  We first apply the method and results in the previous paper to give a new
proof of a result (hold in $ {\bf C}/{\bf Z}$) of Gilkey on the variation of
h-invariants associated to non self-adjoint Dirac type operators. We then give
an explicit local expression of certain h-invariant appearing in recent papers
of Braverman-Kappeler on what they call refined analytic torsion, and propose
an alternate formulation of their definition of the refined analytic torsion. A
refinement in ${\bf C}$ of the above variation formula is also proposed.
