Page:Relativity (1931).djvu/161

$x={\frac {bc}{a}}t$.

If we call $v$ the velocity with which the origin of $K'$ is moving relative to $K$, we then have

$v={\frac {bc}{a}}$

(6).

The same value $v$ can be obtained from equation (5), if we calculate the velocity of another point of $K'$ relative to $K$, or the velocity (directed towards the negative $x$-axis) of a point of $K$ with respect to $K'$. In short, we can designate v as the relative velocity of the two systems.

Furthermore, the principle of relativity teaches us that, as judged from $K$, the length of a unit measuring-rod which is at rest with reference to $K'$ must be exactly the same as the length, as judged from $K'$, of a unit measuring-rod which is at rest relative to $K$. In order to see how the points of the $x'$-axis appear as viewed from $K$, we only require to take a “snapshot” of $K'$ from $K$; this means that we have to insert a particular value of $t$ (time of $K$), e.g. $t=0$. For this value of $t$ we then obtain from the first of the equations (5)

$x'=ax$.

Two points of the $x'$-axis which are separated by the distance $\Delta x'=1$when measured in the $K'$ system are thus separated in our instantaneous photograph by the distance

$\Delta x={\frac {1}{a}}$

(7).