51 



where the Uüler integral is taken over a complete oseilhition of ß... In orih^- lo fix 

 the stationary states, it will now be seen in the first place that, aniony the mulli- 

 tude of states of the perturbed system for which the value of / in the correspondin}^ 

 states of the undisturbed system is equal to nh where n is a given positive integer, 

 the state for which 3 = must beforehand be expected to be a stationary state. In 

 fact, for this value of S, the shape and position of the orbit will nol undergo secular per- 

 turbations but will remain unaltered fora constant external field as well as during a slow 

 and uniform establishment of this field. In contrast to what in general will take place 

 during a slow establishment of the external field, we may therefore expect that, for 

 this special shape and position of the orbit, a direct application of ordinary mechanics 

 will be legitimate in calculating the effect of the establishment of the field, since 

 there will in this case obviously be nothing to cause the coming into play of some 

 non-mechanical process, connected with the mechanism of a transition between 

 two stationarj' states accompanied by the emission or absorption of a radiation of 

 small frequency. With reference to relation (49) we see therefore that, by fixing the 

 stationai'y states of the perturbed system by means of the condition 



3 = u/i, (51) 



where n is an entire number, we obtain a relation between the additional energy 

 ® = f of the system in the presence of the field and the frequencj' p of the secular 

 perturbations, which is exactly of the same type as that which holds between the energy 

 and frequency in the stationary states of a system of one degi-ee of freedom, and 

 which is expressed bv (8) and (10). By means of (51) it is possible, with neglect of 

 small quantities proportional to the square of the perturbing forces, directly to 

 determine the value of the additional energy in the stationary states of a periodic 

 system of two degrees of freedom subject to ah arbitrarily given small external field 

 of force, and consequently with this approximation, by use of the fundamental rela- 

 tion (1), to determine the effect of this field on the frequencies of the spectrum of the 

 undisturbed periodic system. In general this effect will consjst in a splitting up of 

 each of the spectral lines into a number of components which are displaced from 

 the original position of the line by small quantities proportional to the intensity of 

 the external forces. 



, When we pass to perturbed periodic systems of more than two degrees 

 of freedom, the general problem is more complex. For a given external field, 

 however, it may be possible to choose a set of orbital constants a.^, ... «s, 

 /?2, . . . /9s in such a way, that during the motion every of the «'s will depend on 

 the corresponding ß only, while every of the /9's will oscillate between two fixed 

 limits. From analogy with the theory of ordinär;^ conditionally periodic systems 

 which allow of separation of variables, the perturbations may in such a case be 

 said to be conditionally periodic, and, from a calculation quite analogous to 

 that leading to equation (29) in Part 1 which is based entirely on the use of the 

 Hamiltoniau equations, we get for the difference in '/for two slightly diiferenl states 



