  Sylvester doubles sums, introduced first by Sylvester are symmetric
expressions of the roots of two polynomials. Sylvester's definition of double
sums makes no sense in the presence of multiple roots, since the definition
involves denominators that vanish when there are multiple roots. The aim of
this paper is to give a new definition of Sylvester double sums making sense in
the presence of multiple roots, which coincides with the definition by
Sylvester in the case of simple roots, to prove that double sums indexed by
$(k,\ell)$ are equal up to a constant if they share the same value for
$k+\ell$, as well a proof of the relationship between double sums and
subresultants, i.e. that they are equal up to a constant. In the simple root
case, proofs of these properties are already known. The more general proofs
given here are using generalized Vandermonde determinants and symmetric
multivariate Hermite interpolation as well as an induction on the length of the
remainder sequence of $P$ and $Q$.
