^\ 6 
 
 ON THE 
 
 ^^^P^' ADJUSTMENT AND TESTING 
 
 TELESCOPIC OBJECTIVES. 
 
 T. COOKE & SONS, 
 
 BmKlNGHAM WORKS, YORK. 
 
 YORK : 
 Printed by Be-n Johnson and Co., ioo and loi, Mtckt.f.gate. 
 
BOUGHT WITH THE INCOME 
 FROM THE 
 
 SAGE ENDOWMENT FUND 
 
 THE GIFT OF 
 
 Henrg M. Sage 
 
 1891 
 
 ,A7.^Z£d VVgf^^ 
 
Cornell University Library 
 °!3...*.'i!S,,3/?iHi?An?.?PA.?!ni;!.testing of telesco 
 
 ,. 3 1924 031 273 109 
 
Cornell University 
 Library 
 
 The original of tliis bool< is in 
 tlie Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924031273109 
 
ON THE 
 
 ADJUSTMENT AND TESTING 
 
 TELESCOPIC OBJECTIVES. 
 
 T. COOKE & SONS, 
 
 BUCKINGHAM WORKS, YORK. 
 
 YORK : 
 Printed by Ben Johnson and Co., ioo and ioi, Micklegate. 
 
PREFACE. 
 
 In the following pages we hope to supply a want which 
 
 we have every reason to suppose exists largely among the 
 
 numerous and increasing class of astronomical observers, both 
 
 amateur and professional, but more especially those amateurs 
 
 who have few or no opportunities of getting that full 
 
 information about the adjustment and testing of their objectives 
 
 which we shall try to impart. 
 
 T. Cooke &^Sons. 
 
INDEX TO FRONTISPIECE. 
 
 Fig. loa. — Eccentric appearance of interference rings, due to the objective 
 being out of adjustment. See page 8. 
 
 i c. The focussed image of a star, when the maladjustment is about as 
 much as in the last case. 
 
 b. The focussed image, as visible when the objective is moderately out 
 of square. 
 
 J'ig. ir. — A section of the cone of rays taken closer to the focus, exhibiting 
 a more moderate degree of maladjustment. See page 9. 
 
 Hg. 12, — a, b, c, d, & d' are out of focus sections, as will be seen when 
 the objective is correctly "squared on," and quite irrespective of 
 other faults. See page 12. 
 
 a',b' (& d" are appearances of the focussed image corresponding 
 respectively to u, b & d. 
 
 d, d' <& d" are also examples of astigmatism. See pages 19 & 21. 
 
 Fig. 13. — A section taken a very little way within focus, under a high 
 power, exhibiting the fault of astigmatism. 
 
 Fig. 14. — The corresponding appearance to Fig. 13, as shown by a section 
 taken at the same distance beyond focus. See page 21. 
 
 Fig, ij. — Section within focus, showing result of positive spherical aber- 
 ration. 
 
 f 
 
 Fig. i£a. — The corresponding section, taken at the same distance beyond 
 focus. See page 22. 
 
 Fig. 16. — A section taken closer to focus under a high power, exhibiting a 
 slight residual spherical aberration ; the central rings rather weak. 
 
 Fig. i6a. — The corresponding appearance at the same distance beyond 
 focus ; the central rings relatively strong. See page 23. 
 
 Fig. I J. — The spurious disc or image of a star yielded by a perfect objective, 
 and viewed under a very high magnifying power. See page 27. 
 
 Fig. 18. — The spurious disc sometimes yielded by a large objective when 
 resting upon three points, without intermediate supports being 
 supplied to counteract the flexure due to the weight of the lenses. 
 See page 37. 
 
 Figs, zg <& iga. —An example of marked zonal aberration, being sections 
 of the cone of rays taken inside and outside of focus respectively. 
 See page 24. 
 
 Figs. 20 & 3oa. — Another example of zonal aberration. See page 24. 
 
Figs. 21 (& 2ia. — Example of the general figure of an objective being 
 tolerably good, but there is a region in the centre having ii focus 
 somewhat beyond the main focus. See page 24. 
 
 Fi^s, 2Z (& 22a. — Two sections of the cone of rays yielded by a perfect 
 objective, taken very near to and on opposite sides of focus, 
 and viewed under a high power. See page 25. 
 
 Fig. 23. — A section of the cone of rays yielded- by a perfect objective, 
 taken at about ^ inch on either side of focus, and viewed under 
 a moderately high magnifying power. .See page 25. 
 
 Fig. 24a. — Example of violent mechanical strain, due to imperfect mounting 
 or bad annealing. See page 34. 
 
 Fig. 24. — Ditto. 
 
 Fig. 24b. — Example of the effects due to the presence of veins io the 
 material of the objective. 
 
THE ADJUSTMENT AND TESTING OF TELESCOPIC 
 OBJECTIVES. 
 
 IN these days all astronomers, whether amateur or otherwise, 
 must be provided with some form of telescope if their study 
 is to be in any degree satisfactory and fruitful of results, and 
 moreover, such instruments must come up to a certain standard of 
 excellence and be properly adjusted, before they can be expected 
 to yield the best definition attainable by the particular size in 
 question. 
 
 The old established reflecting telescope still remains the 
 favourite with many observers, chiefly by reason of its smaller 
 cost and its perfect achromatism, but there is no doubt that the 
 refracting telescope is generally recognised as the most con- 
 venient and generally reliable form of telescope, principally 
 because it will perform well on nights when the reflector is useless 
 owing to the deposit of dew and the distracting effect of insidious 
 air currents within the tube. 
 
 But there are refractors and refractors, varying greatly in their 
 intrinsic qualities. Among the large number of objectives which 
 we have had submitted to us for our opinion, we have found 
 the greater proportion very indiff"erent in their performance, and 
 very questionable objectives enjoying a reputation for high 
 quality in some quarters, chieflyfor the reason, as it seemsto us, that 
 there is a lack of published information of an authentic and 
 complete character, enabling the amateur to make a sufficiently 
 accurate estimate of the quality of an objective for himself. 
 
 On the other hand, we have had forced upon our attention the 
 fact that occasionally objectives oi first rate quality are neverthe- 
 less looked upon as defective, because those who use them have 
 been insufficiently informed as to the necessary adjustments, 
 and therefore their objectives are put to a more or less serious 
 disadvantage. It is such facts as these which prompt us to publish 
 the following remarks for the purpose of enabling amateurs to 
 gain the very utmost advantage from the use of their instruments. 
 
 In the first place it may be pointed out that portable and pocket 
 telescopes of under three inches aperture and used for terrestrial 
 
purposes, are adjusted by their makers before being sent out, and 
 moreover are not usually supplied with apparatus for adjustment, 
 as the comparatively low magnifying powers used on such telescopes 
 renders a perfect adjustment of the objective unnecessary. Of 
 course, good objectives for astronomical purposes, if supplied with 
 tube and mounting, are also sent out in good adjustment, as far 
 as possible ; nevertheless, after they have been in use for some 
 time, or the objective has been taken out for cleaning, &c., or 
 still more, if they have changed hands and undergone vicissitudes 
 of some sort or another, there will arise need for more or less 
 readjustment of the objective with respect to its tube, or 
 "squaring on" as the operation is called. 
 
 Sometimes the maker has to supply a large telescope to be put up 
 at some distant place (in Australia, for instance), in which case it 
 may be impracticable to send some well qualified person to see to 
 the final adjustment of the objective in its tube ; or again, an 
 objective may be supplied by itself, in which case its adjustment 
 by the maker is not demanded, but is undertaken by its owner or 
 a friend. It is therefore very important that anyone who uses an 
 astronomical telescope of any considerable aperture should be 
 able to perform the operation of " squaring on " the objective to 
 a nicety, for unless this is done the telescope will not perform its 
 best with the highest magnifying powers, especially in the way of 
 separating difficult double stars. 
 
 Technically, the operation called " squaring on '' means the 
 adjustment of the optical axis of the objective until it passes 
 accurately through the centre of the eye-piece. If the objective is 
 not "squared on," its optical axis falls more or less to one side of 
 the centre of the eye-piece, and we have the condition of things 
 shown in Fig, i, where o-a is the optical axis of the objective, 
 falling to one side of the eye-piece, so that the latter is looking 
 obliquely at the objective, as it were. It is impossible for all the 
 rays passing through the objective to focus accurately to a point 
 on the oblique axis o-a, whatever the type of objective may be, 
 although, as will be pointed out further on, there is one type 
 which renders possible a nearer approach to good definition on 
 an oblique axis. If the objective is " squared on " the optical 
 axis o-a will now be transferred to the position o-a^, passing 
 centrally through the eye-piece. 
 
 OrO- — „-. 
 
 Fis. I. 
 
The symptoms resulting from any such maladjustment of the 
 objective are strikingly evident if the objective is tilted to any very 
 serious extent ; but if only to a very slight extent, then the 
 observer will need very careful and discriminating exercise of his 
 eyesight in order to give the final touches to the adjustment. It 
 must be remembered also that an objective must be carefully 
 " squared on " before the observer is in a position to form a just 
 estimate of the quality of the glass, that is unless it is a very bad 
 one. 
 
 , It is supposed throughout that all objectives referred to are 
 composed of two lenses, a positive lens of some sort of crown 
 glass, and a negative lens of some sort of flint glass, for the 
 advantages claimed for the use of more than two lenses in an 
 objective are more than counterbalanced by the practical dis- 
 advantages of the more complex construction, excepting in the case 
 of small objectives for special purposes. 
 
 But it is important to point out that object glasses of two lenses 
 may be divided (and very conveniently for our purpose) into three 
 classes, whose several behaviours when tilted or thrown out of 
 square are different, thus requiring different treatment. 
 
 ist. There is the most important and numerous class, including 
 objectives such as are represented in cross section in Figs. 2, 3, 
 4 and 5. Taking them in their order. Fig. 2 represents a type 
 having a meniscus crown and biconcave flint, thus offering the 
 great practical advantage of three concave surfaces which can be 
 accurately and separately tested. But it has the great practical 
 disadvantage of giving a very limited field of view, the image 
 deteriorating rapidly on leaving the optical axis, and therefore as 
 a corollary from this fault the objective is very sensitive to a slight 
 amount of tilting which would have no appreciable effect upon 
 some of the other forms. 
 
 Fk- 3- 
 
 Fig. 3 represents a form in which the crown lens is arranged 
 to give the minimum amount of spherical aberration consistent 
 with its focal length. It is what is called a lens of the " crossed" 
 form. If of ordinary crown glass its radii of curvature are nearly 
 as I to 6 or 7^. This form has the disadvantages of the last 
 form, but in a minor degree. 
 
 Fig. 4 represents the form of objective which we generally 
 adopt ourselves. The radii of the crown lens are to one anotheir 
 
<as 2 to 3, while the flint curve next to the crown is at the same 
 time represented by 2-815 (provided the indices of refraction for 
 the Dray are i -5 18 and 1-620 respectively), while the fourth surface 
 is a concave of long radius. The advantages of this form are, the 
 least possible sensitiveness to being tilted or thrown out of square, 
 and the largest field consistent with the retention of two concave 
 •surfaces, the latter being an important advantage in practice. 
 Also the correction for spherical aberration is almost wholly due 
 to the inside surface of the flint lens, leaving an exceedingly small 
 residual aberration to be corrected by the refraction at the fourth 
 surface. Thus the radius of the fourth surface can be altered at will 
 to correct for chromatic aberration without disturbing the correc- 
 tion for spherical aberration. 
 
 Fig. 4. Fig. 5. 
 
 Fig. 5 shows a form of objective in which the crown is equi- 
 ■convex or thereabouts, while the fourth surface of the flint is flat or 
 slightly convex. Sometimes the two inside curves are similar and 
 may be cemented together. This form introduces unnecessary 
 practical difficulties, although it is still less sensitive to being 
 thrown out of square, and offers a rather larger field than any of 
 the preceding ; but in large instruments where the field observed 
 generally is very small relatively to the size of the instrument, 
 this ceases to be an advantage of any weight. 
 
 2nd. There is the type of objective shown in Fig. 6. It 
 ■necessarily varies according to the optical characteristics of the 
 glass used, but the chief feature of this form, which puts it in a 
 •class by itself, is the fact that it yields the maximum field of view 
 obtainable and relative excellence of image for some distance 
 from the optical axis, and this condition is most perfectly attained 
 when the curves are designed to fulfil the following law : — Any 
 •two rays, both parallel to the optical axis and falling upon the 
 
 =3 
 
 Fig. 6. 
 
 ^glass, one at the edge of the aperture and the other nearer the 
 centre, must have the sines of their angles of incidence on the 
 •first surface in the same ratio to the sines of their respective angles 
 ■of emergence at the fourth .surface, or, in other words, the ratio 
 
between the sine of the angle of incidence for any ray (falling on 
 the first surface parallel to the optical axis) and the sine of its 
 angle of emergence at the fourth surface, must be a constant 
 quantity for all rays.* 
 
 It is this type of objective which is most eminently suitable for 
 photographic purposes, where the chief desideratum is the largest 
 possible field of view consistent with the aperture. But if a 
 large objective of such a type is used for visual purposes where 
 high magnifying powers are used, it will be attended by certain 
 serious mechanical disadvantages (to be noticed further on) over 
 and above the extra difficulty involved in making a perfect 
 objective with three convex surfaces. As an instance of the ratios 
 of curves it may be noted that if a certain peculiar dense and 
 colourless, yet low refractive crown, made by Schott and Gen. of 
 Jena, is used for the crown lens, and a certain light flint for the 
 flint lens, then the curves are in the following ratios, in order to 
 fulfil very closely the above mentioned condition for large field 
 and to bring the chemical rays about as nearly as possible to one 
 focus, the ratio of aperture to focal length being j-^|. If the 
 radius of first surface is 5, then that of second ditto is 3, the third 
 3*05, and fourth 25. 
 
 3rd. There is a form of objective rather resembling the last 
 type, but its curves are carried to a still further extreme of bulge 
 towards the eye-piece, see Fig. 7. This form has not the 
 advantage of a large field in the same degree as the second type, 
 while its disadvantages are more pronounced, so that it is 
 difficult to see why such forms are made, unless it is that the 
 makers are really aiming at an objective of the second type (Fig. 
 ■6) and fail to realise it. 
 
