136 
PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
denote elements of the normal, drawn from the surface on the two sides. Since 
dqjdv l = 2ct, we find without difficulty 
<r _P [ 1 1 \ La 2 - Mb 2 vsxy 
K ~ 6 V« 2 _ by L 2 - M 2 ‘ a 2 b° ‘ ° 
3. Some particular cases of the formula (9) may be noticed. For a spherical shell 
we have L = -|-7r, and thence 
47r ea 
T — ~ - a, 
3 P 
which is right. For an ellipsoid of revolution (a = h) 
(13) 
when the currents are symmetrical round the axis ; whilst, in the case of currents in 
planes parallel to the axis (say (f> = Cx), 
47r — N ea 
= (477 — L) 
ea c~a 
p c~ + cr 
(14) 
For the prolate form we have 
L -.= M = 
2 t 
l-e\ 1 + e 
[ °gy-_ 
2e 3 
N = 
— 1 
1 i 1+e 
- Io O' - 
2e & 1 - e 
— 1 
md for the oblate for 
m 
(15) 
r u o /VO -d) . 1 — e 2 
L = M = 27t (---- - arc sm e 
e~ 
XT „ 0 s/n-d) 
IN =r 4 tt - — - - -arc sm e 
y 
• • (V* 
e denoting in each case the excentricity of the meridian section. 
Again, if we make c — oo, we get an elliptic “ homoeoidal' cylinder, 
have 
L = 
47 rb 
a + V 
. i 47 ra 
M — — 7 . 
a + h 
N — 0 
We then 
. ( 17 ) 
* Maxwell’s ‘ Electricity,’ § 438. 
