Page:Scientific Papers of Josiah Willard Gibbs.djvu/146

We see again by (165) that the general condition requires that if we regard $t,v,m_{1},m_{2},...m_{n}$ as having the constant values indicated by accenting these letters, $\mu _{n}$ shall be an increasing function of $m_{1}$ when the variable phase differs sufficiently little from the fixed. But as the fixed phase may be any one within the limits of stability, $\mu _{1}$ must be an increasing function of $m_{1}$ (within these limits) for any constant values of $t,v,m_{1},m_{2},...m_{n}$. That is,

$\left({\frac {\Delta \mu _{1}}{\Delta m_{1}}}\right)_{t,v,m_{2},...m_{n}}>0.$

(167)

When this condition is satisfied, as well as (166), $\Phi$ will have a minimum value, for any constant values of $v,m_{2},...m_{n}$, when $t=t'$ and $\mu _{1}=\mu _{1}'$; so that in applying the general condition of stability we need only consider the phases for which $t=t'$ and $\mu _{1}=\mu _{1}'$.

In this way we may also obtain the following particular conditions of stability:

$\left({\frac {\Delta \mu _{2}}{\Delta m_{2}}}\right)_{t,v,\mu _{1},m_{3},...m_{n}}>0,$

(168)

$\left({\frac {\Delta \mu _{n}}{\Delta m_{n}}}\right)_{t,v,\mu _{1},...\mu _{n-1}}>0.$

(169)

When the $n+1$ conditions (166)–(169) are all satisfied, the value of $\Phi$, for any constant value of $v$, will be a minimum when the temperature and the potentials of the variable phase are equal to those of the fixed. The pressures will then also be equal and the phases will be entirely identical. Hence, the general condition of stability will be completely satisfied, when the above particular conditions are satisfied.

From the manner in which these particular conditions have been derived, it is evident that we may interchange in them $\eta ,m_{1},...m_{n}$ in any way, provided that we also interchange in the same way $t,\mu _{1},...\mu _{n}$. In this way we may obtain different sets of $n+1$ conditions which are necessary and sufficient for stability. The quantity $v$ might be included in the first of these lists, and $p$ in the second, except in cases when, in some of the phases considered, the entropy or the quantity of one of the components has the value zero. Then the condition that that quantity shall be constant would create a restriction upon the variations of the phase, and cannot be substituted for the condition that the volume shall be constant in the statement of the general condition of stability relative to the minimum value of $\Phi$.

To indicate more distinctly all these particular conditions at once, we observe that the condition (144), and therefore also the condition obtained by interchanging the single and double accents, must hold