In Fig. [49], A represents a curved surface, such as that of a concave mirror, the centre of curvature being C. Now we can consider that this curved surface is made up of an infinite number of small plane surfaces, and since all lines drawn from the centre, C, to the mirror, will be at right angles to the surface at the points where they meet it, we find, from our experiment with the plane mirror, that rays falling on the mirror at these points will be reflected so that the angles on either side of each of these lines shall be equal; so, for instance, in Fig. [49], we wish to find to what point the upper ray will be reflected, and we draw a line from the centre, C, to the point where it falls on the mirror, and then draw another line from that point making the angle of reflection equal to that made by the incident ray, and we can consider the small surface concerned in reflection flat, so that the ray will in this case be reflected to F. If now we take any other ray, and perform the same operation we shall find that it is also reflected nearly to F, and so on with all other parallel rays falling on the mirror; and this point, F, is therefore said to be the focus of the mirror. If now the rays, instead of falling parallel on the mirror, as if they came from the sun or a very distant object, are divergent, as if they came from a point S, Fig. [50], near the mirror, the rays approach nearer to the lines drawn from the centre to the mirror, one of which is represented by the dotted line; or, in other words, the angles of incidence become reduced, and so the angles of reflection will also be reduced, and the focus of the rays from S will approach the centre of the mirror, and be at s; just so it will be seen that if an illuminated point be at s, its focus will be at S, and these two points are therefore called conjugate foci.
Fig. 51.—Formation of Image of Candle by Reflection.
Fig. 52.—Diagram explaining Fig. [51].
If a candle is held at a short distance in front of a concave mirror, as represented in Fig. [51], its image appears on the paper between the candle and the mirror, so that the rays from every point of the flame are brought to a focus, and produce an image just as the image is produced by a convex lens. If we study Fig. [52] the formation of this image will be clearly understood. First we must note that the rays A, C, a, and B, C, b, which pass through the centre of curvature of the mirror C, will fall perpendicularly on the surface, and be reflected back on themselves, so that the focus of the part a of the arrow will be somewhere on A a, and that of B on B b, and by drawing another ray we shall find it reflected to a, which will be the focus of the point A, and so also by drawing another line from B, we shall find it is reflected to b, which is the focus of the part B; and we might repeat this process for every part of the arrow, and for every ray from those parts. We now see that since the rays A a and B b cross each other at C, the distance from a to b bears the same proportion to the distance from A to B as their respective distances from the point C; or, in other words, the image is smaller than the object in the same proportion as the distance from the image to C is smaller than the distance from the object to C. Now, in dealing with the stars, which are at a practically infinite distance, the rays are parallel, and will be brought to a focus half-way between the mirror and its centre of curvature. In this case, therefore, the distance from the image to the mirror is equal to that from the image to the centre, so that we can express the size of the image by saying that it is smaller than the object, in proportion as its distance from the mirror is smaller than the distance of the object from C; and as it makes little difference whether we measure the distance of the stars from C or from the mirror, and as C is not always known, we can take the relation of the distances of the object and image from the mirror as representing the proportionate sizes of the two.
We will now consider the case of rays falling on a mirror curved the other way, that is, a convex mirror. Let us consider the ray impinging at D, Fig. [53], which would go on to C, the centre of the mirror. Now, as C D is drawn from the centre, it is at right angles to the mirror at D, and the ray L D, being in the same straight line on the opposite side, will also be at right angles, and will be reflected back on itself. Now take the ray I A, draw C E through A, then E A will be perpendicular to the surface at A, and I A E will be the angle of incidence, and E A G the angle of reflection, so that this ray A G will be reflected away from L D, and so will all the other rays falling on the mirror as K B: and if we continue the lines G A and H B backwards, they will meet at M, and therefore the rays diverge from the mirror as if they came from a point at M, and this point is called the virtual focus.
Fig. 53.—Reflection of Rays by Convex Mirror.
So much for parallel rays. Next let us consider another case which happens in the telescope, namely, where converging rays fall on a convex mirror, as in Fig. [53], where we consider the light proceeding to the mirror from a converging lens along the lines H B and G A, these will be made parallel, at B K and A F, after reflection, and it is manifest that by making the mirror sufficiently convex, these rays, tending to come to a focus at M, could be rendered divergent; and if the curvature is decreased by making the centre of curvature at a certain distance beyond C, it will be seen at once by the diagram that these rays will after reflection, converge towards L and will come to a focus in front of the mirror at a point further in front than C is behind it, so that they have been rendered less convergent only by the mirror in this supposed case.