PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
135 
These values of F, G, H, satisfy (1) ; they are continuous at the film, and their 
normal derivatives satisfy (2).* 
We now find that the equations (5) are satisfied, provided we assume for the value 
of the electric potential within the substance of the film 
xp = A xy, 
• ( 8 ) 
and properly determine A. If we write, for shortness, 
L — 27t abc 
dq 
0 ( a 3 + q) Q 
. M = 2ir abc . n ’ N = 2lr abc f A 
J n (0“ + Q ) O, J n >C" -4- 
dq 
the equations in question reduce to 
_ pG/eb* = 
pC/ea 2 = 
whence for the “persistency” we have 
J 0 ( lr + q) Q 
AMC - A, 
ALC - A, 
(c 2 4- q) Q 
' O i 7 •> 
ar f (r 
Also 
v _ i e L + M e , arb 2 
T — X — ~p 1/a 2 + 1/5 2 ~ ~ 
e * (L + M) a%~ 
(9) 
• • (io) 
The value (8) of i p will obtain throughout the internal cavity of the ellipsoid, but 
in the external space we shall have 
xp = A jxy f 
dq 
J q (a 3 + q) (b 2 + q ) Q 
the lower limit being defined by (7). The continuity of xp at the outer surface of the 
film requires 
A, = A + f — dq 
J o (c 2 + q) (b 2 + q) Q 
7 a 2 — b 2 . 
27r abc . —-- • A. 
M - L 
Unless a = b, there will be a distribution of electricity over the outer surface, the 
density cr being determined by 
47TCT d-^r d\p 
K d v dv x 
(U) 
where K is the specific inductive capacity of the surrounding medium, and dv, dv x , 
* See, for instance, Ferrers’ ‘ Spherical Harmonics,’ chap. vi. 
