(6)
The displacement is thus expressed in terms of the six independent quantities ξ, η, ζ, λ, μ, ν. The points whose displacements are in the direction of the resultant axis of rotation are determined by δx : δy : δz = ξ : η : ζ, or
(λ + ηz − ζy)/ξ = (μ + ζx − ξz)/η = (ν + ξy − ηx)/ζ.
(7)
These are the equations of a straight line, and the displacement is in fact equivalent to a twist about a screw having this line as axis. The translation parallel to this axis is
lδx + mδy + nδz = (λξ + μη + νζ)/ε.
(8)
The linear magnitude which measures the ratio of translation to rotation in a screw is called the pitch. In the present case the pitch is
(λξ + μη + νζ) / (ξ2 + η2 + ζ2).
(9)