362 Mr. Megh Nad Saba on the 



We can now express R in terms of the original system of 

 coordinates. : 



E2 = ( ,, o _ rt)2+( , /o _ 6)2 + (ro _ 6 . )2 + (/o _ x) 2 



+ [(.i'o-«M + (?/o— ^)w 2 + (-o — c)w z + (l —\)w 4 ] 2 , 



where (a? , j/ , c , / ) are the coordinates of the centre of the 

 electron, (a, b, c, \) those of the external point. 



N.B. The Scalar and Vector potentials due to the motion 

 of an electron were first obtained by Lienard and Wiechert * 

 about 1898. They were expressed in the forms 



*-r(l-u r /*)' [F,G,H]_ r(1 _ lW6)) , (12) 



where r is the distance of the external point from the 

 point occupied by the electron at a time (t — u r /c), etc. 

 (u l5 u 2 , u 3 ) are the velocity components at the time (t — r/c), 

 u r is the component of this velocity along the line of r. 



The expression (11) is in fact equivalent to the expression 

 (12), as the following reasoning will show. Suppose the 

 time-coordinates are so chosen that 



(.i' -a) 2 +(7/ -^ + (e -6') 2 + (/o-X) 2 = 0, 



i. e. c(t — t') = — r, 



r 



or t = t + -. 



We are in fact estimating the effect at the external point 

 r/c seconds after the electron had been in the position 



(#0» #o> ~o)- 

 Then, since 



w^^-ay+iy-by+te-cy+xi-xy 



■4- [ {% — a)w l + (y — b)w 2 + (z- c)w z + (I — \)w{\ 2 , 



we can, denoting by R' the four-vector with the components 

 {(*-a),fy-ft),(*-rt,a-X)}, 



write R 2 = R' 2 + (R'w) 2 , 



where (Ww) denotes the scalar product of the four- vectors 

 R/ and iv 



* EEclairage Electrique, vol. xvi. (1898) ; Wiechert, Ann. d. Physik, 

 vol. iv. 



