ject, OP, placed between them. The consequence of this arrangement is, that several images will be observed arranged within the circumference of a circle. Thus we observe in the diagram that the image, O P, in the mirror, A C, is p o, while its image in B C is op; and therefore the reflection of p o in B C will be op, while the image of op in A C will be p o. It therefore appears that po is the image of both, P O, in the mirror, b C, and of o p in the mirror, a C, one of the images covering the other, if the angle B C A be b (or the sixth part of a circle), as in the diagram. If the angle be greater or less, the image, p O, will be twofold.

Concave and convex mirrors are commonly ground to a spherical figure, though other figures, such as ellipsoids, paraboloids, &c., are occasionally used for special purposes. It is the properties of spherical concaves that we shall first describe.

S

B

The general action of a spherical mirror inay be understood by regarding it as made up of a number of small plane mirrors, as A, B, C, D, E, F, G, Fig. 218. On such a combination of small mirrors let rays emanating from R impinge. The different degrees of obliquity under which they fall upon the mirrors cause them to follow new paths after reflection, so that they converge to the point S as to a focus.

Fig. 218.

The problem of determining the path of a ray after it has been reflected is solved by first drawing a perpendicular to the surface at the point of impact, and then drawing a line on the opposite side of this perpendicular, making with it an angle equal to that of the angle of incidence of the

S

incident ray. Thus, let r s, Fig. 219, be an incident ray falling on any reflecting surface at s. To find the path it will take after reflection, we first draw s c, a perpendicular to the surface at the point of impact, s; and then draw the line s fon the opposite side of the perpendicular es, such that the angle cs f is equal to the angle c s r. This is nothing but an application of the general law of reflection, that the angles of incidence and reflection are equal to one another, and are on opposite sides of the perpendicular.

Fig. 219.

When rays of light diverge from the centre of a spherical concave mirror after reflection they converge back to the same point; for, from the nature of such a surface, lines drawn from its centre are perpendicular to the point to which they are drawn. Every ray, therefore, impinges perpendicularly upon the surface, and returns to the centre again.

When parallel rays of light fall on the surface of a spherical mirror, the aperture or diameter of which is not very large, they are reflected to a point half way between the surface and centre of the mirror. Thus, let rsrs' be parallel rays falling on the mirror, s s', the aperture, s s', of which is only a few degrees, these rays, after reflection, will be found converging to the point f, which is called the principal focus, half way between the vertex of the mirror, v, and its centre, c; for if we draw the radii, c s c' s', these lines are perpendiculars to the mirror at the points on which they

fall; then make the angles c sf equal c sr, and c's ƒ equal c s' r', and it is easy to prove that the point f is midway between v and c.

But if the aperture, s s', of the mirror exceeds a few degrees, it may be proved geometrically that the rays no longer converge to the focus, ƒ; but, as the aperture increases, they are found nearer and nearer to the vertex, v, until finally, were it not for the opacity of the mirror, they would fall at the back of it. As this deviation is dependent on the spherical figure of the mirror, it is termed aberration of sphericity.

Fig. 220.

Conversely, if diverging rays issue, from a lucid point, f, Fig. 219, half way between the vertex and centre of a spherical mirror of limited aperture, they will be reflected in parallel lines.

Rays coming from any point, r, Fig. 220, at a finite distance beyond the centre of the mirror,

will be reflected so as to fall between the focus, f, and the centre, c.

Rays coming from a point, r, Fig. 221, between the focus, ƒ, and the vertex, v, will diverge after reflection. Under such circumstances a virtual focus, f, exists at the back of the mirror.

Concave mirrors give rise to the formation of images in their foci. This fact may be shown experimentally by placing a candle at a certain distance in front of such a mirror, and a small screen of paper at the focus. On this paper will be seen an image of the flame, beautifully clear and distinct, but inverted. The relative size and position of this image varies according to the distance of the object from the vertex of the mirror.

Fig. 221.

f

The second variety of curved mirrors is the convex; their chief properties are as follow :—

When parallel rays fall on the surface of a convex mirror, they become

divergent after reflection; for let ss' be such a mirror, and r s r's' rays parallel to its axis falling on it; let c be the centre of the mirror, and draw c s c s', which will be respectively perpendicular to the mirror at the points s and s'; then for the reflected rays, make the angle, t s p, equal to p s r, and the angle, t's' p', equal to p' s' r' ; it may then be demonstrated, that not only do these reflected rays diverge, but if they be produced through the mirror till they intersect, they will give a virtual focus at f, half-way between the vertex

of the mirror, v, and its centre, c, so long as the mirror is of a limited aperture.

In a similar manner it may be proved that diverging rays falling on a convex mirror become more divergent.

To avoid the effect of spherical aberration, it has been proposed to give to mirrors other forms than the spherical. Some are ground to a paraboloidal, and others to an ellipsoidal figure. Of the properties of such surfaces I have already spoken, under the theory of undulations, in Chapter XXXI.; and the effects remain the same, whether we consider light as consisting of innumerable small particles shot forth with great velocity, or of undulations arising in an elastic ether. In both cases parallel rays, falling on a paraboloidal mirror, are accurately converged to the focus, whatever the aperture of the mirror may be; and in ellipsoidal ones, rays diverging from one of the foci are collected together in the other. Occasionally, for the purposes of amusement, mirrors are ground to cylindrical or conical figures; they distort the appearance of objects presented to them, or reflect, in proper proportions, the images of distorted or ludicrous paintings.

CHAPTER XXXVII.

