120 J^ W. Gibbs — Equilibrmm of Heterogeneous Substances. 



But if any of the substances S^, S.^ . . . S„ are only possible com- 

 ponents of some parts of the given mass, the variation 6m of the 

 quantity of such a substance in such a part cannot have a negative 

 value, so that the general condition of equilibrium (15) does not 

 require that the potential for that substance in that part should be 

 equal to the potential for the same substance in the parts of which it 

 is an actual component, but only that it shall not be less. In this 

 case instead of (21) we may write 



for all parts of which ^S'j is an actual component, and 



for all parts of which S^ is a possible (but not actual) com- i 

 ponent, ' 



Ih = ^^2 y (22) 



for all parts of which iS'g is an actual component, and 



for all parts of which S2 is a possible (but not actual) com- 

 ponent, 



etc., 



J/j, M2, etc., denoting constants of which the value is only deter- 

 mined by these equations. 



If we now suppose that the components (actual or possible) of the 

 various homogeneous parts of the given mass are not the same, the 

 result will be of the same character as before, provided that all the 

 different components are indej^endeyit, (i. e., that no one can be made 

 out of the others,) so that the total quantity of each component is 

 fixed. The general condition of equilibi'ium (15) and the equations 

 of condition (16), (17), (18) Avill require no change, except that, if 

 any of the substances S^ , S2 . . . S„ is not a component (actual or 

 possible) of any part, the term fx dm for that substance and part will 

 be wanting in the former, and the 6m in the latter. This will require 

 no change in 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 sub- 

 stances were components of all the parts Whenever, therefore, each 

 of the different homogeneous 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: 



