  We give a detailed derivation of the spherical harmonics which correspond to
the spectrum in the tables of the 1985 paper by Kim, Romans and the author on
the Kaluza-Klein dimensional reduction of IIB supergravity on $S_5$ to
5-dimensional maximal anti de Sitter supergravity. We show how all methods that
have been used in the literature to obtain spectra if the internal space is
$S_n$ can be derived from embedding $S_n$ in $R^{n+1}$. A crucial observation
for spin 1/2 spherical harmonics is that one also needs a local Lorentz
rotation if one transforms to polar or stereographic coordinates. The relation
to the vector spherical harmonics of electromagnetism is also worked out.
