= Solution
Let $r_0=|x_0|$ with $0<r_0<a$. The <Dirichlet Green function inside a sphere> uses the exterior inverse point $x_*=a^2x_0/r_0^2$ and strength $s_*=-a/r_0$. Thus
$$
\boxed{G(x,x_0)=-\frac1{4\pi}\left[\frac1{|x-x_0|}-\frac{a/r_0}{|x-x_*|}\right],\quad x_*=\frac{a^2x_0}{r_0^2},\quad N=1.}
$$
For $|x|=a$, the distance identity $|x-x_*|=(a/r_0)|x-x_0|$ makes the bracket zero. The image is outside the sphere, so it adds no interior singularity. This is the radial weight associated with the <Kelvin transform>.
\b[The printed unit-strength restriction must be relaxed in this part:] the required single image has strength $-a/r_0$, whose magnitude is generally not one. A reflected unit charge does not give the spherical boundary cancellation. For a centered source, the limiting formula is
$$
\boxed{G(x,0)=-\frac1{4\pi|x|}+\frac1{4\pi a}.}
$$
Its correcting term is constant and corresponds to the weighted image moving to infinity, rather than a finite unit point image.
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