169 



Qifc=27r 



I 2 pdq=z njc . h ^) 



and is therefore in agreement with the above. 



4. The following points have still to be mentioned : 

 ((. Probably it will be found sufficient that in passing from a ^ 

 constant to^ a =^ a given function of the time, the Hamiltonian 

 equations remain unchanged onl)' if we neglect terms of the 2"*^ 

 and higher orders in a. This has yet to be in\estigaled. 



b. In the present paper it has been supposed that tlje mean motions 

 (O/ are all incommensurable. The toj are, however, functions of the 

 parameters. Hence if the a are varied, the co; change too, and their 

 ratios pass through rational values. It has still to be investigated, 

 whether this may give rise to difficulties. (This applies also to the 

 demonstrations given in the preceding papers). 



SUMMARY. 



If a mechanical system of ?i degrees of freedom possesses solutions 

 which can be expressed by means of multiple trigonometric series, 

 proceeding by the sine's and cosines of n angular variables, between 

 the mean motions of which no relations of commensurability exist, 

 it is possible to determine the canonical momenta corresponding to 

 these angular variables in such a way that they are adiabatic 

 invariants for an infinitely slow change of the parameters of the 

 system. — (The fact that during an adiabatic disturbance the mean 

 motions change and that their ratios pass through rational values 

 has to be further inquired into.) 



1) P. S. Epstein, Verb. d. D. Physik. Ges. 18 (1916) p. 411. 



12 



Proceedings Royal Acad. Amsterdam. Vol. XX. 



