Classification of Electromagnetic Fields. 145 



§5. The electromagnetic fields of sections 3 and 4 may 

 evidently be generalized by writing ,r + a, y + b, z + c, t-\-e 

 in place of a, y, z, t and integrating over some domain of 

 the point (a, b, c, e). It should be possible in this way to 

 get rid of the point singularities and replace them by 

 material particles throughout which the electric and mag- 

 netic forces are finite, but do not necessarily satisfy Maxwell's 

 equations. 



The electromagnetic fields of sections 3 and 4 may also be 

 generalized by adopting the method of section 2, but the 

 analysis is not of sufficient interest to be given in detail. 



The discussion of the singularities which has been given 

 here is not exhaustive, for we have omitted the case of 

 moving singularities at infinity ; the fields which are ob- 

 tained in this way, however, may be regarded as limiting 

 cases of those which have already been discussed. The case 

 of a primary singularity which moves with a velocity greater 

 than that of light has been mentioned only very briefly. 

 The investigation given in Schott's c Electromagnetic Radia- 

 tion ' will give an idea of what is to be expected in this case. 



Summing up the results of our investigations we are able 

 to enumerate four distinct types of elementary electro- 

 magnetic fields. 



In a field of the first type there is one point singularity 

 which moves with a velocity less than that of light, and a 

 constant electric charge is associated with the singularity. 

 An electromagnetic field which can be obtained by super- 

 posing elementary fields of the first type will be said to 

 belong to class A. We shall include under class A' fields 

 which can be obtained by superposing the fields due to 

 Hertzian oscillators in motion, an elementary field of this 

 type being represented by the potentials * 



A = curir-^+grad¥, 3> = divO-^, . (37) 



where the vectors F, O and the scalar "SP* are functions of the 



f(r) 

 type 4/p, t and M having the same meaning as in §1. 



It is probable that all fields of class A' can be regarded as 

 belonging to class A. 



In a field of the second type there is a point singularity 

 (the gun) which moves with a velocity less than that of light, 

 and point singularities are fired out from the gun with 



* These are immediate generalizations of the potentials used by 

 Prof. E. T. Whittaker, Proc. London Math. Soc. ser. % vol. i. (1903). 



Phil. Mag. S. 6. Vol. 27. No. 157. Jan. 1014. L 



