We remark here that _e__{_x_} - _qm__{_y_}, _e__{_y_} + _qm__{_x_} are components of the vector _e_ + [_vm_], where _v_ is a vector in the direction of the positive Z-axis, and | _v_ | = _q_, and [_vm_] is the vector product of _v_ and _m_; similarly -_qe__{_x_} + _m__{_y_}, _m__{_x_} + _qe__{_y_} are the components of the vector _m_ - [_ve_]. The equations 6) and 7), as they stand in pairs, can be expressed as. _e′__{_x′_} + _im′__{_x′_} = (_e__{_x_} + _im__{_x_}) cos _i_ψ + (_e__{_y_} + _im__{_y_}) sin _i_ψ, _e′__{_y′_} + _im′__{_y′_} = - (_e__{_x_} + _im__{_x_}) sin _i_ψ + (_e__{_y_} + _im__{_y_}) cos _i_ψ, _e′__{_z′_} + _im′__{_z′_} = _e′__{_z_} + _im__{_z_}. If φ denotes any other real angle, we can form the following combinations:— (_e′__{_x′_} + _im′__{_x′_}) cos. φ + (_e′__{_y″_} + _im′__{_y′_}) sin φ = (_e__{_x_} + _im__{_x_}) cos. (φ + _i_ψ) + (_e__{_y_} + _im__{_y_}) sin (φ + _i_ψ), = (_e′__{_x′_} + _im′__{_x′_}) sin φ + (_e′__{_y′_} + _im′__{_y′_}) cos. φ = - (_e__{_x_} + _im__{_x_}) sin (φ + _i_ψ) + (_e__{_y_} + _im__{_y_}) cos. (φ + _i_ψ). § 4. Special Lorentz Transformation. The rôle which is played by the Z-axis in the transformation (4) can easily be transferred to any other axis when the system of axes are subjected to a transformation about this last axis. So we came to a more general law:— Let _v_ be a vector with the components _v__{_x_}, _v__{_y_}, _v__{_z_}, and let | _v_ | = _q_ < 1. By _ṽ_ we shall denote any vector which is perpendicular to _v_, and by _r__{_v_}, _r__{_ṽ_} we shall denote components of _r_ in direction of _ṽ_ and _v_. Instead of (_x_, _y_, _z_, _t_), new magnetudes (_x′_ _y′_ _z′_ _t′_) will be introduced in the following way. If for the sake of shortness, _r_ is written for the vector with the components (_x_, _y_, _z_) in the first system of reference, _r′_ for the same vector with the components (_x′_ _y′_ _z′_) in the second system of reference, then for the direction of _v_, we have (10) _r′__{_v_} = (_r__{_v_} - _qt_)/√(1 - _q²_) and for the perpendicular direction _ṽ_, (11) _r′__{_ṽ_} = _r__{_ṽ_} and further (12) _t′_ = (-_qr__{_v_} + _t_)/√(1 - _q²_). The notations (_r′__{_ṽ_}, _r′__{_v_}) are to be understood in the sense that with the directions _v_, and every direction _ṽ_ perpendicular to _v_ in the system (_x_, _y_, _z_) are always associated the directions with the same direction cosines in the system (_x′_ _y′_ _z′_). A transformation which is accomplished by means of (10), (11), (12) with the condition 0 < _q_ < 1 will be called a special Lorentz-transformation. We shall call _v_ the vector, the direction of _v_ the axis, and the magnitude of _v_ the moment of this transformation. If further ρ′ and the vectors _u′_, _e′_, _m′_, in the system (_x′_ _y′_ _z′_) are so defined that,