  The cross sections for inelastic photoproduction of J/psi and psi' mesons
have been measured in ep collisions with the ZEUS detector at HERA, using an
integrated luminosity of 38.0 pb-1. The events were required to have 0.1 < z <
0.9 and 50 < W < 180 GeV, where z is the fraction of the incident photon energy
carried by the J/psi in the proton rest frame and W is the photon-proton
centre-of-mass energy. The psi' to J/psi cross-section ratio was measured in
the range 0.55 < z < 0.9. The J/psi data, for various ranges of transverse
momentum, are compared to theoretical models incorporating colour-singlet and
colour-octet matrix elements. Predictions of a next-to-leading-order
colour-singlet model give a good description of the data, although there is a
large normalisation uncertainty. The J/psi helicity distribution for z > 0.4 is
compared to leading-order QCD predictions.
