PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
141 
or, by (9), 
— ip t I 
1 + ip t L + M 
(27) 
The retardation of phase of the induced currents relatively to the electromotive 
forces of the field is arc tan pr, as usual. When p is very great in comparison 
with t —1 this is equal to 7t/2. Since the magnetic force in the interior of the shell, 
due to the currents alone, is given by 
dG 
dx 
~ = (L + M) C e ipt , 
ay ' 
we see that in this case the currents just neutralise, in the interior, the magnetic 
action of the field, in accordance with a well-known principle. 
7. Take next the case where the shell rotates with constant angular velocity p 
about a principal axis (z) in a uniform and constant magnetic field. It is shown in 
Maxwell’s ‘ Electricity,’ § 600, that the problem is the same if we suppose the shell 
to be fixed, and the field to rotate in the opposite direction, provided we add to the 
electric potential the function 
xfi' = p (yY — a:G).(28) 
First let us suppose the lines of force to be perpendicular to the axis of rotation, so 
that we may write for the components of the field 
a = I cos pt, /3 = — I sin pt, y = 0 ; 
whence 
F = 0, G = 0, H = I (x sin pt + y cos pt) = I (y — ix) e ipt , . (29) 
if, as usual, we retain in the end only the real parts. Hence the solution of our 
problem follows by superposition from the results of the preceding section. Omitting 
the time-factor e ipl , we assume for the current-function 
which gives 
(j) = Cx + D y, 
u — 
\ 
. xssy ^ _ 
w'= ~ c — — D 
lr a " 
( 30 ) 
