(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.