  We give explicit formulas for the number of meromorphic differentials on
$\mathbb{CP}^1$ with two zeros and any number of residueless poles and for the
number of meromorphic differentials on $\mathbb{CP}^1$ with one zero, two poles
with unconstrained residue and any number of residueless poles, in terms of the
orders of their zeros and poles. These are the only two finite families of
differentials on $\mathbb{CP}^1$ with vanishing residue conditions at a subset
of poles, up to the action of $\mathrm{PGL}(2,\mathbb{C})$. The first family of
numbers is related to triple Hurwitz numbers by simple integration and we show
its connection with the representation theory of $\mathrm{SL}_2(\mathbb{C})$
and the equations of the dispersionless KP hierarchy. The second family has a
very simple generating series, and we recover it through surprisingly involved
computations using intersection theory of moduli spaces of curves and
differentials.
