454 Mr. Gr. A. Scbott on Frequencies of Free Vibrations 
2(v-n)e> c^p . . 2(r-tt)e 2 
-. and outside is equal to / ; 
ca a ^ c(t — p+ ^(c^pY'-cr) 
ve 
The radial force due to the central charge being - 2 , the 
action of the swarm is only negligible when its mean radius 
(c) is large compared with the radius (p) of the ring. 
§ 23 . On the assumption that the velocities are small 
enough to allow us to neglect the effects of radiation, we 
find:- 
(1) for the equation of steady motion, 
<o»=<^; (10) 
mp° K ' 
(2) for the frequency equation for axial vibrations, 
mp z 
(ii) 
(3) for the frequency equation for orbital vibrations, 
, f , (N-L-2K>n , Me 2 
1 \_ mp 6 J x mp 6 
(L-f 2K> 2 ~\ _ /M^f\ 2 _ (12) 
_Kg 2 
mp* 
By means of (10) equations (11) and (12) may be written 
in the forms 
a, 2 v-K' 
p 4 tf f N-L-2K\ . v 4M 
or 
w v — K 
_JX _NCL+_2K )t M 2 = 
v-K (^-K) 2 ( - 
These equations no longer involve p explicitly, and the 
coefficients of — are functions of v and n alone. This cir- 
&> 
cumstance makes Nagaoka's system extremely simple ; but it 
must be distinctly noticed that the simplification is only 
obtained at the expense of neglecting the effect of the negative 
swarm (3), and therefore applies only to rings which are small 
compared with the whole system (§ 22). 
§ 24. In order to calculate the roots of these equations 
Nagaoka assumes that n/v is so small that the roots can be 
expanded in ascending powers of the quantities J/v, . . . 
