Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/184

This page needs to be proofread.

152

AMPÈRE'S THEORY.

[512.

given their relative position is as completely determined as if they formed part of the same rigid body.

512.] If we use rectangular coordinates and make $x$ , $y$ , $z$ the coordinates of $P$ , and $x^{'}$ , $y^{'}$ , $z^{'}$ those of $P^{'}$ , and if we denote by $l$ , $m$ , $n$ and by $l^{'}$ , $m^{'}$ , $n^{'}$ the direction-cosines of $P Q$ , and of $P^{'} Q^{'}$ respectively, then

$\frac{d x}{d s} = l$ ,		$\frac{d y}{d s} = m$ ,		$\frac{d z}{d s} = n$ ,
$\frac{d x^{'}}{d s^{'}} = l^{'}$ ,		$\frac{d y^{'}}{d s^{'}} = m^{'}$ ,		$\frac{d z^{'}}{d s^{'}} = n^{'}$ ,

(2)

and

$l (x^{'} - x) + m (y^{'} - y) + n (z^{'} - z) =$	$r \cos θ$ ,
$l^{'} (x^{'} - z) + m^{'} (y^{'} - y) + n^{'} (z^{'} - z) =$	$- r \cos θ^{'}$ ,
$l l^{'} + m m^{'} + n n^{'} = \cos ϵ$ ,

(3)

where $ϵ$ is the angle between the directions of the elements themselves, and

\cos ϵ = - \cos θ \cos θ^{'} + \sin θ \sin θ^{'} \cos η

.

(4)

Again

r^{2} = (x^{'} - x)^{2} + (y^{'} - y)^{2} + (z^{'} - z)^{2}

,

(5)

whence

$r \frac{d r}{d s}$	$= - (x^{'} - x) \frac{d x}{d s} - (y^{'} - y) \frac{d y}{d s} - (z^{'} - z) \frac{d z}{d s}$ ,
	$= - r \cos θ$ .

(6)

Similarly

$r \frac{d r}{d s^{'}}$	$= (x^{'} - x) \frac{d x^{'}}{d s^{'}} + (y^{'} - y) \frac{d y^{'}}{d s^{'}} + (z^{'} - z) \frac{d z^{'}}{d s^{'}}$ ,
	$= - r \cos θ^{'}$ ;

and differentiating $r \frac{d r}{d s}$ with respect to $s^{'}$ ,

$r \frac{d^{2} r}{d s d s^{'}} + \frac{d r}{d s} \frac{d r}{d s^{'}}$	$= - \frac{d x}{d s} \frac{d x^{'}}{d s^{'}} - \frac{d y}{d s} \frac{d y^{'}}{d s^{'}} - \frac{d z}{d s} \frac{d z^{'}}{d s^{'}}$
	$= - (l l^{'} + m m^{'} + n n^{'})$
	$= - \cos ϵ$ .

(7)

We can therefore express the three angles $θ$ , $θ^{'}$ , and $η$ , and the auxiliary angle $ϵ$ in terms of the differential coefficients of $r$ with respect to $s$ and $s^{'}$ as follows,

$\cos θ$	$= - \frac{d r}{d s}$ ,
$\cos θ^{'}$	$= - \frac{d r}{d s^{'}}$ ,
$\cos ϵ$	$= - r \frac{d^{2} r}{d s d s^{'}} - \frac{d r}{d s} \frac{d r}{d s^{'}}$ ,
$\sin θ \sin θ^{'} \cos η$	$= - r \frac{d^{2} r}{d s d s^{'}}$ .

(8)