J. W. Gihhs—Eqailibrli(m of Heterogeneous Substances. 135 



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

 all bodies which can l)e formed out of the actual components of the 

 fluid, (which will ahvaj-s 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 Avhatever 

 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 xS„ . . . >% be the actual components of the solid, and aS'^, . . . S,, 

 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) 



cW = t d)j - ^y civ' -f- //,/ dm J . . . H- //; dm.J 



+ pi/dm^' . . . i-jutdn^. (58) 



By this equation the potentials j.ij . . . /u^.' are perfectly defined. 

 But the difierentials dm„' . . . dmi.', considered as independent, evi- 

 dently express variations w^hich are not possible in the sense required 

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

 into the genei-al condition of equilibrium, if we should express the 

 dependence between them by the j^roper 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 com- 

 ponent S^, of Avhich the nature is represented by the solid itself. We 

 may then write 



6e'=t' dif — p' dv' -f- jjj 6niJ. (59) 



In regard to the relation of the potential /^/ to the potentials occur- 

 ring in equation (58) it will be observed, that as we have by integra- 

 tion of (58) and (59) 



a' =: t' if - p' v' -\- /.(„' mj . . . + pij nij, (60) 



and e' = t' ?/ — p' v' + /jJ mj ; (61) 



therefore /.tj jt/J = /.tj mj . . . -\-f.i,'m,'. (62) 



Now, if the fluid has besides S^, . . . S,, and *S/, . . . S^. the actual 

 components S/ . . . /S„, we may write for the fluid 



