Inclusive jet differential cross sections for the reaction ep --> jet
+ X at Q(2) below 4 GeV2 have been measured with the ZEUS detector at
HERA using an integrated luminosity of 0.55 pb(-1). These cross sectio
ns are given in the kinematic region 0.2 < y < 0.85, for jet pseudorap
idities in the ep-laboratory range -1 < eta(jet) < 2 and refer to jets
at the hadron level with a cone radius of one unit in the eta - phi p
lane. These results correspond to quasi-real photoproduction at centre
-of-mass energies in the range 130-270 GeV and, approximately, for jet
pseudorapidities in the interval -3 < eta(jet)(gamma p CMS) < 0. Thes
e measurements cover a new kinematic regime of the partonic structure
of the photon, at typical scales up to similar to 300 GeV2 and photon
fractional momenta down to x(y) similar to 10(-2). Leading logarithm p
arton shower Monte Carlo calculations, which include both resolved and
direct processes and use the predictions of currently available param
etrisations of the photon parton distributions, describe in general th
e shape and magnitude of the measured eta(jet) and E(T)(jet) distribut
ions.