296 EQUILIBKIUM OF HETEKOGENEOUS SUBSTANCES. 



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

 But this cannot affect the validity of our conclusion with respect ta 

 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 

 Pv 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 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 8 W y will be positive. 

 The same will be true for less values of _p D . For greater values of p^ r 

 the value of W s TT Y , which measures the stability with respect to 

 the kind of change considered, diminishes. It does not vanish, accord- 

 ing to our equations, for finite values of ^ 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 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 

 the six distances of four points in space (see pages 288, 289), a con- 

 dition 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 a 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 8 = %w y . From this we obtain w a w v = ^w y . This 

 is evidently positive when > D is greater than the other pressures. 

 But it diminishes w r ith increase of jp D , as easily appears from the 



* 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 sur- 

 rounding 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. 



