This chapter returns to the study of a general bundle (E, π, M) and the various vector bundles and modules of sections associated with it. These vector bundles are constructed from the tangent and cotangent bundles to E and to M. The main theme of the chapter comes from isolating those tangent vectors and vector fields on E which are tangent to the fibres of π, and those cotangent vectors and differential forms which annihilate them. Many of the definitions in the early part of the chapter are restatements of examples from the previous chapter, but are examined here in more detail.