220 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



Let the portion of S which is included by the closed surface be 

 denoted by S, and the area of this portion by a. Moreover, let the 

 mass contained within the closed surface be divided into three parts 

 by two surfaces, one on each side of S, and very near to that surface, 

 although at such distance as to lie entirely beyond the influence of 

 the discontinuity in its vicinity. Let us call the part which contains 

 the surface S (with the physical surface of discontinuity) M, and the 

 homogeneous parts M' and M", and distinguish by e, e', e", q, rf, q", 

 m v ra/, ra/', m 2 , m 2 ', m 2 ", etc., the energies and entropies of these 

 masses, and the quantities which they contain of their various 

 components. 



It is necessary, however, to define more precisely what is to be 

 understood in cases like the present by the energy of masses which 

 are only separated from other masses by imaginary surfaces. A part 

 of the total energy which belongs to the matter in the vicinity of the 

 separating surface, relates to pairs of particles which are on different 

 sides of the surface, and such energy is not in the nature of things 

 referable to either mass by itself. Yet, to avoid the necessity of 

 taking separate account of such energy, it will often be convenient to 

 include it in the energies which we refer to the separate masses. 

 When there is no break in the homogeneity at the surface, it is 

 natural to treat the energy as distributed with a uniform density. 

 This is essentially the case with the initial state of the system which 

 we are considering, for it has been divided by surfaces passing in 

 general through homogeneous masses. The only exception that of 

 the surface which cuts at right angles the non-homogeneoiis film 

 (apart from the consideration that without any important loss of 

 generality we may regard the part of this surface within the film as 

 very small compared with the other surfaces) is rather apparent than 

 real, as there is no change in the state of the matter in the direction 

 perpendicular to this surface. But in the variations to be considered 

 in the state of the system, it will not be convenient to limit ourselves 

 to such as do not create any discontinuity at the surfaces bounding 

 the masses M, M', M"; we must therefore determine how we will 

 estimate the energies of the masses in case of such infinitesimal 

 discontinuities as may be supposed to arise. Now the energy of 

 each mass will be most easily estimated by neglecting the discon- 

 tinuity, i.e., if we .estimate the energy on the supposition that 

 beyond the bounding surface the phase is identical with that within 

 the surface. This will evidently be allowable, if it does not affect 

 the total amount of energy. To show that it does not affect this 

 quantity, we have only to observe that, if the energy of the mass on 

 one side of a surface where there is an infinitesimal discontinuity of 

 phase is greater as determined by this rule than if determined by 



