58 



in the formula (54), and represent the small frequencies of the secular perturbations 

 of the shape and position of the orbit, the quantity lOp may be considered as 

 representing the mean frequency of revolution of the particles in their approximately 

 periodic orbit. As regards the total energy of the perturbed system, it may next be 

 proved that, looking apart from small quantities proportional to the square of the 

 intensity of the external forces, the difference in the total energy in two slightly 

 difï'erent states of the perturbed system, for which the values of 7, S] , • • • 3s-i diifer 

 by ÔI, d'^^, ... o3s-i I'espectively, is given by the relation.') 



-1 



dE =^ cüpdl+ ^ o;c à S/c , (66) 



which coincides with (52) if dl = 0, and which will be seen to be completely 

 analogous with formula (29) in Part I, holding for] an ordinary conditionally periodic 

 system which allows of separation of variables in a fixed set of positional coordinates; 

 just as (65) is analogous to formula (31) representing the displacements of the par- 



') From a comparison with formula (8), holding for the energy difference between two neighbouring 

 states of the undisturbed system, and with formula (52), it will he seen that (66) implies the condition 

 Wp ^ u) -{- Ô'FJd I, where m is the frequency of revolution in the corresponding state of the undisturbed 

 system characterised by the given value of /, and where, in the partial differential coefficient, '/' is con- 

 sidered as a function of / and 3i, . . , 3s_i. This relation can be verified by means of a consideration 

 based on the perturbation equations (44), which takes into account the simple relation between a^ and 

 / for the undisturbed system, as well as the relation between the mean rate of variation of ßi with 

 the time and the difference between wp and w. We shall not enter, however, on the details of the 

 rather intricate calculations involved in such a consideration, since the problems in question allow of 

 a more elegant treatment by means of another analytical method. Thus it will be shown by Mr. H. A. 

 Kbamers, in the paper mentioned in the end of § 4, that, quite independent of the possibility of separa- 

 tion of variables for the perturbed system in a fixed set of positional coordinates, the theory of secular 

 perturbations exposed in this section offers — if these perturbations as determined by (46) are of con- 

 ditionally periodic type — a means of disclosing a set of angle variables, which may be used to 

 describe the m'otion of the perturbed system with the same degree of approximation as that involved 

 in the preceding calculations. According to the definition of angle variables, mentioned in the Note on 

 page 29 in Part 1, this means that it is possible, in stead of the positional coordinates q'l, ... ç^ of the 

 perturbed system and their conjugated momenta p^, ... p^, to introduce a new set of s variables in 

 such a way, that the gs and p's are periodic in everj' of the new variables with period 1, when they are 

 considered as functions of these variables and of their canonically conjugated momenta. These momenta 



will just coincide with the quantities denoted above by /, Si, ... 3j y and the corresponding angle 



variables may conveniently be denoted by w, Uij, ... li.'s_i respectively. Introducing the new variables, 

 the total energy of the perturbed S3'stem will be a function of 7, Si, . .. S,_j only, if we look apart 

 from small quantities proportional to /.-. With this approximation we get consequently by a calculation, 

 analogous to that given in the Note referred to, that the angle variables w, Uij, . , . U'^^j^ may be repre- 

 sented as linear functions of the time within an interval of the same order as <t/^. Denoting the rates 

 of variation of w, iiij, ... lU5_j bj' Wp, o^, ... Oj ^ respectively, the formulae (65) and (66) are there- 

 fore directly obtained, just as the corresponding formulae (31) and (29) in Part I. In this connection it 

 will be observed that, due to the possibility of introduction of angle vafiables, the conditions (67) appear 

 in the same form as that in which the conditions, which fix the stationarj' states of ordinarj' condition- 

 ally periodic systems which allow of separation of variables, have been formulated by Schwarzschild, 

 and which, as mentioned in the Note in Part I, has already been applied by Burgers to certain systems 

 for which such a separation cannot be obtained. 



