EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 75 



question formed a part 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 

 formation must be infinitely small, the same assumption is evidently 

 entirely inadmissible, as the surfaces must be regarded as infinitely 

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

 modifications are necessary in our formulae in order to include the 

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

 be on the assumption, 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) homo- 

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

 But although the formulae thus modified will apply with sensible 

 accuracy to masses (occurring within masses of a different 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 sufficient 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 

 mutual action of the new and the original parts, which can affect the 

 result. It will be easy to give such a meaning to the expressions 

 De, Dr\, Dv, Dm v Dm 2 , . . . Dm n that this shall be evidently the case. 

 It will be observed that the quantities represented by these expressions 

 have not been perfectly defined. In the first place, we have no right 

 to assume the existence of any surface of absolute discontinuity to 

 divide the new parts from the original, so that the position given 

 to the dividing surface is to a certain extent arbitrary. Even if 

 the surface separating the masses were determined, the energy to 

 be attributed to the masses separated would be partly arbitrary, 

 since a part of the total energy depends upon the mutual action 

 of the two masses. We ought perhaps to consider the case the 

 same in regard to the entropy, although the entropy of a system 

 never depends upon the mutual relations of parts at sensible dis- 

 tances from one another. Now the condition (52) will be valid if 

 the quantities De, Dq, Dv, Dm v Dm 2 , . . . Dm n are so defined that 



