Page:Relativity (1931).djvu/160

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

$x'-ct'=0$

(2).

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

$(x'-ct')=\lambda (x-ct)$

(3).

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

$(x'+ct')=\mu (x+ct)\,$

(4).

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

$a={\frac {\lambda +\mu }{2}}$

and

$b={\frac {\lambda -\mu }{2}}$

we obtain the equations

$\left.{\begin{aligned}x'=&ax-bct\\ct'=&act-bx\end{aligned}}\right\}$

(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)