The path of the rays through an ordinary telescope has been shown in Fig. 5. In principle all the rays from a point in the distant object should unite precisely in a corresponding point in the image which is viewed by the eyepiece. Practically it takes very careful design and construction of the objective to make them meet in such orderly fashion even over an angular space of a single degree, and the wider the view required the more difficult the construction. We have spoken in the account of the early workers of their struggles to avoid chromatic and spherical aberrations, and it is chiefly these that still, in less measure, worry their successors.

The first named is due to the fact that a prism does not bend light of all colors equally, but spreads them out into a spectrum; red refracted the least, violet the most. Since a lens may be regarded as an assemblage of prisms, of small angle near the centre and greater near the edge, it must on the whole and all over bend the blue and violet rays to meet on the axis nearer the rear surface than the corresponding red rays, as shown in Fig. 47. Here the incident ray a is split up by the prismatic effect of the lens, the red coming to a focus at r, the violet at v.

One can readily see this chromatic aberration by covering up most of a common reading glass with his hand and looking through the edge portion at a bright light, which will be spread out into a colored band.

If the lens is concave the violet rays will still be the more bent, but now outwards, as shown in Fig. 48. The incident ray a' is split up and the violet is bent toward v, proceeding as if coming straight from a virtual focus v' in front of the lens, and nearer it than the corresponding red focus r'. Evidently if we could combine a convex lens, bending the violet inward too much, with a concave one, bending it outward too much, the two opposite variations might compensate each other so that red and violet would come to the same focus—which is the principle of the achromatic objective.

If the refractive powers of the lenses were exactly proportional to their dispersive powers, as Newton erroneously thought, it is evident that the concave lens would pitch all the rays outwards to an amount which would annul both the chromatic variation and the total refraction of the convex lens, leaving the pair without power to bring anything to a focus. Fortunately flint glass as compared with crown glass has nearly double the dispersion between red and violet, and only about 20% greater refractive power for the intermediate yellow ray.

Hence, the prismatic dispersive effect being proportional to the total curvature of the lens, the chromatic aberration of a crown glass lens will be cured by a concave flint lens of about half the total curvature, and, the refractions being about as 5 to 6, of ? the total power.

Since the “power” of any lens is the reciprocal of its focal length, a crown glass convex lens of focal length 3, and a concave flint lens of focal length 5 (negative) will form an approximately achromatic combination. The power of the combination will be the algebraic sum of the powers of the components so that the focal length of the pair will be about 5/2 that of the crown lens with which we started.

To be more precise the condition of achromatism is

S?dn + S?'dn' = 0

where ? is the reciprocal of a radius and dn, or dn', is the difference in refractive index between the rays chosen to be brought to exact focus together, as the red and the blue or violet.

This conventional equation simply states that the sum of the reciprocals of the radii of the crown lens multiplied by the dispersion of the crown, must equal the corresponding quantity for the flint lens if the two total dispersions are to annul each other, leaving the combination achromatic. Whatever glass is used the power of a lens made of it is

P( = 1/f) = S?(n - 1)

so that it will be seen that, other things being equal, a glass of high index of refraction tends to give moderate curves in an objective. Also, referring to the condition of achromatism, the greater the difference in dispersion between the two glasses the less curvatures will be required for a given focal length, a condition advantageous for various reasons.

The determination of achromatism for any pair of glasses and focal length is greatly facilitated by employing the auxiliary quantity ? which is tabulated in all lists of optical glass as a short cut to a somewhat less manageable algebraic expression. Using this we can figure achromatism for unity focal length at once,

P = ?/(?-?')P' = ?'/(?-?')? = (n_D-1)/dn

being the powers of the leading and following lenses respectively. The combined lens will bring the rays of the two chosen colors, as red and blue, to focus at the same point on the axis. It does not necessarily give to the red and blue images of an object the same exact size. Failure in this respect is known as chromatic difference of magnification, but the fault is small and may generally be neglected in telescope objectives.

