  For a manifold with boundary, the restriction of Chern's transgression form
of the Euler curvature form over the boundary is closed. Its cohomology class
is called the secondary Chern-Euler class and used by Sha to formulate a
relative Poincar\'e-Hopf theorem, under the condition that the metric on the
manifold is locally product near the boundary. We show that the secondary
Chern-Euler form is exact away from the outward and inward unit normal vectors
of the boundary by explicitly constructing a transgression form. Using Stokes'
theorem, this evaluates the boundary term in Sha's relative Poincar\'e-Hopf
theorem in terms of more classical indices of the tangential projection of a
vector field. This evaluation in particular shows that Sha's relative
Poincar\'e-Hopf theorem is equivalent to the more classical Law of Vector
Fields.
