63 



effect on the spectrum of the perturbed system produced by an arbitrary second 

 external field, which is small compared with the first, will consist in the splitting 

 up of every component into a number of separate components, just as the effect of 

 an arbitrary small external field on the lines of the spectrum of a simple periodic 

 system of two degrees of freedom. We will meet with applications of the above 

 considerations when considering the elïect on the hydrogen spectrum of the com- 

 bined action of different external fields and when considering the effect of an ex- 

 ternal field on the spectra of other elements, which latter problem will be dis- 

 cussed in Part III. 



§ 3. The fine structure of the hydrogen lines. 



An instructive application of the calculations in the last section may be made 

 in connection with the fine structure of the 'hydrogen lines, which, according to 

 Sommerfeld's theory mentioned in Part I on page 18, may be explained by taking into 

 account the small variation of the mass of the electron with its velocity, claimed by 

 the theory of relativity. In this connection it must first of all be remarked that all 

 the general considerations in the preceding sections, as regards relations between 

 energy and frequency and as regards the mechanical transformability of the sta- 

 tionary states, hold unaltered if the relativity modifications are taken into account. 

 This follows from the fact that the Hamiltonian equations (4), which are taken 

 as a basis for all the previous calculations, may be used to describe the motion 

 also in this case. If, when the relativity modifications are taken into account, the 

 motion of the system is simply periodic independent of the initial conditions, we 

 shall consequently expect that the stationary states are characterised by the condi- 

 tion / = iih only, and that the energy and frequency are the same for all states 

 corresponding to a given value of n in this equation. Further the stationary states 

 will also in the relativity case be fixed by (22), if the system is conditionally 

 periodic and allows of separation of variables; while the stationary states of a 

 perturbed periodic system, also in the relativity case, will be characterised by the 

 conditions (67j, if the secular perturbations are of conditionally periodic type. 



Now, when the relativity modifications are taken into account, the motion of 

 the particles in the hydrogen atom will not, as assumed in § 1, be exactly periodic, but 

 the orbit of the electron will be of the same type as that, which would appear on 

 ordinary Newtonian mechanics, if the law of attraction between the particles differed 

 slightly from that of the inverse square. If, for the moment, we consider the mass 

 of the nucleus as infinite, the system will allow of a separation of variables in polar 

 coordinates, and the stationary states may consequently be fixed by the conditions 

 (16). In this way Sommerfeld obtained an expression for the total energy in the 

 stationary states, which, with neglect of small quantities of higher order than the 



y 



