440 Mr. G. A. Schott on Frequencies of Free Vibrations 
due to waves of small amplitude travelling round the ring. 
When the displacements of the ith electron, measured from 
its position in the quasi-permanent motion, are proportional 
to the real part of exp. — ict+y/ — 1 . Iqt— k- — )L the 
forces of the field consist of an infinite series of harmonic 
terms, the frequencies of which are q + ^ -, where s 
takes all integral values from -co to +00; of these the 
most intense are given by 5 = 0, and to these the radiation of 
energy is almost entirely due. For the case of a circular 
orbit the rate at which energy is lost by radiation through 
the sth harmonic is relatively of order J 2m l -=-*- I, where p is 
the radius of the orbit, \ the wave-length of the harmonic, 
and m=k-\-sn. For the harmonics m = 0, m= +1, the order 
is about the same ; for greater values of m it is much less, 
diminishing, as m increases, faster than a geometrical pro- 
gression whose ratio is (-.--) , that is, about 1 : 1,000,000 
For the general case the intensities have not been worked 
out, but there is every reason for expecting essentially the same 
result. We conclude that in general any disturbance of 
allowable amplitude (not greater than the distance between 
neighbouring electrons), for which &=0, +1, gives rise to 
appreciable radiation. 
§ 5. Let us now consider a system of rings or groups of 
electrons. As we have seen, each ring, on account of its 
permanent motion, emits waves which disturb the other rings; 
and similarly is disturbed by them. It executes forced 
vibrations and radiates energy. If the system be a solitary 
one, the whole of this energy is lost to it, and hence the 
system cannot be permanent ; but if the structure is such 
that the forced vibrations of each ring correspond to values 
h other than the values 0, ±1, . . ., the loss of energy is small, 
and the system is nearly permanent. We must bear in mind 
that for spectroscopic purposes an absolutely permanent 
system is not necessary; it is sufficient if we account for the 
degree of homogeneity and the fineness of spectrum-lines 
as we know them. Lummer and Gehrcke estimate that 
interference with one million wave-lengths difference of path 
can be realized with the red cadmium line. Further, the 
width of this line when at its finest is of the order •01A.U., 
of which the greater part is perhaps due to Doppler effect ; 
