132 DISTRIBUTION OF ELECTRICITY ON MOVING CONDUCTORS
111
two boundary conditions give, when we equate factors of
cosio and siniw, four linear equations for the four constants
A, B, A', B'. If these latter are satisfied, so also will be the
former. Using the contractions x/2T = a, r/Re, we get for
these equations-
AninnA-(2n+1)aiB+(n+1)e"+¹A'
0 =(2n+1)aiA -nB * +(n+1)en+¹B',
Ani.ne" - ne"A
=
0
=
*
+(n+1)A' +(2n+1)aiB',
-ne"B-(2n+1)aiA'+(n+1) B'.
These equations determine the four constants uniquely.
Without actually performing the somewhat cumbrous solu-
tion it is easy to recognise the correctness of the following
remarks:-
1. When a = 0, A- Ani, A'B' =B = 0, as must be the
case for a sphere at rest.
2. If a be finite but very small, then A+Ani and A' are
of order a², B, B' are of order a. Hence it follows that the
chief points of the phenomenon are these. The distribution
of the charge on the outer surface (the form of the lines of
constant density) is not changed by the rotation (of course
only for the separate terms of the development); but the
charge appears rotated in the direction of the rotation of the
sphere through an angle of order a, and the density has
diminished by a small quantity of order a². In addition a
charge makes its appearance on the sphere forming the inner
boundary, and its type is similar to that of the first charge;
its density is of order a, and it is turned relatively to the first
charge through a small angle π/2i. In the substance of
the shell as well as inside we get differences of potential of
order a.
3. If a be large, B, B' are of order 1/a, A, A' of order
1/a². As the velocity increases the charge on the external
surface finally appears turned through the angle π/2i; its
density is small, of order 1/a, and the charge on the inner
spherical surface is like it as regards type, position, and density.
In the ultimate state = 0 everywhere, and then we have
the external potential in the substance and the interior of the
spherical shell; the currents everywhere flow in the lines of