474 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The last equality is obtained when any k4 axis has been arbitrarily 

 chosen. Then v is the velocity of the electron and ^4— l^'V is the 

 distance FQ in Figure 22, that is, the projection of the distance from 

 the point of observation to the contemporaneous position of the 

 electron (if assumed to be moving uniformly) upon the line 1^ joining 

 the "retarded" position of the electron to the point of observation. 

 We may call m the extended electromagnetic vector potential. 

 Its projections on space and on the time-axis are respectively the 

 vector potential a and the scalar potential (p, 



a = J -. — . </> = -. , — ' (92) 



h — h'V h — h'V 



precisely in the form first obtained by Lienard.*^ From (69) we have 



O-m = (v - k4^^y(a + ct>k,) = 0. 



Hence ^ , dcf) 



V • a + - - = 0. 

 at 



We see therefore that the Lienard potentials are connected by the 

 same familiar equation as connects the ordinary ^'ector and scalar 

 potentials. Assuming that vector fields produced by two or more 

 electrons are additive, these equations are true for the general case. 



The 2-\'ector field produced by an electron, whether in uniform or 

 accelerated motion, is obtained immediately from (81)-(83). 



nxw. 



1,93) 



(94) 



The first term in this expression vanishes when the curvature is zero. 

 The fact that this term is a singular vector has already been pointed 

 out, and the great importance of this fact in electromagnetic theory 

 will be pointed out later. In the second term nxw is the unit 2-vector 

 determined by the line w and the point Q where the field is being dis- 

 cussed. 



49. In case the electron is unaccelerated the equation assumes the 

 simple form 



M = Inxw. (95) 



43 Eclairage electrique, 16, 5 (1S9S). 



