1113 



paper: on this occasion it will be left out of account and we shall 

 only deal with condition (2). 



This condition imposes on us the task to find the adiabatic in- 

 variants of a given mechanical system and to look for a general 

 method of solving the "adiabatic" pioblem.^) A method of that kind, 

 was unknown so far; the adiabatic invariants had to be guessed at 

 and their adiabatic invariability had to be tested a posteriori. In this 

 way the following invariants were found: 



a. the quantity V of statistical mechanics, which measures the 

 phase-extension limited by the "energy-surface"'); 



b. the "action" calculated for a full period of a periodical system; 



V = C2 T dt ') ; 



c. the quantum integrals of the "conditionally periodic" systems; 



hi 

 Vi = I pi dqi = 2 I ^, dqi . *) 



« Oi 



In what follows I shall sketch out a general method of finding 

 adiabatic invariants and apply it to certain special cases, viz. 



a. Cyclic systems. Properly speaking these systems come under 

 the head of conditionally periodic systems; but as the conditions are 

 particularly simple in this case and bring out the \^\'y natural 

 character of the method, 1 shall discuss them separately; 



/?. conditionally periodic systems; 



y. ergodic systems. 



Under (/?) I shall only consider the limiting case, in which there 

 are no commensurable relations between the periodicity-moduli. To 

 the further cases and in particular their relation to the third con- 

 dition stated above I hope to return on a later occasion. 



1) This point was specially emphasized by Ehrenfest. Compare for instance 

 P. Ehrenfest Phil. Mag. VI Vol. 33. p. 513 (1917). 



«) P. Hertz. Ann. d. Phys. 33 (1910) p. 544. 



') L. BoLTZMANN. Prinz. d. Mechanik II p. 181. P. Ehrenfest Ann. d. Phys. 

 51 (1916) p. 327 Anhang. 



*) J. M. Burgers. Ann. d. Phys. 52 (1917) p. 195. To the papers in the Pro- 

 ceedings of the Amst. Acad, referred to by the author 1 had unfortunately no 

 access. 



72* 



