Page:Thomson1881.djvu/12

From Wikisource
Jump to: navigation, search
This page has been proofread, but needs to be validated.


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

\frac{1}{8\pi}\int\int\int2\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\ dz.

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.