Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/328

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296

PARALLEL CURRENTS.

[693.

(5) It is not necessary that the two figures should be different, for we may find the geometric mean of the distances between every pair of points in the same figure. Thus, for a straight line of length $a$ ,

	$\log R$	${}=\log a-{\frac {3}{2}}$ ,
or	$R$	${}=ae^{-{\frac {3}{2}}}$ ,
	$R$	${}=0.22313a$ .

(6) For a rectangle whose sides are $a$ and $b$ ,

	$\log R=\log {\sqrt {a^{2}+b^{2}}}-{\frac {1}{6}}{\frac {a^{2}}{b^{2}}}\log$	${\sqrt {1+{\frac {b^{2}}{a^{2}}}}}-{\frac {1}{6}}{\frac {b^{2}}{a^{2}}}\log {\sqrt {1+{\frac {a^{2}}{b^{2}}}}}$
		${}+{\frac {2}{3}}{\frac {a}{b}}\tan ^{-1}{\frac {b}{a}}+{\frac {2}{3}}{\frac {b}{a}}\tan ^{-1}{\frac {a}{b}}-{\frac {25}{12}}$ .

When the rectangle is a square, whose side is $a$ ,

$\log R$	${}=\log a+{\frac {1}{3}}\log 2+{\frac {\pi }{3}}-{\frac {25}{12}}$ ,
$R$	${}=0.44705a$ .

(7) The geometric mean distance of a point from a circular line is equal to the greater of the two quantities, its distance from the centre of the circle, and the radius of the circle.

(8) Hence the geometric mean distance of any figure from a ring bounded by two concentric circles is equal to its geometric mean distance from the centre if it is entirely outside the ring, but if it is entirely within the ring

$\log R={\frac {a_{1}^{2}\log a_{1}-a_{2}^{2}\log a_{2}}{a_{1}^{2}-a_{2}^{2}}}-{\frac {1}{2}}$ ,

where $a_{1}$ and $a_{2}$ are the outer and inner radii of the ring. $R$ is in this case independent of the form of the figure within the ring.

(9) The geometric mean distance of all pairs of points in the ring is found from the equation

$\log R=\log a_{1}-{\frac {a_{2}^{4}}{(a_{1}^{2}-a_{2}^{2})^{2}}}\log {\frac {a_{1}}{a_{2}}}+{\frac {1}{4}}{\frac {3a_{2}^{2}-a_{1}^{2}}{a_{1}^{2}-a_{2}^{2}}}$ .