  The dependence of the photon structure on the photon virtuality, Q^2, is
studied by measuring the reaction e^+p\to e^+ + {\rm jet} + {\rm jet} + {\rm X}
at photon-proton centre-of-mass energies 134 < W < 223 GeV. Events have been
selected in the Q^2 ranges \approx 0 GeV^2, 0.1-0.55 GeV^2, and 1.5-4.5 GeV^2,
having two jets with transverse energy E_T^{jet} > 5.5 GeV in the final state.
The dijet cross section has been measured as a function of the fractional
momentum of the photon participating in the hard process, x_gamma. The ratio of
the dijet cross section with x_gamma < 0.75 to that with x_gamma > 0.75
decreases as Q^2 increases. The data are compared with the predictions of NLO
pQCD and leading-order Monte Carlo programs using various parton distribution
functions of the photon. The measurements can be interpreted in terms of a
resolved photon component that falls with Q^2 but remains present at values of
Q^2 up to 4.5 GeV^2. However, none of the models considered gives a good
description of the data.
