71 



energy in the slalionary stales of the system in the presence of the held, with neglect 

 of small quantities proportional to F-, will be equal to £„ + '/'; where E„ is the energy 

 of the hydrogen atom in its undisturbed stationary state. Since both j and cos ^ arc 

 numerically smaller than one, we obtain therefore at once from (74) a lower and 

 an upper limit for the possible variations of the energy in the stationary states, due 

 to the field. Introducing from (41) the values of /i„ and an, and neglecting, here 

 as well as in the following calculations in this section, the small correction due 

 to the finite mass of the nucleus — not only in the expression for the additional 

 energy but, for the sake of brevity, also in the main term — we get for these limits 



E=-^-^I^^^'}y^F, (75) 



which formula coincides with the expression previously deduced by the writer by 

 applying the condition / = ji/i to the two (physically not realisable) limiting cases, 

 corresponding to £ = 1 and cos cp = + 1 , in which the orbit remains periodic in 

 the presence of the field. \) 



In order to obtain further information as to the values of the energy in the 

 stationary states in the presence of the field, it is necessary to consider more closely 

 the variation of the orbit during the perturbations. Since the external forces possess 

 axial symmetry, the problem of the stationary states might be treated by means of 

 the procedure indicated in § 2 on page 55. In the present special case, however, 

 the stationary states of the atom may be very simply determined, due to the fact 

 that the secular perturbations are simply periodic independent of the initial shape 

 and position of the orbit, so that we are concerned with a degenerate case of a 

 perturbed periodic system. This property of the perturbations follows already from 

 some calculations given by Schwarzschild'-') in a previous attempt to explain the 

 Stark effect of the hj'drogen lines, without the help of the quantum theory, by 

 means of a direct consideration of the harmonic vibrations into wlrich the motion 

 may be resolved, according to the analytical theory of conditionally periodic 

 systems. Starting from the above result, that the electrical centre moves in a plane 

 perpendicular to the direction of the external field, the periodicity of the perturba- 

 tions may also be proved in the following way, by means of a simple consideration 

 of the variation of the angular momentum of the electron round the nucleus, due 

 to the effect of the external electric force. 



Using again rectangular coordinates with the nucleus at the origin and the :-axis parallel 

 to the direction of the electric force, and calling the coordinates of the electrical centre f, ;;. 

 f, we have according to formula (74) 



>) See N. BoHK, Pliil. Mag. XXVII, p. 506 (19U) and XXX, p. 39-1 i,191ö\ Compare also E. Warburg, 

 Verli. d. D. Pliys. Ges. XV, p. 1259 (1913), where it was pointed out, for the first time, that the effect of 

 an electric field on the hydrogen lines to be expected on the quantum theory was of the same order 

 of magnitude as the effect observed by Stark. 



=) K. Schwarzschild, Verb. d. D. Phys. Ges. XVI, p. 2U U'Jli)- 



10* 



