Page:Grundgleichungen (Minkowski).djvu/10

can easily be transferred to any other axis when the system of axes are subjected to a transformation about this last axis. So we came to a more general law: —

Let ${\mathfrak {v}}$ be a vector with the components ${\mathfrak {v}}_{x},\ {\mathfrak {v}}_{y},\ {\mathfrak {v}}_{z}$, and let $\left|{\mathfrak {v}}\right|=q<1$. By ${\mathfrak {\bar {v}}}$ we shall denote any vector which is perpendicular to ${\mathfrak {v}}$, and by ${\mathfrak {r_{v}}}$, ${\mathfrak {r_{\bar {v}}}}$ we shall denote components of ${\mathfrak {r}}$ in direction of ${\mathfrak {\bar {v}}}$ and $\left|{\mathfrak {v}}\right|$.

Instead of x, y, z, t, new magnetudes x,' y,' z,' t' will be introduced in the following way. If for the sake of shortness, ${\mathfrak {r}}$ is written for the vector with the components x, y, z in the first system of reference, ${\mathfrak {r}}'$ for the same vector with the components x', y', z' in the second system of reference, then for the direction of ${\mathfrak {v}}$ we have

(10)	${\mathfrak {r'_{v}}}={\frac {r_{v}-qt}{\sqrt {1-q^{2}}}}$,

and for every perpendicular direction ${\mathfrak {\bar {v}}}$

(11)	${\mathfrak {r'_{\bar {v}}}}={\mathfrak {r_{\bar {v}}}}$,

and further

(12)	$t'={\frac {-q{\mathfrak {r_{v}}}+t}{\sqrt {1-q^{2}}}}$

The notations ${\mathfrak {r'_{v}}}$ and ${\mathfrak {r'_{\bar {v}}}}$ are to be understood in the sense that with the directions ${\mathfrak {v}}$, and every direction ${\mathfrak {v}}$ perpendicular to ${\mathfrak {\bar {v}}}$ in the system x, y, z are always associated the directions with the same direction cosines in the system x', y', z' ,

A transformation which is accomplished by means of (10), (11), (12) with the condition $0<q<1$ will be called a special Lorentz-transformation. We shall call ${\mathfrak {v}}$ the vector, the direction of ${\mathfrak {v}}$ the axis, and the magnitude of ${\mathfrak {v}}$ the moment of this transformation.

If further $\varrho '$ and the vectors ${\mathfrak {w}}',\ {\mathfrak {e}}',\ {\mathfrak {m}}'$, in the system x', y', z' are so defined that,

(13)	$\varrho '={\frac {\varrho (-q{\mathfrak {w_{v}}}+1)}{\sqrt {1-q^{2}}}}$,