140 Dr. H. Bateman on tlte 



see this we remark that there is a relation of the form 



cra — 6icw , ....... (23) 



where 6 is some function of r. The existence of a relation 

 of this type is easily realized by considering the particular 

 case whenpo = ^? ^0=^ anc ^ 



1 



\ = Y , ^o=0, v = - - x / r -/r ; }> (24) 



I = q, m = \/p 2 — q 2 -> n=Q : 



^0 = ?, - "0 = — \Zf — q 2 > }, o = o. 



The relation (23) is then verified at once by using (6). 



Since the general values of X, jjl, v, &c, may be derived 

 from these particular ones by a suitable orthogonal sub- 

 stitution, it is easy to see that a relation of type (23) holds 

 universally. 



We shall now assume that X, fi, v, vj } A, , /^o> v , «r are all 

 real, then it is easily seen that <r is zero when 



x — g y—y z—K t—r 



Hence the electromagnetic field has a singularity which 

 starts from the point (f , 97, f, t) and moves with the speed of 

 light along a straight line whose direction cosines are pro- 

 portional to (X , fjb , v ). Taking each point (f, 77, f, t) in 

 turn, we obtain all the singularities of the field in this way. 

 The moving point (f, 97, f, t) may be described as a ^w 

 which moves about in an arbitrary prescribed manner and 

 fires out " bullets " which travel with the speed of light. 

 The direction in which the gun points at any instant can be 

 chosen arbitrarily provided the differential coefficients of the 

 functions A,, //,, v . . . are continuous. 



After a long calculation we find that the component of 

 the electric force along the radius from (f, rj, f, t) to the 

 point (#, y, z, t) at which the force is required is 



where 



jp (A* + M +v? -„ ) - ^- - jp-, 



Po=W+OT V + Wof— p Q . 



