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OO'=2 OA =2 a

,

OO ″=2 O'B - O'A = O'B + OB =2 AB =2 c

,

OO ‴=2 A'O″ - OO″ = AO″ + AO =2 OO ″+2 AO =2 c +2 a

,

⋮:

OO' =2 OB = b

,

OO ″ =2 AO' - OO' = AO' + AO =2 AB =2 c

,

OO ‴ =2 BO ″ - OO ″ = BO ″ + BO = OO ″ +2 BO =2 c +2 b

.

⋮:

The angular distances between the object and the images, that is, the angles $OEO'$ , $OEO ″$ , &c. may be calculated by means of their tangents; thus, if $EN$ be perpendicular to $AB$ ,

$OEO'$	$=$	$NEO′−NEO=tan−1.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .nobar .num{display:block;line-height:1em;margin:0.0em 0.1em}.mw-parser-output .sfrac:not([class*="nobar"]) .num{border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠NO′/EN⁠−tan−1⁠NO/EN⁠$ ,
$OEO ″$	$=$	$NEO ″+ NEO =tan -1 ⁠ NO ″ / EN ⁠ +tan -1 ⁠ NO / EN ⁠$ ,
$OEO ‴$	$=$	$NEO ‴- NEO =tan -1 ⁠ NO ‴ / EN ⁠ -tan -1 ⁠ NO / EN ⁠$ , &c.

Thus supposing the distance $AB$ is $5$ inches,

that $AO$	$=$	$2$ ,
$BO$	$=$	$3$ ,
$EN$	$=$	$6$ ,
$NO$	$=$	$1$ .

Then $OO'=4$ ;	$OO ″=10$ ;	$OO ‴=14$ , &c.
$OO' = b$ ;	$OO ″ =10$ ;	$OO ‴ =16$ , &c.

$⁠ NO / EN ⁠ = ⁠ 1 / 6 ⁠ =.1666\dots$	$NEO =tan -1 .1666\dots=9°27' ⁠ 1 / 2 ⁠$ nearly,
$⁠ NO' / EN ⁠ = ⁠ 5 / 6 ⁠ =.8333\dots$	$NEO'=tan -1 .8333\dots=39°48' ⁠ 1 / 2 ⁠$ ;
$∴ OEO'=30°21'$ ,