PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
139 
where dS is an element of the thickness of the film, directed towards the outer surface. 
If cr 0 , cr l5 be the densities of free electricity at the inner and outer surfaces respectively, 
we then have 
4:7r <t 0 dylr dy\r 
K = dv 0 do 
TttcTj dr^r 
”kT = ^ “ IE J 
(25) 
dv 0 , dv±, denoting, as before, elements of the normal, drawn from the film, on the inside 
and outside respectively. 
For the case of a homoeoidal cylinder this process leads to the result already 
obtained. For other laws of thickness there will, in general, be a distribution of 
electricity on both surfaces of the film." 
5. Another case of interest is obtained by supposing c infinitesimal, so that the 
conductor may be taken to be an elliptic disk whose resistance per unit area varies 
according to the law 
© 
P = P o ✓ 
In the case of a circular disk the formula (13) is replaced by 
where we may put 
Hence 
= (477 — N) 
cc 
4 pA 
N 
477 
77 
9 
<■ 
<7T = 
77 “Of 
• (26)} 
* I find, however, that in the case of a film bounded by confocal cylinders the inner surface alone 
becomes electrified. 
t The symbol p here refers to the disk. Since this is the limit of a double film, its resistance at any 
point is half that of the corresponding portion of the film on either side, 
t For an elliptic disk 
N/4tt =1 — 
Ei(e) c 
v/ (1 — e 2 ) a 
where a is the semi-major axis, e the 
kind. This gives 
excentricity, and E x the complete elliptic integral of the second 
277 ab 
r 2 + b 2 
E 
z % a 
i ( e )-—,• 
So 
T 2 
