Page:Relativity (1931).djvu/163

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

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

{{numbered equation||$$\left. \begin{align} y' = y \\ z' = z \end{align} \right \}$$|(9).}}

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

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

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 $$K'$$—in accordance with the corresponding formula

or,