140 PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
The current at any point is proportional to 
r 
It would be interesting for many reasons to have a solution for the case of a uniform 
disk, but at all events the above result shows that the time-constant of a disk of 
radius a and uniform resistance p must be considerably less than 4'93 a/p'. I find, 
by methods similar to those employed in Lord Rayleigh’s ‘ Sound ’(§§ 89, 305, &c.) s 
that the true value lies between 7 ra/p and 2"26 a/p, the latter value being probably 
not far removed from the truth.* For a disk of copper (p = 1600 C.G.S.) whose radius 
is a decimetre and thickness a millimetre this lower limit gives ‘0014 sec. For disks 
of different dimensions the result will vary as the radius and the thickness conjointly. 
6. Let us next calculate the currents induced in a homoeoidal shell when situate in 
a uniform magnetic field (a, /3, y) of varying intensity. It is sufficient to consider the 
case where the lines of force are parallel to a principal axis. Also the expression for 
the magnetic force may be supposed resolved, as regards the time, into a series of 
simple harmonic terms, each of which may be taken separately. Putting, then, 
a = 0, (3=0, y- = I e ipl , 
and denoting by F, G, H, the components of vector potential due to the field, we 
may write 
F = — \I y e ipt , G = x e pt , H = 0. 
The induced currents will be of the type 
u = 
W C & 
V ’ 
v= -;c s pt , w'= 0 , 
and the corresponding components of electric momentum at the film will be 
F = — MO y e ipt , G = LC x e ipt , H = 0. 
Assuming 
xjj = A xy e ipl , 
and substituting in the equations 
we find 
whence 
d F clF dyfr 
P u = -Jt~lu 
- pC/eb~ = ip MC + 4 ip I - A, 
pC/ecr = — ip LC — -g ip 1 — A. 
{f (k, +| s j + ( L + M )*>| C = - iy I, 
* [See ‘ Roy. Soc. Proc.,’ vol. 42, 1887, p. 294.] 