 ^ 
 
 Fig. 7. 
 
 It may be said that the type of objective, shown in Fig. 6, 
 fulfilling the law of constant ratio of sines of first incidence and 
 fourth emergence, occupies a place by itself, dividing the first 
 
 * If two parallel rays strike the crown at points distant from the axis by 
 ape^ and ffi5^ respectively, it will be found that the points where they 
 emerge at the fourth surface will be distant from the axis by amounts which 
 are no longer in the ratio 2:1, but in the ratio 2 to i minus a quantity which 
 is slightly different for different curves, but almost independent of the thick- 
 nesses of the lenses. Were it not for this disproportion, it would be 
 impossible to fulfil the above condition of constant ratio of sines, unless the 
 radius of the fourth surface were invariably equal to the focal length. 
 
type, characterised by outward bulge of the curves, from the 
 third type, characterised by excessive inward bulge of the curves 
 towards the eye-end. 
 
 Now the appearances to be looked for when an objective is 
 out of square vary according to which of the three types the 
 objective in question happens to belong, and the adjustment is 
 different in each case. In five cases out of six an objective will 
 doubtless be found to belong to the first type, and the following 
 is the method to be pursued for all such objectives. 
 
 In the first place it is supposed that all good objectives of over 
 three inches in aperture are provided with certain screws whereby 
 the cell or mounting may be tilted until the optical axis is caused 
 to pass centrally through the eye-piece. The method generally 
 adopted for ordinary sizes is shown in Figs. 8 and g, which are 
 respectively a cross section and a plan of a cell and its attach- 
 ments such as we generally supply ourselves. In Fig. 8 the 
 shaded section represents the cell holding the object glass. It is 
 attached to the counter-cell c by means of the three bayonet 
 joints b. When the cell is slipped over the three screws b and 
 then rotated so as to bring the narrow ends of the bayonet slots 
 under the screw-heads, the latter are tightened up and the cell is 
 fixed. But the counter-cell c-c, carrying the cell within it, is 
 capable of a tilting movement with respect to the fixed flange f-f 
 by means of the three pairs of counter-screws- shown at j, j„ j,„. 
 On examination it will be found that, of each pair of counter- 
 screws, one (i ) is threaded through the flange /and pushes against 
 the counter cell c, its office being to keep them separated by any 
 interval, while the other {2) passes loosely through the flange /, 
 but is threaded into the counter-cell c, its office thus being to draw 
 the counter-cell nearer to the flange f. It is obvious that when 
 both screws are tolerably tight they exert opposite effects and hold 
 the counter-cell rigidly at any fixed distance (within limits) from 
 the fixed flange /. For convenience we will call (i) the pushing 
 screw, for when tightened it tends to push the counter-cell further 
 out at that point, and the other (2) the binding screw, as it tends, 
 when tightened, to draw the counter-cell flange nearer to the fixed 
 flange. 
 
 If it is desired to bring the counter-cell a little nearer the 
 flange at any point near a pair of counter-screws, then the 
 pushing screw will have to be relaxed (by a lever), and the binding 
 screw tightened or screwed home until the resistance of the 
 pushing screw, again coming into operation, causes the binding 
 screw to feel tolerably tight again, when further tightening should 
 be desisted from. To increase the separation between the 
 
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counter-cell and flange at any point at or close to a pair of 
 counter-screws will, of course, require the binding screw to be 
 sufficiently relaxed or unscrewed first, the pushing screw being 
 then screwed home until of the proper tightness. 
 
 By referring to Fig 9 it will be seen that if the objective has 
 to be pushed invyards (towards eye-end), near the screws s,,,, it 
 may be done either by bringing the flanges nearer together at s„, 
 by means of those counter-screws, or, if a nearer approach of the 
 flanges is impracticable, then the same effect will be produced by 
 pushing out the objective at a point opposite to j„,, and this may 
 of course be effected by separating the flanges by means of both 
 pairs of counter-screws s, and j,„ the adjustment being equally 
 divided between them. 
 
 Or, again, if the counter-cell should require pushing off at or 
 near s„ it may be done either by means of the screws s, or by 
 pushing in by means of the two pairs of counter-screws j,, and s,,,, 
 dividing the adjustment between them-. In order to get at these 
 counter-screws it may sometimes be necessary to remove the 
 dew-cap. 
 
 If it is desired to see whether the objective is correctly 
 adjusted or not, the telescope should be directed to some 
 moderately bright star at a good altitude and its image brought 
 to the centre of the field of the eye-piece, which should be of 
 medium power. However good the objective may really be, no 
 really sharp image of the star will be obtained if it is much out 
 of adjustment, the best possible focus will show a fan shaped 
 appearance like Fig. 10 {c)* of very appreciable size instead of a 
 minute spurious disc. And if the e^'^e-piece be racked in and out 
 of focus alternately the star's light will then be noticed to spread 
 out into a more or less pear shaped luminosity, the smaller end 
 being the brightest. This appearance is similarly situated 
 whether the eye-piece is within focus or beyond focus, and is 
 caused by the objective being too near to the eye-piece at that 
 side corresponding to the smaller and brighter end of the pear 
 shaped appearance. Thus if the latter shows towards the 
 observer's left hand, as in Fig. 10 (a), the fault may be corrected 
 by the counter cell being pushed further out at the corresponding 
 left hand side or by being pushed in on the side opposite. The 
 pear shaped luminosity depends on the fact that if an objective 
 of the first type is tilted out of square, then the rays passing 
 through near the edge which is nearest the eye-piece (say the 
 point s, in Fig. 9) will have the shortest focus, while the focus 
 
 * Figs, 10 to 24, inclusive, are represented by photography in the frontispiece. 
 
will lengthen successively according to the parallel chords 
 shown in dotted lines until the opposite edge is reached where 
 the objective has its maximum tilt outwards, and the rays pass 
 through to the longest focus. But the best thing to direct the 
 attention to, especially during the more deHcate stages of 
 squaring on, is to note carefully the position of the star when in 
 the best possible focus, and then, on racking out of focus 
 suddenly (either outwards or inwards or both) note that the more 
 or less round disc of luminosity which results, does not spread out 
 symmetrically round the position of the star, when in focus, as a 
 centre, but expands itself more or less to one side of the star's 
 position. In Fig. lo (a), the bulk of the luminosity has developed 
 itself towards the right hand of the position of the focussed star, 
 which is marked by a small cross. Therefore the objective needs 
 pushing inwards (towards the eye-piece) on the right hand or else 
 outwards on the left hand (looking from the eye-end). 
 
 One whole turn of each of the counter-screws operated will 
 perhaps be found about sufficient to correct the error. The 
 telescope should again be directed to a star, preferably a smaller 
 one and near the zenith, and another examination made of the 
 image when in the centre of the field. 
 
 It will most probably be found that there still exists a little 
 outstanding error. Using a high magnifying power it may now 
 be found that the luminosity still expands itself somewhat to one 
 side of the position of the focussed image, as in Fig. ii, and if 
 the fault is slight, it will require very careful and discriminating 
 use of the eyesight to observe it and it will also be necessary to 
 rack in and out of focus only to a very small amount, as a little 
 practice will suggest. Moreover, a much more delicate manipula- 
 tion of the counter-screws will be necessary when making the last 
 adjustment. 
 
 The least experienced of observers will have noticed before 
 this, that the luminous disc, visible when a star is thrown out of 
 focus, is not of continuous brightness but is broken up into a 
 system of interference rings ; the farther out of focus and larger 
 the luminous disc, the greater is the number of rings visible. 
 During the last stages of squaring on, it is important to bear in 
 mind that the remaining want of adjustment in the objective will 
 be most readily detected if the observer is careful not to throw the 
 image more out of focus than will cause one or two rings to 
 become visible when using a high-power eye-piece. The least 
 disposition of the rings to expand somewhat to one side of the 
 focussed star may then be detected and the corresponding minute 
 adjustment of the proper counter-screws effected. With care. 
 
the observer will still be able to notice that that part of the 
 luminous disc (or rather ring) which expands the farthest away 
 from the star's place is least luminous, while the opposite side 
 which corresponds more nearly to the star's place is the brightest 
 part {see Fig. 1 1). However, the characteristic of eccentric expansion 
 of the rings is the best thing to observe in practise. 
 
 For objectives of the first type then, the rule is, — the objective 
 requires pushing in nearer to the eye-end on that side of it 
 corresponding to the direction of eccentricity in which the 
 luminosity or ring system yielded by a star out of focus expands 
 relatively to the position of the same star when in focus. Or, as 
 an alternative, the objective may be pushed off, away from the 
 eye-end, on the side opposite to the direction of eccentricity. 
 
 It is usual in some Text Books on Astronomy, when dealing, 
 far too shortly, with this subject of squaring on an objective, tO' 
 instruct the observer to note very carefully the conformation of 
 the star image when in focus under a high power, when, if the 
 objective is out of square, the spurious disc will not be round, but 
 oval with the two or three slight diffraction rings, which should 
 surround it, thrown to one side only of the star disc. This would 
 indicate that the objective needs pushing in on that side towards 
 which the diffraction rings show themselves. For instance, if 
 the star shows an appearance like Fig. lo ifi) with the rings lying 
 towards the right, the inference is that the objective needs 
 pushing in towards the eye-end on the same side, or right hand 
 when looking from the eye-end. This method may certainly be 
 followed, but in practise it is far inferior to the method we have 
 described above, for the principal reason that it requires an 
 exceptionally good observing night, at any rate with large 
 instruments, to enable a close scrutiny of the spurious disc and 
 its rings to be made. The observer might get his objective 
 approximately right on an ordinary night, but on a much finer 
 night ensuing, he would most likely find that his squaring on 
 was not final after all. Whereas, if the observer puts the method 
 which we recommend carefully into practise on an average 
 observing night, he will be able to adjust the objective in much 
 less time, and when it is done, he will find that it will success- 
 fully pass the other test when applied on a really fine night and 
 will show the spurious disc and the diffraction rings symmetrically 
 surrounding it, as in Fig. 12 (aj, that is, if the objective is a 
 really good one. 
 
 If the telescope is a large one and the tube has to be turned 
 down, eye-end upwards, every time the objective is to be 
 adjusted, the observer may easily riiake a mistake and turn the 
 
II 
 
 wrong counter-screws unless he takes the precaution, when he 
 is examining the star image and finds it expand eccentrically 
 when out of focus, of noting in which direction the eccentricity- 
 falls with regard to some fixture upon the telescope tube. For 
 instance, the direction of eccentricity may be towards the finder, 
 away from the finder, at right angles to the same or towards the 
 declination axis, or a direction half-way between the finder and a 
 declination clamping handle, &c. Thus, when the tube is turned 
 object-end downwards, the observer has only to look for guidance 
 to such prominent parts of the telescope in order to see where- 
 abouts the objective needs pushing in or off. 
 
 When the objective has been finally adjusted it will be found, 
 on examining the image of a small star, that the luminous disc or 
 ring system will expand itself concentrically with regard to the 
 position of the star when in focus, as in Figs. 12, a, b, c, d ^ e/^, 
 where a little cross marks the position of the star when in focus. 
 This point is the only one to be regarded at present, for it may be 
 found that although the luminous disc expands itself con- 
 centrically with regard to the focussed image, nevertheless the 
 disc of light is not round but appears oval [d ^ d^, or even of 
 some irregular shape (c). Such appearances are indicative of 
 defects eifher in the objective or in the eye of the observer, 
 or both. Further on we will give full directions for finding out 
 where such defects really lie, and for enabling an observer to 
 form a very good idea of the optical quality of his instrument. 
 
 We have now given directions for the adjustment of all 
 objectives included in the first type. It may be pointed out 
 again that an objective like Fig. 2 will need a more delicate 
 adjustment in squaring on than Fig. 3, still more than Fig. 4, and 
 more still than Fig. 5. An amount of tilting of say -oi inch 
 would cause a less and less serious effect at the foci of these four 
 varieties of objective according to their numerical order. 
 
 But on coming to our second type of objective, shown in Fig. 
 6, and specified above, it will be found that it does not follow the 
 same rule as the first type does. For, if it is considerably tilted,' 
 it will be found that the luminous disc yielded by a star when out 
 of focus, will still expand itself symmetrically round the position of 
 the star when in focus, but it will appear more or less oval in 
 shape, as in Figs. 12 [d &l d^, which represent the appearance at 
 opposite sides of the focus), the major axis of such oval, when 
 inside focus, being parallel to the axis about which the objective 
 is tilted, while, beyond focus, the axis of the resulting oval is at 
 right angles to the last position. This defect of astigmatism for 
 obUque pencils is, however, not so great as to prevent this type of 
 
 B 
 
objective being the most eminently suited for photographic work 
 where a relatively large field of view is required. If an objective 
 of this type is constructed for visual purposes, and the observer 
 finds, when using a high power on a star, that the luminous disc 
 is an upright oval inside focus, and a horizontal oval outside focus, 
 and if he has satisfied himself that such appearance does not 
 result from a defect in his eye (by a simple method yet to be 
 explained), but is caused by the objective, then either the 
 objective is inherently defective, or else it requires " squaring on." 
 If the latter is the case, it must be effected by tilting the objective 
 about a vertical axis, by pushing in on the right and pushing off 
 on the left, or vice versa, for nothing but actual trial will Show 
 which of these opposite movements is required. If the appear- 
 ance inside focus were a horizontal oval then the tilling would 
 have to be done about a horizontal axis, and so on. 
 
 But if the astigmatism cannot be removed by any such tilting 
 movement, the conclusion is that the objective itself is at fault. 
 
 As to the third type of objective, as Fig. 7, it will be found 
 that its behaviour, when thrown out of square, is just the opposite 
 of that exhibited by objectives of the first type. Therefore, when 
 the luminous disc is seen to expand itself eccentrically with regard 
 to the focussed star, the defect is to be remedied by the objective 
 being pushed off on that side of it towards which the eccentric 
 expansion takes place. If the luminous disc expands towards the 
 right of the star's position, then the objective must be pushed out 
 away from the eye-end on the same side, or else pushed in on the 
 opposite side ; that is, it requires just the opposite treatment to 
 that required for the first type. 
 
