the Lorentz Transformation. 251 



"veution they thereby set up measures of velocity of their own. 

 But are these measures which are set up by the second and 

 third observers such that their relative velocity is the same 

 to' both? We must therefore examine if the convention 

 involves any contradiction : and if not, under what conditions 

 it is permissible. 



Consider three observers A, B, C : let the velocity of B 

 relative to A be u, so that the velocity of A relative to 

 B is — u : and let the velocity of C relative to B be v, so 

 that the velocity of B relative to C is — v. Then the 

 velocity of C relative to A is some function of a and v which 

 we denote by f(u, v), and if the convention is permissible 

 the velocity of A relative to C will be —f(u, v). Now 

 let the direction of the axes for each observer be reversed: 

 then the velocity of B relative to C is v, and of A relative 

 to B is u : thus the velocity of A relative to C is f(v, u), 

 or if we revert to the original orientation, —f(v, u). Thus, 

 the convention involves no contradiction if, and only if 



f(u, v)=f(y, u), 



or the laiv of composition of velocities is commutative. 



We are now in a position to attack the main problem. 

 Let a?, t be A's system, and x', t', x" , t" those of B and C. 

 Then the relation of x, t to x' , t' is of the form 



l x' = 0t(u)x + j3(u)t 



\ t ! = y{u)x-\-8{u)t, 



where a, ft, 7, S are functions whose form is to be determined. 

 Some properties of these functions are evident Since the 

 space is isotropic the transformation must remain the same 

 if we change the signs of x, x', u but not of t, t' ; and this 

 requires that a, 8 are even functions and /3, 7 odd functions 

 of u. Again, u is the velocity of B as measured by A, so 

 x' = must yield 00 = ut, whence f3(u)= —uu(u). Thus the 

 transformation reduces to 



X' = u{ll) (X — Itt) 



t'=y{u)x + 8(u)t. 

 The transformations from B to C and from A to C are 

 a}" = u(vXx' — vt r ) 



r =700*'.+ 8(0)*', 



j *"=««(uj(*-uo 

 1 r=7(uv+S(UK, 



8 2 



{ 



{ 



