We first introduce a product of arbitrary non-holonomic distributions of higher order on a fibred manifold. Then we specialize this operation t o higher order connections on a principal fibre bundle and we show the relation to an analogous operation for higher order connections on the associated groupoid. I n the vector bundle case, we obtain as an interesting special result the construction of the splitting determined by the prolongation of a higher order connection in the sense of EHRESMANN. Then we study the prolongation p ( P , A ) of a first order connection I' with respect to a linear connection 11 on the base manifold. We extend our remark of [lo] that the higher order absolute differentiation can be naturally performed in two different ways and we characterize the difference between the corresponding algorithms. After that, we present a comparison of the connection p ( r , A ) with a related construction by GOLLEK, [3]. At this occasion, we develop another point of-view to some of his investigations.
Our considerations are in the category C". The standard terminology and notations of the theory of jets are used throughout the paper, see [12]. I n addition, j ! denotes the canonical projection of r-jets onto Ic-jets, k < r.