PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
137 
so that for the case of currents circulating round the cylinder 
47re « 2 5 2 . 
r = 'TT. ( 18 
p cc~ + o 2 
The surface density of free electricity is then given by 
( 0 70 
47Tcr p a - 6- ri , 
. (1 ») 
For a circular cylinder (18) gives 
ea 
T = *2 tt • ■ «, 
p 
which is right. 
For currents parallel to the axis of the cylinder (say <fi — Cx), 
r = — • — t .(20) 
p ct 4- b 
If in (18) or (20) we make a infinite, we get the case of two uniform parallel plane 
sheets at a distance 2 b apart. The persistency of uniform parallel straight currents 
flowing in opposite directions in the two planes is then 
4. Such special results as these may, of course, be obtained more easily by inde¬ 
pendent processes. Thus for a cylindrical shell of any form, if a current of strength C 
circulate round each unit length, the magnetic induction in the interior is parallel to 
the axis and equal to 4-77-0. Hence, if It be the resistance of unit length to currents 
circulating round it, 
RC = — | (4ttCS), 
where S is the area of the cross-section. This gives 
r = 47rS/R.(22) 
For a “ homoeoidal ” cylinder we have, if ds be an element of the elliptic contour, 
and £ the “ excentric angle,” 
p ds —- p ds/ezz = p ab d^/enr' 1 , 
T 
MDCCGLXXXYII. —A. 
