| P_{_x_} P_{_y_} P_{_z_} P_{_l_} | | | | U_{_x_}^* U_{_y_}^* U_{_z_}^* U_{_l_}^* | | | | V_{_x_}^* V_{_y_}^* V_{_z_}^* V_{_l_}^* | Leaving aside the first column we obtain D_{_x_} = P_{_y_}(U_{_z_}^* V_{_l_}^* - U_{_l_}^* V_{_z_}^*) + P_{_z_}(U_{_l_}^* V_{_y_}^* - U_{_y_}^* V_{_l_}^*) + P_{_l_}(U_{_y_}^* V_{_z_}^* - U_{_z_}^* V_{_y_}^*) = P_{_y_} φ_{_z_ _y_}^* + P_{_z_}^* φ_{_l_ _y_} + P_{_l_} φ^*_{_y_ _z_}. = P_{_y_} φ_{_x_ _y_} + P_{_z_} φ_{_x_ _z_} + _P__{_l_} φ_{_x_ _l_}, which coincides with (Pφ_{_x_}) according to our definition. Examples of this type of vectors will be found on page 36, Φ = wF, the electrical-rest-force, and ψ = 2wf^*, the magnetic-rest-force. The rest-ray Ω = iw[Φψ]^* also belong to the same type (page 39). It is easy to show that Ω = -_i_ | w₁ w₂ w₃ w₄ | | Φ₁ Φ₂ Φ₃ Φ₄ | | ψ₁ ψ₂ ψ₃ ψ₄ | When (Ω₁, Ω₂, Ω₃) = 0, w₄ = _i_, Ω reduces to the three-dimensional vector | Ω₁, Ω₂, Ω₃ | = | Φ₁ Φ₂ Φ₃ | | | | ψ₁ ψ₂ ψ₃ | Since in this case, Φ₁ = w₄ F₁₄ = _e__{_n_} (the electric force) ψ₁ = -_i_w₄ f₂₃ = _m__{_x_} (the magnetic force) we have (Ω) = | _e__{_x_} _e__{_y_} _e__{_z_} | | _m__{_x_} _m__{_y_} _m__{_z_} | [M. N. S.] Note 16. The electric-rest force. (Page 37.) The four-vector φ = wF which is called by Minkowski the electric-rest-force (elektrische Ruh-Kraft) is very closely connected to Lorentz’s Ponderomotive force, or the force acting on a moving charge. If ρ is the density of charge, we have, when ε = 1, μ = 1, _i.e._, for free space ρ₀φ₁ = ρ₀[w₁ F₁₁ w₂ F₁₂ + w₃ F₁₃ + w₄ F₁₄] = ρ₀/(√(1 - V²/_c²_)) [_d__{_x_} + 1/_c_ (_v₂_ _h₃_ - _v₃_ _h₂_)] Now since ρ₀ = ρ√(1 - V²/_c²_) We have ρ₀φ₁ = ρ[_d__{_x_} + 1/_c_ (_v₂_ _h₃_ - _v₃_ _h₂_)] N. B.—We have put the components of _e_ equivalent to (_d__{_x_}, _d__{_y_}, _d__{_z_}), and the components of _m_ equivalent to _h__{_x_} _h__{_y_} _h__{_z_}), in accordance with the notation used in Lorentz’s Theory of Electrons. We have therefore ρ₀ (φ₁, φ₂, φ₃) = ρ (_d_ + 1/_c_ [_v_·_h_]), _i.e._, ρ₀ (φ₁, φ₂, φ₃) represents the force acting on the electron. Compare Lorentz, Theory of Electrons, page 14.