That is, corresponding rays, before entry and at emergence, meet the principal planes in points equidistant from the axis.

This gives a useful method of drawing the paths of corresponding rays. Suppose a ray through the point P on the axis to cut the axis again at Q after passing through the lens. Draw PR to meet the first principal plane in R. Draw RS parallel to the axis to meet the second principal plane in S. Then SQ is the path of the emergent ray.

The

Suppose an object in the first principal plane. This may be a real object if the plane is outside the lens; or may be a virtual object formed by converging rays in any case. image will be in the second principal plane. For if in the equation

we make u = o, then v = o. By what we have just seen, the rays through any point of the object diverge on emerging as if they came from a point at the same distance from the axis. Thus the image is of the same size as the object. The principal planes are called, in consequence, planes of unit magnification.

We may also prove these properties of the principal planes as follows: u and v being distances of image and object, measured from the principal points, the magnification is

I

ย

u

Therefore when u and v vanish, the magnification becomes

unity.

It follows that a ray going before entry to any point on the first principal plane, proceeds on emergence from a point at an equal distance from the axis on the second principal plane. That is, the line R S, in the above figure, is parallel to the axis.

It should be noticed that the principal points are a pair of conjugate foci, since when u = 0, v = 0.

A ray proceeding to the first principal point emerges from the second parallel to its original direction. For if Pp, Q q are object and image,

=

HP: H'Q.

Pp Qq
Thus pH and H'q are
parallel.

From what we have

seen, it follows that graphic constructions are to be made in the same way as for thin lenses, only with the addition of the space between the principal planes. If this space were removed, the points H, H' would coincide; and so would the points R and S in the last figure.

If the media on the two sides of a lens are any whatever, not necessarily the same, a pair of conjugate foci can be found such that if a ray proceeds towards one before entering the lens, it emerges from the other in a direction parallel to its original direction. These points are called nodal points. When the lens is situated in air, the nodal points coincide with the principal points.

The formula for the focal length has been given above; the principal focus is at this distance from the second principal point; and just as for a thin lens, there is another principal focus, for rays coming in the opposite direction. This is at a

distance which is numerically the same from the first principal focus.

Except so far as it involves t, the thickness, the focal length will be just the same as for a thin lens of the same substance and the same curvatures. When is small, the focal length will, as a rule, be positive for a lens thinnest in the middle, and negative for a lens thickest in the middle. But it may happen that may decide the sign of f. For suppose t(μ-I) rs negative, and numerically less than The focal length of the lens is then of opposite sign to that of a thin lens of the same substance and curvatures.

r =

μ

Suppose a lens has equal and opposite curvatures, so that — s; has small thickness; and is made of crown glass,

Thus the principal points are the points of trisection of the portion of the axis intercepted by the surfaces.

Suppose two thick lenses, of focal lengths ff, are placed with their axes coincident, and so that their principal points, when taken in order, are a1, ß1; a ß2 Let aß1 = x.

Let an object at a distance u from a give an image by the first lens at a distance from ẞ1; and then an image by the second lens at a distance v from ẞ Then

(f1+f2+ x)uv + (ƒiƒ2+fix)v — (ƒiƒ1⁄2 +ƒ2x)u - f1fx = 0. Now, if the combination is equivalent to a single lens of focal length F, and principal points at distances y and z from a and ẞ1⁄2, we have—

uv + (F − y)v − (F + z)u + F( y − z) + z = 0. Equating coefficients, we have

=

=

=

ƒ1 + £2 + x fif2+x) = f(f1 + x)
fi
These equations are satisfied by writing—

F=

xf1

; 2 =

fifax

xf2

fi + ƒ2+ x; y = √1 + ƒ2+ x; ƒ1 + ƒ2 + x*

Thus a single lens can be found optically equivalent, for direct centrical pencils, to the given combination of two.

Again, it follows that if we have three coaxal lenses, two of these can be replaced, as above, by a single lens; and then this can be combined with the third, and replaced by a single one.

And in the same way any number of coaxal lenses can be replaced by a single lens with definite principal points and principal foci, as far as the action on small axial pencils is concerned. This result is important in the theory of optical instruments.

CHAPTER VIII.

EYE-PIECES-MICROSCOPES-TELESCOPES-OTHER

OPTICAL APPARATUS.

Vision through a Lens.—A single convex lens is sometimes used as a magnifying-glass. It is sometimes arranged on a stand in a manner suitable for viewing small objects. This arrangement is called a simple microscope. Suppose the lens has focal length f; if object and image are at distances u and v from the lens, we have—

Suppose the lens used to produce an erect, virtual image. Then is positive; and ƒ is negative. Thus the image is larger the further off it is. But the image is required to subtend a large angle at the eye rather than to be large itself. We shall consider, therefore, how this result is to be obtained.

Let the least distance of distinct vision be denoted by D. This is the least distance at which the object can be held from the eye so that it may be seen distinctly, or so that the lens of the eye can produce of it a true image on the retina. Or, in practice, rather it means the least distance at which the object may be placed so as to be seen distinctly without over-straining the eye. For a normal eye this distance is generally taken to be about 10 inches, or 25 centimetres. For this distance or for any greater, a normal eye can see an object distinctly and easily, but not at any less distance.

If, then, an image of given size is formed at distance D from the eye, it subtends the greatest possible visual angle at the eye. If the eye is held close up to the lens (the distance between them being negligible), for the greatest visual angle the image must be formed at distance D from the lens; for as its distance is increased its size is increased, but in a smaller ratio. And if the eye is held at a distance from the lens, to bring the image to a distance, D, from the eye, it must be brought nearer to the lens, and so its size diminished. Thus to get the greatest visual angle at the eye, the eye must be held close to the lens, and the image formed at a distance, D, from either. The visual angle thus obtained, supposing object and image

K

to be very small, is size of image÷D. The greatest visual angle that the object can subtend at the eye for distinct vision is size of object D. The ratio of these is called the magnifying power of the lens, and is

The same result may easily be obtained algebraically. Suppose the eye at a distance x from the lens. Then, since

size of image = size of object × ( 1 − 3), and distance of

image from eye = x+v; we want to have

I

で

x+v

as large as

Let x + v = y, the distance of the image from the eye, and this expression becomes

In this x can have any value, and y any value not less than D. It is clear, then, that for any value whatever of y, this is greatest when x = o ; and then it is greatest when y is as small as possible, or

= D.

Eye-pieces. If a single lens is used to look at an object, or at an image that has been formed by an optical system, we have seen that the image formed is defective: it is distorted, the marginal portions undergoing too much or too little displacement from the axis, on account of the spherical aberration produced by the lens. The field of view thus produced is said to want flatness. It used to be supposed that this appearance of the image was due to curvature, so that the image of an object in a plane at right angles to the axis is not formed entirely in another such plane. This effect of curvature is produced; but the appearance of linear distortion is not due to it, but is something independent (see pp. 118-120).

To remedy the defect of linear distortion in the image, instead of using a single lens to view the image with, in a