  In this paper, we define the eta cochain form and prove its regularity when
the kernel of a family of Dirac operators is a vector bundle. We decompose the
eta form as a pairing of the eta cochain form with the Chern character of an
idempotent matrix and we also decompose the Chern character of the index bundle
for a fibration with boundary as a pairing of the family Chern-Connes character
for a manifold with boundary with the Chern character of an idempotent matrix.
We define the family $b$-Chern-Connes character and then we prove that it is
entire and give its variation formula. By this variation formula, we prove
another noncommutative family Atiyah-Patodi-Singer index theorem. Thus, we
extend the results of Gezler and Wu to the family case.
