57 



lions is secured if Ihe stationary stales in llie presence of a small external ciniial 

 field are characterised by the condition 



S = 27r«2 == n/i, (64) 



where u is an entire number. This condition, which is ecjuivalenl with the second 

 of Sommerfeld's conditions (16), corresponds to (58) and is seen to coincide 

 witli the first of tlie conditions (61), while the second of the latter conditions in the 

 special case under consideration looses its validity corres])ondinf^ to the fad liiat 

 the orientation of the plane of the orbit in space is obviously arbitrary. Since, for 

 a Keplerian motion, the major axis of the orbit depends on the total energy only 

 while the minor axis is proportional to the angular momentum, it will be seen from 

 (64) that the presence of a small external field imposes the restriction on the motion 

 of the atom in the stationary stales, that the minor axis ot the orbit of the electron 

 must be equal to an entire multiple of the n"' part of the major axis, which was 

 given by 2a„ in (41). This result has been pointed out by Sommerfeld as a con- 

 sequence of the application of the conditions (16). 



In the preceding it has been shown how it is possible to attack the probj^em 

 of the stationary states of a perturbed periodic system by an examination of the 

 secular perturbations of the shape and position of the orbit, and to fix these states 

 if the pertui-bations are of periodic or conditionally periodic type. While these con- 

 siderations allow to determine the possible values for the total energy of the per- 

 turbed system and thereby the frequencies of the components into which the 

 lines of the spectrum of the undisturbed system are split up in the presence of the 

 external field, it is necessary, however, for the discussion of the intensities and 

 polarisations of these components to consider more closely the motion of the 

 particles in the perturbed system and the relation of the total energy of this system 

 to the fundamental frequencies which characterise the motion. In the first place it 

 will be seen that, if the secular perturbations as determined by the equations (46) 

 are of conditionally periodic type, the disjjlacemenls of the particles of the system 

 in any given direction may, with neglect of small quantities proportional to the 

 intensity of the external forces, be represented, within a time interval sufficiently 

 large for these forces to produce a considerable change in the shape and position 

 of the orbit, as a sum of harmonic vibrations by expressions of the type: 



f = 2"Ct, ti, .. t^_i cos2;r((rü;p-f tiO^H- ... is-iOs-0 I + CT,i„ ... t,_, }, (65)- 



where the summation is to be extended over all positive and negative entire values 

 of -, ti, ... ts-i, and where the C's and c's are two sets of constants, the former of 

 which depend only on the values of the quantities 3^, •■■ 3s i defined by (53) and 

 on the value of the quantity /, which characterises the corresponding stale of the 

 undisturbed system which would appear if the external field vanished at a slow and 

 uniform rate. While the quantities i\, ... Os^i are the same as those which appear 



