The hadronic structure function of the photon 
$F_\mathrm{L}^{\gamma}(x,Q^2)$ is measured as a function of 
Bjorken $x$ and of the photon virtuality $Q^2$ using 
deep-inelastic scattering data taken by the OPAL detector at LEP 
at e$^+$e$^-$ centre-of-mass energies from 183 to 209~GeV. 
Previous OPAL measurements of the $x$ dependence of 
${F_{2}^{\gamma}}$ are extended to an average $Q^2$
of $\langle Q^2 \rangle = 780$~GeV$^2$ using data in the 
kinematic range $0.15<x<0.98$.
The $Q^2$ evolution of $F_2^\gamma$ is studied for 
$12.1 < \langle Q^2 \rangle < 780$~GeV$^2$ using three ranges of $x$.
As predicted by QCD, the data show positive scaling violations in
${F_{2}^{\gamma}}$ with 
$F_2^\gamma(Q^2)/\alpha = 0.08 \pm 0.02 ^{~+0.05}_{~-0.03}
+ (0.13 \pm 0.01 ^{~+0.01}_{~-0.01}) \ln Q^2$
where $Q^2$ is in GeV$^2$, for the central $x$ region $0.10-0.60$.
Several parameterisations of ${F_{2}^{\gamma}}$ are in qualitative 
agreement with the measurements whereas the quark-parton model 
prediction fails to describe the data.