We have now seen how an objective may be made achromatic and of determinate focal length, but the solution is in terms of the sums of the respective curvatures of the crown and flint lenses, and gives no information about the radii of the individual surfaces. The relation between these is all-important in the final performance.

For in a convex lens with spherical surfaces the rays striking near the edge, of whatever color, are pitched inwards too much compared with rays striking the more moderate curvatures near the axis, as shown in Fig. 49. The ray a' b' thus comes to a focus shorter than the ray a b.

This constitutes the fault of spherical aberration, which the old astronomers, following the suggestions of Descartes, tried ineffectually to cure by forming lenses with non-spherical surfaces.

Fig. 50 suggests the remedy, for the outer ray a is pitched out toward b as if it came from a focal point c, while the ray nearer the center a' is much less bent toward b' as if it came from c'. The spherical aberrations of a concave lens therefore, being opposite to those of a convex lens, the two must, at least to a certain extent, compensate each other as when combined in an achromatic objective.

So in fact they do, and, if the curves that go to make up the total curvatures of the two are properly chosen, the total spherical aberration can be made negligibly small, at least on and near the axis. Taking into account this condition, therefore, at once gives us a clue to the distribution of the total curvatures and hence to the radii of the two lenses. Spherical aberration, however, involves not only the curvatures but the indices of refraction, so that exact correction depends in part on the choice of glasses wherewith to obtain achromatization.

In amount spherical aberration varies with the square of the aperture and inversely with the cube of the focal length i.e. with a²/f³. It is reckoned as + when, as in Fig. 49, the rim rays come to the shorter focus, as-, when they come to the longer focus.

In any event, since the spherical aberration of a lens may be varied in above the ratio of 4:1, for the same total power, merely by changing the ratio of the radii, it is evident that the two lenses being fairly correct in total curvature might be given considerable variations in curvature and still mutually annul the axial spherical aberration.

Such is in fact the case, so that to get determinate forms for the lenses one must introduce some further condition or make some assumption that will pin down the separate curvatures to some definite relations. The requirement may be entirely arbitrary, but in working out the theory of objectives has usually been chosen to give the lens some real or hypothetical additional advantage.

The commonest arbitrary requirement is that the crown glass lens shall be equiconvex, merely to avoid making an extra tool. This fixes one pair of radii, and the flint lens is then given the required compensating aberration choosing the easiest form to make. This results in the objective of Fig. 51.

Probably nine tenths of all objectives are of this general form, equiconvex crown and nearly or quite plano-concave flint. The inside radii may be the same, in which case the lenses should be cemented, or they may differ slightly in either direction as a, Fig. 51 with the front of the flint less curved than the rear of the crown, and b where the flint has the sharper curve. The resulting lens if ordinary glasses are chosen gives excellent correction of the spherical aberration on the axis, but not much away from it, yielding a rather narrow sharp field. Only a few exceptional combinations of glasses relieve this situation materially.

The identity of the inner radii so that the surfaces can be cemented is known historically as Clairault’s condition, and since it fixes two curvatures at identity somewhat limits the choice of glasses, while to get proper corrections demands quite wide variations in the contact radii for comparatively small variations in the optical constants of the glass.

When two adjacent curves are identical they should be cemented, otherwise rays reflected from say the third surface of Fig. 51 will be reflected again from the second surface, and passing through the rear lens in almost the path of the original ray will come to nearly the same focus, producing a troublesome “ghost.” Hence the curvatures of the second and third surfaces when not cemented are varied one way or the other by two or three per cent, enough to throw the twice reflected rays far out of focus.

In this case, as in most others, the analytical expression for the fundamental curvature to be determined turns up in the form of a quadratic equation, so that the result takes the form a ± b and there are two sets of radii that meet the requirements. Of these the one presenting the gentler curves is ordinarily chosen. Fig. 52 a and c shows the two cemented forms, thus related, for a common pair of crown and flint glasses, both cleanly corrected for chromatic and axial spherical aberration.

