  Let xi be a smooth oriented vector bundle, with n-dimensional fibre, over a
smooth manifold M. Denote by xi-hat the fibrewise one-point compactification of
xi. The main purpose of this paper is to define geometrically a canonical
element Upsilon(xi) in H^n(xi-hat,Q) (H^n(xi-hat,Z) tensor 1/2, to be more
precise). The element \Upsilon(\xi) is a secondary characteristic class to the
Euler class in the fashion of Chern-Simons.
