PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
133 
for the case where two of the axes of the ellipsoid are equal, when the Lame’s func¬ 
tions which naturally present themselves in such an investigation reduce to spherical 
harmonics, and so can be handled with comparative facility. The solution of the 
problem of induced currents can then be obtained in a very simple manner. 
Of the special forms which the conducting shell may assume, the most interesting 
is that in which the third axis (that of symmetry) is infinitesimal, so that we have 
practically a circular disk whose resistance varies as (cr — r 2 ), where r is the 
distance of any point from the centre, and a the radius. In view of the physical 
interest attaching to the question, it would be desirable to have a solution for the case 
of a uniform circular disk rotating in any magnetic field; but, in the absence of this, 
the solution for the more special kind of disk here considered may not be uninstructive. 
It appears that, except in the case of currents symmetrical about the axis, when the 
ellipsoid is one of revolution, there is always a surface distribution of electricity in 
the problems considered in this paper. 
I. 
1. If u , v , w, be the components of electric current at any point of a thin con¬ 
ducting film ; F, G, II, those of electric momentum at any point (x , y, z) of space ; the 
following conditions must be satisfied. At all points external to the film we must 
have 
V 2 F — 0, v 3 G=0, v 3 H = 0,.( 1 ) 
where V 2 = d 2 /dx 2 + d 2 /dy 2 + d 2 /dz 2 . The functions F, G, H, are everywhere con¬ 
tinuous, but their derivatives are discontinuous at the film ; viz., we have 
dF df 
dv 0 dv l 
4tt u', 
clG_ 
dv 0 
+ 
dG 
dv x 
47 TV , 
<m r/H 
dv 0 dv j 
Attiv, . (2) 
where dv 0 , dv y , are elements of the normal drawn from the film on the two sides. If 
u, v', w', satisfy the solenoiclal condition over the film, these conditions ensure that 
— _p — 
dx dy dz 
( 3 ) 
everywhere. The electric potential xp satisfies the equation 
V 3 1/> = 0 
at all points external to the film ; it is everywhere continuous, but its normal deriva¬ 
tives may be discontinuous at either or both of the surfaces of the film. 
If p be the resistance of the film, per unit area, the equations of electromotive 
force are 
