80 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



of equilibrium, even in the case of compounds of variable proportions, 

 i.e., even when bodies can exist which are compounded in proportions 

 infinitesimally varied from those of the solids considered. (Those 

 solids which are capable of absorbing fluids form of course an 

 exception, so far as their fluid components are concerned.) It is true 

 that a solid may be increased by the formation of new solid matter 

 on the surface where it meets a fluid, which is not homogeneous with 

 the previously existing solid, but such a deposit will properly be 

 treated as a distinct part of the system (viz., as one of the parts 

 which we have called new). Yet it is worthy of notice that if a homo- 

 geneous solid which is a compound of variable proportions is in 

 contact and equilibrium with a fluid, and the actual components of 

 the solid (considered as of variable composition) are also actual com- 

 ponents of the fluid, and the condition (53) is satisfied in regard to 

 all bodies which can be formed out of the actual components of the 

 fluid (which will always be the case unless the fluid is practically 

 unstable), all the conditions will hold true of the solid, which would 

 be necessary for equilibrium if it were fluid. 



This follows directly from the principles stated on the preceding 

 pages. For in this case the value of (57) will be zero as determined 

 either for the solid or for the fluid considered with reference to their 

 ultimate components, and will not be negative for any body whatever 

 which can be formed of these components ; and these conditions are 

 sufficient for equilibrium independently of the solidity of one of the 

 masses. Yet the point is perhaps of sufficient importance to demand 

 a more detailed consideration. 



Let S a , . . . S g be the actual components of the solid, and S ht ... S k 

 its possible components (which occur as actual components in the 

 fluid); then, considering the proportion of the components of the 

 solid as variable, we shall have for this body by equation (12) 



de' = t'drf p f dv f + ^dm^. . . -f fj. g 'dm g ' 



+ jm h 'dm h '. . . + fr' dm*'. (58) 



By this equation the potentials // a ', ... fa' are perfectly defined. But 

 the differentials dm a ', . . . dm^, considered as independent, evidently 

 express variations which are not possible in the sense required in 

 the criterion of equilibrium. We might, however, introduce them 

 into the general condition of equilibrium, if we should express the 

 dependence between them by the proper equations of condition. 

 But it will be more in accordance with our method hitherto, if we 

 consider the solid to have only a single independently variable 

 component S m of which the nature is represented by the solid itself. 



We may then write 



(59) 



