1195 
give an explanation of certain recent experimental results of RurnEr- 
ForD’s’). The question raised by the author?) before as to the 
quantization of non-periodic motions is therefore put once more 
and discussed in a different manner (§ 4, 7). 
§ 2. The apparent boundary of the phase-plane p= p,. 
The relativistic Kepler-motion is given by the following equation 
between the polar coordinates r,@ (cf. lc. p. 819). 
1 B Vp—p, 
tS =| te eo 5 me p=) | nen 
r P —Po P ~ 
with the abbreviations 
a 
BS spe dre 1, ssi SS aia are idl 
A=a(S+ 2m) ; ie 
c 
a represents the energy of the system, c the velocity of light, 
m the mass of the moving particle. The positive sign of B refers 
to the case of attraction, the negative sign to repulsion; p‚ is the 
azimuth of the radius vector with respect to the aphelion. 
For p >p, with negative energies (A < 0) and attracting forces 
(B > 0) the orbit is an ellips with perihelion-motion. The procession 
of the perihelion increases in speed, the smaller the difference p?— p,’, 
and in the limiting case p =p, the orbit converges on the nucleus 
in a manner similar to an Archimedian spiral’): 
ay 
ree vie : 
an PP, EB 5 . ° ‘ 5 ° . (5) 
But nothing prevents us from now taking p< p,; the expression 
(3) then assumes the form: 
2 2 
5 Bat a nde et ey ies ae rand cng 
eM Dd at sare | 4 
The right-hand side of this expression for a very large positive 
or negative value of ~ becomes exponentially infinite independently 
of the value of the excentricity «. The two extremities of the orbit 
thus approach logarithmic spirals. It further follows from (4) that 
') E. RuTHERFORD. Phil. Mag. 37, p. 537, 1919. 
4) P. S. Epstein. Ann. d. Phys. 50, p. 815, 1916. This paper will be 
quoted here as l.c. 
5) Comp. A. SOMMERFELD Ann. d. Physik. 51, p. 50 1916. 
Ta 
