J. W. Gibbs — Equilibrium of Heterogeneous Substances. 385 



continuity, the vahie of the i)()tenti:U /<, will not be determined by 

 any e([uation, but cannot be greater tliau the potential for the same 

 substance in either of the homogeneous masses in which it may be a 

 possible component. 



It appears, therefore, that the particular conditions of equilibrium 

 relating to temperature and the potentials which we have before 

 obtained by neglecting the influence of the surfaces of discontinuity 

 (])p. 119, 120, 12y) are not invalidated by the influence of such dis- 

 continuity in their application to homogeneous parts of the system 

 bounded like M' and M" by imaginary surfaces lying within the 

 limits of homogeneity, — a condition which may be fulfilled by sur- 

 faces very near to the surfaces of discontinuity. It appears also that 

 similar conditions will apply to the non-homogeneous films like M', 

 which separate such homogeneous masses. The properties of such 

 films, which are of course difterent from those of homogeneous 

 masses, require our farther attention. 



The volume occujned by the mass M is divided by the surface s 

 into two parts, which w^e wnll call v'" and v"", v'" lying next to M', 

 and v"" to M", Let us imagine these volumes filled by masses hav- 

 ing throi;ghout the same temperature, pressure and potentials, and 

 the same densities of energy and entropy, and of the various com- 

 ponents, as the masses M' and M" respectively. We shall then have, 

 by equation (12), if we regard the volumes as constant, 



6e"' = t' chf" + /// S>n,"' + M2' ^^^^2'" + etc, (485) 



6e"" = t" dif" + ///' 67n^"" + ).i^" dm.J'" + etc. ; (486) 



whence, by (482)-(484), we have for reversible variations 



6t"' = t 611" + //j 6m /" -f yuo SmJ" + etc., (48V) 



6t"" = t6jf"'-\-f.i^ 6m,"" + //g 6m.^"" + etc. (488) 



From these equations and (4V7), we have for reversible variations 



S{e-t"'-t"")-t6{if- v'" -n"") 



+ ^i, 6{m , - m,"' - m,'"') + //a ^("^2 " ^^^2'" " »"."")+ etc. (489) 



Or, if we set* 



«« = e - £'" - £"", if = 7/ - if - ii"\ (490) 



m\ =.mi —m, '" — ?«,"", m| = m^ —m^" — m.^'"^ etc., (491) 



* It will be understood that the '^ here used is not an algebraic exponent, but is 

 onl3Mntended as a distinguishing mark. The Roman letter 8 has not been used to 

 denote an_v quantity. 



