  We investigate the continuum limit in $SO(N)$ Principal Chiral Models
concentrating in detail on the $SO(4)$ model and its covering group
SU(2)xSU(2). We compute the mass gap in terms of Lambda_MS and compare with the
prediction of Hollowood of $m/\L_\MS = 3.8716$. We use the finite-size scaling
method of L\"uscher et al. to deduce $m/\L_\MS$ and find that for the $SO(4)$
model the computed result of $m/\L_\MS \sim 14$ is in strong disagreement with
theory but that a similar analysis of the SU(2)xSU(2) yields excellent
agreement with theory. We conjecture that for $SO(4)$ violations of the
finite-size scaling assumption are severe forthe values of the correlation
length, $\xi$, investigated and that our attempts to extrapolate the results to
zero lattice spacing, although plausible, are erroneous. Conversely, the
finite-size scaling violations in the SU(2)xSU(2) simulation are consistent
with perturbation theory and the computed $beta-$function agrees well with the
3-loop approximation for couplings evaluated at scales $L/a \le \xi$, where
$\xi$ is measured in units of the lattice spacing, $a$. We conjecture that
lattice vortex artifacts in the $SO(4)$ model are responsible for delaying the
onset of the continuum limit until much larger correlation lengths are achieved
notwithstanding the apparent onset of scaling. Results for the mass spectrum
for SO(N) m, N=8,10 are given whose comparison with theory gives plausible
support to our ideas.
