the surface of a sphere vanishes, we may substitute in the integral for ; then, transforming to polars, the integral
for values of r < R,
Now is a solid harmonic of the nth order; hence is a solid harmonic of the (n-2)th order; and in particular is a solid harmonic of the second order; and, by the same reasoning as before, we may substitute in the integral for . Now
So for values of r < R the integral becomes
Adding this to the part of the integral for r > R, we get for the coefficient of uu' ,. The coefficients of uv' and uw' vanish by inspection.
The coefficient of vv'
Now when r > R we may, by the same reasoning as before, substitute for , in the integral, and it becomes