The next term in the multipolar expansion is the term:
Of course, you can get it directly from J9.9 (to a lower approximation) as well, but that does not show you what to do if the small approximation is not valid (in step 2 above) and it neglects part of the outgoing wave!
There are two important and independent pieces in this expression. One of the
two pieces is symmetric in
and the other is antisymmetric
(get a minus sign when the coordinate system is inverted). Any vector
quantity can be decomposed in this manner so this is a very general step: