Electron Theory of Matter. 177 
inward radius and axis, the values 
(o, -pelt cos 0, -^elt sin 6 cos (at + 8+ ^)\ • (6) 
Since the disturbances produced are o£ the same types as 
the disturbing forces, with the same frequencies and damping, 
but, it may be, displaced in phase, we have only to consider 
two types of disturbance : — 
(1) A disturbance due to the radial force —/3eh cos 0. q, k 
are both zero ; f. 77, f are constant and the same for 
each electron of the ring. 
(2) A disturbance due to the axial force 
/ 27ri\ 
— /3eh sin cos I at + 8+ - — -I, q = o), £=— 1. 
In each case we have q + kco = 0, that is, the disturbance, 
like the force, is stationary for an outside observer. Never- 
theless it is periodic relative to the electrons of the ring, and 
like any other forced vibration may exhibit phenomena of 
resonance. This is a very important fact ; from it follows as 
a consequence that a ring of revolving electrons may, and 
actually does, possess magnetic properties much greater than 
would be the case if its electrons were all independent. 
Further, it follows from the stationary character of the 
disturbance with reference to space, expressed by the equa- 
tion q + k(a=0, that the radiation of energy into space is 
exceedingly small, in fact only of the same order as the 
radiation due to the steady motion ; hence the displacement 
in phase is almost zero, which means that the maximum of 
the disturbance lies very nearly in the plane through the axis 
and the magnetic force. 
§ 9. By means of the expressions of § 6 and the equations 
(4) -(6) we find the following expressions for the electric 
and magnetic moments of the whole ring, on the supposition 
that squares and products of the displacements may be 
neglected : — 
(a) Steady motion. 
p=0, p=0 . . . . ^1 
m=(0, 0, hie/3 P ) . . J ' ' ( 7 ) 
(b) Disturbance due to the axial magnetic force, and 
therefore radial mechanical force ( — /3eh cos 0). 
8 Pl =n<0, 0, ?), 8^=^(0, 0, 5)=0 . "I 
Sm^JncCO, 0, B(fip) J * ' W 
It must not be forgotten that this force not merely 
displaces the ring but also changes its velocity. 
Phil. Maa. S. 6. Vol. 15. No. 85. Jan. 1908. N 
