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 $\eta$-invariants associated to non self-adjoint Dirac type operators. We then give an explicit local expression of certain $\eta$-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.