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

Generalized weight properties of resultants and discriminants, and applications to projective enumerative geometry