60 



the conditions (öl), tlie angular momentum of tlie system round the axis in the 



stationary states is equal to n, „ , it will be seen moreover that, also in the present 



case, these conclusions obtain an independent support from a consideration of con- 

 servation of angular momentum during the transitions (Compare Part I page 34) '). 

 In the following we will meet with applications of these considerations when dis- 

 cussing the effect of electric and magnetic fields on the hydrogen lines. In the latter 

 case, however, the preceding considerations need some modifications due to the 

 fact, that the external forces acting on the electron cannot be derived from a 

 potential expressed as a function of its positional coordinates; to this point we shall 

 come back in § 5. 



Before leaving the general theory of perturbed periodic systems we shall still 

 consider the problem of the ellect on the spectrum of a periodic system, under- 

 going secular perturbations of conditionally periodic type under the influence of a 

 given small external field, if this system is further subject to the influence of 

 a second external field which is small compared with the first field, 

 but the perturbing effect of which is yet large compared with the small effects 

 on the motion, proportional to the square of the intensity of the first perturb- 

 ing field, which were neglected in the preceding calculations. This problem is 

 closely analogous to the problem, briefly discussed in Part I, of the effect of a 

 small perturbing field on the spectrum of an ordinary conditionally periodic 

 system which allows of separation of variables. As mentioned on page 34, we have 

 in this case, quite independent of the possibility of separation of variables for the 

 perturbed system, that in general the motion under the influence of the external 

 field may still be represented as a sum of harmonic vibrations by a formula of 

 the type (31), if we look apart from small terms proportional to the square of the 

 perturbing forces. Corresponding to this we have in the case under consideration 

 that, independent of the nature of the second external field, the resultant secular 

 perturbations may in general be expressed as a sum of harmonic vibrations of small 

 frequencies of the type (54), if we look apart from small terms of the same order 

 as the product of the secular perturbations produced by the first external field with 

 the square of the ratio between the intensities of the forces due to the first and 

 those due to the second external field. Let us denote this ratio by ii. and let, as 

 above, k represent a small constant of the same order as the ratio between the ex- 



') Note added during the proof. In an interesting paper by A. Rubinowicz (Pliys. Zeitschr. 

 XIX, p. 441 and p. 465 (1918)) which has just been published, a similar consideration of conservation 

 of angular momentum lias been used to draw conclusions, as regards the possibility of transitions 

 between the stationary states of a conditionally periodic system possessing an axis of symmetry, and 

 as regards the cliaracter of the polarisation of the radiation accompanying these transitions. In this way 

 Rubinowicz has arrived at several of the results discussed in the present paper; in tliis connection, 

 however, it may be remarljed that, from a consideration of conservation of angular momentum, it is 

 not possible, even for systems possessing axial symmetry, to obtain as complete information, as regards 

 the number and polarisation of the possible components, as from a consideration based on the resolu- 

 tion of the motion of the electron in Iiarmonic vibrations. 



