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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

image045

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

image046

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.

image047

Thus we have obtained the Lorentz transformation for events on

the x-axis. It satisfies the condition

x′2 - c2t′2 = x2 - c2t2 . . . . . . (8a).

The extension of this result, to include events which take place

outside the x-axis, is obtained by retaining equations (8) and

supplementing them by the relations

image048