406 Prof. S. R. Milner : Does an Accelerated 



comes to rest at £= +Jc when t = 0, and moves back towards 

 |=4- oo, acquiring again at £=+co the velocity of light. 

 The motion is that which would be produced, according to 

 the ordinary principles of mechanics, by the action, on a 

 particle of mass 



of a constant force F, such that 



F = ^p (2) 



To express the electromagnetic field associated with a 

 point-charge e moving in this way, at any point let ^ be the 

 angle included between two lines, lengths ?\ and r 2 , respec- 

 tively drawn from it to the instantaneous position of the 

 point-charge and to that of its image in the plane # = 0, 

 and let yjr= log?^/?^. Then %, -yjr are related to the 

 cylindrical co-ordinates a?, y by the equations 



x= gsinlnfr g sin X ^ _ ( . 6) 



cosh yjr — cos ^' cosher — cos^' 



The third co-ordinate (£, the angle through which the plane 

 containing the point has rotated about the axis of x from a 

 fixed position, is the same in both systems. 



The electric and magnetic forces at any point of the field 

 are given by 



^_ i(l-/3 2 )(cosh^-cos % )' 2 o-„ v ,r\ 



The nature of the field will be made clear by a reference 



to figs. 1 and 2. These show meridian sections at the 



_ k 

 moments £= + - and t = 0, when the charge is at the 



distance a/2/j and k respectively from the origin. The lines 

 of force in each case form the arcs of circles (%= const.) 

 passing through the charge and through its image in the 

 median plane x = : in the figures they are drawn so that 

 i of the total flux of induction is enclosed by adjacent lines. 

 The changes which the field undergoes can be pictured by 



supposing that each line of force in fig. 1 at t = is moving 



normally inwards with a velocity /3c sin %. The velocity 

 gradually decreases until the line comes to rest momentarily 



