Page:Thomson1881.djvu/4

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;

Now , where Y2 is a surface harmonic of the second order. And when ρ > R,

;

and when ρ < R,

;

where Q1, Q2, &c. are zonal harmonics of the first and second orders respectively referred to OP as axis.

Let Y'2 denote the value of Y2 along OP. Then, since , integrated over a sphere of unit radius, is zero when n and m are different, and when n=m, Y'n being the value of Yn at the pole of Qn, and since there is no electric displacement within the sphere,

,

or, as it is more convenient to write it,

.

By symmetry, the corresponding values of G and H are

,

.

These values, however, do not satisfy the condition

.

If, however, we add to F the term , this condition will be satisfied; while, since the term satisfies Laplace's equation, the other conditions will not be affected: thus we have finally