  The Hochschild and cyclic homology groups are computed for the algebra of
`cusp' pseudodifferential operators on any compact manifold with boundary. The
index functional for this algebra is interpreted as a Hochschild 1-cocycle and
evaluated in terms of extensions of the trace functionals on the two natural
ideals, corresponding to the two filtrations by interior order and vanishing
degree at the boundary, together with the exterior derivations of the algebra.
This leads to an index formula which is a pseudodifferential extension of that
of Atiyah, Patodi and Singer for Dirac operators; together with a symbolic term
it involves the `eta' invariant on the suspended algebra over the boundary
previously introduced by the first author.
