585 
limit between the pendulum motions and the rotations. It must 
therefore be investigated, how the invariants of both kinds of motion 
are connected. *) 
$ 7. Connexion with SOMMERFELD's quantum hypothesis for systems 
with more than one degree of liberty. 
We want to show, that the adiabatic law is satisfied by the 
quantum hypothesis recently given by SOMMERFELD for the plane motion 
of a point about a centre of attraction according to Newton's law. 
Let 4 (r, a,,a,,-..) be the potential of a central attracting force. 
Then the differential equations of the plane motion of a point are 
in polar coordinates: r—q,, P=, 
= ; dy : 
(pang i aie ale ie aE) 
dy 
(mr? p) = 0 (185) 
mn Ed — . . . . . . ° ) 
Tt Le 
(184) expresses directly — which is very plausible — that the 
moment is invariant with respect to a change of the parameters 
ee ee 
mrp =p, adiabatically invariant . .- . - (19) 
By eliminating p by means of (19) from (18a) we obtain 
mre dy 
Pz 
This differential equation has the same structure as if it described 
the motion of a point oscillating under the influence of a force 
with the potential: 
mr 
20 
4 dr Saba 
2 
Dp : 
er doy) fa eons. AE) 
2mr? 
over a fixed straight line between two extreme values of 7 ("4 > 
rp >), For this periodic motion (of one degree of liberty) however 
we have according to §§ 4 and 5 the adiabatic invariant: 
1 
N= 
1 . . . ° ee 
== fan dp, = adiabatically invariant. . - (22) 
’ 
‘ 
1 
1 
Equation (19) may be formulated in an analogous way: 
27, 
Vr, 
= f fen. dp, = adiabatically invariant. - « (28) 
1) Perhaps this will be possible by considering the conica! pendulum or a system 
acted upon by a magnetic force. As to this uncertainty comp. § 9 of this com- 
munication and § 3 of communication C. 
