138 Dr. H. Bateman on the 



velocity greater than or equal to that of light. This case 

 has been discussed at some length by Sornmerfeld and 

 Schott (/. c). In the case of the electromagnetic field 

 defined by the potentials (9) there are primary singularities 

 distributed along the curve ®=%(u), y=v{ u )i z=£(u) at 

 time £ = 0, and these give rise to secondary singularities 

 which at time t lie on a tubular surface having the given 

 curve as axis. If we write x— a, y — b, z — c, t — r in place 

 of a;, y, z, t, and integrate over a suitable domain of the 

 variables a, b, c, r, we may get rid of the awkward infinities 

 of the electric and magnetic vectors, but the derivatives of 

 these vectors will not all be continuous over the regions 

 occupied by these infinities. The chief peculiarity of an 

 electromagnetic field of this kind is that a portion of matter 

 sends out radiations for a finite interval of time, and the 

 radiations, which seem to be partly of a material nature, travel 

 outwards with the velocity of light. The radiated matter, 

 however, seems generally to fill the spaces between a number 

 of pairs of moving surfaces, and so the present type of 

 electromagnetic field is essentially different from any of the 

 fields which have so far been observed in nature. 



An electromagnetic field of a more promising nature is 

 obtained by writing 



A>=2^, A y =2^, A 2 =2^, *=S^, . (12) 



W ID Z ID W V J 



where the summation extends over some of the values of u 

 for which equation (10) is satisfied and 



l 2 +m 2 + n 2 = l, (13) 



l?+iriy'+n? = 0, (14) 



w =l(<i>-Z) + m(y- V ) + n(z-Z)-L . . (15) 



If I, m, n are real, the quantity to vanishes only when 



'-T 1 -*?-'-?-!- • • • ^ 



The singularities of the electromagnetic field may be 

 described by saying that there are guns distributed along 

 the curve 



*=£(«), y=*(«), *=««)■ • • • (17) 



Each gun points in a direction at right angles to the 

 curve, and fires out a singularity or bullet at time £ = 0. The 

 bullets all move in straight lines with the velocity of light. 



