Deriving Planck's Law of Radiation. 231 



Hence for the mode whose limiting energy is 7i/we have 



N 



kO\l-e *) 



(3) 



the N molecules per unit volume being all in energy range 

 (0, hf\ 



From (3) and (1), for the total energy of the N molecules 

 in unit volume in mode/ we easily obtain the value 



*«-f^& w 



[l-e' ke ) 



But the "law of equipartition of energy" (assumed applic- 

 able to the combined system of matter and aether in unit 

 volume, the free modes of each being identical, — and not to 

 matter or aether separately) requires that the total energy 

 per unit volume in each mode be independent of/. Accord- 

 ingly the radiant energy contained in unit volume of aether 

 must be 



(5) 



[1-e ke ) 



No additional terms involving 6 alone can enter, in accord- 

 ance with Wien's law. The same point is clear from the 

 consideration that the total kinetic energy in an}^ mode must 

 equal one-third of the kinetic energy of the translational 

 motion of the centroids of the N molecules,} (N£#/2), as stated 

 in Lord Rayleigh's discussion. Hence the mean energy per 

 molecule contained in mode /in unit volume of aether is 



¥ /-> 



— W ^> (6) 



(e*°-l) 



and the number of modes of vibration in the frequency 

 range df, as given by Lord Rayleigh's paper referred to 

 above with the correction indicated by Professor Jeans, is 

 ■STrf 2 df/Y 3 , where V is the velocity of light in aether. We 

 have therefore as the total energy in aether in the frequency 

 range df 



^/^/• r /-x ■• ( 7 ) 



[e>*-l) 



