= Solution
Substitute the <grating equation> into the preceding result:
$$
\boxed{L=W(\sin\alpha+\sin\beta).}
$$
Moving across the illuminated grating by one groove spacing changes the incident-plus-outgoing <optical path length> by $d(\sin\alpha+\sin\beta)=m\lambda$. Across length $W$, the total path difference is therefore $L$. Thus \b[$L$ is the optical path difference between contributions from the two ends of the illuminated grating.]
The number of coherent phase cycles across it is $L/\lambda=mN$, which is the intrinsic diffraction-limited <spectral resolving power> of the <diffraction grating> under the usual first-minimum criterion. In the slit-limited regime, $R\theta D=L$ also states how much sky angle can be accepted at a given resolution and aperture. Holding slit angle and resolution fixed while increasing telescope diameter requires a larger optical path span. The geometry gives $|L|\leq2W$; large incidence and diffraction angles increase resolution per unit grating length, though grazing beams become impractical.
The slit equations do not imply unlimited resolution when $\theta$ tends to zero. Once $\theta D$ is comparable to $\lambda$, finite-aperture <diffraction> matters and the actual resolving power is bounded by about $mN$.
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