(59) F_{ρσ} = ∂φ_{ρ}/∂_x__{σ} - ∂φ_{σ}/∂_x__{ρ}. From (59), it follows that the system of equations (60) ∂F_{ρσ}/∂_x__{τ} + ∂F_{στ}/∂_x__{ρ} + ∂F_{τρ}/∂_x__{σ} = 0 is satisfied of which the left-hand side, according to (37), is an anti-symmetrical tensor of the third kind. This system (60) contains essentially four equations, which can be thus written:— { ∂F₂₃/∂_x₄_ + ∂F₃₄/∂_x₂_ ∂F₄₂/∂_x₃_ = 0 { { ∂F₃₄/∂_x₁_ + ∂F₄₁/∂_x₃_ ∂F₁₃/∂_x₄_ = 0 (60a) { { ∂F₄₁/∂_x₂_ + ∂F₁₂/∂_x₄_ ∂F₂₄/∂_x₁_ = 0 { { ∂F₁₂/∂_x₃_ + ∂F₂₃/∂_x₁_ ∂F₃₁/∂_x₂_ = 0. This system of equations corresponds to the second system of equations of Maxwell. We see it at once if we put { F₂₃ = H_{_x_} F₁₄ = E_{_x_} { (61) { F₃₁ = H_{_y_} F₂₄ = E_{_y_} { { F₁₂ = H_{_z_} F₃₄ = E_{_z_} Instead of (60a) we can therefore write according to the usual notation of three-dimensional vector-analysis:— { ∂H/∂_t_ + rot E = 0 (60b) { { div H = 0. The first Maxwellian system is obtained by a generalisation of the form given by Minkowski. We introduce the contra-variant six-vector F_{αβ} by the equation (62) F^{μν} = _g_^{μα} _g_^{νβ} F_{αβ}, and also a contra-variant four-vector J^μ, which is the electrical current-density in vacuum. Then remembering (40) we can establish the system of equations, which remains invariant for any substitution with determinant 1 (according to our choice of co-ordinates). (63) ∂F^{μν}/∂_x__{ν} = J^μ If we put { F²³ = H′_{_x_} F¹⁴ = -E′_{_x_} { (64) { F³¹ = H′_{_y_} F²⁴ = -E′_{_y_} { { F¹² = H′_{_z_} F³⁴ = -E′_{_z_} which quantities become equal to H_{_x_} ... E_{_x_} in the case of the special relativity theory, and besides J^1 = _i__{_x_} ... J^4 = ρ we get instead of (63) { rot H′ - ∂E′/∂_t_ = _i_ (63a) { { div E′ = ρ The equations (60), (62) and (63) give thus a generalisation of Maxwell’s field-equations in vacuum, which remains true in our chosen system of co-ordinates. _The energy-components of the electro-magnetic field._ Let us form the inner-product (65) K_{σ} = F_{σμ} J^μ. According to (61) its components can be written down in the three-dimensional notation. { K₁ = ρE_{_x_} + [_i_, H]_{x} (65a) { — — — { K₄ = — (_i_, E).