  Following Gorokhovsky and Lott and using an extension of the
b-pseudodifferential calculus of Melrose, we give a formula for the Chern
character of the Dirac index class of a longitudinal Dirac type operators on a
foliated manifold with boundary. For this purpose we use the Bismut local index
formula in the context of non commutative geometry. This paper uses heavily the
methods and technical results developed by E.Leichtnam and P.Piazza.
