27 



Oj and n^- On the other hand, this probability may be deduced by regarding the 

 motion of the system under consideration as the degeneration of a motion charac- 

 terised by three numbers n^, n^ and n^, as in the general applications of the condi- 

 tions (22) to a system of three degrees of freedom. Such a motion may be obtained 

 for instance by imagining the system placed in a small homogeneous magnetic 

 field. In certain respects this case falls outside the general theory of conditionally 

 periodic systems discussed in this section, but, as we shall see in Part II, it can 

 be simply shown that the presence of the magnetic field imposes the further condi- 

 tion on the motion in the stationary states that the angular momentum round the 



axis of the field is equal to n' ^, where n' is a positive entire number equal to or 



less than Hj, and which for the system considered in the spectral problems must 

 be assumed to be different from zero. When regard is taken to the two opposite 

 directions in which the particle may rotate round the axis of the field, we see 

 therefore that for this system a state corresponding to a given combination of 

 rij^ and n^ in the presence of the field can be established in 2n.2 different ways. 

 The a-priori probability of the different states of the system may consequently for 

 all combinations of Hj and n.^ be assumed to be proportional to n„. 



The assumption just mentioned that the angular momentum round the axis 

 of the field cannot be equal to zero is deduced from considerations of systems for 

 which the motion corresponding to special combinations of the n's in (22) would 

 become physically impossible due to some singularity in its character. In such cases 

 we must assume that no stationary states exist corresponding to the combinations 

 (Hj, Hj, ... Hs) under consideration, and on the above principle of the invariance 

 of the a-priori probability for continuous transformations we shall accordingly 

 expect that the a-priori probability of any other state, which can be transformed 

 continuously into one of these states without passing through cases of degenera- 

 tion, will also be equal to zero. 



Let us now proceed to consider the spectrum ofa conditionally periodic 

 system, calculated from the values of the energy in the stationary states by means of 

 relation (1). If E(n^, ... Hj) is the total energy of a stationary state determined by 

 (22) and if u is the frequency of the line corresponding to the transition between 

 two stationary states characterised by n^. ^ n'j. and n^, = n'^ respectively, we have 



V = A. [£(„; , . . . n;) - £ (n';, . . . n;')]. (26) 



In general, this spectrum will be entirely different from the spectrum to be ex- 

 pected on the ordinary theory of electrodj'namics from the motion of the system. 

 Just as for a system of one degree of freedom we shall see, however, that in 

 the limit where the motions in neighbouring stationary states differ very little from 

 each other, there exists a close relation between the spectrum calculated on the 

 quantum theory and that to be expected on ordinary electrodynamics. As in § 2 



