Where $\nabla^2 V = 0$ in 3 dimensional Euclidean space, it is a well-known fact that $${\int_S V(\vec{r'}) d\Omega'\over 4\pi}=V(\vec{a})$$ where $\vec{a}$ is the center of a sphere $S$ of radius $R$ , $d\Omega'$ is infinitesimal solid angle of surface area of $S$ from the point $\vec{a}$, and the integration is done of the surface of $S$.
But I want to know what happens if the integration is done with respect to some point other than $\vec{a}$ where the sphere $S$ is unchanged. Let us call such point $\vec{r}$. My expectation is that it would turn out to be some function similar to $V(\vec{r})$. But I cannot make any progress in my calcaulation. I am also curious about the dependence of the integration on whether $\vec{r}$ is in $S$ or not. Thnak you!