Page:The principle of relativity (1920).djvu/112

 Among these there is the relation

(53) ωΨ̄ = ω_{1} Ψ_{1} + ω_{2} Ψ_{2} + ω_{3} Ψ_{3} + ω_{4} Ψ_{4} = 0

which can also be written as Ψ_{4} = i (u_{x} Ψ_{1} + u_{y} Ψ_{2} + u_{z} Ψ_{3}).

The vector Ψ is perpendicular to ω; we can call it the Magnetic rest-force.

Relations analogous to these hold among the quantities ωF*, M, E, u and Relation (D) can be replaced by the formula

{ D } -ωF* = μΨ[function]*.

We can use the relations (C) and (D) to calculate F and [function] from Φ and Ψ we have

ωF = -Φ, ωF* = -iμΨ, ω[function] = -εΦ, ω[function]* = -iΨ.

and applying the relation (45) and (46), we have

F = [ω. Φ] + iμ[ω. Ψ]* 55) [function] = ε[ω. Φ] + i[ω. Ψ]* 56)

i.e. F_{1 2} = (ω_{1} Φ_{1} - ω_{2} Φ_{1}) + iμ [ω_{3} [Psi}_{4} - ω_{4} Ψ_{3}], etc. [function]_{1 2} = ε(ω_{1} Φ_{2} - ω_{2} φ_{1}) + i [ω_{3} Ψ_{4} - ω_{4} ψ_{3}]. etc.

Let us now consider the space-time vector of the second kind [Φ Ψ], with the components

[ Φ_{2} Ψ_{3} - Φ_{3} Ψ_{2}, Φ_{3} Ψ_{1} - Φ_{1} Ψ_{3}, Φ_{1} Ψ_{2} - Φ_{2} Ψ_{1} ] [ Φ_{1} Ψ_{4} - Φ_{4} Ψ_{1}, Φ_{2} Ψ_{4} - Φ_{4} Ψ_{2}, Φ_{3} Ψ_{4} - Φ_{4} Ψ_{3} ]

Then the corresponding space-time vector of the first kind ω[Φ, Ψ] vanishes identically owing to equations 9) and 53)

for ω[Φ.Ψ] = -(ωΨ̄)Φ + (ωΦ̄)Ψ

Let us now take the vector of the 1st kind

(57 Ω = iω[ΦΨ]*

with the components Ω_{1} = -i | ω_{2} ω_{3} ω_{4} |
 * Φ_{2} Φ_{3} Φ_{4} |
 * Ψ_{2} Ψ_{3} Ψ_{4} |, etc.