— 3 —
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
According to the analogy of the hyperboloid of two sheets, this consists of two sheets separated by . Let us consider the sheet in the region of , 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 , with its asymptotes (Fig. 1). Then let us draw an arbitrary radius vector of that hyperbola branch from the origin O, the tangent in at the hyperbola to the cutting with the asymptote given at the right, and completing to the parallelogram ; at last for what follows, is drawn to meet the x-axis at . Let us now take and as axes for the parallel coordinates with measuring rods , ; then that hyperbola branch is again expressed in the form 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 .
Now let us increase c to infinity, thus 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 more and more approaches x. With respect to this it is clear that the
