66 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



of these quantities, which we may call the independent variables, is 

 evidently (n+2)v, v denoting the number of homogeneous parts 

 into which the whole mass is divided. All the quantities which 

 occur in (19), (20), (21), are functions of these variables, and may be 

 regarded as known functions, if the energy of each part is known as 

 a function of its entropy, volume, and the quantities of its com- 

 ponents. (See eq. (12).) Therefore, equations (19), (20), (21), may 

 be regarded as (v 1) (n + 2) independent equations between the 

 independent variables. The volume of the whole mass and the total 

 quantities of the various substances being known afford n+1 addi- 

 tional equations. If we also know the total energy of the given 

 mass, or its total entropy, we will have as many equations as there 

 are independent variables. 



But if any of the substances S v S 2 , ... S n are only possible com- 

 ponents of some parts of the given mass, the variation Sm 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 l is an actual component, and 



for all parts of which 8 1 is a possible (but not actual) component, 



['(22) 

 for all parts of which 8 2 is an actual component, and 



for all parts of which 8 2 is a possible (but not actual) component, 



etc., 



M v M 2 , etc., denoting constants of which the value is only determined 

 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 independent (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 equilibrium (15) and the equations 

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

 of the substances 8 V S 2) ... 8 n is not a component (actual or possible) of 

 any part, the term fj. Sm for that substance and part will be wanting 

 in the former, and the Sm in the latter. This will require no change in 



