178 Mr. G. A. Schott on the 
(c) Disturbance due to the radial magnetic force, and 
therefore axial mechanical force 
I —fieh sin 6 cos ( cot + S -f ■ — J )• 
Sp 2 =i™?(A'-B, A + B', 0), Sp 2 =-^w(A, A', 0) 
$m a = ine/3(-C, -(7,0) ....... 
where we have written 
(f 5 ^?) = (A,B,C)cos(^ + S+^) 
}, • (9) 
+ (A', B', C ; ) sin (at + 8 + ^Y 
§ 10. The varying e lectric moment Sp 3 in (c) gives rise to 
a magnetic force rot ( — 2 J ; its components are linear func- 
tions of the direction cosines of the radius vector r with 
respect to axes fixed in the ring, and therefore their values 
vanish on the average for a large number of rings, whose 
axes are distributed equally in all directions, and for a 
distant point. 
The electric moment Sp gives rise to a magnetic moment 
rUp-L where for w we may take the velocity of the centre 
of the ring relative to its undisturbed position (§ 6) ; but 
this velocity, as well as op itself, is small of the order of the 
disturbance, so that the resulting magnetic moment is of 
the second order and may be neglected. 
Thus we need not consider the polarization in finding the 
magnetic field due to the disturbing ring. 
The magnetic moment in the steady motion by (a) gives a 
component in the direction h equal to fyiefip cos 6 ; since this 
must vanish on the average the mean value of cos 6 vanishes, 
as it must do for rings equally distributed. 
It only remains to consider the moments Sm given by (8) 
and (9); by the principle of superposition of small disturbances 
we may find the effect of each separately, and add them 
together to find the total effect. 
§ 11. To find the effect of (8) it is necessary to consider 
the motion of the ring during the variable period accom- 
panying the establishment of the magnetic field h. Since 
the motion of the ring itself is periodic, the simplest method 
of proceeding is to resolve the mechanical force into simple 
harmonic components, either by Fourier's series, or integrals, 
and to determine the corresponding forced vibrations, and 
thence, by summation, the resulting motion. 
Suppose the variable period to extend from t = to t = r ; 
