Affine differential geometry of osculating hypersurfaces

Osculating surfaces of second order have been studied in classical affine differential geometry [1].  In this article we generalize this notion to osculating hypersurfaces of higher order of hypersurfaces in Euclidean n-space. Various geometric interpretations are given. This yields a affinely invariant consideration of the local properties of a given hypersurface which depend on the derivatives of higher order.