660 Mr. G. H. Livens on the 



offered in section 3, since according to the views there 

 expressed, 



a being ultimately zero : so that, however, 



T e he 



Ve= Lt -==- • 

 e-o <* A* 



It thus appears that a suitable modification of" Planck's 

 theory of radiation, involving a suitable weighting of the 

 probabilities of specified complexions among the degrees of 

 freedom of the various types considered, which is suggested 

 by considerations of the applicability of the Fourier series 

 analysis, entirely removes all difficulties in the formulation 

 of the problem given by Planck. Moreover, since in the 

 above analysis e must be very small, the essential condition 

 is satisfied for a final approach to a continuous analysis, so 

 that apparently no implication is now contained as to a finite 

 atomic structure for the elements of energy forming the 

 basis of the statistics*. 



A difficulty may be felt as to the implied relation between 

 e and Nx g , viz. 



XT he 



A s 



but such a relation is necessarily involved in a statistical 

 theory of the present type, since we must choose ~N\ 8 (ulti- 

 mately infinite, of course) so that the extents or capacities 

 of the particular type of cell are such that e is the smallest 

 amount, other than zero, they can accommodate. Of course, 

 in a theory of the present type, where no less than three of 

 the quantities involved are dependent on the scale of the 

 statistics, some such relations as those indicated here and in 

 the previous section must necessarily hold, otherwise it would 

 be hopeless to proceed further : this latitude of indefiniteness 

 is however probably the saving factor of the analysis, indi- 

 cating that in reality no conclusions can be drawn as to a 

 definite limit of subdivision such as are involved in Planck's 

 exposition. 



A similar modification also applies to the Payleigh- Jeans 

 theory, and it is shown to lead also to the same general 

 formula for the distribution of the energy. Of course the 

 actual system examined by these authors cannot possibly be 



* Nor as to the finite limit to the fineness of structure in the spectrum 

 which is involved in Planck's theory. 



