86 BUTLER AKT. u 



Planck has also made use of the second function, which has the 

 same properties in a system at constant temperature and pres- 

 sure as the entropy at constant energy and volume. 



8. Differentials of e, \p and f . The variations with temperature 

 and pressure of the quantities i/' and f , for- a homogeneous body 

 of fixed composition, are obtained by differentiating (14) and 

 (15) and comparing with (3). Thus 



but since 



we have 

 and 



Similarly, 



so that 



dyp = de — tdr] — -qdt, (35) 



de = tdrj — pdv, 

 d4/ = —pdv — -qdt, (36) 



(f).=-. (a=- ^3. 



d^ = de — tdr] — 77c?/ + pdv + vdp 



= - ndt + vdp; (38) 



Now, if the system is heterogeneous, the quantity of matter 

 in some of its parts may increase at the expense of that in other 

 parts and we shall need to express the effect of such variations 

 on the energy and on the quantities yp, f and x- Consider a 

 single homogeneous mass containing the quantities Wi, m2, 

 W3, . . . m„ of substances ^1, S2, Sz,... Sn- It is usually 

 possible to express the composition of a mass in a number of 

 different ways. It is immaterial which way is chosen, provided 

 that the components are such that every possible independent 

 variation in the composition of the mass can be expressed in 

 terms of them. For example, possible variations in the com- 

 position of a solution of sulphuric acid in water may equally 



