442 J. W. (Tthhs — Efjnllibriimi of Hetevo(/eneoiis tSuhstances. 



we remember always to treat the component as a different substance 

 according as it is found oii one side or the other of the impermeable 

 film. 



When there is free passage for the component specified by the suf- 

 fixes J and g through other [)arts of the system, (or through any flaws 

 in the film,) we shall have in case of equilibrium //^rz/Yg- If we wish 

 to obtain the fundamental equation for the surface when satisfying 

 this condition, without reference to other possible states of the sur- 

 face, we may set a single symbol for /<j and //g in the raoi'e general 

 form of the fundamental equation. Cases may occur of an impermea- 

 bility which is not absolute, but which renders the transmission of 

 some of the components exceedingly slow. In such cases, it may be 

 necessary to distinguish at least two different fundamental equations, 

 one relating to a state of approximate equilibrium wdiich may be 

 quickly established, and another relating to the ultimate state of 

 complete equilibrium. The former may be derived from the latter by 

 such substitutions as that just indicated. 



Tfte Conditions of Internal Equilihrixmi for a System of Hetero- 

 geneous Fluid Masses without neglect of the Influence of the 

 Surfaces of Discontinuity or of (Travity. 



Let us now seek the complete value of the variation of the energy 

 of a system of heterogeneous fluid masses, in which the influence of 

 gravity and of the surfaces of discontinuity shall be included, and 

 deduce from it the conditions of internal equilibrium for such a sys- 

 tem. In accordance with the method which has been developed, the 

 intrinsic energy, {i. e., the part of the energy which is independent of 

 of gravity,) the entropy, and the quantities of the several compon- 

 ents must each be divided into two parts, one of which we regard as 

 belonging to the surfaces which divide approximately homogeneous 

 masses, and the other as belonging to these masses. The elements 

 of intrinsic energy, entropy, etc., relating to an element of surface 

 Ds will be denoted by De^^ Dif'-, Dni\^ Dm^, etc., and those relating 

 to an element of volume I>v, by Dt""', Dif ^ Dm\^ D^^^2i ^^^- ^® 

 shall also use Ihix^ or V I)s and DnC' or y Di^ to denote the total 

 quantities of matter relating to the elements Ds and Dv respectively. 

 That is, 



Dru^ — r Ds — D)n\ + Dm\ -f etc., (597) 



Dm^ = y Dv = Dm\ + iJml + etc. (598) 



The part of the energy which is due to gravity must also be divided 



