84 



turbations are simply periodic. From the point of view of separation of variables, 

 this degenerate character of the system is in the present case, in contrast to the 

 analogous case of the Stark effect, also directly revealed by the fact, that a separa- 

 tion can be obtained, not only in polar coordinates, but in any set of axial ellip- 

 tical coordinates for which one focus is placed at the nucleus and the other at 

 some point on the axis of the field. Just as in the case of the Stark effect, how- 

 ever, the system is no more degenerate as soon as the relativity modifications are 

 taken into account, in which case a separation of variables will still be possible 

 but only in polar coordinates. To this point we shall come back below. 



The observations on the Zeeman effect of the hydrogen lines show that, if the 

 fine structure is neglected, each line is in the presence of a magnetic field split up 

 in a normal Lorentz triplet; i. e. each line is resolved in three components of 

 which the one is undisplaced and polarised parallel to the direction of the field, 

 while the two other components possess frequencies, which differ from that of the 

 original line by Oh, and are circularly polarised in opposite directions in a plane 

 perpendicular to the direction of the field. As pointed out by Sommerfeld and by 

 Debye, the frequencies of a Lorentz triplet are included among the frequencies of the 

 components deduced from (81) by application of relation (1). In addition to the 

 observed components, however, we might from (81) and (1) expect the appearance 

 of a number of components, displaced from the original positions of the lines by 

 higher multipla of o^- For the non-appearance of these components the theories of 

 Sommerfeld and Debye offered no explanation, no more than for the polarisation of 

 the components observed; except that Sommerfeld in this connection draws atten- 

 tion to the føct, that the law governing the observed polarisations exhibits a certain 

 analogy to the emperical rule of Epstein concerning the observed polarisations of 

 the components ot the Stark effect of the hydrogen lines (see page 76). On the other 

 hand, just as in case of the latter effect, an explanation of the number of the com- 

 ponents observed and their characteristic polarisations is directly obtained on the 

 basis of the general formal relation between the quantum theory of line spectra and 

 the ordinary theory of radiation. In the first place we have at once from Larmor's 

 theorem, denoting the frequency of revolution of the electron in a stationary state of 

 the undisturbed hydrogen atom by co, that the motion of the electron, in a corres- 

 ponding stationary state of the atom in the presence of the field, may be resolved 

 in a number of linear harmonic vibrations parallel to the direction of the magnetic 

 force with frequencies tw, where r is a positive integer, and in a number of circular 

 harmonic rotations perpendicular to this direction with frequencies t(o + Dh or zw — üh, 

 according as the direction of rotation is the same as or the opposite of that of the 

 superposed rotation. Next, with neglect of small quantities proportional to H^, we 

 have for the difference in the total energy between two neighbouring states of the 

 perturbed system under consideration 



âE = âEf, + ,î® = wâl + OflrîS, (82) 



