The fourth component φ₄ when multiplied by ρ₀ represents _i_-times the rate at which work is done by the moving electron, for ρ₀ φ₄ = _i_ρ [_v__{_x_}_d__{_x_} + _v__{_y_}_d__{_y_} + _v__{_z_}_d__{_z_}] = _v__{_x_} ρ₀φ₁ + _v__{_y_} ρ₀φ₂ + _v__{_z_} ρ₀φ₃. -√(-1) times the power possessed by the electron therefore represents the fourth component, or the time component of the force-four-vector. This component was first introduced by Poincare in 1906. The four-vector ψ = _i_ωF^* has a similar relation to the force acting on a moving magnetic pole. [M. N. S.] Note 17. Operator “Lor” (§ 12, p. 41). The operation | ∂/∂_x₁_ ∂/∂_x₂_ ∂/∂_x₃_ ∂/∂_x₄_ | which plays in four-dimensional mechanics a rôle similar to that of the operator (_i_∂/∂_x_, + _j_∂/∂_y_, + _k_∂/∂_z_ = ▽) in three-dimensional geometry has been called by Minkowski ‘Lorentz-Operation’ or shortly ‘lor’ in honour of H. A. Lorentz, the discoverer of the theorem of relativity. Later writers have sometimes used the symbol □ to denote this operation. In the above-mentioned paper (Annalen der Physik, p. 649, Bd. 38) Sommerfeld has introduced the terms, Div (divergence), Rot (Rotation), Grad (gradient) as four-dimensional extensions of the corresponding three-dimensional operations in place of the general symbol lor. The physical significance of these operations will become clear when along with Minkowski’s method of treatment we also study the geometrical method of Sommerfeld. Minkowski begins here with the case of lor S, where S is a six-vector (space-time vector of the 2nd kind). This being a complicated case, we take the simpler case of lor _s_, where _s_ is a four-vector = | _s₁_, _s₂_, _s₃_, _s₄_ | and _s_ = | _s₁_ | | _s₂_ | | _s₃_ | | _s₄_ | The following geometrical method is taken from Sommerfeld. Scalar Divergence—Let ΔΣ denote a small four-dimensional volume of any shape in the neighbourhood of the space-time point Q, _d_S denote the three-dimensional bounding surface of ΔΣ, _n_ be the outer normal to _d_S. Let S be any four-vector, P_{_n_} its normal component. Then Div S = Lim ∫ P_{_n_}_d_S/ΔΣ. ΔΣ = 0 Now if for ΔΣ we choose the four-dimensional parallelopiped with sides (_dx₁_, _dx₂_, _dx₃_, _dx₄_), we have then Div S = ∂_s₁_/∂_x₁_ + ∂_s₂_/∂_x₂_ + ∂_s₃_/∂_x₃_ + ∂_s₄_/∂_x₄_ = lor S.