Abstract
We present geometry and kinematic expressions for a kinematical system of multiple relative motions and containing a differentiable line-system using dual vector-matrix methods. This procedure enables us to obtain a generalization of the well-known three-pole theorem. In particular, for one-parameter spatial motions, the pitch of the helical motion is the translation multiplied by the tangent of the screw motion rotation. As a result, theoretical expressions for the direction of a line complex were revealed and used to examine a ruled surface traced out by a fixed line in a rigid body undergoing a screw motion of constant pitch.