MOTIONS IN A SPHERICAL CONDUCTOR, 
53 L 
The simplest and most important case is when n= 1. This may easily be examined 
by making Xi— x - The l' nes of motion are then all circles having the axis of a? as a 
common axis. 
Second Type. It follows from (45) that we must now have 
^(Mt) = 0.(52). 
In the cases n= 1, n=2, the first few roots of this equation are given by (48), (49), 
respectively. The values of the modulus of decay corresponding to the various values 
of k are to be found from (50). In the most persistent mode of the present type the 
value of r for a sphere of copper is 
r= '000379 R 3 second. 
As regards the nature of the motion inside the sphere we remark in the first place 
that since the radial flow is zero at the surface the electric currents form closed 
circuits. The flow at any point may be resolved into two components, one along, 
the other at right angles to, the radius vector. The radial component is 
The second or transversal component is perpendicular to that cone of the series 
(o„/r , ‘= const, which passes through the point in question; and its amount is 
£{fo^„(fo.)+(»+l)^(fc-)}^.(54), 
! where da denotes as before an elementary angle at the centre of the sphere in a plane 
perpendicular to the above-mentioned cone. 
When the harmonic a) n is zonal, having the axis of x (say) as axis, the nature of the 
motion can be very simply expressed by means of a stream-function TL The motion 
then takes place in a series of planes through the axis of x and is the same in each 
such plane. If u, be the components of current parallel and perpendicular to x, 
viz., 'o=(yv-\-zw)/‘nr , where 7x=</(y 2 -{-z 2 ), we have 
_ 1# 
ot rfcr’ dx 
where 2-rr^ is the total flux through the circle whose coordinates are ( x , nr). Integra¬ 
ting (53) over the segment of the sphere of radius r= ^/(x z -\- nr 3 ) bounded by this 
circle w r e find 
(55), 
(53). 
