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the surface of the ellipsoid, shall vanish at infinity, and shall satisfy (7). We see at once that if satisfies , then satisfies (7). Now from electrostatics we know that

where is connected with by the relation

satisfies .

Hence

, | (8) |

where is connected with by the relation

, | (9) |

satisfies (7).

Writing for , (8) and (9) become

; | (10) |

. | (11) |

This value of is constant over the surface of the ellipsoid , for at all points of this surface; it also vanishes at infinity, and it satisfies (7). It is therefore the value of required. To find the constant A we make have its proper value at the end of axis .

Now

at the end of the axis.

But by (5)

Again, at we have and consequently

Hence