 The operation of squaring on an objective need not occupy 
 much more than an hour in the case of large instruments, 
 provided the weather allows uninterrupted views of stars of 
 considerable elevation. For telescopes without clock-work, 
 perhaps there is no more favourable star for the purpose of 
 squaring on and subsequent testing than the Pole Star. The 
 operation is easy in practice, at any rate up to a tolerable degree 
 of exactness. As it requires a very careful use of the eyesight, 
 especially in the nicer stages, the observer should try to be as 
 comfortable as possible when examining the focus, a strained 
 position being so much against exactitude of observation. Again, a 
 residual amount of maladjustment of the objective will generally 
 escape the notice of a beginner, let him exert his eyesight as- 
 he will ; yet, after he has grown used to his instrument and trained 
 his eye to look out for indications of maladjustment, he will 
 become very much quicker to notice them, and, if he takes pride 
 
13 
 
 in a good instrument, he will also be quick to correct them, for a 
 really good objective cannot give its best definition under high 
 powers while any noticeable error in squaring on exists. 
 
 After an objective has been correctly " squared on," the 
 observer may then turn his attention, if he likes, to forming a 
 judgment of the optical quality of his instrument, and we will now 
 give directions and illustrations of the methods which will enable 
 hitn to do so, at any rate with substantial accuracy. All the 
 following remarks apply impartially to all types of objectives with- 
 out distinction, so long as they ate used for ordinary astronomical 
 observations where the image is viewed by the eye through 
 eye-pieces of considerable magnifying power. But they would 
 not apply to an opera-glass objective at all, and only in a less strict 
 degree to an objective used for photographic purposes, in which 
 case the image is not submitted to the scrutiny of such high 
 powers. 
 
 ACHROMATISM. 
 
 -Perhaps the first thing to direct the attention to in testing an 
 objective is the state of its achromatic correction. With com- 
 paratively few exceptions the eye-pieces used with astronomical 
 telescopes are either Ramsden's or Huygenian eye-pieces, generally 
 the latter, and it is necessary first to explain that the power 
 of an eye-piece exercises a quite perceptible effect upon the 
 apparent achromatic correction of the telescopic image viewed 
 through it. A very common misconception exists as to the real 
 nature of the so called achromatism of such eye-pieces. They 
 are only achromatic in the sense of treating pencils passing 
 laterally through them from the edge of the field in the same 
 manner as an axial pencil ; they treat central and marginal 
 pencils of light impartially, and so far they are achromatic. 
 When such an eye-piece is used upon an absolutely achromatic 
 image, such as that yielded by a reflector, the effect is this. In 
 the first place an axial pencil passing perpendicularly through 
 the centres of the lenses consists on emergence of variously 
 coloured pencils having a common axis, that is, all the coloured 
 pencils seem to come from the same direction, but they do not 
 emerge as though all coming from the same point on their 
 common axis, as they would do were the eye-piece achromatic in 
 the ordinary sense. For instance, if the blue green rays about 
 F emerge stricdy parallel then the orange red rays about C 
 will at the same time emerge in a state of divergence. In the 
 same way, a white pencil forming a point near the edge of the 
 image, will, after passing through near the edges of the eye-piece 
 
14 
 
 lenses, emerge in such a manner that the coloured pencils of 
 which it is composed, seem, as before, to come froni the same 
 direction, but again they do not emerge as if coming from the 
 same point in that axis or direction ; if F rays are parallel, as 
 if coming from a point at an infinite distance, then C rays will be 
 divergent, as though coming from a near point in the same 
 direction. Therefore the image from a reflector viewed through 
 such eye-pieces will not look perfectly achromatic but more or less 
 under-corrected and such want of achromatism will show itsellf 
 almost impartially all over the field, and it is just this latter feature 
 of its behaviour which gives the eye-piece its title to being called 
 achromatic. For it is obvious that if the image were viewed through 
 a single lens of a focal length equivalent to that of the Huygenian 
 eye-piece considered, and the eye placed at such a distance 
 behind such lens as to command a view of the whole field at once, 
 then the image of a star in the centre of the field would show 
 just about the same amount of colour as the same star would do 
 when viewed through the Huygenian eye-piece, but the images of 
 stars towards the edge of the field would appear very indistinct 
 through the single lens, being drawn out into coloured spectra of 
 a length increasing as the edge of the field is approached, and all 
 longitudinally pointing their red ends to the centre of the field. 
 
 Supposing the eye to be achromatic, it can be shown that the 
 colouring effects due to Huygenian or Ramsden eye-pieces, as 
 seen by the eye, are very slight indeed, as are the equal colouring 
 effects of simple lenses of focal lengths equivalent to such eye- 
 pieces (a central pencil only being here considered). It can also 
 be shown that such colouring effects, as perceptible by the 
 achromatic eye, will be independent of the focal length of 
 such eye-pieces, or, in other words, be a constant amount 
 subjectively considered, and independent of the varying magnifying 
 powers used. 
 
 But the most cogent proofs may be brought forward to show 
 that the human eye falls far short of being achromatic. Now 
 this imperfect achromatism of the eye is a factor of great importance, 
 often overlooked, in its bearing upon the apparent achromatism 
 of telescopes under varying magnifying powers. If an observer 
 will examine the image of a star yielded by a reflecting telescope, 
 using a series of Huygenian eye-pieces, the lowest power being 
 such as to transmit a pencil of light into the eye which shall have 
 a breadth sufficient to fill the pupil of the eye (the magnifying 
 power necessary to do this is obtained by dividing the aperture of 
 the mirror by the diameter of the pupil of the eye), he will fmd 
 there is a very perceptible amount of red fringe round the image 
 
15 
 
 of the star as seen under the low power, and it is specially 
 noticeable if the eye-piece is racked a little way within focus. If 
 successively higher powers are applied, it will be noticed that this 
 red fringe or appearance of under correction for the most part 
 disappears, so that under a very high power, only a very small 
 trace relatively of red fringe will remain, and this relatively small 
 amount is about what is really due to the eye-pieces themselves 
 when used with a mirror of that particular angular aperture. 
 The larger amount of colouration noticeable with the low power 
 is really due to the eye. Given the fact that the eye is not 
 achromatic, it is evident that its dispersive effect upon any ray 
 will be greater if it is refracted through near the edges of the pupil 
 and the lenses of the eye than if it is refracted through nearer to 
 the centre of the eye, and therefore a pencil of light such as 
 emerges from a low power eye-piece and is broad enough to fill 
 the pupil, will suffer far more from the eye's want of achromatism 
 than will the very narrow pencil (perhaps under a fiftieth of an 
 inch in breadth) which emerges from a high-power eye-piece. If 
 the observer will then dispense with an eye-piece altogether and 
 look at the image of the star direct and then approach his eye 
 towards the image until it goes out of focus and swells out into a 
 disc of light, he will then see much the same amount of red 
 fringe as he saw when using the low power eye-piece; this will 
 give some idea of the amount of colouration due to the eye alone 
 (ifor there can be no question concerning the perfect achromatism 
 of the star image yielded by a reflector). 
 
 It is of course highly important in the case of large refractors 
 that the best and most achromatic image should be thrown upon 
 the retina when the higher powers are used, as when making 
 delicate observations on stars and planets ; it is, therefore, usual to 
 aim at such a colour correction in the objective as will permit of 
 this result. With a magnifying power of 50 to 70 times the 
 aperture (in inches) a good objective should give a retinal image 
 as perfectly achromatic as is theoretically possible with the 
 materials at command. It is evident that the objective must be 
 absolutely a trifle over-corrected for colour in order to counteract 
 the want of achromatism in an eye-piece of such power combined 
 with that inherent in the eye, for, as we have seen, their 
 combined effect is to cause a truly- achromatic pencil to appear 
 under-corrected for colour. But it is not quite so evident that this 
 amount of over-correction in the objective will not suffice to 
 counteract the effects of eye-piece and eye when lower powers 
 are used, and that it will be more than enough to counteract the 
 same when a higher power is used. But such is the case. If the 
 
i6 
 
 conditions of vision and the effect of the eye-piece are taken into 
 account, it can be theoretically proved that the most perfect 
 achromatism of the retinal image can be obtained witli only 
 one magnifying power, which may be selected. And this fact 
 has to be allowed for when testing an objective for its achromatic 
 correction, as we will shortly explain. 
 
 An objective should be so corrected that the brightest rays of 
 the spectrum lying at and between the line G (orange red) and F 
 (blue green) shall focus simultaneously upon the retina of the 
 eye, when an eye-piece magnifying 50 to 70 times the aperture 
 is used. If this is the case, then the darker red rays beyond C 
 have their focus a little beyond the principal focus, while the more 
 refrangible rays beyond F, which are so unduly susceptible to the 
 influence of the flint, focus so far beyond, as to become very 
 considerably diffused and comparatively unperceived, especially 
 under high powers. 
 
 There is, perhaps, no better star for testing the colour correction 
 of objectives of moderate and larger sizes than the Pole Star. 
 If the telescope is directed to this star and the image examined 
 with the magnifying power we have specified, then the appearances 
 to be looked for, if the objective is nicely corrected, are a 
 yellowish white disc or ring system surrounded by a very narrow 
 red fringe when the eye-piece is racked inside focus sufficiently 
 far for two- or three rings to be visible. The corresponding 
 appearance when outside focus are the same yellowish white disc, 
 but without any trace of a red fringe. Moreover, if the observer 
 racks out of focus with great care, he will be able to notice a 
 little bright red star disc, which forms itself just as the main focus 
 is beginning to perceptibly expand. This is the focus for the less 
 refrangible red rays beyond C. 
 
 On racking outwards a little further, an indefinite blue focus 
 will begin to form itself in the centre. And on going further out 
 still, until six or more rings can be counted, there will be a blue 
 flare superimposed upon the yellowish white ring system ; it will 
 about cover all the inner rings, scarcely reaching to the outer 
 ring, while it grows brighter and more violet towards the centre 
 of the system. 
 
 . If the telescope is now directed to a bluish white star, like 
 Vega, still keeping the same eye-piece in use, it will be found that 
 there is now scarcely a trace of a red fringe to be seen when inside 
 focus. The light from stars of this type is deficient in the less 
 refrangible of the red rays. On the other hand, if the telescope is 
 directed to a ruddy star such as a Orionis, it will be, found that 
 the red fringe visible when inside focus is very strongly marked, 
 
I? 
 
 very much more marked than in the case of Polaris. These 
 ruddy stars are comparatively rich in the less refrangible of the 
 red rays. This is mentioned chiefly because many observers, 
 looking at ruddy stars when trying their telescopes, might other- 
 wise conclude that the objective was under -corrected, when 
 really as achromatic as possible. 
 
 Turning to Pplaris again, let the effect of changing the eye- 
 piece be observed. Supposing the achromatism to have been as 
 perfect as possible under a magnifying power of 50 times the 
 aperture, let an eye-piece magnifying about 100 times aperture 
 be substituted. On racking inside focus it will be found that not 
 a trace of a red fringe is to be seen ; indeed, its place is taken by 
 a just perceptible greenish fringe, and the objective would seem 
 to be over-corrected (as in fact it is). If lower powers are 
 substituted it will be found that the red fringe when inside focus 
 will grow more and more conspicuous in proportion to the 
 focal length of the eye-piece, and the objective will appear more 
 and more under-corrected, until a magnifying power equal to 
 the aperture divided by the diameter of the pupil, is reached, 
 when the apparent under correction will be at its maximum. For, 
 as we have explained 'before, this effect, apparently due to the 
 eye-pieces, is really due to the eye, the subjective effect due to 
 the eye-pieces themselves being a constant quantity. 
 
 Therefore it is not fair to test the achromatism of an objective 
 with any eye-piece that happens to come to hand, or upon any 
 star taken at random, unless the observer knows, at least 
 approximately, how much to allow for the apparent effect of such 
 eye-piece, or for the prevailing colour of the star. 
 
 FALSE CENTERING. 
 
 When the two lenses composing an objective are either untruly 
 edged, or so badly fitted in the cell that their optical centres are 
 thrown out of coincidence, then the effects at the focus are 
 serious, and may easily be detected. For, if the eye-piece is 
 racked somewhat inside of focus, it will be noticed that more red 
 shows upon one side of the ring system than another, and more- 
 over, any bright object in focus will show a red edging m.ore or 
 less marked upon one side, and possibly a greenish fringe on the 
 opposite edge. In either case the implication is that the centre 
 of the flint lens is displaced relatively to that of the crown, in a 
 direction corresponding to where the red fringe shows the most 
 strongly. For instance, if the red fringe shows to the right, it 
 means that the centre of the flint lens is displaced somewhat to 
 the right of the centre of the crown lens. If the error is very 
 
serious the image of a star will be drawn out into a sort of 
 spectrum. When testing for this fault it is highly important to 
 direct the instrument to a star near the zenith, so as to guard 
 against atmospheric dispersion. For a perfectly centred obj ective, 
 if directed towards a star near the horizon, will not give an 
 achromatic image, but it will appear drawn out into a vertical 
 spectrum, with its red end showing uppermost. And even at 
 intermediate altitudes a large telescope, although correctly 
 centred, will show a star image which is perceptibly more red at 
 its upper margin. If required, this atmospheric dispersion may 
 easily be corrected by looking through a very slight prism, placed 
 behind the eye-piece, with its thin edge uppermost. 
 
 ASTIGMATISM. 
 
