Q_{\rm min}$, $Q_B > Q_{\rm min}$ with $Q_{\rm min}^2 = 5\ {\rm GeV}^2$, in order to keep $\alpha_S$ small and 2) $x_A\, x_B s_{ee} > 100\ Q_A Q_B$ in order that the high energy approximation be valid. We base the numerical value of this limit on a demand that the two gluon exchange graph be dominant over the (lower order) quark exchange graph when we chose $\alpha_S = 0.2$. Performing the integrations numerically and adding a similar contribution from longitudinally polarized virtual photons, as described in Ref.~\cite{11}, we obtain % \begin{equation} \label{ratevallep} \sigma \simeq 1 \, {\mbox {pb}} \hspace*{2 cm} ( \sqrt{s} = 200 \, {\mbox {GeV}} ) \end{equation} % at LEP200 energies, and % \begin{equation} \label{ratevalnlc} \sigma \simeq 4 \, {\mbox {pb}} \hspace*{2 cm} ( \sqrt{s} = 500 \, {\mbox {GeV}} ) \end{equation} % at a future next linear collider (NLC). These cross sections would give rise to about $500$ events at LEP200 for a value of the luminosity $L = 500 \, {\mbox {pb}}^{-1}$, and about $2 \times 10^5$ events at the NLC for $L = 50 \, {\mbox {fb}}^{-1}$. For $Q_{\rm min}^2 = 36\ {\rm GeV}^2$, corresponding to a minimum electron scattering angle of 24 mrad, the number of events at the NLC would be about $10^3$. While this looks rather marginal at LEP200, it appears that measuring off-shell photon scattering at the NLC could be a viable way of studying short distance pomeron effects. Even with a modest luminosity, one can examine experimentally how the perturbative pomeron emerges from the soft pomeron as $Q_A$ and $Q_B$ are increased. For small $Q_A$ and $Q_B$ a simple Regge model should apply. For on-shell photons, Regge factorization gives % \begin{equation} \sigma_{ {\gamma} {\gamma} } \approx { \sigma_{ {\gamma} {p} } \; \sigma_{ {\gamma} {p} } / {\sigma_{ p \, p }} }\,. \hspace*{0.8 cm} \end{equation} % Assuming the values $\sigma_{ {\gamma} \, {p} } \approx 0.1 \; {\mbox {mb}}$ and $\sigma_{ p \, p } \approx 40 \; {\mbox {mb}}$, one gets $ \sigma_{ {\gamma} {\gamma} } \approx 250 \; {\mbox {nb}} $. For virtual photons with small $Q_A$ and $Q_B$, the fall-off of the cross section can be estimated from vector meson dominance: % \begin{equation} \label{vmd} \sigma_{ {\gamma^{*}} {\gamma^{*}} } \approx \left( {{M^2_\rho} \over {{M^2_\rho} + Q_A^2}} \right)^2 \;\, \left( {{M^2_\rho} \over {{M^2_\rho} + Q_B^2}} \right)^2 \;\, \sigma_{ {\gamma} {\gamma} } \hspace*{0.8 cm} . \end{equation} % \begin{figure}[htb] \centerline{ \DESepsf(loglog.eps width 12 cm) } \bigskip \caption{$Q^2$-behavior of the vector meson dominance and perturbative cross sections in lowest order, with $Q_A^2 = Q_B^2 \equiv Q^2$.} \label{fig:loglog} \end{figure} Fig.~\docLink{slac-pub-7218.tcx}[fig:loglog]{3} shows a log-log plot of the curves corresponding to the soft and the perturbative formulas (in lowest order) for the $Q^2$-behavior of the cross section. In the region $Q^2 \lesssim 1\ {\rm GeV}^2$, one may expect the formula (\docLink{slac-pub-7218.tcx}[vmd]{15}) based on vector meson dominance to apply. As one goes above this region, the cross section, instead of continuing to fall like $1/Q^8$, should begin to fall more slowly. At large photon virtualities $Q^2 \gtrsim 10\ {\rm GeV}^2$ the cross section should exhibit the perturbative scaling behavior in Eq.~(\docLink{slac-pub-7218.tcx}[gareprsi]{5}), $\sigma \propto 1/Q^2$ at fixed $(s/Q^2)$. With a large luminosity, as at a future high energy linear collider, one can explore virtual photon scattering to higher $Q^2$ and thus probe experimentally the effects of pomeron exchange in the region where summed perturbation theory should apply. One should be able to investigate this region in detail by varying $Q_A$, $Q_B$ and $\hat s = x_A x_B s_{ee}$ independently. We thank D.\ Strom for useful advice. 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