470 J. W. Gibbs — Equilibriitm of Heterogeneous Substances. 



2r,r,,^,'^=-Xy,dy, + Xr,dy, - 2;/, dF,,,, (645) 



s 



If we set r = ll, (646) 



we have j^_/jt -/2^ ^/2-/ .^ (64Y) 



and 2r2<,, — = — A;/j^r — 2f7r2<,, (648) 



With this equation we may eliminate ds from (643). We may also 

 eliminate do' by the necessary relation [see (514)] 



dG = — I\^,^,d/.i.^. 

 This will give 



4 r,,,,^ dM2 = E (A y, f?r + 2 dP^,,,), (649) 



or 



*^:z=Ar,^+2^>, (650) 



where the differential coefficients are to he determined on the condi- 

 tions that the temperature and all the potentials except /j^ and /J2 

 are constant, and that the pressure in the interior of the film shall 

 remain equal to that in the contiguous gas-masses. The latter con- 

 dition may be expressed by the equation 



in which y^' and y2' denote the densities of /S\ and S^ in the con- 

 tiguous gas-masses. [See (98).] When the tension of the surfaces 

 of the film and the pressures in its interior and in the contiguous gas- 

 masses are known in terms of the temperature and potentials, equa- 

 tion (650) will give the value of E in terms of the same variables 

 together with A. 



If we write G^ and G^ for the total quantities of Sy and S.^ per 

 unit of area of the film, we have 



G,= Xy,, (652) 



G2=?iy2+2r2,,^, (653) 



Therefore, 



G,= G,r+2r2,,„ 



/dG,\ ^ ;v ;/, -^ + 2 ^2111, (654) 



\dfxjG^ dfA^ dp.2 



where the differential coefficients in the second member are to be 

 determined as in (650), and that in the first member with the addi- 

 tional condition that G ^ is constant. Therefore, 



