of an Electrified Ellipsoid. 331 



made by Mr. Oliver Heaviside, F.R.S. *, that a distribution 

 of electricity on the surface of a charged body such as to 

 give zero disturbance at all points inside the surface is an 

 equilibrium distribution. Since F satisfies curl F = and F 

 vanishes inside the surface, it follows that on the outside of 

 the surface F is perpendicular to the surface. This implies 

 that M* is constant over the surface. But as neither the 

 electric force E nor the mechanical force experienced by each 

 part of the charged surface (calculated from the Maxwell 

 stress) is normal to the surface, I felt unable to accept the 

 validity of Mr. Heaviside's assumption until I discovered 

 {§ 15} that F is the mechanical force on an isolated moving- 

 unit charge, and that the term — VG-D, which appears in the 

 expression for the force experienced by the surface, has no 

 influence in causing convection of electricity from one part 

 of the surface to another. Here D is the electric displacement, 



dS. 



and G- the " magnetic current " fi — . 



Since ^ is a true potential for the mechanical force F, I 

 have called M* 1 the " electric convection potential." 



When there has been established the boundary condition 

 that *& is constant over the surface, with its consequence that 

 there is zero disturbance within the surface, it is very easy to 

 show that the distribution on an ellipsoid is the same for 

 motion as for rest. Suppose the ellipsoid to have the same 

 distribution as when it is at rest, so that a^qp/iirabc, where 

 q is the charge, a, b, c the axes of the ellipsoid, and p the 

 perpendicular from the centre upon the tangent- plane at the 

 point. Through any internal point M as vertex draw a 

 slender double cone intercepting two areas N, W on the 

 surface. Now the electric force due to a moving point-charge 

 is still radial and still varies inversely as the square of the 

 distance, although it alters with change of direction of the 

 radius vector. Thus it follows just as in electrostatics, since 

 <r x p, that the effects at M of N and W are exactly equal and 

 opposite. The whole surface can be treated in the same 

 manner, and thus it follows that E = at all internal points. 

 Hence H = also. Thus the assumed distribution is in 

 equilibrium and is therefore the actual distribution. Thus 

 the motion has no influence upon the distribution, and this 

 result is true whatever the direction of motion with respect 

 to the axes of the ellipsoid. 



In order to find the state of the field near a charged 

 ellipsoid moving with velocity u parallel to the axis of x, it is 

 necessary to find a value of \P which shall be constant over 

 * l Electrical Papers,' vol. ii. p. 514. 



