72 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



T(2 Srj + 2 Drf) P(2 Sv 4- 2 Dv) from the first member of the 

 general condition of equilibrium (37), T and P being constants 

 of which the value is as yet arbitrary. We might proceed in the 

 same way with the remaining equations of condition, but we may 

 obtain the same result more simply in another way. We will first 

 observe that 



which equation would hold identically for any possible values of the 

 quantities in the parentheses, if for T of the letters & v @ 2 , . . . ^> n were 

 substituted their values in terms of the others as derived from equations 

 (38). (Although @ x , <2> 2 , . . . @ n do not represent abstract quantities, 

 yet the operations necessary for the reduction of linear equations 

 are evidently applicable to equations (38).) Therefore, equation (42) 

 will hold true if for @ 1} @ 2 , . . . n we substitute n numbers which 

 satisfy equations (38). Let M v M%, . . . M n be such numbers, i.e., let 



ttjifj 



M n = 0, \ T equations, (43) 

 etc. 

 then 



^(2 (Smj + 2 Dm^) + ML Sm 2 + 2 



+ J/ n (2 Sm n + 2 Dm n ) = 0. (44) 



This expression, in which the values of n r of the constants M v 

 M z , . . . M n are still arbitrary, we will also subtract from the first 

 member of the general condition of equilibrium (37), which ' will 

 then become 



-Jf 1 2Dm 1 ...-Jlf w 2Dm w ^O. (45) 



That is, having assigned to T, P, M v M 2 , . . . M n any values con- 

 sistent with (43), we may assert that it is necessary and sufficient for 

 equilibrium that (45) shall hold true for any variations in the state 

 of the system consistent with the equations of condition (39), (40), 

 (41). But it will always be possible, in case of equilibrium, to assign 

 such values to T, P, M I} M 2 , . . . M n , without violating equations (43), 

 that (45) shall hold true for all variations in the state of the system 

 and in the quantities of the various substances composing it, even 

 though these variations are not consistent with the equations of con- 

 dition (39), (40), (41). For, when it is not possible to do this, it 

 must be possible by applying (45) to variations in the system not 

 necessarily restricted by the equations of condition (39), (40), (41) to 



