of Quasi-permanent Systems of Electrons. 445 
the ring must be considered in every calculation of the ratio 
of the wave-length to the radius. 
§ 13. We shall now study the frequency vibrations of a 
circular ring of equidistant electrons, which forms part oi 
a system of rings of the type specified in § 9, with a view 
to estimating the wave-lengths, and comparing them with 
those of light-waves. For this purpose it is not necessary to 
prove that the system is stable ; it is sufficient to assume its 
stability and permanence. If it be found that a sufficient 
number of wave-lengths are comparable with those of light- 
waves to account for a reasonably large number of spectrum- 
lines, we may accept the system provisionally as a satisfactory 
model and examine its stability; if not, it must be rejected, 
whether stable or not, as by itself incapable of accounting for 
spectra by means of its free vibrations. If no system be 
found which satisfies all these conditions by itself, we need 
not necessarily reject the electron models as useless ; for we 
never have to deal with isolated atoms of any element, but 
only with the element in bulk. It is quite possible that the 
emission of spectra, as we know them, is a property of 
complexes of atoms, and not of the individual atom. 
§ 14. We shall for the sake of simplicity begin with 
the study of the models proposed by Nagaoka and by 
J. J. Thomson, in which no account is taken of those terms 
in the frequency equations, which are due to radiation, an 
omission implying velocities small compared with that of 
light. We shall then extend our investigations to the general 
case, where the effects of radiation are taken into account, 
and no special hypothesis is made as to the nature of the field 
in which the ring moves. 
Before proceeding to the study of special systems we shall 
mention a few points which concern all systems equally. 
dotation. 
Number of electrons in the ring =n. 
Radius of the ring =p. 
Angular velocity =co. 
Velocity of light =C. 
Velocity of electron/velocity of light =cop/C=i3. 
The azimuth at time t of the ith electron is given by 
2th' 
6=cot-\-8+ — ■; 
T n 
its coordinates in steady motion by 
x = pcos (f>, y = p sin cf>, w = 0. 
Phil Mag. S. 6. Vol. 15. No. S$. April 1908. 2 H 
