Theory of Radiation. 47 



involved in dN are extended over the range of the variables 

 «i, a 2 , &c. in the undisturbed motion. 



and &iv, &c. are functions o£ (a']) , &c. and of the time 

 which appears explicitly in the expressions of the disturbing- 

 forces given in (2)v 



o\%, &c 2 j &c. can therefore also be expressed as functions 

 of #!, w 2 ... a: n , &c. and £. It is evident that 



j80dN = 8j£rfN-j>8(rfN). . . . (10) 



(10) is true whether or no the motion is continuous. 



The disturbing radiation may precipitate a transformation 

 of the variables. Then &(<2N) involves a finite change in 

 #i, # 2 , &c, but only an infinitesimal change in 6?N itself. 



In the ordinary mechanical systems 



SJ^N=0 (11) 



The variables are here coordinates of position and momenta. 

 The presence of radiation does not affect the range of values 

 they can assume. Hence (11), since <j> is a definite function 

 of the variables and the limits of integration are not dis- 

 turbed. I shall throughout use the result (11). It is thus 

 assumed that if there are discontinuous changes they are 

 governed by conditions not directly dependent on the 

 presence of radiation. For example, they may take place 

 when the velocities or coordinates reach certain given values. 

 (10) now becomes 



JS^N=-j^N, (12) 



and this result is in the present state of our knowledge no 

 less general than (10) itself. For d$ still contains the 

 arbitrary function p, which reduces to a constant only in 

 the classical dynamics. 



An important consequence follows at once from (12). 

 The average value of %<f> is zero if §(rfN) =0. 



If the disturbing forces leave unaltered the canonical dis- 

 tribution in phase, to use Willard Gibbs's term, they produce 

 no observable effects at all. It is so with magnetic force in 

 the ordinary theory. Niels Bohr has pointed out * that 

 S(dN) = when the only disturbing force is due to a steady 



* Studier over Metallemes Elektrontheori, p. 106. 



