to the Motion of Electrification through a Dielectric. 335 



But (33) are not limited to the case of a single point-charge, 

 being true outside the electrification when there is symmetry 

 with respect to the s-axis, and the electrification is all moving 

 parallel to it at speed u. 



When u=v, Ei = 0, and E2 = E=)avH, so that we reduce to 



i|rm=0 (34) 



a all 



outside the electrification. Thus, if the electrification is on 



the axis of z, we have 



'Eilixv=^K = 2qvlr, (35) 



difi"ering from (31) only in that q, the linear density, may be 

 any function of z. 



14. If, in the solutions (29), we terminate the fields inter- 

 nally at r = a, the perpendicularity of E and the tangentiality 

 of H to the surface show that (29) represents the solutions^ in 

 the case of a perfectly conducting sphere of radius a, moving 

 steadily along the ^;-axis at the speed u, and possessing a total 

 charge" q. The energy is now finite. Let U be the total 

 electric and T the total magnetic energy. By space-integra- 

 tion of the squares of E and H we find that they are given by 



ujv 



f tan~* 





u/v 



f_ 1-u-lv- r 2uyv^-^ (^^>'-^)^"^ " \l-t^V.^)n ,37. 

 •""2^' 4 L 1-w'K (u/vXl-uVv'f -J 



in which u<v. When u = v, with accumulation of the charge 

 at the equator of the sphere, we have infinite values, and it 

 appears to be only possible to have finite values by making a 

 zone at the equator cylindrical instead of spherical. The 

 expression for T in (37) looks quite wrong ; but it correctly 

 reduces to that of equation (2) when u/v is infinitely small. 



15. The question now suggests itself. What is the state of 

 things when u>v? It is clear, in the first place, that there can 

 be no disturbance at all in front of the moving charge (at a point, 

 for simplicity). Next, considering that the spherical waves 

 emitted by the charge in its motion along the sr-axis travel at 

 speed V, the locus of their fronts is a conical surface whose 

 apex is at the charge itself, whose axis is that of z, and whose 

 semiangle is given by 



sin6=v/u (38) 



