Page:Thomson1881.djvu/12

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

${\frac {1}{8\pi }}\int \int \int 2\mu e\left[\left(r{\frac {d}{dy}}-q{\frac {d}{dz}}\right){\frac {1}{R}}\left({\frac {dH'}{dy}}-{\frac {dG'}{dz}}\right)\right]$ $+\left(p{\frac {d}{dz}}-r{\frac {d}{dx}}\right){\frac {1}{R}}\left({\frac {dF'}{dz}}-{\frac {dH'}{dx}}\right)$ $+\left(q{\frac {d}{dx}}-p{\frac {d}{dy}}\right){\frac {1}{R}}\left({\frac {dG'}{dx}}-{\frac {dF'}{dy}}\right)dx\ dy\ dz$ $-{\frac {1}{8\pi }}\int \int \int \mu e4\pi \left[A(r{\frac {d}{dy}}-q{\frac {d}{dz}}\right){\frac {1}{R}}+B\left(p{\frac {d}{dz}}-r{\frac {d}{dx}}\right){\frac {1}{R}}$ $\left.+C\left(q{\frac {d}{dx}}-p{\frac {d}{dy}}\right){\frac {1}{R}}\right]dx\ dy\ dz$

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

$-{\frac {\mu ep}{4\pi }}\int \int F'\left({\frac {d}{dx}}{\frac {1}{R}}dy\ dz+{\frac {d}{dy}}{\frac {1}{R}}dx\ dz+{\frac {d}{dz}}{\frac {1}{R}}dx\ dy\right)$ $+{\frac {\mu ep}{4\pi }}\int \int {\frac {1}{R}}\left({\frac {dH'}{dx}}dy\ dz+{\frac {dG'}{dx}}dx\ dz+{\frac {dF'}{dx}}dz\ dy\right)$ $+{\frac {\mu ep}{4\pi }}\int \int \int {\frac {1}{R}}{\frac {d}{dx}}\left({\frac {dF'}{dx}}+{\frac {dG'}{dy}}+{\frac {dH'}{dz}}\right)$ $-F'\left({\frac {d^{2}}{dx^{2}}}+{\frac {d^{2}}{dy^{2}}}+{\frac {d^{2}}{dz^{2}}}\right){\frac {1}{R}}dx\ dy\ d$z.

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

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