where the Newmann Green's function is written

(383) |

Here, is solution of Laplace's equation (i.e., ) which is chosen so as to ensure that

and

The latter constraint holds when (or ) lies on . Note that we have chosen the arbitrary constant to which the potential is undetermined such that . It again follows from Sections 3.4 and 3.5 that

(386) |

where the and the are chosen in such a manner that the constraints (385) and (386) are satisfied.

As a specific example, suppose that the volume lies inside the spherical surface . The physical constraint that the Green's function remain finite at implies that the are all zero. Applying the constraint (385) at , we get

(387) |

Similarly, the constraint (386) leads to

(388) |

for . Hence, the unique Green's function for the problem becomes

Finally, expanding and in the forms (378) and (379), respectively, Equations (383) and (390) yield

(390) |

and

(391) |

for .