= Solution
\b[$V_d$ is the <Abbe number>], a measure of inverse relative <optical dispersion>. Ordinary <crown glasses> have comparatively large $V_d$ and lower <refractive index>, while ordinary <flint glasses> have smaller $V_d$ and often higher index. The <Abbe diagram> below spans typical optical-glass values; the shaded family ranges are schematic, and the marked examples are catalogue data. Modern glass compositions broaden and overlap the simple crown/flint picture.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-338-abbe-diagram.png]
{title=Refractive index and Abbe number for representative optical glasses}
{description=Markers use refractive indices and Abbe numbers from the SCHOTT Optical Glass pocket catalogue. Shaded regions are schematic family ranges rather than a catalogue boundary. The horizontal axis follows the traditional convention of decreasing Abbe number toward the right.}
For a positive doublet with $V_{1d}>V_{2d}$, the preceding formulas require $f_{1d}>0$ and $f_{2d}<0$: a converging <crown glass> element and a diverging <flint glass> element. Their dispersion corrections cancel, while their net <optical power> is positive. A small difference in Abbe numbers requires large opposing component powers, which makes the design harder to correct for other <optical aberrations>.
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