the lens in A'. The ray represented by this line will after refraction, pass through the principal focus F; and its intersection with the secondary axis A O

determines the position of a, the focus conjugate to A. We can in like manner determine the position of b, the focus conjugate to B, another point of the object; and the joining line ab will then be

the image of the line

Fig. 724.-Real and Diminished Image.

A B. It is evident that if ab were the object, A B would be the image.

Figs. 724, 725 represent the cases in which the distance of the

object is respectively

greater and less than

twice the focal length of the lens.

1008. Size of Image. -In each case it is evi

Fig. 725.-Real and Magnified Image.

their distances from the centre of the lens. Again, since by equation (6)

and

p-f

f
ab-AB;

(10)

from which formula the size of the image can be calculated without finding its position.

1009. Example.-A straight line 25mm long is placed perpendi

cularly on the axis, at a distance of 35 centimetres from a lens of 15 centimetres' focal length; what are the position and magnitude of the image?

To determine the distance p' we have

1010. Image on Cross-wires.-The position of a real image seen in mid-air can be tested by means of a cross of threads, or other convenient mark, so arranged that it can be fixed at any required point. The observer must fix this cross so that it appears approximately to coincide with a selected point of the image. He must then try whether any relative displacement of the two occurs on shifting his eye to one side. If so, the cross must be pushed nearer to the lens,. or drawn back, according to the nature of the observed displacement, which follows the general rule of parallactic displacement, that the more distant object is displaced in the same direction as the observer's eye. The cross may thus be brought into exact coincidence with the selected point of the image, so as to remain in apparent coincidence with it from all possible points of view. When this coincidence has been attained, the cross is at the focus conjugate to that which is occupied by the selected point of the object.

By employing two crosses of threads, one to serve as object, and the other to mark the position of the image, it is easy to verify the fact that when the second cross coincides with the image of the first, the first also coincides with the image of the second.

1011. Aberration of Lenses. In the investigations of §§ 1004, 1005, we made several assumptions which were only approximately true. The rays which proceed from a luminous point to a lens are in fact not accurately refracted to one point, but touch a curved surface called a caustic. The cusp of this caustic is the conjugate focus, and is the point at which the greatest concentration of light occurs. It is accordingly the place where a screen must be set to obtain the brightest and most distinct image. Rays from the central parts of the lens pass very nearly through it; but rays from the circumferen

tial portions fall short of it. This departure from exact concurrence is called aberration. The distinctness of an image on a screen is improved by employing an annular diaphragm to cut off all except the central rays; but the brightness is of course diminished.

By holding a convex lens in a position very oblique to the incident light, a primary and secondary focal line can be exhibited on a screen perpendicular to the beam, just as in the case of concave mirrors (§ 975). The experiment, however, is rather more difficult of performance.

1012. Virtual Images.-Let an object A B be placed between a convex lens and its principal focus. Then the foci conjugate to the points A, B are virtual,

and their positions can be found by construction from the consideration that rays through A, B, parallel to the principal axis, will be refracted to F, the principal focus on the other side. These refracted rays, if produced backward, must meet the secondary axes O A, O B in the required points. An eye placed on the other side of the lens will accordingly see a virtual image, erect, magnified, and at a greater distance from the lens than the object. This is the principle of the simple microscope. The formula for the distances D, d of object and image from the lens, when both are on the same side, is

ƒ denoting the principal focal length.

1013. Concave Lens.-For a concave lens, if the focal length be still regarded as positive, and denoted by f, and if the distances D, d be on the same side of the lens, the formula becomes

which shows that d is always less than D; that is, the image is nearer to the lens than the object.

In Fig. 727, AB is the object, and a b the image. Rays incident from A and B parallel to the principal axis will emerge as if they

image ab, which is always virtual, erect and diminished.

1014. Focometer.-Silbermann's focometer (Fig. 728) is an instrument for measuring the focal lengths of convex lenses, and is based

on the principle (§ 1006) that, when the object and its image are equidistant from the lens, their distance from each other is four

P

Fig. 729.

B

times the focal length. It consists of a graduated rule carrying three runners M, L, M'. The middle one L is the support for the lens which is to be examined; the other two, M M', contain two thin plates of horn or other translu

cent material, ruled with lines, which are at the same distance apart in both. The sliders must be adjusted until the image of one of these plates is thrown upon the other plate, without enlargement or

REFRACTION AT SPHERICAL SURFACE

1025

diminution, as tested by the coincidence of the ruled lines of the image with those of the plate on which it is cast. The distance between M and M' is then read off, and divided by 4.

1015. Refraction at a Single Spherical Surface. Suppose a small pencil of rays to be incident nearly normally upon a spherical surface which forms the boundary between two media in which the indices are μ, and respectively. Let C (Fig. 729) be the centre of curvature, and CA the axis. Let P, be the focus of the incident, and P, of the refracted rays. Then for any ray P, B, C B P1 is the angle of incidence and C B P2 the angle of refraction. Hence by the law of sines we have (§ 993)

2

2

M1 sin CBP1 = μg sin CBP2.

Dividing by sin B C A, and observing that

1

ultimately;

CP ultimately;

(13)

sin C B P1

CP1

C P1
API

which expresses the fundamental relation between the positions of the conjugate foci.

Let A C=r, A P1=P1, A. P2-P2, then equation (13) becomes

Again, let CA=p, CP1==91, CP2=92, then equation (13) gives

an equation closely analogous to (14) and leading to the result (analogous to (15))

The signs of P1, P2, r, in (14) and (15) are to be determined by the