 The perfect fulfilment of the condition that all rays striking the 
 objective at equal distances from its centre shall be equally 
 refracted, is ot primary importance. If the objective refracts 
 two rays, incident at opposite points on the same diameter, and 
 equidistant from the centre, more strongly than rays incident 
 upon corresponding points along a diameter at right angles to the 
 former, then the objective is said to be astigmatic. The violation 
 of the above condition is more serious than any other. Let that 
 diameter of the objective be selected, at the extremities of which 
 the strongest refraction takes place, and a plane be imagined 
 which shall include both this diameter and the optic axis (we will 
 call this the primary plane), and another (secondary) plane be 
 imagined at right angles to the former, including the optic axis, 
 and that diameter of the objective along which the weaker 
 refraction takes place (and which is at right angles to the first 
 diameter). , 
 
 It will be seen that the effect at the focus, when the telescope 
 is directed to a star, is a very peculiar one. The shortest focus 
 is for rays in the primary plane, while, in the secondary 
 plane, the rays have not yet come to focus. Therefore the 
 effect is the formation of a short focal line in the secondary 
 plane, whereas if the focus of rays in the secondary plane is 
 considered, it will be seen that since the point where they 
 come to focus is beyond the point where rays in the primary 
 plane have come to focus, therefore the rays in the primary plane 
 will have diverged again, and there is formed another focal 
 line, about equal to the former one, only this time it is in 
 the primary plane. So that the image of a star instead of 
 being a minute round disc, like Fig. 12 a', may show either 
 
19 
 
 as an elongated line in the primary plane, or a similar line at 
 right-angles to former, and in the secondary plane. Fig. 12 d" 
 represents such an elongated star image. Which of these focal 
 lines is seen depends simply upon which is focussed. It can 
 be shown that a cross-section of the rays half-way between the 
 two focal lines will be a circle, whose diameter is equal to half the 
 length of either of the two focal lines ; this is the circle of least 
 confusion, and will in practise be the point actually focussed upon. 
 As an example of the mischievous effect of astigmatism, let an 
 objective be taken six inches aperture, and 90 inches focal 
 length, and let it be supposed that the rays in the primary plane 
 come to focus at a point -02 inches shorter than the rays in the 
 secondary plane. Then the length of either of the two focal 
 lines will be •00133, 3^""^ '^^ diameter of the circle of least 
 confusion will be '00066, and these amounts are respectively at 
 least three and one and a half times the diameter which the 
 spurious disc ought to have, were the telescope faultless. Fig. 
 12 d" represents the star image which would be seen, if one of 
 these focal lines were focussed upon, its larger diameter being 
 about four times the lesser. With the same amount of astig- 
 matism, the least circle of confusion would show somewhat like 
 a star disc of over twice the proper size. Therefore such 
 an amount of astigmatism would render the instrument use- 
 less for close double star observations. But, as we saw before 
 in the case of achromatism, it is the perfection of the image 
 as thrown on the retina, which is all important, and there- 
 fore, even if the objective is perfectly free from astigmatism, 
 it by no means follows that the observer will see perfectly 
 well with it, because his eye may be astigmatic. As a matter 
 of fact, astigmatism in a serious degree is a very common 
 fault in the human eye, and, if present, may be easily detected 
 by the observer himself. And now we will describe the symptoms 
 of astigmatism, both in the telescope and in the eye, so as to 
 enable the observer to discriminate between them. 
 
 It is best to select a star near the zenith, and of moderate 
 brightness, so that there may not be too much glare. First use 
 a very low power, and get the star into the centre of the field. 
 On racking inside and outside focus, the luminous disc or ring 
 system should appear perfectly circular. If it does not do so, 
 but appears distinctly oval, with the major axis of the oval, when 
 inside focus, at right-angles to the major axis of the oval, seen 
 when outside of focus, then the inference is, most probably, that 
 the observer's eye is astigmatic. Whether that is the case or not 
 may be decided in this way. Supposing that the observer is lying 
 with his head towards the north, and when inside focus the major 
 
axis of the oval lies in the direction of his two eyes (for example), 
 he has then only to alter his position so as to turn his head 
 towards east or west, and notice whether the major axis of the 
 oval, inside focus, has gone round with him or not, whether it still 
 lies in a direction joining his two eyes. If it does so, and the 
 amount of astigmatism appears just as much as before, and has 
 distinct reference to his own position, then it is conclusive 
 evidence that the fault is subjective, and in his own eye. But if 
 the appearances do not move round with his eye, but appear 
 fixed with regard to the objective, then the fault lies in the 
 telescope. As an extra precaution, the eye-piece had better be 
 rotated to see whether the fault goes round with it or not. If it 
 should do so, which is unlikely, then the astigmatism is in the 
 eye-piece. If neither the eye-piece nor eye is at fault, then the 
 objective is very defective ; and its astigmatism must be very serious 
 to be noticeable with so low a power. Or, again, the observer may 
 find that the image appears to him to be strongly astigmatic when 
 he is in a certain position, while, when he places himself at right 
 angles to that position, the astigmatism no longer shows, and the 
 image seems round and faultless on both sides of focus. This 
 means that there is astigmatism both in the objective and in the 
 eye, but their relative amounts are such as to neutralize one 
 another when the eye is in a certain position relatively to the 
 telescope, while, at right angles to that position, the two amounts 
 are superadded and the astigmatic effect is doubled. Or astig- 
 matism may show itself in degrees varying with the relative 
 position of the observer, in one position seeming very bad, and 
 in another only moderately bad. This indicates that the effects 
 of astigmatic errors of the eye and the telescope are unequal, and 
 therefore they never neutralize one another. The use of a very 
 low power, as we have recommended above, is specially adapted 
 for detecting any astigmatism in the eye, for the simple reason 
 that the astigmatism of the compound lenses of the eye is the 
 more evident, the greater is the aperture of the eye in use. 
 When a low power is used the pencil of light entering the eye is 
 perhaps large enough to fill the pupil and its whole aperture is 
 put to the test, whereas if the magnifying power is a high one, 
 the diameter of the pencil of light may not exceed one-fiftieth of 
 an inch, and that amount will therefore represent the eye-aperture 
 in use. But a low power is of little use for detecting astigmatism 
 in the objective, unless it exists in a very large amount. There- 
 fore as high a power as the night will allow should be used for 
 testing the objective for the same fault, and the same star near the 
 zenith will be best for the purpose. 
 
21 
 
 On throwing the image in and out of focus in the centre of 
 the field, the luminous disc may appear oval. For instance, 
 when inside focus, the appearance may be like Fig. 13, while 
 Fig. 14 represents the corresponding appearance when outside 
 focus. If the observer will carefully note the direction of 
 the length of the oval on one particular side of focus, and 
 then alter his position to one at right angles to the first, he 
 will then be able to see whether it rotates with him, thus 
 indicating astigmatism in his eye, or keeps in a constant direc- 
 tion with respect to the telescope, in which case of course the 
 objective is at fault. The astigmatism in his eye must be very 
 bad in order to show any appreciable effect with so high a 
 power, and in most cases any astigmatism will be found to exist 
 in the objective, if at all. If such is the case the observer is not 
 in a position to eliminate it, as a general rule ; for it may be caused 
 by bad annealing, sometimes defective material, very often bad 
 figuring and sometimes by an imperfect method of mounting the 
 objective in its cell, so that it is subject to mechanical strains. 
 If the latter is the case, the observer may be competent to deal 
 with it. Sometimes astigmatism exists both in crown and flint 
 lenses, and if to about equal extents, then by turning one lens 
 round with respect to the other a position may be found so that 
 the objective is almost, or altogether free from astigmatism. Many 
 objectives are marked upon their edges, indicating that they will 
 perform at their best when corresponding marks come together. 
 
 We have particularly called attention to the means of 
 detecting astigmatism in the eye because it is a very much more 
 common defect than most people are aware of, and we have 
 .good reason to suppose that many observers are disappointed 
 with good instruments, because this defect, of which they are 
 unconscious, prevents them obtaining that clearness of vision 
 and seeing those things in the heavens which they had been led 
 to expect. It is, therefore, highly important that astronomical 
 ■observers should deliberately test their eyes for this fault, and, 
 if they find it present, go to a competent oculist for appropriate 
 glasses, which can be used either as spectacles or mounted in a 
 cap to fit over the eye-pieces and capable of rotation according to 
 'the position of the observer. Defects of eyesight can thus be 
 neutralised, and the utmost clearness of definition obtained by 
 the use of any really good telescope. We ought also to point out 
 the fact that in many individuals astigmatism is found to be 
 variable in amount, one day the eye may be almost free from the 
 defect and on another day it is present to a serious degree. If 
 an observer finds this to be the case, it can be provided against 
 
by the use of a selection of cylindrical lenses of varying powers 
 in the eye-piece caps. 
 
 SPHERICAL ABERRATION. 
 
 Of course any objective worthy of the name must have the 
 spherical aberration of the crown lens completely counteracted 
 by that of the flint lens, and whether this correction is made 
 with great perfection or only imperfectly, may make all the 
 difference between a really good objective and a third rate one. * 
 
 For detecting spherical aberration it is best to direct the 
 telescope, after the objective has thoroughly cooled down (see 
 page 35), to a star of moderate brightness, and examine the- 
 behaviour of the image under a moderately high power when 
 it is thrown out of focus. If the observer racks sufificiently far 
 from focus as to cause three or four rings to be visible, that will 
 be best for the purpose of detecting any well marked aberration, 
 which will show itself in the form of a greater massiveness or 
 brightness of the two outside rings, especially the outermost one, 
 when on one particular side of focus. 
 
 If, on going within focus by racking towards the objective, it 
 is found that the central rings look very feeble, while the edge 
 rings, and especially the outer one, look massive and luminous, 
 while, on racking out of focus, away from the objective, the 
 appearances are complementary to the above, the central rings 
 looking relatively brighter and the outer rings looking weak and 
 poor in comparison to what they appeared when within focus, 
 then the inference is that the edge rays fall short or come to focus 
 at a point nearer to the objective than the focus for the central 
 rays, or, in other words, there is positive aberration. Fig. 15 
 represents the appearance within focus in this case, and Fig. 150: 
 the complementary appearance to be seen outside focus. 
 
 On the other hand, if the central rnigs, when inside focus, look 
 about as luminous or even more so than the outer ring, which is 
 thin and weak (like Fig. 15 a), while on racking outside focus, the 
 complementary effect of a massive and luminous outside ring 
 enclosing comparatively feeble central rings is observed (like 
 Fig. 15), then the inference is that the edge rays cross the axis 
 and come to focus at a point further from the objective than the 
 focus for the central rays, and this is the fault of negative 
 aberration. If the amount of the spherical aberration is very 
 small it will best be detected by using a very high power and 
 taking care not to rack further out of focus than will cause two- 
 rings, or even one ring, to be visible. If there is residual 
 aberration it may be detected by the same symptoms as before,, 
 
 * See page 32 for maximum aberration permissible. 
 
23 
 
 but they will be less marked. For instance, Fig. i6 may 
 represent the appearance on one side of focus, and 1613; the 
 corresponding appearance on the other side, whereas, if the 
 objective were perfectly corrected for aberration, the appearances 
 at both sides of focus would be exactly alike (if differences of 
 colour are neglected), as in Figs. 22 and 2212!. As regards colour, 
 it is as well to remember that the diffusion of a blue flare over 
 the more central rings, when outside of focus, tends to increase 
 their apparent brightness somewhat, so that if an objective is 
 perfect, the outer ring, seen insidt focus when several rings are 
 visible, will appear in somewhat greater contrast to the interior 
 rings than the same ring will do when the eye-piece is racked 
 outside focus. If the images are viewed through strong yellow 
 glass, which absorbs the stray blue light, then the appearances 
 should be exactly the same inside and outside of focus, with the 
 exception of a slight red fringe in the former case. Then again, 
 if the image appears perfectly free trom aberration with a very 
 high power, it will scarcely do so when a low power is applied, 
 for the effect of using a large eye-piece is to cause an appearance 
 of positive spherical aberration, the edge rays of a pencil 
 appearing to fall rather short, but this effect is only just 
 perceptible. 
 
 Thus aberration may exist in very various degrees. It may be 
 so bad as to be evident to the merest tyro at observing, or in 
 such minute quantity that the greatest experience in testing 
 telescopes will be needed in order to say positively that it exists. 
 Still, an ordinary careful observer should be able to detect it if it 
 exists in a really serious degree. 
 
 ZONAL ABERRATION. 
 
 Apart from the aberration resulting from the edge rays and 
 central rays not focussing together, there is another serious fault, 
 which perhaps is the one most commonly met with in objectives. 
 The fault is caused by an objective being divided into zones, 
 greater or less in number, having different foci ; thus several foci 
 are distributed along the optic axis, causing the general focus to 
 be indefinite and confused according to the amount of the evil, 
 the effect upon definition being much the same as that produced 
 by spherical aberration, although the latter is the worse fault of 
 the two. The amount of mischief done to the definition may be 
 measured by the length of that part of the optic axis along which 
 the foci are distributed, that is, if the relative arrangement of the 
 zones and their foci are the same. And an objective which is 
 divided into ten zones, whose foci range over one-thirtieth of an 
 
24 
 
 inch of the axis, may yield far superior definition to one which is 
 divided into three zones, whose foci vary by one-tenth of an inch. 
 So that the existence of a large number of zones in an objective 
 need not greatly interfere with definition, unless their foci vary 
 very considerably, though, of course, the finest definition cannot 
 be expected if they exist in any perceptible degree. 
 
 In order to detect zonal aberration, it is best to direct the 
 telescope to a very bright star, with a moderately high power, 
 and rack in and out of focus as before, only it is best to rack out 
 until 8 to 20 interference rings can be counted, for the irregular 
 zonal effect is most easily detected under such conditions. If 
 the fault is present the interference rings will not appear regular 
 or in harmonious gradation from the centre to edge of the system. 
 Counting from the edge inwards, it may be noticed, for instance, 
 that the outer ring is poor and weak, while the next one or two 
 appear disproportionately strong, the next two or three weak, 
 while those close about the centre are strong again (see Fig. 19), 
 while, on going to an equal distance on the other side of focus 
 the complementary appearance, shown in Fig. 19a, will be seen. 
 Of course, there is great variety possible in such defects. 
 Perhaps the commonest form of it is one in which the objective 
 is divisible into three zones, the outer zone and the central part 
 focussing pretty nearly together, while there is a zone lying 
 between the centre and edge, and generally nearest to the latter, 
 which focusses short of the focus for the rest of the objective. 
 This effect is shown in Figs. 20 and 20a, which represent the 
 appearances to be seen inside of focus and outside of focus 
 respectively. Another common fault is indicated by the appear- 
 ances shown in Figs. 21 and 210:, vvhich are seen inside and 
 outside of focus respectively. Here the objective is pretty well 
 figured in its outer parts, but the rays about the centre focus 
 distinctly beyond the focus for the rest. This is caused by one 
 or more of the surfaces of the crown having a flattened area in 
 the centre, or by the opposite fault in the flint. 
 
