In all these operations the operator lor plays the part of a space-time vector of the first kind. If _f_ represents a space-time vector of the second kind,—lor _f_ denotes a space-time vector of the first kind with the components ∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_, ∂_f₂₁_/∂_x₁_ + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_, ∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + ∂_f₃₄_/∂_x₄_, ∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_ So the system of differential equations (A) can be expressed in the concise form {A} lor f = -_s_, and the system (B) can be expressed in the form {B} log F* = 0. Referring back to the definition (67) for log _ṡ_, we find that the combinations lor ([=(lor _f_)=]), and lor ([=(lor F*)]) vanish identically, when _f_ and F* are alternating matrices. Accordingly it follows out of {A}, that (68) (∂_s₁_/∂_x₁_) + (∂_s₂_/∂_x₂_) + (∂_s₃_/∂_x₃_) + (∂_s₄_/∂_x₄_) = 0, while the relation (69) lor (lor F*) = 0, signifies that of the four equations in {B}, only three represent independent conditions. I shall now collect the results. Let ω denote the space-time vector of the first kind (_u_/√(1 - _u²_}), _i_/√(1 - _u²_)) (_u_ = velocity of matter), F the space-time vector of the second kind (M,-_i_E) (M = magnetic induction, E = Electric force, _f_ the space-time vector of the second kind (_m_,-_ie_) (_m_ = magnetic force, _e_ = Electric Induction. _s_ the space-time vector of the first kind (C, _i_ρ) (ρ = electrical space-density, C - ρ_u_ = conductivity current, ε = dielectric constant, μ = magnetic permeability, σ = conductivity, then the fundamental equations for electromagnetic processes in moving bodies are[26] {A} lor _f_ = -_s_ {B} log F* = 0 {C} ω_f_ = εωF {D} ωF* = μω_f_* {E} _s_ + (ω_ṡ_), _w_ = - σωF. ω ῶ = -1, and ωF, ω_f_, ωF*, ω_f_*, _s_ + (ω_s_)ω which are space-time vectors of the first kind are all normal to ω, and for the system {B}, we have lor (lor F*) = 0. Bearing in mind this last relation, we see that we have as many independent equations at our disposal as are necessary for determining the motion of matter as well as the vector _u_ as a function of _x_, _y_, _z_, _t_, when proper fundamental data are given. § 13. The Product of the Field-vectors _f_ F. Finally let us enquire about the laws which lead to the determination of the vector ω as a function of (_x_, _y_, _z_, _t_.) In these investigations, the expressions which are obtained by the multiplication of two alternating matrices