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

Rh 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, 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