Page:Scientific Papers of Josiah Willard Gibbs.djvu/298

262 of insensible magnitude. {The diminution of the radii with increasing values of $$p_{C} - p_{A}$$ is indicated by equation (565).} Hence, no mass of phase $$C$$ will be formed until one of these limits is reached. Although the demonstration relates to a plane surface between $$A$$ and $$B$$, the result must be applicable whenever the radii of curvature have a sensible magnitude, since the effect of such curvature may be disregarded when the lentiform mass is sufficiently small.

The equilibrium of the lentiform mass of phase $$C$$ is easily proved to be unstable, so that the quantity $$W$$ affords a kind of measure of the stability of plane surfaces of contact of the phases $$A$$ and $$B$$. Essentially the same principles apply to the more general problem in which the phases $$A$$ and $$B$$ have moderately different pressures, so that their surfaces of contact must be curved, but the radii of curvature have a sensible magnitude.

In order that a thin film of the phase $$C$$ may be in equilibrium between masses of the phases $$A$$ and $$B$$, the following equations must be satisfied:— where $$c_{1}$$ and $$c_{2}$$ denote the principal curvatures of the film, the centers of positive curvature lying in the mass having the phase $$A$$. Eliminating $$c_{1} + c_{2}$$, we have or It is evident that if $$p_{C}$$ has a value greater than that determined by this equation, such a film will develop into a larger mass; if $$p_{C}$$ has a less value, such a film will tend to diminish. Hence, when the phases $$A$$ and $$B$$ have a stable surface of contact.