358 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



then, by (5), 



-f // 2 dm 2 . . . -f fjL n dm n . (8) 



If, then, f is known as a function of t,p, m x , m 2 , ... m n , we can find 

 q, v y /Zj, /* 8 , .../ n in terms of the same variables. By eliminating 

 we may obtain again 7i+3 independent relations between the same 

 271+5 variables as at first.* 



If we integrate (5), (6) and (8), supposing the quantity of the 

 compound substance considered to vary from zero to any finite value, 

 its nature and state remaining unchanged, we obtain 



n , (9) 



, (10) 



(11) 



If we differentiate (9) in the most general manner, and compare the 

 result with (5), we obtain 



vdp + ridt+m 1 djUi l +m 2 d[ji 2 ...+m n diui n = Q, (12) 



or 



n Jj. , m i 7 , m 2 J , m 7 A /10\ 



= -dt-\ 1 du, + ^ du.* . . . H ^du n = 0. (13) 



1 n 



V V V V 



Hence, there is a relation between the 7i + 2 quantities t, p, fjL 1} 

 /j. 2 , ... fji n , which, if known, will enable us to find in terms of these 

 quantities all the ratios of the T&+2 quantities TJ, v, m 1? m 2 , ...m n . 

 With (9), this will make 7i+3 independent relations between the same 

 2n -f 5 variables as at first. 



Any equation, therefore, between the quantities 



e, q, v, m 1? m 2 , ...m n , 

 or i/r, ^, v, m 15 m 2 , ...m n , 



or ^, ^, p } mj, m 2 ,...m n , 



or , p, JUL I} // 2 , ... 



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

 any other. 



Coexistent phases. In considering the different homogeneous bodies 

 which can be formed out of any set of component substances, it is 

 convenient to have a term which shall refer solely to the composition 



* The properties of the quantities - \f/ and - f regarded as functions of the tempera- 

 ture and volume, and temperature and pressure, respectively, the composition of the 

 body being regarded as invariable, have been discussed by M. Massieu in a memoir 

 entitled "Sur les fonctions caract&istiques des divers fluides et sur la th^orie des 

 vapours" (M6m. Savants Etrang., t. xxii). A brief sketch of his method in a form 

 slightly different from that ultimately adopted is given in Comptes Eendus, t. Ixix (1869), 

 pp. 868 and 1057, and a report on his memoir by M. Bertrand in Comptes Rendm, t. Ixxi, 

 p. 257. M. Massieu appears to have been the first to solve the problem of representing 

 all the properties of a body of invariable composition which are concerned in reversible 

 processes by means of a single function. 



