Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/327

 

(2) For two lines (Fig. 42) of lengths $a$ and $b$ drawn perpendicular to the extremities of a line of length $c$ and on the same side of it.

⁠	$ab(2\log R+3)={}$	$(c^{2}-(a-b)^{2})\log {\sqrt {c^{2}+(a-b)^{2}}}+c^{2}\log c$	⁠
		${}+(a^{2}-c^{2})\log {\sqrt {a^{2}+c^{2}}}+(b^{2}-c^{2})\log {\sqrt {b^{2}+c^{2}}}$
		${}-c(a-b)\tan ^{-1}{\frac {a-b}{c}}+ac\tan ^{-1}{\frac {a}{c}}+bc\tan ^{-1}{\frac {b}{c}}$.

Fig. 42.

(3) For two lines, $PQ$ and $RS$ (Fig. 43), whose directions intersect at $O$.

$PQ.RS(2\log R+3)$	${}=\log PR(2OP.OR\sin ^{2}O-PR^{2}\cos O)$
	${}+\log QS(2OQ.OS\sin ^{2}O-QS^{2}\cos O)$
	${}-\log PS(2OP.OS\sin ^{2}O-PS^{2}\cos O)$
	${}-\log QR(2OQ.OR\sin ^{2}O-QR^{2}\cos O)$

Fig. 43.

(4) For a point $O$ and a rectangle $ABCD$ (Fig. 44). Let $OP$, $OQ$, $OR$, $OS$, be perpendiculars on the sides, then

$AB.AD(2\log R+3)$	${}=2.OP.OQ\log OA+2.OQ.OR\log OB$
	${}+2.OR.OS\log OC+2.OS.OP\log OD$
	${}+OP^{2}.{\widehat {DOA}}+OQ^{2}.{\widehat {AOB}}$
	${}+OR^{2}.{\widehat {BOC}}+OS^{2}.{\widehat {COD}}$.

Fig. 44.