718 BELL SYSTEM TECHNICAL JOURNAL 



As I mentioned earlier, the idea of compartments in the momentum 

 space may be replaced by the idea of "permitted energy- values," 

 at least in some situations. Let us for convenience return to the 

 latter conception. Then we may say that when a magnetic field is 

 applied to an electron-gas of which the particles are magnets, each of 

 the permitted energy-values is split into two. To any previously- 

 permitted value e correspond a pair of new ones, e -(- A and e — A; 

 or let me say e + mA, using m as a symbol which may have only 

 the values + 1 and — 1. 



Pauli assumed that the most probable distribution of the electrons 

 among this doubled set of energy-values is to be determined by the 

 new statistics, including Fermi's postulate so modified as to state 

 that not more than one electron may possess any one of the permitted 

 values. 



Previously I used the symbol Zis to denote the number of compart- 

 ments or permitted energy-values which lie in the shell .y and are 

 occupied by i electrons apiece; and the symbol Qs to denote the total 

 number of permitted energy-values in the shell s. Now there are Qs 

 permitted values which are shifted upward by A from the original 

 ones, and Qs more which are shifted downward by A from the original 

 ones. Let Zis+\ stand for the number in this upward-shifted group 

 which are occupied by i electrons apiece, and Zis_i for the corre- 

 sponding number in the downward-shifted groups; Zism shall be the 

 general symbol for the two. 



Now consider the distribution: 



7 = n p-'iB-t{t,+Tn.^)ltT ni = 4- 1 — 1 



^^ ism <^sm<-- , '"- T^ ^1 -I) /OT\ 



«■ = 0, 1. ^ ^ 



This answers the standard requirements for thermal equilibrium. 

 For if we define W* in Bose's way, and then say that the entropy is 

 k log W*, we find that when the energy of the assemblage is varied 

 by bE — the total number of cells and the total number of particles 

 remaining constant — the first variation of the entropy is 8E/T, as it 

 should be. I leave it to the reader to prove this statement as corre- 

 sponding statements for other distributions (e.g. (47)) were earlier 

 proved, and to determine the values of the quantities asm', on doing 

 which, and substituting the results into this distribution, he should 

 obtain 



Q. 



Q^ <?. = (;^|3(2™)»(«.)'". (83) 



