= V 
« = s 
_ v 
442 Mr. G. A. Schott on Frequencies of Free Vibrations 
using polar coordinates, we may write 
R = y/ r * + }•'* — 2 n J sin0 cos (<//— <£), 
with r' = constant, cf)' = Q)t / + S, and easily find on changing 
the variable of integration from £' to % = <£>' — $, 
-cos »<©*+ 5-^)1 ^ ^cosx^% 
+ 2 — sin »«(©* + 8-<J>)| g- — ^cos^x, • (3) 
where R = \/r 2 + r /2 — 2rr t sin cos %. 
Thus t occurs only in the circular functions, so long as we 
are dealing with a fixed point (r, 0, <£). To find the mecha- 
nical force on an electron we differentiate as usual with 
respect to the coordinates (r, 0, <£>) and the time (t), as the 
case may be. When the electron is moving, the coordinates 
are given functions of t ; but these values of the coordinates 
as functions of t are only to be substituted after all the 
differentiations necessary in deriving the forces have been 
performed. 
§ 8. When the electron in question belongs to a second 
circular ring of n' electrons, rotating with angular velocity co f 
about the same axis as the first, we substitute $=(dH + 8' + — r , 
and thus find for the potentials and forces series of harmonic 
terms of the form g j n ns j (w — o)')t— — j- > . The pertur- 
bations produced are of the same type, which is obviously 
that of § 4 with /c = 0, q=ns(<a— to 1 ), k = ns ; and the emitted 
waves are given by m = ns + n's', where s' is any positive or 
negative integer. The radiation is small provided ws + wV 
cannot take the values 0, +1, zero values of s, s' being- 
omitted, since they imply zero frequency and no wave at all. 
It is at once obvious that the values 0, + 1 will occur ; if 
P n 
n, n' be incommensurable, and ~ be the last convergent to -y 
we need only make s=n', s / = —n to get m=0, and s= +Q, 
s ' = ipp to get m= +1. Thus we cannot entirely avoid the 
dangerous harmonics in question. 
We can, however, ensure that their amplitudes are small. 
