1194 
dinates the one nearest the axis of abscissae) is equal to /,, that of 
the (n + 1) st stationary orbit will be 
[rig = hy + nh oo Lae ay ee aie ne 
h, has therefore to be determined, in order that all the stationary 
curves be fixed. 
For this purpose Pranck') lays down the principle, that the 
narrowest orbit must coincide with the natural boundary of the 
phase-plane; i.e. if on any grounds, connected with the nature of 
the system, the integral (1), which is essentially positive, cannot 
fall below a definite value, the latter has to be taken as /,. In 
most cases a lower limit of that kind does not exist and the integral 
may be taken equal to zero, whence 
Krk nd es 
In his treatment of the relativistic Kepler-motion SOMMERFELD ”) 
found the case to be different; he there gave a lower limit p, = xe°/c *) 
for the constant azimuthal impulse; this would give h,=—= 2a p,. It 
would therefore, as pointed out by PranckK, be necessary to take (2) 
as the fundamental relation, whereas experiment (the Balmer-series) 
can only be reconciled with supposition (2’). SOMMERFELD ‘) tried to 
remove this contradiction by pointing out, that when the motion of 
the nucleus is taken into account the numerical value of the limit- 
ing impulse is smaller than xe°/c. In what follows we hope to prove 
that the limitation of the phase-plane by the value p=p, is only 
an apparent one, even if the motion of the nucleus is left out of 
account, and that p can very well fall below this value: at the 
same time the character of the motion is then essentially changed. 
The admissibility of stationary orbits of azimuthal impulse p = 0 
which on SOMMERFELD’s theory seemed to be excluded is thereby 
proved in principle. As long as we are dealing with attractive forces 
(nucleus and electron) these orbits are hardly of practical importance, 
as they must lead to a collision of electron and nucleus. But the 
case changes, when the forces are repulsive (nucleus and a-particle) ; 
the orbits are then hyperbolic. If the quantization of such orbits is 
admitted, interesting physical conclusions follow which appear to 
1) M. PrANCK. Ann. d. Phys. 50, p. 385. 1916. 
%) A. SOMMERFELD. Ann. d. Phys. 51, p. 57. 1916. 
8) Here e is the charge of an electron, ze that of the atomic nucleus and 
c the velocity of light. 
4) A. SOMMERFELD. Miinchener Ber., p. 137, 1916. 
