= Solution
The telescope forms the sky image at the entrance slit. A <collimator> makes the transmitted beam parallel, a reflection <diffraction grating> disperses it, and a camera’s <optical lens> focuses each <wavelength> to a different detector position.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-338-spectrograph.png]
{title=Reflection-grating spectrograph and slit-limited line profile}
With a uniformly illuminated slit, ideal imaging and negligible slit <diffraction>, each monochromatic slit image has an approximately top-hat intensity profile of physical width $p$. A small <wavelength> separation $\Delta\lambda$ displaces two such images by $q\Delta\lambda$, using the positive magnitude of <grating dispersion>. In the geometrical, slit-limited convention, resolution occurs when the displacement is of order the slit-image width. Hence
$$
\boxed{\Delta\lambda_{\rm slit}=\frac pq,\qquad R=\frac{\lambda q}{p},\qquad p=q\Delta\lambda_{\rm slit}.}
$$
Here $p/q$ is the slit’s apparent wavelength extent. A finite <diffraction grating> broadens the sharp edges; the top-hat sketch and the following identities presume <slit-limited resolving power of a grating>, rather than the regime where the grating’s own diffraction determines the line width. A precise resolution criterion for nonideal profiles can change order-unity factors.
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