Page:Thomson1881.djvu/18

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. For values of r<R we may, as before, substitute for in the integral. Now

the integral

.

By transforming to polars, this may be shown to be . Adding this to the part of the integral due to values of r > R, we get for the coefficient of vv',

.

As before, the coefficients of uv', vu', uw', &c. disappear by inspection.

The coefficient of ww'

;

substituting, for values of r > R, as before for for in the integral, it becomes

,

which, by transforming to polars, may be shown to be . For values of r < R we may, as before, substitute for in the integral. Now

.

On making this substitution, the integral

.

Adding this to the part obtained before, we get for the coefficient of ww',

, or .

From the part of which arises from that part of H due to e and that part of due to e', we can see,