THERMODYNAMIC AL SYSTEM OF GIBBS 91 



a homogeneous mass. If the variations of n + 1 of these 

 quantities are given any arbitrary values, the variation of the 

 remaining quantity can be determined by (56). A single 

 homogeneous mass is therefore capable of only n + 1 inde- 

 pendent variations of state. 



Additional Relations 



It will be convenient to give here some additional relations 

 which are easily obtained from the equations of the last section. 

 By (37) or (45) we have, for a body of fixed composition and 

 mass (indicated by the subscript m), 



or 



This equation, which has been found a very convenient expres- 

 sion of the relation between \p and e, was first given explicitly 

 by Helmholtz* and is known as the Gibbs-Helmholtz equation. 

 An equivalent equation between f and x is obtained from (39) 

 or (43), viz: 



(S).,. = 



Further, since 



M 37 = - 'J^ = r - X. (59) 



d{yP/t) # 



^'~dr ^^jt-"^' 



we may write (58) as 



/d{m\ ^ _ 1 



\ (II / V, m t 



(60) 



and similarly (59) becomes 



mm ^ _x 



y ai y p, m V 



* Sitzungsber preuss. Akad. Wiss., 1, 22 (1882); cf. Gibbs, I, 412 



(61) 



