= Solution
Use the signed-angle convention in which the reflection <grating equation> is $m\lambda=d(\sin\alpha+\sin\beta)$, with positive <diffraction order> $m$. For fixed incidence,
$$
\frac{d\beta}{d\lambda}=\frac{m}{d\cos\beta},\qquad
q=f_{\rm cam}\frac{d\beta}{d\lambda}=\frac{mf_{\rm cam}}{d\cos\beta}.
$$
The second identity is the local focal-plane scale, measured about the camera axis aligned with the central diffracted ray. Since $A_{\rm cam}=W\cos\beta$, the projected beam size cancels the cosine in the <grating dispersion>:
$$
\boxed{L=\frac{\lambda qA_{\rm cam}}{f_{\rm cam}}=\frac{Wm\lambda}{d}.}
$$
Writing $N=W/d$ for the number of illuminated grooves also gives $L=mN\lambda$. This connects the geometrical instrument invariant to the phase span of the illuminated <diffraction grating>.
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