and α' will be the same Function of (-v) that a' is of v.
But the transformation shows that if x1x2 be two points fixed relative to A and ξ1 ξ2 their coordinates in B at any time τ,
i. e. a line of length l as seen by A appears to be of length , as seen by B moving relatively to it. But this will be the same whichever be the direction of B's motion along the axis of x, so that if , i.e. .
Hence
, i. e. .
Thus the transformation is finally
Now let points not on the axis of x be considered. Since the axes of x and ξ coincide at all time, y and z always vanish when η and ζ vanish.
Hence and , and λ and μ will not change if the velocity of motion of B be changed from v to -v ; thus if ; .
But since by reversing the transformation
, and therefore λ=1.
Similarly μ=1.
The general transformation between x y z t and ξ η ζ τ is therefore