  In this paper we continue the study of the path-integral formulation of
classical mechanics and in particular we better clarify, with respect to
previous papers, the geometrical meaning of the variables entering this
formulation. With respect to the first paper with the same title, we {\it
correct} here the set of transformations for the auxiliary variables
$\lambda_{a}$. We prove that under this new set of transformations the
Hamiltonian ${\widetilde{\cal H}}$, appearing in our path-integral, is an exact
scalar and the same for the Lagrangian. Despite this different transformation,
the variables $\lambda_{a}$ maintain the same operatorial meaning as before but
on a different functional space. Cleared up this point we then show that the
space spanned by the whole set of variables ($\phi, c, \lambda,\bar c$) of our
path-integral is the cotangent bundle to the {\it reversed-parity} tangent
bundle of the phase space ${\cal M}$ of our system and it is indicated as
$T^{\star}(\Pi T{\cal M})$. In case the reader feel uneasy with this strange
{\it Grassmannian} double bundle, we show in this paper that it is possible to
build a different path-integral made only of {\it bosonic} variables. These
turn out to be the coordinates of $T^{\star}(T^{\star}{\cal M})$ which is the
double cotangent bundle of phase-space.
