50 



other hand, from the point of view of the general formal relation between the 

 quantnm theory and the ordinary theory of radiation, we must be prepared to find 

 that the motion and the energy in the stationary states of a perturbed periodic 

 system, for which we only know that the secular perturbations as determined by 

 (46) are of conditionally periodic type, will not be as sharply defined as the motion 

 and the energy in the stationary states of a conditionally periodic system for which 

 the equations of motion allow of a rigorous solution by means of the method of 

 separation of variables. Thus, if we consider a large number of similar atomic 

 systems of the type in question, we may be prepared to find that the values of the 

 additional energy in a given stationary state will for the diffei-ent systems deviate 

 from each other by small quantities; but it must be expected that the values of 

 the additional energy for the large majority of systems will differ from the value 

 of '/; as determined by the method indicated above, only by small quantities pro- 

 portional to P, and that only for a small fraction (at most of the same order as 

 P) of the systems the values of the additional energy will show deviations from 

 this value of '/; which are of the same order as L 



As to the application of the preceding considerations to special problems, it will 

 be seen in the first place that in case of a perturbed periodic system pos- 

 sessing two degrees of freedom, as for instance that considered in the example 

 on page 43, the problem of the fixation of the stationary states of the perturbed 

 system in the presence of a small external field allows of a general solution on 

 the basis of the method developed above, because in this case the secular per- 

 turbations will in general be simply periodic. In fact, in this case the shape and 

 position of the orbit are characterised by two constants u^ and ß^, and from the 

 equations (46), which will be analogous to the equations of motion of a system of 

 one degree of freedom, it follows directly that during the perturbations a.-^ will be 

 a function of ß^ and that in general these quantities will be periodic functions of 

 the time with a period 5 which, besides on «j, will depend on the value of '/ only. 

 Considering two slightly different states of the perturbed system for which the 

 corresponding states of the undisturbed system (i. e. the states which would appear 

 if the external forces vanished at a slow and uniform rate) possess the same energy 

 and consequently the same value for the quantity / defined by (5), we get therefore 

 by a calculation completely analogous to that leading to relation (8) in Part I, which 

 was deduced directly from the Hamiltonian equations, for the difference in the 

 values of the function f for these two states 



d¥= 0^3, (49) 



where o = - is the frequency of the secular perturbations, and where the quantity 

 3 is defined by 



(50) 



= ^«, ^-^dt = y,Dß,, 



