460 J. W. Gihbs — Equilibrmm of Heterogeneous Substances. 



A-B, B-C, C-A as fixed, is that T^ — Wv shall be a minimum under 

 the same conditions. [See (549).] Now for any constant values of 

 the tensions and of jo^, p^^ Pc, we may make v^ so small that when it 

 varies, the system remaining in eqiiilibrium, (which will in general 

 require a variation of jOd,) we may neglect the curvatures of the lines 

 da, db, d c, and regard the figure abed as remaining similar to 

 itself. For the total curvature (^. e., the curvature measured in 

 degrees) of each of the lines a b, be, ca may be regarded as con- 

 stant, being equal to the constant difierence of the sum of the angles 

 of one of the curvilinear triangles a db, b d c, c da and two right 

 angles. Thei-efore, when v^ is very small, and the system is so 

 deformed that equilibrium would be preserved if pj, had the proper 

 variation, but this pressure as well as the others and all the tensions 

 remain constant, W^ will vary as the lines in the figure abed, and 

 Wv as the square of these lines. Therefore, for such deformations, 



This shows that the system cannot be stable for constant pressures 

 and tensions when v^ is small and Wy is positive, since Ws — Wy 

 will not be a minimixm. It also shows that the system is stable 

 when Wy is negative. For, to determine whether Wl— Wy is a 

 minimum for constant values of the pressures and tensions, it will 

 evidently be suflicient to consider such varied forms of the system as 

 give the least value to Wl— fVy for any value of Vd in connection 

 with the constant pressures and tensions. And it may easily be 

 shown that such forms of the system are those which would pre- 

 serve equilibrium if p^ had the proper value. 



These i-esults will enable us to determine the most important ques- 

 tions relating to the stability of a line along which three homogene- 

 ous fluids A, B, C meet, with respect to the formation of a difierent 

 fluid D. The components of D must of course be such as are found 

 in the surrounding bodies. We shall regard p^) and o'da, Cdb, o'pc as 

 determined by that phase of D which satisfies the conditions of equi- 

 librium with the other bodies relating to temperature and the 

 potentials. These quantities are therefore determinable, by means 

 of the fundamental equations of the mass D and of the surfaces D-A, 

 D-B, D-0, from the temperature and potentials of the given system. 



Let us first consider the case in which the tensions, thus deter- 

 mined, can be represented as in figure 15, and jOj, has a value con- 

 sistent with the equilibrium of a small mass such as we have been 

 considering. It appears from the preceding discussion that when v^ is 