REFRACTION OF LIGHT,

Refractive Action described-Law of the Sines.-Relation of the Refractive Power with other Qualities-Total Reflection-Rays on plane SurfacesThe Prism-Action of the Prism on a Ray-The Multiplying-glass.

WHEN a ray of light passes out of one medium into another of a different density, its rectilinear progress is disturbed, and it bends into a new path, This phenomenon is designated the refraction of light.

Thus, if a sunbeam, entering through a small hole in the shutter of a dark room, falls on the surface of some water contained in a vessel, the beam, instead of passing on in a straight line, as it would have done had the water not intervened, is bent or broken at the point of incidence, and moves in the new direction.

In the same way, also, if a coin or any other object, O, Fig. 223, be placed at the bottom of an empty bowl,

A B C D, and the eye at E so situated that it cannot perceive the coin, the edge of the vessel intervening, if we pour in water the object comes into view; and the cause of this is the same as in the former illustration: for while the vessel is empty the ray_is obstructed by the edge of the bowl, as at O G E, but when water is poured into the height F G, refraction at the.point L, from the perpendicular, PQ, ensues;

Fig. 223.

and now the ray takes the course O L E, and entering the eye at E, the

object appears at K, in the line E L K. For the same reason oars or straight sticks immersed in water look broken, and the bottom of a stream seems at a much less depth than what it actually is.

The same result ensues under the circumstances represented in Fig 224, in which E represents a candle, the rays of which fall on a rectangular box, ABCD, under such circumstances as to cast the shadow of the side A C so as to fall at D. If the box be now filled with water, everything remaining as before, the shadow will leave the point D, and go to d, the rays undergoing refraction as they enter the liquid; and if the eye could be placed at d, it would see the candle at e, in the direction of d A produced.

Fig. 224,

Let N O, Fig 225, be a refracting surface, and C the point of incidence of a ray, BC, CE the course of the refracted ray, and C K the course the ray would have taken had not refraction ensued. With the point of incidence, C, as a centre, describe a circle, N MOG, and from A and R draw the lines A D, R H at right angles to the perpendicular M G to the point C. Then AC M will be the angle of incidence, R C G the angle of refraction; A D is the sine of the angle of incidence, and H R the sine of the angle of refraction. Now in every medium these lines have a fixed relation to one another, and the general law of refraction is as follows:

Fig. 225.

In each medium the sine of the angle of incidence is in a constant ratio to the sine of the angle of refraction; the incident, the perpendicular, and the refracted ray are all in the same plane, which is always at right angles to the plane of the refracting medium.

To a beginner, this law of the constancy of sines may be explained as follows:-Let CD, Fig. 226, be a ray falling on a medium, A B, in the

point D, where it undergoes refraction, and
takes the direction D E. Its sine of incidence,
as just explained, is C g, and its sine of refrac-
tion Ee; and let us suppose that the medium
is of such a nature that the sine of refraction is
one-half the sine of incidence-that is, Ee, is
half C g.
Moreover, let there be a second ray,
HD, incident also at the point D, and re-
fracted along DF; Hh will be its sine of
incidence, and Ff its sine of refraction; and
by the law F f will be exactly one-half H h.
The proportion or relation between these sines
differs when different media are used, but for

the same medium it is always the same. Thus, in the case of water, the proportion is as 1.366 to 1; for flint-glass, 1·584 to 1; for diamond, 2·487 to 1. These numbers are obtained by experiment. They are called the indices of refraction of bodies, and tables of the more common substances are given in the larger works on optics.

No general law has as yet been discovered which would enable us to predict the refractive power of bodies from any of their other qualities; but it has been noticed that inflammable bodies are commonly more powerful than incombustible ones, and those that are dense are more energetic than those that are rare.

When a ray of light passes out of a rare into a dense medium, it is refracted toward the perpendicular. Fig. 224 is an illustration-the rays passing from air into water. But when a ray passes from a dense into a rarer medium, it is refracted from the perpendicular. Fig 223 is an example -the rays passing from water into air.

In every case, when a ray falls on the surface of any medium whatever, it is only a portion which is transmitted, a portion being always reflected. If in a dark room we receive a sunbeam on the surface of some water, this division into a reflected and a refracted ray is very evident; and when a ray is about to pass out of a highly refractive medium into one that is less so, making the angle of incidence so large that the angle of refraction is equal to or exceeds 90°, total reflection ensues. This may be readily shown by allowing the rays from a candle, f, or any other object, to fall on the second face, b c, of a glass prism, a b c, Fig. 227; the eye placed at d will receive the reflected ray, de, and it will be perceived that the face, bc, of the glass, when exposed to the daylight, appears as

α

Fig. 228.

though it were sil-
vered, reflecting
perfectly all objects
exposed to its front,

tion of light, so with refraction-it is to be considered as taking place on plane, convex, and concave surfaces.

When parallel rays fall upon a plane refracting surface they continue parallel after refraction. This must necessarily be the case, on account of the uniform action of the medium.

If divergent rays fall upon a plane of greater refractive power than the medium through which they have come, they will be less divergent than before. Thus, from the point a let the rays a b, a b' diverge; after suffering refraction, they will pass in the paths bc, b' c', and if these lines be projected, they will intersect at a', but a' b, a' b' are less divergent than a b, a b'.

If, on the contrary, rays pass from a medium of greater to one of less refractive power, they will be more divergent after refraction. For this reason bodies under water appear nearer the surface than they actually are. When parallel rays of light pass through a medium bounded by planes