  We formulate a refined theory of linear systems, using the methods of a
previous paper, "A Theory of Branches for Algebraic Curves", and use it to give
a geometric interpretation of the genus of an algebraic curve. Using principles
of duality, we prove generalisations of Plucker's formulae for algebraic
curves. The results hold for arbitrary characteristic of the base field L, with
some occasional exceptions when characteristic(L)=2, which we observe in the
course of the paper.
