Page:Relativity (1931).djvu/160

 relative to the system $$K'$$ will be represented by the analogous formula

Those space-time points (events) which satisfy (1) must also satisfy (2). Obviously this will be the case when the relation

is fulfilled in general, where $$\lambda$$ indicates a constant; for, according to (3), the disappearance of $$(x - ct)$$ involves the disappearance of $$(x' - ct')$$. If we apply quite similar considerations to light rays which are being transmitted along the negative $$x$$-axis, we obtain the condition

By adding (or subtracting) equations (3) and (4), and introducing for convenience the constants $$a$$ and $$b$$ in place of the constants $$\lambda$$ and $$\mu$$ where

and

we obtain the equations

{{numbered equation||$$\left. \begin{align} x' = & ax - bct \\ ct' = & act - bx \end{align} \right \}$$|(5).}}

We should thus have the solution of our problem, if the constants $$a$$ and $$b$$ were known. These result from the following discussion.

For the origin of $$K'$$ we have permanently $$x' = 0$$, and hence according to the first of the equations (5)