166 
THE QUANTUM [xn. 4 
dimensions. This means that the velocity of the point D 
(Fig. 23) approaches the velocity of light, so that v = aco = c. 
Consequently, co being 2nv, 2nva — c and thus 
c c a 2 c „ 
a — — = = = 3-86 x 10 11 cm. 12:14 
271V 4^33 4^00 T 
A more convenient expression for the radius is obtained in 
terms of the innermost orbit in Bohr’s theory of the hydrogen 
atom. This may be written 
ac 
a 2 — 
4^00 
Therefore a = aa y 12:15 
Assuming e ■= 4774 X io -10 E.S.U., the numerical value for a 
is 1/137. The radius of the electron is 1/137 times the radius 
of this innermost orbit. 
From the relation E = KMco the magnetic moment of the 
elementary magnet is found to be 
m _ I2;i6 
2 nm 
This is twice the magnetic moment of the Bohr magneton, and 
it may be mentioned that it corresponds very closely with the 
value derived from the saturation intensity of magnetization of 
iron, which according to Ewing * is nearly 2 x io -20 c.g.s. units 
per atom of iron. 
The ratio of the angular momentum to the magnetic moment 
for the electron here considered has the value 2m/fie, which is 
equal to the gyromagnetic ratio deduced from the theory of 
electron orbits by O. W. Richardson.+ 
Case II.— In the preceding calculations we have neglected 
the energy associated with the frequency characteristic of the 
tube itself, energy which may correspond, possibly, to an internal 
vorticity. When this is taken into account we have to deal with 
an amount of energy hv r due to the rotation of the whole tube 
about an axis with frequency v T , and also an amount of energy 
\hv a which is due to the internal condition of the tube, and would 
be present in the tube when not rotating, v $ being the frequency 
appropriate to the stationary tube. These quantities are addi 
tive, and as there are two possible directions of rotation we have 
for the energy in one case hv T -f \hv s , and in the other hv r — \hv s . 
As the first amount is clearly greater than the second let us 
* J. A. Ewing, Proc. Roy. Soc. Edin., vol. 42, p. 124, 1922. 
t O. W. Richardson, Phys. Rev., vol. 26, p. 248, 1908; Proc. Roy. 
Soc., vol. 102, p. 538, 1923.