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 in space to do with this perfect freedom of the time-axis towards the upper direction?

To establish this connection, let us take a positive parameter c, and let us consider the figure

$c^{2}t^{2}-x^{2}-y^{2}-z^{2}=1.\,$

According to the analogy of the hyperboloid of two sheets, this consists of two sheets separated by $$t=0$$. Let us consider the sheet in the region of $$t>0$$, and let us now conceive the transformation of x, y, z, t into four new variables x', y', z', t', and the expression of this sheet in the new variables will be equivalent. Clearly the rotations of space round the null-point belongs to this group of transformations. We can already have a complete idea of the transformations, when we look upon one of them, in which y and z remain unaltered. Let us draw the cross section of that sheet with the plane of the x-and t-axes, i.e., the upper branch of the hyperbola $$c^{2}t^{2}-x^{2}=1$$, with its asymptotes (Fig. 1). Then let us draw an arbitrary radius vector $$OA'$$ of that hyperbola branch from the origin O, the tangent in $$A'$$ at the hyperbola to the cutting $$B'$$ with the asymptote given at the right, and completing $$OA'B'$$ to the parallelogram $$OA' B'C'$$; at last for what follows, $$B' C'$$ is drawn to meet the x-axis at $$D'$$. Let us now take $$OC'$$ and $$OA'$$ as axes for the parallel coordinates $$x', t'$$ with measuring rods $$OC'=1$$, $$OA'=1/c$$; then that hyperbola branch is again expressed in the form $$c^2t'^2-x'^2=1, t'> 0$$ and the transition from x, y, z, t to x' y z t' is one of the transitions in question. Let us add to those characteristic transformations an arbitrary displacement of the space- and time-nullpoints; by the we form a group of transformations still depending on the parameter c, which I may denote by $$G_c$$.

Now let us increase c to infinity, thus $$1/c$$ converges to zero, and it appears from the above figure, that the branch of the hyperbola gradually approaches the x-axis, the asymptotic angle extends becomes more obtuse, that the special transformation in the limit changes into one where the t-axis can have any direction upwards, and $$x'$$ more and more approaches x. With respect to this it is clear that the