in Conducting Sheets and Solid Bodies. 13 



the one probably affects that of the other, so that we could not 

 get the actual value of such an electrification, but it seems 

 that this electrification represents the total effect produced. 

 It is to be noticed that our argument is independent of axes 

 of coordinates altogether ; and that for any motion of a con- 

 stant electromagnetic system, the value of <&' at any point is 

 the scalar product of the vector potential and the relative 

 velocity of the conductor with respect to the medium at the 

 point. 



We can now simplify the equations which give the electric 

 currents when the conductor is in motion, as we can reduce it 

 to rest, and solve the corresponding relative problem, where 

 motion across the lines of force is replaced by a variation of 

 the field itself. 



(1) Let us first take the case of a spherical shell rotating 

 with angular velocity co in a field given, as before, by the 

 function P . We found for the case of the shell at rest the 

 equation 



R*=-a|(P + P ); 



in this case it becomes 



Bft=-q»^(P + P ) (8) 



To obtain a solution, let us take a harmonic term of order n 

 and type s, as follows: — 



and therefore 



p =A (r)Y^, 



P s = A (^)" + ¥•*•*; 



4z7T 



On substituting, we have 



therefore 

 where 



AT > 2n-f 1 ,. . N 



— AR.— -: = — tacos(A + A ); 



i+ip 



(2n + l)R 



^ 4:7rao)s * 



