the diverging rays 1, 2, 3, 4, form a focus at the point o, whereas had they been parallel, their focus would have been at a. That is, the actual focus is at the centre of the sphere, instead of being half way between the centre and circumference, as is the case when the 4 incident rays are parallel. The real focus therefore is beyond, or without the principal focus of the mirror.

Fig. 145.

By the same law, converging rays will form a point within the principal focus of a mirror.

Thus, were the rays falling on the mirror, fig. 144, parallel, the focus would be at a; but in consequence of their previous convergency, they are brought together at a less distance than the principal focus and meet at o.

The images of objects reflected by a convex mirror we have seen, are smaller than the objects themselves. But the concave mirror, when the object is nearer to it than the principal focus, presents the image larger than the object, erect, and behind the mirror.

If the incident rays are divergent, where will be the focus? If the incident rays are convergent, where will be the focus? When will the image from a concave mirror be larger than the object, erect, and behind the mirror? Explain fig. 145, and show why the image is larger than the object.

n, as seen behind the mirror; but by reflection they are made to diverge less than before, and consequently to make the angle under which they meet more obtuse at the eye b, than it would be if they continued onward to e, where they would have met without reflection. The result, therefore, is to render the image h, upon the eye, as much larger than the object a, as the angle at the eye is more obtuse than the angle at e.

On the contrary, if the object is placed more remote from the mirror than the principal focus, and between the focus and the centre of the sphere of which the reflector is a part, then the image will appear inverted on the contrary side of the cenFig. 146.

tre, and farther from the mirror than the object; thus, if a lamp be placed obliquely before a concave mirror, as in fig. 146, its image will be seen in

verted in the air on the contrary side of a perpendicular line through the centre of the mirror.

From the property of the concave mirror to form an inverted image of the object suspended in the air, many curious and surprising deceptions may be produced. Thus, when the mirror, the object, and the light, are placed so that they cannot be seen, (which may be done by placing a screen before the light, and permitting the reflected rays to pass through a small aperture in another screen,) the person mistakes the image of the object for its reality, and not understanding the deception. thinks he sees persons walking with their heads downwards and cups of water turned bottom upwards without spilling a drop. Again, he sees clusters of delicious fruit, and when invited to help himself, on reaching out his hand for that purpose, he finds that the object either suddenly vanishes from his sight, owing to his having moved his eye out of the proper range, or that it is intangible.

This kind of deception may be illustrated by any one who has a concave mirror only of three or four inches in diameter, in the following manner:

When will the image from the concave mirror be inverted and before the mirror? What property has the concave mirror by which singular deceptions may be produced? What are these deceptions?

Concave Mirror. The shape of the concave mirror, is exactly like that of the convex mirror, the only difference between them being in respect to their reflecting surfaces. The reflection of the concave mirror takes place from its inside, or concave surface, while that of the convex mirror is from the outside, or convex surface. Thus the section of a metallic sphere, polished on both sides, is both a concave and convex mirror, as one or the other side is employed for reflection.

The effect and phenomena of this mirror will therefore be, in many respects, directly the contrary from those already detailed in reference to the convex mirror.

From the plane mirror the relation of the incident rays are not changed by reflection; from the convex mirror they are dispersed; but the concave mirror renders the rays reflected from it more convergent, and tends to concentrate them into a focus.

The surface of the concave mirror, like that of the convex, may be considered as a great number of minute plane mirrors, inclined to each other at certain angles, in proportion to its concavity.

The laws of incidence and reflection, are the same when applied to the concave mirror, as those already explained in reference to the other mirrors.

Fig. 141.
α b

In reference to the concave mirror, let us, in the first place, examine the effect of two plane mirrors inclined to each other, as in fig. 141, on parallel rays of light. The incident rays, a and b, being parallel before they reach the reflectors, are thrown off at unequal angles in respect to each other, for b falls on the mirror more obliquely than a, and consequently is thrown off more obliquely in a contrary direction, therefore, the angles of reflection being equal to those of incidence, the two rays meet at c. Thus we see that the effect of two plané mirrors inclined to each other, is to make parallel rays converge and meet in a focus.

What is the shape of the concave mirror, and in what respect does it differ from the convex mirror? How may convex and concave mirrors be united in the same instrument? What is the difference of effect between the concave, convex, and plane mirrors, on the reflected rays? In what respect may the concave mirror be considered as a number of plane mirrors?

The same result would take place, whether the mirror was one continued circle, or an infinite number of small mirrors inclined to each other in the same relation as the different parts of the circle.

The effect of this mirror, as we have seen, being to render parallel rays convergent, the same principle will render diverging rays parallel, and converging rays still more convergent.

The focus of a concave mirror is the point where the rays are brought together by reflection. The centre of concavity in a concave mirror, is the centre of the sphere, of which the mirror is a part. In all concave mirrors, the focus of parallel rays, or rays falling directly from the sun, is at the distance of half the semi-diameter of the sphere, or globe, of which the reflector is a part.

Fig. 142.

a

-1

Thus, the parallel rays 1. 2, 3, &c. fig. 142, all meet at the point o, which is half the distance between the centre a of the whole -2 sphere, and the surface of the reflector, and therefore one quarter the diameter of thewhole sphere, of which the mirror is a part.

3

In concave mirrors of all dimensions, the reflected rays follow the same law; that is, parallel rays meet and cross

each other at the distance of one fourth the diameter of the sphere of which they are sections. principal focus of the reflector.

This point is called the

But if the incident rays are divergent, the focus will be removed to a greater distance from the surface of the mirror, than when they are parallel, in proportion to their divergency.

This might be inferred from the general laws of incidence and reflection, but will be made obvious by fig. 143, where

What is the focus of a concave mirror?

At what distance from its surface is the focus of parallel rays in this mirror? What is the principal focus of a concave mirror?

Suppose the tumbler a, to be filled with water, and placed beyond the principal focus of the concave mirror, fig. 147, and so managed as to be hid from the eye c, by the screen b. The lamp by which the tumbler is illuminated must also be placed behind the screen, and near the tumbler. To a person placed at c, the tumbler with its contents will appear inverted at e, and suspended in the air. By carefully moving forward, and still keeping the eye in the same line with respect to the mirror, the person may pass his hand through the shadow of the Fig. 147.

a

tumbler; but without such conviction, any one unacquainted with such things, could hardly be made to believe that the image was not a reality.

By placing another screen between the mirror and the image, and permitting the converging rays to pass through an aperture in it, the mirror may be nearly covered from the eye, and thus the deception would be increased.

The image reflected from a concave mirror, moves in the same direction with the object, when the object is within the principal focus; but when the object is more remote than the principal focus, the image moves in a contrary direction from the object, because the rays then cross each other. If a man place himself directly before a large concave mirror, but farther from it than the centre of concavity, he will see an inverted image of himself in the air, between him and the mirror, but less than himself. And if he hold out his hand towards the mirror, the hand of his image will come out toward his hand, and he may imagine that he can shake hands

Describe the manner in which a tumbler with its contents may be made to seem inverted in the air. Why does the image move in a contrary direction from its object, when the object is beyond the principal focus ?