78 



motion ol' the electron can be resolved. The examination of this problem has been 

 undertaken by Mr. H. A. Kramers, who has deduced complete expressions for these 

 amplitudes, by means of which it was found possible, for each of the hydrogen 

 lines Ha, Hß, Hy and Hå, to account in a convincing way for the apparently capricious 

 laws which govern the intensities of the components observed by Stark. ^) This agree- 

 ment offered at the same time a direct experimental support for the conclusions 

 mentioned above: that there exist no stationary states corresponding to n„ = 0, while 

 the stationary states corresponding to other values of n., are a-priori equally pro- 

 bable; and that transitions can only take place between pairs of stationary states 

 for which n„ is the same or differs by one unit. A general discussion of these pro- 

 blems will be given by Kramers in the paper, mentioned on page 69 in the last 

 section, in which also the problem of the intensity of the fine structure components 

 is treated in detail. 



In the former section and in the present we have seen, how the problems of 

 the influence of the relativity modifications on the lines of the hydrogen spectrum 

 and of the influence of an external electric field on this spectrum can be treated, 

 by regarding the motion of the electron as a perturbed periodic motion, and by 

 fixing the stationary states on the basis of the relation between the energy and the 

 frequencies of the secular perturbations. As it was done originally by Sommerfeld 

 and Epstein, both these problems can also be treated by means of the theory of 

 the stationary states of conditionally periodic systems which allow of separation 

 of variables in a fixed set of positional coordinates. If, however, we consider the 

 problem of the simultaneous influence on the hydrogen spectrum of 

 the relativity modifications and a homogeneous electric field of any 

 given intensity, there does not exist a set of coordinates for which a separation of 

 variables can be obtained. On the other hand it is possible, also in this case, to 

 apply the general considerations about perturbed periodic systems developed in the 

 preceding. In fact, with reference to the treatment given in § 3 of the problem of 

 the fine structure of the hydrogen lines, it will be seen that the deviations of the 

 orbit of the electron from a Keplerian ellipse in the problem under consideration 



>) Note added during the proof. In recent papers H. Nyquist (Pliys. Rev. X, p. 226 (1917)) 

 and J. Stark (Ann. d. Physik, LVI, p. 569 (191S)) have published measurements on the elïect of an 

 electric field on certain lines of tlie helium spectrum whicli is given by (35), if in (40) we put IV ^ 2. 

 As will be seen from v78), tlie differences between the frequencies of tlie components into which these 

 lines are split up will, for the same Intensity of the external electric field, be smaller than for the 

 hydrogen lines. Jn conformity with this it was not possible, with the experimental arrangement used 

 by the authors mentioned, to observe separately the numerous components to be expected on the 

 theory, but only to obtain certain rough features of the resolution of the lines in question. For the 

 interpretation of these observations a detailed consideration of the relative intensities to be expected 

 for the different theoretical components is therefore essential; and, as it will be shown in Kramers' 

 paper, it is possible, on tlie basis of the calculation of the amplitudes of the harmonic vibrations into 

 which the motion of the electron in the stationary states can be resolved, to account satisfactorilj' for 

 Nyqoist's and Stark's results. 



