Page:Optics.djvu/88

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89. The equation

${\frac {1}{\Delta ''}}={\frac {1}{F}}+{\frac {1}{\Delta }},$ or $\Delta ''={\frac {\Delta F}{\Delta +F}},$

when put into geometrical language, gives rise to the following proportion, (Fig. 88.)

$Aq:AQ::AF:AQ+AF,$

or if $Af=AF,$ that is, if $f$ be the principal focus for rays incident on the contrary side of the lens to $Q,$

$Aq:AQ::Af:fQ,$

which it is more convenient to state thus

$Qf:fA::QA:Aq.$

From this we derive another useful proportion,

$Qf:QA::QA:Qq.$

From either the equations or the proportions it will be easy to prove that when the distance of $Q$ from the lens is varied, that is, when the place of $Q$ is changed, the lens remaining fixed, the two foci move in the same direction.

The following are corresponding values of $\Delta$ and $\Delta '',$ for a concave lens:

$\infty ..2F..F..{\frac {F}{2}}...0....-{\frac {F}{2}}..-F..-2F..-3F..-\infty$

$F..{\frac {2}{3}}F..{\frac {F}{2}}..{\frac {F}{3}}...0....-F....\infty ....2F....{\frac {3}{2}}F....F.$

The following are for a convex one

$\infty ,2F,F,{\frac {9}{10}}F,{\frac {F}{2}},{\Bigl (}{\frac {F}{n}}{\Bigr )},0,-{\frac {F}{2}},{\Bigl (}-{\frac {F}{n}}{\Bigr )},-F,-3F,-\infty$

$-F,-2F,\infty ,9F,F,{\Bigl (}{\frac {F}{n-1}}{\Bigr )}0,-{\frac {F}{3}},{\Bigl (}-{\frac {F}{n+1}}{\Bigr )},-{\frac {F}{2}},-{\frac {3}{4}}F,-F.$

90. The distance $Qq$ between the foci is represented by $\Delta -\Delta '',$ or $\Delta +\Delta '',$ according as the lens is concave or convex,