Page:SearleEllipsoid.djvu/4

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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

.