EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 69 



of others, all such relations can be expressed by equations such as 

 a@ ft + j8@6 + etc. = ic fc + X@ I + ete. (30) 



where <S a , <5 6 , @ t> etc. denote the units of the substances S a , $ b , S k , etc., 

 (that is, of certain of the substances S v S 2 ,... S n ,) and a, /#, K, 

 etc. denote numbers. These are not, it will be observed, equations 

 between abstract quantities, but the sign = denotes qualitative as 

 well as quantitative equivalence. We will suppose that there are 

 r independent equations of this character. The equations of con- 

 dition relating to the component substances may easily be derived 

 from these equations, but it will not be necessary to consider them 

 particularly. It is evident that they will be satisfied by any values 

 of the variations which satisfy equations (18); hence, the particular 

 conditions of equilibrium (19), (20), (22) must be necessary in this 

 case, and, if these are satisfied, the general equation of equilibrium 

 (15) or (23) will reduce to 



M^ dm, + 3/ 2 2 8m 2 . . . + M n 1 Sm n > 0. (31) 



This will appear from the same considerations which were used in 

 regard to equations (23) and (27). Now it is evidently possible to 

 give to 2<Sm a , 2<Sm 6 , 2<$m fc , etc. values proportional to a, /3, K, 

 etc. in equation (30), and also the same values taken negatively, 

 making 2 Sm = in each of the other terms ; therefore 



a^ a + /W 6 + etc. ...-/clffc-X^-etc. = 0, (32) 



or, a^ a + /W& + etc. = /cJlf* + X^ + eta (33) 



It will be observed that this equation has the same form and coeffi- 

 cients as equation (30), M taking the place of @. It is evident that 

 there must be a similar condition of equilibrium for every one of the 

 r equations of which (30) is an example, which may be obtained 

 simply by changing in these equations into M. When these 

 conditions are satisfied, (31) will be satisfied with any possible values 

 of 2 6m v 2 Sm 2 , ... 2 8m n . For no values of these quantities are 

 possible, except such that the equation 



(2Sm 1 ) l + (2Sm 2 ) z ...+(28m n ) n = (), (34 



after the substitution of these values, can be derived from the r equa- 

 tions like (30), by the ordinary processes of the reduction of linear 

 equations. Therefore, on account of the correspondence between (31) 

 and (34), and between the r equations like (33) and the r equations 

 like (30), the conditions obtained by giving any possible values to 

 the variations in (31) may also be derived from the r equations like 

 (33); that is, the condition (31) is satisfied if the r equations like 

 (33) are satisfied. Therefore the r equations like (33) are with 

 (19), (20), and (22) the equivalent of the general condition (15) 

 or (23). 



