428 J. W. Gibbs — Equilibrimn of Heterogeneous 8i(bstances. 



change before the film of phase C, or (as we shall see in the following 

 paragraph) a lentiforni mass of that phase, can be formed. 



The work which must be expended in order to form on the surface 

 between indefinitely large masses of phases A and B a lentiform mass 

 of phase C in equilibrium, may evidently be represented by the 

 formula 



- Pc Vc +7>A Fa + Pb Fb, (573) 



where *Sac, -^bc denote the areas of the surfaces formed between A and 

 C, and B and C, S^b tlie diminution of the area of the surface between 

 A and B, Fc the volume formed of the phase C, and Va, Vb the 

 diminution of the volumes of the phases A and B. Let us now sup- 

 pose (Tacj o'bc, o'ab, Pa, Pb to remain constant and the external bound- 

 ary of the surface between A and B to remain fixed, while pc 

 increases and the surfaces of tension receive such alterations as are 

 necessary for equilibrium. It is not necessary that this should be 

 physically possible in the actual system ; we may suppose the changes 

 to take place, for the sake of argument, although involving changes 

 in the fundamental equations of the masses and surfaces considered. 

 Then, regarding TF simply as an abbreviation for the second member 

 of the preceding equation, we have 



- Pc d T^c + Pa d Fa + Pb d Vb - Vc dpc. (574) 



But the conditions of equilibrium require that 

 O'ac d/Sj^c -\- Cbc <^^'bc — O'ab ^^'^'ab 



—pcdVc-\-pAdVj,-^PBdVB=0. (575) 



Hence, 



dW= - Vcdpc. (576) 



Now it is evident that T'^ will diminish as pc increases. Let us 

 integrate the last equation supposing p^ to increase from its original 

 value until Vc vanishes. This will give 



W" — W r= a negative quantity, (577) 



where W and W" denote the initial and final values of IF. But 

 TF"=0. Hence W is positive. But this is the value of W in the 

 original system containing the lentiform mass, and expresses the 

 work necessary to form the mass between the phases A and B. It is 

 therefore impossible that such a mass should form on a surface be- 



