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when the magnifying power of the instrument is large, the emergent pencil never fills the pupil. When the telescope is directed towards a bright surface the emergent pencil fills the eye-ring. Let r be the radius of the eyering, and p the radius of the pupil; then, as has been remarked, r is usually smaller than p, and the apparent brightness will be less than the brightness of the object in the proportion of the areas of the eye-ring to that of the pupil. The brightness is therefore given by the equation

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But if m be the magnifying power, m = b/r, where b is the semi-aperture of the object-glass. Hence

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(mp).

Thus the brightness depends on the magnifying power and on the aperture of the object-glass; and if the magnifying power be large, the aperture of the object-glass must be large too, otherwise the brightness of the image will be impaired.

Ex. In making with an astronomical telescope an observation for which it is essential that the brightness of the image on the retina should be at least a hundredth part of that of the object, show that if the diameter of the object-glass be 25 inches and that of the pupil inch, the greatest magnifying power that can be used is 1000. What is the highest magnifying power that can be used without any diminution of brightness?

In Galileo's telescope the eye is placed close to the eye-lens, and the pupil is filled when points are seen by full pencils, and therefore the brightness of the image is very nearly equal to that of the object, and it does not depend on the aperture of the object-glass. But in this instrument the field of view depends on the aperture of the object-glass. This aperture, however, cannot be made very large, because the refraction through the lens is excentrical, and if the aperture be large, the extreme pencils will be refracted at such a distance from the axis as to make the chromatic aberration considerable.

137. Object-glasses are usually made of two lenses, a convex lens of crown glass being combined with a concave lens of flint glass. The pencils of light are incident centrically on the first lens, and if there were an interval between the lenses, the incidence on the second lens would be excentrical; this would be disadvantageous, and the two lenses are placed close together.

We have therefore four quantities at our disposal, namely, the radii of curvature of the four surfaces of the two lenses.

The focal lengths of the two lenses are immediately determined by two essential conditions. These are, that the combination must have a given focal length, and must be achromatic. Let f, f' be the focal lengths of the lenses, and F the focal length of the combination. Then

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These two equations determine f and f', so that no other condition can be satisfied which involves relations between the focal lengths.

The radii of curvature of the surfaces are chosen so as to eliminate as many as possible of the defects due to aberration.

Eye-pieces.

138. In the Astronomical telescope instead of a single eye-glass it is usual to use a combination of two lenses separated by an interval. The introduction of a third lens between the object-glass and the eye-glass will increase the field of view of the instrument. For this reason it is usually called the field-glass.

The incidence of the pencils on the field-glass is not centrical, so that no advantage is gained by placing it

close to the eye-glass. The two lenses of an eye-piece are therefore separated by an interval.

We have therefore five quantities at our disposal, namely, the four radii of curvature of the four surfaces of the lenses, and the distance between the lenses.

If f, f' be the focal lengths of the two lenses, a the distance between them, the focal length of the equivalent lens will be given by the equation

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The focal length F of the combination will be a given quantity, so this is to be considered as one relation between. the constants.

By far the most important defect of the image given by a single lens is that due to chromatic aberration. For a combination of two lenses separated by an interval, it is not possible to remove entirely the defects of this chromatic aberration. The defects of the image are twofold, the coloured images are not in the same plane perpendicular to the axis of the telescope, and they are not of the same magnitude. Either of these defects can be removed, but not both; and the first defect is of the less consequence and is therefore neglected. It is best to make the lenses of the same kind of glass, for then if the combination be achromatic as regards two colours, it will be perfectly achromatic, because there will be no irrationality of dispersion.

It has been shown in § 100 that the condition for this imperfect achromatism for two lenses of the same kind of glass is

a = {} (ƒ+ƒ').

This is a second relation between the constants.

139. The errors of spherical aberration are very complicated in eye-pieces. Without entering into details connected with these defects, it will be understood that the errors will, in general, be reduced by diminishing the aberrations of extreme pencils, and that if the forms of

the lenses be given, this effect will be produced by increasing their number and dividing the refraction. The resulting aberration, other things being equal, will be least when the whole bending of the ray is equally divided among the lenses.

The condition for equal refraction is easily obtained. We shall confine our attention to two lenses. Let a ray, originally parallel to the axis, meet the two lenses at distances y, y' from the axis. Then the deviations produced by the lenses are y/f, and y'f', so that we must have y/f=y'f'. But if 0 be the inclination to the axis of the ray between the lenses

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This condition, expressed in words, is that the interval between the lenses must be equal to the difference of their focal lengths. This is the principle on which Huyghens' eye-piece was constructed.

The preceding conditions only relate to the focal lengths and positions of the lenses, and are independent of their particular forms. The lenses employed are almost invariably plano-convex or equi-convex lenses.

140. If we combine the condition of achromatism with the condition for equal refraction at the two lenses, we get the two equations

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The eye-piece will therefore consist of two lenses, the

field-glass having a focal length equal to three times that of the eye-glass, and the distance between them equal to twice the focal length of the eye-glass. This is the construction of Huyghens' eye-piece, invented by him to diminish the effects of aberration, by making the deviations of the rays at the two lenses equal. It was afterwards pointed out by Boscovich, that it also possessed the advantage of being achromatic.

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The eye-piece is usually made with plano-convex lenses, the plane faces being next the eye. Rays proceeding from the object-glass would meet in q, qp being in the principal focal plane of the object-glass; the rays are caught by the field-glass before reaching q, and are brought to a focus at q', which is in the focus of the eye-glass, so that the rays will emerge parallel to each other. Let A, B be the centres of the lenses, AF the focal length of the lens A; then since AF=3ƒ', AB = 2f', the point F is also the principal focus of the lens B. Since q'p' is in the focus of the lens B, Bp'AB. Also, since p, p' are conjugate foci with respect to the lens A,

=

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and Ap' f'; therefore Ap=3f'=AB. Thus p is the middle point of AF.

Thus the field-glass must be placed between the objectglass and its principal focus, at a distance equal to half its own focal length from the latter.

This eye-piece cannot be used in telescopes where measurements by means of spider-lines or fine wires are to be made. For the principal focus of the object-glass

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