356 Mr. Megh Nad Saha on the 



The result can also be easily proved if we introduce a 

 Lorentz-transformation, by which the axis of motion becomes 

 the new-time-axis. Then in the expression (4), the four- 

 vector p (w) becomes 



Po(0, 0, 0, v/~l), 



and the problem is reduced to one at rest. The denominator 

 becomes equivalent to R 2 + /' 2 , where R is the perpendicular 

 from P on the axis of motion. 

 We have therefore 



.-if 



IT 1_ 



Po(i)dV _ P() (0, 0, 0, i) 

 R 2 + /'-~ R 



Now (0, 0, 0, a 4 ') are the components, in the transformed 

 system, of potential-four-vector a', whose components in the 

 original system are (a,, a 2 , a ;} , a 4 ). Re-transforming to the 

 original system, we have 



[a 1 ,a 2 ,a 3 ,a 4 ] = ^"-"^'"-^.. . . (6) 



Otherwise — When by means of an orthogonal Lorentz- 

 transformation, we transform from the system (#, y, 2, I) to 

 the system (a?', y\ z\ Z'), the generalized Laplacian D 2 & is 

 transformed to 



(^\2 ^2 ^2 ^\2 \ 



5^ + W* + 5?* + PV a=0 ' or ~ inS '- 



In the present case, the distribution on an infinite line is 

 along the Z'-axis. Therefore a must be independent of V , 

 from which 



S' 



a= 



vVHf + s' 



\/x' 2 +y' 2 + z' 2 is easily seen to be equivalent to. what we 

 have called R previously. 



Thus according to this method of investigation also, the 

 potential-four- vector 



p (w u w 2 , w 3 , u> 4 ) . 



a- s , .... (6) 



where R = perpendicular distance from the external point 

 («, b, c, X) on the axis of motion of the point-charge : — 

 direction cosines \/ — l(io 1 , w 2 , w 3 , w 4 ). 



