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

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

and further

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 -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,