74 



where n^. n.,, n, are the positive entire numbers which occur as factors to Planck's 

 constant on the right sides of the mentioned three conditions. 



As regards the possible values of the total energy of the hydrogen atom in 

 the presence of the electric field, it will be seen that (78i coincides with 1 77) if we 

 put n, — n., — /J3 = n and ;?., — n^ = n. At the same time it will be observed, 

 however, that the motion in the stationary states, as fixed by the procedure followed 

 by Epstein, is more restricted than was necessary in order to secure the right 

 relation between the additional energy and the frequency of the secular perturba- 

 tions. Thus, in addition to the condition which fixes the plane in which the elec- 

 trical centre moves. Epstein's theory involves the further condition, that the angular 

 momentum of the electron round the axis of the perturbed system is equal to an 

 entire multiple of h 2-: which multiple is seen to be even or uneven, according as 

 /! — n is an even or an uneven number respectively. This circumstance is intimately 

 connected with the fact that, although the perturbed system under consideration is 

 degenerate if we look apart from small quantities proportional to the square of the 

 intensity of the external force, the degenerate character of the system does not reveal 

 itself from the point of view of the theory of stationary states based on the con- 

 ditions (22), because the system under consideration allows of separation of vari- 

 ables only in one set of positional coordinates. On the other hand, this degenerate 

 character of the system has been emphasised by Schwarzschild ^) on the basis 

 of the theory of stationary states based on the introduction of angle variables, in 

 which the periodicity properties of the motion play an essential part. In a later 

 discussion of this point Epstein -j calls attention to the fact that, if small quantities 

 proportional to the square of the electric force are taken into account, the system 

 appears no more as degenerate; and he finds therein a justification of the fixation of 

 the stationary states by means of i22i. From the point of view of perturbed systems, 

 this would mean that the motion in the stationary states of the system in question, 

 as fixed by |22), would certainly be stable for infinitely small disturbances, but that 

 we should expect finite deviations from the motion in these states, already if the 

 system was exposed to a second perturbing field, the intensité' of which was only 

 of the same order as the product of the external electric force with the ratio between 

 this force and the attraction from the nucleus. A closer consideration, however, in 

 which regard is taken to the influence of the relativity modifications, learns that 

 the degree of stability of the motion in the stationary states, as determined by (22), 

 actually is often much higher, the order of magnitude of the external force, necessary 

 to cause finite deviations from this motion, being of the same order as the product 

 of the attraction from the nucleus with the square of the ratio of the velocity of 

 the electron and the velocity of light. To this point we shall come back at the end 

 of this section, when considering the simultaneous perturbing influence on the 



^) K. Schwarzschild. Ber. Âkad. Berlin. 1916. p. 548. 

 =! P. Epstein. Ann. d. Phys. LI. p. 168 (1916). 



