466 Mr. G-. A. Schott on Frequencies of Free Vibrations 
by the number of waves in a train. If this be supposed to be 
only ten thousand in place of one hundred thousand, the limit 
for A is increased to 10" 650 . 
§ 38. By the ordinary theory of resonance we know that 
a periodic disturbing force excites every free vibration of the 
system on which it acts to a greater or less extent, unless it 
be localized at the nodes of one of the free vibrations, which 
in that case is not excited at all. But the waves which act 
on our system are not so localized, and necessarily excite the 
vibrations of instability as well as ail the others; so that 
amplitudes greater than 10 — Bo0 °, or for that matter greater 
than 10 _6o °, are certain to occur. It follows that Nagaoka's 
model cannot give trains of as many as 10 5 or even 10 4 waves; 
and that interference phenomena with large phase differences 
are impossible for the waves emitted by it. Hence it cannot 
account for optical phenomena by means of its free vibra- 
tions, quite apart from the difficulty of accounting for fine 
spectrum- lines in sufficient number. Nevertheless its study 
is extremely instructive, for on account of its simple struc- 
ture it is possible to obtain numerical results and deduce 
definite conclusions. Besides, the interesting properties 
studied by Nagaoka in his later papers for the most part are 
not due to his particular assumption of a central positive 
charge, but to the arrangement of the negative electrons in 
rings, and so may be expected to belong to other ring systems 
also. 
It should be particularly noted that the arguments of 
§§ 33-37 apply to all systems possessing frequency equations 
with complex roots. We may therefore conclude that in 
general no system can account for longwave-trains unless the 
imaginary part of each of its frequencies is vanishingly small 
(of order 10 ~ 5 or so) in comparison with the real part. 
§ 39. It will be convenient here to summarize the con- 
clusions to which we have been led during the course of this 
investigation. 
In the first place, we have been led to assign new meanings 
to the terms "permanence" and "stability." Xo system 
which includes electric charges in orbital motion can be 
absolutely permanent, for orbital motions always imply 
radiation of energy. By the introduction of the hypothesis 
of an expanding electron we can, it is true, supply the loss of 
energy at the expense of the intrinsic energy of the electrons, 
and so give the system, as it were, a permanence of a higher 
order, which we may call secular; but obviously sooner or 
later the structure of the system must be changed. Therefore 
