﻿82Q Messrs. 0. (x. Darwin and R. H. Fowler on 



weight and e r the energy in the rth cell, we define as 

 the partition function 



summed over all the cells. The partition function is thus a 

 function of 3, x^ x 2 , .... For systems of classical mechanics 

 the extensions of the cells, that is the p's, all tend to zero, 

 and the sum is replaced by an integral. 



As an example of the systems treated we may mention 

 first Planck's line vibrator, which can take energy in 

 multiples of e. This has the partition function 



(2-1) 



1-S 



More important is the free monatomic molecule. If it is 

 of mass m and is confined in a volume V, its partition 

 function is 



(27rm)3/2V 



A 8 (Jogl/3) 8/a { ' J ^ J 



The partition function of a diatomic molecule is simply 

 given by multiplying (2-2) by the partition function corre- 

 sponding to the quantized rotations (assuming that the 

 atoms cannot vibrate relatively to one another). 



Now suppose that we have any number of types of 

 systems together in an assembly ; let there be, say, M A 

 of type A with partition function f A . Then it was shown 

 that the average energy among the systems of this type 

 was given by 



E A =M A S^log/ A , .... (2-3) 



where $ is uniquely determined in terms of E the total 

 energy by 



E = 2 A M A $^log/ A . . . . (2-4) 



It is easy to show that 3 is always less than unity. The 

 average number of systems A in the rth cell is also easily 

 found, and is 



«<■ = M A p A /v// A (2-5) 



In connexion with the relation of entropy to probability, 

 we must also recapitulate some of the work from which the 

 above results were derived. The statistical state of an 



