462 J. W. Gibhs — Equilibrimn of Heterogeneous Substances. 



the tensions satisfy the condition supposed and Vq is small, it follows 

 that when p^ has a less value, the line where the fluids A, B, C meet 

 is stable with respect to the formation of the fluid D, Whenp^ has 

 a greater value, if such a line can exist at all, it must be at least 

 practically unstable, i. e., if only a very small mass of the fluid D 

 should be formed it would tend to increase. 



Let us next consider the case in which the tensions of the 

 new surfaces are too small to be represented as in figure 15. If 

 the pressures and tensions are consistent with equilibrium for any 

 very small value of Vj), the angles of each of the curvilinear tri- 

 angles adb, bdc, cc?awill be together less than two right angles, 

 and the lines ab, be, ca, will be convex toward the mass D. For 

 given values of the pressures and tensions, it will be easy to deter- 

 mine the magnitude of Vj). For the tensions will give the total 

 curvatures (in degrees) of the lines a b, be, ca; and the pressures 

 will give the radii of curvature. These lines are thus completely 

 determined. In order that ?Jd shall be very small it is evidently 

 necessary that p^ shall be less than the other pressures. Yet if the 

 tensions of the new surfaces are only a very little too small to be 

 represented as in figure 15, Vi-, may be quite small when the value 

 of pu is only a little less than that given by equation (636). In any 

 case, when the tensions of the new surfaces are too small to be repre- 

 sented as in figure 15, and Vj) is small, TFy is negative, and the equi- 

 librium of the mass D is stable. Moreover, W^ — Wy, which repre- 

 sents the woT'k necessary to form the mass D with its surfaces in 

 place of the other masses and surfaces, is negative. 



With respect to the stability of a line in which the surfaces A-B, 

 B-C, C-A meet, when the tensions of the new surfaces are too small to 

 be represented as in figure 15, we first observe that when the pressures 

 and tensions are such as to make v^ moderately small but not so 

 small as to be neglected, [this will be when p^ is somewhat smaller 

 than the second member of (636), — more or less smaller according as 

 the tensions diflfer more or less from such as are represented in 

 figure 15,] the equilibrium of such a line as that supposed (if it is 

 capable of existing at all) is at least practically unstable. P^'or greater 

 values of po (with the same values of the other pressures and the 

 tensions) the same will be ti'ue. For somewhat smaller values of pn, 

 the mass of the phase D which will be formed will be so small, that 

 we may neglect this mass and regard the surfaces A-B, B-C, C-A as 

 meeting in a line in stable equilibrium. For still smaller values of 

 Pp, we may likewise regard the surfaces A-B, B-C, C-A as capable 



