254 Temperature-Radiation and Partition of Energy. 



31. Apart from this, the identity of k and R can, I think,, 

 be established by an argument of a very general nature, 

 which does not depend on these special assumptions. 



The continuous spectrum emitted by a solid must bft 

 granted to proceed from the motions of free electrons. A 

 calculation, based on our knowledge of the total kinetic 

 energy of these electrons and of the rate at which they 

 radiate energy, will show that their rate of radiation of 

 energy is very slow, when the time-scale is taken to be the 

 average interval between two collisions. It follows that the 

 kinetic energies of these electrons must be distributed in 

 accordance with Maxwell's law about the mean value §RT. 

 This theoretical result can be obtained independently of any 

 assumptions as to the nature of free-paths, collisions, or 

 forces by which the electrons are acted upon. Its truth has 

 recently been confirmed in a very striking manner by the 

 experiments of Richardson and Brown*. 



It follows that the motion of the electrons can be analysed 

 into the motion of trains of waves by the analysis of § 19. 

 Each of the trains of waves, into which the motion of the- 

 electrons can be analysed, will have kinetic energy appro- 

 priate to the temperature T. There will obviously be ex- 

 tremely rapid transfer of energy between these waves of 

 electrons and the aether in which they are imbedded. Thus 

 the aether in the interior of matter or in a cavity made in the 

 matter, will immediately take up its equilibrium partition of 

 energy appropriate to temperature T, namely, that given by 

 the formula 



£7rRT*,- 4 d\ (38) 



There is a limit to the applicability of this argument. The 

 analysis of the electron-motion into regular trains of waves 

 holds only for wave-lengths great compared with the dis- 

 tances apart of the nearest electrons. Thus formula (38) 

 will hold only for values of A- which are great compared with 

 a, where a is given by equation (36). This is exactly what 

 is given by Planck's formula, if the h of his formula is- 

 identified with R. 



Princeton, Oct. 8, 1908. 



* Phil. Mag. xvi. p. 353. 



