takes place in the plane of the great circle drawn through the luminous point and the point of incidence.

Let MAM' be the intersection of the mirror by the plane Fig. 9. in question. In this plane at S we place the luminous point, and designate by SI the incident ray which we are to consider. From the point I to the centre of the sphere we draw the perpendicular IC, and taking the angle CIR equal to CIS, we have IR for the direction of the reflected ray.

Since we wish to confine ourselves to the incident rays which are nearly perpendicular, the angles CIR, CIS, must be very small, even for the rays which fall near the edge of the mirror. This requires too things; (1.) That the surface of the mirror should comprehend but a small number of degrees of the sphere of which it is a part. (2.) That the incident ray SA, drawn from the radiant point S to the centre of figure of the mirror A, should make a very small angle SAC with the central perpendicular AC, which is called the axis of the mirror.

13. These two conditions being fulfilled, if we apply to each ray, proceeding from the point S, the construction which we have just given for S1, we shall find by the figure, as well as by calculation, that the reflected rays will pass very near each other, within a small space ƒ, called the focus, which may be considered as a point; so that if it be situated before the mirror, it will give the image of the point S. This is in fact confirmed by experiment, as we shall presently see, but we must first determine the situation of ƒ for every given situation of the luminous point.

Fig. 10.

11.

For this purpose, we begin with the case where this point is situated in the axis AC produced at so great a distance that all the rays proceeding from it may be considered, for the extent embraced by the mirror, as parallel to this axis. We now find that the focus F is situated precisely at the middle point of the axis of curvature AC of the mirror, at equal distances from its surface and centre. The interval AF thus determined is called the principal focal distance of the mirror, and the point F is called the principal focus. When the mirror is concave toward the Fig. 10. radiant point, the principal focus is situated on the same side with this point, before the mirror, and in this focus is a real concentration of light. But if the mirror is convex, the principal Fig. 11. focus falls behind its surface, and the reflected rays not being

13.

able to pass through the mirror to arrive at it, the focus F only indicates the imaginary point of meeting of these rays produced.

14. This case being resolved in a general manner, nothing is more easy than to find the place of the focus for all other posFig. 12. sible situations of the radiant point. For example, let S be this point. Among all the rays proceeding from it, draw SI parallel to the axis AC of the mirror. SI being reflected will pass into the focus F of parallel rays, situated at the middle point of AC; so that the reflected ray will be IF. Drawing now SA to the centre of figure of the mirror, this ray will evidently be reflected on the opposite side of the axis AC, forming with it equal angles on the two sides; and this reflected ray will be f. Produce IF and Af till they meet; the point ƒ will be their focus, and it will be the focus also of all the other rays which proceed from S. The results obtained by construction being expressed algebraically, we shall have a general formula for determining all the successive values of the focal distances, for all possible curvatures of the mirror, and for all the different distances of the radiant point from its surface. Thence we easily determine the distance, position and form of the images, given by all objects of a sensible extent; for each point of the object sends to the mirror a cone of luminous rays which has its focus at the point indicated by the formula, and the assemblage of these foci forms the image of the object.

Fig. 14.

In order to verify these observations by experiment, we must take for the object a body of small extent, so that even if it be brought within a small distance of the mirror, the inclinations of the rays which proceed from it may not exceed the limits supposed in our approximations. This object must also be very luminous, and of such a form that it may be easily known if its image is erect or inverted. Nothing better fulfils all these conditions than the flame of a taper, held nearly in the axis of the mirror, and presented successively at different distances from the surface.

15. We begin with supposing the mirror concave, and place the taper SS' at a considerable distance compared with the diameter of its concavity. Then, since the incident rays may be considered as sensibly parallel, the image ff' will be formed nearly in the principal focus of the mirror at the middle point between its surface and its centre of curvature. The taper is not to be

