502 Prof. P. Ehrenfest on Adiabatie Invariants 



(2) To demonstrate the importance of the " adiabatie in- 



variants " for the theory of quanta. In this respect 



2T 

 the discussion of the invariant — mentioned above 



v 



gives the connexion between the adiabatie hypothesis 



and the formulae by which Planck, Debye, Bohr, 



Sommerfeld and others have introduced the quanta. 



(3) To indicate the difficulties which arise in the application 



of the hypothesis, if the reversible adiabatie trans- 

 formation leads through singular motions. 



(4) To indicate the connexion between the adiabatie problems 



and the statistical-mechanical roots of the Second Law 

 of Thermodynamics. Boltzmann's deduction of this 

 law is based upon a statistical principle which has been 

 destroyed by the introduction of the quanta. At the 

 present time we possess a statistical deduction of this 

 law for some special systems (e. g. for systems with 

 simple harmonic motions) but not for general systems*. 

 I take the liberty of publishing my considerations, in the 

 hope that others may overcome the difficulties which I could 

 not solve. Perhaps on closer examination it will appear 

 that the adiabatie hypothesis is not generally valid ; in any 

 case, the correctness of Wien's displacement law seems to 

 indicate that the reversible adiabatie processes take a pro- 

 minent place in the theory of quanta — it seems that they may 

 be treated in a " classical " way. 



§ 1. Definition of a reversible .adiabatie affection of a system. 

 Motions /3(a) and fi(a') which are adiabatically related to each 

 other. 



Let the coordinates of the system be denoted by q 1 . . . q n . 

 The potential energy <X> may contain besides the coordinates q 

 certain " parameters " a 1? a 2 . . . , the values of which can 

 be altered infinitely slowly. The kinetic energy T may be a 

 homogeneous quadratic function of the velocities qi . . . q n , 

 the coefficients of which are functions of the q and may be 

 of the «i, a 2 . . . . By changing the parameters from the 

 values a l5 a 2 . . . to the values a/, a 2 ' . . . in an infinitely 

 slow way, a given motion /3(a) may be transformed into 

 another motion /3(a'). This special type of influencing 

 upon the system may be called " a reversible adiabatie 

 affection" the motions /3(a) and /3(a f ) " adiabatically related 

 to each other." 



* P. Ehrenfest, Physik. Zeitschr. vol. xv. (1914) p. 657 (paper D). 

 Also § 8 of this paper. 



