Page:Thomson1881.djvu/16

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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

,