﻿the Partition of Energy. 837 



Then 



dQ = dIH+X 1 da: 1 + X a da: 2 + ..., 



= 2 r M,{[llog/ r+ ^ a log/,]^ 



• • '• (7-4) 



It follows that log (1/3) c?Q is a perfect differential. We 

 can therefore at once define the absolute temperature scale, 

 so as to make dQ/T a perfect differential, by the equation 



log 1/3 = 1/AT (7-5) 



No function only of «3, except log 1/3, can be an integrating 

 factor of dQ, and therefore the absolute temperature so 

 defined is unique, apart from k, the constant undetermined 

 factor which it always contains. Moreover, by definition 

 of Sth., dQ/T = dSth., and therefore, except for one arbitrary 

 additive constant, 



Sth. = ECBk Aog/r + log 1/3 . 3 |L log/r\ 



= Sst., (7-6) 



where S s t. is defined by (5*1). The identification of our 

 statistical entropy with the entropy of thermodynamics is- 

 complete, and the rest of thermodynamics follows in due 

 course, so long as the assemblies considered are of such: 

 types as to be representable by partition functions. 



§ 8. The characteristic function of Planck. 



We have presented the formal proof in its most familiar 

 form, but we can now make the presentation mathematically 

 much simpler. The expression (5'2) invites us to make 

 our fundamental definition not that of entropy but of 

 the ' ; characteristic function ; ' of Planck. This function ^, 

 which is closely allied to the " free energy," is defined in 

 thermodynamics by 



^= S-E/T (8'1> 



