EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 67 



the form of the particular conditions of equilibrium as expressed by 

 (19), (20), (22); but the number of single conditions contained in (22) 

 is of course less than if all the component substances were components 

 of all the parts. Whenever, therefore, each of the different homo- 

 geneous parts of the given mass may be regarded as composed of some 

 or of all of the same set of substances, no one of which can be formed 

 out of the others, the condition which (with equality of temperature 

 and pressure) is necessary and sufficient for equilibrium between the 

 different parts of the given mass may be expressed as follows : 



The potential for each of tlie component substances must have a 

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

 is an actual component, and have a value not less than this in all 

 parts of which it is a possible component 



The number of equations afforded by these conditions, after elimi- 

 nation of M v M 2 , ... M n , will be less than (n + 2)(v 1) by the number 

 of terms in (15) in which the variation of the form 8m 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, we may write an equation 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 v 8 2 , ... 8 n 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> 3 be 

 composed of 8 l and $ 2 combined in the ratio of a to b, S 1 and 8 2 

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

 8 B in other parts, which do not contain 8 l and $ 2 as separately 

 variable components. The general condition of equilibrium will still 

 have the form of (15) with certain of the terms of the form ju.8m 

 omitted. It may be written more briefly 



^(t8r)) ^ l (p8v)-\-^(fj. l 8m l )-\-^ l (juL 2 8m z } ... + Z(/z w <5m n )=0, (23) 

 the sign S denoting summation in regard to the different parts of 

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



2 8m, =0, 2 ($?fto = 0, 2 8m.> = 0, (24) 



A * * O * \ . / 



we shall have the two, 



a 

 i dm 3 = U, 



(25) 



The other equations of condition, 



= 0, etc., (26) 



