the Lorentz Transformation. 252 



Thus the transformation is 



I x' = X(u)(x—ut) 



\ t' = \(u)(t—ux/c 2 ), 



which is the transformation of Lorentz. 



(In the above we have assumed that 7 is not zero : if 7 is 

 zero, c is infinite, and we get the Newtonian transformation 



/ x' = X — lit 

 \ t'=--t. ) 



The transformations for the transverse coordinates are easily obtained. 

 For y' (say) cannot depend on x, t without violating- the hypothesis that 

 the space-time is isotropic, and we suppose the axes so oriented that 

 it does not depend on z. Thus y' = e(u)y, and the condition of con- 

 sistency gives the functional equation 



e(U) = e(u)€(iS) (9) 



Putting v= —u we have 



The first member is a constant, and therefore e(u) is a constant. This 

 constant is e(0), and the equation gives e(0) = l. Thus y'=y, and 

 similarly z' —z. 



The FitzGerald contraction and the invariance of the 

 velocity c follow at once from the transformation. The law 

 of composition of velocities 



tanh -1 U/e — tanh -1 u/c -f tanh --1 v/'c 



shows that the resultant of any number of velocities less 

 than c is itself less than c. c is thus an upper limit to the 

 velocities of moving bodies. Since these results, the 

 FitzGerald contraction and the invariance of the limiting 

 velocity, follow from the transformation, they must be 

 implied in our fundamental assumption of the consistency 

 of the transformations among themselves. But it must be 

 remembered that the Newtonian case (when c is infinite) 

 also satisfies the conditions ; and it is only r the experimental 

 discovery that a finite velocity is invariant that decides 

 against it. 



The Electromagnetic Variables. 



The transformation of the electromagnetic vectors can 

 be obtained by similar reasoning. The method generally 

 adopted is to seek a linear transformation which leaves 

 Maxwell's Equations unchanged in form. This of course 



