SURFACES OF DISCONTINUITY 633 



is zero) p c would have to be equal to Pa or pa, and if in the form 

 of a lentif orm mass p c would have to be greater than Pa or pa. 

 Hence A and B in contact would be quite stable as regards the 

 formation of C in such a range of values of t, ni, IJ.2, . . . If we 

 now consider the range of values of these quantities for which 

 Pc(t, m) ^ PA{t, ijl), we have to deal with the two cases 

 which arise; (1) when ct^bC^ m) = (^Acit, m) + <rBc{t, m); 

 (2) when CAsit, m) < <^Ac{t, fx) + (Tscit, /x). 



(1) If pc(t, m) = PA{t, m) there would just be equilibrium with 

 a plane sheet of C between A and B, since the surface tensions 

 between A and C, and B and C would just balance the surface 

 tension between A and B in the portion where A and B meet. 

 On the other hand if we varied t, ni, 1x2, ■ ■ • to values t', ^i/, 

 H2, ... such that pc(^', mO > ??A(i', m')> (PsCi'jM') still remaining 

 equal to Pa (f, n') as postulated originally) , then equilibrium could 

 not be maintained unless the surfaces separating A and B from 

 C became concave towards the latter phase, tending towards a 

 lens form. This would upset the balance of the surface ten- 

 sions at the edge where the surface A-B meets the surfaces 

 A-C and B-C, The conditions of this equilibrium can, for 

 purely mathematical purposes, be regarded as equivalent to the 

 equilibrium of three forces. Now the directions of the forces 

 equivalent to cac and cbc are no longer opposite to that equiv- 

 alent to (Jab- The force equivalent to (Tab is greater than the 

 resultant of the inclined forces equivalent to <tac and cbc since 

 (Tab = (Tac + (Jbc* Hence the edge tends to move outward, 

 i.e., the mass C tends to increase and in so doing draws on the 

 masses A and B for material, and so alters the phases in such a 

 way as to bring them to such values that the equality of p c to 

 Pa will be restored. We see that in this case there is a tendency 

 for the mass C to form between A and B. 



(2) If (Tab < (J AC + (Tbc the argument of the previous para- 

 graph breaks down. Clearly, no plane sheet of C can form 

 between A and B when pc = Pa, the force equivalent to ctab 

 being too small to pull it out, as it were, against the force equiv- 



* As is well-known, this is a convenient way of dealing with the fact 

 that if an outward displacement of the edge were made there would be a 

 diminution of free surface energy. 



