and the Theory of Quanta. 507 



Its vibrations are not harmonical, and the frequency v^v 

 does not depend only on the values of the parameters a h a 2 . . ., 

 but also on the exciting force. For the special values of 

 the parameters 



a Y = a 2 — . . . =0, 

 it passes into Planck's resonator. Hence from the adiabatic 

 hypothesis (comp. the formulation in § 3) follows : also for 

 non-harmonically vibrating resonators only those motions 

 are allowed for which 



~= Udqd 2 ^0,h.2h. . . . (14) 



So b} r means of the adiabatic hypothesis we have derived 

 Debye's hypothesis on the values of U dq dp for non- 

 harmonical vibrations * from Planck's hypothesis of energy 

 elements. 



An electrical doublet with the electrical moment a 1? the 

 moment of inertia a 2 , is suspended in such a way that it can 

 turn freely about the z axis f . An electrical field of intensity 

 a 3 acts parallel to the axis of x. As the coordinate q we 

 choose the position-angle of the doublet. We will begin 

 with very great values of a 1? a 3 , and also of a 2 ; then even 

 tor great values of the exciting energy we may consider the 

 vibrations as infinitely small and harmonical — resonator of 

 Planck's type. By diminishing infinitely slowly the values 

 of a 2 and a 3 we can pass in a reversible adiabatic way to 

 vibrations of finite amplitude, and then make the pendulum 

 " turn over " ; if now the moment of inertia a 2 is no more 

 changed, but the directing field a 3 is diminished to zero, we 

 arrive at a molecule which rotates uniformly, uninfluenced by 

 any force. For ail the motions considered, which are related 

 adiabatically to each other, the adiabatic invariant 



2T (V 



has to retain its original values 0, h, 2h, ... If for the 

 uniform rotation we identify the frequency v with the 

 number of rotations of the doublet in unit of time 



v=±ql'27r, . (15) 



and observe that 



2T = 2T=ra, (16) 



* P. Debye, Quantenhypothese (Gfittinger Vorl. Teubner, 1913). Comp. 

 also S. Boguslawski, Physik. Zeitschr. xv. (1914) p. 569. 



t Comp. the treatment and the use of this example in papers JB and C. 

 Comp. especially the diagram given in C, § 3. 



2X2 



