89. The equation

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

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

${\displaystyle Aq:AQ::AF:AQ+AF,}$

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

${\displaystyle Aq:AQ::Af:fQ,}$

which it is more convenient to state thus

${\displaystyle Qf:fA::QA:Aq.}$

From this we derive another useful proportion,

${\displaystyle Qf:QA::QA:Qq.}$

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

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

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

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

The following are for a convex one

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

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

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