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