48 



total variation in «j within the interval in question will, just as in case of a con- 

 stant perturbing field, be only a small quantity of this order. We get therefore 

 the result, that, for the same shape and position of the original orbit, the cjcle of 

 shapes and positions passed through by the orbit during the increase of the external 

 field will be the same as that which would appear for a constant perturbing field, 

 and that, with neglect of small quantities proportional to /-, the value of the 

 function '/' will consequenth' remain constant during the establishment of the field. 

 With this approximation we get therefore from 48i, putting / ^ U, 



J,jaj-^i?,j 



J^5d^ = r, 



which shows that the change in the total energy of the system, due to the slow and 

 uniform estabhshment of the external field, is just equal to the value of the func- 

 tion W, and consequently equal to the mean value of the potential of the external 

 forces taken over an approximate period of the perturbed motion. This result may 

 also be expressed by stating, that, with neglect of small quantities proportional to the 

 square of tlie external forces, the mean value of the inner energ}- taken over an 

 approximate period of the perturbed motion will be equal to the energy possessed 

 by the system before the establishment of the perturbing field. 



Returning now to the problem of the fixation of the stationary' states of a 

 periodic system subject to the influence of a small external field of constant potential, 

 we shall base our considerations on the fundamental assumption that these states 

 are distinguished' betw-een the continuous multitude of mechanically possible states 

 by a relation between the additional energ\" of the system due to the presence of 

 the external field and the frequencies of the slow variations of the orbit produced 

 by this field, which is analogous to the relation discussed on page 42 in the special case 

 in which the perturbed system allows of separation of variables in a fixed set of 

 coordinates. On this assumption we shall expect in the first place that, apart from 

 small quantities proportional to /., the cycles of shapes and positions of the orbit 

 belonging to the stationary,' states of the perturbed system will depend only on the 

 character of the external field, but not on its intensity. Since now-, as shown above, 

 such a cycle will remain unaltered during a slow and uniform increase of the in- 

 tensity of the external field if the effect of the external forces is calculated by means 

 of ordinarj' mechanics, we are therefore, with reference to the principle of the 

 mechanical transformabUity of the stationär}- states, led to the conclusion that it is 

 possible by direct application of ordinary- mechanics, not only to follow the secular 

 perturbations of the orbit in the stationan.- states corresponding to a constant ex- 

 ternal field, but also to calculate the variation in the energy of the system in the 

 stationär}' states which results from a slow and uniform change in the intensif}' of 

 this field. If we denote the energ}' in the stationär}' states of the perturbed system 

 by £„ \ 6, where £„ is the value of the energy in the stationarv state of the un- 



