FIG. 48.—Experimental Proof that the Angle of Incidence = Angle of Reflection. ]
In order to find the direction of the reflected luminous rays, we must turn the telescope on its axis, until the rays reflected by the surface of the mercury bath enter it and produce an image of the star. When the image is brought to the centre of the telescope, it is found that the angle R´ I´ N´ is equal to the angle of reflection N, I, R. Thus, in reading the measure on the graduated circle of the theodolite the angle of reflection can be compared with the angle of incidence.
Now, whatever may be the star observed, and whatever its height above the horizon, it is always found that there is perfect equality between these angles. Moreover, the position of the circle of the theodolite which enables the star and its image to be seen evidently proves that the ray which arrives directly from the luminous point and that which is reflected at the surface of the mercury are both in the same vertical plane.
Now this demonstrates one of the most important laws of reflection. The laws of refraction do not deal directly with the angles themselves, but with the _sines_ of the angles; in reflection the _angles_ are equal; in refraction the _sines_ have a constant relation to each other.
So far we have dealt with plane surfaces, but in the case of telescopes we do not use plane surfaces, but curved ones, so we will proceed at once to discuss these.
FIG. 49.—Convergence of Light by Concave Mirror. ]
FIG. 50.—Conjugate Foci of Convex Mirror. ]
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.