of Quasi-permanent Systems of Electrons. 453 
§ 21. Nagaoka's model *. 
The model consists of three parts : 
(1) The ring of n equidistant negative electrons each of 
charge e and mass m. 
(2) A central positive charge ve of mass M. To ensure 
the limited stability required by § 5, v must be very large 
compared with n. 
(3) A swarm of negative electrons sufficient to make the 
system neutral on the whole. According to § 9 perma- 
nence requires them to be grouped in rings coaxial with the 
ring (1). Stray electrons will either escape from the system, 
to be shortly replaced by others, or will owing to radiation 
lose their kinetic energy and coalesce to form rings, or join 
rings already present. Since the system cannot long 
remain charged without attracting negative electrons, when 
it is positive, and repelling them, when it is negative, it 
must on the average during its existence be very nearly 
neutral. 
Nagaoka considers it likely that his system forms a flat 
disk or ring, all the negative electrons crowding towards the 
invariable plane. He appears to assume that the system as a 
whole can be permanent ; but there appears to be some 
doubt, whether a flat disk or ring of many electrons moving 
with such small velocities as are necessary for permanence, 
under the influence of forces acting inversely as the distance, 
can be stable at allf. 
§ 22. It must be noted that in deriving his results Nagaoka 
neglects the field due to the swarm of negative electrons (3). 
This is strictly correct when the swarm forms an elliptic 
homoeoid completely enclosing the ring ; but JSTagaoka con- 
siders it more probable that it forms a flat ring, approxi- 
mately in the plane of the ring (1) . The total charge of the 
swarm is that of v=?? negative electrons, the whole system 
being neutral, and is herefore comparable in magnitude with 
the central positive charge. Hence its action cannot be 
neglected without further investigation. In order to form 
same idea of its effect we may treat it as an elliptic cylinder 
of the same cross-section, the long axis of the section lying 
in the plane of the ring. 
Let the long axis of the section be 2a, the mean radius of 
the swarm c, the short axis negligibly small. We easily find 
for the radial force on a negative electron, at a distance p 
from the axis in the plane of the ring, a repulsion from the 
mean line (c) of the swarm, which inside the swarm is equal to 
* Xagaoka, Phil. Maof. [6] toI. Tii. p. 445 : Tokyo Proc. yoL ii. 
nos. 17-21. Schott, Phil. Mag. [6] Vol. viii. p. 384. 
t Pellat, Comptes Rendus, March 4 and April 8, 1907. 
