276 EQUILIBKIUM OF HETEROGENEOUS SUBSTANCES. 



surface. On the same principle, we may use I\ and T 2 to denote the 

 values of mf and m| per unit of surface, and m/, m 2 ", y/, y 2 " ^ 

 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 

 were all really different. The superficial tension will be a function 

 of fa and fj. 2 , with the temperature and the potentials for the 

 other components, and 1\ , F 2 will be equal to its differential 

 coefficients with respect to fa and // 2 . In a word, all the general 

 relations which have been demonstrated may be applied to this 

 case, if we remember always to treat the component as a different 

 substance according as it is found on one side or the other of the 

 impermeable film. 



When there is free passage for the component specified by the 

 suffixes l and 2 through other parts of the system (or through any 

 flaws in the film), we shall have in case of equilibrium fa = fa. ^ 

 we wish to obtain the fundamental equation for the surface when 

 satisfying this condition, without reference to other possible states 

 of the surface, we may set a single symbol for fa and fa in the 

 more general form of the fundamental equation. Cases may occur 

 of an impermeability which is not absolute, but which renders the 

 transmission of some of the components exceedingly slow. In such 

 cases, it may be necessary to distinguish at least two Different 

 fundamental equations, one relating to a state of approximate 

 equilibrium which may be quickly established, and another relating 

 to the ultimate state of complete equilibrium. The latter may be 

 derived from the former by such substitutions as that just indicated. 



The Conditions of Internal Equilibrium for a System of Hetero- 

 geneous Fluid Masses without neglect of the Influence of the 

 Surfaces of Discontinuity or of Gravity. 



Let us now seek the complete value of the variation of the energy 

 of a system of heterogeneous fluid masses, in which the influence of 

 gravity and of the surfaces of discontinuity shall be included, and 

 deduce from it the conditions of internal equilibrium for such a 

 system. In accordance with the method which has been developed, 

 the intrinsic energy (i.e. the part of the energy which is independent 

 of gravity), the entropy, and the quantities of the several components 

 must each be divided into two parts, one of which we regard as 

 belonging to the surfaces which divide approximately homogeneous 



