1887.] 



On Ellipsoidal Current Sheets. 



197 



tion of the shell in a constant field, it being known from Maxwell's 

 ' Electricity,' § 600, that the induced currents are the same if we 

 suppose the conductor to be fixed, and the field to rotate in the 

 opposite direction. When the conductor is symmetrical about the 

 axis of rotation, the current-function of any normal type contains as 

 a factor cos sw or sin sw, where w is the azimuth, and s is integral (or 

 zero). When we apply Maxwell's artifice, the corresponding time- 

 factor is e is ^, where p is the angular velocity of the rotation ; and 

 we easily find that the system of nduced currents of any normal 

 type is fixed in space, but is displaced relatively to the field though 

 an angle 



- arc tan psr 



s 



in azimuth, in the direction of the rotation. 



In the most important normal types the distribution of current 

 over the ellipsoid is one which has been indicated by Maxwell 

 ('Electricity,' § 675) as giving a uniform magnetic field throughout 

 the interior. For instance, the axes of coordinates being along the 

 principal axes, a, 6, c, we may have 



= C* (1.) 



and the corresponding persistency is 



where ^^^^^ . . . (3.) 



p denoting the specific resistance of the material, and e the small 

 constant ratio of the thickness of the shell to the perpendicular on 

 the tangent plane. There is a difference of electric potential over the 

 shell, viz., we have 



yjr = Axy, (4.) 



where A = 0, 



L, M being obtained from (3) by interchanging a and c, or b and c, 

 respectively. This implies a certain distribution of electricity over 

 the outer surface of the shell. 



Some special forms of the ellipsoid (e.g., a sphere, or an elliptic 

 cylinder) are considered, and the formula (2) shown to agree with 

 the results obtainable, in these cases, in other ways. 



The problem of induced currents due to simple harmonic variation 



