RECENT STATISTICAL THEORIES 689 



between the various proposed statistical laws. We shall see that in 

 passing from one to another sometimes one of the factors is changed, 

 sometimes the other, sometimes both. 



In particular, we may pass from the Maxwell- Boltzmann law to a 

 distribution like that which Planck derived for oscillators, simply by 

 changing the factor Qs. We have been dividing the momentum-space 

 into compartments of equal volume, so that the number comprised 

 in a shell .y between spheres Cs and eg + des is proportional to es^'-dcs. 

 Let us instead divide it into compartments of which the volumes 

 increase steadily from the origin outward, at such a rate that the 

 number in a shell s is proportional to des without the factor e^^'-. 



This is, of course, not the way in which Planck's postulate is ha- 

 bitually stated, though it is substantially the way in which Planck 

 stated it himself. Usually it is said, that Planck restricted the energy 

 of the particles of the assemblage to a set of "permitted values" 

 spaced at equal intervals: say the values a, a -\- b, a -{- 2b, a -{- 3b, 

 • • • where a and b stand for constants. Each of these permitted 

 values corresponds to a sphere in the momentum-space. In the shell 

 s there are approximately dejb of these "permitted spheres"; the 

 approximation being closer, the larger es and des are in comparison 

 to b. Now whether we conceive that the des/b sets of particles in 

 the shell 5 are located on the surfaces of as many permitted spheres, 

 or alternatively that they are scattered through as many compart- 

 ments, is for the statistical results of no importance. There may be 

 other reasons for preferring one picture to the other; but the pre- 

 dictions of the statistical theory are the same, whichever is adopted. 

 I will therefore alternate between the two pictures, retaining for the 

 moment that of a subdivision of the momentum-space into compart- 

 ments; but now it will be expedient to think of these as thin spherical 

 films, centered at the origin and increasing in volume from the inner- 

 most outward at the specified rate. 



If the shell s is large enough to contain many of these compartments 

 or permitted spheres, we may use the first approximation for the 

 number which it contains: 



Qs = de/b, (28) 



and putting the expression (26) for the number of particles per com- 

 partment, we get: 



Ms = QsNs = |exp (- esfkT)des = F{es)des (29) 



for the number of particles having energy-values between e^ and 



