where we have used the following identity:

Using (3.120) and (3.142), we obtain a series expansion
of the fundamental solution as follows:

where the function is given by (3.121) and the functions and are defined as

In (3.144) and (3.145) the function is given by (3.122).

Now we compute the integral on the right-hand side of
(3.140) over a subset of *S* denoted by *S*_{y} for an
which is away from *S*_{y}. Using (3.143) we obtain

where

In (3.146) we have assumed that the inequality ( ) holds. Notice that

The multipole moments are translated according to the following formulae
as the centre of the multipole expansion is shifted from *O* to *O*':

where we have used (3.123).

One can evaluate the integral on the right-hand side of
(3.140) with the local expansion in the following manner:

where and are the coefficients of the local expansion centred at defined as

and the functions and are obtained by replacing with in (3.144) and (3.145). In the derivation of (3.152) and (3.153) we have used (3.124) and have assumed that the inequality holds.

The coefficients of the local expansion are translated according to the
following formulae as the centre of the local expansion is shifted from
to
:

where we have used (3.123).

Notice that M2M translations in (3.149) and (3.150), M2L translations in (3.152) and (3.153) and L2L translations in (3.154) and (3.155) have the same form as the M2M translation in (3.129), M2L translation in (3.131) and L2L translation in (3.132) in Helmholtz's equation. Also, the derivation of translation formulae in elastodynamics is similar to that in Helmholtz's equation.

We have thus prepared all the formulae needed for FM-BIEM. Using formulae presented above and the algorithm described in chapter 2, one can implement FM-BIEM.