to the Motion of Electrification through a Dielectric. 337 



which is one half the value due to an infinitely long (both 

 ways) straight current of strength qu. The notable thing is 

 the independence of the ratio u/v. 



But if w = u in (40), the result is zero, unless Vj = l, when 

 we have the result (41). But if P be still further to the left, 

 we shall have to add to (41) the solution due to the electrifi- 

 cation which is ahead of P. So when the line is infinitely 

 long both ways, we have double the result in (41), with 

 independence of u/v again. 



But should qhe a function of z, we do not have indepen- 

 dence of u/v except in the already considered case of u = v, 

 with plane waves, and no component of electric force parallel 

 to the line of motion. 



17. Next, let the electrified line be in steady motion per- 

 pendicularly to its length. 

 Let q be the linear density 

 (constant), the ^-axis that of 

 the motion, the l^'-axis coin- 

 cident with the electrified 

 line and that of 3/ upward 

 on the paper. Then the A 

 at P will be 



{l-uyv')i ^X2 + {x2^+f + z^l-uyv')-'}i' ^ ' 



where y and z belong to P, and Xi, x^ are the limiting values of 

 X in the charged fine. From this derive the solution in the 

 case of an infinitely long line. It is 



,E = g-g.- ^^~f(ff , H=cEmv, .... (43) 

 r 1 — V M / V ' 



where v=sin Q ; understanding that E is radial, or along ^P in 

 the figure, and H rectilinear, parallel to the charged line. 



Terminating the fields internally at r^a, we have the case 

 of a perfectly conducting cylinder of radius a, charged with 

 q per unit of length, moving transversely. When u=v there 

 is disappearance of E and H everywhere except in the plane 

 = ^'n; as in the case of the sphere, and consequent infinite 

 values. It is the curvature that permits this to occur, i. e. 

 producing infinite values ; of course it is the self-induction 

 that is the cause of the conversion to a plane wave, here and 

 in the other cases. There is some similarity between (43) and 

 (29). In fact, (43) is the bidimensional equivalent of (29). 



18. Coming next to a plane distribution of electrification, 

 Phil. Mag. S. 5. Vol. 27. No. 167. April 1889. Z 



