu _{2},...\mu _{n-1}$ constant values

1. This term is used to characterize a series having a single degree of extension.