Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/47

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MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING.

47

are solutions of the equation

.

For putting , this equation becomes

,

that is, is a solution of Laplace's equation.

§ 2. Expansion of the Functions.—Let the plane through the axis of the ring and a point , cut the ring in the two circles whose centres are and .

Let

;;;

and the angle

.

Also, for convenience, put

,

and

.

When is less than the above integrals may be expanded in ascending powers of or .

[Cayley, 'Ell. Func.,' p. 54.]