  By means of an {\it ad hoc} modification of the so-called
``Castelnuovo-Harris analysis" we derive an upper bound for the genus of
integral curves on the three dimensional nonsingular quadric which lie on an
integral surface of degree $2k$, as a function of $k$ and the degree $d$ of the
curve. In order to obtain this we revisit the Uniform Position Principle to
make its use computation-free. The curves which achieve this bound can be
conveniently characterized.
  Keywords: Castelnuovo-Harris Bound, Uniform Position Principle, Low
Codimension, Linkage
