i 3 6 THE QUANTUM [ x . i 
* H. S. Allen, Proc. Roy. Soc. Edin., vol. 41, p. 34, 1921. 
Thus, as we found previously in dealing with the ring 
electron, the number of magnetic tubes is n (^’ an d is an in 
tegral number of times the constant quantity h/e, which we take 
to define the unit quantum tube. 
It may be pointed out that here, as in the electronic theory 
of magnetism developed by Langevin, it is assumed that the 
classical electron circulates with very high frequency, and may 
be treated as equivalent to a continuous current giving rise to 
a magnetic field similar to that produced by a steady current. 
In a paper read before the Royal Society of Edinburgh in 
1921 the author * gave a more general proof of the principle of 
Bernoulli, with the object of avoiding as far as possible par 
ticular assumptions as to the character of the electrical distribu 
tion. A system was considered composed of a number of point 
charges rotating about an axis with the same uniform angular 
velocity, co. 
Although the model employed may differ widely from an 
actual atomic or molecular system, the method of calculating 
the angular momentum is of some interest and will now be 
indicated. 
The equivalent mass per unit volume of an electrostatic tube 
is 4^D 2 sin 2 0, where D is the electric polarization or displace 
ment, and 6 is the angle between the direction of the tube and 
its velocity. Hence the angular momentum for unit volume of 
the tube is (471/nD 2 sin 2 6)r 2 co. The moving Faraday tubes are 
accompanied by a magnetic field, at right angles to their length 
and to their direction of motion, given by H = (4ttD sin 6)rco. 
Hence (D sin 6)r — H/4nco, and the angular momentum for unit 
volume of the tube may be written 
/ H \ 2 _ 2 f/tI 2 
co 8n 
dJCUCOl ■ ) 
\A7ZCO/ 
10 :7 
Hence the total angular momentum of the system takes the form 
2 uH 2 
w ¡StT’ summa ^ on extending over the whole space occupied 
by the magnetic tubes. 
mH 2 
The sum — represents the electrokinetic energy, and may 
on 
be expressed in the form + . . + M ia «Vi + • • •» where 
Lj is the self inductance for the circuit carrying a current i x = 
M la the mutual inductance for the circuits i lf i 2 , etc.