146 On the Classification of Electromagnetic Fields. 



velocities equal to that of light, an electromagnetic field 

 which is built up from elementary fields of this type will be 

 said to belong to class B. 



In a field of the third type the primary singularities lie 

 along a curve at some instant t, at any subsequent time 

 there are singularities distributed all over a tubular surface 

 having the curve as axis. An electromagnetic field which is 

 built up from elementary fields of this type will be said to 

 belong to class C. 



In a field of the fourth type the primary singularity moves 

 with a velocity which is sometimes (or always) greater than 

 that of light, and secondary singularities are fired out in 

 various directions with the velocity of light. An electro- 

 magnetic field which is built up from elementary fields of 

 this type will be said to belong to class D. 



It is possible that all these different types of electro- 

 magnetic fields can be obtained from a single fundamental 

 type by superposition. The fundamental solution of the 

 equation of wave-motion which seems to be the most suitable 

 for such a purpose is of the form 



a = 



1 



J\j(x-a) + m(y-b)+n(z--c)-p(t-e)\> ' [ ^ J 



where a, b, c, e, I, m, n, p, A,, //,, v, vr are constants satisfying 

 equations (18) and (19). To obtain from this a solution of 

 the form (21) we must either regard a, b, c as functions of 

 the complex variable e and integrate round a closed contour 

 in the complex plane *, or else regard a, />, c as functions of 

 the real variable e and take the principal value of an integral. 

 The difficulties of the analysis are, however, so great that it 

 seems better to retain the four different classes of fields 

 with the understanding that it may be necessary to supple- 

 ment them. 



When the list is complete any real field which can be 

 derived by superposition from potentials of type (38) and 

 which is such that the electric and magnetic forces never 

 become infinite (although they may not satisfy Maxwell's 

 equations over the whole domain of the real variables 

 x, y, z, t), ought to be obtainable by superposition from 

 fields belonging to the different classes. A question which 



* This method has been used by Prof. A. W. Conway to obtain 

 Lienard's potentials. Proc. London Math. Soc. ser. 2, vol. i. 



