Impulsive Motion of Electrified Systems. 119 



is the same as if the particle had been at rest at for an 

 infinite time. 



More generally, suppose that both the direction and the 

 magnitude of the velocity are instantaneously changed so 

 that the velocity of the particle is v 1 from £=— cc> to t = 0, 

 and v 2 from t = onwards. Then, if a sphere of radius vt 

 be described about 0, the point where the impulse occurs, the 

 field outside the sphere is the same as if the particle had con- 

 tinued to move with velocity u x , and the field inside the sphere 

 is the same as if the particle had been in steady motion with 

 velocity u 2 for an infinite time. 



These results, which will be assumed in the present paper, 

 can be established by the method of. the " retarded potential" 

 for points either inside or outside the sphere of radius vt*. 

 For external points they can also be established by Poisson's 

 solution of the general equation d 2 (f>/dt 2 = v 2 ^ 2 (f). 



§ 3. A point charge, q 9 in steady rectilinear motion, pro- 

 duces an electromagnetic field which travels with the charge. 

 In this field the lines of electric force are straight and. pass 

 through <7, while the lines of magnetic force are circles in 

 planes normal to the direction of motion. If R and <£, 

 measured from q as an origin and from the line of motion 

 respectively, be the coordinates of any point, then, as was 

 first shown by Dr. Oliver Heaviside f , the electric force is 

 given by 



KR 2 (l-n 2 sin 2 £)t' ' " ' ' KJ 

 where u is the velocity of the charge and 



n = u/v, (2) 



and K is the specific inductive capacity. 

 The magnetic force is given by 



H=K M Esin<|> (3) 



The principles stated in § 2, together with these values of 

 E and H, are sufficient for the solution of every problem 

 connected with the motion of charged bodies. 



* The most direct proof of the formulae of the " retarded potential " is 

 perhaps that given by Sommerfeld in Lis paper " Simplified deduction 

 of the Field and the Forces of an Electron moving" in any given way." 

 Proceedings of the Koninldijke Academie van "Wetenschappen te 

 Amsterdam. Xov. 26, 1904. 



t 'The Electrician,' Dec. 7, 1888, p. 148, or ' Electrical Papers,' vol. ii. 

 p. 511. 



