= Solution
One instrument is a <diffraction grating> spectrometer. Its principal maxima satisfy $d\sin\theta=m\lambda$ at normal incidence. N coherently illuminated grooves give a narrow principal peak; applying the <Rayleigh criterion> to the next wavelength yields the <resolving power of a diffraction grating>
$$
\boxed{\lambda/\delta\lambda=mN.}
$$
Higher order and more illuminated grooves improve resolution, but finite entrance-slit width, aberrations, detector sampling and overlap between spectral orders limit useful resolution. Angular dispersion $d\theta/d\lambda=m/(d\cos\theta)$ is distinct from the ability to resolve two finite-width peaks.
A second instrument is a <Fabry–Pérot interferometer>. For plate spacing L, refractive index n and internal angle theta, transmission resonances satisfy $2nL\cos\theta=m\lambda$. Multiple coherent reflections produce an Airy transmission profile. For identical lossless mirrors of intensity reflectivity $\mathcal R$, the high-finesse limit gives
$$
\boxed{\mathcal F\simeq\frac{\pi\sqrt{\mathcal R}}{1-\mathcal R},\qquad
\lambda/\delta\lambda\simeq m\mathcal F.}
$$
The frequency free spectral range is approximately $c/(2nL\cos\theta)$ for nondispersive n. A larger optical spacing narrows a line but also shrinks the unambiguous frequency interval; nearby orders need sorting. Mirror flatness, absorption, alignment and aperture broadening lower the effective finesse. This resolving-power formula assumes separated narrow lines and a stated line-separation convention, rather than equating transmission peak spacing with peak width.
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