128 J, W. Gibhs — Equilibrium of Heterogeneous Substances. 



<Sj, ®2, • • • ©n, as before, to denote the units of the various com- 

 ponents. Thus understood, the equation expresses that the vahies 

 of the quantities in the parentheses are such as are consistent with 

 the equations of condition (41). The change in the system, there- 

 fore, which we are considering, is not one which violates any of the 

 equations of condition, and as (45) does not hold true for this change, 

 and for all values of 2\ P, J/^, J/2, • • • ^^A which are consistent 

 with eqs. (43), the state of the system cannot be one of equilibrium. 

 Therefore it is necessary, and it is evidently sufficient for equilibrium, 

 that it shall be possible to assign to 2\ P, 31^, 31^, . . . M„ such values, 

 consistent with eqs. (43), that the condition (45) shall hold true for 

 any change in the system irrespective of the equations of condition 

 (39), (40), (41). 



For this it is necessary and sufficient that 



t=2\ p = P, (50) 



for each of the original parts as previously defined, and that 



Be - TDi]^PDr - J/, X*;//, - M^Dm^ ... - 3I,,Dm„^ 0, (52) 



for each of the 7iew parts as previously defined. If to these condi- 

 tions we add equations (43), we may treat 1\ P, J/^, J/g, . . . J/„ 

 simply as unknown quantities to be eliminated. 



In regard to conditions (51), it will be observed that if a sub- 

 stance, 6' J, is an actual component of the part of the given mass 

 distinguished by a single ac<^ent, dtn^' may be either positive or 

 negative, and we shall have fi^' = M^ ; but if S^ is only a possible 

 component of that part, (Sm^' will be incapable of a negative value, 

 and we will have /^^'^ M^. 



The formula^ (S*^)- (51), ^^^^^ (43) express the same particular con- 

 ditions of equilibrium which we have before obtained by a less gen- 

 eral process. It remains to discuss (52). This formula must hold 

 true of any infinitesimal mass in the system in its varied state which 

 is not approximately homogeneous with any of the surrounding 

 masses, the expressions i>£, J9//, i>y. Dm ^, J)in2, . . . Dm„ denoting 

 the energy, entropy, and volume of this infinitesimal mass, and the 

 quantities of the substances S^, S.y, . . . *S'„ which we regard as com- 

 posing it, (not necessarily as independently variable components). 

 If there is more tlian one way in which this mass may be considered 

 as composed of these substances, we may choose whichever is most 

 convenient. Indeed it follows directly from the relations existing 

 between J/j, J/g, . . . and J/„ that the result Avould be the same in 



