156 PROFESSOR H. LAMB OIST ELLIPSOIDAL CURRENT-SHEETS. 
the left-hand side being obtained as the electromotive force necessary to balance the 
resistance, and the right-hand side as the electromotive force of induction due to the 
decay of the currents. If r be the modulus of decay of the type in question, 
T — —" J/I p. 
Now let (J) represent a fictitious distribution of current over the ellipsoid, which shall 
have the same magnetic effect in the interior as the actual inducing field. This 
distribution is found at once from (69). The equation of induced currents will then be 
w = j(#+f) 
or 
* = -'(#+#).W 
When the free currents have died away all our functions will vary as where p 
measures the rapidity of the changes in the field. Substituting in (90), we find 
— ipr 
I + ipr 
(/>. 
When pj is very great this becomes 
(91) 
The result (91) may be verified in the problem of § 6. 
15. Let us next consider the rotation of the shell in a constant field. There will 
be no currents due to those terms in the expansion of O which are zonal solid 
harmonics ; the only effect of these being a certain surface electrification. We may 
complete the investigation of § 8 by finding the density of this electrification in the 
case of an oblate ellipsoid of revolution rotating in a uniform field about an axis 
parallel to the lines of force. We have to find a function xjt which shall have the 
value 
^ = \vy (^ 3 + l f) + C .( 92 ) 
at the surface of the conductor (£ = £ 0 ), and shall satisfy v~xp= 0 throughout the 
external space. Denoting by a the equatorial radius, (92) may be put in the form 
xfj = C + \pya* — \pya z . Po (/x).(93) 
Hence in the external space 
+=< c +^)f^-*^wi§ 
