15 



for the frequency u) in two states corresponding to clifTerent values of n in (10) in 

 general are different, we see at once that we cannot expect a simple connec- 

 tion between the frequency calculated by (1) of the radiation corresponding to a 

 transition between two stationary states and the motions of the system in these 

 states, except in the limit where n is very large, and where the ratio between the 

 frequencies of the motion in successive stationary states differs very little from unity. 

 Consider now a transition between the state corresponding to n = n' and the stale 

 corresponding to n ^ n" , and let us assume that n' and n" are large numbers and 

 that n' — n" is small compared with n' and n" . In that case we may in (8) for 

 oE put E' — E" and for 31 put /' — /", and we get therefore from (1) and (10) for 

 the frequency of the radiation emitted or absorbed during the transition between 

 the two states 



V = ^ (E' — E") = ^ (I' -I") = (n'-n")cu. (13) 



Now in a stationary state of a periodic system the displacement of the par- 

 ticles in any given direction may always be expressed by means of a Fourier- 

 series as a sum of harmonic vibrations: 



Ç = 2'CrCOs2n-(ra;f + Cr), (14) 



where the C's and c's are constants and the summation is to be extended over all 

 positive entire values of r. On the ordinary theory of radiation we should there- 

 fore expect the system to emit a spectrum consisting of a series of lines of fre- 

 quencies equal to toj, but, as it is seen, this is just equal to the series of frequencies 

 which we obtain from (13) by introducing different values for n' — n". As far as 

 the frequencies are concerned we see therefore that in the limit where n is large 

 there exists a close relation between the ordinary theory of radiation and the theory 

 of spectra based on (1) and (10). It may be noticed, however, that, while on the 

 first theory radiations of the different frequencies -o> corresponding to different 

 values of r are emitted or absorbed at the same time, these frequencies will on 

 the present theory, based on the fundamental assumption I and II, be connected 

 with entirely different processes of emission or absorption, corresponding to the 

 transition of the system from a given state to different neighbouring stationary states. 

 In order to obtain the necessary connection, mentioned in the former section, 

 to the ordinary theory of radiation in the limit of slow vibrations, we must further 

 claim that a relation, as that just proved for the frequencies, will, in the limit of 

 large n, hold also for the intensities of the different lines in the spectrum. Since 

 now on ordinary electrodynamics the intensities of the radiations corresponding 

 to different values of r are directlj' determined from the coefficients C^ in (14), we 

 must therefore expect that for large values of n these coefficients will on the quantum 

 theory determine the probability of spontan uous transition from a given 

 stationary state for which n = n' to a neighbouring state for which n = n" ^ n — - 

 Now, this connection between the amplitudes of the different harmonic vibi-atious into 



