2h 



309 



lively, while the frequency of revolution is proportional to the intensity of the field 

 and equal to 0/ . The variation of the plane of the orbit during this motion of the 

 electrical centre may be found by observing that the angular momentum of the 

 electron round the z-axis will remain constant, from which it follows that the area 

 of the projection of the orbit on the x-y-plane remains constant. It is easily seen 

 that the plane of the orbit is perpendicular to the plane through the major axis 

 and the z-axis every time the electrical centre passes one of the apses of the ellipse 

 which it describes. In Part II of Bohr's paper the appearance of the small frequency 

 0/ has been discussed from the point of view of the theory of perturbations. 



For the sake of the latter applications it will be of interest to examine the 

 special form which the equations (70) assume when one of the quantities ly and /, 

 becomes equal to zero. If for instance we assume == 0, it will be seen that the 

 fundamental frequency w., does not appear at all in the motion of the electron. In 

 fact, denotes the mean frequency with which the electron oscillates between two 

 paraboloids of revolution which are characterised by the roots of the integrand in 

 the expression for /.^ given by (45). For /., ^ these roots coincide, so that the 

 amplitude of these oscillations has become equal to zero, which means that the 

 frequency all present in the motion. Introducing the value ^ in the 



equations (70) we have, since in this case, as seen from (62), a.^ = c.^ = 0, -= 1, 

 '23 = '^1 '= 'i ^ï^^ since J„(0) = 1, 



r = ^ / — P 2' J't { Tiy) cos 2 ;r , / , | 



â ' ^ 



where the summations are to be extended over all entire values of r except r - 0. 

 The equations (71) representing the motion of the electrical centre become 



showing that the electrical centre will move in a circle and that tin- Iveplerian 

 ellipse which the electron at any moment may be considered to describe possesses 



a constant eccentricity equal to - | . The plane of the orbit remains perpen- 

 dicular to the plane through the major axis and the z-axis, while it rotates uniformly 

 round the latter axis with frequency o,.. The projection of the orbit on the a--i/-plane 

 is at any moment a circle while the cosine of the angle between the plane of the 

 orbit and the z-axis is equal to the eccentricity c^. It will be observed that in the 

 present simple case the equations (72) could have been obtained from the expression 

 (39) for the motion of an electron in a Keplerian ellipse by imagining the orbil 

 placed in a position relative to the z-axis as that just described, and by giving it ;i 

 uniform rotation of frequency 0/ round this axis, applying the same method ol 

 calculation as that followed on page 15. 



I). Iv. IJ. ViUuiisk. brUli.Ski-.. iiuturviaeiisk. iiiulhciii. AfU.. S. Ki<.-kki:, 111. 3. 



40 



