Page:Great Neapolitan Earthquake of 1857.djvu/110

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Trigonometrical

that produced the fractures at $n$ and $w$ ; $f''$ the vertical component corresponding to these; $u$ the intersection of the polar (or direction of emergence of the wave), with the plane passing through $n$ , $w$ and $p$ .

The angles made by $op$ and $ab$ = 90°.

Calling the angle		$qem$	=	$\phi$
"	"	$neo$	=	$\theta$
"	"	$weo$	=	$\theta '$
"	"	$ueo$	=	$\theta ''$ = $epo$ = angle of emergence
"	"	$wne$	=	$\sigma$
"	"	$nwe$	=	$\tau$
"	"	$\sigma +\tau$	=	$\rho$ = 180° − $\phi$
The arc of		$\phi -\phi '$	=	$\psi$
The line		$ne$	=	$\mathrm {L} =f$
"		$we$	=	$\mathrm {M} =f'$
"		$pe$	=	$\mathrm {N}$
"		$no$	=	$z=wo$
"		$oe$	=	$y$
"		$re$	=	$z={\frac {\mathrm {R} }{2}}$

R being the common resultant of $f$ , $f'$ , $f''$ , in $ab$ .

Then $\mathrm {L} +\mathrm {M$

$\tan {\frac {1}{2}}(\theta -\theta ')={\frac {\tan {\frac {1}{2}}\phi (\mathrm {L} -\mathrm {M} )}{\mathrm {L} +\mathrm {M} }}$

\theta ={\frac {\phi -\psi }{2}}

\theta '={\frac {\phi +\psi }{2}}

But ${\displaystyle \sin \theta$
as the diagonals $se$ and $wn$ mutually bisect

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