Since ξ2 + η2 + ζ2, or ε2, is necessarily an absolute invariant for all transformations of the (rectangular) co-ordinate axes, we infer that λξ + μη + νζ is also an absolute invariant. When the latter invariant, but not the former, vanishes, the displacement is equivalent to a pure rotation.
If the small displacements of a rigid body be subject to one constraint, e.g. if a point of the body be restricted to lie on a given surface, the mathematical expression of this fact leads to a homogeneous linear equation between the infinitesimals ξ, η, ζ, λ, μ, ν, say
Aξ + Bη + Cζ + Fλ + Gμ + Hν = 0.
(10)
The quantities ξ, η, ζ, λ, μ, ν are no longer independent, and the body has now only five degrees of freedom. Every additional constraint introduces an additional equation of the type (10) and reduces the number of degrees of freedom by one. In Sir R. S. Ball’s Theory of Screws an analysis is made of the possible displacements of a body which has respectively two, three, four, five degrees of freedom. We will briefly notice the case of two degrees, which involves an interesting generalization of the method (already explained) of compounding rotations about intersecting axes. We assume that the body receives arbitrary twists about two given screws, and it is required to determine the character of the resultant displacement. We examine first the case where the axes of the two screws are at right angles and intersect. We take these as axes of x and y; then if ξ, η be the component rotations about them, we have
λ = hξ, μ = kη, ν = 0,
(11)
where h, k, are the pitches of the two given screws. The equations (7) of the axis of the resultant screw then reduce to
x/ξ = y/η, z(ξ2 + η2) = (k − h) ξη.
(12)