Frequencies of Free Vibrations of Electrons. 439 
At a distance from the electron large compared with its 
radius 
(1) 
where R is measured from the position of the electron at 
time t', and u is its velocity at that time. Thus 
R = V^(«-*') s + (y-y)»+(*- z'f, 
where the coordinates (a/, y\ z') of the electron are assigned 
functions of t' of period T. 
The electric and magnetic forces due to a single electron 
thus consist of series of harmonic terms, no member of which 
in general is missing. The energy radiated away is therefore 
very considerable, and the motion can only be permanent for 
a very small velocity. We know from Earnshaw's Theorem 
that systems of discrete electric charges at rest are unstable : 
in general a minimum velocity is necessary for stability, but 
this is usually inconsistent with permanence. Hence we 
conclude that a system including stray electrons, each de- 
scribing its own orbit independently of the others, is not 
permanent and stable. Ultimately these electrons either 
escape from the sjstem or combine into groups, describing 
the same orbits. 
§ 3. In order that a group may be quasi-permanent a 
number of the harmonic terms in the forces due to its several 
electrons must annul each other by interference. The con- 
dition for this is easily seen to be, that the coordinates of the 
ith electron of the group be given by equations of the form 
*'='C + ?> *'-'(" + ?} --"('' + ?> 
The potentials then reduce to series of the form 
*=« ne f T 1 2irns / R \ . 
4> = 2 y) r C0S ~TV c — t ) dt ^ - - ( 2 ) 
where R is now measured from any one of the n electrons. 
The first n — 1 periodic terms are now missing from the 
potentials ; the largest periodic term left is that given by 5 = 1, 
or by j = n. 
§ 4. A similar interference occurs when the field is that 
