EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 293 



The condition of stability for the system when the pressures and 

 tensions are regarded as constant, and the position of the surfaces 

 A-B, B-C, C-A as fixed, is that W 8 W y shall be a minimum under 

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

 the tensions and of p A , p*,pc> w ^ may make i> D so small that when 

 it varies, the system remaining in equilibrium (which will in general 

 require a variation of ^D), we may neglect the curvatures of the 

 lines da, db, dc, and regard the figure abed as remaining similar 

 to itself. For the total curvature (i.e., the curvature measured in 

 degrees) of each of the lines ab, be, ca may be regarded as constant, 

 being equal to the constant difference of the sum of the angles of 

 one of the curvilinear triangles adb, bdc, cda and two right angles. 

 Therefore, when V D is very small, and the system is so deformed 

 that equilibrium would be preserved if jp D had the proper variation, 

 but this pressure as well as the others and all the tensions remain 

 constant, W B will vary as the lines in the figure abed, and TT V 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 D is small and TF V is positive, since W B W y 

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

 when TF V is negative. For, to determine whether W 8 TF V is a 

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

 evidently be sufficient to consider such varied forms of the system 

 as give the least value to W 8 W v for any value of Vj> in connection 

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

 shown that such forms of the system are those which would 

 preserve equilibrium if p^ had the proper value. 



These results will enable us to determine the most important 

 questions relating to the stability of a line along which three 

 homogeneous fluids A, B, C meet, with respect to the formation of 

 a different fluid D. The components of D must of course be such 

 as are found in the surrounding bodies. We shall regard p^ and 

 "DA> O"DB> O-DO as determined by that phase of D which satisfies 

 the conditions of equilibrium 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-C, 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 p D has a value 

 consistent with the equilibrium of a small mass such as we have 

 been considering. It appears from the preceding discussion that 



