446 Mr. Gr. A. Schott on Frequencies of Free Vibrations 
When the ring is slightly disturbed from steady motion 
the displacements of the ith electron, in the direction of 
motion, towards the centre and parallel to the axis, are denoted 
by (£, 77, J) ; they are measured from the position which the 
electron would have occupied at the same time in the steady 
motion, not from a point fixed in space. Thus when disturbed 
the coordinates of the electron are 
js = (p—ti)cos(<l> + l;/p), y = (p-v)sin (<£ + £//>), £=£ 
Squares and products of the displacement are neglected in 
the equations of vibration (not of course in the energy). 
This is substantially the method of representation used by 
Maxwell in his paper on Saturn's Rings, and by Nagaoka 
and J. J. Thomson in their investigations. 
§ 15. We may resolve the disturbance (?, y, f) into a 
series of harmonic components proportional to terms of the 
form exp. i(pt — k- — J, where k is an integer, and p is a 
complex constant of the form q + lk. 
This harmonic represents a wave with 2k nodes and 2k 
loops, travelling round the ring with angular velocity q/k 
relative to the rotating ring ; for, apart from the damping 
factor, the displacements are unaltered when we increase i 
by n, and t by 2irk/q. The angular velocity of the wave 
relative to fixed space is co + q/k; accordingly q is the fre- 
quency relative to the rotating ring, as it appears to an observer 
revolving with it ; q + kto is the frequency relative to fixed space, 
as it appears to a stationary observer. We may speak of q as 
the relative frequency, of q + k<o as the absolute frequency, 
or the frequency simply. 
Accordingly we notice that the waves emitted by the 
disturbed ring into the surrounding medium on account of 
the disturbance (q, k), consist of a series of simple harmonic 
waves with frequencies given by the formula q + (k + sn) co, 
s any integer (§4). 
It is obvious that we can obtain all the different types of 
disturbance possible, either by giving k every integral value 
and making s = 0, or more conveniently, by giving k n inde- 
pendent integral values, and s all values in turn. We shall 
select the values 
, n — 2 7i — 4 ., . ., n — 2 n . 
k — g-, g-,... — 1, 0, +1,.. + -j-> + 2' n n 1S even ' 
7 n — 1 n — 3 -1^1 . n — 3 w — 1 , .11 
£= — — ^— , y~ , . .. — 1, 0, +1, . . . + — ^— , + —^j when n is odd. 