Nearly a century ago Sir John Herschel proposed another defining condition, that the spherical aberration should be removed both for parallel incident rays and for those proceeding from a nearer point on the axis, say ten or more times the focal length in front of the objective. This condition had little practical value in itself, and its chief merit was that it approximated one that became of real importance if the second point were taken far enough away.

A little later Gauss suggested that the spherical aberration should be annulled for two different colors, much as the chromatic aberration is treated. And, being a mathematical wizard, he succeeded in working out the very intricate theory, which resulted in an objective approximately of the form shown in Fig. 53.

It does not give a wide field but is valuable for spectroscopic work, where keen definition in all colors is essential. Troublesome to compute, and difficult to mount and center, the type has not been much used, though there are fine examples of about 9½ inches aperture at Princeton, Utrecht, and Copenhagen, and a few smaller ones elsewhere, chiefly for spectroscopic use.

It was Fraunhofer who found and applied the determining condition of the highest practical value for most purposes. This condition was absence of coma, the comet shaped blur generally seen in the outer portions of a wide field.

It is due to the fact that parallel oblique rays passing through the opposite rims of the lens and through points near its center do not commonly come to the same focus, and it practically is akin to a spherical aberration for oblique rays which greatly reduces the extent of the sharp field. It is reckoned + when the blur points outwards,-when it points inwards, and is directly proportional to the tangent of the obliquity and the square of the aperture, and inversely to the square of the focal length i.e. it varies with a²tan(u)/f².

Just how Fraunhofer solved the problem is quite unknown, but solve it he did, and very completely, as he indicates in one of his later papers in which he speaks of his objective as reducing all the aberrations to a minimum, and as Seidel proved 30 years later in the analysis of one of Fraunhofer’s objectives. Very probably he worked by tracing axial and oblique rays through the objective form by trigonometrical computation, thus finding his way to a standard form for the glasses he used.[10]

Fraunhofer’s objective, of which Fig. 54a is an example worked by modern formulÆ for the sine condition, gives very exact corrections over a field of 2°-3° when the glasses are suitably chosen and hence is invaluable for any work requiring a wide angle of view.

With certain combinations of glasses the coma-free condition may be combined successfully with Clairault’s, although ordinarily the coma-free form falls between the two forms clear of spherical aberration, as in Fig. 52, b, which has its oblique rays well compensated but retains serious axial faults.

Fraunhofer’s objective has for all advantageous combinations of glasses the front radius of the flint longer than the rear radius of the crown hence the two must be separated by spacers at the edge, which in small lenses in simple cells is slightly inconvenient. However, the common attempt to simplify mounting by making the front flint radius the shorter almost invariably violates the sine condition and reduces the sharp field, fortunately not a very serious matter for most astronomical work.

The only material objection to the Fraunhofer type is the strong curvature of the rear radius of the crown which gives a form somewhat susceptible to flexure in large objectives. This is met in the flint-ahead form, developed essentially by Steinheil, and used in most of the objectives of his famous firm. Fig. 54b shows the flint-ahead objective corresponding to Fig. 54a. Obviously its curves are mechanically rather resistant to flexure.[11]

Mechanical considerations are not unimportant in large objectives, and Fig. 55, a highly useful form introduced by the Clarks and used in recent years for all their big lenses, is a case in point. Here there is an interval of about the proportion shown between the crown and flint components.

This secures effective ventilation allowing the lenses to come quickly to their steady temperature, and enables the inner surfaces to be cleaned readily and freed of moisture. Optically it lessens the deviation from the sine condition otherwise practically inseparable from the equiconvex crown, reduces the chromatic difference of spherical aberration, and gives an easy way of controlling the color correction by slightly varying the separation of the lenses.

