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and are incident on the small mirror, and are brought to a focus atq; as before, the points Q, q lie on the same line through the centre of the small mirror. The eyelens is placed in such a position that qp is in its principal focal plane, and therefore the rays of the pencil after passing through the eye-lens emerge parallel to each other.

In the original description of the instrument the large reflector was a paraboloid of revolution, and the smaller, a prolate spheroid whose foci are at P and p, the positions of the two images. With reflectors formed of these surfaces, there would be no aberration for rays in the centre of the field. It was for some time deemed hopeless to prepare mirrors having these forms, and the instrument was never constructed till after that of Newton.

Gregory's telescope is generally preferred to Newton's. Its superiority seems to arise from the fact that the two specula may be matched and their irregularities of form made to counteract each other; whereas in Newton's telescope there is nothing to compensate any defect in the form of the object-mirror, and experience shows that such mirrors can seldom be made truly spherical.

151. Let F, F" and ƒ be the focal lengths of the two mirrors and the eye-lens, respectively; e and e' the distances of the object-mirror and the eye-lens from the smaller mirror, and x, x' the distances of the two images. from the same mirror. Then when the instrument is arranged for distant objects so as to suit normal eyes,

But, are conjugate focal distances with respect to the smaller mirror, and therefore,

If we eliminate x, x', we get the equation

This is the equation of condition for distinct vision in Gregory's telescope. The equation is similar to that previously obtained for a refracting telescope with three lenses.

The eye-glass is usually fixed in position and the adjustment to distinct vision effected by moving the smaller mirror by a fine screw.

Let B, B' be the linear magnitudes of the first and second images. Then the angle subtended at the eye by the object is equal to -B/F, and the angle under which the last image is seen is equal to 'f. The magnifying power of the instrument is therefore

But in § 38 it was shown that in reflexion at a spherical surface the relation between the linear magnitudes of an object and its image is expressed in the equation

The values of e, e' as obtained from these equations are

e=F+F+

FF
fm'

e' = F" +ƒ +

F" fm

F›

which determine the intervals between the mirrors and

the lens when the focal lengths and the magnifying power are given.

The first form for m gives a simple approximate value of the magnifying power. For, since the first image is nearly at the principal focus of the smaller mirror, and the second nearly at the vertex of the greater, e'-F"'—ƒ = F', very nearly, and therefore we get

The second image being inverted with respect to the first, and the first with respect to the object, the image as seen through the telescope is erect.

152. The smaller mirror must be of such dimensions as to receive the whole cone of rays converging from the large mirror to its principal focus; if it be greater than this, it will intercept more than is necessary of the incident pencil. The aperture of the smaller mirror is therefore determined by the equation

The aperture in the vertex of the object-mirror must not exceed the aperture of the smaller mirror, for otherwise some of the incident light would fall directly upon the eye-lens; it is usual therefore to make the aperture equal to that of the small mirror, in order that the aperture of the eye-glass, and therefore the field of view, may be as large as possible.

153. The extreme field of view in Gregory's telescope is found by joining the corresponding extremities of the small mirror and the eye-glass by the line B'b; the intercepted portion of the image qp, will determine the field. If a be the semi-aperture of the eye-glass, and B' half the linear magnitude of the second image thus determined, it will be seen that

nearly, since f is small in comparison with x'. already been seen that A'= Ax/F, and therefore

Also, using the same value of m as before,

It has

where denotes half the field of view; and therefore

The second term within the brackets is, in general, small compared with the first, and therefore the field of view is approximately

154. Another reflecting telescope was invented some years after Gregory's and Newton's telescopes by a Frenchman named Cassegrain, probably without any knowledge of what had been done in England. Cassegrain's telescope only differs from Gregory's in having its small mirror convex instead of concave, and placed between the large mirror and its principal focus. The arrangement of the mirrors and images is shown in the figure.

The investigations to find the position of the mirrors and lenses so as to admit of distinct vision, and the magnifying power and the field of view in Gregory's telescope are all applicable to Cassegrain's; we have only to

change the sign of F', the focal length of the small mirror, throughout.

The image will appear inverted, just as in the astronomical telescope.

Ex. The focal lengths of the larger and smaller mirrors of a Gregorian telescope are 32 inches and 3 inches, and the distance between their principal foci inch; it is fitted with a Huyghenian eye-piece, the focal lengths of whose lenses are 3 and 1 inches. Prove that when the instrument is adjusted for normal vision the distance between the field-glass and the smaller mirror is 371⁄2 inches, and that the magnifying power is 256. If the instrument be adjusted for vision at a distance of 12 inches, by altering the position of the smaller mirror, prove that the distance through which the latter must be moved is 001 inch towards the large reflector.

Let e, e' be the distances of the object-mirror and the field-lens, respectively, from the small mirror, and x, x' the distances of the first two images from the same mirror. Then the second image must lie between the lenses of the eye-piece at a distance from the field-lens equal to the distance between the lenses, in order that the rays may emerge from the eye-lens parallel to each other. Hence

Also the distance between the principal foci of the mirrors is inch, and therefore