Page:Thomson1881.djvu/15

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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