322 
PROFESSOR C. NIYEN ON THE INDUCTION OF ELECTRIC 
> —P 
outside inside 
dP\ /rfpN 
dz 
outside 
dz 
■*/ inside 
( 33 ) 
when z—-±b\ and it must be remembered that, outside the plate, the vector potential 
is determined by 
r/P dV 
F= —~r, G= ; , H=0, V 2 P=0.(34) 
dy dx v ' 
If we employ semipolar coordinates the equation in P in the substance of the plate 
may be written 
<r /d~P 1 dP 1 <PP <P?\ dP 
47 r\dp 2 p dp p 2 dcf) 2 dz 2 ) dt 
To satisfy it, put 
P= cos m(f)J m (i<p)(A cos nz-j -B sin nz)e~ M .(35) 
where J,*(/cp) is Bessel’s function of the degree satisfying the equation 
and 
d*J 1 dJ 
dp 2 p dp 
(36) 
*=£(**+»*) 
(37) 
We observe also that k is a constant, which the problem does not enable us to 
determine : it must therefore be supposed to have all values from 0 to oo ; m is 
necessarily a positive integer. 
Outside the plate P is given by v 2 P=0, and is satisfied by 
P = e kf cos m<f)J m (Kp).Ce % 
P = e~ w cos m(f)J m (Kp).T)e +Kg , 
positive 
: negative 
it being observed that P must vanish when z = cc . 
To determine n, we have, by equations (33), 
(1) when z=-\-b. 
A cos nb -\-B sin nb =Ce~ Kb 
— n (A sin nb — B cos nb) — —kC€~ k,j , 
(2) when z— —b 
A cos nb —B sin nb =De~ Ki 
+ n(A sin nb -\-B cos nb) = /cDe“A 
(38) 
