# Page:Thomson1881.djvu/15

The kinetic energy

$=\frac{1}{2}\int\int\int\left(F\frac{df}{dt}+G\frac{dg}{dt}+H\frac{dh}{dt}\right)dx\ dy\ dz$.

Now

 $F=\frac{\mu}{5}\left[e\left(u\frac{d^{2}}{dx^{2}}\frac{1}{R}+v\frac{d^{2}}{dx\ dy\ }\frac{1}{r}+w\frac{d^{2}}{dx\ dz\ }\frac{1}{r}\right)\left(\frac{5r^{2}}{6}-\frac{a^{2}}{2}\right)+e\frac{10}{3}\frac{u}{r}\right.$ $\left.+e'\left(u'\frac{d^{2}}{dx^{2}}\frac{1}{r'}+v'\frac{d^{2}}{dx\ dy\ }\frac{1}{r'}+w'\frac{d^{2}}{dx\ dz\ }\frac{1}{r'}\right)\left(\frac{5r'^{2}}{6}-\frac{a'^{2}}{2}\right)+\frac{e'10}{3}\frac{u'}{r'}\right]$,

with similar expressions for G and H.

 $\frac{df}{dt}=\frac{1}{4\pi}\left[e\left(u\frac{d^{2}}{dx^{2}}\frac{1}{r}+v\frac{d^{2}}{dx\ dy\ }\frac{1}{r}+w\frac{d^{2}}{dx\ dz\ }\frac{1}{r}\right)\right.$ $\left.+e'\left(u'\frac{d^{2}}{dx^{2}}\frac{1}{r'}+v'\frac{d^{2}}{dx\ dy\ }\frac{1}{r'}+w'\frac{d^{2}}{dx\ dz\ }\frac{1}{r'}\right)\right]$,

with similar expressions for $\tfrac{dg}{dt}$ and $\tfrac{dh}{dt}$. Since the particles are supposed to be very small, we shall neglect those terms in F which depend on and a'².

The part of the kinetic energy we are concerned with involves the product ee': let us first calculate that part of it arising from the product of that part of F due to e with that part of $\tfrac{df}{dt}$ due to e'. We shall take the line joining the particle as the axis of x; and for brevity we shall denote $\tfrac{\mu ee'}{24\pi}$ by σ.

The coefficient of uu' in the part of the kinetic energy we are considering

:$=\sigma\int\int\int\left(\frac{d^{2}}{dx^{2}}\frac{1}{r}+\frac{4}{r^{3}}\right)r^{2}\frac{d^{2}}{dx^{2}}\frac{1}{r'}dx\ dy\ dz$.

Now, for values of r > R,

 $\frac{1}{r'}=\frac{1}{r}-R\frac{d}{dx}\frac{1}{r}+\frac{R^{2}}{2\ !}\frac{d^{2}}{dx^{2}}\frac{1}{r}-\dots$; $\therefore\frac{d^{2}}{dx^{2}}\frac{1}{r'}=\frac{d^{2}}{dx^{2}}\frac{1}{r}-R\frac{d^{3}}{dx^{3}}\frac{1}{r}+\dots$.

Now, since

$\frac{d^{n}}{dx^{n}}\frac{1}{r}=(-)^{n}\frac{n\ !}{r^{n+1}}Q_{n}$,

where Qn is a zonal harmonic of the nth order; and since the product of two harmonics of different degrees integrated over