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It is well-known that if p is a homogeneous polynomial of degree k in n variables, p∈Pk, then the ordinary derivative p(∇) ( r2−n ) has the form An,kY(x)r where An,k is a constant and where Y is a harmonic homogeneous polynomial of degree k, Y∈Hk, actually the projection of p onto Hk. Here we study the distributional derivative p ( ∇ )( r2−n ) and show that the ordinary part is still a multiple of Y, but that the delta part is independent of Y, that is, it depends only on p−Y. We also show that the exponent 2−n is special in the sense that the corresponding results for p(∇)(rα) do not hold if α 6=2−n. Furthermore, we establish that harmonic polynomials appear as multiples of r2−n−2k−2k ′ when p(∇) is applied to harmonic multipoles of the form Y′ (x)r2−n−2k for some Y∈Hk.

Harmonic Polynomials Via Differentiation