﻿of Radiation in any Material System. 541 



the statistical method applies to aether with its infinity of 

 degrees of freedom. As in this paper the ethereal energy 

 fails into the normal vibrations determined by the volume 

 enclosed. Where there is no matter, 



p= m nf(H 1 )f(H,)...f(m- ■ • (100) 



Here H m is the energy of the mth a3thereal degrees of 

 freedom p m , q m , where 



apm 2 dQm __ rj ^' . 2 _i_ 1 2 2 



7, — K m ?m; ~%~AT — TP m '> -Hm ^y pm ~r 2 K m $m • 



Where there is no matter present H^ H 2 ... H m , &c. are all 

 constant, for the normal vibrations do not interact, and 

 there is nothing to determine f\H m ). That function does 

 not, however, necessarily re-duce to a constant unless all the 

 laws of: motion are deduced from a single action formula. 

 As in this paper we may suppose a very large volume of 

 " aether " containing matter only for the purpose of bringing 

 about a temperature distribution. (100) now no longer 

 gives the complete expression for the function p. But li 

 suppose the material system defined by 2n coordinates 



ow 



P = F(M) X 3> ( M, A) X n / (H m ) • 

 m=l 



And for the purely material interactions 



Here F M is a function containing only the material co- 

 ordinates. <I>(m, a) contains both material and aethereal degrees 

 of freedom, and since these interact only slightly when the 

 volume of aether is made very large, we may make ^(m.a) 

 tend towards unity by sufficient increase of that volume. 

 If the ordinary statistical argument be now applied, it can 

 be shown (see Jeans' ' Dynamical Theory of Gases ') that 

 the infinitely probable distribution of energy is one in 

 which 



P , = Jle-hH m f(H m ) } 

 m—l 



and the average value of H m say H m is given by 



