Classification of Electromagnetic Fields. 139 1 



This type o£ electromagnetic field may be generalized by 

 integration just like the previous one. In the resulting 

 electromagnetic field material particles are fired out from a 

 material wire for a certain interval of time, arid these 

 particles move along straight lines with the velocity of light. 

 The way in which the shape and size o£ such a particle varies 

 during its motion has not yet been ascertained. 



§ 3. We shall now discuss another class of electromagnetic 

 fields in which the radiated energy is concentrated round 

 certain moving points which travel with the speed of light, 

 but in this case the difficulty with regard to the roots of (10) 

 is absent. 



Defining t as before by means of equations (6) and (7), 

 we choose 16 functions of t which satisfy the equations 



7 2 + m 2 + ri 2 =pr. l 2 -\-m 2 +n 2 —p 2 , X 2 + fj? + v 2 — vr 2 , ) 



V + /V + V = W> ) 



IX -f nifju+ liv—pvj, / X+ m /jL+ n^v—pQ^y, l\ + m/jL + nv =pvr Q , i 

 l \ + m {M + n v =povr ; J 



and write 



iv = l( ai -£) + m(y—n) + n(z-Q-p(t-T t ), ) 

 w = k(x—t;)+m (y—r)) + n Q (z—g)-p (t-T), \ 



cr u = \ Q (x — %)+fA {y — v)+V Q (z— £)^-<sr (t — T).) 



If M is defined by equation (8), it is easy to prove that 

 when / and F are arbitrary functions the expressions 



satisfy the wave-equation (2), and that the potentials 

 A _ \l_ o- lg jr 2\ _ m <r m Q a 2fi ~) 



y (22) 



A = - - j- ^o — 2v &-P- a. t° a ?? 



2 M w + M iv - M ' M w + M iv " M ' J 



which consequently satisfy (2), are connected by the re- 

 lation (3). 



The electromagnetic field specified by these potentials has 

 singularities at space-time points for which <r is zero. To 



(18) 



(19) 