 The rule in interpreting such appearances is that a bright zone 
 or area noticed when inside focus corresponds in position to a 
 zone or area in the objective, which focusses short, while a bright 
 zone or area noticed when outside of focus corresponds in 
 position to a zone or area which focusses beyond the general 
 focus. 
 
 All these different sorts of zonal aberration are due to imperfect 
 figuring of one or more of the surfaces of the objective. But 
 aberration, or the regular graduation of the error from centre to 
 edge, may or may not be due to imperfect figuring. For, on the 
 
■25 
 
 one hand, the radii of the spherical curves may have been 
 calculated and worked to with the utmost nicety, with a view ta 
 neutralising all spherical aberration, and yet a failure result owing 
 to imperfect figuring, which, while producing a regular curve, may 
 yet result in something in the way of a parabola, or, on the 
 contrary, an oblate ellipsoid. 
 
 On the other hand, it is possible that all the surfaces may be 
 accurately figured and spherical, but, owing to a miscalculation of 
 the radii, there is a very serious residual aberration, perhaps so 
 serious that it is impossible to neutralise it by any deliberate 
 deviation from the spherical curves in the direction of a parabola, 
 or vice versa, which is practically attainable. As a matter of fact 
 it seldom happens that the calculated radii are so accurately 
 carried out as not to necessitate slight deviation from truly 
 spherical curves in order to neutralise very small amounts of 
 residual aberration. 
 
 PERFECT FIGURE. 
 
 If all the conditions necessary to perfection are fulfilled in an 
 objective and in the eye of the observer, it will be found that the 
 ring systems, seen when a bright star is thrown out of focus, are 
 perfectly circular in outline, while the individual rings grow 
 gradually and regularly stronger and further apart as the outside 
 ring is approached, this outer ring running a little out of propor- 
 tion seemingly in its brightness and breadth. 
 
 Above all, the appearance and arrangement of the rings should 
 be exactly the same on both sides of focus, if allowance is made 
 for the blue flare which somewhat disguises and enhances the 
 brightness of the more central rings when outside focus. Figs. 
 2 2 and 2 2.2 represent the appearances, inside and outside focus 
 respectively, which should be seen under a high power, when the 
 eye-piece is racked just sufficiently out of focus for two rings to- 
 become distinctly visible, while another is on the point of 
 expanding in the centre. 
 
 Figs. 23 represents the appearance which will be seen when 
 the eye-piece is racked so far in or out of focus as to render 
 eight rings visible or thereabouts. It is under such conditions 
 that any irregularity in the formation of the rings will be most 
 easily detected. Fig. 23 scarcely does full justice to that exquisite 
 softness and regular gradatiofl of the interference rings, which 
 should be visible when viewing' a large star (out of focus) through^ 
 a perfect objective, when th eatmosphere is clear and steady. 
 
 As an observer gains experience in the use of an instrument,, 
 and especially if he often directs his attention to the essentiaL 
 
26 
 
 points indicative of quality, he will in the course of time grow 
 far more critical, and little indications of imperfections, casual or 
 inherent, will become plainly visible to him, which were too 
 subtle to strike his attention at first, for the testing of an 
 objective needs the most careful and discriminating eyesigh't and 
 close attention, although any cautious observer, guided by proper 
 directions, may gain a very good general idea of the quality of 
 an objective. 
 
 We can imagine the question being asked, " What is the need 
 for all this throwing the star image iu and out of focus, and 
 examining the appearance of the interference rings, &c., when 
 what we are chiefly concerned with is the quality of the image when 
 in focus ; as long as the objective is seen to give fine definition, 
 why should we care about the appearances of the image when it 
 is out of focus, and consequently useless ? " 
 
 This question admits of several answers, the principal one 
 lying in the fact that no objective will )'ield the most perfect 
 possible star image unless the various tests which we have 
 described are conformed to ; if an objective successfully fulfils 
 these out of focus conditions, then the most perfect definition 
 which is theoretically possible may be considered assured. Then 
 again, most of the faults which we have alluded to are more 
 visible to a trained eye, if the out of focus appearances are 
 examined, than if attention is confined only to the actual focus. 
 The principal reason for this is, that the weather does not, on the 
 average, permit of a critical examination of the star image under 
 a high power, the same flickerings, which are sufficient to make 
 it impossible to see the focussed star image distinctly, are not 
 sufficient to obliterate the much larger ring systems which are 
 visible when out of focus. They will appear unsteady to be sure, 
 but the eye learns to discriminate between permanent features 
 and casual distortions in the form and distribution of the rings. 
 
 Of course, if an observer is trying a telescope on a night fine 
 enough to enable him to closely examine the star image with a 
 high power, and he finds it faultless and answering to the 
 description which we shall presently give, he may rest satisfied 
 at once that he is looking through a first class instrum.ent, pro- 
 vided that he sees the same appearance in all positions of his 
 eye. But if there seems to be something wrong with the star 
 image, it will not be possible to sdy what the fault is, or where it 
 lies, without making an examination of the out of focus appear- 
 ances, unless the fault lies in astigmatism, maladjustment, or want 
 of true centering of the lenses, for these three faults may be 
 identified at the actual focus on a steady night. 
 
27 
 
 THE STAR IMAGE. 
 
 Fig. 17 represents the " spurious disc " which is formed at the 
 focus of a perfect objective when directed to a star, a star being 
 practically equivalent to a mathematical point of light, in being 
 devoid of apparent dimensions. And yet the image of such star 
 Shows dimensions which are quite measurable under high powers. 
 Its size depends principally or almost wholly upon the relation 
 between the average length of a wave of luminous light and the 
 aperture of the objective. Thus no amount of perfection in 
 the objective can make the star image any smaller, although the 
 presence of small imperfections will cause it to appear un- 
 necessarily large or deformed. If the star image is examined 
 ■carefully with a very high power, and on a really fine night, it will 
 be found to consist of a neat round little disc of Ught, with a 
 softened outline, and distinctly edged with red. Surrounding it, 
 after a dark interval, is a thin bright ring, which is only seen by 
 glimpses in a complete state, and after another dark interval is 
 another ring, which rarely seems complete and is much less bright. 
 If the star is a very bright one, three or four rings may be 
 counted, the outer ones being visible with difficulty and by 
 glimpses. Theoretically, these rings should show colour, green on 
 their inside edges and red on their outside edges, but it can rarely 
 be seen except in the disc itself and the next and brightest ring. 
 The wave theory of light offers a complete explanation of the 
 formation of the spurious disc and the surrounding rings. 
 
 In a thoroughly well corrected object glass, with its optic axis 
 directed to a star, the rays after passing through the lenses all 
 proceed, as exactly as possible, towards the focal point / (Fig. 
 25). And if any spherical surface of any radius less than the 
 focal length is imagined to be described about / as a centre, then 
 such surface, where it cuts the cone of rays, will be a wave- 
 surface, or a surface in which the ether particles are all exactly in 
 the same phase of wave motion at any instant. Let a-b represent 
 a diametrical section of such a wave surface taken close to the 
 object glass. Then, according to the wave theory of light, every 
 point or ether particle along the spherical surface a-b is itself a 
 centre of light, tending to radiate light in all directions. Now, 
 just provisionally, suppose that the aperture of the objective is 
 square (Fig 25a) with the diameter a-b. Then the spherical wave 
 surface will be bounded by or contained in a square. It is 
 evident that the ether vibrations, which start from all points of 
 the spherical surface a-b simultaneously and in the same phase, 
 will reach the point / simultaneously and in the same phase, 
 since / is in the centre of such spherical surface. Hence the 
 o 
 
28 
 
 great intensity of the light at the focal point /, resulting from the 
 coalescence of all the rays arriving there in similar phase. But let 
 a point d be taken at such a distance from / that the distance 
 b-d is greater than a-d by an amount equal to a whole wave 
 length of the light in question. That is, if a spherical surface is 
 described about d as centre and with d-a as radius, this surface 
 will cut d-b at c, such that the distance b-c is equal to a whole 
 wave length, and it follows, from the fact that d-f is perpen- 
 dicular to the optic axis, that the line of intersection of the two 
 spheres (which line is shown in section at the point a) is a circle 
 lying in a plane perpendicular to the plane of the paper and 
 parallel to the optic axis i-f. Similarly, all lines of constant 
 intervals between the two spheres will be circles in planes parallel 
 to this first plane. 
 
 The wave surface a-b, if viewed in elevation from a distant 
 point on the optic axis, will be the square shown in Fig. 25 (a). 
 The side g-m is the line of intersection of the two spherical 
 surfaces described about /and d^ and although part of a circle, 
 it must appear a straight line in Fig. 25 (a), because it is viewed 
 from a point in its own plane. Similarly i-g passing through the 
 centre, is a line along which the interval between the two 
 intersecting spheres is equal to half a wave length, for in Fig. 25 
 it will be seen that the intervals of separation between the two 
 spheres are in direct proportion to the distance from a, because 
 a-c is equivalent to the curve a-b rotated about the point a by an 
 angle equal to || ; a-i being half of a-b, then the separation at / 
 must be half of b-c, that is, half a wave length. 
 
 From the above it follows that light arriving at the point d 
 from the strip h,-b (Fig. 25), or the rectangle /-/ (Fig. 2512) at the 
 same instant as light arriving there from the strip i-h, or the 
 rectangle i-i, will be half a phase of vibration behind the latter, 
 and consequently, as the intensities of both are equal, they will 
 exactly interfere with one another and produce darkness at d. In 
 the same way the light from the rectangle h^-h will exactly 
 neutralise the light from h-h, also the rectangle g^-g, will have its 
 illuminating power on d cancelled by that of the rectangle g-g. 
 Therefore d will be a point of complete darkness. The same 
 result will be obtained if the effect at d of the spherical wave 
 surface a^-b^ is considered, d-b^ will still exceed d-a, by one whole 
 wave length (5/,), and the conditions for complete interference at 
 d will be found to prevail wherever we may consider the spherical 
 wave surface a^-b, to be situated. It is plain, from the above 
 conditions, that the luminous focus at / cannot extend in radius 
 quite so far as the point d. The point of maximum brightness 
 
29 
 
 is at /and there is a gradual falling off of brightness before d is 
 reached. The angular diameter of the interval /to d, subtended 
 at the centre of the objective, is evidently equal to '-if, Sor to 
 ' TpTrt'urT"' ; for wc supposed a-b to be equal to the aperture,Jand 
 i-f to be equal to the focal length. ■ -i.-ir3| 
 
 Therefore, in the case of an objective of square aperture 
 {l-n-g-m) the exact position of the first dark space d next to the 
 focal point is easily arrived at, its linear measurement or 
 distance from/ being equal to ~, or the wave-length multiplied by 
 the ratio of focal-length to aperture. The result is a " square " 
 spurious disc, whose diameter lies within twice ■^. 
 
 But supposing the aperture is altered from a square form to a 
 circular one having the same diameter, as the circle shown in Fig. 
 25«, then it is evident that the conditions of interference are very 
 much modified, and the difficulty of finding exactly the first 
 point of interference dis immensely increased. For now only a 
 portion of the rectangular strip of wave surface g-g is available 
 for interfering with g^-g,, and only a portion of /-/ for interfering 
 with i-i, although h,-h, will almost neutrahse h-h. Therefore, at 
 the point d, there will no longer be darkness, for there will be a 
 balance of wave motion, due to the excess of the area of g,-i 
 over the combined arc bounded areas within /-/ and g-g. To find 
 the point d correctly therefore involves very careful analysis, but 
 it is evident that drf must'be increased. 
 
30 
 
 Sir George Airy, as the result of careful mathematical investi- 
 gation, arrived at the result that d will be a point of perfect 
 darkness, provided that the excess of d-b over d-a is equal to 
 I '2 19,7 wave lengths, or nearly one wave length and a fifth. 
 Thus if a-b is divided into six parts, then the position of the line 
 where the interval between the two spherical wave surfaces is 
 equal to one wave-length will fall at about h,. Fig. 25, or the thick 
 line to the left in Fig. 25^. 
 
 In the latter figure the strips lettered above the diameter are 
 severally in opposite phases of wave motion to those lettered 
 below the line, and the sum total of their interferences at the 
 point d is zero. If, in Fig. 25, ^ is supposed to be so placed 
 that b-c = 1*2 2 wave lengths, then, if the figure is rotated about 
 the axis i-f the point d will trace out a circle about f, which 
 corresponds to a dark ring which embraces the spurious disc of 
 light within it, as shown in the figure. Outside the dark ring 
 will be a bright ring, the points along which correspond to where 
 the light frpm any wave surface (a,-^,) again leaves a balance of 
 luminosity. 
 
 The linear diameter of this first dark ring will then be equal to 
 ' '' V " " > where F = the focal length, A the aperture, and A the 
 wave length. 
 
 The angular diameter, as viewed from the optical centre of the 
 objective, will be equal to iiL^i*-. 
 
 In the case of an objective of 6 inches aperture and 90 inches 
 focal length, the linear diameter of the first dark ring will be 
 equal to 30 X i'22 A, and if we take the wave length of the 
 most luminous light (citron green) as about -^\-^^ inch, then 
 this amounts to -0008 1 inches, or i^Vc>t^> while the angular 
 diameter of the same is "000009, which is the circular measure 
 for I '86 seconds of arc. 
 
