the Lorent": Transformation. 257 



and the transformation is completely determined save for 

 the constants a and k. 



The determination of these constants involves, as we should 

 anticipate, some further convention to coordinate the electro- 

 magnetic measure in the two systems. The convention we 

 adopt is that both observers choose the same unit of charge. 

 Then a simple special case will serve to determine the con- 

 stants a and k. For example, let a unit charge be at rest at 

 the origin in B's system at time t 1 = : so that in A's system 

 it is passing the origin with velocity u at time t = 0. Consider 

 now the point «r = 0, y — 1, £ = 0, t — 0, which has the same 

 coordinates in both systems. For this point 



P' = l, Q'=0, 



P = A,, Q = iX u/c, 



from the known formulae for the field of a moving charge. 

 Hence 



1 = a(u) X-\-k 2 y(u) . iX ujc 



= <y(u) X -f <x(it) . iX u/c, 



and these two equations serve to determine a and k, for they 

 give 



1 2 •' ,2 



{u(u) + ky(u)\\x(u)-ky(u)\ =-- fj^^2> 

 and from (19) 



{ ot(u) + k y(u) } \ a{u) — k y(u) } = 1. 



Hence k 2 =— 1, k=zhi> Taking k=+i 



=\{l + u/c) 



V 1— u/c" 

 a=l. 

 We have therefore 



a(u) =8(u) =x 



iy(u) = —ifi(u) ==Xu/c. 



The other value £=— i leads to a= — 1 and yields the 

 same formula?. 



