the Lorentz Transformation. 255 



where the functions a, /3, 7, 8 contain real and imaginary 

 parts. Moreover, since we can change the sign of w, P, P', 

 and not of Q, Q' without altering the transformation, 01(11), 

 8(u) must be even functions, and j3(it), y(u) odd functions of 

 the argument u. 



The conditions for the consistency of these equations are 



r«(U) = «(«)«(«)+/3(«) 7 («J . . . (12) 

 j/3(U) = «(r)/3(»)+/3(>)S(«) . . . (13) 

 y(U) = y{v)*(u)+8(v)j(u) . . . (14) 



L S(U) = 7 (r) /?(«)! S(r)«(«) • • • (15) 



where U= -— *- — — ^: and these are to hold for all sufficiently 

 1 + uv/cr J 



>/< 



small real values of u and v. 



From (12) interchanging u and v we have 



whence -~r = —7^ = = k\ a constant. 



y(u) y(v) 



Similarly, from (13) we have 



«(w) j8(H)+j8(r) S(w) =a(tt) /3(v)+/3(u) S(v) 



or /8(t*)i«W-8W^^(i;)i*(i«)-«(u)}. 



«(m) — 8 {u) . , . ., . „ 



Hence — ^—- is constant ; and similarly rrom (14), 



p(u) J v ' 



— r — - is constant. But oL(it), 8(u) are even functions 

 y{u) 



and ft{u) 9 y(u) odd functions of u, and therefore a(uj—S(u) 



is zero for all values of the argument — unless, indeed, both 



f}(u) and y(u) are identically zero. Thus ot(u)=8(u). 



Our equations (12)... (15) now reduce to the two 



symmetrical equations 



a (V)=u(u)*(v) + k*y(u)y(v) . . . (16) 

 y(U) =*{u) y(v) +y{u) u{v), 



I «(m)=8(w) 

 /8(u)=*» 7 (u) 



and U= r^Tl =c tanh (tanh" 1 14 V 4- binh" 1 r c). 



1 + wv/c- 



with 1 



