Page:Scientific Papers of Josiah Willard Gibbs.djvu/329

The condition of stability for the system when the pressures and tensions are regarded as constant, and the position of the surfaces ${\text{A-B, B-C, C-A}}$ as fixed, is that $W_{S}-W_{V}$ shall be a minimum under the same conditions. (See (549).) Now for any constant values of the tensions and of $p_{A},p_{B},p_{C}$, we may make $v_{D}$ so small that when it varies, the system remaining in equilibrium (which will in general require a variation of $p_{D}$), we may neglect the curvatures of the lines $da,db,dc$, and regard the figure $abcd$ as remaining similar to itself. For the total curvature (i.e., the curvature measured in degrees) of each of the lines $ab,bc,ca$ may be regarded as constant, being equal to the constant difference of the sum of the angles of one of the curvilinear triangles $adb,bdc,cda$ and two right angles. Therefore, when V D is very small, and the system is so deformed that equilibrium would be preserved if $v_{D}$ had the proper variation, but this pressure as well as the others and all the tensions remain constant, $W_{S}$ will vary as the lines in the figure $abcd$, and $W_{V}$ as the square of these lines. Therefore, for such deformations,

$W_{V}\propto W_{S}^{2}.$

This shows that the system cannot be stable for constant pressures and tensions when $W_{V}$ is small and $W_{V}$ is positive, since $W_{S}-W_{V}$ will not be a minimum. It also shows that the system is stable when $W_{V}$ is negative. For, to determine whether $W_{S}-W_{V}$ is a minimum for constant values of the pressures and tensions, it will evidently be sufficient to consider such varied forms of the system as give the least value to $W_{S}-W_{V}$ for any value of $v_{D}$ in connection with the constant pressures and tensions. And it may easily be shown that such forms of the system are those which would preserve equilibrium if $p_{D}$ had the proper value.

These results will enable us to determine the most important questions relating to the stability of a line along which three homogeneous fluids $A,B,C$ meet, with respect to the formation of a different fluid $D$. The components of $D$ must of course be such as are found in the surrounding bodies. We shall regard $p_{D}$ and $\sigma _{DA},\sigma _{DB},\sigma _{DC}$ as determined by that phase of $D$ which satisfies the conditions of equilibrium with the other bodies relating to temperature and the potentials. These quantities are therefore determinable, by means of the fundamental equations of the mass $D$ and of the surfaces ${\text{D-A, D-B, D-C}}$, from the temperature and potentials of the given system.

Let us first consider the case in which the tensions, thus determined, can be represented as in figure 15, and $p_{D}$ has a value consistent with the equilibrium of a small mass such as we have been considering. It appears from the preceding discussion that