Andrew Hassell, Rafe Mazzeo and Richard B. Melrose
We present a signature formula for manifolds with corners
of codimension two endowed with complete b-metrics. To prove it, we apply
the results of our previous work on the
limiting behaviour of the eta invariant under analytic surgery degeneration
to the 
Atiyah-Patodi-Singer index theorem for an exhaustion by
manifolds with boundary and examine the limiting behaviour. There is an
extra contribution at the corners which is determined by the scattering
data for the induced signature operator on the codimension one boundaries
meeting at that corner. We also discuss some product formulas for the
b-eta invariant.