For thin lenses in contact, <optical powers> add. At the two reference wavelengths, the <lensmaker's equation> gives total power
$$
\Phi_F=\rho_1(n_{1F}-1)+\rho_2(n_{2F}-1),\qquad\Phi_C=\rho_1(n_{1C}-1)+\rho_2(n_{2C}-1).
$$
An <achromatic lens> has $\Phi_F=\Phi_C$. Subtracting and solving for the curvature-factor ratio yields
$$
\boxed{\frac{\rho_1}{\rho_2}=-\frac{n_{2F}-n_{2C}}{n_{1F}-n_{1C}}.}
$$
Opposite curvature-weighted powers compensate the two materials' <optical dispersion>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 338 / 3 / b / i / Solution

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An achromatic lens combines elements with different <Abbe numbers> to make two specified <wavelengths> share a <focal length>. Other wavelengths can retain secondary chromatic error.

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 Incoming links: Achromatic lens