RECENT STATISTICAL THEORIES 693 



freedom) for the number of distinct coordinates q required to describe 

 the individual member of whatever assemblage one may be considering; 

 this is also the number of distinct momenta p, there being one p for 

 each q. Then the principle generalized out of Planck's postulate for 

 oscillators is this: For an assemblage of individuals with n degrees of 

 freedom the phase-space is to he divided into compartments of volume h^. 



We will now see what the classical statistics, supplemented by 

 this principle, proposes for an assemblage of particles for which the 

 relation between energy and momentum is e = cp as it is for corpuscles 

 of light, instead of e = p^/lm as it is for corpuscles of matter. 



Different energy-values correspond as before to different spheres 

 all centred at the origin of the momentum-space, but the numerical 

 relations are changed. Instead of equations (24), we have: 



Cs = crs, des = cdrs, 



. , (36) 



dV = 4TrMrs = i47r/c')ej'd€s, 



dV standing for the volume of the shell 5 covering the energy-range 

 between e^ and e^ + des. Divide the momentum-space into compart- 

 ments of equal volume //. We derive the "smoothed-over" distri- 

 bution-function for the case in which Ni varies so little from one 

 compartment to the next that even when the shell 5 is thick enough 

 to comprise very many compartments the values of Ni for all of 

 them may be equated to a mean value N^. Under these conditions 

 we may write for the number of compartments in the shell s, 



Qs = dV/H = {4T/cm)e-'des. (37) 



Putting down the classical value (15) for the number of particles in 

 any of these compartments, remembering that Ni is identified with 

 Ng, we obtain for Ms the number of particles in the shell s: 



Ms = QsNs = a(4w/c'H)es'e~'^"''des = F(e,)(^e, (38) 



and evaluate a by the same procedure as before. The result is: 



This is the smoothed-over distribution-in-energy predicted for the 

 radiation-gas by the classical statistics, it being assumed that the 

 momentum-space is to be divided into compartments of equal volume. 

 Experiment however supplies a quite different distribution-in-energy, 

 to wit: 



^W=(|J),-;^- (40) 



45 



