Equations (i) and (ii) become when expanded into Cartesians:—
∂mz/∂y - ∂my/∂z - ∂ex/∂τ = ρνx }
∂mx/∂z - ∂mz/∂x - ∂ey/∂τ = ρνy } ... (1·1)
∂my/∂x - ∂mx/∂y - ∂ez/∂τ = ρνz }
and ∂ex/∂x + ∂ey/∂y + ∂ez/∂z = ρ (2·1)
Substituting x₁, x₂, x₃, x₄ and x, y, z, and iτ; and ρ₁, ρ₂, ρ₃, ρ₄ for ρνx, ρνy, ρνz, iρ, where i = √(-1).
We get,
∂mz/∂x₂ - ∂my/∂x₃ - i(∂ex/∂x₄) = ρνx{ = ρ₁ }
- ∂mz/∂x₁ + ∂mx/∂x₃ - i(∂ey/∂x₄) = ρνy = ρ₂ } ... (1·2)
∂my/∂x₁ - ∂mx/∂x₂ - i(∂ez/∂x₄) = ρνz{ = ρ₃ }