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
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