We should thus have the solution of our problem, if the constants a and b were known. These result from the following discussion.
For the origin of K′ we have permanently x′ = 0, and hence according to the first of the equations (5)
If we call v the velocity with which the origin of K′ is moving relative to K, we then have
The same value v can be obtained from equations (5), if we calculate the velocity of another point of K′ relative to K, or the velocity (directed towards the negative x-axis) of a point of K with respect to K′. In short, we can designate v as the relative velocity of the two systems.
Furthermore, the principle of relativity teaches us that, as judged from K, the length of a unit measuring-rod which is at rest with reference to K′ must be exactly the same as the length, as judged from K′, of a unit measuring-rod which is at rest relative to K. In order to see how the points of the x′-axis appear as viewed from K, we only require to take a “snapshot” of K′ from K; this means that we have to insert a particular value of t (time of K), e.g. t = 0. For this value of t we then obtain from the first of the equations (5)
x′ = ax
Two points of the x′-axis which are separated by the distance Δx′ = 1 when measured in the K′ system are thus separated in our instantaneous photograph by the distance