1887.] On Ellipsoidal Current Sheets. 199 



The problem of induced currents is then discussed, and I consider 

 more particularly the case of a circular disk, of the kind indicated, rota- 

 ting in any constant magnetic field. In view of the physical interest 

 attaching to the question, it would be interesting to have a solution 

 for the case of a uniform disk ; but in the absence of this, the 

 solution for the more special kind of disk here considered may not be 

 uninstructive. 



As in all our calculations relating to ellipsoids of revolution, we 

 employ elliptic coordinates ; viz., seeking the origin at the centre of 

 the disk, and the axis of z perpendicular to its plane, we write 



x = a</(l -ft) v/(£3 + 1) cos u ^ 



y = ay(l-A* 8 ) + sin* I. . . , (8.) 



z = ap$. J 



where fi may range from 1 to 0, and £ from zero (its value at the disk) 

 to oo. The magnetic potential Q due to the field may be supposed 

 expanded, for the space near the disk, in a series of terms of the 

 form 



^KMF^i^^S}., , , (9.) 



where P w is the zonal harmonic, and p n a similar function in which 

 all the terms are +, instead of alternately + and — .* 



The terms for which s = are symmetrical about the axis, and 

 produce no currents, but only a certain superficial electrification. 

 The density of this is calculated for the particular case n = 1, i.e., 

 for the case of a disk rotating in a uniform field about an axis parallel 

 to the lines of force. 



The only terms of the expansion (9) which produce sensible currents 

 in a rotating disk are those tessaral solid harmonics for which n—s 

 is odd. The induced current-function is found to be (taking, say, 

 cos siv in (9) ) 



2.4... (n+s-1) A . n 2 xi,^P«W . f . 



(10.) 



where v = arc tan spr, 



t having the value (7). 



The most important type of induced currents is when n == 2, 

 s = 1 ; in which case _ 



Q oc xz, 



* See Ferrers, c Spherical Harmonics,' chap. vi. 



