464 Mr. Gr. A. Schott on Frequencies of Free Vibrations 
distance between neighbouring electrons of the undisturbed 
ring. Even for values of v as great as 100,000 we saw in 
§ 29 that rings of more than 450 electrons can hardly occur ; 
we must suppose the radius to be comparable with 10~~ 7 cm. if 
they are to give observable spectrum-lines. This makes the 
distance between neighbouring electrons equal to 10~ 9 cm. at 
least. Hence the disturbance must be supposed to be appre- 
ciably less than this value. 
§ 36. On this assumption we may neglect the effect of the 
squares of the small terms, f 2 , . . . in the small terms of P, 
and altogether in T. Taking the averages of the equations of 
motion for one period we find 
«V-2»^ + £-:=-, .... (23) 
p p nv 
1^+2^ = 0, (24) 
where the cross-bar denotes average values. 
In averaging we may obviously neglect the variation of the 
factor e Kt during one period, particularly if we suppose t to 
refer to the middle of that period. Hence 
Jv=Tv = 0; 
so that (24), and therefore (22), is still satisfied. 
Further 
ft = -J^ABe 2 * p = i#*BV« 
v _qve 2 C 2 ^ dt ve 2 
27r J o P 2 { l ~ AeKt sin (.& + a ) I 2 p \ X ~ AV**)! ' 
so far as it depends on the central charge. 
Ke 2 
JJ; The part due to the ring is — —^ as before. Thus the 
equation (23) becomes 
a, 2 . ( 1 + 2 ABe 2W + £;Bh"A = 3n ^^ " ^. 
Now equation (22) gives to a first approximation 
2 9 . M 
B 
CO v — K 
A 
q 2 N 
a> 2 v-K 
