  A Riemannian metric is of constant curvature if and only if it is locally
projectively flat. There are infinitely many locally projectively flat Finsler
metrics of constant curvature, that are special solutions to the Hilbert's
Fourth Problem.
  In this paper, we use the technique in the paper titled "Finsler metrics with
K=0 and S=0" (math.DG/0109060) to construct infinitely many Finsler metrics on
the 2-sphere with constant curvature K=1 and infinitely many Finsler metrics on
the 2-disk with constant curvature K = -1. These metrics are not projectively
flat. So far, the classification of Finsler metrics of constant curvature has
not been completely done yet. These examples are important to the
classification problem.
