74 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



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 = T, p = P, (50) 



fi l Sm l ^M l Sm lt ]UL 2 Sm 2 ^ M 2 Sm 2 , ... [j. n Sm n ^M n Sm n (51) 

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



De-TDq+PDv-M^m^MzDmt... ~^Dm n ^O, (52) 



for each of the new parts as previously defined. If to these con- 

 ditions we add equations (43), we may treat T, P, M v M 2 ,...M n 

 simply as unknown quantities to be eliminated. 



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

 S v is an actual component of the part of the given mass distinguished 

 by a single accent, Sm^ may be either positive or negative, and we 

 shall have fj.^ = M^\ but if S l is only a possible component of that 

 part, (Sm/ will be incapable of a negative value, and we will have 



The formulae (50), (51), and (43) express the same particular con- 

 ditions of equilibrium which we have before obtained by a less general 

 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 De, Dq, Dv, Dm l3 Dm 2 , . . . Dm n denoting the 

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

 quantities of the substances S v $ 2 > S n which we regard as comppsing 

 it (not necessarily as independently variable components). If there 

 is more than 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 M v M 2 , . . . and M n that the result would be the same in 

 any case. Now, if we assume that the values of -De, Dr\, Dv, Dm v 

 Dm 2 , . . . Dm n are proportional to the values of e, ?/, v, m v ra 2 , . . . m n for 

 any large homogeneous mass of similar composition, and of the same 

 temperature and pressure, the condition is equivalent to this, that 



e-Tri + Pv-M^-M^ ... -M n m n ^0 (53) 



for any large homogeneous body which can be formed out of the 

 substances S v S 2 , ... S n . 



But the validity of this last transformation cannot be admitted 

 without considerable limitation. It is assumed that the relation 

 between the energy, entropy, volume, and the quantities of the 

 different components of a very small mass surrounded by substances 

 of different composition and state is the same as if the mass in 



