132 
PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
if 
// T> 
E 
It + ip L E (1 + ip t) 
E cos 6 
K 
,—iB 
tan 6 pr. 
Hence the phase of the currents lags behind that of the inducing electromotive force 
by an amount arc tan pr. This remark, obvious as it is, is of some importance in 
relation to the practically interesting question of the rotation of a conductor about an 
axis of symmetry in a constant magnetic field. The magnetic potential of any normal 
type will be proportional to cos soj or sin sco, where oj is the azimuth about the axis 
of symmetry, and 6’ is integral (or zero). If now, as in Maxwell’s ‘ Electricity,’ 
§ GOO, we employ coordinate axes moving with the conductor, the electromotive forces 
relative to these will vary as e lspt , where p is the angular velocity of rotation. On 
account of the symmetry about the axis, the retardation of phase above spoken of 
comes to this, that the system of currents of any normal type is, owing to its inertia, 
displaced relatively to the field through an angle 1/s arc tan spr, where r is the 
modulus of decay proper to the type. (See §§ 7, 16, below.) 
For other than linear conductors the problem above stated was first solved by 
Maxwell in the case of an infinite plane sheet of uniform conductivity. The cases 
of solid spherical and cylindric conductors, and of thin spherical and cylindric shells, 
have been treated by Prof. C. Niven,* Lord Rayleigh,! and the writer .\ It is 
remarkable that, with a certain exception, § no difference of electric potential, and 
consequently no surface distribution of electricity, is called into existence diming the 
decay of free currents in conductors of the forms mentioned, 
In § 675 of his ‘Electricity and Magnetism,’ Maxwell has indicated a certain 
arrangement of currents over the surface of an ellipsoid, which produces a uniform 
magnetic field in the interior. I do not know that it has yet been noticed that this 
arrangement fulfils the conditions for a natural mode of decay of free currents in a 
thin ellipsoidal film whose conductivity (per unit area) varies as the perpendicular 
from the centre on the tangent plane ; or, say, in a thin shell of uniform material 
bounded by similar and coaxial ellipsoids. This is proved in Part I. of the following 
paper ; and we thence easily find the currents induced in such a shell when situate in 
a uniform magnetic field of varying intensity ; or, again, the currents induced by 
rotation of the shell in a uniform and constant field. 
I have attempted to generalise these results and to ascertain the remaining normal 
types of currents in a shell of the kind indicated. In Part II. is given the complete 
solution of this problem, including the determination of the corresponding persistencies, 
* ‘ Phil. Trans.’, 1882. 
t ‘ Brit. Assoc. Rep.’, 1882. 
I ‘Phil. Trans.’, 1883 ; ‘London Math. Soc. Proo.’, vol. 15, pp. 139 and 270. 
§ That of the currents of the “ Second Type ” in a spherical conductor. Such currents cannot, 
however, be excited by any electromagnetic operations outside the sphere. 
