Electromagnetic Theory of Moving Charges. 489 



making an angle 6 with the line of motion is proportional to 



1 



(l-^sin^) 1 



where u is the velocity of the moving charge and V the 

 velocity of light. The lines of magnetic force are circles 

 round the line of motion. 



2. This solution of course represents the state of affairs at 

 a great distance from a small charged conductor of any shape. 

 It would also give us the distribution of charge on a moving 

 sphere if it were correct to assume that the lines of displace- 

 ment meet the charged surface at right angles. This 

 assumption was made by Prof. Thomson and, at first, by Mr. 

 Heaviside, but the latter, quoting a suggestion of Mr. G. F. C. 

 Searle, subsequently pointed out that when there is motion 

 the electric force is no longer derived from a potential 

 function, and as a consequence does not meet the equilibrium 

 surface at right angles. Substituting, the correct surface 

 condition, he showed that the charged conductor, whose motion 

 would give at all points the radial distribution found for a 

 point charge, was not a sphere but a spheroid of certain 

 ellipticity. 



3. It seemed of some interest to inquire what the distri- 

 bution of charge on a moving sphere would be. The surface- 

 density at a point of the surface is now the normal component 

 of the displacement at that point. By carrying the investiga- 

 tion a step further I have found that, if the conductor be a 

 sphere or any ellipsoid, the ordinary static arrangement of 

 charge is unaltered by the motion ; i. e. the number of tubes 

 of displacement leaving each element of the surface is 

 unchanged, but the tubes no longer leave the surface at right 

 angles. We may imagine that the motion has the effect of 

 deforming the tubes, keeping their ends on the conductor 

 fixed. The proof of this, involving a consideration of the 

 general case, is here given and is followed by a note on the 

 energy of a moving charge in a magnetic field. 



4. Suppose we have any distribution of charge moving 



with uniform velocity u parallel to the axis of z, and that the 



field has assumed its steady configuration. We shall denote 



u 2 

 !— y2°y &> V Dei °g tne velocity of light. Then since we 



have a steady state, 



d_ d 

 -.-'■■ dt" U Tz 



