591 
introduced for arbitrary central forces, starting from the hypothesis 
of the energy steps for harmonic vibrations. 
Here we must also mention the difficulties which we meet when 
we want to apply the notion: reversible adiabatic change, adiabatic 
invariant ete. to an ensemble of non-periodic motions, as e.g. the 
hyperbolic motions of a point under the influence of a force of 
Newton or CovromB: here too the change of the energy and the 
moment of momentum depend on a double boundary passage on 
the course of the complete motion from t—=—o to t= + o and 
on the infinitely slow adiabatic change. 
$ 10. Conclusion. The purpose of this paper was to show, that 
the adiabatic hypothesis and the notion of the adiabatic invariants 
are important for the generalization of the quantum theory to 
an always increasing number of classes of motion ($$ 6, 7). 
further they throw some new light on the question, for which con- 
ditions BonrzMann’s relation between entropy and probability remains 
valid (§ 8). The analyzation of the difficulties occurring at the passage 
of singular motions will perhaps lead to a completion of the adiabatic 
hypothesis. But at any rate I believe that in view of Wien’s law 
it must be given in the quantum theory a special place to the 
reversible adiabatic processes. 
AP PEND Xx “FE. 
Proof, that — is an adiabatic invariant for a system 
Yr 
with periodic motions. 
Let 
Belg: 0: a) — ® (q, a) 
be the function of LaAGRANGR of the system (the motions of which 
are for the moment not yet supposed to be periodic). And let us 
consider two infinitely near systems, for which the parameters have 
the values: a,,a,,... and a, + Aa,, a, + Aa,,...'), further the 
moments ¢4, tz and t4 + Aty, tg + Aty. We shall consider: 
I a continuous passage of the system from the configuration 
QiAs+++QnA at the time #4 to the configuration qiz,... QnB at the 
time ¢g with the values of the parameters « (change I). 
IL a continuous passage of the system from the configuration 
fra + Oqia... at the time ¢4-+ At, into the configuration MB+ 
') For the sake of implicity we shall further use only one parameter. It is easily 
seen that at each point of the discussion we can return to the case of more 
parameters, 
