Page:Encyclopædia Britannica, Ninth Edition, v. 24.djvu/462

436 WAVE T H E R Y These numbers represent the influence of aberration upon the intensity at the central point, upon the understanding that the focusing is that adapted to a small aperture, for which h might be neglected. If a readjustment of focus be permitted, the num bers will be sensibly raised. The general conclusion is that an aberration between the centre and circumference of a quarter period has but little effect upon the intensity at the central point of the image. Disturb- As an application of this result, let us investigate what amount ance due of temperature disturbance in the tube of a telescope may be to varia- expected to impair definition. According to Biot and Arago, the tion of index /t for air at t C. and at atmospheric pressure is given by temper- 00029 ature. *~ l =T If we take C. as standard temperature, Images by simple aper tures. Simple versus achro matic lens. Thus, on the supposition that the irregularity of temperature t extends through a length I, and produces an acceleration of a quarter of a wave-length, or, if we take A = 5 3 x 10 - 5 , the unit of length being the centimetre. We may infer that, in the case of a telescope tube 12 cm. long, a stratum of air heated 1 C. lying along the top of the tube, and occupying a moderate fraction of the whole volume, would pro duce a not insensible effect. If the change of temperature pro gressed uniformly from one side to the other, the result would be a lateral displacement of the image without loss of definition ; but in general both effects would be observable. In longer tubes a similar disturbance would be caused by a proportionally less dif ference of temperature. We will now consider the application of the principle to the formation of images, unassisted by reflexion or refraction. 1 The function of a lens in forming an image is to compensate by its vari able thickness the differences of phase which would otherwise exist between secondary waves arriving at the focal point from various parts of the aperture (OPTICS, vol. xvii. p. 802). If we sup pose the diameter of the lens to be given (2R), and its focal length / gradually to increase, the original differences of phase at the image of an infinitely distant luminous point diminish without limit. When / attains a certain value, say f lt the extreme error of phase to be compensated falls to $. But, as we have seen, such an error of phase causes no sensible deterioration in the definition ; so that from this point onwards the lens is useless, as only improving an image already sensibly as perfect as the aperture admits of. Throughout the operation of increasing the focal length, the resolving power of the instrument, which depends only upon the aperture, remains unchanged ; and we thus arrive at the rather startling conclusion that a telescope of any degree of resolving power might be constructed without an object-glass, if only there were no limit to the admissible focal length. This last proviso, however, as we shall see, takes away almost all practical importance from the proposition. To get an idea of the magnitudes of the quantities involved, let us take the case of an aperture of ^ inch, about that of the pupil of the eye. The distance / 1; which the actual focal length must exceed, is given by so that (4). = riT we nlu /! = 800 inches. The image of the sun thrown upon a screen at a distance exceeding 66 feet, through a hole inch in diameter, is therefore at least as well defined as that seen direct. As the minimum focal length increases with the square of the aperture, a quite impracticable distance would be required to rival the resolving power of a modern telescope. Even for an aperture of 4 inches,/! would have to be 5 miles. A similar argument may be applied to find at what point an achromatic lens becomes sensibly superior to a single one. The question is whether, when the adjustment of focus is correct for the central rays of the spectrum, the error of phase for the most extreme rays (which it is necessary to consider) amounts to a quarter of a wave-length. If not, the substitution of an achromatic lens will be of no advantage. Calculation shows that, if the aper ture be i inch, an achromatic lens has no sensible advantage if the focal length be greater than about 11 inches. If we suppose the focal length to be 66 feet, a single lens is practically perfect up to an aperture of 17 inch. Some estimates of the admissible aberration in a spherical lens have already been given under OPTICS, vol. xvii. p. 807. In a similar manner we may estimate the least visible displacement of 1 Phil. May., March 1SS1. the eye-piece of a telescope focused upon a distant object, a Accu- question of interest in connexion with range-finders. It appears - racy of that a displacement Sf from the true focus will not sensibly focusing. impair definition, provided 5/</-A/R- , ....... (5), 2R being the diameter of aperture. The linear accuracy required is thus a function of the ratio of aperture to focal length. The formula agrees well with experiment. The principle gives an instantaneous solution of the question of Delicacy the ultimate optical efficiency in the method of " mirror-reading," of mirroi as commonly practised in various physical observations. A rotation reading. by which one edge of the mirror advances % (while the other edge retreats to a like amount) introduces a phase-discrepancy of a whole period where before the rotation there was complete agreement. A rotation of this amount should therefore be easily visible, but the limits of resolving power are being approached ; and the conclusion is independent of the focal length of the mirror, and of the employ ment of a telescope, provided of course that the reflected image is seen in focus, and that the full width of the mirror is utilized. A comparison with the method of a material pointer, attached to Compari the parts whose rotation is under observation, and viewed through son witl a microscope, is of interest. The limiting efficiency of the micro- material scope is attained when the angular aperture amounts to 180 pointer. (MICROSCOPE, vol. xvi. p. 267; OPTICS, vol. xvii. p. 807); and it is evident that a lateral displacement of the point under observation through ^A entails (at the old image) a phase-discrepancy of a whole period, one extreme ray being accelerated and the other retarded by half that amount. We may infer that the limits of efficiency in the two methods are the same when the length of the pointer is equal to the width of the mirror. An important practical question is the amount of error admis- Admis sible in optical surfaces. In the case of a mirror, reflecting sible at nearly perpendicular incidence, there should be no deviation from truth (over any appreciable area) of more than -|A. For glass, /j. 1 nearly; and hence the admissible error in a refracting surface of that material is four times as great. In the case of oblique reflexion at an angle (j>, the error of retardation due to an elevation BD (fig. 7) is errors of optic; surfaces from which it follows that an error of given magnitude in the figure of a surface is less important in oblique than in perpendicular reflexion. It must, however, be borne in mind that errors can sometimes be compensated by altering adjustments. If a surface intended to be flat is affected with a slight general curvature, a remedy may be found in an alteration of focus, and the remedy is the less complete as the reflexion is more oblique. The formula expressing the optical power of prismatic spectro- Optical scopes is given with examples under OPTICS, vol. xvii. p. 807, and power o may readily be investigated upon the principles of the wave theory, prisms. Let A B (fig. 8) be a plane wave-surface of A the light before it falls upon the prisms, AB - the corresponding wave-surface for a parti- J - VN cular part of the spectrum after the light has passed the prisms, or after it has passed the eye-piece of the observing telescope. The path of a ray from the wave-surface A B to A or B is determined by the condition that the optical distance, fj. ds, is a minimum (OPTICS, vol. xvii. p. 798) ; and, as AB is by supposition a wave-surface, this optical distance is the same for both points. Thus fl^ds (tor A) =fnds(foi- B) ..... (6). We have now to consider the behaviour of light belonging to a neighbouring part of the spectrum. The path of a ray from the wave-surface A B to the point A is changed ; but in virtue of the minimum property the change may be neglected in calculating the optical distance, as it influences the result by quantities of the second order only in the changes of refrangibility. Accordingly, the optical distance from A B to A is represented by J~(/jL + 5fj.)ds, the integration being along the original path A . . . A ; and similarly the optical distance between A B and B is represented l>yf(/j.+ 8/j.)ds, the integration being along B . . . B. In virtue of (6) the difference of the optical distances to A and B is fSpds (along B . . . l}}-fSfj.ds (along A . . . A) . (7). The new wave-surface is formed in such a position that the optical distance is constant ; and therefore the dispersion, or the angle through which the wave surface is turned by the change of refrangibility, is found simply by dividing (7) by the distance AB. If, as in common flint-glass spectroscopes, there is only one dispers ing substance,y~5 J uc?s=S,u.s, where s is simply the thickness tra versed by the ray. If t. 2 and t 1 be the thicknesses traversed by the

- Phil. Mag., xx. p. 354, 1885.