# Page:Thomson1881.djvu/15

The kinetic energy

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

Now

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

with similar expressions for G and H.

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

with similar expressions for ${\displaystyle {\tfrac {dg}{dt}}}$ and ${\displaystyle {\tfrac {dh}{dt}}}$. 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 ${\displaystyle {\tfrac {df}{dt}}}$ due to e'. We shall take the line joining the particle as the axis of x; and for brevity we shall denote ${\displaystyle {\tfrac {\mu ee'}{24\pi }}}$ by σ.

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

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

Now, for values of r > R,

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

Now, since

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

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