where the Dirichlet Green's function is written

(372) |

Here, is solution of Laplace's equation (i.e., ) which is chosen so as to ensure that when (or ) lies on . Thus, it follows from Sections 3.4 and 3.5 that

(373) |

where the and the are chosen in such a manner that the Green's function is zero when lies on .

As a specific example, suppose that the volume lies between the two spherical surfaces and . The constraint that as implies that the are all zero. On the other hand, the constraint when yields

(374) |

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

Furthermore, it is readily demonstrated that

It is convenient to write

It follows from Equation (311) that

(379) | ||

(380) |

Thus, Equations (372), (376) and (377) yield

(381) |