Page:Scientific Papers of Josiah Willard Gibbs.djvu/294

258 sign, is always equal to two-thirds of the first, as appears from equation (550) and the geometrical relation $$v' = \tfrac{1}{3}rs$$. We may therefore write

Let $$A, B$$, and $$C$$ be three different fluid phases of matter, which satisfy all the conditions necessary for equilibrium when they meet at plane surfaces. The components of $$A$$ and $$B$$ may be the same or different, but $$C$$ must have no components except such as belong to $$A$$ or $$B$$. Let us suppose masses of the phases $$A$$ and $$B$$ to be separated by a very thin sheet of the phase $$C$$. This sheet will not necessarily be plane, but the sum of its principal curvatures must be zero. We may treat such a system as consisting simply of masses of the phases $$A$$ and $$B$$ with a certain surface of discontinuity, for in our previous discussion there has been nothing to limit the thickness or the nature of the film separating homogeneous masses, except that its thickness has generally been supposed to be small in comparison with its radii of curvature. The value of the superficial tension for such a film will be $$\sigma_{AC} + \sigma_{BC}$$, if we denote by these symbols the tensions of the surfaces of contact of the phases $$A$$ and $$C$$, and $$B$$ and $$C$$, respectively.

This not only appears from evident mechanical considerations, but may also be easily verified by equations (502) and (93), the first of which may be regarded as defining the quantity or. This value will not be affected by diminishing the thickness of the film, until the limit is reached at which the interior of the film ceases to have the properties of matter in mass. Now if $$\sigma_{AC} + \sigma_{BC}$$ is greater than $$\sigma_{AB}$$, the tension of the ordinary surface between $$A$$ and $$B$$, such a film will be at least practically unstable. (See page 240.) We cannot suppose that $$\sigma_{AB} > \sigma_{AC} + \sigma_{BC}$$, for this would make the ordinary surface between $$A$$ and $$B$$ unstable and difficult to realize. If $$\sigma_{AB} = \sigma_{AC} + \sigma_{BC}$$ we may assume, in general, that this relation is not accidental, and that the ordinary surface of contact for $$A$$ and $$B$$ is of the kind which we have described.

Let us now suppose the phases $$A$$ and $$B$$ to vary, so as still to satisfy the conditions of equilibrium at plane contact, but so that the pressure of the phase $$C$$ determined by the temperature and potentials