of Quasi-permanent Systems of Electrons. 467 
we call a system " permanent " when it lasts sufficiently long 
to satisfy the observed conditions as to constancy of mass and 
of other properties. 
Such a system may be stable or not in the ordinary sense ; 
that is to say when disturbed in any way, the disturbance 
may diminish and the system come back to its original con- 
figuration, except in so far as that configuration has been 
altered by radiation ; or the disturbance may increase until 
the system falls away altogether from that configuration. 
Whichever happens, we call the system " stable/'' when the 
progressive change in it, produced by terms of the second 
order in the disturbance, is so slow as to permit of the emission 
of trains of many waves and of the production of spectrum- 
lines of the fineness actually observed. 
When the structure of the undisturbed system is fixed by 
the hypothesis (A) of the expanding electron, its free periods 
of vibration are determinate, although subject to secular 
changes. If it be stable it can produce fine spectrum-lines 
when disturbed, and the observed width of the lines must be 
accounted for by means of secondary causes, Doppler effect 
and the like. But if the hypothesis (A) be not adopted, then 
a part of the width must be ascribed to progressive changes 
of period due to unbalanced radiation. We have seen that in 
any case a degree of permanence sufficient to allow of the 
production of fine spectrum-lines can only be expected from 
systems built up of coaxal circular rings of equidistant 
electrons, because other systems involve radiation, on account 
of mutual perturbations of their electrons, which cannot be 
balanced by the energy set free by the expansion of these 
electrons. 
As regards the particular model of ISTagaoka, we have seen 
that it can give a large number of spectrum-lines, but not 
sufficient to account for a whole spectrum by means of the 
free vibrations of a single system, however numerous its 
electrons may be. Further, this model is far too unstable to 
permit of the production of trains of waves long enough to 
account for the observed phenomena of interference with 
great path differences. 
Physical Institute, Bonn, 
" April 29th, 1907. 
