130 J W. Gibbs — Eqailibriain of Heterogeneous Substances. 



+ 11," Sm," . . . + /.It" 6m,r-{-pi/' dm/' . . . + //„" f^?«„", (63) 

 and as by suppusition 



nij ®^ = mj ©„...+ »'*</' ®.v (6*) 



equations (43), (oO), and (51) will give in this ease on elimination of 

 the constants T, P, etc., 



t'=:t", p'=p", (65) 



and 



mj M.' = '".,' I-'.." ■ • • +w^; //,/'• (66) 



Equations (65) and (66) may be regarded as expressing the condi- 

 tions of equilibrium between the solid and the fluid. The last con- 

 dition may also, in virtue of (62), be expressed by the equation 



w,,'//,,' . . . -j-n,; /.i; = mj /j„" . . . +';/*,>;'. (67) 



But if condition (53) holds true of all bodies which can be formed 



of «S'„ . . . S^, S,„ . . . iSi; S, . . . /8„, we may write for all such bodies 



£ — t" ?/-\-p" V — //„" m„ ... — //,/' m„ — //;," nh 



. . . — /V'w'i- — l-h' nil . . . M„"m„^ 0. (68) 



(In applying this formula to various bodies, it is to be observed that 

 only the values of the unaccented letters are' to be determined by 

 the different bodies to which it is applied, the values of the accented 

 letters being already determined by the given fluid.) Now, by (60), 

 (65), and (67), the value of the first member of this condition is zero 

 when applied to the solid in its given state. As the condition must 

 hold true of a body differing infinitesimally from the solid, we shall 

 have 



dt' — t" dif -\-p" di^' — l^i„" dnij . . . ^" dnij 



— f.i,," dm,! ... - /V'fW= 0, (69) 



or, by equations (58) and (65), 



{l-i,,' — l^a') dm,; . . . -[_(//,/-;/;') c?;/,; 



+ {Ih'-^u") dm,; ... 4- (/V-yWi") dm,'^ 0. (70) 



Therefore, as these differentials are all independent, 



^,; = Ma", ■ ■ • mJ = mJ\ M>'= /■</'', ' • • Mh'^ /'x"; (71) 

 which with (65) are evidently the same conditions which we would 

 have obtained if we had neglected the fact of the solidity of one of 

 the masses. 



