(13) ρ′ = ρ[(-_qu__{_v_} + 1)/√(1 - _q²_)], ρ′_u_′_{_v_} = ρ(_u__{_v_} - _q_)/√(1 - _q²_), ρ′_u__{_ṽ_} = ρ′_u__{_v_}, further (14) (_e′_ + _im′_)_{_ṽ_} = ((_e_ + _im_) - _i_[_v_, (_e_ + _im_])']_{_ṽ_})/√(1 - _q²_). (15) (_e′_ + _im′_)_{_v_} = (_e_ + _im_) - _i_[_u_, (_e_ + _im_)]_{_v_}. Then it follows that the equations I), II), III), IV) are transformed into the corresponding system with dashes. The solution of the equations (10), (11), (12) leads to (16) _r__{_v_} = (_r′__{_v_} + _qt′_)/√(1 - _q²_), _r__{_ṽ_} = _r′__{_ṽ_}, _t_ = (_qr′__{_v_} + _t′_)/√(1 - _q²_), Now we shall make a very important observation about the vectors _u_ and _u′_. We can again introduce the indices 1, 2, 3, 4, so that we write (_x₁_′, _x₂_′, _x₃_′, _x₄_′) instead of (_x′_, _y′_, _z′_, _it′_) and ρ₁′, ρ₂′, ρ₃′, ρ₄′ instead of (ρ′_u′_{_x′_}, ρ′_u′_{_y′_}, ρ′_u′_{_z′_}, _i_ρ′). Like the rotation round the Z-axis, the transformation (4), and more generally the transformations (10), (11), (12), are also linear transformations with the determinant + 1, so that (17) _x₁²_ + _x₂²_ + _x₃²_ + _x₄²_ _i. e._ _x²_ + _y²_ + _z²_ - _t²_, is transformed into _x₁′²_ + _x₂′²_ + _x₃′²_ + _x₄′²_ _i. e._ _x′²_ + _y′²_ + _z′²_ - _t′²_. On the basis of the equations (13), (14), we shall have (ρ₁² + ρ₂² + ρ₃² + ρ₄²) = ρ²(1 - _u__{_x²_}, -_u__{_y²_}, -_u__{_z²_}) = ρ²(1 - _u²_) transformed into ρ²(1 - _u²_) or in other words, (18) ρ√(1 - _u²_) is an invariant in a Lorentz-transformation. If we divide (ρ₁, ρ₂, ρ₃, ρ₄) by this magnitude, we obtain the four values (ω₁, ω₂, ω₃, ω₄) = (1/√(1 - _u²_))(_u__{_x_}, _u__{_y_}, _u__{_z_}, _i_) so that ω₁² + ω₂² + ω₃² + ω₄² = -1. It is apparent that these four values are determined by the vector _u_ and inversely the vector _u_ of magnitude < 1 follows from the 4 values ω₁, ω₂, ω₃, ω₄; where (ω₁, ω₂, ω₃) are real, -_i_ω₄ real and positive and condition (19) is fulfilled. The meaning of (ω₁, ω₂, ω₃, ω₄) here is, that they are the ratios of _dx₁_, _dx₂_, _dx₃_, _dx₄_ to (20) √(-(_dx₁²_ + _dx₂²_ + _dx₃²_ + _dx₄²_)) = _dt_√(1 - _u²_). The differentials denoting the displacements of matter occupying the spacetime point (_x₁_, _x₂_, _x₃_, _x₄_) to the adjacent space-time point. After the Lorentz-transformation is accomplished the velocity of matter in the new system of reference for the same space-time point (_x′_ _y′_ _z′_ _t′_) is the vector _u′_ with the ratios _dx′_/_dt′_, _dy′_/_dt′_, _dz′_/_dt′_, _dl′_/_dt′_, as components. Now it is quite apparent that the system of values _x₁_ = ω₁, _x₂_ = ω₂, _x₃_ = ω₃, _x₄_ = ω₄ is transformed into the values _x₁′_ = ω₁′, _x₂′_ = ω₂′, _x₃′_ = ω₃′, _x₄′_ = ω₄′ in virtue of the Lorentz-transformation (10), (11), (12).