= Solution
At fixed incidence angle, <differentiation> of the <grating equation> gives $d\cos\beta\,d\beta=m\,d\lambda$. Near the camera axis, the focal-plane displacement is $dX\simeq f_{\rm cam}d\beta$, so the local <grating dispersion> is
$$
\boxed{q=\frac{dX}{d\lambda}=\frac{mf_{\rm cam}}{d\cos\beta}.}
$$
It has units of distance per unit <wavelength>. If the camera axis is at $\beta_0$ and the flat focal-plane coordinate is retained exactly as $X=f_{\rm cam}\tan(\beta-\beta_0)$, then
$$
q=\frac{mf_{\rm cam}\sec^2(\beta-\beta_0)}{d\cos\beta}.
$$
The boxed expression is the local value at the optical axis.
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