of Quasi-permanent Systems of Electrons. 441 
let us assume a width of one millionth of a wave-length in 
round numbers. Then we may say, that if the red cadmium 
line be due to one of the free vibrations of a system of rings 
of electrons, the structure of the system, as determined by 
the velocities of its rings, can alter by as much as one 
millionth during the emission of the line, that is during 
one million periods ; otherwise the line would be broader than 
is really is. Thus the greatest change of velocity consistent 
with the observed homogeneity and fineness of spectrum-lines 
is of the order 10 ~ 12 in one period of the red cadmium line, 
that is in 2 . 10~ 15 second ; in all probability it is much 
smaller. The perturbations admissible in our system of rings 
must be consistent with this upper limit, that is, the rate of 
loss of energy caused by them must be less than 10 ~ 12 of the 
energy of the system in one period. 
§ 6. It is to be noted that the rate of change of structure 
just calculated is the greatest possible consistent with ob- 
servation. If we assume that it actually exists, we thereby 
make a special hypothesis as to the emission of spectrum- 
lines. For the loss of energy implies a diminution of the 
orbital velocities of the rings of electrons, and a corresponding 
diminution of the free periods of the system. When the 
system begins to radiate it emits the most refrangible part of 
the line, the light emitted becomes gradually less and less 
refrangible, and after one million vibrations or so the emission 
ceases entirely. The system cannot again emit red cadmium 
light until by some external agency its internal energy has 
been restored to its original value. Thus we are compelled 
to suppose (1) that only a fraction of all the atoms in a 
radiating gas at any instant emit any one spectrum-line ; 
(2) that the atoms begin to emit red cadmium light, and only 
begin to do so, when- their internal energy reaches a perfectly 
determinate upper limit, and cease to do so when it falls to a 
certain lower limit ; while they emit no energy whatever for 
a considerable range outside these limits. There is sufficient 
evidence in favour of the first supposition to make it appear 
reasonable : but it is so difficult to construct a mechanism in 
accordance with the second, that for the present we shall 
assume the radiation from our system to be much smaller 
than the limit just calculated. 
§ 7. The question now arises: What types of groupings 
of electrons, and what arrangements of groups are consistent 
with small radiation, implying the absence of all perturbations 
for which £=0, ±1? 
Let us consider the field due to a circular ring, which is of 
course given by the general expression (2). In this case, 
