32 



we must therefore conclude that, in the Hmit of large values for the n's, the pro- 

 bability of spontaneous transition between two stationary states of a conditionally 

 periodic system, as well as the polarisation of the accompanying radiation, can be 

 determined directly from the values of the coefficient C-....7^ in (31) corresponding 

 to a set of r's given by -^. = n'j. — n'^, if n[, ■ ■ ■ n'^ and n", . . . n'^ are the numbers which 

 characterise the two stationary states. 



Without a detailed theory of the mechanism of transition between the stationary 

 states we cannot, of course, in general obtain an exact determination of the pro- 

 bability of spontaneous transition between two such states, unless the n's are 

 large numbers. Just as in the case of systems of one degree of freedom, however, 

 we are naturalh' led from the above considerations to assume that, also for 

 values of the n's which are not large, there must exist an intimate connection 

 between the probability of a given transition and the values of the corresponding 

 Fourier coefficient in the expressions for the displacements of the particles in the 

 two stationary states. This allows us at once to draw certain important conclusions. 

 Thus, from the fact that in general negative as well as positive values for the 

 r's appear in (31), it follows that we must expect that in general not only such 

 transitions will be possible in which all the n's decrease, but that also transitions 

 will be possible for which some of the n's increase while others decrease. This 

 conclusion, which is supported by observations on the fine structure of the hydrogen 

 lines as well as on the Stark effect, is contrary to the suggestion, put forward 

 by Sommerfeld with reference to the essential positive character of the I's, that 

 every of the n's must remain constant or decrease under a transition. Another 

 direct consequence of the above considerations is obtained if we consider a system 

 for which, for all values of the constants «j, ... a,, the coefficient Cr,....-^ corres- 

 ponding to a certain set rj, . . . zs of values for the r's is equal to zero in the ex- 

 pressions for the displacements of the particles in everj' direction. In this case we 

 shall naturally expect that no transition will be possible for which the relation 

 n'l. — n'^ = tI is satisfied for every k. In the case where Ct^, • • • t° is equal to zero 

 in the expressions for the displacement in a certaiç direction only, we shall expect 

 that all transitions, for which n'j. — 7!^^ = ~^. for every A-, will be accompanied by 

 a radiation which is polarised in a plane perpendicular to this direction. 



A simple illustration of the last considerations is afforded by the system men- 

 tioned in the beginning of this section, and which consists of a particle executing 

 motions in three perpendicular directions which are independent of each other. In 

 that case all the Fourier coefficients in the expressions for the displacements 

 in any direction will disappear if more than one of the r's are different from 

 zero. Consequently we must assume that only such transitions are possible for 

 which only one of the n's varies at the same time, and that the radiation corres- 

 ponding to such a transition will be linearly polarised in the direction of the dis- 

 placement of the corresponding coordinate. In the special case where the motions 

 in the three directions are simply harmonic, we shall moreover conclude that none 



