Richard B. Melrose 
and Paolo Piazza 
A version of the Atiyah-Patodi-Singer index theorem is proved for
general families of Dirac operators on compact manifolds with boundary.
The vanishing of the analytic index of the boundary family, in 
of the
base, allows us to define, through an explicit trivialization, a smooth
family of boundary conditions of generalized Atiyah-Patodi-Singer type. The
calculus of b-pseudodifferential operators is then employed to establish
the family index formula. A relative index formula, describing the effect
of changing the choice of the trivialization, is also given. In case the
boundary family is invertible the form of the index theorem obtained by
Bismut and Cheeger is recovered.