462 Mr. G. A. Schott on Frequencies of Free Vibrations 
TT 
occurs for & = 1, and is equal to 2K — \ cot — , that is, approxi- 
/ 0'12\ 
mately equal to 0*733 . ni Log n— 0*162 ^ -f- J. Using the 
minimum values of n given in the table of § 28 we find, for 
v = 10, 100, 1000, 10,000, 100,000, 
a/^>*90. -38, -15, -07, -05. 
The method of reckoning here adopted is the one which 
is most convenient for our purpose ; we compare the increases 
in amplitude of the waves, not after one revolution of the 
ring, but after one period of each wave, the period being 
of course less for the waves with the greater number of 
loops. With the former method of reckoning the waves 
with the greatest value of k are the most dangerous, with 
our present method those for which k = l. The latter is 
obviously more favourable to the ring than the former. 
The physical interpretation of the numbers given is obvious; 
in a system of 10 electrons an initial disturbance of the type 
considered is at least doubled in '77 period, in a system of 
100,000 electrons in 13*9 periods. In the usual sense we 
should say that the ring is unstable. 
§ 34. To this conclusion Nagaoka * raises two objections i 
I 2M\ 
(1) Since the frequency I w ) is small, the vibration in 
question does not enter into the system in general. 
(2) The analysis only holds for small oscillations. 
2M 
As to the first objection, we must remember that co 
measures the frequency relative to the ring ; the frequency 
of the corresponding emitted wave is the absolute frequency 
/ 2M\ 
«i f k + - — ) ; the external waves which most strongly excite 
the dangerous vibration have the same absolute frequency. 
Any external wave of nearly the same absolute frequency 
will excite it to a certain extent, the more strongly the more 
nearly coincidence of frequencies is approached. Now the 
system emits orbital waves of frequency more or less nearly 
equal to keo (§ 25), which are in fact assumed by Nagac-ka 
to account for spectrum-lines. Hence when the system is 
not isolated, but surrounded by a large number of similar 
systems all emitting these waves, it is necessarily subject 
* Nagaoka. Tokyo Proc. vol. ii. No. 17, p. 4. 
