J. W. Gihbs — Equilihrmm of Heterogeneous Substances. 121 



The potentUd for each of the component substances must luioe a 

 constant value in all parts of the given mass of iddch that substance 

 is an actual conxponeyit^ and have a value not less than this in all 

 parts of which it is a possible componetit. 



The number of equations aftbrded by these conditions, after elimina- 

 tion of J/j, iT/g, . . . Jf„, will be less than {n +• 2) (k - 1) by the num- 

 ber of terms in (15) in which the variation of the form dm is either 

 necessarily nothing or incapable of a negative value. The number of 

 variables to be determined is diminished by the same number, or, if 

 we choose, Ave may write an equaticm of the form m — for each of 

 these terms. But when the substance is a possible component of the 

 part concerned, there will also be a condition (expressed by ^ ) to 

 show whether the supposition that the substance is not an actual 

 component is consistent with equilibrium. 



We will now suppose that the substances S-^^, 8^, . . . iS„ are not 

 all independent of each other, i. e., that some of them can be formed 

 out of others. We will first consider a very simple case. Let S^ be 

 composed of S^ and So combined in the ratio of a. to b, S^ and S2 

 occurring as actual components in some parts of the given mass, and 

 /S'g in other parts, which do not contain S^ and S2 as separately 

 A^ariable components. The general condition of equilibrium will 

 still have the form of (15) with certain of the terms of the form 

 /< dm omitted. It may be written more briefly [(23) 



^{tSi/) - 2{pdv)-^:::^{/.i,(hn^)-^2{/'2dm2) ■ ' .-\-^^{Mn<^'n„)^0, 

 the sign ^ denoting suumiation in regard to the difierent parts of 

 the given mass. But instead of the three equations of condition, 



2 6m 1=0, 2" dm2 = 0, 2 6m^ — 0, (24) 



we shall have the two, 



2Sm,+^^2Sm, = 0,] 



The other equations of condition, 



2 Sij = 0, :2 dv = 0, ^ Sm^ = 0, etc., (26) 



will remain unchanged. Now as all values of the variations which 

 satisfy equations (24) will also satisfy equations (25), it is evident 

 that all the particular conditions of equilibrium which we have 

 already deduced, (19), (20), (22), are necessary in this case also. 

 When these are satisfied, the general condition (23) reduces to 

 M, 2 6)n , -f 3f, 2 6m 2+ M^ 2 6m 3^0. (27) 



Trans. Conn. Acad. 16 October. 1875. 



;. (25) 



