53 



freedom undergoes in the presence of :i given exlernnl liekl, cimnol he expected 

 to be conditionally periodic and to exhibit periodicity properties of the type ex- 

 pressed by formula (54). In such cases it seems impossible to define stationary 

 states in a way which leads to a complete fixation of tlie total energy in these 

 states, and we are therefore led to the conclusion, that the etTect of the external 

 field on the spectrum will not consist in the splitting up of the spectral lines of the 

 original system into a number of sharp components, but in a diffusion of these 

 lines over spectral intervals of a width proportional to the intensity of the external 

 forces. 



In special cases in which the secular perturbations of a perturbed periodic 

 system of more than two degrees of freedom are of conditionallj' periodic type, it 

 may occur that these perturbations are characterised by a number of fundamental 

 frequencies, which is less than s — 1. In such cases, in which the perturbed periodic 

 system from analogy with the terminology used in Part I may be said to be 

 degenerate, the necessary relation between the additional energy and the frequencies 

 of the secular perturbations is secured by a number of conditions less than that 

 given by (55), and the stationary states are consequently characterised by a number 

 of conditions less than s. With a typical example of such systems we meet il", for 

 a perturbed periodic system of more than two degrees of freedom, the secular per- 

 turbations are simply periodic independent of the initial shape and position of 

 the orbit. In direct analogy to what holds for perturbed periodic systems of two 

 degrees of freedom, the difference between the values of f in two slightly dilferent 

 states of the perturbed system, corresponding to the same value of /, will in the 

 present case be given by 



â'f= 0^3, (56) 



where o is the frequency of the secular perturbations, and where 3 is defined by 





«A-^r'^'' ^^'^ 



where § = i/„ is the period of the perturbations. We may therefore conclude that 

 the stationary states of the perturbed system, corresponding to a given stationary 

 state of the undisturbed system, will be characterised by the single condition 



S = nh, (58) 



in which u is an entire number, and which will be seen to be completely analogous 

 to the condition which fixes the stationary states of ordinary' periodic systems of 

 several degrees of freedom. 



In the following sections we shall apply the preceding considerations to the 

 problem of the fixation of the stationary states of the hydrogen atom, 

 when the relativity modifications are taken into account, and when the atom is ex- 

 posed to small external fields. In this discussion we shall for the sake of simpli- 



D. K. D. Vidensk. Selsk. Ski-.,natui'Vidensk,oy*niatheni. Afd., 8. lljukUe, IV. 1. y 



