J. W. Glhbs — Equilibrium of Heterogeneous Siibstances. 457 



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 tlie 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. Fig. 15. Fig. 16. 



Let v-o denote the volume of the mass D per unit of length or the area 

 of the curvilinear triangle a b c. Equilibrium is evidently possible for 

 any values of the surface-tensions (if only o'ab, (^bc? '5'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 t^p, if the edges of D ai'e 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 D-A, 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 Avhen it is rigid, and no forces external to the system would 

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

 equilibrium. But it is evident from the points of application and 

 directions of these forces that they cannot have a zero resultant unless 

 each force is zero. We may therefore introduce a fourth mass D 

 without disturbing the parts which remain of the surfaces A-B, B-C, 

 C-D. 



It will be observed that all the angles at a, b, c, and d in figure 14 

 are entirely determined by the six surface-tensions Cab? <>bc5 c^caj o'da, 

 ^DB5 ^Dc- [►^e^ (615).] The angles maybe derived from the tensions 



