Physics. — “On the determination of quanta-conditions by means 
of adiabatic invariants.” By G. Krurkow. (Communicated by 
Prof. P. EHRENFEST.) 
(Communicated at the meeting of September 25, 1920). 
In a series of papers Enrenrest has shown that only such functions 
of the general co-ordinates of a mechanical system may be quanti- 
cized as are adiabatic invariants’). These functions can always be 
found ®). Moreover, as we shall see, theory may answer the question 
as to the number of essential adiabatic invariants, which in accord- 
ance with the quanta-hypothesis have to assume discontinuous values. 
If we suppose that the “density of probability” of the motion ot 
the system, when not adiabatically acted upon, does not depend 
explicitly on the time, and if then by means of some hypothesis or 
some theorem which is derived from the properties of the system, 
we replace the time-mean of a phase-function by a numerical mean, 
it follows immediately that the number of essential invariants is 
equal to the number of determining quantities of the system which 
is left after the numerical mean has been determined (comp. 
equations (12) sqq. below). By the determination of the adiabatic 
invariants and the separation of the essential ones the uncertainty 
as to the choice of the forms of motion which are admissible on 
the quanta-hypothesis, becomes materially lessened. Still we must 
not expect that the adiabatic invariants which we have found are 
necessarily those which have to be quanticized: any arbitrary fune- 
tion of those quantities is again an adiabatic invariant and has thus 
equal claims to being selected. However, this liberty of choice can 
be somewhat restricted; there is a further condition to which we 
may subject the quania-functions. This condition is of the nature of 
a hypothesis, but we may give it a simple statistical interpretation. 
In every case, where the theory of quanta has been applied with 
success *), the condition is fulfilled. It was introduced by PLaNcK as 
a fundamental theorem for a complete determination of the quantities 
1) P. EHRENFEST. These Proc. XIX N°. 3, p. 576. Ann. der Phys. 51 (1916) 
Deel. 
*) G. Krutxow. Proc. Amst XXI p. 1112. 1919. 
3) My knowledge of the literature of the subject does not, however, extend 
beyond the beginning of 1917. 
