and the Theory of Quanta. 503 



Remarks. — A. I£ some of the motions considered are 

 distinctly non-periodic (e. g. the hyperbolic motion in the 

 case of a Newtonian attraction) , the addition " reversible " 

 loses its original meaning. 



B. The definition given above must be generalized in a 

 suitable manner, if the system is affected by an (infinitely 

 slowly increasing) magnetic field (Zeeman effect), or if the 

 mechanical system is replaced by an electrodynamical one 

 (reversible adiabatic compression of radiation). 



§ 2. Formulation of the adiabatic hypothesis for systems 

 with periodical or quasi-periodical motions. 



Consider the system first when the parameters have some 

 given values a 10 , a 2 o • • • • The theory of quanta will not 

 allow every motion /3(a ), which is possible with these values 

 of the parameters according to the equations of the classical 

 mechanics, but only some distinct special motions *. Con- 

 sequently we speak of the " allowed " motions B{a } 

 belonging to the values a 10 , a 20 . . . of the parameters. To 

 any other values a 1? a 2 . . . belong other " allowed " motions 

 B{a}. Now our hypothesis asserts : 



For general values a l5 a 2 . . . of the parameters, those and 

 only those motions are allowed which are adiabatically 

 related to the motions which were allowed for the special 

 values a 10 , a %Q . . . (i. e. which can be transformed into them, 

 or may be derived from them in an adiabatic reversible 

 way). 



Remarks. — A. "Whether it be possible to extend the 

 hypothesis to non-periodic motions, and how this should be 

 done, I am not able to tell on account of some difficulties, 

 which are mentioned in § 9. 



B. Some forms of adiabatic affections may be realized 

 physically — for instance, the strengthening of an electric or 

 a magnetic field surrounding an atom (Stark and Zeeman 

 effect). Others have more the character of a mathematical 

 fiction (e. g. the change of a central field of force). 



§ 3. The Adiabatic Invariants and their application. 



Each application of the adiabatic hypothesis forces us to 

 look for " adiabatic invariants " — that is, for quantities which 



* In the newer form of his radiation theory, Planck speaks only of 

 11 critical " motions, besides which other motions are " allowed " too. In 

 order not to become not diffuse, we will leave this form of the theory of 

 quanta out of consideration. The suitable adaptation of the con- 

 siderations given in this paper is easily to be found. 



