Now , where Y_{2} is a surface harmonic of the second order. And when ρ > R,

and when ρ < R,

where Q_{1}, Q_{2}, &c. are zonal harmonics of the first and second orders respectively referred to OP as axis.

Let Y'_{2} denote the value of Y_{2} 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 Y_{n} at the pole of Q_{n}, 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