260 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



In the preceding paragraph it is shown that the surface of contact 

 of phases A and B is stable under certain circumstances, with respect 

 to the formation of a thin sheet of the phase C. To complete the 

 demonstration of the stability of the surface with respect to the 

 formation of the phase C, it is necessary to show that this phase 

 cannot be formed at the surface in lentiform masses. This is the 

 more necessary, since it is in this manner, if at all, that the phase 

 is likely to be formed, for an incipient sheet of phase 

 C would evidently be unstable when CAB <0- A o+0"Bc> 

 and would immediately break up into lentiform 

 masses. 



It will be convenient to consider first a lentiform 

 mass of phase C in equilibrium between masses of 

 phases A and B which meet in a plane surface. Let 

 figure 10 represent a section of such a system through 

 the centers of the spherical surfaces, the mass of phase 

 A lying on the left of DEH'FG, and that of phase B 

 on the right of DEH"FG. Let the line joining the 

 centers cut the spherical surfaces in IT and H", and the 

 plane of the surface of contact of A and B in I. Let 

 the radii of EH'F and EH"F be denoted by r', r", and the segments 

 IH', IH", by x', x". Also let IE, the radius of the circle in 

 which the spherical surfaces intersect, be denoted by R. By a 

 suitable application of the general condition of equilibrium we may 

 easily obtain the equation 



r -x' 



r"-x 



(561) 



which signifies that the components parallel to EF of the tension 

 (TAG & n d <T BO are together equal to O- A B- If we denote by TFthe amount 

 of work which must be expended in order to form such a lentiform 

 mass as we are considering between masses of indefinite extent having 

 the phases A and B, we may write 



W=M-N, (562) 



where M denotes the work expended in replacing the surface between 

 A and B by the surfaces between A and C and B and C, and N 

 denotes the work gained in replacing the masses of phases A and B 

 by the mass of phase C. Then 



-O-ABAB> ( 563 ) 



where s A c> S BO> S AB denote the areas of the three surfaces concerned; 

 and 



JV= V'(p G -p A ) + V"(p -p B ), (564) 



