where

are the polar coordinates of the point ,

In FMM, Greengard uses the spherical harmonics making his formulation somewhat complicated. Therefore we use solid harmonics to make the formulae needed for FMM concise. The use of solid harmonics has been suggested by White and Head-Gordon[81] and Perez-Jorda and Yang[66].

We now compute the integral on the right-hand side of
(3.15) over
a subset of *S* denoted by *S*_{y} for
which is away from
*S*_{y}. Using (3.17) we obtain

where

In (3.22) we have assumed that the inequality ( ) holds.

The multipole moment is transformed according to the following formula
when the centre of the multipole moment is translated from *O* to *O*'

where we have used (3.21) and (3.23) (See Appendix E.1). The procedure given by (3.24) is called ``M2M translation''. Noting (A.5) and the fact that is real-valued, one finds that the multipole moment has the following property:

The integral in (3.22) can be evaluated with the coefficient of the local expansion

where is expressed in terms of

In the derivation of this formula we have used (3.20) and have assumed that the inequality holds (See Appendix E.2). The procedure given by (3.27) is called ``M2L translation''. Also, the coefficient of the local expansion has the following property:

The coefficient of the local expansion is translated according to the following formula when the centre of the local expansion is shifted from to

where we have used (3.21) and (3.26) (See Appendix E.3). The procedure given by (3.29) is called ``L2L translation''.