13 
PROFESSOR H. LAMB OR ELLIPSOIDAL CURRENT-SHEETS. 
, , rlF d-yfr 
pU = ~ ~dt ~ He’ 
p v = — 
dQr d'yjr 
dy 
dt 
/ / 
) w — 
d.YY ddr 
dt 
dz 
• (4] 
In these equations \b is supposed to have the value appropriate to the space included 
between the two surfaces of the film, which may differ in form from the values which 
it has in the external space on either side, 
[n any natural mode of decay the time occurs through a factor of the form e~ xt , 
where A is real and positive. The preceding equations then become 
P 
V = AF - 
dx 
f r \ d dr 
p v = AG — -f- 
r dy 
mddf 
- -T-. 
' dz 
( 5 ) 
2. Let us apply this to the case of an ellipsoidal shell v/hose thickness varies as the 
)erpendicular ar from the centre on the tangent plane; say it equals eny, where e is a 
mall numerical constant. If p be the specific resistance of the material, w T e then have 
ear. 
Let the semi-axes of the shell be a, b, c, and let the axes of coordinates be taken 
along these. In the most important type of free currents the lines of flow are in 
planes perpendicular to a principal axis. If this axis be that of 2 , the current- 
function over the surface of the ellipsoid is of the form 
(f) — Cz. 
The corresponding values of u, v, w', are 
u — -— (J, v = — (J, iv = 0. 
b- cc~ 
The values of F, G, H, in the internal space are 
F = — 2?r abc C . y f° 7 , - 1 — 1 
Jo( & ' + <7 
G = 27t abc C . x\ 
J n 
H = 0 
where 
, 0 (Jr + q) Q | 
^ » 
(6) 
I o O 3 + 2) Q 
J 
Q = { (a 3 + q) (Ir + q) (c 3 + q) 
The corresponding values for the external space are obtained by replacing the lower 
limit of the integrals by the positive root of 
