EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 223 



the variations being subject to the equations of condition 



(481) 

 -\-6 / m 2 V> 



etc. 



It may also be the case that some of the quantities Sm^, Sm^', 

 #m 2 ", etc., are incapable of negative values or can only have the 

 value zero. This will be the case when the substances to which these 

 quantities relate are not actual or possible components of M' or M". 

 (See page 64.) To satisfy the above condition it is necessary and 



sufficient that 



t = t' = t", (482) 



2 ' , etc., (483) 



// 2 ''(Sm 2 ''^jM 2 <$m 2 '', etc. (484) 



It will be observed that, if the substance to which JUL V for instance, 

 relates is an actual component of each of the homogeneous masses, 

 we shall have A4 = /*/ = /*i"- If it is an actual component of the 

 first only of these masses, we shall have /^ 1 = /w 1 / . If it is also a 

 possible component of the second homogeneous mass, we shall also 

 have /*! = ///'. If this substance occurs only at the surface of dis- 

 continuity, the value of the potential // x will not be determined by 

 any equation, but cannot be greater than the potential for the same 

 substance in either of the homogeneous masses in which it may be a 

 possible component. 



It appears, therefore, that the particular conditions of equilibrium 

 relating to temperature and the potentials which we have before 

 obtained by neglecting the influence of the surfaces of discontinuity 

 (pp. 65, 66, 74) are not invalidated by the influence of such dis- 

 continuity in their application to homogeneous parts of the system 

 bounded like M' and M" by imaginary surfaces lying within the limits 

 of homogeneity, a condition which may be fulfilled by surfaces very 

 near to the surfaces of discontinuity. It appears also that similar 

 conditions will apply to the non-homogeneous films like M, which 

 separate such homogeneous masses. The properties of such films, 

 which are of course different from those of homogeneous masses, 

 require our farther attention. 



The volume occupied by the mass M is divided by the surface 3 

 into two parts which we will call v'" and v"", v'" lying next to M', 

 and v"" to M". Let us imagine these volumes filled by masses having 

 throughout the same temperature, pressure and potentials, and the 

 same densities of energy and entropy, and of the various components, 



