The fact that in this special case, the relation is satisfied, suffices to establish the theorem (45) generally, for this relation has a covariant character in case of a Lorentz transformation, and is homogeneous in (ω₁, ω₂, ω₃, ω₄). After these preparatory works let us engage ourselves with the equations (C,) (D,) (E) by means which the constants ε μ, σ will be introduced. Instead of the space vector _u_, the velocity of matter, we shall introduce the space-time vector of the first kind ω with the components. ω₁ = _u__{_x_}/√(1 - _u²_), ω₂ = _u__{_y_}/√(1 - _u²_), ω₃ = _u__{_z_}/√(1 - _u²_), ω₄ = _i_/√(1 - _u²_). (40) where ω₁² + ω₂² + ω₃² + ω₄² = -1 and -_i_ω₄ > 0. By F and _f_ shall be understood the space time vectors of the second kind M - _i_E, _m_ - _ie_. In Φ = ωF, we have a space time vector of the first kind with components Φ₁ = ω₂F₁₂ + ω₃F₁₃ + ω₄F₁₄ Φ₂ = ω₁F₂₁ + ω₃F₂₃ + ω₄F₂₄ Φ₃ = ω₁F₃₁ + ω₂F₃₂ + ω₄F₃₄ Φ₄ = ω₁F₄₁ + ω₂F₄₂ + ω₃F₄₃ The first three quantities (φ₁, φ₂, φ₃) are the components of the space-vector (E + [_u_, M])/√(1 - _u²_), and further (φ₄ = _i_[_u_ E]/√(1 - _u²_). Because F is an alternating matrix, (49) ωΦ = ω₁ φ₁ + ω₂ Φ₂ + ω₃ Φ₃ + ω₄ Φ₄ = 0. _i.e._ Φ is perpendicular to the vector ω; we can also write Φ₄ = _i_[ω_{x} Φ₁ + ω_{y} Φ₂ + ω_{z} Φ₃]. I shall call the space-time vector Φ of the first kind as the _Electric Rest Force_.[24] Relations analogous to those holding between -ωF, E, M, U, hold amongst -ω_f_, _e_, _m_, _u_, and in particular -ω_f_ is normal to ω. The relation (C) can be written as {C} ω_f_ = εωF. The expression (ω_f_) gives four components, but the fourth can be derived from the first three. Let us now form the time-space vector 1st kind, ψ - _i_ω_f_*, whose components are ψ₁ = -_i_(ω₂ _f₃₄_ + ω₃ _f_₄₂ + ω₄ _f₂₃_) ψ₂ = -_i_(ω₁ _f_₄₃ + ω₃ _f_₄₄ + ω₄ _f₃₁_) ψ₃ = -_i_(ω₁ _f₂₄_ + ω₂ _f_₄₁ + ω₄ _f₁₂_) ψ₄ = -_i_(ω₁ _f_₃₂ + ω₂ _f_₁₃ + ω₃ _f_₂₁) Of these, the first three ψ₁, ψ₂, ψ₃, are the _x_, _y_, _z_ components of the space-vector 51) (m - (_ue_))/√(1 - _u²_) and further (52) ψ₄ = _i_(_u_m)/√(1 - _u²_). Among these there is the relation (53) ωψ = ω₁ ψ₁ + ω₂ ψ₂ + ω₃ ψ₃ + ω₄ ψ₄ = 0 which can also be written as ψ₄ = _i_ (_u__{_x_} ψ₁ + _u__{_y_} ψ₂ + _u__{_z_} ψ₃). The vector ψ is perpendicular to ω; we can call it the _Magnetic rest-force_. Relations analogous to these hold among the quantities ωF*, M, E, _u_ and Relation (D) can be replaced by the formula { D } -ωF* = μψ_f_*. We can use the relations (C) and (D) to calculate F and _f_ from Φ and ψ we have ωF = -Φ, ωF* = -_i_μψ, ω_f_ = -εΦ, ω_f_* = -_i_ψ.