70 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



For determining the state of a given mass when in equilibrium 

 and having a given volume and given energy or entropy, the condi- 

 tion of equilibrium affords an additional equation corresponding to 

 each of the r independent relations between the n component sub- 

 stances. But the equations which express our knowledge of the 

 matter in the given mass will be correspondingly diminished, being 

 n r in number, like the equations of condition relating to the 

 quantities of the component substances, which may be derived from 

 the former by differentiation. 



Conditions relating to the possible Formation of Masses Unlike any 



Previously Existing. 



The variations which we have hitherto considered do not embrace 

 every possible infinitesimal variation in the state of the given mass, 

 so that the particular conditions already formed, although always 

 necessary for equilibrium (when there are no other equations of con- 

 dition than such as we have supposed), are not always sufficient. 

 For, besides the infinitesimal variations in the state and composition 

 of different parts of the given mass, infinitesimal masses may be 

 formed entirely different in state and composition from any initially 

 existing. Such parts of the whole mass in its varied state as 

 cannot be regarded as parts of the initially existing mass which 

 have been infinitesimally varied in state and composition, we will 

 call new parts. These will necessarily be infinitely small. As it is 

 more convenient to regard a vacuum as a limiting case of extreme 

 rarefaction than to give a special consideration to the possible for- 

 mation of empty spaces within the given mass, the term new parts 

 will be used to include any empty spaces which may be formed, 

 when such have not existed initially. We will use De, Dq, Dv, 

 Dm v Dm 2 , . . . Dm n to denote the infinitesimal energy, entropy, and 

 volume of any one of these new parts, and the infinitesimal quantities 

 of its components. The component substances 8 lt S 2 , ... S n must 

 now be taken to include not only the independently variable com- 

 ponents (actual or possible) of all parts of the given mass as initially 

 existing, but also the components of all the new parts, the possible 

 formation of which we have to consider. The character S will be 

 used as before to express the infinitesimal variations of the quantities 

 relating to those parts which are only infinitesimally varied in state 

 and composition, and which for distinction we will call original parts, 

 including under this term the empty spaces, if such exist initially, 

 within the envelop bounding the system. As we may divide the 

 given mass into as many parts as we choose, and as not only the 

 initial boundaries, but also the movements of these boundaries during 



