  In 1853 Sylvester introduced a family of double sum expressions for two
finite sets of indeterminates and showed that some members of the family are
essentially the polynomial subresultants of the monic polynomials associated
with these sets. A question naturally arises: What are the other members of the
family? This paper provides a complete answer to this question. The technique
that we developed to answer the question turns out to be general enough to
charactise all members of the family, providing a uniform method.
