  Let C be a curve over a non-singular base variety S. We study algebraic
cycles on the symmetric powers C^[n] and on the Jacobian J. The Chow homology
of C^[*], the sum of all C^[n], is a ring using the Pontryagin product. We
prove that this ring is isomorphic to CH(J)[t]<u>, the PD-polynomial algebra
(variable: u) over the usual polynomial ring (variable: t) over the Chow ring
CH(J). We give two such isomorphisms that over a general base are different.
Further we give some precise results on how CH(J) sits embedded in CH(C^[*])
and we give an explicit geometric description of how the derivations with
regard to t and u act.
  Our results give rise to a new grading on the Chow ring of the Jacobian.
After tensoring with Q the associated descending filtration coincides with the
one coming from Beauville's decomposition. The grading we obtain is in general
different from Beauville's.
  Finally we give a version of our main result for tautological classes, and we
show how our methods give a very simple and geometric proof of some relations
obtained by Herbaut and van der Geer-Kouvidakis.
