Fundamental Law of Electrical Action. 361 



We shall now introduce again the above-mentioned 

 Lorentz-transformation. Then we can write 



d% drj d%dv for dx dy dz dl, 



p (0, 0, 0, i) for p (w u w 2i iv 3 , w 4 ), 



and (f _f' ) 2+ ( ^_^ )2+(r _ ? / )2+(v ^ ) 2 



for (>-a) 2 -4- (y- Z>) 2 + (^-c) 2 + (Z -X) 2 . 



Now a', the transformed of a becomes 



"^JJJJ(f-r) 2 +(^-v) 2 +(f-r') 2 +(^-v / ) 2, 



integrated over the world-spnce 



(£-?o) 2 + (*-*,)»+ (?-5,) 2 < »■* (A). 



We shall first integrate over the new time-axis. The limits 

 are then from — ao loco. 



jYj ' ft,(0.0,0,i)rff<Mr 



'/\2' 



over the Spherical Volume (A). This is a three-dimensional 

 potential problem, and is easily seen to be 



g(0, 0, 0, t) 



w 



vtfo-f ) 2 + (%-V) 2 + (?.-r) 2 ' 



here 0= li f^o rff ^17 c/f, 



integrated over the spherical volume (A). 



Now ^(?o-r) 2 +(%-V) 2 +(?-?o) 2 



is the perpendicular distance from the external point 

 (f',77'5 ?'? O on the axis of motion ; we can denote this by R. 



Then , (0, 0, 0, i)e 



a ~ R~~ " 



Now a' is what the potential-four-vector a with the 

 components (a 1? a 2 , a 3 , a 4 ) becomes when the transformation 

 is introduced. Retransforming to the original coordinates, 

 we have 



[a 1 ,a 2) a 3 ,a 4 ]=(^^^. . . (1.) 



