eq. 38:  file eq38.gif

Thus we have obtained the Lorentz transformation for events on the x-axis.  It satisfies the condition

         x’2 — c^2t’2 = x2 — c^2t2 . . . (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

eq. 39:  file eq39.gif

In this way we satisfy the postulate of the constancy of the velocity of light in vacuo for rays of light of arbitrary direction, both for the system K and for the system K’.  This may be shown in the following manner.

We suppose a light-signal sent out from the origin of K at the time t = 0.  It will be propagated according to the equation

eq. 40:  file eq40.gif

or, if we square this equation, according to the equation

x2 + y2 + z2 = c^2t2 = 0 . . . (10).

It is required by the law of propagation of light, in conjunction with the postulate of relativity, that the transmission of the signal in question should take place —­ as judged from K1 —­ in accordance with the corresponding formula

r’ = ct’

or,

x’2 + y’2 + z’2 — c^2t’2 = 0 . . . (10a).

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

x’2 + y’2 + z’2 — c^2t’2 = s (x2 + y2 + z2 — c^2t2) (11).

Since equation (8a) must hold for points on the x-axis, we thus have s = I. It is easily seen that the Lorentz transformation really satisfies equation (11) for s = I; 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 K1 be chosen so that they are spatially parallel to those of K. It is also not essential that the velocity of translation of K1 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’, x’, t’, in terms of linear homogeneous functions of x, y, x, t, of such a kind that the relation

     x’2 + y’2 + z’2 — c^2t’2 = x2 + y2 + z2 — c^2t2 (11a).

is satisficd identically.  That is to say:  If we substitute their expressions in x, y, x, t, in place of x’, y’, x’, t’, on the left-hand side, then the left-hand side of (11a) agrees with the right-hand side.