Fundamental Law of Electrical Action. 355 



in the previous section, the potential-four-vector a is given 

 by the integral 



p (w)dl' 



I 



0,-a) 2 + (y-6) 2 +(^- C ) 2 + G-X) 2 ' 



where p (iv) = p(u v u 2 , u 3 , \/— 1), 



and therefore p = p^/(l — u 2 ), the rest-density, which 

 is an invariant according to Lorentz and Einstein, 



d/' = an element of length along the axis of 

 motion ; dV is easily seen to be equivalent to dly/(l — u 2 ) . 



Now (x-a) 2 +(y-b) 2 +{z-c) 2 + (l-X) 2 



=■ Z 2 (l-u 2 ) + 2«7(u 1 a + u 2 ?> + u 3 c4-2X)- r -a 2 + 6 2 + c 2 +X 2 

 = I' 2 + 2il'(w l a + icj) + w z c + w±\) + a 2 + b 2 + c 2 -f- X 2 . 

 Putting r = V(l-u 2 ), 



this integral is easily seen to be equivalent to 



p (w) 



[a 2 H- b 2 + c 2 + X 2 + (ai^'i + 6w 2 + cw 3 + X«; 4 ) 2 ] * 



(5) 



With the aid of four-dimensional geometry, we can give 

 an interesting interpretation to this expression . The direction 

 of motion of the charge (p) is given by the line 



IV x IV 2 W 3 W± ' 



P (a,b,c,A) 



0/ 



(x,y,z,l) 



Let P be the point («, b, c, X). Thert we have 

 PN 2 =OP 2 -ON 2 



= (a 2 + b 2 + c 2 -f X 2 ) 4- (aztfi + 6w 2 + cw 3 + Xw 4 ) 2 , 



for ON = projection of OP on OA = i(w l a + iv <2 b + w 3 c-hw i \). 

 Thus the denominator in the expression (5) is seen to be 

 equivalent to R, where R is the perpendicular distance from 

 the external point on the axis of motion. 



2C2 



