154 J. W. Gihbs — Equilibrium of Heterogeneous Substances. 



Let one of these equations be 



v' dp=z 7]' dt + mj djj>a + "Tin-i df.ii + etc., (124) 



and by the proposed elimination let it become 



v' dp= If dt + A^' dj-i^ + A^ df.i2 . . . + A„' d/.j„. (125) 

 It will be observed that //„, for example, in (124) denotes the poten- 

 tial in the mass considered for a substance *S'„ which may oi- may not 

 be identical with any of the substances S^, S2, etc. to which the 

 potentials in (125) relate. Now as the equations between the poten- 

 tials by means of which the elimination is performed are similar to 

 those which subsist between the units of the corresponding sub- 

 stances, (compare equations (38), (43), and (51),) if we denote these 

 units by (Sa, ©4, etc., ©i, ©g, etc., we must also have 



m„'(S„ + m;®,, + etc. = .4j'(5i-|-^2'®2 • • • +^J„'®„- (126) 

 But the first member of this equation denotes (in kind and quantity) 

 the matter in the body to which equations (1 24) and (125) relate. As 

 the same must be true of the second memV)er, we may regard this same 

 body as composed of the quantity A ^' of the substance aS,, with the 

 quantity A^' of the substance 1S2, etc. We will therefore, in accord- 

 ance with our general xisage, write m^' tn^', etc. for A^', A2', etc. in 

 (125), which will then become 



v' dp = 7/ dt -f »2i' d/.i^ + m^ dyi^ • • • + "*«' d^^- (127) 

 But we must remember that the components to which the m/, mg', 

 etc. of this equation relate are not necessarily independently variable, 

 as are the components to which the similar expressions in (9V) and 

 (124) relate. The rest of the /i + 1 equations may be reduced to a 

 similar form, viz., 



v" dp = 7f dt -\- m^" di-i^ ■^-m^'dj.i^ . ■ . +m„"dp„, (128) 



etc. 

 By elimination of f?/< j, d/^i^, . . . dfi„ from these equations we obtain 



v m 

 v" m 

 v'" rn 



dt. (129) 



In this equation we may make v', v", etc. equal to unity. Then 

 m,', mg', m/', etc. will denote the separate densities of the compo- 

 nents in the different phases, and //', ?/', etc. the densities of entropy. 