One further special case is worth noting, that of annulling the spherical aberration for rays passing through the lens in both directions. By proper choice of glass and curvatures this can be accomplished to a close approximation and the resulting form is shown in Fig. 56. The front of the crown is notably flat and the rear of the flint conspicuously curved, the shape in fact being intermediate between Figs. 52b and 52c. The type is useful in reading telescopes and the like, and for some spectroscopic applications.

There are two well known forms of aberration not yet considered; astigmatism and curvature of field. The former is due to the fact that when the path of the rays is away from the axis, as from an extended object, those coming from a line radial to the axis, and those from a line tangent to a circle about the axis, do not come to the same focus. The net result is that the axial and tangential elements are brought to focus in two coaxial surfaces touching at the axis and departing more and more widely from each other as they depart from it. Both surfaces have considerable curvature, that for tangential lines being the sharper.

It is possible by suitable choice of glasses and their curvatures to bring both image surfaces together into an approximate plane for a moderate angular space about the axis without seriously damaging the corrections for chromatic and spherical aberration. To do this generally requires at least three lenses, and photographic objectives thus designed (anastigmats) may give a substantially flat field over a total angle of 50° to 60° with corrections perfect from the ordinary photographic standpoint.

If one demands the rigorous precision of corrections called for in astronomical work, the possible angle is very much reduced. Few astrographic lenses cover more than a 10° or 15° field, and the wider the relative aperture the harder it is to get an anastigmatically flat field free of material errors. Astigmatism is rarely noticeable in ordinary telescopes, but is sometimes conspicuous in eyepieces.

Curvature of field results from the tendency of oblique rays in objectives, otherwise well corrected, to come to shorter focus than axial rays, from their more considerable refraction resulting from greatly increased angles of incidence. This applies to both the astigmatic image surfaces, which are concave toward the objective in all ordinary cases.

Fortunately both these faults are negligible near the axis. They are both proportional to tan²{u}/f where u is the obliquity to the axis and f the focal length; turn up with serious effect in wide angled lenses such as are used in photography, but may generally be forgotten in telescopes of the ordinary F ratios, like F/12 to F/16. So also one may commonly forget a group of residual aberrations of higher orders, but below about F/8 look out for trouble. Objectives of wider aperture require a very careful choice of special glasses or the sub-division of the curvatures by the use of three or more lenses instead of two. Fig. 57 shows a cemented triplet of Steinheil’s design, with a crown lens between two flints. Such triplets are made up to about 4 inches diameter and of apertures ranging from F/4 to F/5.

In cases of demand for extreme relative aperture, objectives composed of four cemented elements have now and then been produced. An example is shown in Fig. 58, a four-part objective of 1 inch aperture made by Tolles years ago for a small hand telescope. Its performance, although it worked at F/4, was reported to be excellent even up to 75 diameters.

The main difficulty with these objectives of high aperture is the relatively great curvature of field due to short focal length which prevents full utilization of the improved corrections off the axis.

Distortion is similarly due to the fact that magnification is not quite the same for rays passing at different distances from the axis. It varies in general with the cube of the distance from the axis, and is usually negligible save in photographic telescopes, ordinary visual fields being too small to show it conspicuously.

Distortion is most readily avoided by adopting the form of a symmetrical doublet of at least four lenses as in common photographic use. No simple achromatic pair gives a field wholly free of distortion and also of the ordinary aberrations, except very near the axis, and in measuring plates taken with such simple objectives corrections for distortion are generally required.

At times it becomes necessary to depart somewhat from the objective form which theoretically gives the least aberrations in order to meet some specific requirement. Luckily one may modify the ratios of the curves very perceptibly without serious results. The aberrations produced come on gradually and not by jumps.

A case in point is that of the so-called “bent” objective in which the curvatures are all changed symmetrically, as if one had put his fingers on the periphery and his thumbs on the centre of the whole affair, and had sprung it noticeably one way or the other.

