49 



disturbed system chaiaclerised hy a given entire value of ;i in liie condition / n h, 

 we may therefore conclude from the above thai the additional energy tt in 

 the stationary states of tiie perturbed system will be ecjual to the 

 value in these states of the function '/'" defined by (45), it we look 

 apart from small quantities proportional lo the square of the in- 

 tensity of the external forces. It will be seen that this result is equivalent 

 to the statement, that the mean value of the inner energy taken over an approximate 

 period of the perturbed motion will be equal to the value /s„ of tbe energy in the 

 corresponding stationarj' state of the undisturbed system. In case of the perturbed 

 system allowing of separation of variables in a fixed set of coordinates, this result 

 may be simply shown to be a 'direct consequence of the fixation of the stationary 

 states by means of the conditions (22). In fact, if we assume that the undis- 

 turbed motion, considered in (47), corresponds to some stationary state, satisfying (24) 

 for a given value of n, and that the perturbed motion is also stationary and satis- 

 fies (22), we see that the right side of (47) will be zero, and we get the result that 

 the mean value of the inner energy in the stationary states of the system, with the 

 approximation mentioned, will not be altered in the presence of the external field. 

 Due to the above result that the additional energy ß in the stationary staijes 

 of the perturbed system, with neglect of small quantities proportional to /-', may 

 be taken equal to the value in these states of the function 'f entering in the 

 equations (46) which determine the secular perturbations of the orbits, we are 

 now able to draw further conclusions from the fact, mentioned above, that these 

 equations are of the same type as the Hamiltonian equations of motion for a 

 mechanical system of s — 1 degrees of freedom. In fact, we see that the fixation 

 of the stationary states of the perturbed system is reduced to a pro- 

 blem which is formally analogous to the fixation of these states for 

 a mechanical system of less degrees of freedom. As it will appear from 

 the following applications this problem may, quite independent of the possibility of 

 separation of variables for the perturbed system, be treated directly on the basis of 

 the fundamental relation between energy and frequency in the stationary states of 

 periodic or conditionally periodic systems, discussed in Part I, if only the solution 

 of the equations (46) is of a periodic or conditionallj' periodic character. In this con- 

 nection it may once more be emphasised that these equations, according to the 

 manner in which they were deduced, allow to follow the secular perturbations only 

 through a time interval of the same order of magnitude as that sufficient for the 

 external forces to produce a finite alteration in the shape and position of the orbil. 

 With reference to the necessary stability of the stationary states of an atomic system, 

 it seems justified, however, to conclude that any possible small discrepancy between 

 the motion to be expected from a rigorous application of ordinary mechanics and that 

 determined by a calculation of the secular perturbations, based on the equations 

 (46), cannot cause a material change in the character of the stationary states as 

 fixed by a consideration of the periodicity properties of these perturbations. On the 



