There exists a rich literature on systems of connections and systems of vector fields, stimulated by the irimportance in geometry and physis. In the previous papers [T1], [T2] we examined a simple type of systems of vector fields, called parameter dependent vector fields, and established their varionational equation.
In this paper we generalize the above equation to the projectable system of vector fields. The material is organized as follows: in the first section the geometry of the product bundle is presented. In the second we introduce the notion of derivative along a direction and prove Theorem 1. The third section is devoted to Theorem
2, which represents the main result of the paper. Some examples are presented in the last section. In a further paper we will apply the results in order to investigate some special systems as strong systems, “nice” systems and systems of connections generated by systems of vector fields.