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under which the physical laws come forth, gain in intelligibility, as I shall presently show. Above all, the idea of acceleration becomes much more striking and clear.

I shall again use the geometrical method of expression, which presents itself by tacitly neglecting x from the triple x, y, z. Let us call any world-point O as a space-time-null-point. The cone

$c^{2}t^{2}-x^{2}-y^{2}-z^{2}=0\,$

Fig. 2

consists of two parts with O as apex, one part having $t<0$, the other having $t>0$ (Vide fig. 2).. The first, the fore-cone consists of all those points which "send light towards O", the second, the aft-cone, consists of all those points which "receive their light from O". The region bounded by the fore-cone may be called the "fore-side of O", and the region bounded by the aft-cone may be called the "aft-side of O". On the aft-side of O we have the already considered hyperboloidal shell

$F=c^{2}t^{2}-x^{2}-y^{2}-z^{2}=l,\ t>0$.

The region between the cones will be occupied by the hyperboloid forms of one sheet

$-F=x^{2}+y^{2}+z^{2}-c^{2}t^{2}=k^{2}\,$,

to all constant positive values $k^{2}$. The hyperbolas which lie upon this figure with O as center, are important for us. For the sake of shortness the individual branches of this hyperbola will be called the interhyperbola to center O. Such a hyperbolic branch, when thought of as a world-line of a substantial point, would represent a motion which for $t=-\infty$ and $t=+\infty$ asymptotically approaches the velocity of light c.

If, by way of analogy to the idea of vectors in space, we call any directed length in the manifoldness x, y, z, t a vector, then we have to distinguish between a time-like vector directed from O towards the sheet $+F=1$, $t>O$ and a space-like vector directed from O towards the sheet $-F=1$. The time-axis can be parallel to any vector of the first kind. Any world-point between the fore and aft cones of O, may by means of the system of reference be regarded either as synchronous with O, as well as later or earlier than O. Every world-point on the fore-side of O is necessarily