EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 295 



A, B, C meet is stable with respect to the formation of the fluid D. 

 When p D 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 V D , the angles of each of the curvilinear triangles 

 adb, bdc, cda will 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 determine 

 the magnitude of V D . For the tensions will give the total curvatures 

 (in degrees) of the lines ab, be, ca; and the pressures will give 

 the radii of curvature. These lines are thus completely determined. 

 In order that v^ shall be very small it is evidently necessary that 

 Pv 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, V D may be quite small when the value of > D 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 represented as 

 in figure 15, and v^ is small, TF V is negative, and the equilibrium 

 of the mass D is stable. Moreover, W B W y , which represents the 

 work 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 D moderately small 

 but not so small as to be neglected (this will be when p^ is some- 

 what smaller than the second member of (636), more or less smaller 

 according as the tensions differ more or less from such as are repre- 

 sented 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. 

 For greater values of _p D (with the same values of the other pressures 

 and the tensions) the same will be true. For somewhat smaller 

 values of > D , 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 p^ , we may likewise regard the surfaces A-B, B-C, 

 C-A as capable of meeting in stable equilibrium. It may be observed 

 that when t> D , as determined by our equations, becomes quite insensible, 

 the conception of a small mass D having the properties deducible 

 from our equations ceases to be accurate, since the matter in the 

 vicinity of a line where these surfaces of discontinuity meet must be 



