of Quasi-permanent Systems of Electrons. 461 
If M be the mass of the positive charge and A the atomic 
weight of the atom, we find at once 
A=-^-+ M . . (20) 
1700 1700m' [ } 
so that 1700 A is an tipper limit to the value o£ v. 
Thus we find that a system, representing the H atom in 
mass, cannot give more than 92 lines (1), or 130 lines (2), 
in all. 
Similarly, a system representing the Fe atom, for which 
v< 95,200, cannot give more than 376 lines (1), or 1560 
lines (2). 
These numbers include all the lines that can be given by 
the system under all possible conditions, whether they occur 
in series, or in bands, or as stray lines. When we consider 
that they are upper limits for the selected values of p and v, 
while these selected values are themselves in all probability 
chosen too high, we are driven to the conclusion that a single 
system, constructed on Nagaoka's model, cannot account for 
spectra as we know them ; but we cannot on these grounds 
reject the possibility that such a system may account for a 
part of a spectrum, e. g. a band or series, or even for a single 
spectrum of the element. 
To decide this question we must consider the stability of the 
svstem. 
§ 33. In § 21 we saw that each of the n sets of orbital 
vibrations includes a vibration of instability, that is a vibration 
whose component vibrations in the plane of the orbit (£, 77) 
'. , , Kt I . 2nd \ .,, /3N 
are proportional to e f cosl qt — k +a , with k = (o\/ — • 
The frequency of the emitted wave is q=(olh+ - — ), prac- 
tically kco ; its period (T) is 'Ivr/ha. Hence /c=\/— -.— 
which means that during each period the amplitude is 
multiplied e^3N/*-v times. 
By § 20 we have 
n/* 2 = 2K = 2~ ' (; -;r i)2 cot fot 1 ^ 
s-0 k" 2;i 
for h >0, zero for k = 0. The greatest value of N/& 2 obviously 
Phil. Mag. S. 6. Vol. 15. No. 88r April 1908. 2 I 
