and the Theory of Quanta. 513 



of h, whereas in the former it can be equal to every integral 

 multiple ofh. 



From these considerations it appears that the adiabatic 

 hypothesis wants a special complement in order that in this 

 case (and also in the analogous cases of a passage through 

 singular motions) the double limiting process should lead 

 to a definite value. If such a complement could be found, 

 it would be possible to deduce the quantum formulae for 

 arbitrary central forces from the hypothesis of energy 

 elements for harmonically vibrating resonators. 



At this place we must also mention the difficulties which 

 arise if we try to extend the notions of " reversible adiabatic 

 affection," "adiabatic invariant/'' &c, to families of motions 

 which are essentially unperiodic — as, for instance, the hyper- 

 bolic motions of a point in a Newtonian field of force. In 

 this case, too, the change of the energy and of the moment 

 of momentum of the motion depend on a double limiting 

 process : the course of the whole motion from t= — oo to 

 £=+co,and the infinitely slow change of the parameters 

 <2 1? a 2 . . . . 



§ 10. Conclusion. 



The problems discussed in this paper show, as I hope, 

 that the adiabatic hypothesis and the notion of adiabatic 

 invariants are of importance for the extension of the theory 

 of quanta to still more general classes of motions (§§ 6, 7) ; 

 furthermore, that they throw some light on the question : 

 What conditions are necessary that Boltzmann's relation 

 between probability and entropy may remain valid (§ 8) ? 



Hence it would be of great interest to develop a systematic 

 method of finding adiabatic invariants for systems as gener- 

 ally as possible. 



The difficulties which arise by the passage through singular 

 motions are yet awaiting their solution; perhaps it will be 

 necessary to seek for some complement of the adiabatic 

 hypothesis. In any case, it seems to me that the validity 

 of Wien's displacement law shows that reversible adiabatic 

 ajfections take a prominent place in the theory of quanta. 



Postscriptum. — The beautiful researches of Epstein (Ann. 

 d. Phys. l.pp. 489 & 815, 1916), Schwarzschild (Silzungsber. 

 Berl. Akad. 1916, p. 548), and others which have appeared 

 in the meantime, show the great importance the cases which 

 are integrable by means of Stackel's method of "separation 

 of the variables" have for the development of the theory of 

 quanta. Hence the question arises : How far are the different 

 parts into which these authors separate the integral of action 

 according to Stackers method adiabatic invariants ? In the 

 problem treated by Sommerfeld this is the case, as is shown 

 in § 7. 