 Since the spurious disc is brightest in the centre, and really 
 shades ofi' into the dark ring, it is evident that its apparent linear 
 extension will depend very intimately upon the brightness of the 
 star in question, that the spurious disc formed when a bright star 
 is viewed will appear larger than in the case of a dim one, 
 although the maximum size can never amount to as much as the 
 diameter of the first dark ring. To this must also be added the 
 effect of irradiation in the case of the brighter stars. As a matter 
 of fact, it is notorious how much smaller the star-discs appear to 
 be in the case of small stars than in the case of bright ones. On 
 the average perhaps, the diameter of a star disc may be 
 considered as one half that of the first dark rine. In all 
 
31 
 
 objectives having their focal lengths equal to 15 times the 
 aperture, then the linear diameter of the spurious disc may be 
 said to average -0004 inches, or about ^^^ inch. With 6 inches 
 aperture this corresponds to an angular diameter of 0-9 seconds, 
 and in a 1 2 inch aperture to 0*45 seconds. So these respectively 
 represent the dividing powers of such apertures upon double stars 
 of average brightness. 
 
 From the above theory of the formation of the spurious disc 
 and rings, it follows that the disc and ring's formed by those red 
 rays which come exactly to focus, will be larger than the disc and 
 rings formed by the green blue rays in the proportion of their 
 relative wave lengths, and they will consequently not coincide, 
 and thus the fact is accounted for that the spurious disc, if 
 examined under favourable conditions, is seen to be edged with 
 red and be more greenish in tint at the centre, while the first 
 bright ring is greenish inside and red outside. And since the 
 angular radius (in circular measure) of the first dark ring should 
 be equal to ^^^, it should therefore vary in inverse ratio to the 
 aperture. This is fully borne out by the fact. 
 
 It is most interesting and instructive to observe the image of 
 a bright star through a large telescope furnished with an iris 
 diaphragm, using a high magnifying power. While the full 
 aperture is in use, the usual spurious disc is seen, but, on working 
 the aperture down; the disc and its rings will be seen to spread 
 out in the most remarkable manner, until, if the aperture is cut 
 down to one quarter, it will be four times as large in every way 
 as before, moreover the coloured fringes and the general structure 
 of the disc and rings will have become more evident, although 
 less bright. 
 
 We have availed ourselves of the property of the spurious 
 disc varying in size in inverse ratio to the aperture, to make 
 some careful micrometer measurements of the diameter of the 
 first dark ring when the aperture was purposely cut down, 
 yielding a spurious disc and rings which are much more easily 
 and accurately measurable than when the full aperture is in 
 use. 
 
 A six-inch objective, of 91 inches focal-length, was directed 
 to a bright star, and the objective cut down, in the first place, 
 to a square aperture, i"5 inches diameter. The mean of four 
 measurements gave the diameter of the first dark ring (in this 
 case square in shape) as "0027 inch, while the formula "-^ 
 (where A = ^^\i,(^ inch) gives •00266 as the theoretical value. 
 
 A circular aperture, diameter i"22 inches, was then placed in 
 front of the objective, when the mean of four measurements 
 
32 
 
 gave a diameter of "0039 for the first dark ring, while the formula 
 g F x^i-g2 \ giygs a value of '0040. 
 
 In both cases the image was viewed through an eye-piece 
 magnifying about 450 times, and, in order to ensure having to deal 
 with a somewhat definite wave length, a piece of green glass was 
 placed behind the eye-piece. According to a spectroscopic 
 examination, this glass allowed to pass with freedom only those 
 rays lying between D and E, the maximum transparency being 
 rather nearer to E than to D, and having a wave-length of about 
 ^5^0() inch. Since this is about the brightest part of the 
 spectrum, this wave-length is a good one to take as a basis for 
 calculating the size of the first dark ring. 
 
 The theoretical values, and the actual measurements, are 
 therefore in as close an accordance as can possibly be expected, 
 when the wave-length is only an approximation. The diameter 
 of the spurious disc itself was apparently about two-thirds of 
 that of the first dark ring, and its outline shaded off into the 
 latter. 
 
 The diameter of the first dark ring, as depicted with the whole 
 aperture of six inches in use, was also measured as nearly as its 
 minute size would allow, the measurement obtained ranging 
 about 'oooS (subject to an error of perhaps 10%), while the 
 value given by the formula ° ^ '*^''" *• is -00081. Thus the theory 
 accords perfectly well with the facts. 
 
 For some very interesting information concerning the beautiful 
 interference phenomena to be seen at the focus of a telescfipe, 
 when compound apertures of various shapes are placed in front 
 of the objective, we would refer the reader to Sir John Herschel's 
 article on " Light," in the Encyclopaedia Metrbpolitana. 
 
 It will be plainly seen that in order that an objective shall 
 form at its focus a neat round spurious disc and rings, so well 
 defined as to leave the observer in no doubt as to whether he has 
 got the exact focus or not, it must be corrected for aberration 
 with great nicety and exactness. For instance, if an objective, 
 whose focal length is fifteen times the aperture, were so deficient 
 in this correction as to cause the edge rays to focus at a point one 
 eightieth of an inch within or beyond the focus for the central 
 rays, it can be shown that the least circle of aberration (or the 
 smallest circle through which all the rays could pass at the focus) 
 would be -jifVff of ^"^ 'v!i(^ in diameter, or equal to more than half 
 the size of the spurious disc which ought to be formed. Doubtless 
 then the effect would be to make the disc perceptibly larger than 
 it ought to be in the case of small stars at any rate, and any 
 
spherical aberration greater than this would be detrimental to a 
 good objective, and become more serious as the relative aperture 
 is increased. 
 
 MECHANICAL STRAINS. 
 
 All strains which distort the lenses in any way may affect the 
 •definition to a very serious extent, although, as we will presently 
 point out, the degree of mischief done depends very intimately 
 upon the forms of the curves and also upon the relative 
 thicknesses of the crown and flint lenses. 
 
 Strains may be brought about in the following ways,: — 
 
 I St, by pressure of the cell upon the objective, either in its own 
 plane or at right angles to that plane. 
 
 2nd, by unevenness of temperature when the objective is 
 cooling down or vice versa. 
 
 3rd, by defective annealing of the material of the lenses. 
 
 4th, by the sagging of the lenses between their points of 
 support owing to their own weight, this occurring to a serious 
 extent only in the larger sizes, over 6 inches aperture for instance. 
 
 (i.) An objective should never be embraced tightly by the 
 cell in the direction of its own plane, but just sufficient slackness 
 of fit between the lenses and the cell should be allowed at 
 ordinary temperatures, as will permit of the cell just fitting the 
 lenses (without nipping) at the lowest temperature at which the 
 objective is likely to be used. Not only so, but it is better that 
 the objective should only be allowed to come in contact with the 
 cell at three equidistant points on its edge, the cell being 
 provided with three projections or studs on its inner cylindrical 
 face. If these are fixed, then the degree of slackness of fit above 
 mentioned should be allowed, but if the instrument is one of 
 great precision, such as a transit instrument, it may be necessary 
 that one of the three studs should press, by means of a yielding 
 spring, the objective continually up against the other two, in 
 order to prevent any lateral shifting which would disturb the 
 accuracy of collimation. The pressure necessary to do this need 
 not be sufficient to do any harm, moreover the moderate pressure 
 of three equidistant points can never do as much harm as the 
 violent pressure at irregular intervals round the edge, which 
 would take place at low temperatures were the cell unprovided 
 with studs and made too close a fit. Fig. 24 is a specimen of 
 the sort of appearance caused by such distortion. 
 
 It is still more necessary, in objectives over 4 or 5 inches 
 aperture, that the edge of the flint lens should not be allowed to 
 
34 
 
 bed itself anyhow and at random upon a simple flange, but the 
 latter should be provided with three slight projections P, P ^ P, 
 Fig. 9, corresponding to the positions of those which confine the 
 lens laterally. For even supposing that the flange of the cell were 
 turned with mathematical accuracy, it would even then be next 
 to impossible to make sure that the flint actually touched it all 
 round; as a matter of fact the flint would really rest itself upon 
 chance particles of dust, and if the particles on which it rested 
 happened to lie nearly at opposite ends of a diameter of the 
 objective, it is evident that the lens would sag down on each side 
 of such line, and there would be seen close to the focus the rough, 
 astigmatic effect shown in Fig. 24a:. Thus there is no certainty 
 in such a method of bedding a lens, owing to the existence of 
 dust. Moreover, apart from dust, however true the flange may be, 
 it is next to impossible to ensure that it will not be somewhat 
 distorted when the cell is fixed in its place. Hence then the 
 necessity for adopting three equidistant fixed points for the lenses 
 to rest upon and also for confining them. Nor again should the 
 crown lens be allowed to find its bed anyhow on the edge of the 
 flint. There should be three projections, made of tinfoil, paper 
 or very thin card, pasted on the edge of the flint at positions 
 directly over the projections which support the flint. The weight 
 of the crown is taken by these projections and transmitted direct 
 through the flint into the three points supporting the latter. And 
 lastly, the counter ring or upper flange, which confines the crown 
 from above should also be provided with three slight projections,, 
 which must lie just over the two sets of bearing points above 
 described. This upper ring should not bear down upon the 
 crown with more pressure than is requisite for preventing the 
 objective turning round when wiped or otherwise handled. It 
 will be seen, that all such pressure, being exerted at those points 
 of the objective which are supported immediately below, can 
 have little or no effect in distorting the lenses. They are held 
 and supported at three equidistant points, while they are entirely 
 free from contact with anything along the circumference lying 
 between. But in objectives of 12 inches aperture and upwards, 
 even the pressure 01 the counter ring is better avoided, and a 
 little play perpendicular to the objective allowed, the weight of 
 the lenses being sufficient to prevent rotation. 
 
 (2.) It is of the greatest importance to know that when an 
 objective is CBoling down, as in the case of a telescope being 
 brought out of a house into the cold night air, the best perform- 
 ance is not to be expected from it. A 6-inch objective should be 
 allowed at least half an hour for settling down into an even 
 
35 
 
 temperature before the observer should expect to use the highest 
 powers with advantage. 
 
 When an objective is in its tube and exposed to a lower 
 temperature, it tends to cool most rapidly upon the outside of 
 the crown, while the back of the flint cools, if anything, more 
 rapidly than its inside surface. Thus the curvature of the first 
 surface flattens somewhat, while the second surface is deepenedf 
 and the effect upon the flint is to cause the third surface to flatten,, 
 and the fourth surface deepen to a minute extent. The combined 
 effect at the focus is just the same as if the objective were 
 under - corrected for spherical aberration. The effect, when 
 inside focus will approach to Fig. 15, and outside focus to 
 Fig. 15a, and may look actually worse than this on very cold 
 nights. For a given amount of cooling-, the effect increases with 
 the size of the objective ; but as larg-e telescopes are almost 
 invariably mounted in observatories, where internal temperature 
 more nearly corresponds with the temperature of the outside air, 
 the relative amount of cooling down of the objective when 
 commencing- work is not so large, and they may be put to use 
 at once. Still it may happen that a suddenly cold night coming 
 after a very hot day may cause a very considerable distortion in 
 the objective at nightfall, and it may be an hour or more after 
 opening the shutters before the image looks in good form. 
 Besides, the effects of cooling upon the tube and its enclosed air 
 is to cause peculiar slow flickerings of the image, and sometimes 
 to produce a pronounced astigmatic effect, owing to the gathering 
 of the warmer air towards the upper side of the tube. 
 
 (3.) There are very few large discs which do not show a more 
 or less marked black cross when examined by polarized light, but 
 if such cross is symmetrical and its centre coincides with the centre 
 of the disc or lens, then the defective annealing indicated will 
 not be hkely to appreciably affect the definition, because the 
 alteration of density or refractive power of the disc is progressive 
 from centre to edge, and consequently the error introduced closely 
 corresponds to spherical aberration, and may be fully neutralised 
 by a slight modification in the figuring. If, on the other hand, 
 the polarized light test shows a very irregular and malformed 
 black cross, or an irregular patchy appearance, the inference is 
 that the annealing is very irregular and defective, and likely to 
 produce stray wings and brushes of light at and near the focus, 
 such appearances as are shown in Fig. 12^. Defects due to 
 bad anneahng will generally show far worse while the objective 
 is cooling down. But before a maker of repute would work up 
 discs of any size for an astronomical objective, he would first 
 
36 
 
 examine them by polarized light, and if not deemed satisfactory, 
 either reject them or have them reannealed. 
 
 (4.) There are now the important effects of the flexure of the 
 lenses by their own weight to be considered. Almost needless to 
 say, the flint lens is much more liable to sagging from this cause 
 than the crown, not only is its material heavier, but its section is 
 ill adapted for rigidity, and then there is another fact against it, 
 which will be mentioned subsequently. 
 
 The flexure, or sagging, of each lens may be regarded from two 
 points of view, (i.) There is the flexure from edge to centre, 
 or from centre to edge ; this would be the only sort of flexure 
 present if a lens were perfectly supported at all points around its 
 edge, and its effect at the focus, if anything, is simply of the 
 nature of spherical aberration, positive or negative. Consequently 
 the effect of such symmetrical flexure can be neutralised by 
 appropriate figuring, at any rate for vertical or nearly vertical 
 positions of the telescope, when gravity acts on the lenses with 
 nearly its full effect. (2.) But the flexure which takes the form of 
 a sagging of those parts of the edge which are unsupported, is 
 necessarily different in its action. If the crown lens is supported 
 upon three points, as above described, then the unsupported 
 portions between bend downwards somewhat under their own 
 weight, while the three supported portions are bent upwards 
 relatively, and the effect of this at the focus, provided the flint is 
 supposed to be perfectly free from sagging, is to cause those rays 
 which pass through near the unsupported parts of the edge to fall 
 beyond the true focus, while those rays which pass through at or 
 near the supported points are caused to fall rather short of the 
 true focus. Thus if the image were examined when inside focus, 
 the three-cornered ring system, shown in Fig. 12b, would be 
 noticed, the projecting corners corresponding to the unsupported 
 parts of the crown. If, on the other hand, the crown is imagined 
 *o be perfectly free from sagging, while the sagging of the flint 
 is considered, it will be seen that in its case also, the unsupported 
 parts of the edge sag downwards, while the supported parts are 
 bent upwards, the result at the focus being that those rays which 
 pass through at or near the unsupported parts of its edge fall 
 short of the proper focus, while those rays which pass through at 
 or near the supported points fall beyond the proper focus. But 
 the supported parts of the flint lens coincide with the supported 
 parts of the crown. Hence the effect at the focus of the sagging 
 of the flint is in direct opposition to the effect arising from the 
 sagging of the crown, and therefore, if their amounts are equal, 
 they will neutralise one another, and there will be no effect at the 
 
37 
 
 focus. That is, given two discs, a certain relative thickness of 
 the crown and flint might be hit upon, such that the flexures of 
 the two lenses might neutralise one another. But the attainment 
 of this condition cannot be reckoned on with confidence, since 
 the amount of flexure in the lenses is a function of several 
 factors, among which is the highly uncertain one of elasticity, 
 which depends so much on the annealing. As a matter of fact, 
 in many moderate sized objectives, 6 to 8 inches aperture, the 
 two flexures certainly seem to completely neutralise one another, 
 while in other cases, on a fine night, a distinctly perceptible 
 tendency to a three-cornered effect can be made out at the 
 focus, and on turning the lenses round it will be found that this 
 has distinct reference to the points of support, moreover, it may 
 be seen which flexure is the predominating one, that of the crown 
 •or that of the flint. The effect when a little out of focus is 
 like Fig. lib, and when in focus the spurious disc will show 
 somewhat like Fig. \2b', or Fig. i8. But in the case of larger 
 apertures, a relative predominance of the flexure of one lens 
 •over that of the other, which would scarcely matter in a 6-inch 
 ■objective, would very likely be enough to spoil a 12-inch or 
 larger size for delicate double star work. 
 
