RECENT STATISTICAL THEORIES 697 



in each shell which contain each of the permissible quotas of particles. 

 We must translate it into a distribution-function-in-energy such as 

 we used to express the results of the old statistics. 



Before undertaking this translation, we compute the values of the 

 constants as by summing the numbers Zis over all values of i for 

 each shell separately, and equating the sum to the total number Qs 

 of compartments in the shell. We obtain: 



Qs = HZis = asZe-'''"'' = «s(l - e-''"'')-K (48) 



i i 



On substituting these values of as into (46) we get something which 

 begins to look familiar. 



Next for the number Ms of the particles in the shell s, we compute: 



Ms = ZiZis = ase-'-i'^'^il - e-^'i''^)-^ 



i 

 ~ *?« gUlkT _ I ' (4^) 



which begins to look very familiar indeed. 



Now for Qs, the number of compartments in the shell s, we put 

 the value already stated in equation (37), derived from the assumption 

 that the compartments are all of the same volume H: 



^^=% ,.//_! ^^^ - ^(^^)^^- (50) 



Here the function F{es) is the "smoothed-over" distribution-in- 

 energy in which the new statistics culminates. Unlike those which 

 we earlier derived from the old statistics, it involves the volume of 

 the elementary compartments directly. Whereas from the old sta- 

 tistics we obtained formulae involving the quantum only by avoiding 

 the approximation, here we obtain a quantum-formula even when 

 we admit the approximation — a contrast on which, I think, it is 

 worth while to insist. 



Let us then, in preparing for the final assumption, accept the 

 principle generalized from Planck's assumption about oscillators: let 

 the elementary cell of phase-space be given the volume h^. This is 

 not yet an assumption about the compartment of momentum-space. 

 Supplement it, then, by supposing that the compartment of phase- 

 space h^ is the product of the compartment // of momentum-space 

 and the entire volume V occupied by the radiation-gas. Then : 



H = h'JV. (51) 



