Physics. — "Adinbatic Invariants of Mechanical Sy^ite^ns". II. By 

 J. M. Burgers. Supplement N". 41^/ to the Communications 

 from the Physical Laboratory at Leiden. (Communicated by 

 Prof. H. Kamerlingh Onnes). 



(Communicated in the meeting of December 21, 1916). 



Systems betioeen the mean motions of luhich relations of 

 com'ïnensurahility exist. 



In the 1^^ part of this paper ^) it was shown that for mechanical 

 systems, possessing the following properties: 



1. each momentum pji can be expressed as a function of the form: 



pk = ^F]c {qk ,«'...««, a) 



2. the motion of each coordinate is a libration ; 

 the 71 phase-integrals : 



Ik = I pk dqk 



are all of them adiabatic myrt?'/«?ife, provided no relations of commen- 

 surability exist between the mean motions w-h of the "angular 

 variables" r.^. As remarked in the paper quoted, this supposition 

 was necessary in order that the system might pass consecutively 

 through all the states which are represented by the points of a 

 period-cell, so that an integral with respect to the time might be 

 replaced by an integral over the volume of a period-ce'll. 



In this section we shall consider the case that relations of commen- 

 surability do exist between the mean motions, and it will be shown 

 that if the adiabatic disturbances are limited to such as do not violate 

 these relations, at least certain detinite linear combinations of the 1^ 

 (with integral coefficients) are invariants. If the system is rigorously 

 periodic, so that the mean motions are all equal, the only combi- 

 nation of this character is found to be the sum of all the phase- 

 integrals (in other words: the integral of action, extended over a 

 full period of the system), the invariancy of which has already been 

 demonstrated by Ehrenfest"). 



We shall describe the motion of the mechanical system in the 



1) These Proceedings, p. 149. 



2) P. Ehrenfest, ibidem XIX (1) p. 576, 1917. 



