Page:The Meaning of Relativity - Albert Einstein (1922).djvu/51

The factor $\lambda$ therefore measures the lateral contraction of the cylinder, and can thus, from the above, be only an even function of $v$.

If we introduce a third system of co-ordinates, $K'$ which moves relatively to $K'$ with velocity $v$ in the direction of the negative $x$-axis of $K$ we obtain, by applying (29) twice,

${\begin{aligned}&x_{1}''=\lambda (v)\lambda (-v)&x_{1}\\&\vdots &\vdots \\&l''=\lambda (v)\lambda (-v)&l.\end{aligned}}$

Now, since $\lambda (v)$ must be equal to $\lambda (-v)$, and since we assume that we use the same measuring rods in all the systems, it follows that the transformation of $K''$ to $K$ must be the identical transformation (since the possibility $\lambda =-1$ does not need to be considered). It is essential for these considerations to assume that the behaviour of the measuring rods does not depend upon the history of their previous motion.

Moving Measuring Rods and Clocks. At the definite $K$-time, $l=0$, the position of the points given by the integers $x_{1}'=n$, is with respect to $K$, given by $x_{1}=n{\sqrt {1-v^{2}}}$; this follows from the first of equations (29) and expresses the Lorentz contraction. A clock at rest at the origin $x_{1}=0$ of $K$, whose beats are characterized by $l=n$, will, when observed from $K'$ have beats characterized by

$l'={\frac {n}{\sqrt {1-v^{2}}}};$

this follows from the second of equations (29) and shows