  We present an alternate definition of the mod {\bf Z} component of the
Atiyah-Patodi-Singer $\eta$ invariant associated to (not necessary unitary)
flat vector bundles, which identifies explicitly its real and imaginary parts.
  This is done by combining a deformation of flat connections introduced in a
previous paper with the analytic continuation procedure appearing in the
original article of Atiyah, Patodi and Singer.
