EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 261 



where V and V" denote the volumes of the masses of the phases 

 A and B which are replaced. Now by (500), 



JO-JA-^, and Po -p e = ^>. (565) 



We have also the geometrical relations 



F'=f^'-^(r' -*'),! 

 V" = |*yi a" - %TrR*(r" - a"). J 



By substitution we obtain 





= -JTT (TAG rV - |7T^ 2 <TAO 



x 



O-BO r' V - 



OBC 



and by (561), 



Since 



we may write 



2-TrrV = 



2?rr V = S 



BO , 



= S 



AB 



(567) 

 (568) 



(569) 



(The reader will observe that the ratio of M and N is the same as that 

 of the corresponding quantities in the case of the spherical mass 

 treated on pages 252-258.) We have therefore 



^ r =(o-Ao s AO + o-BC s BC o*AB SAB)- (570) 



This value is positive so long as 



since SAC > SAB > and S B C>SAB- 



But at the limit, when 



we see by (561) that 



and therefore 



SAO = SAB > and S B C = 

 TF=0. 



It should however be observed that in the immediate vicinity of 

 the circle in which the three surfaces of discontinuity intersect, the 

 physical state of each of these surfaces must be affected by the 

 vicinity of the others. We cannot, therefore, rely upon the formula 

 (570) except when the dimensions of the lentiform mass are of sensible 

 magnitude. 



We may conclude that after we pass the limit at which p becomes 

 greater than p A an d PB (supposed equal) lentiform masses of phase C 

 will not be formed until either O-AB = "AC + O"BC> or Pop becomes so 

 great that the lentiform mass which would be in equilibrium is one 



