  We study properties of evolution equations which are first order in time and
arbitrary order in space (FTNS). Following Gundlach and Mart\'in-Garc\'ia
(2006) we define strong and symmetric hyperbolicity for FTNS systems and
examine the relationship between these definitions, and the analogous concepts
for first order systems. We demonstrate equivalence of the FTNS definition of
strong hyperbolicity with the existence of a strongly hyperbolic first order
reduction. We also demonstrate equivalence of the FTNS definition, up to N=4,
of symmetric hyperbolicity with the existence of a symmetric hyperbolic first
order reduction.
