144 
PROFESSOR H. LAMB OR ELLIPSOIDAL CURRENT-SHEETS. 
47 rf 
d^ 
K = 
~ dx 
47 TCj 
df 
K = 
dy j 
47 rh 
d"\Jr j 
IT = 
dz J 
( 38 ) 
where f g, h, are the components of dielectric polarisation, and K the specific inductive 
capacity. We have also the solenoidal condition 
df_ , d <] , dJi 
dx dy dz 
It is to be carefully borne in mind that nothing is known of the function ip beyond 
what is contained in these equations, except that it is everywhere continuous. The 
familiar electrostatic relations of \Jj may be deduced from these equations by putting 
p = 0. In the present problem we have 
xp = \p y (x 2 + y~) -f const.(40) 
throughout the conductor, whilst in the external space xp satisfies the equation 
V 2 xp=0, 
with the conditions that its value at the surface shall agree with (40) and its first 
derivatives vanish at infinity. The surface density cr is then given by 
a-=lf+mg+nh= .(41) 
where l, m, n, are the direction-cosines of the element dv 1 of the normal drawn 
outwards. 
The solution of this problem for an ellipsoidal conductor is obtained by an adapta¬ 
tion of the results given by Ferrers (‘ Spherical Harmonics,’ chapter 6, §§ 29, 30). I 
do not think it worth while to transcribe these, as the result for the more specially 
interesting case of an ellipsoid of revolution, and in particular for a circular disk, is 
given below in § 15. 
