42 



of the perturbed system are charaoteriseil by a greater number of extra-mechanical 

 conditions than the stationary states of tlie undisturbed system. On the other hand, 

 we were led to assume from the general formal relation between the quantum theorj' 

 of line spectra and the ordinary theory of radiation, that it is possible to obtain 

 information about the stationary states of the perturbed system from a direct 

 consideration of the slow variations which the periodic orbit undergoes as a 

 consequence of the mechanical effect of the external field on the motion. Tlius, if 

 these variations are of periodic or conditionally periodic type, we may expect that, 

 in the presence of the external field, the values for the additional energy of the 

 system in the stationary states are related to the small frequency or frequencies of 

 the perturbations, in a manner analogous to the relation between energy and frequency 

 in the stationary states of an ordinary periodic or conditionallj' periodic system. 



If the equations of motion for the perturbed system can be solved by means 

 of separation of variables, it is easily seen that the relation in question is fulfilled 

 if the stationary states are determined by the conditions (22). Consider thus a sj'stem 

 for which every orbit is periodic, and let us assume that in the presence of a given 

 small external field a separation of variables is possible in a certain set of coordi- 

 nates q^, . . . qs- For the undisturbed system we have then, according to equation 



(23), that the quantity 1, defined by (5), is equal to x^I^-\- — xsis, where 



/j, ... /s are defined by (21) and calculated with respect to the set of coordinates 

 just mentioned, and where the z's are a set of entire positive numbers without a 

 common divisor. For simplicity let us assume that at least one of the x's, say Xs, 

 is equal to one, and that consequently, as mentioned on page 22, the number n in 

 (24), which characterises the stationary states of the undisturbed system, may take 

 all positive values. This condition will be fulfilled in case of all the applications to 

 spectral problems discussed below: it will be seen, however, that the extension to 

 problems where this condition is not fulfilled will only necessitate small modifica- 

 tions in the following considerations. Bj' use of (29) we get now for the difference 

 in the total energy of two slightly different states of the perturbed system 



s s s— 1 



oE = ^y (01,0 Ik = ws y' y.koh— y^ (cok — ■/.kws)dlk- (42) 



1 1 1 



Since for the undisturbed system wk = y.k(^s, the differences cuu — y-ktos appearing 

 in the last term will, for the perturbed system, be small quantities which will just 

 represent the frequencies of the slow variations which the orbit undergoes in the 

 presence of the external field. These quantities will in the following be denoted by 



s 



Oa-. Consider now the multitude of states of the perturbed system for which I'xkh 

 is equal to nh, where n is a given entire positive number. This multitude will be 

 seen to include all possible stationary states of the perturbed system, which satisfy 

 (22), and the motion of which differs at any moment only slightly from some 

 stationary motion of the undisturbed sj'stem, satisfying (24) for the given value of 



