Let B, B', " be the linear magnitudes of the first three images. Then the angle subtended by the object to the eye is -B/F, and that subtended by the final image is 8"/1. The magnifying power of the instrument is therefore

Next, let the instrument be arranged for an eye seeing at a distance of 12 inches, the adjustment being made by altering the position of the small mirror. Since the position of the eye-piece is unchanged, the value of e' -e is unchanged, and therefore

Let u, v represent the distances of the first and second images in front of and behind the field-lens, respectively, and u', v' similar quantities with reference to the eye-lens. Then we have

or

But

x'=e' - u

=x+34 +1.68,

x=x+35.93.

And since x, x' are conjugate focal distances, this gives

This is a quadratic to find x, and if we solve it we find

x=3·249, very nearly.

But the value of x in the previous arrangement of the instrument was 3.25.

Hence the small mirror must be moved through a distance 001 inch nearer to the larger mirror.

The Compound Microscope.

155. In its simplest form the compound microscope, like the astronomical telescope, consists of two lenses, an object-glass or objective, as it is usually called, and an eye-glass or eye-piece. The objective has a very short focal length, and the object is placed at a distance from it slightly greater than the focal length; the objective then forms a real inverted image of the object, which is viewed through the eye-piece.

The objective is usually made up of a system of lenses, designed to diminish chromatic and spherical aberration. Very generally, there are three doublets, each consisting of a double-convex lens of crown-glass cemented to a plano-concave lens of flint, arranged to be achromatic for centrical pencils; these doublets are placed with their plane faces towards the incident light, the lens of shortest focal length being next the object, and their apertures increasing from the first outwards. In this way the apertures can be chosen that a pencil filling the first lens will just fill the other lenses in succession, so that diaphragms are unnecessary; this is a great advantage, because diaphragms will always introduce diffraction fringes

which interfere with the definition in the outer parts of the field.

Magnifying power of the Microscope.

156. To obtain a correct measure of the magnifying power of an instrument, we must compare the magnitudes of the retinal images, first when the eye is used in combination with the instrument, and secondly when the eye. is used alone. But before this comparison can be definite, we must say where the object and the image formed by the lens system must be placed, in order that the retinal images formed may be fit for the determination of the magnifying power. To make this comparison correct, the eye, and the combination of the eye and the instrument, must be compared as much as possible under analogous circumstances; this may be realised by comparing them while working as favourably as possible, that is, when they give the largest possible images on the retina. For the eye alone, the object must therefore be placed at the nearest point for distinct vision. But the smallest distance for distinct vision is very different for different persons; whereas the magnifying power ought to give an idea of the amplification of the instrument for the eye in general. It has therefore been agreed to place the object at a distance conventionally fixed, a distance not too great for the retinal images to be near their greatest dimensions, and which is large enough for the great majority of eyes to remain accommodated for it during a long time. The distance chosen is 10 inches, and is generally called the "distance of distinct vision." The phrase is not a happy one, for at every distance at which an eye can accommodate itself, it sees equally distinctly. The distance chosen for the position of the image formed by the lens system is the same; for then the retinal images will be proportional to the linear magnitudes of the object and image themselves.

Let x, x be the distances of the object and final image from the first and second principal points of the system, and f the principal focal length. Then

Let B, B' be the linear magnitudes of the object and image, then

B'
x

Also if the distance of the eye from the second principal point be §, the angle under which the image will be seen is given by the equation

Now f is very small, and x' will be negative, and the eye will be not far from the principal point, so that / may be neglected, and

Also supposing the image at the conventional image distance x, x = -λ, and therefore

tan 0 = 8 (+1)

When the object is viewed by the eye at the distance λ, it is seen under an angle 6,, where

and therefore the measure of the magnifying power will be

In general ƒ is very small compared to λ, and therefore the magnifying power is

Ex. The lenses of a common astronomical telescope whose magnifying power is 16, and length from object-glass to eye-glass 8 inches, are arranged as a microscope to view an object placed inch from the object-glass; if the distance of vision be taken to be 8 inches, show that the magnifying power will be 8.

EXAMPLES.

1. The object-glass of an astronomical telescope has a focal length of 50 inches, and the focal length of each lens of the Ramsden's eye-piece is 2 inches; show that when adjusted for normal vision the distance between the field-glass and the objectglass is 50.5 inches, and that the magnifying power is 100/3.

Show that, in order to adjust the instrument for vision at a distance of 10 inches, the eye-piece must be pushed inwards through a distance of an inch, and that then the magnifying power is increased to 35.

2. The focal length of the object-glass of an astronomical telescope is 40 inches, and the focal lengths of four convex lenses forming an erecting eye-piece are, respectively,,,,inches, reckoning backwards from the object-glass. The intervals between the first and second and between the second and third being one inch and half-an-inch, respectively, show that when the instrument is in adjustment for eyes which can see with parallel rays, the distance of the eye-lens from the object-glass is 41 inches, and the magnifying power of the instrument 88.

3. The object-glass of an astronomical telescope has an aperture of 1 foot and a magnifying power of 240. Show that if the aperture of the pupil be inch, the brightness of the image is to that of the object as 1 : 25.

4. A Galileo's telescope, the focal lengths of whose lenses are 6 inches and 1 inch has an objective aperture of 11⁄2 inches; find the field of view as determined by the axes of the extreme pencils. Show that if the instrument be directed towards a graduated rod distant 100 yards, the length of rod visible through the instrument is 10 yards. If the instrument be arranged for vision at a distance of 10 inches, show that the length of rod visible is increased to about 11:36 yards.