_x₁′_ = _x₁_, _x₂′_ = _x₂_, _x₃′_ = _x₃_ cos _i_ψ + _x₄_ sin _i_ψ, (1) _x₄′_´ = - _x₃_ sin _i_ψ + _x₄_ cos _i_ψ, Putting $$ - i \tan i\psi = \frac{e^{\psi} - e^{-\psi}}{e^{\psi}+e^{-\psi}} = q $$ , $$ \psi = \frac{1}{2} \log \frac{1 + q}{1 - q′} $$ (2) We shall have cos _i_ψ = 1/√(1 - _q²_), sin _i_ψ = _iq_/√(1 - _q²_) where -1 < _q_ < 1, and √(1 - _q²_) is always to be taken with the positive sign. Let us now write _x′₁_ = _x′_, _x′₂_ = _y′_, _x′₃_ = _z′_, _x′₄_ = _it′_ (3) then the substitution 1) takes the form _x′_ = _x_, _y′_ = _y_, _z′_ = (_z_ - _qt_)/√(1 - _q²_), _t′_ = (-_qz_ + _t_)/√(1 - _q²_), (4) the coefficients being essentially real. If now in the above-mentioned rotation round the Z-axis, we replace 1, 2, 3, 4 throughout by 3, 4, 1, 2, and φ by _i_ψ, we at once perceive that simultaneously, new magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄, where ρ′₁ = ρ₁, ρ′₂ = ρ₂, ρ′₃ = ρ₃ cos _i_ψ + ρ₄ sin _i_ψ, ρ′₄ = - ρ₃ sin _i_ψ + ρ₄ cos _i_ψ), and _f′__{1 2} ... _f′__{3 4}, where _f′__{4 1} = _f__{4 1} cos _i_ψ + _f__{1 3} sin _i_ψ, _f′__{1 3} = - _f__{4 1} sin _i_ψ + _f__{1 3} cos _i_ψ, _f′__{3 4} = _f__{3 4}, _f′__{3 2} = _f__{3 2} cos _i_ψ + _f__{4 2} sin _i_ψ, _f′__{4 2} = - _f__{3 2} sin _i_ψ + _f__{4 2} cos _i_ψ, _f′__{1 2} = _f__{1 2}, _f__{_k h_} = - _f′__{_k h_}, must be introduced. Then the systems of equations in (A) and (B) are transformed into equations (A´), and (B´), the new equations being obtained by simply dashing the old set. All these equations can be written in purely real figures, and we can then formulate the last result as follows. If the real transformations 4) are taken, and _x´_ _y´_ _z´_ _t´_ be taken as a new frame of reference, then we shall have (5) ρ´ = ρ [(-_qu__{_z_} + 1)/√(1 - _q²_)], ρ´_u__{_z_}´ = ρ[(_u__{_z_} - _q_)/√(1 - _q²_)], ρ´_u__{_x_}´ = ρ_u__{_x_}, ρ´_u__{_y_}´ = ρ_u__{_y_}. (6) _e´__{_x´_} = (_e__{_x_} - _qm__{_y_})/(√(1 - _q²_)), _m´__{_r´_} = (_qe__{_x_} + _m__{_y_})/(√(1 - _q²_)), _e´__{_z´_} = _e__{_z_}. (7) _m´__{_x´_} = (_m__{_x_} - _qe__{_y_})/(√(1 - _q²_)), _e´__{_y_´} = (_qm__{_x_} + _e__{_y_})/(√(1 - _q²_)), _m_´_{_z_´} = _m__{_z_}. Then we have for these newly introduced vectors _u´_, _e´_, _m´_ (with components _u__{_x_}´, _u__{_y_}´, _u__{_z_}´; _e__{_x_}´, _e__{_y_}´, _e__{_z_}´; _m__{_x_}´, _m__{_y_}´, _m__{_z_}´), and the quantity ρ´ a series of equations I´), II´), III´), IV´) which are obtained from I), II), III), IV) by simply dashing the symbols.