Page:Elementary Text-book of Physics (Anthony, 1897).djvu/430

416 surface. Hence $$p'$$ is always negative and the focus virtual when $$L$$ is real.

338. Images formed by Mirrors.—In Fig. 105 let $$ab$$ represent an object in front of the concave mirror $$mn.$$ We know from what precedes that if we consider only the light incident near $$c,$$ the light reflected will be concentrated at some point $$a'$$ on the axis $$ac$$ at a distance from the mirror given by equation (109).

$$a'$$ is a real image of $$a.$$ In the same way $$b'$$ is an image of $$b.$$ If axes were drawn through other points of the object, the images of those points would be found in the same way. They would lie between $$a'$$ and $$$$b', and $$a'b'$$ is therefore a real image of the object. It is inverted, and lies between the axes $$ac, bd,$$ drawn through the extreme points of the object. The ratio of its size to that of the object is seen from the similar triangles $$abC, a'b'C',$$ to be the ratio of the distances from $$C.$$ From equation (109) we obtain $$\frac{p}{p'} = \frac{r}{2p - r} = \frac{r - p'}{p - r}\cdot$$

Since $$r - p'$$ and $$p - r$$ are respectively the distances from the centre of the image and object, we have $$\frac{a'b'}{ab} = \frac{r - p'}{p - r} = \frac{p'}{p};$$ or, the image and object are to each other in the ratio of their respective distances from the mirror. As the object approaches, the image recedes from the mirror and increases in size. At the centre of curvature the image and object are equal, and when the object is within the centre and beyond the principal focus the image is outside the centre and larger than the object. When the object is between the principal focus and the mirror, the image is virtual