In this article "Reflections on Maxwell’s Treatise", Section 4.2, it says:
He replaces $\mathbf{m}$ with a volume element of magnetization $\mathbf{M}\ dV$ , integrates over $V$ , and lets the same integral define the magnetic potential inside as well as outside the magnetization to get in Art. $385$ equation $8$ (of Maxwell's treatise):
$\displaystyle \psi_m (\mathbf {r})=\dfrac{1}{4 \pi} \int_V \mathbf{M(r')}.\nabla' \left( \dfrac{1}{\left| \mathbf{r}-\mathbf{r'} \right|} \right) dV' \tag{22}$
Why is the singularity in equation $(22)$ being ignored?