 It may happen that the flexures may neutralise one another 
 pretty nearly, but it is by far the best to render the objective 
 independent of chance by introducing intermediate points of 
 support for the edges, such points being borne up by proper 
 counterpoises in such a manner that the weight of each lens shall 
 be equally distributed among its bearing points. It is evident 
 that the use of more than three fixed or rigid points would be 
 incompatible with this necessary condition. 
 
 THE RELATION BETWEEN THE FORM OF THE CURVES AND 
 THE OPTICAL EFFECT OF FLEXURE. 
 
 It is a well-known fact that if a prism is caused to refract a ray 
 with the minimum deviation, then the amount of the deviation of 
 that ray will not be appreciably altered if the prism is rotated on 
 its base by a small amount, such as one degree. 
 
 On the other hand if the same prism is so placed with regard to 
 the ray, as to be very considerably out of the position of minimum 
 deviation, then one degree of rotation will cause a very per- 
 ceptible alteration in the amount of the deviation. This broad 
 afct has a very important bearing upon objectives, for it shows 
 -conclusively that if the curves can be so arranged that a ray 
 passing through near the edge of the objective shall be refracted 
 with minimum deviation, then ordinary amounts of flexure or 
 
38 
 
 distortion can have no effect in'altering the amount of deviation 
 of the ray. A prism gives minimum deviation when a ray is- 
 equally refracted at both surfaces, and so does a lens. Therefore, 
 if the curves of a crown lens are arranged for minimum deviation 
 for the edge ray, then the focus will not suffer from any distor- 
 tions arising either from its own weight or from mechanical 
 strains, should it be adopted for a very large objective. In order 
 that a lens of ordinary crown shall give the minimum deviation 
 of the edge rays, or refract them equally at both surfaces, when 
 the incident rays are parallel, it must have the radii of its curves 
 in about the ratio 8 to 25, the deeper curve being turned out- 
 wards to receive the parallel rays. 
 
 As an illustration, let an objective be considered whose crown 
 lens is of the above form for minimum deviation of the edge ray. 
 Fig. 26 represents the section of such a lens, which we will 
 suppose to be a 12 -inch one. If this forms part of an objective 
 having a focal length of about 15 feet, then the focal length of 
 the crown will be about 69 inches. The parallel ray r-r, falling 
 upon the lens at a point 6 inches from the axis, will be refracted 
 equally by both surfaces, and after emergence will pass to a point 
 on the axis about 69 inches away, a deviation of nearly 5 degrees. 
 
 Fig. 26. 
 
 If two tangents be drawn, one to each curve at the point wh^e 
 the ray passes it, then we have the prism, i-a-c, which is exactly 
 equivalent to the lens in its action on the ray r-r. Now suppose 
 the lens to bend towards the right at its edge, as it would do 
 under the strain of its own weight if laid horizontally ; the two- 
 tangents would then take up the position of the dotted lines, 
 and the prism 6-a-c will have rotated by a small angle. Let the 
 angle of the prism be 9° 30', and the refractive index for the ray 
 be 1-52. Then the minimum deviation would be 4° "57' "42" '64. 
 Now let the lens be bent towards the right by an amount 
 which would cause the tangents d-a and a-c to rotate by 30- 
 
39 
 
 minutes of arc. This would represent an impossibly large 
 amount of distortion of the lens ; nevertheless, the only effect 
 upon the ray rr-r would be to increase its deviation by only one 
 second of arc. 
 
 Fig. 26a. 
 
 And now let the same prism, refracting angle 9° "30', be 
 placed in the position shown in Fig. 26a, so that the angle of 
 incidence upon the first surface is only about a third of the angle 
 of emergence at the second surface. The prism then corres- 
 ponds to a crown lens such as would be used in an objective 
 approaching to the second type, Fig. 6, useful for its large field, the 
 radii being as 5 to 3, or thereabouts, and the second and deeper 
 surface exerting a refractive power on the ray, which is about 
 three times that exerted by the first surface. If this form of lens 
 is supposed to be bent towards the right by an amount which 
 would cause the taxigents 6-a and a-c, forming the prism d-a-c, to 
 rotate by 30 minutes of arc, as in the last case, then the deviation 
 of the ray r-r will be altered by no less than 18 seconds of arc. 
 That is, an amount of distortion which would alter the deviation of 
 the ray by only one second, in the case of a lens of least deviation 
 (Fig. 3), would alter the deviation of the same ray by eighteen 
 seconds, if it took place in a lens of the radii 5 to 3, but of the 
 same focal length. The discrepancy would come out very much 
 stronger, if, instead of supposing the enormous distortion 
 expressed by 30 minutes rotation, the more likely amount of one 
 minute had been supposed. It should here be pointed out that 
 since the star-disc is only about half a second in diameter in 
 the case of a 12-inch objective, therefore a deviation of any 
 rays from their true path (by distortion of the objective) by any 
 amount exceeding ^ second or so, would begin to cause serious 
 defects in the star-image. 
 
 Hence, if in the case of very large objectives, the curves could 
 be arranged so that each lens refracted the ray equally at both 
 surfaces, there would then be no reason for fearing the ill effects 
 
40 
 
 of distortion, whether brought about by the great weight of the 
 lenses, I or by inequalities of temperature. It would be necessary 
 to subject them to very forcible strains indeed, before the effect 
 would become visible at the focus. But unfortunately, although 
 it is easy enough to fulfil this condition in the case of the crown, 
 it cannot be done for the flint, unless an extra dense variety is 
 resorted to, which introduces other objections and practical 
 difficulties. But at any rate it is of great advantage to do away 
 with the effects of distortion, if only in the crown, for the flint 
 is perhaps more easily counterpoised round the edges than the 
 crown, moreover, distortions due to cooling are less serious in the 
 flint, owing to its taking place more evenly and gradually, partly 
 because of its shape and partly because it is not in direct contact 
 with the outside air. Besides, when the crown is in the form for 
 minimum deviation for the edge ray, the flint at any rate makes 
 almost the nearest approach practicable to the form which would 
 yield the minimum deviation. 
 
 But it has been pointed out before that an objective of this 
 form, which is intermediate between those shown in Figs. 3 and 4, 
 only yields a comparatively limited field of distinct vision. This 
 is somewhat against it, but it must be remembered that, in the 
 case of very large objectives, the actual field of view embraced 
 by even the lowest powers of eye-pieces is small in relation to the 
 size of the telescope, so that this objection need not amount to 
 much in practise. But since such a form of objective is very 
 sensitive to being thrown out of square in the slightest degree, 
 it would be advisable to make it possible to adjust it in that 
 respect, without leaving the eye- end of the telescope. On the 
 other hand, in proportion as objectives differ from such a type as 
 Fig, 3, and approach to the types shown in Figs. 6 and 7, so do 
 they become more and more susceptible at their foci to the effects 
 of distortions and strains, however brought about, and these 
 facts are fully borne out by practical experience. Most observers 
 can have no idea of the difficulty there is in mounting even 
 smaller objectives in such a manner that no distorting effects 
 upon the surfaces can be observed, when tested by reflection. 
 Luckily the refractive effect of a distortion is only about ^^th 
 part of the effect on a reflected ray, even in that form of crown 
 lens which is about the worst in this respect ! * 
 
 * An alteration of 30 minutes in the perpendiculars to the surfaces of the 
 above prism causes 18 seconds alteration in the refracted ray, while a reflected 
 ray would be altered by 2 x 30 minutes, or one degree. 
 
 rpT, alte ration of ray by refraction — 18^^ — 1 4-t, 
 ■l""S alteration of ray by reflection 'S'B'Ur" — ^TRJ'-"- 
 
41 
 
 TERRESTRIAL TELESCOPES. 
 
 In the case of refracting telescopes, made for viewing terrestrial 
 objects, the quahty of their objectives may of course be tested in- 
 the same manner as in the case of astronomical ones, provided 
 that their own erecting eye-pieces are used for that purpose, for 
 it should be pointed out that an objective must be considerably 
 over-corrected for colour, in order that it may appear achromatic 
 with its erecting eye-piece. Therefore it is useless to expect good 
 results by testing such an objective with a Huygenian or 
 Ramsden eye -piece, or good definition on the stars if high 
 power astronomical eye-pieces are applied to them. If the 
 definition yielded by such a telescope, with its erecting eye-piece, 
 seems defective, the objective can be more searchingly tested 
 either on a real star or an artificial one. The latter is arranged 
 by placing a bright thermometer bulb in full sunshine, and 
 viewing the same through the telescope from a distance of not less 
 than fifty yards. The minute image of the sun, formed virtually 
 within the bulb, will be found to exhibit all the characters of a 
 real star, showing a spurious disc and rings. The image of the 
 sun in the bulb subtends at the telescope an angle very much 
 smaller than the angular diameter of the spurious disc, at any 
 rate unless the bulb is unusually large, or else is viewed from- 
 too near a point. 
 
 REFLECTORS. 
 
 The reflecting telescope may be tested for quality in precisely 
 the same manner as a refractor. The same spurious disc is seen 
 at the focus and the same systems of rings are visible when the 
 eye-piece is racked inside and outside of the focus. If faults in the 
 figuring or any strains exist they will show themselves in the same 
 manner as in the case of the refractor. 
 
 THE KNIFE-EDGE METHOD OF TESTING. 
 
 We do not know to whom this method of testing is originally 
 due, but a complete exposition of its theory and use was given by 
 Mr. Wassail, before the Liverpool Astronomical Society, together 
 with a partial and exaggerated statement of its superiority to any 
 other test. As we believe there are many amateurs, especially 
 those who construct reflecting telescopes for themselves, who seem 
 to incline to the opinion of Mr. Wassail, we will proceed to make 
 a comparison between the theory of the two methods and their 
 respective advantages and disadvantages. 
 
 The theory of the direct focussing method of testing, which we 
 have advocated in the foregoing pages is easily understood by 
 
42 
 
 referring to Figs. 27 and 28, where 0-0 represents the aperture of 
 the objective, / a simple eye-piece, e-e-e the eye, and b the focal 
 point where a star image is supposed to be formed. The rays 
 diverging from b are refracted through the lenses of the eye-piece 
 and eye and come to a focus again exactly on the retina, that is 
 provided the eye-piece is correctly focussed upon b, and that both 
 are free from appreciable spherical aberration, which is the case 
 with high powers though perhaps not with low ones. 
 
 Next suppose that the eye-piece /and the eye are racked inwards 
 by the distance b-a. The rays from b will now enter the eye in 
 such a state of divergence that they only come to a focus at / 
 behind the retina, thus painting a luminous disc upon it instead 
 of a mere point, but this is not all. The plane upon which the 
 eye and eye-piece can focus correctly has been transferred to where 
 it makes a cross section through the cone of rays, and it follows 
 from the laws of optics that the disc of light on the retina is a 
 true image and reproduction of this cross section. 
 
 On the other hand, let the eye-piece and the eye be racked 
 outwards by the distance a-c. Then the plane of correct focus 
 will be transferred to c. Fig. 28. There will be a luminous disc 
 on the retina caused by the rays from b coming to a focus before 
 reaching it and diverging again. At the same time it can be 
 shown that the disc upon the retina is a correct image of the section 
 of the cone of rays made by the plane c. But if the rays from 0-0 
 focus with accuracy at b it is evident that sections taken through 
 the cone of rays at opposite sides of b and at equal distances from 
 it will be similar in all respects excepting coloration, absolutely 
 similar, were the objective perfectly achromatic. Again, the 
 luminous disc upon the retina must be of a certain size before it 
 can appear large enough to be scrutinized in its details, and that 
 size bears a fixed ratio to the size of the sections at a and c, when 
 a given eye-piece is used ; but if the power of the eye-piece is 
 doubled, it is evident that the ratio between the size of the disc 
 on the retina and the diameter of the sections at a or c will 
 be doubled, and therefore the eye and eye-piece may now focus 
 upon a section of the cone of rays taken half-way between b 
 and c and half the size of that at c, and yet the image upon the 
 retina will be of the same size as it was in the first instance. 
 That is, successively higher powers enable the eye to scrutinise 
 sections of the cone of rays taken nearer and nearer to the 
 focal point b in inverse proportion to the powers used, and it 
 is obvious that if the rays do not pass accurately through 
 the focal point b, then these closer sections seen under higher 
 powers will furnish the most distinct evidence of the fault. 
 
