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M2M translation formula

The multipole moment centred at O' is given by (See (3.23))
$\displaystyle M_{n,m}(O') = \int_{S_y} \frac{\partial R_{n,m}(\overrightarrow{O'\mbox{\boldmath$ y $ }})}
{\partial n_y}\phi(\mbox{\boldmath$ y $ }) dS_y.$     (E.1)

From (B.11) we obtain
$\displaystyle R_{n,m}(\overrightarrow{O'\mbox{\boldmath$ y $ }}) = \sum_{n'=0}^...
R_{n-n',m-m'}(\overrightarrow{O\mbox{\boldmath$ y $ }}).$     (E.2)

Substituting (E.2) into (E.1) we have

\begin{eqnarray*}M_{n,m}(O') &=& \int_{S_y} \frac{\partial R_{n,m}(\overrightarr...
...{m'=-n'}^{n'} R_{n',m'}(\overrightarrow{O'O})

Ken-ichi Yoshida