VeVi Meg aces eee eee 
For a system with one degree of liberty (10) and (8) are evidently 
identical; for a system with more degrees of liberty this is however 
not the case *). 
§ 6. Connection between the adiabatic hypothesis and the quantuin- 
hypotheses of PrasckK, Desir, and others for systems with one degree 
of liberty. PLANCK’s hypothesis of tbe energy steps (1901) says, that 
a harmonically vibrating resonator of the frequency », can only have 
one of the following energies ¢: *) 
Se Ba, eal oa a! ee tee eee 
Therefore the adiabatic invariant of the resonator can only have 
ar ier | 
ee mld dad By et a (AN 
ro Po 
Let us now consider a resonator with the non-linear equation of 
motion 
the values 
Ga eg Gage fa, ge a gd ayia tng or (EN) 
This does not execute harmonic vibrations; the frequency »v =|= v° 
of its oscillations depends not only on a,,a,.... but also on the 
intensity with which they are excited. 
For the special parameter values a, = a,...=O it becomes the 
resonator of Ptanck. Therefore the adiabatic hypothesis (comp. the 
formulation in § 3) becomes: For these non-harmonic resonators too 
only those motions are possible for which 
zee = | {et = Oise es} ae tS ie ae Ce) 
Y 
From the hypothesis of PLANcK’s energy-steps we have thus 
1) In the deduction of his theorem P. Hertz has to calculate the mean with 
respect to the time of the force acting on the parameter «. He replaces this mean 
with respect to the time by a corresponding numerical mean in a microcanonical 
ensemble. It is known that BoLrZMANN and MAXWELL have only been able to 
justify this way of proceeding by the assumption that the considered system is ergodic. 
A resonator with one degree of Jiberty is really ergodic. This is however not 
the case for molecules with two or more degrees of liberty. Therefore a special 
investigation is needed here whether the above defined quantity Vo is adiaba 
tically invariant. If for all degrees of liberty of a molecule only harmonic vibrations 
can occur, Vo is really an adiabatic invariant: 
*) Comp, the note in § 2, 
