384 J. W. Gibbs — Equillbrmm of Heterogeneous /Substances. 



It will be convenient to have a notation which will enable us to ex- 

 press this by an equation. Let be denote the smallest value (i. e., the 

 value nearest to — oo) of Se consistent with given values of the 

 other variations, then 



"bs = t drf -^ pij^ d))ii -f" A's ^>^h + ^tc. (478) 



For the internal equilibrium of the whole mass which consists of 

 the parts M, M', M", it is necessary that 



Se + 6a' + de" ^ (479) 



for all variations which do not affect the enclosing surface or the 

 total entropy or the total quantity of any of the various components. 

 If we also regard the surfaces separating M, M', and M" as invaria- 

 ble, we may derive from this condition, by equations (478) and (12), 

 the following as a necessary condition of equilibrium : 



t Sr^ -{- /.i^ Sni^ -\- /<2 ^ni^ + etc. 

 + t' Sif -|- /^/ Sm^' -\- i^i^' 6^2 + etc. 

 4- t" dtf + j.i/ dm/ + /,i/ dm/ + etc. ^ 0, (480) 



the variations being subject to the equations of conditions 



6jf-\-6>/' + 6>/' = 0, 1 



6m^ -^ Sin^' -\-6m/ = 0, I 



dni^ + 6m/ -\- 61)12" ^^ ^) I 



etc. J 



It may also be the case that some of the quantities 6nt / 6tn/, 

 6m/ ^ 6m/\ etc., are incapable of negative values or can only have 

 the value zero. This will be the case when the substances to which 

 these quantities relate are not actual or possible components of M' 

 or M". (Seepage 117.) To satisfy the above condition it is neces- 

 sary and sufficient that 



t = t' - t", (482) 



fx^' 6ni^^ lA^ 6m/ j^i/ 6m/ ^ i^i^ 6m/^ etc., (483) 



ix/ 6ni/^l-i^6m/, /.t/ 6m/^ /.(2 6m/, etc. (484) 



It will be observed that, if the substance to which /^,, for instance, 

 relates is an actual component of each of the homogeneous masses, 

 we shall have /u^ =z ju^' = /.(/. If it is an actual component of the 

 first only of these masses, we shall have //j z= ///. If it is also a 

 possible component of the second homogeneous mass, we shall also 

 have jx^ = /x/. If this substance occurs only at the surface of dis- 



