Achromatic doublet power balance

For two thin lenses in contact, achromatism requires ${\Phi }_1/V_1+{\Phi }_2/V_2=0$, where $V_i$ are their Abbe numbers. With total power $\Phi =1/f$, the component powers are ${\Phi }_1=\Phi V_1/(V_1-V_2)$ and ${\Phi }_2=-\Phi V_2/(V_1-V_2)$. A positive doublet using a larger-$V$ crown glass element and smaller-$V$ flint glass element therefore uses a converging crown and a diverging flint.