464 J. W. Gibbs — Equilibrium, of Heterogeneoui<. IStibstances. 



the six distances of four points in space (see page 455), — a condition 

 which evidently agrees with the supposition which we have made. If 

 we denote by 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 

 10^ the work expended in forming the new surfaces in place of the old, 

 it may easily be shown by a method similar to that ixsed on page 459 

 that ««s=f*^v- From this we obtain Ws — w)v:=^^^v- This is evidently 

 positive when jOj, is greater than the other pressures. But it diminishes 

 with increase of />d, as easily appears from the equivalent expression 

 ^s- Hence the line of intersection of the surfaces of discontinuity A-B, 

 B-C, C-A is stable for values of p-o greater than the other pressures 

 (and therefore for all values of />o) so long as our method is to be re- 

 garded 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 

 <5'daj ^dbj (^dc cannot favor the formation of D, and hence cannot im- 

 pair 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 

 p-Q 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 page 

 426.) 



The Condition of IStability for Fluids relating to the Formation 



of a Neio 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-x\-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 ver- 

 tical point with respect to the possible formation of a different fluid 

 mass E. 



