28. A plano-convex lens is in contact with a concavoplane lens on the same axis, the refractive indices being μ, μ', and r the radius of the common spherical surface. A ray which cuts the axis at a small angle e and at a distance d from the compound lens is refracted through it. Prove that d the deviation of the ray is (u-1

29. A hollow globe of glass has a speck on its interior surface; if this be observed from a point outside the sphere on the opposite side of the centre, prove that the speck will appear nearer than it is by a distance"-1

.t, provided that 3μ -1 t the thickness of the glass is equal to the radius of the internal cavity, and μ is the refractive index for glass.

30. A pencil issuing from a given point falls directly on a refracting sphere. If Q, be the focus of the part reflected at the front surface, Q, the focus of the part which emerges after reflection at the back, and Q, the focus of the part which goes straight through, show that

CHAPTER VII.

ON IMAGES AND SIMPLE OPTICAL INSTRUMENTS.

77. WE can now explain the manner in which an image or representation of an object is produced by a lens.

We will first take the case of a convex lens.

Let C be the centre of such a lens, CQ its axis, PQ the object.

Fig. (1).

Fig. (2).

Q

Then a pencil of rays from any point P in this object will fall upon the lens so as to cover the whole face of the lens, and will thus be incident centrically. This pencil after refraction will approximately converge to a point p in PC produced, or diverge from some point p in CP produced, the distance Cp being given by the formula

f being the numerical value of the focal length of the lens. If CP <f, Cp is positive, and p lies to the right of C. If CP >f, Cp is negative and p lies in PC produced.

In the first case there emerges from the lens a pencil of rays apparently diverging from a point p as in fig. (1), and in the other case a pencil of rays converging to a point p as in fig. (2).

The same will be true of the pencils which emanate from other points in PQ.

The assemblage of points from which, in the one case, the pencils after refraction appear to diverge, or to which in the other case they converge, is called the image of the object PQ formed by the lens.

In the former case the image is called a virtual image, in the latter a real image, these terms being defined as follows.

A real image formed by a lens or mirror is an image, through the points of which the pencils of light which form the image do actually pass before diverging from them.

A virtual image is one through the points of which the rays of light do not actually pass.

The image in fig. (1) is called an erect image, that in fig. (2) is an inverted image.

Secondly if the light from any object fall upon a concave lens it is easy to see that a virtual image of the object is formed nearer to the lens than the object. The position of

any point p of the image is determined in terms of that of the corresponding point P of the object by the formula

C being the centre of the lens, and p being on the line CP. In a similar way the formation of an image by a mirror can be explained (Art. 47).

78. In any of these cases an eye suitably placed so as to receive the pencils of light after divergence from the points of the image will be rendered sensible of the apparent existence of an object in the position of the image.

This image will more or less closely resemble the original object. It has however two defects.

(1) Indistinctness, arising from the fact that the pencils which emanate from various points in the original object do not accurately converge to or diverge from points after refraction through the lens; the formulæ we have used being only approximations. The image will thus consist of a number of small overlapping circles or ovals, which will cause the general appearance to be somewhat hazy. With good lenses, if the curvatures of their surfaces be not very large, this defect is not very serious, and can be somewhat alleviated by a proper choice of the form of the lens.

A. G. O.

7

(2) Curvature. It is clear that the formula

will not give Cp in a constant ratio to CP. Hence for instance if PQ be a straight line the image pq will not be a straight but a curved line.

The image of any object will similarly be differently curved from the object itself.

The image is also rendered indistinct and imperfect by the fact that white light is composed of a great number of kinds of light of unequal refrangibility. This subject is treated of in Chapter IX.

79. The preceding Articles furnish a ready means of ascertaining by experiment the focal length of a convex lens. If it be placed so as to form a real image q, of any bright object Q in its axis, and the distances of the point and its image from the lens be measured, the focal length is known from the formula which applies to fig. (2) of Art. 77,

1 1 1

ƒ=CQ + Cq'

where ƒ is the numerical value of the focal length.

It can be more readily found by the following method.

Hence Qq the distance between the point and its image

1 1

CQCq

is greatest.

is least when CQ. Cq is least, or when But it is known that when the sum of two quantities is constant their product is greatest when they are equal. Hence