The foundation of the kinematic geometry of helical motions of order \(k\) in \(E^n\) is discussed. First the author gives a summary of known results about one parametric motion in Euclidean space \(E^n\), which is generated by the transformation \[ {{\mathcal X} \brack 1} = {{\mathcal A} {\mathcal C} \brack 0 1} {\overline {\mathcal X} \brack 1}; \quad {\mathcal A} \cdot {\mathcal A}^T = \mathbf{1}, \] where \({\mathcal A} : J \to \text{SO} (n)\); \({\mathcal C} : J \to \mathbf{1}\mathbb{R}^n\) are functions of differentiability class \(C^r\) \((r \geq 3)\). Then the author defines the notion of fixed and moving axoids and the helical motion of order \(k\) in \(E^n\). In the next part generalized ruled surfaces are studied. In the third part the symmetric helical motion of order \(k\) in \(E^n\) is defined and some results about integral invariants of the pair of axoids under the motions are obtained.