86 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



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 homo- 

 geneous mass containing n independently variable components varies 

 and not its nature or state, the quantities e, r\, v, m^ m 2 , . . . w n 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 

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

 consider that we have to learn from experiment the relation sub- 

 sisting between the n+2 ratios of the 7i+3 quantities e, r\, v, m v m 2 , 



. . m,.. To fix our ideas we may take for these ratios -> -> -> -> 



V V V V 



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



G Tl 



1 and -5 which may be called the densities of energy and entropy. 

 But when there is but one component, it may be more convenient to 



c Yt 1) 



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

 m m m 



tion of Ti + 1 independent variables, of which the form is to be 

 determined by experiment. 



Now if e is a known function of ?/, v, m v m 2 , . . . m n , as by 

 equation (12) 



de = tdrjpdv + im l dm 1 +fjL 2 dm 2 . . . -f jm n dm n , (86) 



> P> /*!> /*2> Vn are 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 + 3 

 independent known relations between the 271 + 5 variables, e, ?/, v, 

 m p m 2 , . . . m n , t, p, JUL I} yu 2 , . . . fJL n . These are all that exist, for 

 of these variables, 7i + 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 

 be deduced we will call a fundamental equation for the substance in 

 question. We shall hereafter consider a more general form of the 

 fundamental equation for solids, in which the pressure at any point 

 is not supposed to be the same in all directions. But for masses 

 subject only to isotropic stresses an equation between e, 77, v, m v 

 ra 2 , . . . m n is a fundamental equation. There are other equations 

 which possess this same property.* 



*M. Massieu (Comptes Rendm, T. Ixix, 1869, p. 858 and p. 1057) has shown how all 

 the properties of a fluid "which are considered in thermodynamics" maj' be deduced 



