456 J. W. Gibhs — Equilihrium of Heterogeneous /Substances. 



than the tension of the corresponding surface, but a certain diagonal 

 is equal to the tension in the polygons constructed for a finite portion 

 of the line, farther investigations are necessary to determine the 

 stability of the system. For this purpose, the method described on 

 page 413 is evidently applicable, 



A similar proposition may be enunciated in many cases with re- 

 spect to a point about which the angular space is divided into solid 

 angles by surfaces of discontinuity. If these surfaces are in equilib- 

 rium, we can always form a closed solid figure without re-enti-ant 

 angles of which the angular points shall correspond to the several 

 masses, the edges to the surfaces of discontinuity, and the sides to 

 the lines in which these edges meet, the edges being perpendicular 

 to the corresponding surfaces, and equal to their tensions, and the 

 sides being perpendicnlar to the corresponding lines. Now if the 

 solid angles in the physical system are such as may be subtended by 

 the sides and bases of a triangular prism enclosing the vei'tical point, 

 or can be derived from such by deformation, the figure representing 

 the tensions will have the form of two triangular pyramids on oppo- 

 site sides of the same base, and the system will be stable or pi-actic- 

 ally unstable with respect to the formation of a surface between the 

 masses which only meet in a point, according as the tension of a sur- 

 face between such masses is greater or less than the diagonal joining 

 the corresponding angular points of the solid representing the ten- 

 sions. This will easily appear on consideration of the case in which 

 a very small surface between the masses would be in equilibrium. 



The Conditions of Stability for Fluids relating to the Formation 



of a Neic Phase at a Line in which Three Surfaces of 



Discontinuity meet. 



With regard to the fonuation of new phases there are particular 

 conditions of stability which relate to lines in which several surfaces 

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

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

 rence, 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 tliree 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 (Tab^ o'bc, Gck 

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



