352 
PROFESSOR C. NIVEN ON THE INDUCTION OF ELECTRIC 
19. We shall briefly verify these results by a direct investigation of the case of an 
infinitely thin shell. 
Putting as before 
u— 0 , v= 
f?g> _dj> 
sin ddcf)’ cW 
(36) 
and denoting the vector potential of the shell by 
F = 0, G= 
cW dP 
sin 0d<f>’ d6 
(37) 
we know (Maxwell’s ‘Electricity and Magnetism,’ Yol. II., Art. 670) that P is the 
scalar potential of a shell of matter coinciding with the surface of the sphere where 
surface density is 
a 
(38) 
The equations of the currents on the sheet are 
wherein, also, 
E d<t> . „ d4r 1 
sm 6 d(f> nw 
K dd~ 
r sm 
in 6d(f)J 
a — ~ 
r 
1 d . v/P + P 0 
r. sm ij- 
sin 6 d$' 
dd 
d 2 P + P n 
ssin 2 6 d<\. i> 2 
as in the foregoing articles, and where also r is to be put equal to a. 
All these equations are satisfied by 
xp=~R‘d sin 0^, R x = co(P + P 0 ) 
t Jj now referring only to the surface of the sheet. 
Outside the sheet P X =C^ e””*P,/, 
Inside „ P 3 =C^ e iM *P,/, 
(39) 
(40) 
P : and P 3 being the values due to the currents in the sheet. At the sheet 