The corrections in general are slightly deteriorated but the field may be in effect materially flattened and improved. An extreme case is the photographic landscape lens. Figure 59 is an actual example from a telescope where low power and very large angular view were required. The objective was first designed from carefully chosen glass to meet accurately the sine condition. Even so the field, which covered an apparent angle of fully 40°, fell off seriously at the edge.

Bearing in mind the rest of the system, the objective was then “bent” into the form given by the dotted lines, and the telescope then showed beautiful definition clear to the periphery of the field, without any visible loss in the centre.

This spurious flattening cannot be pushed far without getting into trouble for it does not cure the astigmatic difference of focus, but it is sometimes very useful. Practically curvature of field is an outstanding error that cannot be remedied in objectives required to stand high magnifying powers, except by going to the anastigmatic forms similar to those used in photography.[12]

Aside from curvature the chief residual error in objectives is imperfection of achromatism. This arises from the fact that crown and flint glasses do not disperse the various colors quite in the same ratio. The crown gives slightly disproportionate importance to the red end of the spectrum, the flint to the violet end—the so-called “irrationality of dispersion.”

Hence if a pair of lenses match up accurately for two chosen colors like those represented by the C and F lines, they will fail of mutual compensation elsewhere. Figure 60 shows the situation. Here the spectra from crown and flint glasses are brought to exactly the same extent between the C and F lines, which serve as landmarks.

Clearly if two prisms or lenses are thus adjusted to the same refractions for C and F, the light passing through the combination will still be slightly colored in virtue of the differences elsewhere in the spectrum. These residual color differences produce what is known as the “secondary spectrum.”

What this does in the case of an achromatic lens may be clearly seen from the figure; C and F having exactly the same refractions in the flint and crown, come to the same focus. For D, the yellow line of sodium, the flint lens refracts a shade the less, hence is not quite powerful enough to balance the crown, which therefore brings D to a focus a little shorter than C and F. On the other hand for A' and G', the flint refracts a bit more than the crown, overbalances it and brings these red and violet rays to a focus a little longer than the joint C and F focus.

The difference for D is quite small, roughly about 1/2000 of the focal length, while the red runs long by nearly three times that amount, the violet by about four. Towards the H line the difference increases rapidly and in large telescopes the actual range of focus for the various colors amounts to several inches.

This difficulty cannot be avoided by any choice among ordinary pairs of glasses, which are nearly alike in the matter of secondary spectrum. In the latter part of the last century determined efforts were made to produce glasses that would give more nearly an equal run of dispersion, at first by English experimenters, and then with final success by Schott and AbbÉ at Jena.

Both crown and flint had to be quite abnormal in composition, especially the latter, and the pair were of very nearly the same refractive index and with small difference in the quantity ? which we have seen determines the general amount of curvature. Moreover it proved to be extremely hard to get the crown quite homogeneous and it is listed by Schott with the reservation that it is not free from bubbles and striÆ.

Nevertheless the new glasses reduce the secondary spectrum greatly, to about ¼ of its ordinary value, in the average. It is difficult to get rid of the spherical aberration, however, from the sharp curves required and the small difference between the glasses, and it seems to be impracticable on this account to go to greater aperture than about F/20.

Figure 61 shows the deeply curved form necessary even at half the relative aperture usable with common glasses. At F/20 the secondary spectrum from the latter is not conspicuous and Roe (Pop. Ast. 18, 193), testing side by side a small Steinheil of the new glasses, and a Clark of the old, of almost identical size and focal ratio, found no difference in their practical performance.

Another attack on the same problem was more successfully made by H. D. Taylor. Realizing the difficulty found with a doublet objective of even the best matched of the new glasses, he adopted the plan of getting a flint of exactly the right dispersion by averaging the dispersions of a properly selected pair of flints formed into lenses of the appropriate relative curvatures.

