396 Mr. 0. Heaviside on Electromagnetic Waves, and the 



If this case be taken literally, then, since it involves an 

 infinite concentration in a geometrical line of a finite amount 

 of vorticity of e, the result for the steady field is infinite close 

 up to that line, and the energy is infinite. But imagine, 

 instead, the vorticity to be spread over a zone at the equator 

 of the sphere r = a, half on each side of it, and its surface- 

 density to hefiv, where /i is finite. Consider the effect pro- 

 duced at a point in the equatorial plane. From time t = to 

 t 1 =r—a (if the point be external) there is no disturbance. 

 But from time t 1 to t 2 = b/v, where b is the distance from the 

 point to the edges of the zone, the disturbance must be iden- 

 tically the same as if the harmonic distribution J\v were 

 complete, viz. by (142), 



H =(fc)K 1+ ^> • • • *"» 



After this moment t 2 , the formula of course fails. Nov nar- 

 row the band to width acW at the equator and simultaneously 

 increase / l5 so as to make f ^0=6, the strength of the shell 

 of impressed force when there is but one. The formula (175) 

 will now be true only for a very short time, and in the limit it 

 will be true only momentarily, at the front of the wave, viz. 



f l a/2ju, Q vr=K = e/2(jL vrtW } . . . , (176) 



going up infinitely as d6 is reduced. To avoid infinities in 

 the electric and magnetic forces we must seemingly keep either 

 to finite volume or finite surface-density of vorticity of e, just as 

 in electrostatics with respect to electrification. 



Instead of a simple shell of impressed electric force, it may 

 be one of magnetic force, with similar results. As a verifica- 

 tion calculate the displacement through circle v on the sphere 

 r=a clue to a vortex circle at v x on the same surface, the 

 latter being of unit strength. It is 



^avce m vQ" m 



2 ~22^TT { } 



due to 2e M Q m , through the circle v. Take then 



(2m + l) yi »Q; 



2m{m + l) 



which represents e m due to vortex line of unit strength at v v 

 Use this in the preceding equation (177) and we obtain 



D=2 ^!f«yjQ4 (I7!I) 



(178) 



