448 Mr. G. A. Schott on Frequencies of Free Vibrations 
Jeans, it is true, obtains the required condition by intro- 
ducing a hypothetical non-electromagnetic force between the 
electrons ; this procedure amounts to giving up the sim- 
plicity which is the greatest advantage of the electron theory. 
But it has another disadvantage ; it is of no use for a system 
in which the electrons are in orbital motion, and is in fact 
employed to avoid the necessity of orbital motions. In con- 
sequence it can only explain spectrum-lines by means of 
dynamical considerations, and gives the relation between the 
frequencies (N) of the lines in the form 
N 2 ==/(m), m an integer (4) 
To account for spectrum series we require the form 
N=/(m), m an integer (5) 
This fact led Lord Rayleigh to suggest that the numerical 
relation between the frequencies of a spectrum series is of a 
kinematical rather than a dynamical character ; for it is by 
no means evident that the equation (4) can, by simple 
extraction of the square root, always be reduced to the form 
(4) in such a way as to agree with observed series to the 
accuracy required by experiment. The reduction has in fact 
been accomplished only in very special cases (cf. the model 
ofW. Kitz*). 
Any system, on the other hand, which admits of the exist- 
ence of orbital motions, for that very reason admits the 
possibility of a kinematical explanation of spectrum series, 
an advantage which ought not lightly to be given up. But 
every such system also requires the hypothesis of expanding 
electrons in order to completely fix its structure. 
§ 18. I have stated elsewhere f that the rate of loss of 
energy due to radiation from a ring of electrons in steady 
motion is approximately equal to 
p 2 
^/gexp. 2re ( 7 -|logi±r), WW^ 
provided n be large ; and to J. J. Thomson's value 
2Cg»n(n+l)(ni8y»+* 
p 2 | 2n + l 
for small values of n, for which /3 also is small. 
* Ritz, Ann. Phys. (4) xii. p. 264. 
t Schott, Phil. Mag. [6] vol. xii. p. 22 ; vol. xiii. p. 194. One or two 
mistakes have corrected been here. 
