_v__{_x_}^* _u__{_x_} + _v__{_y_}^* _u__{_y_} + _v__{_z_}^* _u__{_z_} + _v__{_l_}^* _u__{_l_} = 0 _v__{_x_}^* _v__{_x_} + _v__{_y_}^* _v__{_y_} + _v__{_z_}^* _v__{_z_} + _v__{_l_}^* _v__{_l_} = 0 If we multiply these equations by _v__{_l_}, _u__{_l_}, _v__{_s_}, and subtract the second from the first, the fourth from the third we obtain _u__{_x_}^* φ_{_x_ _l_} + _u__{_y_}^* φ_{_y_ _l_} + _u__{_z_}^* φ_{_z_ _l_} = 0 _v__{_x_}^* φ_{_z_ _l_} + _v__{_y_}^* φ_{_y_ _l_} + _v__{_z_}^* φ_{_z_ _l_} = 0 multiplying these equations by _v__{_x_}^* . _u__{_x_}^*, or by _v__{_y_}^* . _u__{_y_}^*, we obtain φ_{_x_ _z_}^* φ_{_x_ _l_} + φ_{_y_ _z_}^* φ_{_y_ _l_} = 0 and φ_{_x_ _y_}^* φ_{_x_ _l_} + φ_{_z_ _x_}^* φ_{_z_ _l_} = 0 from which we have φ_{_y_ _z_}^* : φ_{_x_ _y_}^* : φ_{_z_ _x_}^* = φ_{_x_ _l_} : φ_{_z_ _l_} : φ_{_y_ _l_} In a corresponding way we have φ_{_y_ _z_} : φ_{_x_ _y_} : φ_{_z_ _x_} = φ_{_x_ _l_}^* : φ_{_z_ _l_}^* : φ_{_y_ _l_}^*. _i.e._ φ_{_i_ _k_}^* = λφ(_{_i_ _k_}) when the subscript (_ik_) denotes the component of φ in the plane contained by the lines other than (_ik_). Therefore the theorem is proved. We have (φ φ*) = φ_{_y_ _z_} φ_{_y_ _z_}^* + ... = 2 (φ_{_y_ _z_} φ_{_z_ _l_} + ...) = 0 The general six-vector _f_ is composed from the vectors φ, φ^* in the following way:— _f_ = ρφ + ρ^* φ^*, ρ and ρ^* denoting the contents of the pieces of mutually perpendicular planes composing _f_. The “conjugate Vector” _f_^* (or it may be called the complement of _f_) is obtained by interchanging ρ and ρ^*. We have _f_^* = ρ^*φ + ρφ^* We can verify that _f__{_y z_}^* = _f__{_x l_} etc. and _f²_ = ρ² + ρ^*², (_f__f_^*) = 2ρρ^*. | _f_ |² and (_f__f_^*) may be said to be invariants of the six vectors, for their values are independent of the choice of the system of co-ordinates. [M. N. S.] Note 12. Light-velocity as a maximum. Page 23, and Electro-dynamics of Moving Bodies, p. 17. Putting _v_ = _c_ - _x_, and _w_ = _c_ - λ, we get V = (2_c_ - (_x_ + λ))/(1 + (_c_ - _x_)(_c_ - λ)/_c²_) = (2_c_ - (_x_ + λ))/(_c²_ + _c²_ - (_x_ + λ)_c_ + _x_λ/_c²_) = _c_ (2_c_ - (_x_ + λ))/(2_c_ - (_x_ + λ) + _x_λ/_c_) Thus _v_ lt; _c_, so long as | _x_λ | > 0. Thus the velocity of light is the absolute maximum velocity. We shall now see the consequences of admitting a velocity W > _c_. Let A and B be separated by distance _l_, and let velocity of a “signal” in the system S be W > _c_. Let the (observing) system S′ have velocity +_v_ with respect to the system S. Then velocity of signal with respect to system S′ is given by W′ = (W - _v_)/(1 - W_v_/_c²_)