828 
we may return from the canonical equations which are so convenient 
in the theory of quanta to the equations on cartesian co-ordinates. 
Here the invariance in question means: invariance with respect to 
the groups of rotations and translations; vector-analysis thus provides 
the means of testing hypothetical quanta-quantities for the new 
postulata *). 
The above mentioned means enable us in special cases to separate 
the quanta-quantities without ambiguity, for instance for the mecha- 
nical systems considered by PLaNnck in the paper quoted. In some 
cases, however, an ambiguity remains, which we may get rid of in 
the following manner: by putting all but one of the quanta-quantities 
equal to nought, a “singular motion” must be obtained. In this manner 
we are able to make a connection between the methods sketched 
out above and Pranck’s theory on the physical structure of the 
phase-space, PLANCK'’s singular motions forming the last step in the 
series. We may recapitulate as follows: 
The quanta-quantities are (a) functions of the integrals of the 
equations of motion (8) adiabatic invariants which (y) must be 
“normalized” and (d) have a meaning which is independent of the 
system of co-ordinates and finally (e) yield singular motions in 
Pianck’s sense of the expression. 
$ 1. The fundamental equation. 
Let a mechanical system of n degrees of freedom be given by 
its canonical equations of motion 
ND 
: 0g . 
We shall consider a number of systems and introduce a function 
o(pigi,t) which may be called the density of probability: @ must 
satisfy the fundamental equation of statistical mechanics *) : 
do % dop. dug. 
—~-+ 2 | — == 0 ae 
or, using (1): 
dg Bui dor do : do 
= = | —P. SS). ) SS eee IP ‘ 
Ot i = (5 urn òg, i) dt (2 ) 
o is therefore a function of the integrals of the equation (1). 
1) In the theory of the ZEEMAN-effect as given by SOMMERFELD and DEBIJE 
(Phys. Zschr. 17. (1916) a difficulty is met with here. This may, I think, be 
evaded in different ways, but I am not able to give a uniquely determined 
solution. 
2) J. W. Gisps. Scientific papers. II’ p. 16; Statistical Mechanics. Chapter I. 
