But the imago formed at op, instead of being a plane, Is nearly on a portion of n spherical surface whose centre is at x ; and, on the other hand, in order that the rays in each pencil may after refraction in c D be parallel to one another, they ought to diverge from a point nearly in the surface of a sphere whose centre is at Y, the two spherical surfaces befog in contact at z : consequently when the distance between the lenses is such that the crossing of the rays in a pencil parallel to the axis takes place exactly at z, the crossing z in one of the oblique pencils will be at a certain distance from the point a', at which it ought to be to permit the rays in it to go out of c it parallel to one another ; the rays of the pencils which proceed from the margin of the object will not then emerge parallel to one another, and consequently that margin will not be distinctly seen. Moreover, from the unequal refrangibility of the different kinds of light, the rays in each pencil will be decomposed in passing through the lens o D, so that though the chromatic aberration were perfectly corrected in the image at p o, it would exist in the image which is formed in the eye by the rays emerging from c D.
The spherical aberration can only be diminished by diminishing the inclination at which the rays in the marginal pencils fall upon the surface of the lens after having crossed at the focus of the object glass; that is, by using a lens of leas convexity or of greater focal length ; adding a second eye-glass in order finally to render the rays in each pencil parallel to one another. Thus, if it be required to preserve the same magnifying power and field of view as might be obtained with any single eye-glass; let, as before, x, fig. 7, be the place of the objectglass, op the image formed by it, and let C n be the place of the single eye glass : then draw a line o q so as to bisect the angle n o Y, which may be considered as the whole refraction produced by the lens c n : let a, on the right or left of op, be the assumed place of what is called the field-glass, and draw a LI perpendicular to x a', the axis of the telescope, meeting x D in H ; also through draw IC parallel to o q, cutting G o, or a o produced, in m : again draw It N perpendicular to the axis of the telescope, and M a parallel to o. Y; also draw R a perpendicular to the axis. Lastly, draw c a parallel to o zs to meet x o In u, and u v perpendicular to the axis. Then, from the principles of optics, If a lens be placed at a, having its focal length equal to c v, and another at n, whose focal leogth is R s ; the ray x o a will by refraction in the first lens take the direction u 8, and by refraction in the second lens it will take the direction S T parallel to 0 Y or ri E : thus the present visual angle a r a will be equal to D E Y, which was obtained with the single eye-glass.
This is called the Iluyghrnian eye-piece, and it is that which is gene rally used for astronomical telescopes : the object seen through it is inverted, as in the last-mentioned telescope.
If the places a and it of the two eyeglasses are given (G a being very near op; its focal length being also known), and it be required to find the focal length of n a so that the red and violet rays in each pencil may emerge from it parallel to one another, that length might be determined in the following manner. In a pencil of rays crossing each other at II, fig. 8,1ct um Lc the direction of a mean ray, and ur, rra those of a red and a violet ray ; these last will make with one another an angle equal to about of the angle D rim, which may be supposed to be known. Now, by optical principles, if these rays are to emerge from is a in directions parallel to one another, the focal lengths of the lens for red and violet rays, namely, a r and nf must be to one another as 28 to 27, and the foci r and f must be in places determined by perpendi culars drawn to the axis from points w and ac, in which the line a w supposed to be drawn parallel to rr' or rv', meets or and sic ; that is, by finding the position of a line to be drawn from R to cut the given lines nr, sic, so that it w may be to nw as 28 to 27. For this purpose, having drawn the straight line a n, the angles R 11W, R 11w will be known; let them be represented by a and also let the angle u u w be represented by 0: then by trigonometry we shall have, after a few reductions, 27 cotan. a-28 cotan. b= cam'. B.
In order to afford a view of objects in the same position as they appear to have when seen by the naked eye, a. telescope may be formed with three lenses besides the object-glass. In the construction of this instrument, if attention is paid only to the rays which suffer a mean refraction, the first eye-glass, or that which is nearest to the object-end of the telescope, may be placed between the image formed by tho object lens and the eye, with the foci of the two lenses in coincidence ; by this means the rays in each pencil will emerge from the first eye glass in directions parallel to one another, those of the pencils which are oblique to the axis of the telescope crossing each other at some point in the latter axis. A second eye-glass is then placed at any convenient distance from the former, beyond the place where the oblique pencils cross each other; and by this lens a second image is formed in a position contrary to that which is formed by the object lens. Lastly, the third lens being placed between this image and the eye at a distance from the former equal to its focal length, the rays in the several pencils will emerge parallel to one another, and an erect image of the object will thus be formed in the eye.