i.e. Φ₁ψ₂ - Φ₂ψ₁ = i(ω₃Ω₄ - ω₄Ω₃) etc.
The vector Ω fulfils the relation
(ωΩ) = ω₁Ω₁ + ω₂Ω₂ + ω₃Ω₃ + ω₄Ω₄ = 0,
(which we can write as Ω₄ = i(ωxΩ₁ + ωyΩ₂ + ωzΩ₃) and Ω is also normal to ω. In case ω = 0, we have Φ₄ = 0, ψ₄ = 0, Ω₄ = 0, and
[Ω₁, Ω₂, Ω₃ = | Φ₁ Φ₂ Φ₃ |
| ψ₁ ψ₂ ψ₃ |.
I shall call Ω, which is a space-time vector 1st kind the Rest-Ray.
As for the relation E), which introduces the conductivity σ we have -ωS = -(ω₁s₁ + ω₂s₂ + ω₃s₃ + ω₄s₄) = (- | u | Cu + ρ)/√(1 - u²) = ρ′.
This expression gives us the rest-density of electricity (see [§8] and [§4]).
Then 61) = s + (ωṡ)ω represents a space-time vector of the 1st kind, which since ωω = -1, is normal to ω, and which I may call the rest-current. Let us now conceive of the first three component of this vector as the (x-y-z) co-ordinates of the space-vector, then the component in the direction of u is