If _a₁₄_ = _a₂₄_ = _a₃₄_ = 0, then _a₄₄_ = 1, and the Lorentz transformation reduces to a simple rotation of the spatial co-ordinate system round the world-point. If _a₁₄_, _a₂₄_, _a₃₄_ are not all zero, and if we put _a₁₄_ : _a₂₄_ : _a₃₄_ : _a₄₄_ = _v__{_x_} : _v__{_y_} : _v__{_z_} : _i_ _q_ = √(_v__{_x_}² + _v__{_y_}² +_v__{_z_}²) < 1. On the other hand, with every set of values of _a₁₄_, _a₂₄_, _a₃₄_, _a₄₄_ 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 (_a₁₄_, _a₂₄_, _a₃₄_, _a₄₄_) as the last vertical column,—and then every Lorentz-transformation with the same last vertical column (_a₁₄_, _a₂₄_, _a₃₄_, _a₄₄_) 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 (ρ₁, ρ₂, ρ₃, ρ₄) with the condition that in case of a Lorentz-transformation it is to be replaced by the set (ρ₁′, ρ₂′, ρ₃′, ρ₄′), where these are the values of (_x₁′_, _x₂′_, _x₃′_, _x₄′_), obtained by substituting (ρ₁, ρ₂, ρ₃, ρ₄) for (_x₁_, _x₂_, _x₃_, _x₄_) in the expression (21). Besides the time-space vector of the 1st kind (_x₁_, _x₂_, _x₃_, _x₄_) we shall also make use of another space-time vector of the first kind (_y₁_, _y₂_, _y₃_, _y₄_), and let us form the linear combination (23) _f₂₃_(_x₂__y₃_ - _x₃__y₂_) + _f₃₁_(_x₃__y₁_ - _x₁__y₃_) + _f₁₂_(_x₁__y₂_ - _x₂__y₁_) + _f₁₄_(_x₁__y₄_ - _x₄__y₁_) + _f₂₄_(_x₂__y₄_ - _x₄__y₂_) + _f₃₄_(_x₃__y₄_ - _x₄__y₃_) with six coefficients _f₂₃_--_f₃₄_. Let us remark that in the vectorial method of writing, this can be constructed out of the four vectors. _x₁_, _x₂_, _x₃_; _y₁_, _y₂_, _y₃_; _f₂₃_, _f₃₁_, _f₁₂_; _f₁₄_, _f₂₄_, _f₃₄_ and the constants _x₄_ and _y₄_, at the same time it is symmetrical with regard the indices (1, 2, 3, 4). If we subject (_x₁_, _x₂_, _x₃_, _x₄_) and (_y₁_, _y₂_, _y₃_, _y₄_) simultaneously to the Lorentz transformation (21), the combination (23) is changed to: (24) _f₂₃′_(_x₂′__y₃′_ - _x₃′__y₂′_) + _f₃₁_(_x₃′__y₁′_ - _x₁′__y₃′_) + _f₁₂_ (_x₁′__y₂′_ - _x₂′__y₁′_) + _f₁₄′_(_x₁′__y₄′_) - _x₄′__y₁′_) + _f₂₄′_(_x₂′__y₄′_ - _x₄′__y₂′_) + _f₃₄′_(_x₃′__y₄′_ - _x₄′__y₃′_), where the coefficients _f₂₃′_, _f₃₁′_, _f₁₂′_, _f₁₄′_, _f₂₄′_, _f₃₄′_, depend solely on (_f₂₃_ _f₂₄_) and the coefficients _a₁₁_ ... _a₄₄_. We shall define a space-time Vector of the 2nd kind as a system of six-magnitudes _f₂₃_, _f₃₁_ ... _f₃₄_, with the condition that when subjected to a Lorentz transformation, it is changed to a new system _f₂₃′_ ... f₃₄, ... which satisfies the connection between (23) and (24).