  We consider the Chow ring with rational coefficients of the Jacobian of a
curve. Assume D is a divisor in a base point free g^r_d of the curve such that
the canonical divisor K is a multiple of the divisor D. We find relations
between tautological cycles. We give applications for curves having a degree d
covering of P^1 whose ramification points are all of order d, and then for
hyperelliptic curves.
