= Solution
$V_d$ is the <Abbe number>, a dimensionless inverse measure of <optical dispersion>. Large $V_d$ means weak dispersion. The <Abbe diagram> below spans a representative visible-light <optical glass> range, roughly $1.45\lesssim n_d\lesssim2.0$ and $20\lesssim V_d\lesssim95$. The shaded families are schematic, not exact catalogue boundaries; modern compositions overlap and include high-index <crown glasses>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-338-glass-map.png]
{title=An Abbe diagram of representative optical glasses}
For a usual positive objective, choose component 1 as a higher-$V_d$ <crown glass> and component 2 as a lower-$V_d$ <flint glass>. Then $f_{1d}>0$, $f_{2d}<0$, and $|f_{2d}|/f_{1d}=V_{1d}/V_{2d}>1$: \b[the <crown glass> is converging, the <flint glass> diverging, and their dispersive powers cancel while leaving positive net power]. The <Abbe numbers> alone do not set their absolute <focal lengths>; the specified total <focal length> does. As a representative pair, $V_{1d}=64.17$, $V_{2d}=36.37$ gives $f_{1d}\simeq0.4332f_d$, $f_{2d}\simeq-0.7644f_d$. Representative index and dispersion data are given in https://www.schott.com/de-de/products/optical-glass-p1000267[SCHOTT's optical-glass data].
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