Note 6. Field Equations in Minkowski's Form.
Equations (i) and (ii) become when expanded into Cartesians:—
[part]m_z/[part]y - [part]m_y/[part]z - [part]e_x/[part]τ = ρυ_x }
[part]m_x/[part]z - [part]m_z/[part]x - [part]e_y/[part]τ = ρυ_y } . . . (1·1)
[part]m_y/[part]x - [part]m_x/[part]y - [part]e_z/[part]τ = ρυ_z }
and [part]e_x/[part]x + [part]e_y/[part]y + [part]e_z/[part]z = ρ (2·1)
Substituting x_1, x_2, x_3, x_4 and x, y, z, and iτ; and rho_1, ρ_2, ρ_3, ρ_4 for ρυ_x, ρυ_y, ρυ_z, iρ, where i = [sqrt](-1).
We get,
[part]m_z/[part]x_2 - [part]m_y/[part]x_3 - i([part]e_x/[part]x_4) = ρυ_x{ = ρ_1 }
- [part]m_z/[part]x_1 + [part]m_x/[part]x_3 - i([part]e_y/[part]x_4) = ρυ_y = ρ_2 } . . . (1·2)
[part]m_y/[part]x_1 - [part]m_x/[part]x_2 - i([part]e_z/[part]x_4) = ρυ_z{} = ρ_3 }
and multiplying (2·1) by i we get
[part]ie_x/[part]x_1 + [part]ie_y/[part]x_2 + [part]ie_z/[part]x_3 = iρ = ρ_4 . . . . . . (2·2)
Now substitute
m_x = [function]_{2 3} = -[function]_{3 2} and ie_x = [function]_{4 1} = -[function]_{1 4}
m_y = [function]_{3 1} = -[function]_{1 3} ie_y = [function]_{4 2} = -[function]_{2 4}
m_z = [function]_{1 2} = -[function]_{2 1} ie_z = [function]_{4 3} = -[function]_{3 4}
