K’k = ∂S′1k/∂x₁′ + ∂S′2k/∂x₂′
+ ∂S′3k/∂x₃′ + ∂S′4k/∂x₄′.
Now for the differentiation of any function of (x y z t) we have the rule ∂/∂xk′ = ∂/∂x₁ ∂x₁/∂xk′ + ∂/∂x₂ ∂x₂/∂xk′ + ∂/∂x₃ ∂x₃/∂xk′ + ∂/∂x₄ ∂x₄/∂xk′ = ∂/∂x₁ a1k + ∂/∂x₂ a2k + ∂/∂x₃ a3k + ∂/∂x₄ a4k.
so that, we have symbolically lor′ = lor A.
Therefore it follows that
lor ′S′ = lor (A A⁻¹ SA) = (lor S)A.
i.e., lor S behaves like a space-time vector of the first kind.
If L is a multiple of the unit matrix, then by lor L will be denoted the matrix with the elements
| ∂L/∂x₁ ∂L/∂x₂ ∂L/∂x₃ ∂L/∂x₄ |
If s is a space-time vector of the 1st kind, then