./ W. Gibbs — Equilibriuiu of Heterogeneous /Substances. rJ9 



any case. Now, if we assume tliat the values of i>f, />;/, JDo, Drn^, 

 Dm.^, . . . Din„ are proportional to the values of f, //, v, m,, jh.^, . . . 

 m„ for any large homogeneous mass of similar composition, and of 

 the same temperature and pressure, the condition is equivalent to 

 this, that 



€ - T?^ + Pv - 3Ij m^ -3I2 in^ ... - iT/„m„ ^ (53) 



for any large homogeneous body which can be formed out of the 

 substances aS'j, S2 . . . S„. 



But the validity of this last transformation cannot be admitted 

 without considerable limitation. It is assumed that the relation 

 between the energy, entropy, volume, and the quantities of the dif- 

 ferent components of a very small mass surrounded by substances 

 of diiferent composition and state is the same as if the mass in ques- 

 tion formed a jaart of a large homogeneous body. We started, 

 indeed, with the assumption that we might neglect the part of the 

 energy, etc., depending upon the surfaces separating heterogeneous 

 masses. Now, in many cases, and for many purposes, as, in general, 

 when the masses are large, such an assumption is quite legitimate, 

 but in the case of these masses which are formed within or among 

 substances of different nature or state, and which at their first forma- 

 tion mi;st be infinitely small, the same assumption is evidently 

 entirely inadmissible, as the siirfaces must be regarded as infinitely 

 large in proportion to the masses. We shall see hereafter what 

 modifications are necessary in our formula in order to include the 

 parts of the energy, etc., which are due to the surfaces, but this will 

 be on the assinnption, which is usual in the theory of capillarity, 

 that the radius of curvature of the surfaces is large in proportion to 

 the radius of sensible molecular action, and also to the thickness of 

 the lamina of matter at the surface which is not (sensibly) homoge- 

 neous in all respects with either of the masses which it separates. 

 But although the formula? thus modified will apply with sensible 

 accuracy to masses (occurring within masses of a diftei'ent nature) 

 much smaller than if the terms relating to the surfaces were omitted, 

 yet their failure when applied to masses infinitely small in all their 

 dimensions is not less absolute. 



Considerations like the foregoing might render doubtful the validity 

 even of (52) as the necessary and sufiicient condition of equilibrium 

 in regard to the formation of masses not approximately homogeneous 

 with those previously existing, when the conditions of equilibrium 

 between the latter are satisfied, unless it is shown that in establishing 

 this formula there have been no quantities neglected relating to the 



Trans. Conn. Acad., Vol. III. 17 October, 1875. 



