(82) L = - ½ ε Φ [=Φ] + ½ μ ψ [=ψ],
(83) Shk = - ½ ε Φ [=Φ] ehk - ½ μ ψ [=ψ] ehk
+ ε (Φh Φk - Φ ([=Φ]) ωh Ωk
+ μ (ψh ψk - Ψ [=ψ] Ω{h} ωk) - ωh ωk - εμ ωh Ωk
(h₁ k = 1, 2, 3, 4).
Here we have
Φ [=Φ] = Φ₁² + Φ₂² + Φ₃² + Φ₄², ψ[=ψ] = ψ₁² + ψ₂² + ψ₃² + ψ₄²
ehh = 1, ehk = 0 (h ≠ k).
The right side of (82) as well as L is an invariant in a Lorentz transformation, and the 4 × 4 element on the right side of (83) as well as Sk h represent a space time vector of the second kind. Remembering this fact, it suffices, for establishing the theorems (82) and (83) generally, to prove it for the special case ω₁ = 0, ω₂ = 0, ω₃ = 0, ω₄ = i. But for this case ω = 0, we immediately arrive at the equations (82) and (83) by means (45), (51), (60) on the one hand, and e = εE, M = μm on the other hand.
The expression on the right-hand side of (81), which equals