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XLIII. On the Frequencies of the Free Vibrations of Quasi- 
permanent Systems of Electrons, and on the Explanation of 
Spectrum Lines. Part I. By G. k. Schott, B.A., B.Sc, 
University College of Wales, Aberystwyth *. 
§ 1. TN a previous communication f I have shown that a 
A ring o£ electrons, rotating in afield due to electrons, 
all of which exert only electromagnetic forces, has a prac- 
cally determinate radius and velocity, provided only each 
electron be expanding at a very slow rate, uniform or 
not. With the aid of expanding electrons we can build up 
a purely electromagnetic system possessing a determinate 
structure, and therefore also determinate free periods. 
Secondly % I have examined the waves emitted by a 
rotating ring of equidistant electrons, when disturbed from 
steady motion in any way, and have shown that of all the 
free vibrations, which can be excited in such a ring even by 
violent disturbances, only a few can produce waves sufficiently 
powerful to give observable spectrum-lines. It follows that 
only a small proportion of the free vibrations of a system of 
electrons can be used to account for spectrum-lines. In all 
probability a similar limitation exists for other vibrating 
systems ; but as they do not so readily lend themselves to a 
calculation of relative intensities, it has hitherto escaped 
notice. It is indeed hardly conceivable that this difficulty is 
peculiar to systems of electrons. 
§ 2. In order to account for known spectra we must study 
more complex systems of electrons ; but if, for the present, 
we confine ourselves to quasi-permanent systems, that is to 
systems which can last for very many periods of vibration 
without appreciable change of structure, the problem is very 
much simplified. In fact such systems are necessarily built 
up of circular rings of equidistant electrons, relatively far 
apart, and each rotating with its own determinate velocity 
about a common axis. This may be seen as follows. 
The electromagnetic field due to an electron which describes 
a closed orbit with period T can be expressed in the usual 
way in terms of a scalar potential <f> and vector potential A 
by the equations 
E=-v$-p^7, H = curlA. 
* Communicated by the Author. 
t Schott, Phil. Mao-. [6] vol. xii. p. 21. 
% Phil. Mag. [6] vol. xiii. p. 189. 
