Page:Relativity (1931).djvu/164

In order that equation (10a) may be a consequence of equation (10), we must have

$x'^{2}+y'^{2}+z'^{2}-c^{2}t'^{2}=\sigma \left(x^{2}+y^{2}+z^{2}-c^{2}t^{2}\right)$

(11).

Since equation (8a) must hold for points on the $x$-axis, we thus have $\sigma =1$. It is easily seen that the Lorentz transformation really satisfies equation (11) for $\sigma =1$; 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.

Mathematically, we can characterise the generalised Lorentz transformation thus:

It expresses $x'$, $y'$, $z'$, $t'$, in terms of linear homogeneous functions of $x$, $y$, $z$, $t$, of such a kind that the relation