J. W. Glhbs — Equilibrium of Heterogeneous Snhstances. 441 



the components of the contiguous masses. Such may be the case, 

 for example, when a film of oil is spread on a surface of water, even 

 when tlie film is too thin to exhibit the properties of the oil in mass. 

 In such cases, if there is communication between the contiguous 

 masses through other parts of the system to which they belong, such 

 that the components in question can pass freely from one mass to the 

 other, the impossibility of a direct passage through the film may be 

 regarded as an immaterial circumstance, so far as states of equilib- 

 rium are concerned, and our formula? will require no change. But 

 when there is no si;ch indirect communication, the potential for any 

 component for which the film is impermeable may have entirely 

 different values on opposite sides of the film, and the case evidently 

 requires a modification of our usual method. 



A single consideration will suggest the proper treatment of such 

 cases. If a certain component which is found on both sides of a film 

 cannot pass from either side to the other, the fact that the part of the 

 component which is on one side is the same kind of matter with the 

 part on the other side may be disregarded. All the general relations 

 must hold true, which would hold if they were really different sub- 

 stances. We may therefore write j.i^ for the potential of the com- 

 ponent on one side of the film, and ja^ for the potential of the same 

 substance (to be treated as if it were a different substance) on the 

 other side ; ^^'^j for the excess of the quantity of the substance on the 

 first side of the film above the quantity which would be on that side 

 of the dividing surface (whether this is determined by the surface of 

 tension or otherwise) if the density of the substance were the same 

 near the dividing surface as at a distance, and m| for a similar quan- 

 tity relating to the other side of the film and dividing surface. On 

 the same principle, we may use I\ and /^g to denote the values of 

 »i^j and rn% P^i' ^"^i^ of surface, and m^\ '^2"-> Vil Y2 ^^ denote the 

 quantities of the substance and its densities in the two homogeneous 

 masses. 



With such a notation, which may be extended to cases in which 

 the film is impermeable to any number of components, the equations 

 relating to the surface and the contiguous masses will evidently have 

 the same form as if the substances specified by the different suffixes 

 wei'e all really different. The superficial tension will l)e a function 

 of fx^ and /Vg, Avith the temperature and the potentials for the other 

 components, and — i"i, —^2 ^^'^^ ^^^ equal to its differential coeffi- 

 cients with respect to /<, and //,. In a word, all the general rela- 

 tions which have been demonstrated may be applied to this case, if 



