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