EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 85 



But independently of any assumption in regard to the permeability 

 of the diaphragm, the following relation will hold true in any case in 

 which each of the two fluid masses may be regarded as uniform 

 throughout in nature and state. Let the character D be used with 

 the variables which express the nature, state, and quantity of the 

 fluids to denote the increments of the values of these quantities 

 actually occurring in a time either finite or infinitesimal. Then, as 

 the heat received by the two masses cannot exceed t'nfi' + tf'DTj", and 

 as the increase of their energy is equal to the difference of the heat 

 they receive and the work they do, 



DC + De" < f Dfl' + tf Dq" -p DV -p"Dv", (83) 



i.e., by (12), 



/I I 'DWI I / +/I I "DW I " + // 2 'Dm 2 '+ju 2 "Dm 2 / '+etc. ^ 0, (84) 



or 



O. (85) 



It is evident that the sign = holds true only in the limiting case in 

 which no motion takes place. 



Definition and Properties of Fundamental Equations. 



The solution of the problems of equilibrium which we have been 

 considering has been made to depend upon the equations which 

 express the relations between the energy, entropy, volume, and the 

 quantities of the various components, for homogeneous combinations 

 of the substances which are found in the given mass. The nature of 

 such equations must be determined by experiment. As, however, it 

 is only differences of energy and of entropy that can be measured, or 

 indeed, that have a physical meaning, the values of these quantities 

 are so far arbitrary, that we may choose independently for each 

 simple substance the state in which its energy and its entropy are 

 both zero. The values of the energy and the entropy of any com- 

 pound body in any particular state will then be fixed. Its energy 

 will be the sum of the work and heat expended in bringing its 

 components from the states in which their energies and their entropies 

 are zero into combination and to the state in question; and its 



entropy is the value of the integral l~ for any reversible process 



by which that change is effected (dQ denoting an element of the 

 heat communicated to the matter thus treated, and t the temperature 

 of the matter receiving it). In the determination both of the energy 

 and of the entropy, it ia understood that at the close of the process, 

 all bodies which have been used, other than those to which the deter- 

 minations relate, have been restored to their original state, with the 



