638 Prof. 0. W. Richardson on Distribution of the 



natural, following Debye's ideas, to regard these deflexions 

 as arising from the effect of the elastic vibrations of the 

 substance. In that case, in order that the conductivity 

 should become infinite at T = 0, it is necessary that Planck's 

 zero-point energy should not contribute to the elastic 

 vibrations or, at any rate, should not affect the mean 

 free path of the electrons. It is not easy to see why 

 this part of the energy should be exceptional in this 

 respect ; but Wien * has worked out in detail a theory of 

 metallic conduction along lines similar to those just indi- 

 cated, and the results of this theory agree quantitatively 

 with the facts down to very low temperatures. 



There is a point in connexion with equation (16) which 

 may be referred to. At moderate temperatures this equation 

 reduces to 



__ w 2 — a>i — const . 



n 2 = const. xT^xe " ~St 

 very nearly. To be in formal agreement with equation (1), 

 we should have to have n x cc TX approximately. This may 

 be regarded as affording an explanation of the incompre- 

 hensible result f, that the concentration of the free electrons 

 in metals is very nearly proportional to T*, which is obtained 

 when the classical theory is applied to the specific heat of 

 electricity. According to the standpoint of the present 

 theory, for which no appeal is made except in regard to its 

 capacity for explaining the facts, the application of the 

 classical mechanics to both these cases is invalid and leads 

 to exactly the same inconsistency. 



It is well established by experiment that the distribution 

 of kinetic energy among the electrons emitted by hot 

 bodies is in accordance with Maxwell's laws. This result 

 is required by the theory now under consideration. For 

 in any actual case the equilibrium-concentration of the 

 external electrons is so small and Y so large, that the results 

 of the classical theory will always be valid so far as the 

 external electrons are concerned. But from this fact one 

 cannot establish the conclusion, which was drawn by the 

 writer and F. C. Brown % on the basis of the classical 

 dynamics, that the distribution of kinetic energy among the 

 electrons inside the conductor also follows Maxwell's law. 

 The density of the internal free electrons is of an entirely 

 different order of magnitude, and their behaviour will be 

 governed by the considerations of the quantum theory. 



* Columbia University Lectures, p. 29 (New York, 1913). 

 t O. W. Richardson, Phil. Mag. vol. xxiii. p. 276 (1912). 

 % Phil. Mag-, vol. xvi. p. 376 (1908). 



