Motion of Electrification through a Dielectric. 325 



the vector of no divergence whose curl is 47r times the electric 

 current ; or the vector-potential of the curl of the current ; 

 or the curl of the vector-potential of the current, &c. &c. Thus, 

 as found by J. J. Thomson *, the magnetic field of a charge 

 moving at a speed which is a small fraction of that of light 

 is that which is commonly ascribed to a current-element 

 itself. I think it, however, preferable to regard the magnetic 

 field as the primary object of attention; or else to regard 

 the complete system of closed current derived from it by 

 taking its curl as the unit, forming what we may term a 

 rational current-element, inasmuch as it is not a mere mathe- 

 matical abstraction, but is a complete dynamical system 

 involving definite forces and energy. 



2. Let the axis of z be the line of motion of the charge q 

 at the speed u ; then the lines of magnetic force H are circles 

 centred upon the axis, in planes perpendicular to it, and its 

 tensor H at distance r from the charge, the line r making an 

 angle 6 with the axis, is given by 



B.= \umid = cY\uv, (1) 



r 



where v = sin 6, E the intensity of the radial electric force, c 

 the permittivity such that yu-^ct'^ = l, if /Xq is the other specific 

 quality of the medium, its inductivity, and v is the speed of 

 propagation. 



Since, under the circumstance supposed of ujv being very 

 small, the alteration in the electric field is insensible, and the 

 lines of E are radial, we may terminate the fields represented 

 by (1) at any distance r = a from the origin. We then 

 obtain the solution in the case of a charge q upon the surface 

 of a conducting sphere of radius a, moving at speed u. This 

 realization of the problem makes the electric and magnetic 

 energies finite. Whilst, however, agreeing with J. J. 

 Thomson in the fundamentals, I have been unable to corrobo- 

 rate some of his details ; and since some of his results have 

 been recently repeated by him in another place f, it may be 

 desirable to state the changes I propose, before proceeding to 

 the case of a charge moving at any speed. 



3. First, as regards the magnetic energy, say T. This is 

 the space-summation S/XoHVSvr ; or, by (1) j, 



* Phn. Mag. April 1881. 



t ' Applications of Dynamics to Physics and Chemistry,' chap. iv. 

 pp. 31 to 37. 



X < Electiician; Jan. 24, 1885, p. 220. 



