MEMORIAL TO CLASSICAL STATISTICS 127 



Comparing these, we see first of all that 7?/;; V appears in both, as was already 

 stated. It also seems at first glance that { — RliiA + U/T) is to be identi- 

 fied with the integral in (27), and that —Rlii Vo is to be identified with the 

 constant in (27). This however is not necessarily the case, for {—Rln A + 

 U/T) may prove to include constant terms. Indeed they do; and we must 

 proceed to evaluate both .4 and U in terms of T in order to round off the 

 task. 

 I recall equation (10) and write it thus: 



\/A = 2 exp {-Ei/kT) (28) 



This is a summation, to which each cell contributes one term having the 

 value of E appropriate to that cell — Et for the iXh. cell. Of the volumes of 

 these cells I have thus far said nothing, except that all are equal. I con- 

 tinue to say nothing further, but I give to their common volume the symbol 

 //o • Let us now form the integral : 



III exp {-E/kT) dpjpydp., E = {\/2m) (pi ^ pi + pi) (29) 



the range of integration extending over the whole of momentum-space. 

 This integral may be described as follows. Let the momentum-space be 

 divided into cells of unit volume. Each of these cells of unit volume makes a 

 contribution 



exp (—E/kT) 



to the integral, E standing now for the average value of E in the cell in ques- 

 tion. The integral is the sum of all of these contributions. Now let us in- 

 quire how much of a contribution is made by this same cell of unit volume to 

 the summation (28). This second contribution is made up of 1/Ho terms, 

 one for each of the cells of volume Ho which occupy the cell of unit volume. 

 The values Ei corresponding to these cells will not be exactly equal to the 

 value E corresponding to the entire cell of unit volume; but to the degree of 

 approximation which is now being used, the difference may be neglected. 

 The summation (28) is then equal to 1/Ho times the integral (25). Now the 

 value of the integral (29) is given in all tables of definite integrals, and in 

 terms of our symbols it amounts to 



(iTrmkT)"'^ 



so we come to the conclusion: 



In ^ = - In {lirmkTf'^ ^ In Ih 



= -| In r - In (IwrnkY" + In Ho (30) 



