328 



44 



radialion emitted by tlie atom, we may according to Bohr draw the conclusion 

 that also the intensities and polarisations of the spectral lines emitted 

 in the region of large n's will asymptotically be the same as the 

 intensities and polarisations of the corresponding lines which on 

 ordinary electrodynamics would be emitted by the atom. This hypo- 

 thesis is in agreement with the fact, that in the limit of large wave lengths Planck's 

 formula for the intensity distribution in temperature radiation coincides with the 

 formula of Rayleigh and Jeans, which is deduced on the basis of ordinary 

 electrodynamics. Now the radiation energy emitted in unit time by an electron 

 performing in a certain direction a linear harmonic motion which may^be repre- 

 sented by x = C cos 2 71 tot, where C is the amplitude and w the frequency of the 

 vibration, would, according to the laws of electrodynamics, be proportional to the 

 mean -value of the square of the acceleration of the electron and would therefore 

 be given by gC^oj'^, where g» is a universal constant with the value of which we 

 are not concerned here. From the above we may therefore conclude that, for a 

 conditionally periodic system consisting of a single electron moving in a fixed field 

 of force and the stationary states of which are determined by (99), the a-priori 

 probability of spontaneous transition between an initial state characterised by the 

 large integers n\, n'^, n'^ and a final slate characterised by = n[ — r^, = n'^ — r„, 

 = n'^ — where Tj, Tj, -3 are a set of positive or negative integers which are 



small compared to n[, n'^, n',, will be asymptotically given by g Cuo^lhæ = 



where co = rjWi + r^co^ j- r^cu^ represents the frequency of the emitted radiation and 

 C the amplitude of the harmonic vibration of this frequency occurring in the 

 motion of the electron in the initial stale or in the final stale. For simplicity it has 

 in this consideration been assumed that the vibration of frequency r^oi]^ -]- -^(o.^ + tgWg 

 is linear, parallel to a given direction, and we may therefore further conclude that 

 the radiation emitted during the transition in question is linearly polarised in this 

 direction. In the cases where, on ordinary electrodynamics, the radiation of frequency 

 + T2W2 + "s'^^s 11^^ slates under consideration would be circular or elliptical 

 we shall naturally conclude, that the probability of transition can be calculated in a 

 corresponding way, and that the radiation emitted during a transition will be 

 circularly, resp, elliplically polarised, the directions in space characterising these 

 polarisations being the same as those characterising the corresponding harmonic 

 vibrations in the motion of the system. 



Returning now to the region of stationary states where the n's in (99) are 

 small numbers, we may assume, according to Bohr, that there will still exist an 

 intimate connection between the coefficients C appearing in the trigono- 

 metric series of the type (12) by which the motion of the system may 

 be represented and the a-priori probabilities for transitions between 

 these states. Thus, if for the displacements of the particles in all directions in 

 space the coefficient Cr?, -j, corresponding to the frequency r^' -| ... TsOJq. is 



