J. W. Gibbs — Equilibrium of Hetei'ogeiieoiis Substances. 141 



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

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

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

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

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

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

 exception of the sources of the work and heat expended, which must 

 be used only as such sources. 



We know, however, a priori, that if the quantity of any homoge- 

 neous mass containing it. independently variable components varies 

 and not its nature or state, the quantities f, ?/, v, »i,, m^, . . . ni„ will 

 all vary in the same proportion ; therefore it is sufficient if we learn 

 from experiment the relation between all but any one of these quan- 

 tities for a given constant value of that one. Or, we may consider 

 that we have to learn from experiment the relation subsisting 

 between the n i- 2 ratios of the n -{- 3 quantities f, //, v, m^, ra^, 



. . . m„. To fix our ideas we may take for these ratios , -, — ?, — -. 



etc., that is, the separate densities of the components, and the ratios 



£ If 



- and -, which may be called the densities of energy and entropy. 

 But when there is but one comj^onent, it may be more convenient to 



choose — , — , — as the three variables. In any case, it is only a func- 



m ni, ni j 7 ., 



tion of w. -f- 1 independent variables, of which the form is to be deter- 

 mined by experiment. 



Now if £ is a known function of ;/, w, m^, m^, . . . m^, as by equa- 

 tion (12) 



de-=.td)] - p dv + // , dm j -|- /ig ^^2 • • • + /v„ dm„, (86) 



t,p,' 1^1, ^2') • • • A'n ^'"^ functions of the same variables, which may 

 be derived from the original function by differentiation, and may 

 therefore be considered as known functions. This will make n -\- S 

 independent known relations between the 2n + 5 variables, e, /;, v 

 m^, 7712, • • • "^n» t,P, /-^i-, 1^2, ■ ■ ■ /'n- These are all that exist, for 

 of these variables, n + 2 are evidently independent. Now upon 

 these relations depend a very large class of the properties of the 

 compound considered, —we may say in general, all its thermal, 

 mechanical, and chemical properties, so far as active tendencies are 

 concerned, in cases in which the form of the mass does not require 

 consideration. A single equation from which all these relations may 



