Electron Theory of Matter. 179 
then we may write, between these limits, 
7 v tt J.TT8Z 
A= 2 Hs exp. i — — , .... (10) 
where the coefficients are small for large values of s, and the 
series may be supposed differentiable. The axial magnetic 
force, which alone concerns us here, is got by multiplying by 
cos 0, and gives rise to a mechanical force to the centre 
— pelt cos 6 (§ 8, (6)). Since however the magnetic force is 
changing, an induced electric force E is produced along the 
circle, which causes a tangential mechanical force eE. To 
find it we apply .Faraday's Law of Induction to the circuit 
made up of the orbit of the electron from t = to £ = t, 
together with so much of a circle and radius as are necessary 
to make a closed circuit. This circuit differs from the 
original circular orbit, described an integral number of times, 
by quantities of the order of the disturbance ; since it occurs 
on both sides of the equation with 7i, or E, as a factor we 
may substitute the circle for it without making an error of 
more than the second order. Thus we get, from the general 
equation of Maxwell's theory, 
E =-fJ( 7tC0S *> tii) 
§ 12. A great simplification ensues because we may gene- 
rally suppose t very large compared with the time of 
revolution of an electron. In fact r will rarely be as small 
ms 10 -6 second — the value for a coil of radius 1 cm., length 
100 cm., with 1000 turns and of resistance 10 ohm would be 
about 4.10 -5 sec. — while the period for an electron describing 
an orbit of atomic dimensions with a velocity one thousandth 
of that of light is about 2.10 -15 sec. Since such a velocity 
is already very small to be consistent with stability for a ring 
of more than 3 electrons (in J. J. Thomson's well-known 
model the lower limit is in general several hundredths of 
the velocity of light), our assumption appears to be amply 
justifiable on experimental and theoretical grounds for any 
but the smallest rings. 
Hence if q be the frequency of any one of the harmonic 
components of the disturbance, - and a fortiori ^?, is an 
exceedingly small quantity. ^ *° 
We shall now develope the equations of the disturbed 
motion ; as the full investigation is long it will not be pos- 
sible to do more than indicate the method. We first use the 
equations of motion of an electron under any forces * to get 
* Abraham, Eleldrizitiit, vol. ii. p. 123. 
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