Invariants of Mechanical Systems. 519 



where dense *. Now the mean of a quantity Z with respect 

 to the time, L e. the mean value of Z for all states repre- 

 sented by a great length of the £-line, may be replaced by 

 the mean value of Z for all the points of one period-cell f- 

 Hence 



Z=p( • ■ -\dh. . . dtn. Z . . . (11) 



(the integration is effected over the volume of one period- 

 cell). 



Transforming from the t to the q as variables, the formula 

 becomes 



-*H 



d qi ...dq n .F.Z . . (12) 



as according to (9) -=-^ =f ; 



~dqk 



'hi' 



If we use this formula to calculate the quantities 



da ' 



- 2 X dq, ~^r t ' ' ' (13) 



after some transformations we find 



Introducing the values found in equation (13) into 

 equation (8), we find that 8Ia- = 0. Thus it has been demon- 

 strated that the " pliase-integrals " Ik are invariants during 

 an adiabatic disturbance of the system. 



4. Summary. 

 For mechanical systems possessing the following pro* 

 perties : — 



(1) each momentum ph can be expressed as a function of 

 the corresponding coordinate qk together with the 

 constants of integration of the n first canonical 

 integrals and with certain parameters ; 



* This theorem is due to Stackel. It is founded upon theorems by 

 Jacobi and Kronecker. See for instance Kronecker, Werhe, iii. 1, p. 47. 



A necessary condition is the following : between the quantities wji 

 (the so-called " mean motions ") there may not exist any relations of 

 the form 2mjo,j l =0, 



J 

 where the mi are integral numbers. 



t For a demonstration of this the reader ma} r be referred to the paper 

 in the Proc. Acad. Amst. The demonstration is founded upon a 

 development of the function Z in a multiple Fourier-series. 



\ Cf the paper ill the Proc. Acad. Amst. 



