J. W. Gihbs — Equilihrimn of Heterogeveons ASt/ftstancen. 401 



entropy and of the quantities ol" tlio components which we regard 

 as belonging- to the surface. Tlie increments of entropy and of the 

 various components wliich the rest of the system receive will be 

 expressed by 



— Z/?;^, —^i)i^,, -^Aml, etc., — Jm^ , — /Im^ , etc., 



and the consequent increment of energy will be by (12) and (501) 



— t J/f — //„ J;;?^ — //,, A7nl — etc. — jj^ Arn^^ — jA,, Am^, — etc. 



Hence the total increment of energy in the Avhole system will be 



At^ -^ t Aif" — //„ A)iil — //,, A-m^i, — etc. , 



; (-516) 



— //„ J>/?^, — i.t,,AvvJ, — etc. ) 



If the value of this expression is necessarily positive, for finite 

 changes as well as infinitesimal in the nature of the part of the film 

 to which J6*, etc. relate,* the increment of energy of the whole 

 system will be positive for any possible changes in the nature of the 

 film, and the film will be stable, at least with respect to changes in 

 its nature, as distinguished from its position. For, if we write 



Bt\ l))i^, Dml, Dml, etc., Z>m«, Bit^ , etc. 



for the energy, etc. of any element of the surface of discontinuity, we 

 have from the supposition just made 



AD6^ - t ADif -//„ ADinl - //, ADml— etc. 



- //, ADm% — //, AI)m\ - etc. > ; (517) 



and integrating for the w^hole surface, since 



AfDm^y=0, A/Djj>t=0, etc., 

 we have 



A/Be^ - t AfDif- //„ AJ'Bml - }h AfDml ~ etc. > 0. (518) 



Now AfDif is the increment of the entropy of the whole surface, 

 and — AJ'Drf is therefore the increment of the entropy of the two 

 homogeneous masses. In like manner, —AfDni^^, —AfDm\^ etc. 

 are the increments of the quantities of the components in these masses. 

 The expression 



- t A/Dif - iJ, AfDml - l-h AfDml ~ etc. 



* In the case of infinitesimal changes in the nature of the film, the sign A must be 

 interpreted, as elsewhere in this paper, without neglect of infinitesimals of the higher 

 orders. Otherwise, by equation (501), the above expression would have the value 

 zero. 



