  In a book dating back to 1862, Salmon stated a formula giving the first terms
of the Taylor expansion of the discriminant of a plane algebraic curve, and
from it derived various enumerative quantities for surfaces in $\mathbf{P}^3$.
In this text, we provide complete proofs of this formula and its enumerative
applications, and extend Salmon's considerations to hypersurfaces in a
projective space of arbitrary dimension. To this end, we extend reduced
elimination theory by introducing the concept of reduced discriminant, and
provide a thorough study of its weight properties; the latter are deeply linked
to projective enumerative geometric properties. Then, following Salmon's
approach, we compute the number of members of a pencil of hyperplanes that are
bitangent to a fixed projective hypersurface. Some other results in the same
spirit are also discussed.
