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Since equation (8a) must hold for points on the x-axis, we thus

have σ = 1. It is easily seen that the Lorentz transformation

really satisfies equation (11) for σ = 1; for (11) is a

consequence of (8a) and (9), and hence also of (8) and (9). We

have thus derived the Lorentz transformation.

The Lorentz transformation represented by (8) and (9) still

requires to be generalised. Obviously it is immaterial whether

the axes of K′ be chosen so that they are spatially parallel to

those of K. It is also not essential that the velocity of

translation of K′ with respect to K should be in the direction of

the x-axis. A simple consideration shows that we are able to

construct the Lorentz transformation in this general sense from

two kinds of transformations, viz. from Lorentz transformations

in the special sense and from purely spatial transformations.

which corresponds to the replacement of the rectangular

co-ordinate system by a new system with its axes pointing in

other directions.