RECENT STATISTICAL THEORIES 695 



This means that each distribution in the new sense comprises a 

 number of distinct distributions in the old sense. This we shall 

 regard as the number of different ways in which the distribution in 

 the new sense may be realized. Going over entirely to the new 

 terminology: we shall now identify the probability W* of a distribution 

 with the number of arrangements which are included in it. Previously 

 we identified the probability W of an arrangement with the number 

 of permutations included in it, according to equation (7). This we 

 now must forget; we must proceed as if the probability of each arrange- 

 ment were the same. 



The number W* is now to be evaluated. In doing this we must 

 remember that we have to count, not the total number of arrangements 

 yielding the prescribed set of values of the quantities Zi, but the 

 portion of these which give the prescribed value to the total energy 

 of the assemblage. 



As before, we superpose upon the partitioning of the momentum- 

 space into small compartments of equal volume //, another partitioning 

 into spherical shells each of which is sufficiently large to contain many 

 of the compartments, yet sufficiently small so that the same value 

 of e may be assigned to all the compartments within it. The final 

 result is thus to be a " smoothed-over " formula. It is rather singular 

 that whereas Planck introduced the quantum into physics by avoiding 

 the smoothing-over which had been customary in the classical statistics 

 the quantum-formula for radiation is now derived by a method in 

 which it is accepted. 



Consider then any shell at random, say the "shell 5." Denote by 

 Qs or by Z the total number of compartments in it; by Zis the number 

 of these compartments which contain i particles apiece; by Ms the 

 number of particles in the shell. According to the scheme now being 

 tried out, the number of ways of attaining the particular distribution 

 characterized by the numbers Zis is given by the formula: 



Ws"^ = ^ , „ ^;l , , Qs-HZis. (41) 



As the energy-values for all the compartments in the shell are (by 

 hypothesis) approximately the same, these various ways of attaining 

 the distribution Zis all corresponding to approximately the same total 

 energy, as well as the same total number of compartments and the 

 same total number of particles. 



Suppose this process repeated for every one of the shells 5. The 

 total number of ways of attaining the actual distribution, compatible 



