EQUILIBEIUM OF HETEROGENEOUS SUBSTANCES. 277 



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 8 , Drj 9 , Dm\ , Draf , etc., and those relating 

 to an element of volume Dv, by De v , Dif, Dm\, Dnil, etc. We 

 shall also use Dm 8 or F Ds and Dm v or y Dv to denote the total 

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



That is, 



Dm 9 = T Zte = Dm* + Dm 9 + etc., (597) 



Din v = yDv = Dm\ + Dm\ + etc. (598) 



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

 into two parts, one of which relates to the elements Dm 9 , and the 

 other to the elements Dm v . The complete value of the variation of 

 the energy of the system will be represented by the expression 



SfDe y + 8/De 9 + 8 fgz Dm? + 8 fgz Dm 9 , (599) 



in which g denotes the force of gravity, and z the height of the 

 element above a fixed horizontal plane. 



It will be convenient to limit ourselves at first to the consideration 

 of reversible variations. This will exclude the formation of new 

 masses or surfaces. We may therefore regard any infinitesimal 

 variation in the state of the system as consisting of infinitesimal 

 variations of the quantities relating to its several elements, and 

 bring the sign of variation in the preceding formula after the sign 

 of integration. If we then substitute for 8De y , <5De 8 , 8Dm y , 8 Dm 9 , 

 the values given by equations (13), (497), (597), (598), we shall have 

 for the condition of equilibrium with respect to reversible variations 

 of the internal state of the system 



ft 8Dr] v - fp SDv+ffr SDrnl+fjuL 2 8Dm1+etc. 

 +ft 8 Drj 9 + fa- 8 Ds + /X 8 Dm 9 + /> 2 8 Dm 9 + etc. 



+fg 8z Dm v +fgz 8 Dm\ + fgz 8 Dm\ + etc. 



+fg 8z Dm 9 +fgz S Dm\ + fgz 8 Dm\ + etc. = 0. (600) 



Since equation (497) relates to surfaces of discontinuity which are 

 initially in equilibrium, it might seem that this condition, although 

 always necessary for equilibrium, may not always be sufficient. It 

 is evident, however, from the form of the condition, that it includes 

 the particular conditions of equilibrium relating to every possible 

 deformation of the system, or reversible variation in the distribution 

 of entropy or of the several components. It therefore includes 

 all the relations between the different parts of the system which 

 are necessary for equilibrium, so far as reversible variations are 

 concerned. (The necessary relations between the various quantities 

 relating to each element of the masses and surfaces are expressed 



