Page:Scientific Papers of Josiah Willard Gibbs.djvu/362

326 Let us now examine the special condition of equilibrium which relates to a line at which three different masses meet, when one or more of these masses is solid. If we apply the method of pages 316, 317 to a system containing such a line, it is evident that we shall obtain in the expression corresponding to (660), beside the integral relating to the surfaces, a term of the form to be interpreted as the similar term in (611), except so far as the definition of $$\sigma$$ has been modified in its extension to solid masses. In order that this term shall be incapable of a negative value it is necessary that at every point of the line for any possible displacement of the line. Those displacements are to be regarded as possible which are not prevented by the solidity of the masses, when the interior of every solid mass is regarded as incapable of motion. At the surfaces between solid and fluid masses, the processes of solidification and dissolution will be possible in some cases, and impossible in others.

The simplest case is when two masses are fluid and the third is solid and insoluble. Let us denote the solid by $$S$$, the fluids by $$A$$ and $$B$$, and the angles filled by these fluids by $$\alpha$$ and $$\beta$$ respectively. If the surface of the solid is continuous at the line where it meets the two fluids, the condition of equilibrium reduces to If the line where these masses meet is at an edge of the solid, the condition of equilibrium is that

which reduces to the preceding when $$\alpha + \beta = \pi$$. Since the displacement of the line can take place by a purely mechanical process, this