52 



of the perturbed system, lor which the value of / in the corresponding states of the 

 undisturbed system is the same, 



dl'- 



^y^ d/co3a-, 



(52) 



where oa- is the mean frequency of oscillation of ßk + \ between its limits, and where 

 the quantities 5/,- arc defined by 



3a- = \ «A + i -D/Sjt + i, (A- = 1, . . 



1) 



(53) 



where the integral is taken over a complete oscillation of ßk-'-i- I" analogy with 

 the expression (31) for the displacements of the particles of an ordinary condition- 

 ally periodic system which allows of separation of variables, we get further in tlie 

 present case that every of the a's and ß's may be expressed as a function of the 

 time by a sum of harmonic vibrations of small frequencies 



ßf 



-St., . . . t,_ 1 cos 2 - ( ( t^ Li, + . . . ts - 1 us - 1) t — i-t„ . 



)' 



(54) 



where the S's and c's are constants, the former of which, besides on /, depend on 

 the 3's only, and where the summation is to be extended over all positive and 

 negative entire values of the t's. If therefore the secular perturbations are con- 

 ditionally periodic, we may conclude that the stationary states of the perturbed 

 system, corresponding to a given stationary state of the undisturbed system, will be 

 characterised by the s — 1 conditions 



Sa- = "A^!, (A- 



1, 



1) 



(55) 



where iij, ... iij i form a set of entire numbers. In fact, as seen from (52), we obtain 

 in this way a relation between the additional energy and the frequencies of the 

 secular perturbations of exactly the same type as that holding for the energy and 

 frequencies of ordinary conditionally periodic systems and expressed by (22) and 

 (29); moreover we may conclude beforehand that the state in which every of the 

 quantities 3a, defined by (53), is equal to zero must belong to the stationary states of the 

 perturbed system, because in this case the orbit will not undergo secular perturbations 

 for a constant external field, nor during a slow and uniform establishment of this 

 field. Since the conditions (55), with neglect of small quantities proportional to the 

 square of the intensities of the external forces, allow to determine the additional 

 energy of the system due to the presence of the external field, we see therefore that 

 the effect of this field on the spectrum of the undisturbed system, if the secular 

 perturbations are conditionalh' periodic, will consist in a splitting up of each spec- 

 tral line in a number of components, in analogy with the elïect of a perturbing 

 field on the spectrum of a periodic system of two degrees of freedom. In general, 

 however, the perturbations, which a periodic system of more than two degrees of 



