J. W. Gf'bbs — Eq-tiilibriuni of Heterogeneous Huhs^tances. 469 



When the film has two or more components of Avhich the potentials 

 are not maintained constant by the contiguons gas masses, they will 

 not in general exist in the same proportion in the interior of the film as 

 on its surfaces, but those components which diminish the tensions will 

 be found in greater proportion on the surfaces. When the film is ex- 

 tended, there will therefore not be enough of these substances to keep 

 up the same volume- and surface-densities as before, and the deficiency 

 will cause a certain increase of tension. The Aalue of the elasticity of 

 the film ^ (i. e., the infinitesimal increase of the united tensions of its 

 surfaces divided by the infinitesimal increase of area in a unit of sur- 

 face), may be calculated from the quantities which specify the nature 

 of the film, when the fundamental equations of the interior mass, of 

 the contiguous gas-masses,\and of the two surfaces of discontinuity 

 are knoAvn. We may illustrate this by a simple example. 



Let us suppose that the two surfaces of a plane film are entirely 

 alike, that the contiguous gas-masses are identical in phase, and that 

 they determine the potentials of all the components of the film 

 except two. Let us call these components S^ and S2, the latter 

 denoting that which occurs in greater proportion on the surface than 

 in the interior of the film. Let us denote by y ^ and y2 the densities 

 of these components in the interior of the film, by A the thickness of 

 the film determined by such dividing surfaces as make the surface- 

 density of S^ vanish (see page 397), by r^d) ^^® surface-density of 

 the other component as determined by the same surfaces, by and s 

 the tension and area of one of these surfaces, and by ^the elasticity 

 of the film when extended under the supposition that the total quan- 

 tities of /iSj and S^ in the part of the film extended are invariable, as 

 also the temperature and the potentials of the other components. 

 From the definition of JE we have 



2dG=E-, (643) 



s 



and from the conditions of the extension of the film 

 ds ^(^Yx) ^{^ y 2 -\- '^ ^ 2ii)) 



Hence we obtain 



^Vi '^rs +2^^2(1) 



\ y ^^ = - y ^dX^ \ dy ^, 



(644) 



ds 

 {Xy2 + 2 F,!^, ))-;- = — ^2^^^ — Xdy^ — 2dT\^^. 



and eliminating c?A, 



