  A families index theorem in K-theory is given for the setting of Atiyah,
Patodi and Singer of a family of Dirac operators with spectral boundary
condition. This result is deduced from such a K-theory index theorem for the
calculus of cusp, or more generally fibred cusp, pseudodifferential operators
on the fibres (with boundary) of a fibration; a version of Poincare duality is
also shown in this setting, identifying the stable Fredholm families with
elements of a bivariant K-group.
