A screw is determined by its axis and its pitch, and therefore involves five Independent elements. These may be, for instance, the five ratios ξ : η : ζ : λ : μ : ν of the six quantities which specify an infinitesimal twist about the screw. If the twist is a pure rotation, these quantities are subject to the relation
λξ + μη + νζ = 0.
(4)
In the analytical investigations of line geometry, these six quantities, supposed subject to the relation (4), are used to specify a line, and are called the six “co-ordinates” of the line; they are of course equivalent to only four independent quantities. If a line is a null-line with respect to the wrench (X, Y, Z, L, M, N), the work done in an infinitely small rotation about it is zero, and its co-ordinates are accordingly subject to the further relation
Lξ + Mη + Nζ + Xλ + Yμ + Zν = 0,
(5)
where the coefficients are constant. This is the equation of a “linear complex” (cf. § 8).
Two screws are reciprocal when a wrench about one does no work on a body which twists about the other. The condition for this is
λξ′ + μη′ + νζ′ + λ′ξ + μ′η + ν′ζ = 0,
(6)