EQUILIBEIUM OF HETEKOGENEOUS SUBSTANCES. 297 



equivalent expression %w a . Hence the line of intersection of the 

 surfaces of discontinuity A-B, B-C, C-A is stable for values of 

 greater than the other pressures (and therefore for all values of 

 so long as our method is to be regarded as accurate, which will be so 

 long as the mass D which would be in equilibrium has a sensible size. 

 In certain cases in which the tensions of the new surfaces are much 

 too large to be represented as in figure 15, the reasoning of the two 

 last paragraphs will cease to be applicable. These are cases in which 

 the six tensions cannot be represented by the sides of a tetrahedron. 

 It is not necessary to discuss these cases, which are distinguished by 

 the different shape which the mass D would take if it should be 

 formed, since it is evident that they can constitute no exception to 

 the results which we have obtained. For an increase of the values 

 of o- DA , <r DB , <TDC cannot favor the formation of D, and hence cannot 

 impair the stability of the line considered, as deduced from our equa- 

 tions. Nor can an increase of these tensions essentially affect .the 

 fact that the stability thus demonstrated may fail to be realized when 

 Pv is considerably greater than the other pressures, since the a priori 

 demonstration of the stability of any one of the surfaces A-B, B-C, C-A, 

 taken singly, is subject to the limitation mentioned. (See pages 

 261, 262.) 



The Condition of Stability for Fluids relating to the Formation of 

 a New Phase at a Point where the Vertices of Four Different 

 Masses meet. 



Let four different fluid masses A, B, C, D meet about a point, so as 

 to form the six surfaces of discontinuity A-B, B-C, C-A, D-A, D-B, 

 D-C, which meet in the four lines A-B-C, B-C-D, C-D-A, D-A-B, these 

 lines meeting in the vertical point. Let us suppose the system stable in 

 other respects, and consider the conditions of stability for the vertical 

 point with respect to the possible formation of a different fluid mass E. 



If the system can be in equilibrium when the vertical point has 

 been replaced by a mass E against which the four masses A, B, C, D 

 abut, being truncated at their vertices, it is evident that E will have 

 four vertices, at each of which six surfaces of discontinuity meet. 

 (Thus at one vertex there will be the surfaces formed by A, B, C, 

 and E.) The tensions of each set of six surfaces (like those of the 

 six surfaces formed by A, B, C, and D) must therefore be such that 

 they can be represented by the six edges of a tetrahedron. When 

 the tensions do not satisfy these relations, there will be no particular 

 condition of stability for the point about which A, B, C, and D meet, 

 since if a mass E should be formed, it would distribute itself along 

 some of the lines or surfaces which meet at the vertical point, and it 



