PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
143 
If we write 
p t x = tan coj, p To tan co.~ 
2’ 
and restore the time-factor, the expression for the current-function becomes, on 
discarding the imaginary part, 
4 > = 
I sin ojj • \ i I sin Wo 
M + N 
x sin (pt — ojj) ~b 
L + N 
y cos (pt — to 2 ). 
. . (36) 
The currents flow at any instant in a system of ellipses whose planes are parallel to 
one another and to the axis of rotation. When the ellipsoid is one of revolution 
about z we have L = M, oj 1 = &> 3 . The planes of the currents are dragged round, as 
it were, in the direction of the rotation of the shell, through a constant angle from 
the direction of the magnetic force in the inducing field, in accordance with a general 
principle pointed out at the beginning of this paper.* 
8. When the lines of force are parallel to the axis of rotation there are no induced 
currents, but only a superficial distribution of electricity. The calculation of this 
distribution involves assumptions which vary with the particular theory of electro¬ 
magnetism adopted; and even Maxwell’s theory has been differently interpreted 
in this respect by different writers. It may be well, therefore, to state with some 
care the view here taken. 
Considering, for the sake of simplicity, the case of a solid conductor rotating in a 
field of uniform intensity y about an axis (z) parallel to the lines of force, and 
supposing the axes of x, y, to move with the solid, then, on the hypothesis that there 
are do currents, we have, throughout the interior. 
0 
0 
0 
y.px 
y-py 
elf 
dx 
d^r 
dy 
d'p 
~dz 
(37) 
whilst in the surrounding dielectric (taken to be sensibly at rest)— 
* The case of a spherical shell has been discussed by C. Niven ( loc. cit.) and J. Larmor, ‘ Phil. Mag.,’ 
Jan., 1884. 
Maxwell has considered the currents induced by rotation of a solid ellipsoid (see Stewart and Tait, 
‘Roy. Soc. Proc.,’vol. 15,1867, p. 291), leaving out of account, however, the mutual action of the currents 
themselves. This is equivalent to supposing the period of rotation to be long in comparison with the 
modulus of decay of free currents. 
