EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 71 



any variation in the state of the system are arbitrary, we may so 

 define the parts which we have called original, that we may consider 

 them as initially homogeneous and remaining so, and as initially con- 

 stituting the whole system. 



The most general value of the variation of the energy of the whole 



system is evidently 



2&+2D6, (35) 



the first summation relating to all the original parts, and the second 

 to all the new parts. (Throughout the discussion of this problem, the 

 letter 8 or D following 2 will sufficiently indicate whether the sum- 

 mation relates to the original or to the new parts.) Therefore the 

 general condition of equilibrium is 



S<5e+2De^O, (36) 



or, if we substitute the value of Se taken from equation (12), 

 2I)e+2(^77)-2(^^)+S(// 1 ^m 1 )+2(// 2 5m 2 )...H-2(/z n ^m n )^0. (37) 



If any of the substances S v S 2 , ... S n can be formed out of others, 

 we will suppose, as before (see page 69), that such relations are 

 expressed by equations between the units of the different substances. 

 Let these be 



oA + <^2 +* = <>1 



&ii + & 2 2 + 6 n = 1 r equations. (38) 



etc. 



The equations of condition will be (if there is no restriction upon the 

 freedom of motion and composition of the components) 



0, (39) 



0, (40) 



and n r equations of the form 





etc. 

 Now, using Lagrange's " method of multipliers," t we will subtract 



*In regard to the relation between the coefficients in (41) and those in (38), the 

 reader will easily convince himself that the coefficients of any one of equations (41) 

 are such as would satisfy all the equations (38) if substituted for S lt 8 2 , ...S n ; and 

 that this is the only condition which these coefficients must satisfy, except that the 

 n-r sets of coefficients shall be independent, i.e., shall be such as to form independent 

 equations ; and that this relation between the coefficients of the two sets of equations is 

 a reciprocal one. 



tOn account of the sign ^ in (37), and because some of the variations are incapable 

 of negative values, the successive steps in the reasoning will be developed at greater 

 length than would be otherwise necessary. 



