hand, it is the left which seems raised in the mirror; and if we raise the left hand the right seems raised. We should falsely express this transposition of the parts of the image in reference to the object if we merely say that the image was reversed; if it were nothing else than the object reversed, in raising the right hand the image should also raise the right hand, while it really is the left which is raised.

This special equality which exists between an object and its image is expressed by saying, that the image is symmetrical in reference to the object; that is, that any point of the image is arranged behind the mirror in identically the same manner as the corresponding point of the object in front. For it may be shown by geometrical considerations, that these two points are equidistant from the mirror, and on the same right line, which is at right angles to the surface. From the respective distance and position of the different parts of the object and of its image, it is concluded that the latter is of the same magnitude as it, and equidistant from the mirror.

Lastly, images formed in plane mirrors are virtual, by which we mean, that they have no real existence, and are only an illusion of the eyes. For in fig. 213 as well as in fig. 214 the light, as it does not pass behind the mirror cannot form any image there, and that which we see has no existence: this is expressed by the word virtual as opposed to actual or real. Virtual images are only an optical illusion; but we shall soon see that, in concave mirrors and in lenses, real images are produced which can be received on screens; this is not the case with virtual images.

We may thus sum up what we have said: images in plane mirrors are symmetrical in reference to the object, of the same magnitude, at the same distance on the other side of the mirror, and are virtual.

303. Multiple images formed by glass mirrors.-Metallic mirrors which have but one reflecting surface only give one image; it is different with glass mirrors, the two surfaces of which reflect, though unequally. For if we apply any object, the point of a pencil, for instance, against a thick piece of polished glass at first, when it is looked at obliquely a very feeble image is seen in contact with it; then, beyond it, another and far more intense one. The first image is due to the light reflected from the anterior surface of the plate ; that is, on the glass itself, while the second is due to the light which, penetrating into the glass, is reflected from the layer of metal by

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Reflection from Transparent Bodies.

295

which the posterior face is covered. The difference in intensity of the two images is readily explained; glass being very transparent, only a small quantity of light is reflected from the first face of the mirror, which gives the least intense image; while the greater part of the incident light passing into the mass is reflected from the surface of the metal, and gives the most luminous image.

The above experiment furnishes a simple means of measuring the thickness of a glass mirror. For the more intense image should appear behind the layer of metal at the same distance as the point of the pencil in front; and it follows thence, that the distance between the point of the pencil and the point of its image is double the thickness of the mirror. If this distance seems to be the eighth of an inch, it will be concluded that the real thickness is th of an inch. The double reflection from mirrors is prejudicial to the sharpness of the images, so that, in scientific observations, metallic mirrors are preferred to glass ones.

304. Reflection from transparent bodies. We have seen that glass, spite of its transparency, reflects a sufficient amount of light to give images, which, though feeble, are distinct. The same is the case with water and other transparent liquids. Thus, on the borders of a pool, we see formed in the water the reversed image of objects on the opposite bank. We say reversed image, so as to

express the appearance; but rigorously we should say symmetrical, from what we have before said (302).

Fig. 216 represents the phenomenon of reflection from the surface of water; it shows how the reflected rays, reaching the eye in an upward direction, reproduce the image of objects situated above the water, just as they would if reflected from a horizontal mirror.

CURVED MIRRORS.

305. Concave mirrors.-There are many kinds of curved mirrors; those most in use are called spherical mirrors, from their curvature being that of a sphere. They may be either of metal or of glass, and are either concave or convex, according as the reflection is from the internal or the external face of the mirror. A curved watch glass, seen from above, gives an idea of a convex mirror, especially if it is covered by a coating of metal on the inside; the same glass coated externally and seen from the inside becomes a concave mirror.

We shall first investigate concave mirrors, and, to facilitate the investigation, will first consider what is called a section; that is, the figure obtained by cutting it into two equal parts. Let MN be the section of a spherical mirror, and C the centre of the corresponding sphere. In reference to the sphere this point is called the centre of curvature; the point A is the centre of the figure.

The infinite right line, ACX, which passes through A and C, is the principal axis of the mirror: any right line, iCd, which simply passes through the centre C, and not through the centre of figure A, is a secondary axis. The angle MCN, formed by joining the centre and extremities of the mirror, is the aperture. A principal or meridional section is any section made by a plane through its principal axis. In speaking of mirrors those lines alone will be considered which lie in the same principal section. There is only one principal axis, but the number of secondary axes is unlimited.

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Focus of Concave Mirrors.

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The theory of the reflection of light from curved mirrors is easily deduced from the laws of reflection from plane mirrors, by considering the surface of the former as made up of an infinitude of extremely small plane surfaces, all equally inclined to each other so as to form a regular spherical surface. Thus, on this hypothesis, when a ray of light falls upon any point whatever of a curved mirror, it is really from a small plane mirror that it is reflected; the reflection takes place then in accordance with the laws already laid down (296).

306. Focus of concave mirrors.-The small facettes, of which we have assumed concave mirrors to be made up, being all inclined towards a common centre, which is the centre of curvature of the mirror, it follows from this obliquity that the rays reflected by their mirrors tend to unite in a single point, which is called the focus, as we have already seen in the case of heat (201).

To explain this property of curved mirrors let SI be a ray falling

upon such a mirror parallel to the axis AX (fig. 217). From the hypothesis assumed above, the reflection takes place at I, on an infinitely small plane mirror. It can be shown by geometrical con

siderations, that the normal to this small mirror is represented by the right line CI from the centre to the point I. Hence the angle SIC represents the angle of incidence; and if we imagine, on the other side of the normal, a straight line IF, which makes with CI an angle FIC, equal to CIS, this straight line will be in the direction of the reflected ray.

But when the incident rays are parallel to the axis of the mirror, as in the above example, it may be proved by geometrical considerations, that the point F, where the luminous ray cuts the axes, is the middle of AC; that it is equidistant from the centre and the mirror. This property being common to all rays parallel to the axis, it follows that, after reflection, these rays will all coincide in the same focus, F, as shown in the figure.

The focus described above that formed at an equal distance from the centre and from the mirror, is called the principal focus; it is produced whenever the rays falling on the mirror are parallel to its axis. An example of this is seen in fig. 218, which represents a pencil of solar light falling upon a concave mirror. If where the reflected rays tend to concentrate themselves a small ground glass screen be placed, a very luminous point will appear, which is the principal focus.

307. Conjugate focus. In the preceding examples we have considered the case of pencils of parallel rays, which presupposes a luminous object at an infinite, or at all events a very great, distance. Let us now consider the case in which the source of light being at a small distance, the rays falling on the mirror are

divergent, as shown in fig. 218. Here the reflected rays are converged, but less so than in figs. 216 and 217, which results from the