J. TF. Gihhs — Equilihrhmi of ITeterogeneous Substances 443 



into two parts, one of whicli relates to the elements Dm^, and^the 

 other to the elements l)ni^. The complete valne of the variation of 

 the energy of the system will be represented by the expression 



d/De" + dJ'Dt^ + 6fg z Dni" + 8 f g z Drn^, (599) 



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

 ment 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 SDe"', SDt^, 6Dmy, 6IJnt^, 

 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 6I))f' -fp 6Dv +//<! 6Dm\ +///2 SJ^ml + etc. 

 -\-ft6Z>f/^ -^fffSUs+fji^ dlJm\ +.///0 dDml + etc. 



-f- fg Sz Dm^ -\- fg z 6TJni\ -f fg z SDm^ + etc. 



+ /// 6z Dm^-\-fg z 8Dtn\ ^fgz dDinl -f 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 suflicient. It is 

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

 particular conditions of equilibrium relating to every possible deforma- 

 tion 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 neces- 

 sary for equilibrium, so far as reversible variations are concerned. 

 (The necessary relations between the various quantities relating to 

 each element of the masses and sui-faces are expressed by the funda- 

 mental equation of the mass or surface concerned, or may be imme- 

 diately derived from it. See pp. 140-144 and 391-393.) 



The variations in (600) are subject to the conditions which arise 

 from the nature of the system and from the supposition that the 

 changes in the system are not such as to aiFect external bodies. This 

 supposition is necessary, unless we are to consider the variations in 

 the state of the external bodies, and is evidently allowable in seeking 

 the conditions of equilibrium whicli relate to the interior of the sys- 



