= Solution
For $N$ equally spaced illuminated slits, with $N\simeq W/D$, summing a <geometric series> of <complex amplitudes> gives
$$
I(\theta)=I_1(\theta)\left[\frac{\sin(N\beta)}{\sin\beta}\right]^2,\qquad
\beta=\frac{\pi D\sin\theta}{\lambda}.
$$
Here $I_1(\theta)$ is the one-slit <single-slit diffraction> <wave envelope>. The <grating equation> for principal maxima is
$$
D\sin\theta_m=m\lambda,
$$
with height $N^2I_1(\theta_m)$ and much weaker subsidiary maxima between orders. The first zeros around order $m$ are $\beta=m\pi\pm\pi/N$, so the angular distance from a maximum to its first minimum is, for large $N$,
$$
\delta\theta_{\rm zero}\simeq\frac{\lambda}{ND\cos\theta_m}
\simeq\frac{\lambda}{W\cos\theta_m}.
$$
Changing <wavelength> by $\delta\lambda$ moves the same <diffraction order> by
$$
\delta\theta_\lambda\simeq\left|\frac{d\theta_m}{d\lambda}\right|\delta\lambda
=\frac{|m|\delta\lambda}{D\cos\theta_m}.
$$
The <Rayleigh criterion> places the maximum for one wavelength at the first zero for its neighbour. Equating these separations gives
$$
\boxed{\frac{\lambda}{\delta\lambda}=|m|N\simeq\frac{|m|W}{D}.}
$$
This is the ideal <resolving power of a diffraction grating>, for an illuminated and transmitted order with sufficiently narrow source and instrumental broadening. The zeroth order has no <wavelength> separation and therefore no useful spectral resolving power. The distinction between $W=ND$ and the span $(N-1)D$ of the outer slit centres is a small edge convention; the exact state count in the resolving-power formula is $N$.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2022/paper-2-b7-grating.png]
{title=Original <diffraction grating> pattern and the first-zero width used in the <Rayleigh criterion>}
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