  Every Riemannian metric or Finsler metric on a manifold induces a spray via
its geodesics. In this paper, we discuss several expressions for the
X-curvature of a spray. We show that the sprays obtained by a projective
deformation using the S-curvature always have vanishing X-curvature. Then we
establish the Beltrami Theorem for sprays with X=0
