Page:The Meaning of Relativity - Albert Einstein (1922).djvu/49

$K'$ relatively to $K$, instead of the imaginary angle $\Psi$. We have, first,

${\begin{aligned}x_{1}'&=x_{1}\cos \psi -il\sin \psi \\l'&=-ix_{1}\sin \psi +l\cos \psi \end{aligned}}$

Since for the origin of $K'$, i.e., for $x_{1}=0$, we must have $x_{1}=vl$, it follows from the first of these equations that

$v=i\tan \psi$

(27)

and also

$\left.{\begin{aligned}\sin \psi &={\frac {-iv}{\sqrt {1-v^{2}}}}\\\cos \psi &={\frac {1}{\sqrt {1-v^{2}}}}\end{aligned}}\right\},$

(28)

so that we obtain

$\left.{\begin{aligned}x_{1}'&={\frac {x_{1}-vl}{\sqrt {1-v^{2}}}}\\l'&={\frac {l-vx_{1}}{\sqrt {1-v^{2}}}}\\x_{2}'&=x_{2}\\x_{3}'&=x_{3}\end{aligned}}\right\}$

(29)

These equations form the well-known special Lorentz transformation, which in the general theory represents a rotation, through an imaginary angle, of the four-dimensional system of co-ordinates. If we introduce the ordinary time $t$, in place of the light-time $l$, then in (29) we must replace $l$ by $ct$ and $v$ by ${\frac {v}{c}}$.

We must now fill in a gap. From the principle of the constancy of the velocity of light it follows that the equation

$\sum \Delta x_{\nu }^{2}=0$