J. W. Gibbs — Equilibrium of Heterogeneous Substances. 463 



of meeting in stable e<juilibrium. It may be observed that when 

 Vq, as determined by our equations, becomes quite insensible, 

 the conception of a small mass D having the properties deducible 

 from our equations ceases to be accurate, since the matter in the 

 vicinity of a line where these surfaces of discontinuity meet must 

 be in a peculiar state of equilibrium not recognized by our equations.* 

 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 sup- 

 pose that they are not very much too great to be thus represented. 

 When the pressures are such as to make Vq 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-Q is somewhat greater than the second member of (636), — more or 

 less o-reater according as the tensions differ more or less from such as 

 ai*e represented in figure 15,] the line where the surfaces 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 TP^— TFy will be posi- 

 tive. The same will be true for less values of jOd. For greater values 

 of jOu, the value of TFg - TFy, 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 jt>D- But these equa- 

 tions 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 D-A, D-B, D-C meet together (with the surfaces 

 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 



* See note on page 455. "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 sur- 

 rounding masses. The value of the linear tension for the supposed line, would be 

 nearly equal to the value of W^ — W^ 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 equihbrium of such a double bubble we must recog- 

 nize a certain negative tension in the line of intersection of the three surfaces of 

 discontinuity. 



Trans. Conn. Acad., Vol. III. 59 March, 1878. 



