290 



EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



of discontinuity meet. We may limit ourselves to the case in which 

 there are three such surfaces, this being the only one of frequent 

 occurrence, and may treat them as meeting in a straight line. It 

 will be convenient to commence by considering the equilibrium of a 

 system in which such a line is replaced by a filament of a different 

 phase. 



Let us suppose that three homogeneous fluid masses, A, B, and C 

 are separated by cylindrical (or plane) surfaces, A-B, B-C, C-A, which 

 at first meet in a straight line, each of the surface-tensions <r AB , er BC , or CA 

 being less than the sum of the other two. Let us suppose that the- 

 system is then modified by the introduction of a fourth fluid mass D, 

 which is placed between A, B, and C, and is separated from them by 

 cylindrical surfaces D-A, D-B, D-C meeting A-B, B-C, and C-A in 

 straight lines. The general form of the surfaces is shown by figure 14^ 

 in which the full lines represent a section perpendicular to all the 

 surfaces. The system thus modified is to be in equilibrium, as well 

 as the original system, the position of the surfaces A-B, B-C, C-A 

 being unchanged. That the last condition is consistent with equili- 

 brium will appear from the following mechanical considerations. 



FIG. 14. 



Fm. 15. 



FIG. 16. 



Let V-Q denote the volume of the mass D per unit of length or the area 

 of the curvilinear triangle abc. Equilibrium is evidently possible for 

 any values of the surface tensions (if only ar AE , <TBC> O"CA satisfy the con- 

 dition mentioned above, and the tensions of the three surfaces meet- 

 ing at each of the edges of D satisfy a similar condition) with any 

 value (not too large) of %>, if the edges of D are constrained to remain 

 in the original surfaces A-B, B-C, and C-A, or these surfaces extended, 

 if necessary, without change of curvature. (In certain cases one of 

 the surfaces DA, D-B, D-C may disappear and D will be bounded 

 by only two cylindrical surfaces.) We may therefore regard the 

 system as maintained in equilibrium by forces applied to the edges 

 of D and acting at right angles to A-B, B-C, C-A. The same forces 

 would keep the system in equilibrium if D were rigid. They must 

 therefore have a zero resultant, since the nature of the mass D is im- 

 material when it is rigid, and no forces external to the system would 

 be required to keep a corresponding part of the original system in 



