Page:Optics.djvu/62

 When $$P$$ is between $$F$$ and $$A$$, (Fig. 44.) $$c>\frac{r}{2}; \ e<1,$$ and the image is part of an ellipse, namely, all but that part, which we found in the first case.

If we suppose $$P$$ to go outside of the circle beyond $$A$$, (Fig. 46.) we shall be led to the case of a convex mirror.

Our equation will still be

and the image will in all cases be a part of an ellipse, turning its convexity towards $$P$$.

When $$P$$ is on the circle at $$A$$, (Fig. 45.) the image extends from that point both ways to $$f, \ f'$$ the bisection of the radii $$EN, \ EN',$$ which are parallel to $$PQ$$.

When $$P$$ is at an infinite distance from $$A$$, the image is a semi-circle with centre $$P$$, and radius $$EF$$.

45.We may now show how the curve of the image, which we have in different cases found to be a part of a conic section, may be supposed to be completed.

Supposing in all cases the line to be infinite in extent each way.

In the first place, when $$P$$ is at an infinite distance, (Fig. 47.) the semi-circle $$NAN'$$ representing a concave mirror gives a semi-circular image $$fFf';$$ and the convex mirror represented by $$NA'N'$$ gives the image $$fF'f'$$, which completes the circle.

When $$P$$ is at a finite distance outside the circle, the concave and convex parts give together a complete ellipse $$pfp'f'$$, (Fig. 48.).

When $$P$$ is on the circle at $$A'$$, the ellipse is such as represented in Fig. 49, where $$Ep$$ is two-thirds of $$EF.$$

When $$P$$ is between $$A'$$ and $$F'$$, the ellipse cuts the circle, (Fig. 50.)

When $$P$$ is at $$F'$$ the middle point of $$EA'$$, (Fig. 51.) the two parts of the circle divided by the line, unite to produce a complete parabola.