This gives a practical way of measuring the magnifying power of a telescope. The telescope is pointed to a bright surface, and the diameter of the eye-ring is measured by a graduated scale and lens, forming a micrometer. The diameter of the object-glass can also be measured, and the ratio of the latter to the former gives the magnifying power.

Sometimes it happens that the eye-point falls within the telescope, that is, in front of the outer surface of the eye-lens. The eye cannot then be placed at the eye-point, but is placed as close to it as possible; it is therefore placed close to the eye-glass. In estimating the field of view the radius of the pupil must be used instead of the radius of the eye-lens. Also the object-glass ought not to be considered as the common base of the incident pencils which go to form the picture on the retina, because parts of the full pencils are stopped. The image of the pupil of the eye as formed by the instrument, will then be the common base of all the incident pencils; this image has been called the entrance-pupil. The axes of all the incident pencils which go to form the picture pass through the centre of the entrance-pupil, and the field of view is limited by the cone of rays issuing from this point and filling the object-glass.

124. Brightness of images.

The magnitude of the pupil is subject to be varied by various degrees of light; let ON be its semi-diameter when the object PL is viewed by the naked eye from the distance OP. If the breadth of the principal pencil at O be not less than ON, the area of the pupil will be totally illuminated by the pencil that flows from P. Let PtsrŇ be a ray of that pencil cutting the object-glass in t; and supposing the glasses to be removed, let an unrefracted ray PmN cut the line ct in m.

Then the quantity of refracted rays which fall upon NO is to the quantity of unrefracted rays as the angle CPt to the angle CPm; or, in the ratio of the apparent magnitude of ON to the true. And therefore by turning the

figure round about the axis, the quantity of refracted rays which fill the pupil is to the quantity of unrefracted rays

which would fill it as the apparent magnitude of any surface at O seen from P, to the true; or as the apparent magnitude of any surface at P seen from O, to the true; and consequently as the apparent magnitude of the least surface or physical point P, to the true; that is as the picture of the point P formed on the retina by those refracted rays, to its picture formed by the unrefracted rays. These pictures of the point P are therefore equally bright and cause the appearance of P to be equally bright in both cases.

Next let the pupil be larger than the greatest area at O illuminated by the pencil from P; and supposing a smaller pupil equal to this area, we have shown that the pictures of P upon the retina made by refracted and unrefracted rays would be equally bright. Consequently the picture will be less bright than when the larger pupil is filled by unrefracted rays in the proportion of the smaller pupil to the larger.

Thus it is proved that an object seen through lenses may appear as bright as to the naked eye, but never brighter, even though all the incident light be transmitted through the lenses.

125. The magnifying power of any optical instrument formed of media bounded by spherical surfaces whose centres are ranged along an axis may be found by Gauss' methods.

The axes of the extreme pencils which enter the

object-glass, determine the angle under which the object would be seen, were the eye placed at the centre of the object-glass; and in the case of a telescope, this does not differ sensibly from the angle under which it would be seen by the eye in its position when looking through the instrument; for the distance of an object is usually very great compared with the length of the telescope.

2

With the notation which we used in stating Gauss' general theory, let the axis of one of the extreme pencils be determined by the quantities B, b; and after refraction at the several surfaces, by B,, b1, B2, b2 ... B', b'. Then since the axes of all the incident pencils pass through the first nodal point of the object-glass, b=0, very nearly; for the first nodal point will be very close to the surface of the object-glass. Substituting this value of b in the equaB = kb + 18,

tion

and therefore the magnifying power of the instrument is represented by l.

126. To find the magnitude and position of the eyering, we have only to make = 0 in the equation of § 71, and we get

μη
τ

The eye-ring will be outside the instrument unless h/l be positive.

This proves as before that the magnifying power of the instrument is represented by the ratio of the radius of the object-glass to that of the eye-ring.

EXAMPLES.

1. A person who can see distinctly at a distance of three feet, finds that with a pair of plano-convex spectacles he can see distinctly at a distance of one foot. Find the radius of the curved surface, the refractive index of glass being 3. Ans. 9 inches.

2. A wafer is viewed through a convex lens of 8 inches focal length, placed half-way between it and the eye; find the diameter of the lens when the whole is seen, the diameter of the wafer being half an inch, and its distance from the eye 8 inches. Ans.inch.

3. Three-convex lenses of focal lengths fi, f2, f3 are separated by intervals a, b; find the magnifying power of the combination, and prove that it is independent of the position of the object if

(ƒ1− a) ( ƒ3 − b) +ƒ1⁄2 (ƒ1+ƒ3− a−b)=0.

4. The light after passing through an optical instrument symmetrical about an axis is reflected by a plane mirror perpendicular to its axis so as to pass through it again in the reverse direction show that the compound instrument so formed is equivalent in every respect, if spherical aberration be neglected, to a simple spherical mirror, with its vertex in the position conjugate to the plane mirror and its centre of curvature at the corresponding principal focus.

5. If in any optical instrument formed of lenses and mirrors on the same axis, m is the magnifying power when the instrument is adjusted for an eye which sees clearly with the incident light parallel, and if the eye-glass (focal length f) is moved till the instrument is in adjustment for an eye whose distance of distinct vision is d, show that the magnification is increased by mƒ/8.

6. A stereoscope is constructed of two glass prisms (μ=3) with their edges coincident, and placed so that the faces of each are equally inclined to the plane on which the two pictures are placed, and at a distance of 6 in. The eyes of an observer are 2 in. apart; find their distance from the prism when the axes of the pencils from the middle points of the two pictures have minimum deviation and cross at the point half-way between them, the points being 4 in. apart. Show that the angles of the prisms must be nearly tan-13.

CHAPTER VIII.

OPTICAL INSTRUMENTS.

127. WE have already treated the theory of vision through a single lens and its application to spectacles and reading-glasses. The next optical instrument in the order of simplicity is the simple microscope.

We have seen that when an object is placed at the focus of a convex lens, the rays of the several pencils will emerge parallel to each other, and therefore each pencil will be brought to a focus on the retina without effort; and in this position the angle under which it will appear to the eye is the angle it would subtend at a distance equal to the focal length of the lens. Consequently the image will be distinct and magnified. A lens of high power thus used is called a simple microscope.

If B denote the linear dimensions of the object, the tangent of the visual angle will be B/f, while the tangent of the angle under which it would be seen by the naked eye at the least distance of distinct vision is B/λ; the measure of the magnifying power is therefore /f. Single lenses answer very well so long as the focal length is not smaller than one inch; but when higher powers are required, combinations of more than one lens are preferable.

128. A form of simple magnifier, which possesses certain advantages over a double convex lens, is that commonly known as a "Coddington lens." Coddington lens." The lens is spherical, but the rays are made to pass nearly through