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 When $$P$$ is between $$E$$ and $$F',$$, the reflexion of the whole circle gives two complete hyperbolas in Fig. 52.

In the first place, the semi-circle $$NAN'$$ gives the portion of hyperbola $$fpf'$$. The part $$gA'g'$$ gives the infinite branches $$gh, \ g'h,$$ and the conjugate hyperbola $$mp'm';$$ and the former hyperbola is completed by the reflexions at $$Ng, \ N'g',$$ considered as convex mirrors. The part $$kk'$$ of the object has for its images the hyperbola $$mp'm',$$ and part of $$fpf'$$ namely, $$\kappa p \kappa';$$ the infinite parts of the line outside the circle are represented by $$fg, \ f'g',$$ and by $$f \gamma, f' \gamma';$$ the remaining parts $$kg, \ k'g',$$ have for images only $$\gamma \kappa,$$ and $$\gamma' \kappa'.$$

46.In all that has preceded, we have confined our attention to sections of the mirror, object, and image; but of course the reader will not find the smallest difficulty in inferring that the image of a plane object, made by a spherical mirror, is, according to circumstances, a portion of a sphere, a spheroid, a parabolic or hyperbolic conoid, or a plane.

47.By referring to the figures, it will readily be seen that when the mirror is concave, the image is, in most cases, inverted with respect to the object: a convex mirror always gives an erect image.

48.It will also be seen, that when the image is inverted, it is what is called a real image: when erect, it is imaginary.

49.Let the object presented to a concave mirror be a portion of its own sphere, (Fig. 53.)

Since rays proceeding from $$P,$$ the extremity of the diameter, are reflected to $$p,$$ making $$Ep$$ two-thirds of $$EF,$$ and that all points of the object are equally distant from the centre, it will readily be seen that the image of the portion of sphere represented by $$PQ,$$ is a corresponding portion of sphere, $$pq,$$ having its radius one-third of that of the mirror.

50.Suppose now the object be a portion of any other sphere.