Page:Relativity (1931).djvu/162

But if the snapshot be taken from $K'(t'=0)$, and if we eliminate $t$ from the equations (5), taking into account the expression (6), we obtain

$x'=a\left(1-{\frac {v^{2}}{c^{2}}}\right)x$

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

$\Delta x'=a\left(1-{\frac {v^{2}}{c^{2}}}\right)$

(7a).

But from what has been said, the two snapshots must be identical; hence $\Delta x$ in (7) must be equal to $\Delta x'$ in (7a), so that we obtain

$a^{2}={\frac {1}{1-{\frac {v^{2}}{c^{2}}}}}$

(7b).

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

$\left.{\begin{aligned}x'={\frac {x-vt}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\\t'={\frac {t-{\frac {v}{c^{2}}}x}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\end{aligned}}\right\}$

(8).