  We give a geometric proof of a formula, due to Segal and Wilson, which
describes the order of vanishing of the Riemann theta function in the direction
which corresponds to the direction of the tangent space of a Riemann surface at
a marked point. While this formula appears in the work of Segal and Wilson as a
by-product of some non-trivial constructions from the theory of integrable
systems (loop groups, infinite-dimensional Grassmannians, tau functions, Schur
polynomials, ...) our proof only uses the classical theory of linear systems on
Riemann surfaces.
