Page:Scientific Papers of Josiah Willard Gibbs.djvu/332

in a peculiar state of equilibrium not recognized by our equations.^[1] But this cannot affect the validity of our conclusion with respect to the stability of the line in question.

The case remains to be considered in which the tensions of the new surfaces are too great to be represented as in figure 15. Let us suppose that they are not very much too great to be thus represented. When the pressures are such as to make $v_{D}$ moderately small (in case of equilibrium) but not so small that the mass $D$ to which it relates ceases to have the properties of matter in mass (this will be when $p_{D}$ is somewhat greater than the second member of (636),—more or less greater according as the tensions differ more or less from such as are represented in figure 15), the line where the surfaces ${\text{A-B, B-C, C-A}}$ meet will be in stable equilibrium with respect to the formation of such a mass as we have considered, since $W_{S}-W_{V}$ will be positive. The same will be true for less values of $p_{D}$. For greater values of $p_{D}$, the value of $W_{S}-W_{V}$, which measures the stability with respect to the kind of change considered, diminishes. It does not vanish, according to our equations, for finite values of $p_{D}$. But these equations are not to be trusted beyond the limit at which the mass $D$ ceases to be of sensible magnitude.

But when the tensions are such as we now suppose, we must also consider the possible formation of a mass $D$ within a closed figure in which the surfaces ${\text{D-A, D-B, D-C}}$ meet together (with the surfaces ${\text{A-B, B-C, C-A}}$) in two opposite points. If such a figure is to be in equilibrium, the six tensions must be such as can be represented by the six distances of four points in space (see pages 288, 289), a condition which evidently agrees with the supposition which we have made. If we denote by $w_{V}$ the work gained in forming the mass $D$ (of such size and form as to be in equilibrium) in place of the other masses, and by $w_{S}$ the work expended in forming the new surfaces in place of the old, it may easily be shown by a method similar to that used on page 292 that $w_{S}={\tfrac {3}{2}}w_{V}$. From this we obtain $w_{S}-w_{V}={\tfrac {1}{2}}w_{V}$. This is evidently positive when $p_{D}$ is greater than the other pressures. But it diminishes with increase of $p_{D}$, as easily appears from the

1. See note on page 288. We may here add that the linear tension there mentioned may have a negative value. This would be the case with respect to a line in which three surfaces of discontinuity are regarded as meeting, but where nevertheless there really exists in stable equilibrium a filament of different phase from the three surrounding masses. The value of the linear tension for the supposed line, would be nearly equal to the value of $W_{S}-W_{V}$ for the actually existing filament. (For the exact value of the linear tension it would be necessary to add the sum of the linear tensions of the three edges of the filament.) We may regard two soap-bubbles adhering together as an example of this case. The reader will easily convince himself that in an exact treatment of the equilibrium of such a double bubble we must recognize a certain negative tension in the line of intersection of the three surfaces of discontinuity.