Page:The principle of relativity (1920).djvu/111

 The first three quantities (φ prev page and below]_{1}, φ_{2}, φ_{3}) are the components

of the space-vector (E + [u, M])/[sqrt](1 - u^2), and further (φ_{4} = i[u E]/[sqrt](1 - u^2).  Because F is an alternating matrix,  (49) ωΦ̄ = ω_{1} φ?]_{1} + ω_{2} Φ_{2} + ω_{3} Φ_{3} + ω_{4} Φ_{4} = 0.  i.e. Φ is perpendicular to the vector ω; we can also write Φ_{4} = i[ω_{x} Φ_{1} + ω_{y} Φ_{2} + ω_{z} Φ_{3}].  I shall call the space-time vector Φ of the first kind as the Electric Rest Force.  Relations analogous to those holding between -ωF, E, M, U, hold amongst -ω[function], e, m, u, and in particular -ω[function] is normal to ω. The relation (C) can be written as  { C } ω[function] = εωF.  The expression (ω[function]) gives four components, but the fourth can be derived from the first three.  Let us now form the time-space vector  1st kind, Ψ - iω[function]*, whose components are  Ψ_{1} = -i(ω_{2} [function]_{3 4} + ω_{3} [function]_{4 2} + ω_{4} [function]_{2 3}) } Ψ_{2} = -i(ω_{1} [function]_{4 3} + ω_{3} [function]_{4 4} + ω_{4} [function]_{3 1}) } Ψ_{3} = -i(ω_{1} [function]_{2 4} + ω_{2} [function]_{4 1} + ω_{4} [function]_{1 2}) } Ψ_{4} = -i(ω_{1} [function]_{3 2} + ω_{2} [function]_{1 3} + ω_{3} [function]_{2 1}) }  Of these, the first three Ψ_{1}, Ψ_{2}, Ψ_{3}, are the x, y, z

components of the space-vector 51) (m - (ue))/[sqrt](1 - u^2)

and further (52) ψ]_{4} = i(um)/[sqrt](1 - u^2).