therefore, appear inverted if we are using an astronomical telescope. Experiment.-Determine the radius of the given surface, checking the result by the use of the spherometer. Enter results thus : Surface Convex A = 175.6 cm. 1 = 2.06 cm. r = 20'5 cm. Value found by spherometer 2016 cm. Measurement of Focal Lengths of Lenses. The apparatus generally employed to determine the focal length of a lens is that known as the optical bench. It consists simply of a horizontal scale of considerable length, mounted on a substantial wooden beam, along which upright pieces can slide, and to these are severally attached the lens, the luminous object, and a screen on which the image formed by the lens is received. These sliding-pieces carry verniers, by which their position with reference to the scale can be determined. The position of each face of the lens relatively to the zero of the vernier is known or can be found as described on p. 337. 51. Measurement of the Focal Length of a Convex Lens. First Method. For this purpose a long bar of wood is employed, carrying at one end a ground-glass screen, fixed at right angles to the length of the bar. A stand, in which the lens can conveniently be fixed with its axis parallel to the length of the bar, slides along it, and the whole apparatus is portable, so that it can be pointed towards the sun or any other distant object. Place the lens in the stand and withdraw to a dark corner of the laboratory; point the apparatus to a distant well-defined object-a vane seen through a window against the sky is a good object to choose if the sun be not visible— and slide the lens along the bar until a sharply defined image of the object is formed upon the ground glass. Since the object is very distant, the distance of the lens from the screen is practically equal to the focal length, and can be measured either with a tape or by means of graduations on the bar itself. The observation should, of course, be made more than once, and the mean of the measurements taken. 52. Measurement of the Focal Length of a Convex Mount on one of the stands of the bench a diaphragm with a hole in it across which two fine threads are stretched, or, if more convenient, a piece of fine wire grating, or a pin in a vertical position with its point about the centre of the hole. Place a light behind the hole, taking care that the brightest part of the light is level with the hole and exactly behind it, while the light is as close to the hole as may be. In the second stand place the lens, fixing it so that its centre is on the same level as that of the hole in the diaphragm, while its axis is parallel to the length of the bench. In the third stand fix an opaque white screen; a piece of ground glass or unglazed paper is most suitable. For the present purpose the objects can generally be fixed on their respective stands so as to occupy with sufficient accuracy the same relative positions with regard to the zeros of the verniers, and thus the distances between the different objects in question can be obtained at once, by reading the verniers and subtracting. If the distance between the first and third stand be more than four times the focal length of the lens, the latter can be placed so that there is formed on the screen a distinct image of the object in the first stand. Move the stand carrying the lens till this is the case. Then measure by means of the verniers fixed to the stands, or as described on p. 337, the distance, u, between the object and the first surface of the lens and the distance, v, between the image and the second surface. Then if we neglect the thickness of the lens the focal length ƒ is given by the formula' The values of v should be observed for at least three different values of u Experiment.-Determine by the methods of this and the preceding sections the focal length of the given lens. 53. Measurement of the Focal Length of a Convex The methods already described for finding the focal lengths of lenses involve the measurement of distances from the lens surface, and con sequently a certain amount comes very important in FIG. 27 the case of short-focus lenses and of lens combinations 'Glazebrook, Physical Optics, chap. iv. The following method avoids the difficulty by rendering the measurement from the lens surfaces unnecessary. We know that for a convex lens, if u, v are the distances respectively of the image and object from the principal points of the lens E F (fig. 27), and fits focal length; then I I u and v being on opposite sides of the lens. Now, if we have two screens A B, CD a distance / apart, and we place the lens EF, so that the two screens are in conjugate positions with regard to it, then u+v=l, provided we neglect the distance between the two principal points. In strictness, u+v is not equal to 7, as the distances u and v are not measured from the same point, but from the two principal points respectively, and these are separated by a distance which is a fraction of the thickness of the lens. Thus, if be the thickness of the lens, it may be shewn that the distance between the principal points is μ-It, if we neglect terms involving 2; the value of this for μ glass is about it. The image of a cross-wire or a piece of wire-grating at the one screen A B will be formed at the other, CD. Now we can find also another position of the lens, E' F', between the screens, such that the image of the cross-wire or grating is again focussed on the second screen. This will evidently be the case when the lens is put so that the values of u and v are interchanged. Let u' and ' be the values which u and v assume for this new position of the lens, and let the distance u'uor v-v' through which the lens has been moved be a. Then we have 1 See Pendlebury's Lenses and Systems of Lenses, p. 39 et seq. so that the focal length may be determined by measuring the distance between the screens (which must be greater than four times the focal length), and the distance through which the lens has to be moved in order to transfer it from one position in which it forms an image of the first screen on the second, to the other similar position. This latter measurement should be made three or four times and the mean taken. For screens, in this case, we may use small pieces of wire gauze mounted in the circular apertures of two of the stands of the optical bench, or we may fix two pins with their points at the centres of these apertures. The coincidence of the image of the first object with the second may be determined by the parallax method described in §§ 47 and 49; or the following very convenient arrangement may be adopted :-In the apertures of the two stands of the optical bench mount two pieces of gauze, as suggested above, setting one of them with its wires horizontal and vertical, and the other with its wires inclined at an angle of 45° to these directions. On the stand carrying the gauze on which the image is to be received, mount a magnifying glass of high power-the positive eye-piece of a telescope serves the purpose admirably-and adjust it so that the gauze is accurately in focus. To obtain the coincidence of the image of the first gauze with the second, we have now only to move |