138 
PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
whence 
a~ + b 2 
ab ’ 
giving, of course, the same value (18) of r as before. 
If we wish to determine, not merely the persistency, but also the distribution of 
free electricity, we may proceed somewhat as follows. Taking the case of an elliptic 
cylinder, and resolving parallel to the principal axes, we have, as before, 
and, at the surface, 
u = — — G, v = — U, w — 0, 
0" ct 
F = — My . C, G = Lx . C, H = 0, 
where L, M, have the values (17). Resolving in the direction of the current. 
P 
' C = X (F + G~- 
as as 
dxfr 
47 t ab 
v A uT 
ct -\- 0 
The resistance p is here supposed independent of 2 , but is otherwise unrestricted. 
Introducing the excentric angle £ we have, since ~ ds = ab d<~, 
d\]r 
~di 
47 rab 
a + b 
X (a sin 3 £ + b cos'- £) C. 
(23) 
Integrating from £ = 0 to f = 27 r, 
47 t~ abX = J p ds, 
which agrees with (22). The value of xp over the film is found by integration of (23), 
p being supposed a known function of £ We can then find two functions which 
satisfy v :: xp= 0 and are finite, &c, throughout the interior and exterior spaces 
respectively, and coincide at the film with value just indicated. Again, within the 
substance of the film itself, the electromotive force in the direction of the normal must 
be zero. This a - ives 
O 
0 = —^ X F + 
a~ 
d\jr 
Ts ’ 
or 
'G r _ , b n mx// 
7 “17T Av_ » j • 
do a + b ab 
(24) 
