1196 
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for A=0, e=l. Thus with negative energy 7 always remains finite, 
> & 
the particle moves out from the centre and again returns to it. 
When the energy disappears or becomes positive the orbit divides 
into two branches which run from the centre to infinity or vice 
versa. In the limiting case p=—=0 r is only finite for w =p, Le. 
the motion is rectilinear. 
Thus it is seen, that in reality there is not a limit p= p, atall: 
with small positive values of p—p, tbe orbit encircles the centre 
many times, while 7 diminishes, but remains at a finite distance 
from it which passes through a minimum and then increases again. 
For p=p, the curve runs into the centre as an Archimedian spiral. 
The approach to the centre is even more rapid when p< p,, the 
spiral becoming logarithmic. It must not be supposed that the particle 
in its motion on the spiral will permanently remain near the centre: 
for although the spiral encircles the point an infinite number of 
times, its total length is finite and the time to describe it from a 
finite distance, as a simple calculation shows, is also finite and 
practically very small. Therefore the collision will oceur very soon. 
§ 3. Quantization of the spiral orbits. In the last section we have 
shown, that in the relativistic Kepler-motion, even with negative 
energy, besides the ellips-like orbits other forms are possible which 
are of finite length and are only once described. 
The question now arises, whether these motions can be submitted 
to quantum-conditions and in what manner this would have to be 
done. Our answer to the first question is implicitly contained in the 
above discussion: the disappearance of the limiting value /, in 
assumption (2) we have explained by the fact that orbits have to 
be taken into account for which pis less than the azimuthal quantum 
po. It follows that these orbits join on continuously to the others 
and must be equivalent to them from the point of view of the 
quantum theory. Since for p >> p, the stationary motions are given 
by the relation p= nh/2a0 (n=1,2,....), it follows that for p < p, 
the only possible stationary condition is p=QO. This conclusion is 
strengthened by the circumstance that when the movement of the 
nucleus is taken into account (as proposed by SoOMMERFELD) similar 
spiral-shaped orbits have to be considered in order to explain the 
possibility of p= 0: this can be easily shown to be the case. 
We have therefore only to discuss the quantisation of the radial 
impulse: its dependence on the radius vector 7 and on the constants 
of the problem is given by the equation (lc. p. 823). 
