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always earlier, every point on the aft side of O, necessarily later than O. The limit $c=\infty$ corresponds to a complete folding up of the wedge-shaped cross-section between the cones in the plane manifoldness $t=0$. In the figure drawn, this cross-section has been intentionally drawn with a different breadth.

Let us decompose a vector drawn from O towards x, y, z, t into its four components x, y, z, t. If the directions of the two vectors are respectively the directions of the radius vector OR of O at one of the surfaces $\pm F=1$, and additionally a tangent RS at the point R of the relevant surface, then the vectors shall be called normal to each other. Accordingly

$c^{2}tt_{1}-xx_{1}-yy_{1}-zz_{1}=0,\,$

which is the condition that the vectors with the components x, y, z, t and $\left(x_{1}\ y_{1}\ z_{1}\ t_{1}\right)$ are normal to each other.

For the sums of vectors in different directions, the unit measuring rods are to be fixed in the following manner; — a space-like vector from to $-F=1$ is always to have the sum 1, and a time-like vector from O to $+F=1$ , $t>0$ is always to have the sum $1/c$.

Let us now fix our attention upon the world-line of a substantial point running through the world-point P(x, y, z, t); then as we follow the progress of the line, the quantity

$d\tau ={\frac {1}{c}}{\sqrt {c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}}},$

corresponds to the time-like vector-element dx, dy, dz, dt.

The integral $\tau =\int d\tau$ of this sum, taken over the world-line from any fixed initial point $P_{0}$ to any variable endpoint P, may be called the "proper-time" of the substantial point in P. Upon the world-line, we may regard x, y, z, t, i.e., the components of the vector OP, as functions of the "proper-time" $\tau$; let $\left({\dot {x}},\ {\dot {y}},\ {\dot {z}},\ {\dot {t}}\right)$ denote their first differential-quotients with respect to $\tau$, and $\left({\ddot {x}},\ {\ddot {y}},\ {\ddot {z}},\ {\ddot {t}}\right)$ their second differential quotients with respect to $\tau$, and denote the corresponding vectors, i.e. the derivation of the vector OP with respect to $\tau$ the motion-vector in P, and the derivation of this motion-vector with respect to $\tau$ the acceleration-vector in P. There we have

$\left.{\begin{array}{l}c^{2}{\dot {t}}^{2}-{\dot {x}}^{2}-{\dot {y}}^{2}-{\dot {z}}^{2}=c^{2}\\c^{2}{\dot {t}}{\ddot {t}}-{\dot {x}}{\ddot {x}}-{\dot {y}}{\ddot {y}}-{\dot {z}}{\ddot {z}}=0\end{array}}\right\}$

i.e., the motion-vector is the time-like vector in the direction of the world-line at P of sum 1, the acceleration-vector at P is normal to the motion-vector at P, and is in any case a space-like vector.