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-vector directed from O towards the sheet ±F = 1, _t_ > 0 and a space-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 always earlier, every point on the aft side of O, later than O. The limit _c_ = ∞ corresponds to a complete folding up of the wedge-shaped cross-section between the fore and aft cones in the 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 components. If the directions of the two vectors are respectively the directions of the radius vector OR to one of the surfaces ±F = 1, and of a tangent RS at the point R of the surface, then the vectors shall be called normal to each other. Accordingly _c²__tt₁_ - _xx₁_ - _yy₁_ - _zz₁_ = 0, which is the condition that the vectors with the components (_x_, _y_, _z_, _t_) and (_x₁_ _y₁_ _z₁_ _t₁_) are normal to each other. For the _measurement_ of vectors in different directions, the unit measuring rod is to be fixed in the following manner;—a space-like vector from 0 to -F = I is always to have the measure unity, and a time-like vector from O to +F = 1, _t_ > 0 is always to have the measure 1/_c_. Let us now fix our attention upon the world-line of a substantive point running through the world-point (_x_, _y_, _z_, _t_); then as we follow the _progress_ of the line, the quantity _d_τ = (1/_c_) √(_c²__dt²_ - _dx²_ - _dy²_ - _dz²_), corresponds to the time-like vector-element (_dx_, _dy_, _dz_, _dt_). The integral τ = ∫_d_τ, taken over the world-line from any fixed initial point P₀ to any variable final point P, may be called the “Proper-time” of the substantial point at P₀ 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” τ; let ([._x_], [._y_], [._z_], [._t_]) denote the first differential-quotients, and ([.._x_], [.._y_], [.._z_], [.._t_]) the second differential quotients of (_x_, _y_, _z_, _t_) with regard to τ, then these may respectively be called the _Velocity-vector_, and the _Acceleration-vector_ of the substantial point at P. Now we have _c²_ [._t²_] - [._x²_] - [._y²_] - [._z²_] = _c²_ _c²_ [._t_][.._t_] - [._x_][.._x_] - [._y_][.._y_] - [._z_][.._z_] = 0