Page:Optics.djvu/80

56

We may now tabulate our results as follows:

Case.	Refracting Medium.	Surface.	Equation.
1.	Denser,	Concave,	${\frac {1}{\Delta '}}={\frac {m-1}{mr}}+{\frac {1}{m\Delta }}.$
2.	Denser,	Concave,	${\frac {1}{\Delta '}}={\frac {m-1}{mr}}+{\frac {1}{m\Delta }}.$
3.	Rarer,	Convex,	${\frac {1}{\Delta '}}={\frac {m-1}{r}}+{\frac {m}{\Delta }}.$
4.	Rarer,	Convex,	${\frac {1}{\Delta '}}={\frac {m-1}{r}}+{\frac {m}{\Delta }}.$

79.The distance $Aq$ being independent of the angle $RQA,$ provided that angle be extremely small, we may consider $q$ as the focus in which the refracted rays meet when several incident rays proceed from $Q$ in an extremely small pencil nearly coincident with the axis.

80.In order to find the principal focal distance, which we call $f,$ as in Chap. II, we have of course only to make $\Delta$ infinite in the equations just given; we have then in

Case 1,${\frac {1}{f}}={\frac {m-1}{mr}},$ or $f={\frac {m}{m-1}}r.$

2,${\frac {1}{f}}={\frac {m-1}{mr}},$ or $f=-{\frac {m}{m-1}}r.$

3,${\frac {1}{f}}=-{\frac {m-1}{r}},$ or $f=-{\frac {1}{m-1}}r.$

4,${\frac {1}{f}}={\frac {m-1}{r}},$ or $f={\frac {1}{m-1}}r.$

We might of course easily have found this directly; thus, let $QR,$ (Figs. 71—74.) be an incident ray parallel to the axis $AE,\ RS$ the refracted ray cutting the axis in $F$ the principal focus.