of= <i\lr - F. (76) 



Now fioiii Ihe periodic motion of the electrical centre we may conclude thai, in llie 

 presence of the field, the system will be able to emit or absorb a radiation of 

 frequency op, and that accordingly the possible values of the additional energy of 

 the system in the presence of the field will be given directly by Planck's funda- 

 mental formula (9), holding for the possible values of the total energy of a linear 

 harmonic vibrator, if in this formula cu is replaced by the above frequency O/-. Since 

 further a circular orbit, perpendicular to the direction of the electric force, will not 

 undergo secular perturbations during a slow establishment of the field, and there- 

 fore must be included among the stationary states of the perturbed system, we get 

 for the total energy of the atom in the presence of the field 



£ = £;„ + u üf/i = r^. r o ■-. 1,7 F, (77) 



' n^ h' ' 8 Ti' Nem ' ^ ' 



where n is an entire number which in the present case may be taken positive as 

 well as negative. From a comparison between (75) and (77), we see that the presence 

 of the external field imposes the restriction on the motion of the atom in the sta- 

 tionarj' states, that the plane in which the electrical centre of the orbit moves 



must have a distance from the nucleus equal to an entire multiple of the n'^" part 



3 

 of its maximum distance ^ an- 



The result, contained in formula (77), is in agreement with the expression for 



the total energy in the stationary states, deduced by Epstein and Schwarzschild 



by means of the general theory of conditionally periodic systems based on the 



conditions (22). The treatment of these authors rests upon the fact, that, as mentioned 



in Part I, the equations of motion for the electron in the present problem may be 



solved by means of separation of variables in parabolic coordinates (compare page 21). 



Taking for q^ and q., the parameters of the two paraboloids of revolution, which 



pass through the instantaneous position of the electron and which have their foci 



at the nucleus and their axes parallel to the direction of the field, and for q., the 



angular distance between the plane through the electron and the axis of the system 



and a fixed plane through this axis, the momenta p^, p^, p.^ will during the motion 



depend on the corresponding q's only, and the stationary states will be fixed by 



three conditions of the type (22). With neglect of small quantities proportional to 



higher powers of F, the final formula for the total energy, obtained by Epstein in 



this way, is given by 



p_ 2it'-N'e^m 3 /i^ (n^ + n^ -f n^) { n ^ — n .^) 



h^(n, + n,-^n,y 87z"Nem ^ ^' > ^°' 



') P. Epstein, Ann. d. Phys. L, p. 50S (1916) 



