334 Mr. Gr, F. C. Searle on the Steady Motion 



approximate to Heaviside ellipsoids as X is made very great. 



The value of ^ at the surface A, is — 7=-. 



v'x 



Putting c = b so that we have an ellipsoid of revolution, 

 the axis of revolution being the axis of x, we see by taking 

 X,= — b 2 that a uniformly-charged line of length 2 s/lc — I^o. 

 lying along the axis of a? produces exactly the same effect 

 as the ellipsoid a, b, b. It may therefore be called its 

 "image/'' When b = a this length becomes 2au/v. Thus, 

 when a charged sphere is at rest it produces the same effect 

 as a point-charge at its centre. When the sphere is in 

 motion it produces the same effect as a uniformly- charged 

 line whose length bears to the diameter of the sphere the 

 same ratio as the velocity of the sphere bears to the velocity 

 of light. When u = v, so that the sphere moves with the 

 velocity of light, the line becomes the diameter of the 

 sphere ; and the same is true for an ellipsoid. Since when 

 u = v each element of the charged line produces a disturb- 

 ance which is confined to the plane through the element 

 perpendicular to the direction of motion {(46)}, it follows 

 that the disturbance is entirely confined between the planes 

 x= +a. Between them the electric force is radial to the 

 axis of x and has exactly the same value, viz. q/aKp, as if 

 the line had been of infinite length and had had the same 

 line-density q/2a. Here p stands for \y 2 + z' z \*. The mag- 

 netic force is by (3) qv/ap. Hence the field between the 

 planes a?= +a is 'independent of x. There are therefore no 

 displacement-currents except in the two bounding-planes. 

 There is an outward radial current in the front plane and an 

 inward current in the back plane, the total amount of current 

 in each case being qu, equal in amount to the convection- 

 current carried by the ellipsoid. 



It appears, however, that at the velocity of light any 

 distribution on any surface is in equilibrium. For the value 

 of M* at any point near a moving point-charge is {(43)} 



q^a. 



\[/ _ 



K^/a+/ + / 



and this vanishes when u = v (sothata = 0), even when x=0. 

 Thus the value of M* for a point-charge vanishes, and the 

 value of ^ for any distribution being derivable from that for 

 a point-charge by integration, it follows that M^ has the 

 constant value zero everywhere. Hence the charge is in 

 equilibrium however it may be distributed. The same result 

 follows from the expression {§ 19} for the force between two 



