Page:Thomson1881.djvu/12

 The part of the kinetic energy we are concerned with will evidently be

Let us take the first integral first, and take the term depending on p; this is

$$\frac{\mu ep}{4\pi}\int\int\int\frac{d}{dz}\frac{1}{R}\left(\frac{dF'}{dz}-\frac{dH'}{dx}\right)-\frac{d}{dy}\frac{1}{R}\left(\frac{dG'}{dx}-\frac{dF'}{dy}\right)dx\ dy\ dz$$.

Integrating by parts this becomes

The surface-integrals are to be taken over the surface of the sphere; and the triple integral is to be taken throughout all space exterior to the sphere.

If the sphere be so small that we may substitute for the values of F', $$\frac{dF'}{dx}$$, &c. at the surface their values at the centre of the sphere, the first surface-integral $$=\mu epF'_{1}$$, where $$F'_1$$ is the value of $$F'$$ at the centre of the sphere; the second surfaceintegral vanishes, and the triple integral also vanishes, since

$$\frac{d^{2}}{dx^{2}}\frac{1}{R}+\frac{d^{2}}{dy^{2}}\frac{1}{R}+\frac{d^{2}}{dz^{2}}\frac{1}{R}=0$$

and

$$\frac{dF'}{dx}+\frac{dG'}{dy}+\frac{dH'}{dz}=0$$.