of Quasi-permanent Systems of Electrons. 465 
With this value we find that the factor of co 2 becomes 
1 2N + M 2Kt 
very nearly, the second term of which is very small compared 
is small 
(v-K)e 2 
2N + M 
with A 2 e 2 % since -^— — ^ is small. Hence we get verv 
nearly *{v-&) 
77ip 3 (l-A 2 e 2K 0i 
(25) 
Ke 2 
for in the small term — g we may without appreciable error 
put 1 — A 2 6 2K * in place of unity. 
Comparing (25) with (10) of § 23, we see that the effect of 
the second and higher order terms is the same as if the area 
of the ring underwent a progressive diminution, its value 
after a time t from the beginning of the disturbance being 
less by the fraction A 2 e 2 "*. The effect on co is to produce an 
increase of nearly f of this amount ; and since every frequency 
is proportional to co, the frequency of every line increases by 
the same fraction of its initial value. This change is obviously 
not one which can be counteracted by the expansion of the 
electron on hypothesis (A), because it is a progressive 
change while the latter is secular. 
§ 37. At this stage we introduce the condition of fineness 
and homogeneity of spectrum-lines (§ 5). In order to be on 
the safe side let us only assume trains of 100,000 waves, and 
lines of 1 A.U. width, and let us neglect the part of the width 
due to Doppler effect. In other words, we assume that the 
frequency of the lines does not change by more than one 
ten thousandth of its value during one hundred thousand 
periods. Is this consistent with equation (25) ? 
/3N 1 
By § 33 the value of k is \ f -p- 7^, which for a system of 
0*15 
1000 electrons is at least equal to ^~. Hence for t = 10 5 . T 
we nnd 3_A 2 e 2 "* >-2-A 2 . e 3 * 104 > A 2 . 10 13000 . 
This must not exceed 10" . Hence we find 
A<10" 6500 . 
Obviously the value of A is not appreciably affected by the 
fineness of the lines, but only by the value of k, and especially 
