94 



the effects of the two fields on the motion of the electron are of the same order 

 of magnitude we may in this case expect that the hydrogen lines will not be 

 resolved into sharp components but will become diffuse. From the considera- 

 tions on page 60 of the effect on the spectrum of a perturbed periodic system due 

 to a second external field, the perturbing effect of which is small compared with 

 that of the first, we may conclude, however, that, if the effect of one of the fields on 

 the motion of the electron is large compared with that of the other, the hj'drogen 

 lines will still show a resolution in a number of components, the spectral widths 

 of which are small compared with the displacements which they have undergone due to 

 the presence of the weaker of the external fields. In the discussion of this problem 

 ■we shall for simplicity neglect the influence of the relativity modifications, assuming 

 that the effect on the spectrum produced by each external field separately is large 

 compared with the inherent fine structure of the hydrogen lines. Denoting, as in 

 § 2, by /i a small constant of the same order as the ratio between the forces on 

 the electron due to the weaker of the external fields and those due to the stronger 

 of these fields, and by ?. a small constant of the same order as the ratio between 

 the latter forces and the attraction from the nucleus, we have, as shown on page 

 61, that, wilh neglect of small quantities of the same order of magnitude as 'åijl^,^) 

 the change in the additional energy of the atom due to the presence of the weaker 

 field is, in general, directly obtained bj' taking the mean value of the function ¥, 

 corresponding to the weaker field, over the cycle of shapes and positions which 

 the orbit of the electron passes through in the stationary states of the atom in the 

 presence of the stronger field only. In the special case under consideration, how- 

 ever, the perturbed system, formed by the atom in the presence of the stronger field 

 only, is degenerate, the secular perturbations of the orbit of the electron being of a 

 simple periodic character. The mean value in question will therefore not be completely 

 determined, but will be different for the different periodic cycles of shapes and 

 positions of the orbit, which represent the continuous multitude of stationary motions 

 which the electron may perform in each of the stationary states of the atom in 

 the presence of the stronger field only. In order to fix the stationary states in the 

 presence of both fields and the change in the additional energy of the atom due to 

 the presence of the weaker field, it will thus, as mentioned on page 62, be neces- 

 sary to examine the relation between the mean value in question and the frequency 

 of the slow periodic "secular" variations which the cycles under consideration will 

 undergo under the influence of the weaker of the external fields. Now, in the 

 special case under consideration this problem may be treated very simply, if we 

 imagine the weaker field as composed of two homogeneous fields of which the one 

 is parallel and the other perpendicular to the stronger field, and if we consider 

 separately the secular effect due to each of these fields. In fact, due to the symmetry 



^) Rigorously this result holds with neglect of small quantities of the same order of magnitude 

 as the largest of the quantities /i° and ),;jr, but for the sake of simplicity it is here and in the following 

 assumed that ß is not smaller than \X (compare page 61). 



