The Followers #/ Maxwell. 341 



current. The defect, as Fitz Gerald showed, may be immediately 

 removed by assuming that a moving charge itself is to be counted 

 as a current-element : the total current, thus composed of the 

 displacement- currents and the convection-current, is circuital. 

 Making this correction, Fitz Gerald found that the magnetic 

 force due to a sphere of charge e moving with velocity v along 

 the axis of z is curl (0, 0, ev/r) a formula which shows that the 

 displacement-currents have no resultant magnetic effect, since 

 the term ev/r would be obtained from the convection-current 

 alone. 



The expressions obtained by Thomson and Fitz Gerald were 

 correct only to the first order of the small quantity v/c. The 

 effect of including terms of higher order was considered in 1889 

 by Oliver Heaviside,* whose solution may be derived in the 

 following manner : 



Suppose that a charged system is in motion with uniform 

 velocity v parallel to the axis of z ; the total current consists of 

 the displacement- cur rent E/4?rc 2 where E denotes the electric 

 force, and the convection-current pv where p denotes the 

 volume-density of electricity. So the equation which connects 

 magnetic force with electric current may be written 



E/c 2 = curl H - 4:irpv. 

 Eliminating E between this and the equation 



curl E = - H, 

 and remembering that H is here circuital, we have 



H/c 2 - V 2 H = 4?r curl pv. 

 If, therefore, a vector-potential a be defined by the equation 



a/c 3 - V 2 a = 4?rpv, 



the magnetic force will be the curl of a ; and from the equation 

 for a it is evident that the components a x and a y are zero, and 

 that a z is to be determined from the equation 

 a z /c~ - V"a z = 4npv. 



* Phil. Mag. xxvii (1889), p. 324. 



