Adiabatic Invariants and the Theory of Quanta. 501 



to more general motions *. In these researches I especially 

 made use of the following hypothesis, to which Einstein | 

 gave the name " Adiabatenhypothese.'" 



If a sy stem be affected in a reversible adiabatic way, allowed 

 motions are transformed into allowed motions %. 



Suppose that for some class of motions we, for the 

 first time, introduce the quanta. In some cases the hypo- 

 thesis fixes completely which special motions are to be 

 considered as allowed : this occurs if the new class of 

 motions can be derived by means of an adiabatic trans- 

 formation from some class for which the allowed motions 

 are already known (especially if the new motions can be 

 derived from harmonic motions of one degree of freedom) §. 



In other cases the hypothesis gives restrictions to the 

 arbitrariness which exists otherwise in the introduction of 

 the quanta. 



In these applications of the adiabatic hypothesis the so- 

 called " adiabatic invariants " are of great importance, i. e. 

 those quantities which may have the same values before and 

 after the adiabatic affection. Especially I have shown 

 before || that arbitrary periodic motions (of one or more 

 degrees of freedom) possess the adiabatic invariant 



iT 



T (1) 



(v, frequency ; T, mean with respect to time of the 

 kinetic energy), which in the case of harmonic motions of 

 one degree of freedom reduces to 



} wi 



The object of the considerations of this paper is : — 



(1) To formulate as sharply as possible the adiabatic 

 hypothesis, at the same time showing what is wanting 

 in sharpness, especially for non-periodic motions. 



* P. Ehrenfest, Verh. d. phys. Ges. vol. xv. (1913) p. 451 (quoted 

 as B). P. Ehrenfest, "A Theorem of Boltzmann and its connexion with 

 the theory of quanta," Proc. Acad. Amsterdam (quoted as 0). 



t A Einstein, " Beitrage z. Quaatentheorie," Verh. d. phys. Ges. 

 vol.xvi. (1914) p. 826. 



X For the definitions of the expressions used here comp. § 1, 2. 



§ Examples : C, § 3 ; this paper, § 7, 8. 



|| Paper B, § 1. 



i Conip. A, § 2 ; 0, § 2. The existence of this adiabatic invariant may 

 be considered as the root of .Wien's displacement law. 



