(58a) T_{μν} = -_g__{μν} _p_ + _g__{μα} _dx__{α}/_ds_ _g__{μβ} _dx__{β}/_ds_ ρ as well as the mixed tensor (58b) T^α_{σ} = -δ^α_{σ} _p_ + _g__{σβ} _dx__{β}/_ds_ _dx__{α}/_ds_ ρ. If we put the right-hand side of (58b) in (57a) we get the general hydrodynamical equations of Euler according to the generalised relativity theory. This in principle completely solves the problem of motion; for the four equations (57a) together with the given equation between _p_ and ρ, and the equation _g__{αβ} _dx__α/_ds_ _dx__{β}/_ds_ = 1, are sufficient, with the given values of _g__{αβ}, for finding out the six unknowns _p_, ρ, _dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_ _dx₄_/_ds_. If _g__{μν}’s are unknown we have also to take the equations (53). There are now 11 equations for finding out 10 functions _g_, so that the number is more than sufficient. Now it is be noticed that the equation (57a) is already contained in (53), so that the latter only represents (7) independent equations. This indefiniteness is due to the wide freedom in the choice of co-ordinates, so that mathematically the problem is indefinite in the sense that three of the space-functions can be arbitrarily chosen. §20. Maxwell’s Electro-Magnetic field-equations. Let φ_{ν} be the components of a covariant four-vector, the electro-magnetic potential; from it let us form according to (36) the components F_{ρσ} of the covariant six-vector of the electro-magnetic field according to the system of equations (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