A structure theorem for spherically symmetric associated homogeneous
distributions (SAHDs) based on $R^{n}$ is given. It is shown that any SAHD
is the pullback, along the function $\left\vert \mathbf{x}\right\vert
^{\lambda }$,\ $\lambda \in \mathbf{C}$, of an associated homogeneous
distribution (AHD) on $R$. The pullback operator is found not to be
injective and its kernel is derived (for $\lambda =1$). Special attention is
given to the basis SAHDs, $D_{z}^{m}\left\vert \mathbf{x}\right\vert ^{z}$,
which become singular when their degree of homogeneity $z=-n-2p$, $\forall
p\in \mathbb{N}$. It is shown that $\left( D_{z}^{m}\left\vert \mathbf{x}
\right\vert ^{z}\right) _{z=-n-2p}$ are partial distributions which can be
non-uniquely extended to distributions $\left( \left( D_{z}^{m}\left\vert
\mathbf{x}\right\vert ^{z}\right) _{e}\right) _{z=-n-2p}$ and explicit
expressions for their evaluation are derived. These results serve to
rigorously justify distributional potential theory in $R^{n}$.