J. W. Gihbs — Eqidlihrinm of Heterogeyieous Snhstances. 381 



form of this closed surface as to suppose that on each side of S, as far 

 as there is any want of perfect homogeneity in the fluid masses, the 

 closed surface is such as may be generated by a moving normal to S. 

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

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

 nu\ss 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 

 tlie 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 f, f', «", //, if, ?/', 

 »?,, »?/, m,', m„, nij, rn^" , etc., the energies and entropies of these 

 masses, and the quantities which they contain of their various com- 

 ponents. 



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 2:)articles which are on diflerent 

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

 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 sejjarate 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-homogeneous film — 

 (apart from the consideration that without any imj^ortant 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 ci'eate any discontinuity at the surfaces bounding 

 the masses M, M', ]\I" : 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 l)e most easily estimated by neglecting the discon- 

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



