the centre of the lens. The first idea of it is due to Wollaston, who proposed to unite two hemispherical lenses by their plane sides, with a stop interposed, the central aperture of which should be equal to one-fifth of the focal length. Brewster showed that the same end might be attained more satisfactorily by grinding a deep groove round the equatorial part of a spherical lens, and filling it with something opaque. The great advantage of this lens is that the oblique pencils as well as the central pencils, pass normally into the lens, so that it is but little subject to defects of aberration.

129. Wollaston was the first to use a combination of two lenses instead of a single lens; this combination is still known as Wollaston's Doublet. It was suggested by an inverted Huyghens' eye-piece, to be described presently. It consists of two plano-convex lenses whose focal lengths are in the proportion of 1: 3, the plane surfaces being turned towards the object, and the lens of shorter focal length being placed next the object. The distance between the lenses can be adjusted to suit different eyes, but is usually 1⁄2 of the shorter focal length.

Ex. A Wollaston's doublet is composed of two convex lenses of focal lengths 1 and 3 inches, separated by a distance of two inches; show that when the second lens is placed near the eye, the position of the object in order that it may be seen at a distance of 10 inches, must be at a distance of inch from the first lens, and that the magnifying power is 33.

130. The refracting telescopes and the compound microscope, in their simplest forms, consist of two lenses. The lens placed nearer to the object receives rays directly from the object and forms a real inverted image of the object; this lens is called the object-glass, or the objective. The inverted image is viewed by the eye through the other lens, which is called the eye-glass or eye-piece; this eye-glass alters the divergence of the small pencils which form the first image, so that they can be brought to a focus on the retina without effort, and increases the visual angle under which the image is seen. In general, an eye is accommodated for rays emerging parallel to each other;

H. O.

10

the eye-glass is therefore placed so that the first image is in the principal focal plane of this lens. In microscopes, however, where the magnifying power is very important, the instrument is arranged so that the final image is at a distance of about 10 inches from the eye; this distance is conventional, but is chosen once for all, so that the magnifying powers of different instruments may be compared under like circumstances.

131. The common Astronomical telescope, the construction of which was first explained by Kepler, consists primarily of two convex lenses fixed in a tube. In the figure, BAC is the lens which is turned towards the object, and it is therefore called the object-glass. This lens forms an inverted image pq, of the object, corresponding points of image and object lying on the same line through A, the centre of the object-glass. Bq, Aq, Cq are three rays diverging from any one point of the object which, after refraction by the object-glass, are made to meet in q, the corresponding point of the image. These rays after crossing at q, fall upon the convex lens bac, called the eyeglass, and after refraction they are in general made to emerge parallel to each other. This will be effected by adjusting the position of the eye-glass, so that the image pq shall lie in its principal focal plane. When the telescope is directed towards distant objects, pq will also be in the principal focal plane of the object-glass, so that the distance between the lenses must then be equal to the sum of their focal lengths.

Let f f be the focal lengths of the object-glass and eye-glass, respectively. Then the angle qAp is the angle which the object subtends at the centre of the object

glass, and this will not differ sensibly from that subtended at the eye. By the naked eye, therefore, the object is seen under an angle whose tangent is -B/f, where B is the linear dimensions of the image. Also, the image pq will be seen through the lens at an angle whose tangent is B/f, wherever the eye be placed, supposing pq to be in the principal focus of the eye-glass. The magnifying power is therefore

m =

f f''

132. The field of view is defined by the axes of the extreme pencils which are transmitted by the eye-glass. It will therefore be the angle which the eye-glass subtends at the centre of the object-glass. Wherefore, if b' denote the semi-aperture of the eye-glass, and half the field of view,

In order to take in the whole extent of this field the eye must be placed at the point in which the axes of the extreme pencils, diverging from the centre of the object-glass, meet the axis of the telescope on their final emergence. The place of the eye is therefore the focus conjugate to the centre of the object-glass as seen through the eye-glass. If x be the distance of this point outside the eye-glass,

1

=

1

計府

1

so that

In the construction of the instrument the tube is prolonged to the required distance and is there furnished with an eye-stop, and in looking through the instrument the eye is placed close to the end of the tube.

133. The field of view as defined by the axes of the extreme pencils is not the entire extent of the visible field, as determined by any rays whatever transmitted

through both the lenses. For if we join the extremities and the centre of the object-glass to one extreme point of the eye-glass, and let the joining lines meet the common focal line in rqs, all the rays from the object-glass which

fall within ps strike the eye-glass; but only half the rays which meet at q are transmitted by the eye-glass, while only one ray of those meeting in r will meet the eyeglass. Thus all the field within As is seen by full pencils, while that between the lines As and Aq is seen by parts of pencils, the part exceeding half the pencil in each case; and the part of the field between Aq and Ar is seen by parts of pencils, the parts being less than half the pencil in each case. Let O," be the values of half the bright field, and half the total visible field, respectively. Let the line mnb be drawn through the extremity of the eyeglass parallel to the axis of the telescope; then by similar triangles, Cm mb = sn : nb. If we denote ps by y, and the semi-apertures of the lenses by b, b', respectively, this relation becomes

:

If b'/b =ƒ'/f, that is, if the apertures of the lenses are proportional to their focal lengths, ' vanishes; in this case the brightness of the field decreases from the centre to the circumference. If b'/b be less than flf, the value of 'becomes negative, and no part will be illuminated by full pencils.

The field as determined by the axes of extreme pencils is limited by the line Aq, and therefore by elementary geometry, or by the values previously obtained,

The field is limited practically to the bright field O', by means of a circular stop, which is placed at the principal focus of the object-glass, whose radius is

This will exclude the images of all points formed by partial pencils.

In an Astronomical telescope there is usually fixed a network of fine wires, vertical and horizontal, the plane of the wires being the focal plane of the object-glass. The image of the object given by the object-glass will then lie in the plane of the wires, and the image and the wires are viewed together through the eye-lens. By the aid of these wires the position of the image of any point can be accurately measured.

B

Galileo's Telescope.

D

A

p

134. This telescope, called after its inventor, Galileo, was the first whose construction was explained on theo