Nonuniqueness of resolution of hertz vector in presence of a boundary, and the horizontal dipole problem

It is proved that Sommerfeld's resolution of
\Pi, \Pi=(\Pi_{x},0,\Pi_{z})
for the
x
directed dipole source in the classical dipole problem is not unique and is just one of four acceptable resolutions. A method is shown for analytically obtaining one such acceptable resolution, namely,
\Pi=(\Pi_{x},\Pi_{y},0)
. With this resolution, the geometrical optics results for the dipole problem are obtained.