EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 303 



gas -masses are known in terras of the temperature and potentials, 

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

 variables together with X. 



If we write G l and G z for the total quantities of 8 l and 8 2 per 

 unit of area of the film, we have 



Therefore, 



(652) 

 (653) 



(654) 



where the differential coefficients in the second member are to be 

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

 additional condition that G 1 is constant. Therefore, 



E 



, (655) 



the last differential coefficient being determined by the same condi- 

 tions as that in the preceding equation. It will be observed that the 

 value of E will be positive in any ordinary case. 



These equations give the elasticity of any element of the film 

 when the temperature and the potentials for the substances which 

 are found in the contiguous gas-masses are regarded as constant, 

 and the potentials for the other components, // 1 and /z 2 , have had 

 time to equalize themselves throughout the element considered. The 

 increase of tension immediately after a rapid extension will be greater 

 than that given by these equations. 



The existence of this elasticity, which has thus been established 

 from a priori considerations, is clearly indicated by the phenomena 

 which liquid films present. Yet it is not to be demonstrated simply 

 by comparing the tensions of films of different thickness, even when 

 they are made from the same liquid, for difference of thickness does 

 not necessarily involve any difference of tension. When the phases 

 within the films as well as without are the same, and the surfaces of 

 the films are also the same, there will be no difference of tension. 

 Nor will the tension of the same film be altered, if a part of the 

 interior drains away in the course of time, without affecting the 

 surfaces. In case the thickness of the film is reduced by evapor- 

 ation, the tension may be either increased or diminished. (The 

 evaporation of the substance 8 lt in the case we have just considered, 

 would diminish the tension.) Yet it may easily be shown that 



