EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 87 



Let >/. = e-ty, (87) 



then by differentiation and comparision with (86) we obtain 



d\l? = rjdtpdv+/uL 1 dm l + fjL2dm 2 ...+fjL n dm n . (88) 



If, then, \[s is known as a function of t, v, m ly m 2 , . . . m^ we can 

 find rj, p, fi v fa, . . . /j. n in terms of the same variables. If we then 

 substitute for \]s in our original equation its value taken from eq. (87), 

 we shall have again n+3 independent relations between the same 

 271 + 5 variables as before. 



Let x = +pv, (89) 



then by (86), 



d\ = tdq + v dp + fjL 1 dm l + // 2 dra 2 . . . + fi n dm n . (90) 



If, then, x b 6 known as a function of i\, p, m lt ra 2 , . . . ra n , we can find 

 t, v, fJL v /z 2 , ... fji n in terms of the same variables. By eliminating % t 

 we may obtain again n-f 3 independent relations between the same 

 '2n + 5 variables as at first 



Let f=e-ty+2>v, (91) 



then, by (86), 



^f = - 1 dt + v dp + fji^dm^ 4- /* 2 dm 2 . . . + t* n d>m n . (92) 



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

 find ij, v, fj. v /UL Z , ... fji n in terms of the same variables. By eliminating 

 f , we may obtain again n -f 3 independent relations between the same 

 2u + 5 variables as at first. 



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

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

 and state remaining unchanged, we obtain 



e = tn -pv + fji l m 1 + yu 2 ?n 2 ... 4- p n m nt (93) 



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



n n , (95) 



n n . (96) 



The last three equations may also be obtained directly by integrating 

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



from a single function, which he calls a characteristic function of the fluid considered. 

 In the papers cited, he introduces two different functions of this kind, viz., a function 

 of the temperature and volume, which he denotes by ^, the value of which in our 



notation would be - or -^; and a function of the temperature and pressure, 



' t 



which he denotes by ^', the value of which in our notation would be or -. 



t t 



In both cases he considers a constant quantity (one kilogram) of the fluid, which is 

 regarded as invariable in composition. 



