  In this paper, we analyse the question of existence of a natural and
projectively equivariant symbol calculus, using the theory of projective Cartan
connections. We establish a close relationship between the existence of such a
natural symbol calculus and the existence of an \sl(m+1,\R)-equivariant
calculus over \R^{m} in the sense of [15,1]. Moreover we show that the formulae
that hold in the non-critical situations over \R^{m} for the
\sl(m+1,\R)-equivariant calculus can be directly generalized to an arbitrary
manifold by simply replacing the partial derivatives by invariant
differentiations with respect to a Cartan connection.
