  We compute the special values at nonpositive integers of the partial zeta
function of an ideal of a real quadratic field applying an asymptotic version
of Euler-Maclaurin formula to the lattice cone associated to the ideal
considered. The Euler-Maclaurin formula involved is obtained by applying the
Todd series of differential operators to an integral of a small perturbation of
thecone. The additive property of Todd series w.r.t. the cone decomposition
enables us to express the partial zeta values in terms of the continued
fraction of the reduced element of the ideal. The expression obtained uses the
positive continued fraction which yields a virtual decomposition of the cone.
  We apply the expression to some indexed families of real quadratic fields
satisfying certain condition on the shape of the continued fractions. The
families considered include those appeared in \cite{J-L1} and \cite{J-L2} as
well as the Richaud-Degert types. We show that the partial zeta values at a
given nonpositive integer $-k$ in the family indexed by $n$ is a polynomial of
$n$.
  Finally, we compute explicitly the polynomials producing the partial zeta
values at $s=-k$ for small $k$ of some chosen families and compare these with
some previously known results.
