XXXII

which is at rest with reference to $K^{\prime }$ must be exactly the same as the length, as judged from $K^{\prime }$, of a unit measuring-rod which is at rest relative to $K$. In order to see how the points of the $x^{\prime }$-axis appear as viewed from $K$, we only require to take a "snapshot" of $K^{\prime }$

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 eqn:(5) $x^{\prime }=ax.$

Two points of the $x^{\prime }$-axis which are separated by the distance $\Delta x^{\prime }=1$ when measured in the $K^{\prime }$ system are thus separated in our instantaneous photograph by the distance $\Delta x=\frac{1}{a}.(7)$

But if the snapshot be taken from $K^{\prime }$ ($t^{\prime }=0$), and if we eliminate $t$ from the equations eqn:(5), taking into account the expression eqn:(6), we obtain $x^{\prime }=a(1-\frac{v^2}{c^2})x.$

From this we conclude that two points on the $x$-axis and separated by the distance $1$ (relative to $K$) will be represented on our snapshot by the distance $\Delta x^{\prime }=a(1-\frac{v^2}{c^2}).(7a)$

But from what has been said, the two snapshots must be identical; hence $\Delta x$ in eqn:(7) must be equal to $\Delta x^{\prime }$ in eqn:(7a), so that we obtain $a^2=\frac{1}{1-{\displaystyle \frac{v^2}{c^2}}}.(7b)$

The equations eqn:(6) and eqn:(7b) determine the constants $a$ and $b$. By inserting the values of these constants in eqn:(5), we obtain the first and the fourth of the equations given in [chapter:XI]Section XI. $\left.

$$\begin{gathered} x^{\prime }\&=\frac{x-vt}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}, \\ {t}^{\prime }\&=\frac{t-{\displaystyle \frac{v}{c^2}}x}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}. \end{gathered}$$

\right} (8)$

Thus we have obtained the Lorentz transformation

for events on the $x$-axis. It satisfies the condition $x^{\prime 2}-c^2{t}^{\prime 2}=x^2-c^2{t}^2.(8a)$

The extension of this result, to include events which take place outside the $x$-axis, is obtained by retaining equations eqn:(8) and supplementing them by the relations $\left.

\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 $K^{\prime }$. 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 $r=\sqrt{x^2+y^2+z^2}=ct,$

or, if we square this equation, according to the equation $x^2+y^2+z^2-c^2{t}^2=0.(10)$