55 



ciission on pajçe 52, oscillale between (ixed limits, liiil it will dc seen tlial ,i. (Inrinf{ 

 the perturbations may either oseillate l)etween two such iimils or increase (or de- 

 crease) continuously, while /?, will always vary in the latter manner. This con- 

 stitutes, however, only a formal difficulty of the same kind as that mentioned in 

 Part I in connection with the discussion of the conditions (16), which fix the 

 stationary states of a system consisting of a particle moving in a central held of 

 force. Thus from a simple consideration it will be seen that, in cüm|)lete analogv 

 to the relations (52) and (53), we get in the present case lor the dilVerence l)ctween 

 the energy of two slightly dilTerent states of the perturbed system, which correspond 

 to the same value of /, 



d'F=- v\d\ + o^d%, (59) 



where \.\ is the frequency with which the shape of the orbit and its position rela- 

 tive to the axis of the field repeats itself at regular intervals and which is charac- 

 terised by the variation of o.., and ß.,, while o^ is the mean frequency of rotation of 

 the plane of the orbit round this axis characterised by the variation of /?,, and 

 where 3i and ^.^ are defined by the equations 



P 



Si == \«2-ö/?2, % = \a.,Dß^ = 2na.,. (60) 



In case ß.^ varies in an oscillating manner with the lime, the first integral nuist be 

 taken over a complete oscillation of this orbital constant, while, if /J., during the 

 perturbations increases or decreases continuously, the integral in the expression for 

 Si must be taken over an interval of 2t:, just as the integral in the expression 

 for So- By fixing the stationary states of the perturbed system by means of the two 

 conditions ^) 



Si = "i''. S2 = "2/'' (61) 



where ii^ and lu are entire numbers, it will therefore be seen that we obtain the right 

 relation between the additional energy ® = '/' of the perturbed atom and the fre- 



^) Quite apart from the problem of perturbed periodic systems, tlie second of these conditions 

 would also follow directly from certain interesting considerations of Epstein iBer. d D. Pbys. Ges. XIX. 

 p. 116 (1917)) about the stationary states of systems which allow of what may be called "partial separa- 

 tion of variables". In this case it is possible to choose a set of positional coordinates q.^. . . . qs in such 

 a way that, for some of the coordinates, the conjugated momenta may be considered as functions of 

 the corresponding q's only, so that, for these coordinates, quantities / may be defined by 21) in the 

 same way as for systems for which a complete separation of variables can be obtained. From analogy 

 with the theory of the stationary states of the latter systems, Epstein proposes therefore the assump- 

 tion, that some of the conditions to be fulfilled in the stationary states of the systems in question may 

 be obtained b}' putting the /'s thus defined equal to entire multipla of h. It will be seen that, in case 

 of sj'stems possessing an axis of symmetrj', this leads to the second of the conditions (^(il), which ex- 

 presses the condition that in the stationary states the total angular momentum round the axis must 

 be equal to an entire multiple of /i/2jr. As pointed out in Part 1 on page 34, this condition would also 

 seem to obtain an independent support from considerations of conservation of angular momentum 

 during a tiansition between two stationary states. 



8* 



