— 
. mt. 

Particles illustrating the Line and Band Spectrum. 451 
where 
w w(L+N+2K 
ee 
(434 

par fuss UE AR 
Remembering that L, N, and M are all expressible in a 
series of the form 
ay tah +ahit+ ..., 
a, being absent for the last two quantities, we find the value 
of n for + sign in (13) 
Roy t Ae BAAR eee) 
where A>0O and B<0O generally, and @) is nearly equal to 
w. The expression for n can also be put in the form 


@ 
7 ti bse Sette 14’ 
ail JSatbh?+ch'+... Cm 
where a is nearly equal to unity, and <0 and nearly pro- 
: v2 
portional to a 
Here we notice that waves of frequency x travel round the 
ring in opposite senses, so long as the particles are not acted 
on by extraneous forces. The frequency increases as h is 
increased, and the nature of the series shows that the spectral 
lines corresponding to these vibrations will gradually crowd 
together when / is large. The qualitative coincidence of the 
above result with the line-spectrum is at once evident, if h 
be not small. Diagrammatically represented, the arrange- 
ment of the lines will be as in fig. 1 already given, with h 
counted in the opposite sense. 
An important difference between the present result and 
the empirical formule must not be forgotten. When h is 
very great, the difference between successive lines begins to 
diverge, but im the empirical formule given by various 
investigators in this field of research, the frequency ultimately 
tends to a limiting value. It seems doubiful if very large 
values of h will ever be observed. 
In the present case, the particles must be very smal] com- 
pared to the attracting centre, in order that the ring may 
not ccllapse, when disturbances corresponding to large values 
of h are propagated round the ring. 
Let us now compare the line- and the band-spectrum of the 
212 
