Page:Optics.djvu/88

 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,