
‘Particles illustrating the Line and Band Spectrum. 449 
The kinetic energy of the particle at point 7, d, = 1s given 
by ag: 16\? 
ary ( «~\- 
T=3n1 (GF) +7 “(SP) + + (@) \. 
By Lagrange’s equation 
do 
di = = So = ma Ge malo sas ia ii) 
fe NCL me 4 
=—- 39 one 5) 
with similar expressions for ¢ and = 
Thus, we obtain as the equations of motion 





eo 5 fg OF. eK 2eK ‘) 
i ee Stages ma? * ma? | 
ae aK +k MB) cos u} | hs 
> 
ve a oF = ae 1a? —* _(MA-+NB) sin u | 
@e— E+ e | 
d2  ma*>' 2ma? a a J 
From the last of the equations of motion (8), we find that 
the frequency of transversal oscillation n’ is given by 
n= +,/S—pl, a PE i aS me bs 
2 
where S stands for us and yw tor iam From the con- 
ma 2ma 
dition of stability, we suppose that E is very large compared 
toe. When the number of particles v is considerable, we 
notice from (6) that J can be expanded in a series of the 
form 
J= A;h? + B ht + 
Consequently, 
+n'=o,/—ajh?+bhi+... . .«. ~ (10) 
where ,' is the principal constant term, and a;>0 and nearly 
2 
- V - > . 
proportional to —. For small values of h, n’lie very near each 
@ 
other, but as / increases, the interval gradually becomes larger 
and ultimately reaches a maximum. The interval between 
successive frequencies decreases as / is increased. Con- 
structing the frequency lines as functions of 4, we find a 
Phil. Mag. 8. 6. Vol. 7. No. 41. May 1904. 21 

