﻿Theory of Long Wave Radiation. 171 



practice this assumption is justified only to a comparatively 

 rough extent and only when all the other times involved in 

 the analysis are large compared with the usual intra- 

 molecular periods, so that the results obtained can only be 

 applicable in the extreme ultra-red region of the spectrum, 

 where it is o£ course known to apply. As soon as the en- 

 counters between the electrons and atoms are sufficiently 

 long compared with the period of the light discussed, the 

 effect of the collision will make itself felt in modifying the 

 radiation formula, a conclusion drawn some time previously 

 by J. J. Thomson. 



It is just the assumption concerning the shortness of the 

 collisions which is the predominating factor in restricting 

 the general application of the Eayleigh-Jeans formula. 



The fact that the general radiation formula which is to 

 be applicable all along the spectrum must ultimately contain 

 some general account of the actual collisions of the electrons 

 with the atoms can be illustrated in various ways. Let us 

 confine our attention to one of the free electrons in the 

 metal considered in the above analysis ; the average flux of 

 energy through iv' arising as a result of its motion is 



tW0J o V dt ) ~ 8ttW0 J dt lxj ' 



8tt 



Now if the collisions are all of short duration and A[t'] 

 denotes the total change iu \_v~\ during a typical one of these 

 collisions of total duration At, then the energy radiated 

 through w' during this collision is of total amount 



eho' {A[v x \y 



8irW0 At ' 



which is inversely as At. Thus if, as in the above, we 

 assume the duration of all the collisions to be infinitely short, 

 the total amount of energy radiated away will be infinitely 

 large. This merely means of course that as soon as the time 

 of a collision becomes appreciable a closer investigation 

 will be necessary, involving necessarily some account of the 

 nature of the collision. 



It is now no longer surprising that a divergent series is 

 obtained in the expression for the total energy : in fact the 

 shorter the collisions the farther up the spectrum does the 

 agreement between theory and practice hold, but then the 

 bigger is the total energy.' 



These results are still further illustrated by the result 

 of a. general theory developed by Thomson * on molecular 



* Phil. Mag. [5] xiv. p. 217 (1907). 



