252 Mr. L. A. Pars on 



where U =/(?«, v)=f(v, u). These are consistent it' 



f a(J]) = u(u)a(v)-ra(v)y(u) . . . (J) 

 [u«(U)-=M«(M)a(p) + »*(t;)8(w) . . (2) 



.7CU) = *(w)7(^+7(«)W • • • G>) 

 L 8( : U) = -ua(u)y(v) + $(u)8(v), . . (4) 



and these are to hold for all real values of u and v *. 

 From (2) putting #= — w we have, since f(u, ~u) — Q, 



ol(u) = 8(u). 



Then from (1) and (4) (or by interchanging u and v in 

 {1) or (4)) 



z* ot(u) y(v) = v ol(v) y(u), 



whence — t-t-= — ^r =....= — c , a real constant- 



70) 7( v ) 



Our equations (1) .... (4) now reduce to 



( ol(TJ) = *(u)x(v)(1+uv/c 2 ) . . . (5) 

 iU«(U) = «(w)*(t7)(ti + i?) (6) 



We thus obtain the law of composition of velocities 



U=/M = -^ (7> 



From (5) putting w = — u we get 



u(0) = a(ii) 2 {l-u 2 /c 2 ). 



The first member is a constant, and therefore the second 

 member also is constant for all values of u. Putting n~0. 

 we get a(0) = {a(0)} 2 , whence a(0; = l. 



Thus «(„)= J_^ =X(»), 



y(u) = — ^X(t*) 



* Or at any rate for all sufficiently small real val 



