to the Motion of Electrification through a Dielectric. 327 



6. Now if the external field be that of another moving 

 charge, we shall obtain the mutual magnetic energy from (3) 

 by letting Aq be the vector-potential of the current in the 

 second moving system, constructed so as to have no diver- 

 gence. Now the vector-potential of the convection-current ^u 

 is simply qxajr ; this is sufficient to obtain the magnetic force 

 by curling ; but if used to calculate the mutual energy, the 

 space-summation would have to include every element of 

 current in the other system. To make the vector-potential 

 divergenceless, and so be able to abolish this work, we 

 must add on to qu/r the vector-potential of the displacement 

 current to correspond. Now the complete current may be 

 considered to consist of a linear element qu having two poles ; 

 a radial current outward from the + pole in which the current- 

 density is qu/Airri^; and a radial current inward to the — pole, 

 in which the current-density is — qujAirr^ ; where r^ and r^ are 

 the distances of any point from the poles. The vector-potentials 

 of these currents are also radial, and their tensors are ^qu 

 and —^qu. We have now merely to find their resultant 

 when the linear element is indefinitely shortened^ add on to the 

 former qvijr, and multiply by //-oj to obtain the complete diver- 

 genceless vector-potential of qu, viz. : — 



A=^,?(u-i«v|), (5) 



where r is the distance from q to the point P when A is reckoned, 

 and the difierentiation is to s the axis of the convection- 

 current. Both it and the space -variation are taken at P. 

 The tensor of u is u. Though different and simpler in form 

 (apart from the use of vectors) this vector-potential is, I be- 

 lieve, really the same as the one used by J. J. Thomson. 

 From it we at once find, by the method described in § 4, the 

 mutual energy of ,a pair of point-charges g-, and q^ moving at 

 velocities Ui and n2 to be 



M: 



/^o7ig2 f . d'^v 



(d r \ 



when at distance r apart. Both axial diff'erentiations are to 

 be effected at one end of the line r. 



As an alternative form, let e be the angle between Ui and 

 U2, and let the differentiation to ^i be at dsi, that to ^3 at ds^, 

 as in the German investigations relating to current-elements ; 

 then* 



d'^r 

 \ " " ' ^ dsids2 

 * ' Electrician,' Dec. 28, 1888, p. 230. 



M = M^^(eos.H3^). ... (7) 



