MOTION OF ELECTRIFIED SYSTEMS OF FINITE EXTENT, ETC. 187 



Thus the disturbance is not of a vibratory character, but is of a purely exponential 

 type. 



It may be observed that since the tangential component of electric force must 

 vanish at the surface of a conductor, no couple on the conductor can arise from this 

 cause. There remains the contribution to the tangential component of electrodynamic 

 force on account of the magnetic field. If this arises from the motion contemplated 

 it will give terms of the second order in the velocity of rotation, and may safely be 

 neglected. Thus to this order there can be no tether reaction on the conductor due 

 to its rotation. This does not imply independence of the rotation on the whole 

 magnetic field. If, for instance, the external force is due to a rotating magnetic field, 

 surface currents will be set up and a couple produced which will set the sphere in 

 rotation. 



If, however, the sphere is an insulator, the tangential component of electric force is 

 no longer zero, and it will appear that there is a resultant couple on the sphere. 



The equations for the inside of the sphere are, of course, altered by the assumed 

 rotation ; but just as in the problem of linear motion, we require a solution which is 

 small of the first order in the angular velocity of the sphere, and hence neglecting 

 terms involving squares of the angular velocity, the equations for insulators at rest 

 suffice. 



Inside the sphere we have both diverging and converging disturbances represented 

 by fa (CV;- >) and fa (C't + r) respectively. 



Thus for the field inside we assume 



(X, Y, Z) = <J(0, *, -y) {r(fa-fa)+fa>+fa>} 



K-'-(a, & y) = ^(-1, 0, 0) {i*(fa" + fa") + r (fa'-fa) + fa + fa} 



+ ^ (x\ xy, xz) {r 2 (fa" + fa") + 3r (fa' + fa') + 3 (fa + fa)}. 



In these equations K = C 2 /C' 2 and the factor K~ 12 is introduced in the form for 

 magnetic force inside in order to make the units of measurement the same outside 

 and inside. 



Further, ^, fa, and fa are supposed to be small quantities proportional to the angular 

 velocity <a, and squares of w are neglected. 



Since the field must be finite at the origin, we must have 



fa(G't)+fa(C't) = o ......... (i). 



The normal component of magnetic force is continuous at r = a. Thus 



or 



a W-^') + fa + fa ......... (2). 



2 B 2 