placed precisely in the direction of the axis, but a little to the Fig. 15. right or left of it, so that the image may be formed on the opposite side of the axis. If we then place in the focus a plate of ground glass and look at it from behind, we shall see painted upon it a little image of the taper, very bright and moreover inverted, as we are taught to expect both from calcalation and our graphic construction. We may also remove the glass and look simply at the image, placing the eye in the direction of the rays which proceed from it, and at such a distance as would be convenient for seeing a real object distinctly. Now if the object be gradually brought nearer to the mirror, the focal distance increases, and the image recedes towards the centre of curvature, preserving always its inverted position and increasing in size; and when the object reaches the centre, they unite and coincide throughout; if the object be brought still nearer, they separate and the image continues to recede, still increasing in size, and still inverted, of which we may satisfy ourselves by looking at it from behind, either with the naked eye or through a plate of ground glass. These appearances continue till the object reaches the principal focus F when the image is removed to an infinite distance from the mirror and is infinitely great, so that it is impossible to place ourselves in a situation to see it. But if the object continue to approach the surface of the mirror, the image reappears behind the surface on the side opposite to the observer. It is now very large and erect; but as the object ap- Fig. 16. proaches it diminishes in size, still remaining erect, and finally when the object touches the surface, it becomes equal to the image, and is situated on the same surface. In the second series, when the image appears behind the mirror, it is no longer a real collection of light, but the imaginary place from which the reflected rays would diverge, if produced beyond the mirror, each in the direction determined by reflection, as represented in the figure.

16. The appearances produced by convex mirrors have much less variety. The image in this case is always imaginary, and is situated behind the mirror, so that it may be seen with the naked eye, but cannot be received upon a plate of ground glass. When the object is placed at a very great distance from the mirror, the image appears in the principal focus, erect, and much smaller than the object itself would appear at an equal distance;

Fig.

17.

it approaches the surface of the mirror as the object approaches, remaining always erect and increasing in size, till finally they coincide upon this surface.

When we wish to make use of spherical mirrors in experiments requiring great exactness, we must be able to determine their foci. If the mirror is concave, it is placed in a room partially darkened, at some distance from a window, from which may be seen different objects at a sufficient distance. Its concave side is then turned towards the window; and it is inclined a little to the right or left, as in figure 15, so that the focus of reflected rays may be thrown into the dark part of the room. We next place in this direction a plate of polished glass, or a piece of white pasteboard ff, which may be brought nearer or removed further from the mirror, till an image of the proposed object abroad is distinctly painted upon it. When the point required is found, we measure its distance from the centre of figure of the mirror, and take the double of this for the length of the radius of curvature. But this method requires, agreeably to what we have before said, that the surface of the mirror should comprehend but a small part of the sphere to which it belongs; otherwise the reflected rays would not, strictly speaking, rigorously meet at any single focus, and the approximation which we have made use of would be inapplicable.

17. If the mirror is convex, the operation is more difficult. In this case we paste upon one of its diameters a strip of black Fig. 18. paper DO, and make in this two small circular apertures, at

equal distances from the centre of figure. The rays of the sun. are then made to fall upon the mirror; and those which are reflected from the two apertures being produced, their imaginary point of diverging is the focus of parallel rays. Measuring their departure from each other at different distances from the surface of the mirror, we easily calculate their point of meeting; and the distance of this point from the surface is half of the radius of the sphere. This method is evidently not capable of much exactness; but if it were necessary to be very accurate in our measurements, which rarely happens with this kind of mirrors, we must make use of the spherometert. We can still collect by

See note subjoined to this treatise on the spherometer.

a concave mirror the reflected rays, fix the point of their convergence and deduce from them the primitive focus, which will be the focus of the concave mirror proposed.

18. When we wish only to verify by observation the results obtained from calculation, it is of little importance whether we use mirrors of metal or of glass. It is nevertheless necessary to remark, that with the latter substance we have always two images, one reflected from the first surface, the other from the second. The latter of these is usually the brightest, because there is applied to the posterior surface of the glass an amalgam of tin, in order to render the reflection more intense; so that in strictness such a mirror is composed of two others, a first of glass and a second of metal. This doubling of images is not attended with any serious inconvenience when we wish only to observe the general results of reflection; but it would be fatal where much accuracy is required; especially if the images are magnified by lenses placed near the eye, as is the case in astronomical instruments. Therefore, metallic mirrors are the only ones we can make use of. We shall see in one of the following sections, the manner in which they are to be employed in constructing the instruments which are called reflecting telescopes.

Of the Heliostat.

19. MOST optical experiments, especially those which have for their object to determine the physical properties of light, are performed upon solar rays admitted into a dark room through a very small aperture in the window-shutter. But this method is attended with two inconveniences; the first is the obliquity of the sun's rays to the horizon; the second is the continued motion of the sun.

By the oblique direction of the rays, they are determined towards the floor on entering the room, and we are confined in our experiments to a small space. From this same canse also they can be retained only for a small part of the day, and we are thus limited as to time and the convenience of continuing our experiments in the same place. Finally, on account of the motion of the sun the direction of the rays is perpetually changing, and we are every instant obliged to change the situation of the objects we would present to them.