The resulting form of objective is made, especially, by Cooke of York, and also by Continental makers, and carries the name of “photo-visual” since the exactness of corrections is carried well into the violet, so that one can see and photograph at the same focus. The residual chromatic error is very small, not above 1/8 to 1/10 the ordinary secondary spectrum.

By this construction it is practicable to increase the aperture to F/12 or F/10 while still retaining moderate curvatures and reducing the residual spherical aberration. There are a round dozen triplet forms possible, of which the best, adopted by Taylor, is shown in Fig. 62. It has the duplex flint ahead—first a baryta light flint, then a borosilicate flint, and to the rear a special light crown. The two latter glasses have been under some suspicion as to permanence, but the difficulty has of late years been reported as remedied. Be that as it may, neither doublets nor triplets with reduced secondary spectrum have come into any large use for astronomical purposes. Their increased cost is considerable,[13] their aperture even in the triplet, rather small for astrophotography, and their achromatism is still lacking the perfection reached by a mirror.

The matter of achromatism is further complicated by the fact that objectives are usually over-achromatized to compensate for the chromatic errors in the eyepiece, and especially in the eye. As a general rule an outstanding error in any part of an optical system can be more or less perfectly balanced by an opposite error anywhere else in the system—the particular point chosen being a matter of convenience with respect to other corrections.

The eye being quite uncorrected for color the image produced even by a reflector is likely to show a colored fringe if at all bright, the more conspicuous as the relative aperture of the pupil increases. For low power eyepieces the emerging ray may quite fill a wide pupil and then the chromatic error is troublesome. Hence it has been the custom of skilled opticians, from the time of Fraunhofer, who probably started the practice, to overdo the correction of the objective just a little to balance the fault of the eye.

What actually happens is shown in Fig. 63, which gives the results of achromatization as practiced by some of the world’s adepts. The shortest focus is in the yellow green, not far from the line D. The longest is in the violet, and F, instead of coinciding in focus with C as it is conventionally supposed to do, actually coincides with the deep and faint red near the line marked B. Hence the visible effect is to lengthen the focus for blue enough to make up for the tendency of the eye in the other direction. The resulting image then should show no conspicuous rim of red or blue. The actual adjustment of the color correction is almost wholly a matter of skilled judgment but Fig. 63 shows that of the great makers to be quite uniform. The smallest overcorrection is found in the Grubb objective, the largest in the Fraunhofer. The differences seem to be due mainly to individual variations of opinion as to what diameter of pupil should be taken as typical for the eye.

The common practice is to get the best possible adjustment for a fairly high power, corresponding to a beam hardly 1/64 inch in diameter entering the pupil.

In any case the bigger the pencil of rays utilized by the eye, i.e., the lower the power, the more overcorrection must be provided, so that telescopes intended, like comet seekers, for regular use with low powers must be designed accordingly, either as respects objective or ocular.

The differences concerned in this chromatic correction for power are by no means negligible in observing, and an objective actually conforming to the C to F correction assumed in tables of optical glass would produce a decidedly unpleasant impression when used with various powers on bright objects. And the values for ? implied in the actual color correction are not immaterial in computing the best form for a proposed objective.

1 is from Franunhofer’s own hands, the instrument of 9.6 inches aperture and 170 inches focus in the Berlin Observatory.

2 The Clark refractor of the Lowell Observatory, 24 inches aperture and 386 inches focal length. This is of the usual Clark form, crown ahead, with lenses separated by about ? of their diameter.

3 is a Steinheil refractor at Potsdam of 5.3 inches aperture, and 85 inches focus.

4 is from the fine equatorial at Johns Hopkins University, designed by Professor Hastings and executed by Brashear.

The objective was designed with special reference to minimizing the spherical aberration not only for one chosen wave length but for all others, has the flint lens ahead, aperture 9.4 inches, focal length 142 inches, and the lenses separated by ¼ inch in the final adjustment of the corrections.

