(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). K_{σ} is a covariant four-vector whose components are equal to the negative impulse and energy which are transferred to the electro-magnetic field per unit of time, and per unit of volume, by the electrical masses. If the electrical masses be free, that is, under the influence of the electro-magnetic field only, then the covariant four-vector K_{σ} will vanish. In order to get the energy components T_{σ}^ν of the electro-magnetic field, we require only to give to the equation K_{σ} = 0, the form of the equation (57). From (63) and (65) we get first, K_{σ} = F_{σμ} ∂F_{μν}/∂_x__{ν} = ∂/∂_x__{ν} (F_{σμ} F^{μν}) - F^{μν} ∂F_{σμ}/∂_x__{ν}. On account of (60) the second member on the right-hand side admits of the transformation— F^{μν} ∂F_{σμ}/∂_x__{ν} = -½ F^{μν} ∂F_{μν}/∂_x__{σ} = -½ _g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}. Owing to symmetry, this expression can also be written in the form = -1/4 [_g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ} + _g_^{μα} _g_^{νβ} ∂F_{αβ}/∂_x__{σ} F_{μν}], which can also be put in the form - 1/4 ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ} F_{αβ} F_{μν}) + 1/4 F_{αβ} F_{μν} ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ}). The first of these terms can be written shortly as - 1/4 ∂/∂_x__{σ} (F^{μν} F_{μν}), and the second after differentiation can be transformed in the form - ½ F^{μτ} F_{μν} _g_^{νρ} ∂_g__{στ}/∂_x__{σ}.