Statistical Relations of Radiant Energy. 661 



regarded as a model applicable to thermal radiation, owing 

 to the discontinuity in the spectrum of the vibrations ob- 

 tained in it : in fact, the number of vibrations with a wave- 

 length greater than any specified length is finite, although 

 directly related to the dimensions of the space in which they 

 take place. To obtain a more analogous system we must 

 probably dispose of the perfection in the ideal conditions 

 assumed (absolute rigidity and perfect reflexion), or in some 

 other way introduce a means of securing thermal equilibrium 

 directly*. But then the number of oscillations possible in a 

 volume V with wave-lengths between X s and \ s + dX s is no 

 longer simply the number counted by Jeans and Kayleigh, 

 viz. 



X s 4 > 



but is to be obtained by the insertion of a very large number 

 N\* of vibrations between each of those counted, N\ s being 

 ultimately infinite to secure perfect continuity. But then 

 the total amount of energy associated with these wave- 

 lengths is 



which is again Planck's formula if, as above, 

 Lt Nx s e= Lt - = ^. 



6=0 6 = a s X, 



It thus appears that the Rayleigh-Jeans theory of radia- 

 tion is also consistent with the present scheme, and the 

 result to be deduced from it is identical with Planck's 

 formula. The assumption of finite units of energy or of any 

 definite limit in the scale of the statistics is, however, no 

 longer involved in the considerations on which it is based. 



* It is interesting to notice how essential are the imperfections on the 

 ideal conditions usually assumed in any such system, both to secure 

 thermal equilibrium and also complete continuity in the spectrum. 

 Another example of this principle is afforded in Planck's method of 

 discussion. As Larmor points out, such a system of fixed linear vibra- 

 tors of the type assumed by Planck are not effective for the transfor- 

 mation of the relative intensities of the various types of radiation to 

 those corresponding to a common temperature. There would be equi- 

 librium established only between the mean internal vibratory energy in 

 the vibrators of each period and the density of the radiation of that 

 period. In order to obtain a means of interchanging energy between 

 vibrators of different types we must probably do away with their absolute 

 rigidity or else introduce more complex vibrators. 