5 is from the Potsdam equatorial by Grubb, 8.5 inches aperture 124 inches focus.

The great similarity of the color curves is evident at a glance, the differences due to variations in the glass being on the whole much less significant than those resulting from the adjustment for power.

Really very little can be done to the color correction without going to the new special glasses, the use of which involves other difficulties, and leaves the matter of adjustment for power quite in the air, to be brought down by special eye pieces. Now and then a melting of glass has a run of dispersion somewhat more favorable than usual, but there is small chance of getting large discs of special characteristics, and the maker has to take his chance, minute differences in chromatic quality being far less important than uniformity and good annealing.

Regarding the aberrations of mirrors something has been said in Chap. I, but it may be well here to show the practical side of the matter by a few simple illustrations.

Figure 64 shows the simplest form of concave mirror—a spherical surface, in this instance of 90° aperture, the better to show its properties. If light proceeded radially outward from C, the center of curvature of the surface, evidently any ray would strike the surface perpendicularly as at a and would be turned squarely back upon itself, passing again through the center of curvature as indicated in the figure.

A ray, however, proceeding parallel to the axis and striking the surface as at bb will be deflected by twice the angle of incidence as is the case with all reflected rays. But this angle is measured by the radius Cb from the center of curvature and the reflected ray makes an angle CbF with the radius, equal to FCb. For points very near the axis bF, therefore, equals FC, and substantially also equals cF. Thus rays near the axis and parallel to it meet at F the focus half, way from c to C. The equivalent focal length of a spherical concave mirror of small aperture is therefore half its radius of curvature.

But obviously for large angles of incidence these convenient equalities do not hold. As the upper half of the figure shows, the ray parallel to the axis and incident on the mirror 45° away at e is turned straight down, for it falls upon a surface inclined to it by 45° and is therefore deflected by 90°, cutting the axis far inside the nominal focus, at d. Following up other rays nearer the axis it appears that there is no longer a focal point but a cusp-like focal surface, known to geometrical optics as a caustic and permitting no well defined image.

A paraboloidal reflecting surface as in Fig. 65 has the property of bringing to a single point focus all rays parallel to its axis while quite failing of uniting rays proceeding from any point on its axis, since its curvature is changing all the way out from vertex to periphery. Here the parallel rays a, a, a, a meeting the surface are reflected to the focus F, while in a perfectly symmetrical way the prolongation of these rays a', a', a', a' if incident on the convex surface of the paraboloid would be reflected in R, R', R R' just as if they proceeded from the same focus F.

The difference between the spherical and parabolic curves is well shown in Fig. 66. Here are sections of the former, and in dotted lines of the latter. The difference points the moral. The parabola falls away toward the periphery and hence pushes outward the marginal rays. But it is of relatively sharper curvature near the center and pulls in the central to meet the marginal portion. In the actual construction of parabolic mirrors one always starts with a sphere which is easy to test for precision of figure at its center of curvature. Then the surface may be modified into a paraboloid lessening the curvature towards the periphery, or by increasing the curvature toward the center starting in this case with a sphere of a bit longer radius as in Fig. 66a.

Practice differs in this respect, either process leading to the same result. In any case the departure from the spherical curve is very slight—a few hundred thousandths or at most ten thousandths of an inch depending on the size and relative focus of the mirror.

Yet this small variation makes all the difference between admirable and hopelessly bad definition. However the work is done it is guided by frequent testing, until the performance shows that a truly parabolic figure has been reached. Its attainment is a matter of skilled judgment and experience.

The weak point of the parabolic mirror is in dealing with rays coming in parallel but oblique to the axis. Figure 67 shows the situation plainly enough. The reflected rays a', a no longer meet in a point at the focus F but inside the focus for parallel rays, at f forming a surface of aberration. The practical effect is that the image rapidly deteriorates as the star passes away from the axis, taking on an oval character that suggests a bad case of astigmatism with serious complications from coma, which in fact is substantially the case.

