79 



will be Ihe same as the secular perturbations produced on a Kcplerinn iu(;li<jn by 

 the simullaneous iiiHuence oi' an external homogeneous field offeree and an external 

 central force proportional to the inverse cube of tlie distance from the nucleus. 

 Since these two fields together form a perturbing field possessing axial symmetry, 

 it follows therefore that the secular perturbations, when the relativity modications 

 are taken into account, will be conditionally periodic and that the problem of the 

 stationary states may be treated by means of the method mentioned in 5; 2 on 

 page 55. In this way we obtain in the first place the result, that, for any value of 

 the intensity of the external electric field, we must expect that the hydrogen lines 

 will be split up in a number of sharp components. Next, since for any value of 

 this intensity different from zero the system will be non-degenerate, it follows from 

 the conditions (61), that we must assume that the angular momentum round the 

 axis of the field is always equal to an entire multiple of h!2n; in consistence with 

 the assumption of the validity of the analogous condition involved in the fixation 

 of the stationary states by means of the method of separation of variables, when 

 applied to an explanation of the Stark effect with neglect of the relativity modi- 

 fications (compare page 74). On the basis of the conditions (61) it is possible to 

 predict in detail, how the fine structure of the hydrogen lines will be influenced 

 by an increasing electric field until, for a sufficiently large intensity of this field, 

 the phenomenon develops gradually into the ordinary Stark effect. The problem 

 of this transmutation will be treated in a later paper by Mr. H. A. Kramers'), who 

 has kindly drawn my attention to this interesting application of the method of 

 perturbations, and has thereby given a valuable impetus to the detailed elaboration 

 of this method as regard the treatment of more complicate problems. 



§ 5. The effect of a magnetic field on the hydrogen spectrum. 



A theory of the Zeeman effect of the hydrogen lines based on the quantum 

 theory of line spectra has, as mentioned in the introduction, been given indepen- 

 dently by Sommerfeld and by Debye. The calculations of these authors rest upon 

 the fact, that it is possible, also in the presence of a magnetic field, to write the 

 equations of motion of the electron in the canonical Hamiltonian form given by 

 (4), if the momenta Pi-, P-^, Ps-, which are conjugated to the positional coordinates of 

 the electron q^, q.^, q.■^, are defined in a suitable way. In complete analogy to the 

 problem of the fixation of the stationary states of an atomic system when the rela- 

 tivity modifications are taken into account, it follows therefore that, if these equations 



') Besides the discussion of tliis problem, the paper in question will contain a general treatment 

 of the theory of perturbed periodic systems from the point of view of the possibility of describing 

 the motion b}' means of angle variables (compare Note on page 58). 



11* 



