88 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



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

 the result with (86), we obtain 



=Q, (97) 



J Jj. , 1 7 , 9 7 , 7 



or dp = dt + -d^+-^dn 2 ...+-^diJ. n . (98) 



Hence, there is a relation between the n + 2 quantities , j9, /j, // 2 , 

 . . . fjL n , which, if known, will enable us to find in terms of these 

 quantities all the ratios of the n + 2 quantities rj, v, m 1? m 2 , ...m n . 

 With (93), this will make 7i+3 independent relations between the 

 same 2n + 5 variables as at first. 



Any equation, therefore, between the quantities 



e, q, v, mp m 2 , ...m w , (99) 



or \[,, t, v, m v ra 2 , ...m w , (100) 



or x , rj, p, m v m...m n , (101) 



or t, p, m v m 2 ,...m n , (102) 



or t, p, /ZP // 2 , ... p n , (103) 



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

 other.* For any homogeneous mass whatever, considered (in general) 

 as variable in composition, in quantity, and in thermodynamic state, 

 and having n independently variable components, to which the sub- 

 script numerals refer (but not excluding the case in which n = 1 and 

 the composition of the body is invariable), there is a relation between 

 the quantities enumerated in any one of the above sets, from which, if 

 known, with the aid only of general principles and relations, we may 

 deduce all the relations subsisting for such a mass between the 

 quantities e, ^, x , t], v, m v m 2 , ... m n , t, p, // 1? // 2 , ... fj. n . It will be 

 observed that, besides the equations which define i/r, ^, and there is 

 one finite equation, (93), which subsists between these quantities 

 independently of the form of the fundamental equation. 



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

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

 between e, i), v t m^, wi 2 , ... m n > 



with one between e, t, y, m lt m z , . . . m n . 



AB,by(86), *=(;) 



\G"7/tmi 



the second equation may evidently be derived from the first. '" But the first equation 

 cannot be derived from the second ; for an equation between 



is equivalent to one between \:r} > > v m i> '"hi m n> 



which is evidently not sufficient to determine the value of ?? in terms of the other 

 variables. 