Even when the angular aperture is very small the focal surface is nevertheless a sphere of radius equal to one half the focal length, and the aberrations off the axis increase approximately as the square of the relative aperture, and directly as the angular distance from the axis.

The even tolerably sharp field of the mirror is therefore generally small, rarely over 30' of arc as mirrors are customarily proportioned. At the relative aperture usual with refractors, say F/15, the sharp fields of the two are quite comparable in extent. The most effective help for the usual aberrations[14] of the

mirror is the adoption of the Cassegrain form, by all odds the most convenient for large instruments, with a hyperboloidal secondary mirror.

The hyperboloid is a curve of very interesting optical properties. Just as a spherical mirror returns again rays proceeding from its center of curvature without aberration, and the paraboloid sends from its focus a parallel axial beam free of aberration, or returns such a beam to an exact focus again, so a hyperboloid, Fig. 68, sends out a divergent beam free from aberration or brings it, returning, to an exact focus.

Such a beam a, a, a, in fact behaves as if it came from and returned to a virtual conjugate focus F' on the other side of the hyperbolic surface. And if the convex side be reflecting, converging rays R, R', R, falling upon it at P, P', P, as if headed for the virtual focus F, will actually be reflected to F', now a real focus.

This surface being convex its aberrations off the axis are of opposite sign to those due to a concave surface, and can in part at least, be made to compensate the aberrations of a parabolic main mirror. The rationale of the operation appears from comparison of Figs. 67 and 68.

In the former the oblique rays a, a' are pitched too sharply down. When reflected from the convex surface of Fig. 68 as a converging beam along R, R', R, they can nevertheless, if the hyperbola be properly proportioned, be brought down to focus at F' conjugate to F, their approximate mutual point of convergence.

Evidently, however, this compensation cannot be complete over a wide angle, when F' spreads into a surface, and the net result is that while the total aberrations are materially reduced there is some residual coma together with some increase of curvature of field, and distortion. Here just as in the parabolizing of the large speculum the construction is substantially empirical, guided in the case of a skilled operator by a sort of insight derived from experience.

Starting from a substantially spherical convexity of very nearly the required curvature the figure is gradually modified as in the earlier example until test with the truly parabolic mirror shows a flawless image for the combination. The truth is that no conic surface of revolution save the sphere can be ground to true figure by any rigorous geometrical method. The result must depend on the skill with which one by machine or hand can gauge minute departures from the sphere.

Attempts have been made by the late Professor Schwarzchild and others to improve the corrections of reflectors so as to increase the field but they demand either very difficult curvatures imposed on both mirrors, or the interposition of lenses, and have thus far reached no practical result.

References

• Schwarzchild: Untersuchungen 2, Geom., Opt. II.
• Sampson Observatory 36, 248.
• Coddington: “Reflexion and Refraction of Light.”
• Herschel: “Light.”
• Taylor: “Applied Optics.”
• Southall: “Geometrical Optics.”
• Martin: Ann. Sci. de l’Ecole Normale, 1877, Supplement.
• Moser: Zeit. f. Instrumentenkunde, 1887.
• Harting: Zeit. f. Inst., 1899.
• Harting: Zeit. f. Inst., 1898.
• Von Hoegh: Zeit. f. Inst., 1899.
• Steinheil & Voit: “Applied Optics.”
• Collected Researches, National Physical Laboratory, Vol. 14.
• Gleichen: “Lehrbuch d. Geometrische Optik.”

Note.—In dealing with optical formulÆ look out for the algebraic signs. Writers vary in their conventions regarding them and it sometimes is as difficult to learn how to apply a formula as to derive it from the start. Also, especially in optical patents, look out for camouflage, as omitting to specify an optical constant, giving examples involving glasses not produced by any manufacturer, and even specifying curves leading to absurd properties. It is a good idea to check up the achromatization and focal length before getting too trustful of a numerical design.