43 
 
 If there is a slight amount of spherical aberration then the 
 section inside focus (Pig. t6) will show the luminosity growing 
 more dense towards the edg'e of the section than is the case 
 with the corresponding section taken outside focus (Fig. 1612), 
 which latter may appear softened off towards the edge.* 
 
 In Fig. 29 is shown a case of very strong aberration such as 
 would be yielded by a truly spherical mirror 0-0 when tried on 
 a star. The theoretical focus for the central pencil is at /, that 
 for the edge rays at a, so that a-f is the whole amount of the 
 aberration. It follows that the least circle of confusion is at l>, 
 one quarter of a-/ from a, while c, one quarter of a-fhomf, is 
 the point where the rays from a narrow zone d-n, half-way between 
 the centre of the mirror and the edge, will cross the axis. If the 
 eye and eye-piece are focussed upon a, then of course the edge 
 
 * Whenever aberrations of rays from the true focus exist in an objective, it 
 can be shown that a cross-section of the cone of rays, taken at some distance 
 from the general focus for a star, will show variations or irregularities in its 
 brightness (generally in the form of zones), and that the relative luminosity of 
 the different parts of the section will vary in inverse ratio to the squares of the 
 axial distances between the cross-section and those points where any rays cut 
 through cross the optic axis . 
 
 For instance, if an objective has the fault of a zone, the rays from which 
 cross the axis at a point 'OI inch short of the main focus, and a cross-section 
 be taken at '2 inches within the main focus, then the distance of the section 
 within focus for the rays from the faulty zone will be 'ig inches, and the 
 brightness of the section, where it cuts through the zone, compared with the 
 brightness of the rest, will be represented by "J^S^ = U^ = l^. Almost 
 needless to say, such a zone, distinguished by 1 1 % extra brightness, could not 
 escape notice, especially when its existence would be confirmed by a corres- 
 ponding zone of 10 % ^ess than the average brightness revealed by viewing a 
 cross-section taken at 2 inch beyond the main focus. But the above formula is 
 only true when a point of light is focussed upon, such as a star. Should the 
 object focussed upon have any moderate size, the variations of brightness in 
 
 any cross-section of the cone of rays will be represented by /d j. ^\ 2 where 
 
 d is the diameter of the focussed image of the object (a planetary disc, for 
 instance), D is a variable, representing the axial distance from the cross- 
 section to the point where any particular rays, cut through by the section, cross 
 the axis (as instanced above), and r is the ratio of focal-length to aperture of 
 the objective. It is evident that as d grows relatively large, so will differences 
 of brightness in the parts of any cross-section (consequent upon the objective 
 being defective) tend to disappear. Moreover, an additional complication will 
 be introduced, owing to the increasing overlapping of the elementary cones of 
 rays. But if d is only just large enough to cause the systems of interference 
 rings to so overlap one another as to give an out of focus disc of continuous 
 brightness, then it will be found that irregularities due to aberrations will 
 show themselves at least as unmistakably as they would do were the telescope 
 directed to a star and a critical examination made of the configuration of the 
 interference rings. 
 D 
 
45 
 
 rays focussing at a and forming an ill defined bright point there, 
 will again focus to a bright point on the retina, while the other 
 rays through which a cuts will form a diffused halo upon the retina, 
 since they are all diverging from successive points betwen a and/ 
 which are in various degrees so near to the eye that they focus at 
 points beyond the retina. From such considerations alone it will 
 be seen that the image on the retina must resemble the section 
 taken through the cone of rays at a. 
 
 And now, in illustrating the knife-edge test, let it be supposed 
 that a lower power is substituted for /, sufficiently low to enable 
 the eye to focus upon a section of the cone of rays, either on g 
 or h (for example), without such section appearing too large and 
 dim. Under such circumstances the sections will appear 
 tolerably uniform and regular in luminosity, being so far from the 
 pseudo-focus. Or the eye-piece may be dispensed with, and the 
 eye placed somewhere close behind/, when a disc of luminosity 
 should be seen, apparently filling the mirror o-o up to the edge. 
 Suppose a knife-edge, with its edge perpendicular to the plane of 
 the paper, to be advanced edgewise from right to left across the 
 cone of rays in the plane a. Then the first rays to be stopped 
 will be those from about d, and the eye will see a luminous disc 
 A,i„ with a dark patch to the right of the centre. When the 
 knife-edge has reached the centre exactly, the above dark patch 
 will still remain, while the rays from the extreme edge will be 
 simultaneously in course of being cut off and will have half 
 disappeared, at the same time that the rays from about n will 
 escape the knife-edge altogether. Therefore the disc will appear 
 as in A 2. 
 
 If now the knife-edge is transferred to the plane b and made to 
 traverse the least circle of confusion, the first rays to be stopped 
 will be those from d and those from 0, simultaneously, as in ^ i. 
 On the knife just reaching the axis it will be on the point of 
 cutting off simultaneously all the light from that zone which 
 focusses at b, which zone will be |^ths of the full aperture. Rays 
 from d will be cut off entirely, while those from n only will 
 escape. Therefore we have a figure like B 2. If the knife-edge 
 is transferred to the plane c, the first rays to be stopped are those 
 from I?,, and the result is like C, i. On it reaching the axis, all 
 the rays from the zone d-n, which focus at c, will be getting 
 stopped off simultaneously, leaving a half dark ring, while 
 those coming from points between 0, and «, and also those from 
 between d and the centre will be stopped entirely, the only light yet 
 uninterfered with being that from between n and the centre. The 
 result is the arrangement of light and shade shown in C 2. It 
 
46 
 
 should be borne in mind that all these figures are in negative, the 
 shading standing for illuminated portions. 
 
 Turning now to Fig. 27, where all the rays are supposed to 
 focus accurately at the point b, it will be seen at once that if the 
 knife-edge is made to traverse the plane a, within focus, the first 
 rays to be stopped will be from the extreme right of the aperture, 
 and as it advances the rays which escape will always be bounded 
 by a segment of a circle on the left, and the vertical chord of the 
 knife-edge on the right. A i and A 2, Fig. 27, respectively 
 represent the first contact, and the case of the edge having just 
 reached the axis. But if the knife-edge is caused to traverse the 
 plane c, outside focus, it is evident that the appearances will be the 
 opposite of those seen in the former case, for the first rays to be 
 cut off will be those from the left hand side of the aperture, and 
 therefore the shadow will seem to advance from the left hand. 
 Thus the observer can always say whether the knife edge is 
 within or beyond the focus, and by altering its position along the 
 optic axis accordingly, he can find the position where the action 
 of the advancing knife-edge is to cut off all the rays from the 
 objective or the mirror simultaneously, as is necessarily the case, 
 when the edge cuts exactly through the focal point b. If there is 
 no longer the slightest tendency to cut off either the right hand or 
 left hand margin of the luminous disc before any other part, 
 then it is certain that the knife-edge is at the focus within a very 
 small margin of error. There is no more exact method of finding 
 the exact focal point than this. In the case of objectives with 
 their focal length equal to about 15 times the aperture, we find it 
 possible to find the focus with no greater error than -^^th of an 
 inch, if the definition of the star is steady. This is one of the 
 chief uses of the method. 
 
 There is another use to which it has been put by makers of 
 reflectors, and that is for finding the radii of the different zones 
 of paraboUc reflectors of large aperture, when it may be 
 inconvenient or impossible to try them by the best of all methods, 
 that is on a star. A correctly figured reflector has a parabolic 
 curve, the radius of each zone increasing as the edge is 
 approached. By mounting a knife-edge and an artificial star at 
 the centre of curvature, and receiving the light as it is reflected 
 back into the naked eye, the mirror will be apparently filled with 
 light, and the knife-edge may be employed for finding the foci 
 for each successive zone, for if the light from any particular zone 
 can be made to disappear simultaneously, it is then known that 
 the knife-edge is traversing its focus. The differences between 
 the foci of the successive zones should, if the figure is parabolic. 
 
47 
 
 come out just double the theoretical differences which should 
 exist between the respective radii, provided the artificial star is 
 immovable; or equal to the theoretical differences, if the star 
 moves with the knife-edge. 
 
 It is almost needless to point out that the knife-edge must be 
 mounted in rigid attachments, and be capable of an accurate and 
 measurable motion parallel to the optic axis by means of a fine 
 screw. When it is at the focus, a lateral or traversing movement 
 of xjre^th of an inch may make all the difference between the 
 whole light passing and all being stopped, so that a very delicate 
 movement across the axis must be allowed for. This test has 
 been advocated for detecting minute faults in a mirror or 
 objective, by using it at the focus in the manner indicated in 
 Pig. 29, the faults showing themselves in the form of the irregular 
 appearances shown in the FigS'. A, B and C, although perhaps in 
 less pronounced forms. 
 
 Certainly, with care and patience, and the expenditure of 
 considerable time, the faults may be so detected, and after a good 
 deal of thinking, the visual appearances may be interpreted, but 
 all this involves a needless loss of time and exercise of patience, 
 while the other method, which we have advocated in these pages, is 
 very much more direct, expeditious, easily applied, and also more 
 searching and severe, at any rate when applied by one who has 
 had considerable experience. The knife-edge tests has its uses, 
 but we can confidently state, after comparing both methods 
 together, that for the purpose of detecting errors of workmanship, 
 &c., the direct focussing method has the decided advantage. 
 When an objective seems faultless under the direct focussing test, 
 no amount of bothering with the knife-edge will give any other 
 result than to cause the whole disc of light to vanish simul- 
 taneously, as it should do when the rays focus accurately to a 
 point. On the other hand, an objective that shows a little 
 residual fault under the direct focussing test may easily pass 
 muster under the knife-edge test. It is interesting to know that 
 that keenest of observers, the Eev. W. H. Dawes, insisted upon 
 there being no test for an objective so severe as the direct 
 focussing method. 
 
 There is another " method " of testing an objective or mirror 
 for aberration, about which a few words should be said. We have 
 heard of observers stopping down the aperture, to perhaps one 
 third, with a piece of cardboard with a hole in the middle ; then 
 after getting an object into fociys with a certain eyepiece, the 
 cardboard is removed, a disc equ^t to the aperture in the latter is 
 placed over the centre of the objective, thus leaving exposed the 
 
48 
 
 outer zone which was covered before, and then the observer notices 
 carefully how much the eye-piece has to be racked in or out in 
 order to get the image into correct focus. If any alteration has 
 to be made, he forthwith concludes that either positive or negative 
 aberration exists. Nothing could be more grossly misleading than 
 this practice, for the simple reason, that when an objective is 
 stopped down to a small aperture near the centre, the cone of rays 
 painting each point of the image is so narrow that the eye-piece 
 may be racked in and out between considerable limits without the 
 definition seeming to be disturbed, hence it is impossible to fix 
 the focus with that confidence which would warrant the observer 
 in concluding that the centre area had a different focus to the 
 outer zone. The observer may easily convince himself of the 
 futility of this procedure if he will go about it in the opposite way, 
 first covering up the centre and finding the focus accurately for 
 the outer zone, and then, covering up the outer zone and exposing 
 the central part, before covered, look through the eye- piece and 
 see whether the image does not still appear in focus. If it does 
 not, and the eye-piece has to be racked in or out before more 
 distinct vision is obtained, then the objective must be a shocking 
 bad one, and under the ordinary direct focussing test, at full 
 aperture, it ought to exhibit a dense bright patch about the centre 
 of the luminous disc either inside or outside focus, with a corres- 
 ponding vacuity on the other side of focus, and such feature ought 
 not to escape the notice of the most casual observer. 
 
 We have emphasised the words more distinct\v&\. above, because, 
 when an objective is cut down to (say) one third of its aperture, 
 the spurious disc which represents each point of the image is made 
 three times as large as when the full aperture is used, therefore the 
 image must be expected to appear less sharp and well defined, so 
 the observer must not conclude he can necessarily find a better 
 focus. All really good objectives, whether telescopic or micro- 
 scopic, should give the most crisp definition when the full aperture 
 is in use, any contraction of the same being at the expense of 
 the definition, to say nothing of the illumination of the image. 
 
 These dodges with cardboard discs and diaphragms are all apt 
 to mislead ; if there is any zonal aberration at all, nature's laws, 
 as manifested in the exquisite and complex reactions of light waves 
 upon one another, will, to a certainty, reveal such faults in the 
 irregular distribution and brightness of those interference rings 
 which unfold themselves as the observer racks inside and outside 
 of the focus. 
 
 Noi'E. — The theory of interference of light would lead one to expect that 
 any cone of rays, converging to -^ point, must be broken up into a series 
 
49 
 
 of unbroken and exquisitely thin conical (?) surfaces or shells of luminosity 
 alternating with conical (?) shells of darkness, the alternate bright and dark 
 apices of such shells all lying along the optic axis, the apices of the more 
 interior and shorter shells of course falling successively nearer to the objective. 
 Therefore any cross section of the cone of rays must reveal a system of rings, 
 the number visible depending upon the number of shells which are cut through 
 by the plane on which the eye is focussed, that is, upon the distance from the 
 focus. But it seems extremely difficult to account for the fact that these inter- 
 ference lings are apparently achromatic, for the theory of interference would 
 seem to indicate a separate system of shells and rings for each colour that goes 
 to focus. The F rays, for instance, forming a system of shells and rings which 
 would be closer together than the rings formed by the C rays in proportion to 
 their respective wave-lengths, or as 3 : 4, would lead one to expect some- 
 thing like a series of fine Newton's rings ; but there seems to be no marked 
 indication of such an effect. 
 
 Whether the shells of light and darkness are all strictly conical right up to 
 the objective, or whether a longitudinal section through the axis would exhibit 
 some sort of curves, we are not in a position to state, although it is evident 
 that the outer shell must be conical. 
 
 But, by way ot illustrating the fineness of these rings, we may say that with 
 a 5-inch objective, we could count six interference rings when the eye-piece 
 (a high power) was racked within focus by '21 inches. The ratio of the radius 
 of the aperture to the focal -length was -jV and ^-^ih of "21 = -0076 as the 
 radius of the cross section of the cone of rays focussed upon, so that the 
 average interval from centre to centre of the rings was -^M =' •0013, or 
 approximately eityth of ^n inch. The intervals between the more interior 
 rings always grow successively smaller, but the quickest gradation is from 
 the outside one to the 3rd or 4th. 
 
TOEK 
 
 Feinted bt Ben Johnson, & Co., 100, and 101, Micklegate, 
 
 1891.