150 
PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
If we now assume 
4 > — C (1 — /x 2 ) s 
the first member of (63) reduces to 
siz 
dfi s 
sm soj, 
( 64 ) 
L« (« + !) + 
(65) 
by the differential equation of tessaral harmonics. Again 
d / dsn 
dco \ cl/i 
/(S d^-) = 
dfi \ act) 
Hi, 2 
. <ia 
. . ( 66 ) 
where II is the magnetic potential due to the currents (64); for, if we multiply each 
member by d[x dco, the left-hand side is equal to the line-integral of the electric 
momentum round an infinitesimal circuit, and the right-hand side gives the magnetic 
induction through the circuit. To find fi, we assume 
r\ \ / 1 9Ac/2 i ^ \?/2 ^Pdf) • 
n 0 = A (1 - /x 2 f 2 - — (£- - lyi* -jg- sm sco 
n l = B(i — /x 2 ) 
2W2 d^PrcQ) , 1 \ x/ 2 d S Qn(0 
> 
(67) 
d/ji s 
(?-1y 1 *' 
sm I 
d? J 
for the spaces inside and outside the shell respectively. If we write, for shortness, 
T„'(0 = (C 2 -1) 
u/u) = (e -1 r z 
s/2, ^P »(£) 
d%* 
d S Qn(0 
d? 
. . . ( 68 ) 
the conditions to be satisfied at the surface of the shell give 
BU/(£q) — AT/(4) + 4?rC, 
rftV(|o)_ dT n °(Q 
d £o d £ 0 ’ 
\ n ~ s ,,dV n 
whence 
