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

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

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

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.

{{numbered equation||$$\left. \begin{align} x' = \frac{x - vt}{\sqrt{1 - \frac{v^2}{c^2}}} \\ t' = \frac{t - \frac{v}{c^2}x}{\sqrt{1 - \frac{v^2}{c^2}}} \end{align} \right \}$$|(8).}}