But if the snapshot be taken from K′(t′ = 0), and if we eliminate t from the equations (5), taking into account the expression (6), we obtain
From this we conclude that two points on the x-axis separated by the distance 1 (relative to K) will be represented on our snapshot by the distance
But from what has been said, the two snapshots must be identical; hence Δx in (7) must be equal to Δx′ in (7a), so that we obtain
The equations (6) and (7b) determine the constants a and b. By inserting the values of these constants in (5), we obtain the first and the fourth of the equations given in Section XI.
Thus we have obtained the Lorentz transformation for events on the x-axis. It satisfies the condition