424 fj. W. Gibhs — Equilihrhim of Heterogeneous Substances. 



The stability of the system in respect to such a change must therefore 

 extend beyond the point where the pressure of C commences to be 

 less than that of A and B. We arrive at the same result if we use 

 the expression (520) as a test of stability. Since this expression has 

 a finite positive value when the pressures of the phases are all equal, 

 the ordinary surface of discontinuity must be stable, and it must 

 require a finite change in the circumstances of the case to make it 

 become unstable.* 



In the pi'eceding paragraph it is shown that the surface of contact 

 of phases A and B is stable under cei'tain 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 for- 

 mation 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 neces- 

 sary, since it is in this manner, if at all, that the phase is likely to be 

 formed, for an incipient sheet of phase C Avould evidently be unstable 

 when o'ab<Co'ac+ Crc, 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 sur- 

 faces, the mass of phase A lying on the left of D E H' F G, 

 and that of phase B on the right of DEH"FG. Let 

 the line joining the centers cut the spherical surfaces in 

 H' and H", and the plane of the surface of contact of A 

 and B in L Let the radii of EH'F and E H" F be 

 denoted by r', r\ and the segments I H', I H" by x', x". 

 Also let I E, the radius of the circle in which the spher- 

 ical surfaces intersect, be denoted by B,. By a suitable 

 application of the general condition of equilibrium we 

 may easily obtain the equation 



r'-x' , r"-x" ^^^^, 



Cac — -, h o'bc — -r-^ = o-AB, (561) 



* It is true that such a case as we are now considering is formally excluded in the 

 discussion referred to, which relates to a plane surface, and in which the system is 

 supposed thoroughly stable with respect to the possible formation of any different 

 homogeneous masses. Yet the reader will easily convince himself that the criterion 

 (520) is perfectly valid in this case with respect to the possible formation of a thin 

 sheet of the phase C, which, as we have seen, may be treated simply as a different 

 kind of surface of discontinuity. 



