J. W. Gibbs — JEquilibrlum of Heterogeneous Substances. 143 



find If, V, yUj, /.i.j,, • • • Mn i^i terms of the same variables. By elimi- 

 nating C, we may obtain again n -{- S independent relations between 

 the same 2n + 5 variables as at first. 



If we integrate (86), supposing the quantity of the compound sub- 

 stance considered to vary from zero to any finite value, its nature 

 and state remaining unchanged, we obtain 



s=ztff — pv + /^ 1 in J + //^ »« 3 . . . + //„ ?n„, (93 ) 



and by (87), (89), (91) 



Tlie last three equations may also be obtained directly by integrating 

 (88), (90), and (92). 



If we differentiate (93) in the most general manner, and compare 

 the result with (86), we obtain 



— V dp -\- tjdt + m^ dfi^ -\- in^ dji^ . . . + )n„dii„-=. 0, (97) 

 or 



dp=i- dt H i <?/<! H df.i^ . . . H df.1^ = 0. (98) 



Hence, there is a relation between the n + 2 quantities t, p, jli^, fi.^, 

 . . . yt/„, which, if known, will enable us. to find in terms of these quan- 

 tities all the ratios of the n + 2 quantities //, v, m^, m^ . . . m„. 

 With (93), this will make n + S independent relations between the 

 same 2n + 5 variables as at first. 



Any equation, therefore, between the quantities 



is a fundamental equation, and any such is entirely equivalent to any 

 other.* For any homogeneous mass whatever, considered (in gen- 

 eral) as variable in composition, in quantity, and in thermodynamic 

 state, and having n independently variable components, to which 



* The distinction between equations which are, and which are not, fundamental, in 

 the sense in whicli the word is here used, may be illustrated by comparing an equation 



