89 



conclusion mentioned above, that the value of the additional energy in the sta- 

 tionary states of the perturbed system is given by the value of '/' in these stales. 

 From the above considerations it follows that the problem of the stationary 

 states of the hydrogen atom in the presence of external electric and magnetic forces 

 may be treated in a manner, which is exactlj^ analogous to that applied in § 2 

 in case of a periodic system exposed to a small external field of constant potential. 

 Thus, if the secular perturbations as determined by (46) are of conditionally periodic 

 type, we shall expect that, with neglect of small quantities proportional to Å, the 

 cycles of shapes and positions which the orbit of the electron passes through 

 in the stationary states of the perturbed system will be characterised by the 

 conditions (55), and that the possible values of the additional energy of the 

 atom in the stationarj' states will be fixed by these conditions with neglect of small 

 quantities proportional to /.-. We shall therefore conclude that, also in the presence 

 of external magnetic forces, the lines of the hydrogen spectrum will, if only the secular 

 perturbations are of conditionally periodic type, be split up in a number of sharp 

 components, the frequencies of which are determined by means of the conditions (67) 

 together with relation (1). As regards the problem of the intensities and polarisation 

 of these components, we may further proceed in a way quite analogous to that fol- 

 lowed in § 2. In fact, if the secular perturbations are of conditionally periodic type, 

 the displacement of the electron in any given direction may be represented as a 

 sum of harmonic vibrations by an expression of the same type as (65). Moreover it 

 can be proved that the difference in the total energy of two neighbouring states 

 of the perturbed atom will again be given by the expression (66)'). The general 

 considerations in § 2 will therefore apply without alterations to the problem of 

 the intensity and polarisation of the components into which the hydrogen lines 

 are split up in the presence of small external forces, also if these forces are entirely 

 or partly of magnetic origin. Similarlj^, it will be seen that the effect on the spec- 

 trum of a perturbed hydrogen atom, which will be due to the presence of a second 

 external field small compared with the first, also in this case may be discussed 

 directly by means of the considerations at the end of § 2. 



We meet with a direct application of the preceding considerations, if the 

 hydrogen atom is exposed to the simultaneous influence of an external 

 electric and an external magnetic field, which possess axial sym- 

 metry round a common axis through the nucleus. Introducing the same 

 set of orbital constants as described in § 2 on page 54, we get in this case that 

 Wm, as well as 9e, and consequently the function T = ¥e + I'm which enters in the 

 equations (46 j, will, besides on «j, depend on «3, ßo and «„ but not on ß^. The general 

 character of the secular perturbations of the orbit of the electron will therefore be 

 the same as in the case, considered in § 2, where the atom is exposed only to an 



*) Compare Note on page 58. X]so in the presence of small magnetic forces, it will be pos- 

 sible to describe the motion of the perturbed system by means of a suitably chosen set of angle vari- 

 ables, if only the secular perturbations are of conditionally periodic type. 



