where by L, we mean L-times the unit matrix, _i.e._ the matrix with elements | L_e__{_hk_} |, (_e__{_hh_} = 1, _e__{_hk_} = 0, _h_ ≠ _k_ _h_, _k_ = 1, 2, 3, 4). Since here SL = LS, we deduce that, F*_f_*_f_F = (-S - L)(S - L) = -SS + L², and find, since _f_*_f_ = Det^{½}_f_, F*F = Det^{½}F, we arrive at the interesting conclusion (79) SS = L² - Det^{½}_f_ Det^{½}F _i.e._ the product of the matrix S into itself can be expressed as the multiple of a unit matrix—a matrix in which all the elements except those in the principal diagonal are zero, the elements in the principal diagonal are all equal and have the value given on the right-hand side of (79). Therefore the general relations (80) S_{_h_1} S_{1_k_} + S_{_h_2} S_{2_k_} + S_{_h_3} S_{3_k_} + S_{_h_4} S_{4_k_} = 0, _h_, _k_ being unequal indices in the series 1, 2, 3, 4, and (81) S_{_h_1} S_{1_h_} + S_{_h_2} S_{2_h_} + S_{_h_3} S_{3_h_} + S{_h_4} S_{4_h_} = L² - Det^{½}_f_ Det^{½}F, for _h_ = 1, 2, 3, 4. Now if instead of F, and _f_ in the combinations (72) and (73), we introduce the electrical rest-force Φ, the magnetic rest-force ψ, and the rest-ray Ω [(55), (56) and (57)], we can pass over to the expressions,— (82) L = - ½ ε Φ [=Φ] + ½ μ ψ [=ψ], (83) S_{_hk_} = - ½ ε Φ [=Φ] _e__{_hk_} - ½ μ ψ [=ψ] _e__{_hk_} + ε (Φ_{_h_} Φ_{_k_} - Φ ([=Φ]) ω_{_h_} Ω_{_k_} + μ (ψ_{_h_} ψ_{_k_} - Ψ [=ψ] Ω{_h_} ω_{_k_}) - ω_{_h_} ω_{_k_} - εμ ω_{_h_} Ω_{_k_} (_h₁_ _k_ = 1, 2, 3, 4). Here we have Φ [=Φ] = Φ₁² + Φ₂² + Φ₃² + Φ₄², ψ[=ψ] = ψ₁² + ψ₂² + ψ₃² + ψ₄² _e__{_hh_} = 1, _e__{_hk_} = 0 (_h_ ≠ _k_). The right side of (82) as well as L is an invariant in a Lorentz transformation, and the 4 × 4 element on the right side of (83) as well as S_{_k_ _h_} represent a space time vector of the second kind. Remembering this fact, it suffices, for establishing the theorems (82) and (83) generally, to prove it for the special case ω₁ = 0, ω₂ = 0, ω₃ = 0, ω₄ = _i_. But for this case ω = 0, we immediately arrive at the equations (82) and (83) by means (45), (51), (60) on the one hand, and _e_ = εE, M = μ_m_ on the other hand. The expression on the right-hand side of (81), which equals [½ (_m_ M - _e_E)²] + (_em_) (EM), is >= 0, because (_em_ = ε Φ [=ψ], (EM) = μ Φ [=ψ]; now referring back to 79), we can denote the positive square root of this expression as Det^{1/4} S. Since _ḟ_ = -_f_, and Ḟ = -F, we obtain for Ṡ, the transposed matrix of S, the following relations from (78), (84) F_f_ = Ṡ - L, _f_* F* = -Ṡ - L, Then is Ṡ - S = | S_{_h_ _k_} - S_{_t_ _k_} | an alternating matrix, and denotes a space-time vector of the second kind. From the expressions (83), we obtain, (85) S - Ṡ = - (εμ - 1) [ω, Ω], from which we deduce that [see (57), (58)]. (86) ω (S - Ṡ)* = 0,