98 BUTLER ART. D 



the component water and the amount of water contained in the 

 component, hydrate), the equation 



25wi + —7-7 S5m3 = (75) [25] 



must hold. 



Similarly the equation 



25w2 + — n 25m3 = (76) [25] 



a -\- 



expresses the constancy of the sum of the amount of the com- 

 ponent sodium chloride and the amount of sodium chloride 

 present in the hydrate. The other equations of condition, 



2577 = 0, Xdv = 0, 257^4 = 0, etc. (77) [26] 



will remain unchanged. 



We may first consider variations of the system which satisfy 

 (74). Such variations evidently satisfy (75) and (76) and 

 constitute some, but not all of the variations of which the 

 system is capable. Equation (73) must hold for such varia- 

 tions, so that all the conditions of equilibrium, (68), (69) and 

 (72) must apply to this case also. Therefore in (73), /xi, /X2, Ms 

 have constant values Mi, M2, Ms in all parts of the system of 

 which Si, S2 and S3 are actual components. In the general 

 case, when these conditions are satisfied (73) reduces to 



Mi25mi + ikfaSSwa + MsSSms ^ 0*. (78) [27] 



* The proof of the equivalence of (78) with (73), given by Gibbs, may 

 be stated as follows. When conditions (68), (69) and (72) are satisfied, 

 and so long as 5m is zero for every substance in all parts of the system of 

 which that substance is not an actual component, i.e., for all terms in 

 (73) involving a value of m which may be greater than the corresponding 

 value of M, we may write (73) in the form 



tE5v — pSSy + MiS5mi + M225m2 + MzHbrnt + Mi'L&nn . . . + M„S5to„ ^ 0, 



and since 



S67; = 0, 'Lhv = 0, S5m4 = 0, etc., 



