40 



we should exjject on the considerations in Pari I that no transition between two 

 stationary states of the atom would be possible for which n' and ii" ditrer by more 

 than one unit; but this would obviously be inconsistent with the observations, 

 since for instance the lines of the ordinary Balmer series, according to the theory, 

 correspond to transitions for which n" = 2 while n' takes the values 3, 4, 5, . . . 

 In connection with this consideration it may be remarked that, adopting a termino- 

 logy well known from acoustics, we may from the point of view of the quantum 

 theory regard the higher members of the Balmer series (n' = 4,5,...) as the 

 "harmonics" of the first member [n' = 3), although of course the frequencies of 

 the former lines are by no means entire multipla of the frequency of the latter line. 

 While in the above way it was possible to obtain a simple interpretation of 

 certain main features of the hydrogen spectrum, it was not found possible in this way 

 to account in detail for such phenomena in which the deviation of the motion of 

 the particles from a simple Keplerian motion plays an essential part. This is the 

 case in the problem of the fine structure of the hydrogen lines, which is due to 

 the effect of the small variation of the mass of the electron with its velocity, as 

 well as in the problems of the characteristic etTects of external electric and magnetic 

 fields on the hydrogen lines. As mentioned in the introduction, a progress, of 

 fundamental importance in the treatment of such problems was made by Sommer- 

 feld, who obtained a convincing explanation of the fine structure of the hydrogen 

 lines by means of his theory of the stationary stales of central systems, in which 

 the single condition I = nh was replaced by the two conditions (16); and the 

 theory was further developed by Epstein and Schwarzschild, who on this line 

 established the general theory, based on the conditions (22), of the stationary states 

 of a conditionally periodic system for which the equations of motion may be 

 solved by means of separation of variables in the Hamilton-Jacobi partial differential 

 equation. If the hydrogen atom is exposed to a homogeneous electric or to a 

 homogeneous magnetic field, the atom forms a system of this class, and, as shown 

 by Epstein and Schwarzschild as regards the Stark effect and by Sommerfeld and 

 Debye as regards the Zeeman effect, the theory under consideration leads to values 

 for the total energy of the atom in the stationary slates, which together with rela- 

 tion (1) lead again to values for the frequencies of the radiations emitted during 

 the transitions between these states, which are in agreement with the measured fre- 

 quencies of the components into which the hydrogen lines are split up in the pre- 

 sence of the fields. As pointed out in Part I, it is possible moreover lo throw light 

 on the question of the intensities and polarisations of these components on the 

 basis of the necessary formal relation between the quantum theory of line spectra 

 and the ordinary theory of radiation in the limit where the motions in successive 

 stationary states differ very little from each other. In the following sections the 

 mentioned problems will be discussed in detail. As regards the fixation of the 

 stationary states we shall not, however, follow the same procedure as used by the 

 authors just mentioned, which rests upon the immediate application of the conditions 



