342 The Followers of Maxwell. 



Now, let (x, y, ) denote coordinates relative to axes which 

 are parallel to the axes (a;, y, z) , and which move with the 

 charged bodies ; then a z is a function of (x, y, ) only ; so we 

 have 



a a , a a 



5 -IT and *' "Vr 



and the preceding equation is readily seen to be equivalent to 



where 1 denotes (1 - v'/c 2 )'^. But this is simply Poisson's 

 equation, with & substituted for z; so the solution may be 

 transcribed from the known solution of Poisson's equation : it is 



/LV dx' dy d%i' 



the integrations being taken over all the space in which there 

 are moving charges ; or 



_rrr 



jJJ 



If the moving system consists of a single charge e at the point 

 5 = 0, this gives 



ev 



%(1 - tf sin 8 0/c")* ' 

 where sin 2 = (a 8 + y 2 )/r 2 . 



It is readily seen that the lines of magnetic force due to the 

 moving point-charge are circles whose centres are on the line of 

 motion, the magnitude of the magnetic force being 



ev (1 - v 2 /c 2 ) sin 8 



The electric force is radial, its magnitude being 



r 2 (l - v 2 sin 2 0/c 2 )f 



The fact that the electric vector due to a moving point- 

 charge is everywhere radial led Heaviside to conclude that the 

 same solution is applicable when the charge is distributed over 





