Page:The principle of relativity (1920).djvu/89

 this last is quite evident from the continuity of the aggregate x, y, z, t. The last vertical column of co-efficients has to fulfil, the condition 22) a_{14}^2 + a_{24}^2 + a_{34}^2 + a_{44}^2 = 1.  If a_{14} = a_{24} = a_{34} = 0, then a_{44} = 1, and the Lorentz transformation reduces to a simple rotation of the spatial co-ordinate system round the world-point.  If a_{14}, a_{24}, a_{34} are not all zoro, and if we put a_{14} : a_{24} : a_{34} : a_{44} = v_{x} : v_{y} : v_{z} : i

q = [sqrt](v_{x}^2 + v_{y}^2 +v_{z}^2) < 1.

On the other hand, with every set of value of a_{14}, a_{24}, a_{34}, a_{44} which in this way fulfil the condition 22) with real values of v_{x}, v_{y}, v_{z}, we can construct the special Lorentz transformation (16) with (a_{14}, a_{24}, a_{34}, a_{44}) as the last vertical column,—and then every Lorentz-transformation with the same last vertical column (a_{14}, a_{24}, a_{34}, a_{44}) can be 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 ρ_{1}, ρ_{2}, ρ_{3}, ρ_{4}) with the condition that in case of a Lorentz-transformation it is to be replaced by the set ρ_{1}´, ρ_{2}´, ρ_{3}´, ρ_{4}´), where these are the values of x_{1}´, x_{2}´, x_{3}´, x_{4}´), obtained by substituting (ρ_{1}, ρ_{2}, ρ_{3}, ρ) for (x_{1}, x_{2??]}, x_{3}, x_{4}) in the expression (21).

Besides the time-space vector of the 1st kind (x_{1}, x_{2}, x_{3}, x_{4}) we shall also make use of another space-time vector of the first kind (y_{1}, y_{2}, y_{3}, y_{4}), and let us form the linear combination

(23) [function]_{2 3}(x_{2}y_{3} - x_{3}y_{2}) + [function]_{3 1}(x_{3}y_{1} - x_{1}y_{3}) + [function]_{1 2}(x_{1}y_{2} - x_{2}y_{1}) + [function]_{1 4}(x_{1}y_{4} - x_{4}y_{1}) + [function]_{2 4}(x_{2}y_{4} - x_{4}y_{2}) + [function]_{3 4}(x_{3}y_{4} - x_{4}y_{3})