Physics. — "Adiahatic Invariants of Mechanical Systems. III". 

 By J. M. Burgers. Supplement N". 4l6no the Communications 

 from the Physical Laboratory at Leiden. (Communicated by 

 Prof. H. Kamerlingh Onnes). 



(Communicated in the meeting of January 27, 1917). 



In the two preceding papers ') on this subject the question was 

 investigated as to what quantities possess the property of being 

 adiabatic invariants for those mechanical systems in which the 

 variables can be separated, i.e. where the momenta can be expressed 

 by formulae of the form : 



pk = ^Fk iqh «^ . . . «", a) 

 The resul, obtained was that .he "phase-in.egrals": /.=/.%p. 



do not change during an adiabatic disturbance of the system; this 

 conclusion is closely connected with the quantum formulae as intro- 

 duced by Epstein, Debye and Sommerfeld, who put these integrals equal 

 to integral multiples of Planck's constant. Schwarzschii.d^), however, 

 has put the quantum formulae into another form, which is far more 

 general. He supposes that by means of certain transformations it is 

 possible to express the original coordinates and momenta {q,p) as 

 functions of a new system {Q,P), possessing the following properties: 



1. The Q are linear functions of the time; 



2. the P are constants; 



3. the q and p are periodic functions of the Q with a period 

 2/r; hence for instance: 



q (Q, + 2l,7r, ... . Q„ + 2lnJt) = q (Q,, . . Q„). 



These variables Q are the so-called "angular variables" ("Winkel- 

 koordinaten"). He then introduces the quantum formulae: 



dQk . Pk = 2jt Fk=: njc . h + constant (A). 







If the character of the system is such that the variables can be 



P 



^ These Proceedings p. 149 and 158. 



2y K. ScHWARZSCHiLD, Sitz. Ber. Bed. Akad. p. 548, 1916. 



