and applying the relation (45) and (46), we have F = [ω. Φ] + _i_μ[ω. ψ]* 55) _f_ = ε[ω. Φ] + _i_[ω. ψ]* 56) _i.e._ F₁₂ = (ω₁ Φ₁ - ω₂ Φ₁) + _i_μ [ω₃ Ψ₄ - ω₄ ψ₃], etc. _f₁₂_ = ε(ω₁ Φ₂ - ω₂ φ₁) + _i_ [ω₃ ψ₄ - ω₄ ψ₃]., etc. Let us now consider the space-time vector of the second kind [Φ ψ], with the components [ Φ₂ ψ₃ - Φ₃ ψ₂, Φ₃ ψ₁ - Φ₁ ψ₃, Φ₁ ψ₂ - Φ₂ ψ₁ ] [ Φ₁ ψ₄ - Φ₄ ψ₁, Φ₂ ψ₄ - Φ₄ ψ₂, Φ₃ ψ₄ - Φ₄ ψ₃ ] Then the corresponding space-time vector of the first kind ω[Φ, ψ] vanishes identically owing to equations 9) and 53) for ω[Φ.ψ] = -(ωψ)Φ + (ωΦ)ψ Let us now take the vector of the 1st kind (57) Ω = _i_ω[Φψ]* with the components Ω₁ = -_i_ | ω₂ ω₃ ω₄ | | Φ₂ Φ₃ Φ₄ | | ψ₂ ψ₃ ψ₄ |, etc. Then by applying rule (45), we have (58) [Φ.ψ] = _i_[ωΩ]* _i.e._ Φ₁ψ₂ - Φ₂ψ₁ = _i_(ω₃Ω₄ - ω₄Ω₃) etc. The vector Ω fulfils the relation (ωΩ) = ω₁Ω₁ + ω₂Ω₂ + ω₃Ω₃ + ω₄Ω₄ = 0, (which we can write as Ω₄ = _i_(ω_{x}Ω₁ + ω_{y}Ω₂ + ω_{z}Ω₃) and Ω is also normal to ω. In case ω = 0, we have Φ₄ = 0, ψ₄ = 0, Ω₄ = 0, and [Ω₁, Ω₂, Ω₃ = | Φ₁ Φ₂ Φ₃ | |ψ₁ ψ₂ ψ₃ |. I shall call Ω, which is a space-time vector 1st kind the Rest-Ray. As for the relation E), which introduces the conductivity σ we have -ωS = -(ω₁_s₁_ + ω₂_s₂_ + ω₃_s₃_ + ω₄_s₄_) = (- | _u_ | C_{_u_} + ρ)/√(1 - _u²_) = ρ′. This expression gives us the rest-density of electricity (see §8 and §4). Then 61) = _s_ + (ω_ṡ_)ω represents a space-time vector of the 1st kind, which since ωω = -1, is normal to ω, and which I may call the rest-current. Let us now conceive of the first three component of this vector as the (_x_-_y_-_z_) co-ordinates of the space-vector, then the component in the direction of _u_ is C_{_u_} - (| _u_ | ρ′)/√(1 - _u²_) = (_c__{_u_} - | _u_ |ρ)/√(1 - _u²_) = J_{_u_}/(1 - _u²_) and the component in a perpendicular direction is C_{_u_} = J_{_ū_}. This space-vector is connected with the space-vector J = C - ρ_u_, which we denoted in §8 as the conduction-current. Now by comparing with Φ = -ωF, the relation (E) can be brought into the form {E} _s_ + (ω_ṡ_)ω = - σωF, This formula contains four equations, of which the fourth follows from the first three, since this is a space-time vector which is perpendicular to ω. Lastly, we shall transform the differential equations (A) and (B) into a typical form. §12. The Differential Operator Lor.