of *x, y, z, t* in *x', y', z', t'* with essentially real co-efficients, whereby the aggregrate transforms into , and to every such system of values *x, y, z, t* with a *positive t*, for which this aggregate , there always corresponds a *positive t';* this last is quite evident from the continuity of the aggregate *x, y, z, t*.

The last vertical column of co-efficients has to fulfill, the condition

(22) |

If then , and the Lorentz transformation reduces to a simple rotation of the spatial co-ordinate system round the world-point.

If are not all zero, and if we put

On the other hand, with every set of value of which in this way fulfill the condition 22) with real values of , we can construct the *special* Lorentz-transformation (16) with as the last vertical column, — and then every Lorentz-transformation with the same last vertical column supposed to be composed of the special Lorentz-transformation, and a rotation of the spatial co-ordinate system round the null-point.

The totality of all Lorentz-Transformations forms a *group*.

Under a *space-time vector of the 1st kind* shall be understood a system of four magnitudes with the condition that in case of a Lorentz-transformation it is to be replaced by the set , where these are the values obtained by substituting for in the expression (21).

Besides the time-space vector of the 1st kind we shall also make use of another spacetime vector of the first kind , and let us form the linear combination