Classification of Electromagnetic Fields. 141 



We shall assume that p and p vanish so that our expression 

 may be finite for o" = 0. 



Integrating over the surface of a sphere for which t—r 

 is constant, we find that the total charge associated with the 

 singularity (f , 77, f, t) is 



In general the charge varies with t : to make it constant 

 we must introduce a restriction such as 



Xf , + M V + <-^=?' 2 + V 2 +?' 2 -l. . . (25) 

 If, in particular, we take sr = l, this condition implies that 

 the angle between the direction of the gun and the direction 

 of the gun's motion has a cosine equal to i\ where v is the 

 velocity of the gun. We may get rid of the charge at 

 the singularity (f, 77, f, t) by adding suitable multiples of 

 Lienard's potentials (5). To simplify our expressions, we 

 shall add to (19) the additional equations 

 er=z<ar =p_=p =l, Ig + mrj' + nt;' = 1, l %' + m oV ' + n£ = 1, 



. . (26) 

 which, together with (19), lead to (25). The electro- 

 magnetic field specified by the potentials 



. I (T l Q CT 



x ~m.w + m.w 



A - — 



y ~ 2M to 



a 



~ + 



m (T 

 2M w n 



. n (T n cr 



2= 2M^ + 2Mt^ 



M 



p—v' 

 M 



M 



<&: 



(<L 



2MVj 



+ 



10n/ 



(27) 



J 



then has no charge associated with the singularity (f, 77, £ r). 

 and both the electric and magnetic forces at (p, y, z, t) are 

 perpendicular to the radius from (f, 77, f, t). 



If we add to (27) the field specified by the potentials 



a; = 



A v ' = 



a: = 



£ <T <0°0 



2M 10 + 2Mio " 



x -r 



M 



1 

 ' 1 



m a w? o- 

 2M 10 + 2M tc 



Po—v' 

 M 



1 



1 



' 1 



[ 



1 



1 



II <T II (To 



2M to + 2M u? " 



" M 



(28) 



1 /(To (To\ 



2M\w ■ wo J 



J 



