of Quasi-permanent Systems of Electrons. 463 
to waves which are capable of exciting the dangerous vibra- 
tions in question to a greater or less extent. 
As to the second objection, it is of course a valid one ; 
but it can be surmounted by taking higher powers of the 
disturbance into account. We shall find that for our pur- 
pose it is quite unnecessary to pursue the investigation so 
far as to decide whether, after a greater or less excursion 
from the circular shape, the ring returns or does not return 
to its original condition ; that is to say, whether it is stable 
or not in the usual sense. We can apply the condition of 
§ 5, that the ring must give trains of a minimum number 
of waves, whose period does not change during emission by 
more than an assigned small fraction of its value. 
§ 35. The equations of motion of an electron of the ring- 
are, for motions in the plane of the orbit, which alone concern 
us here, 
\dt) df m? l L) 
df ^ dtdt nC V " J) 
where P, T are the mechanical forces towards the centre and 
along the tangent. 
When the ring is disturbed we have 
r = p—7j, 6 = cot + tj/p, 
where 7]=pAe Kt sin (gt + a). 
£=pBe Kt cos (qt + cc), 
for free vibrations on the supposition that linear terms only 
need be taken into account. 
For another electron of the ring, the lib. from the selected 
one, we have 
77 "I 
where f ., rj { involve qt + u — k- — in place of gt+a. 
"- ve 2 
The force P involves a term ? ^ due to the attraction 
of the central positive charge, together with a small term due 
to the electrons of the ring from i = l to i=n^- <1; the force 
T involves the latter term alone. Nagaoka's analysis depends 
on the fact that the latter terms are small in comparison with 
the first ; we may assume that this remains true, provided 
the disturbance never becomes very nearly equal to the 
2 12 
