Page:Optics.djvu/80

 We may now tabulate our results as follows:

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.