﻿392 Mr. G. H. Livens on the Law of 



draw any conclusions regarding the volume of this space as 

 compared with that discussed above, or the size of the element 

 of energy which is used in the usual method of deducing 

 this formula, or even the number of vibrations involved 

 (which, however, on Planck's method, must have a finite 

 limit). The equality of the spaces in the two cases would, 

 however, imply some such relation as that discussed in our 

 previous paper and mentioned above, between the element 

 of energy and extent of cells. 



In conclusion it might be useful to illustrate the restrictions 

 and limitations of Planck's Theory by the alternative deduction 

 given by Jeans. 



Other things being equal, if a vibration can have energies 

 0, e, 2e, ..., then the ratio of the probabilities of these events, 

 as in the usual gas theory calculations^ are 



1 • e ~ 2qe : e~^ e ■ 

 where, however, according to the calculations given in the 

 earlier part of this paper, q is not equal to \y^y v but to 

 2 ^m , « being the value of the probability constant 



corresponding to these vibrations. 



If out of the N vibrations under consideration M have 

 zero energy, then the number which have energy e is M^ -226 , 

 the number having energy 2e is Me~ iq€ and so on. Thus 



N=M(l-e~ 2qe + e- 4 * e +...) 



M 



If E is the total energy of the N vibrations 

 E=Me(> 22e +2*T 426 -f...) 

 M€<T 2g6 _ Ke 



which is Planck's law if 



aie he 



2qe 



and 



R«0 ~ B\0 



„ Uic 



But now e can be taken to be zero if a is sufficiently small 

 and N sufficiently bio-. 



