140 J. W. Gibbs — Equilibrmm of Heteroge)ieous ^Substances. 



f," _t' //' -\-p'v"- /.i^'m," - J^to'ms" . . . - /-'n m„" <^0. (82) 



This relation indicates the instability of tlie fluid to which the single 

 accents refer. (See page 133.) 



But independently of any assumption in regard to the permeability 

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

 which eacli of the two fluid masses may be regarded as unifonn 

 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 actu- 

 ally occurring in a time either flnite or infinitesimal. Then, as the 

 heat received by the two masses cannot exceed t'T>}/ -\-t" v>if', and as 

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

 they receive and the work they do, 



Di' + T>b" -St' litf + «"d//' — />'du'— p"iyv", (83) 



i.e., by (12), 



yu,'Dm,'+/(i"Dm/' + //2'n?;4' + /<2"Dm2" + etc. ^0, (84) 



or 



(///' — ///) r.m/'+ (/^2"-/^2') ^>m,"+ etc. ^0. (85) 



It is evident that the sign = liolds 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 com- 

 ponents 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 / — for any reversible process 



