61 



ternal forces due lo the first lield and the inlunial I'urccs of the system. On Ihr 

 basis of the general rehttion between energy and I'requency in the stalioiKu y slates, 

 we may then expect that it is possible to fix the motion in these stales Ibr Ihe per- 

 turbed periodic system in the presence of both external fields with neglect of small 

 terms of the same order as the largest of the quantities «- and ;,, and to fix the 

 corresponding values for the energy with neglect of small terms of the same order 

 as the largest of the quantities Xii' and /". ') In general, however, the eflect on the 

 spectrum of the perturbed system, produced by the second external field, may be 

 calculated without considering the perturbing effect of this field in detail. In facl, it 

 is in general possible, by means of the principle of the mechanical translbrmability 

 of the stationary states, with the approximation mentioned to determine the altera- 

 tion of the energy of the system, due to the presence of the second external 

 field, directly from the character of the secular perturbations produced by the first 

 external field only. Thus let us assume that the second field is slowly established 

 at a uniform rate within a time interval of the same order of magnitude as 

 that in which the system will ])ass approximately through anj' state belonging 

 to the cycle of shapes and positions, which the orbit passes through in the 

 stationary states in the presence of the first external field only. Denoting a time 

 interval of this order by It and the potential of the first perturbing field by ii and 

 that of the second by Ai^, we get then, by a calculation quite analogous to that 

 given in Part I on page 11 for the alteration in the mean value of the energy of a 

 periodic system during a slow establishment of a small external field, that the 

 alteration in the mean value of a^ -\- Q taken over a time interval of the same 

 order as ë, due to the establishment of the second external field, will be a 

 small quantity of the same order of magnitude as ^ (Jß)"; but with the notation 

 used above this means, in general, a small quantity of the same order as /,«-. It 

 follows consequently that, with this approximation, the alteration in the energy in 

 a given stationary state, due to the presence of the second perturbing field, is equal 

 to the mean value of the potential of this field taken over the cycle of shapes and 

 positions, which the orbit would pass through in the corresponding stationary state 

 of the perturbed system under the intluence of the first external field only. In 

 general, the effect on the spectrum will therefore consist in a small displacement of 

 the original components proportional to the intensity of the forces due to the second 

 perturbing field; and as regards the degree of approximation with which these dis- 

 placements are defined, it will be seen from the above that, if ;i is smaller than VJ, 

 the fixation of the energy in the stationary states in the presence of the second ex- 

 ternal field, and therefore also the determination of the frequencies of the spectral 



^) In analogy with the considerations on page 5Ü it may be expected, however, that these limits 

 for the definition of the energy in the stationary states will hold only for the great majorit3' among 

 a large number of atomic systems. Thus in the present case we must be prepared to find that for a small 

 fraction of the systems of the same order as //'- lif fi-':^X) the energy will differ from that (ixcd by 

 the method under consideration by small quantities of the same order as ij.L 



D, !.. U. Vidcnsk. Selsk. SUr., n.aturvitlcnsli. o^ mnthem. .\M„ 8. K;i-UUf. IV. 1. 9 



