Their Significant Properties

The virtue of the Hansen solutions is that they ``automatically'' work to decompose field components into parts that are mutual curls (as required by Faraday/Ampere's laws for the fields) or divergences (as required by Gauss's laws for the fields): _L & = & 0
_L & = & 0
_L & = & i k f_&ell#ell;(kr) Y_L() Hence $\vM_L$ and $\vN_L$ are divergenceless, while the divergence of $\vL_L$ is a scalar solution to the HHE! $\vL_L$ is related to the scalar field and the gauge invariance of the theory in an interesting way we will develop. Also: _L & = & -ik _L
_L & = & ik _L
_L & = & 0 which shows how $\vM_L$ and $\vN_L$ are now ideally suited to form the components of electric and magnetic multipole fields mutually linked by Ampere's and Faraday's law.