Let V be an algebraic surface in R^3 containing 0.  We investigate the
tangent cone C and the Nash space N of the surface.  We prove a structure
theorem for N analogous to that over the complexes established by Le Dung
Trang:  there are finitely many ``exceptional rays" in C so that N is the
union of N(C) and the set of elements in N containing one of the
exceptional rays.  We show that there is a sharp dichotomy between
exceptional rays in C which are tangent to the singular locus of V and
those that aren't.  In the former case, the set of elements in N
containing the exceptional ray is semialgebraic, but can be disconnected
and have discrete elements. In the latter case, we show that any ray along
which C is singular must be exceptional (with one exception: if C is
locally the union of smooth surfaces tangent along the ray, the ray need
not be exceptional), and the set of elements in N containing the
exceptional ray cannot contain discrete elements---in fact, it is closed,
connected and 1-dimensional, and we can give a lower bound on the size of
this set.