lor ′_ṡ′_ = lor A. Ā_s_ = lor _s_.
_i.e._, lor _s_ is an invariant in a Lorentz-transformation.
In all these operations the operator lor plays the part of a space-time
vector of the first kind.
If _f_ represents a space-time vector of the second kind,—lor _f_
denotes a space-time vector of the first kind with the components
∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_,
∂_f₂₁_/∂_x₁_ + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_,
∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + ∂_f₃₄_/∂_x₄_,
∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_
So the system of differential equations (A) can be expressed in the
concise form
{A} lor f = -_s_,
and the system (B) can be expressed in the form
{B} log F* = 0.
Referring back to the definition (67) for log _ṡ_, we find that the
combinations lor ([=(lor _f_)=]), and lor ([=(lor F*)]) vanish
identically, when _f_ and F* are alternating matrices. Accordingly it
follows out of {A}, that
(68) (∂_s₁_/∂_x₁_) + (∂_s₂_/∂_x₂_) + (∂_s₃_/∂_x₃_) + (∂_s₄_/∂_x₄_) =
0,
while the relation
(69) lor (lor F*) = 0,
signifies that of the four equations in {B}, only three represent
independent conditions.