NRLF 
 
 14 251 DBS 
 
 HlHHB 
 
TELESCOPE OBJECTIVES 
 
THE 
 
 ADJUSTMENT & TESTING 
 
 OF 
 
 TELESCOPE OBJECTIVES 
 
 T. COOKE & SONS, LTD- 
 
 BUCKINGHAM WORKS, YORK 
 
 AND 
 
 3, BROADWAY, WESTMINSTER, S.W. 1. 
 1921. 
 
 All rights reserved. 
 
:>2 
 
 (A 
 
PREFACE TO THIRD EDITION. 
 
 IpNCOU RAGED by the favourable reception accorded to previous 
 editions of this book, in printed reviews, in many private letters 
 received from astronomers, and as attested by the fact that it has 
 been deemed worthy to be translated into several foreign languages, 
 we have thought ourselves justified in issuing a third edition. 
 
 Since the date of publishing the first edition, in the summer 
 of 1891, Mr. H. Dennis Taylor, our optical manager, has entered 
 into many calculations and' experiments on our behalf, with a view 
 to the further improvement of the so-called achromatic telescope, 
 with the result that we were able, in the spring of 1894, to offer 
 to astronomers an entirely new objective practically free from 
 secondary colour aberrations, and capable of being used for celestial 
 photography without any special adjustment. Seeing that our Triple 
 Photo- Visual Objective has in a large number of cases supplanted the 
 old double form, we have considered it necessary to insert a supple- 
 mentary chapter upon its adjustment, while retaining all instructions 
 relating to ordinary double objectives, many of which will, doubt- 
 less, continue to be used. 
 
 By the kind permission of the Council of the Royal A stronomical 
 Society, we are enabled to include a reprint of Mr. Dennis Taylor's 
 paper on "The Secondary Colour Aberrations of the Refracting 
 Telescope in relation to Vision," in ivhich the great detriment suffered 
 by large refractors of the ordinary type, by reason of their imperfect 
 colour corrections, is fully dealt ivith ; and also his paper entitled 
 "A Description of a Perfectly Achromatic Refractor," in which 
 our Photo-Visual Objective is described. The general text of the 
 book is substantially that of the second edition. 
 
 484774 
 

 
 PLATE I. 
 
INDEX TO PLATE I. 
 
 Fig. Wa. Eccentric appearance of interference rings, due to the objective 
 being out of adjustment. (See page 16). 
 
 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. 
 
 Fig. 11. A section of the cone of rays taken closer to the focus, exhibit- 
 ing a more moderate degree of maladjustment. (See page 17). 
 
 Fig, 12. a, b, c, d and d' are out-of-focus sections, as will be seen when 
 the objective is correctly "squared on," and quite irrespective of 
 other faults. (See pages 18 and 19). 
 
 a', b' and d" are appearances of the focussed image correspond- 
 ing respectively to a, b and d. 
 
 d, d' and d" are also examples of astigmatism. (See page 31). 
 
 Fig. 13. A section taken a very little way within focus, under a high 
 power, exhibiting the fault of astigmatism. (See page 33). 
 
 Fig. 14. The corresponding appearance to Fig. 13, as shown by a section 
 taken at the same distance beyond focus. (See page 33). 
 
 Fig. 15. Section within focus, showing result of positive spherical aber- 
 ration. (See page 35). 
 
 Fig. 15a. The corresponding section, taken at the same distance beyond 
 focus. (See page 36). 
 
 Fig. 16. A section taken closer to focus under a high power, exhibiting 
 a slight residual spherical aberration ; the central rings rather 
 weak. (See page 36). 
 
 Fig. 16a. The corresponding appearance at the same distance beyond 
 focus ; the central rings relatively strong. (See page 36). 
 
 Fig. 17. The spurious disc or image of a star yielded by a perfect objec- 
 tive, and viewed under a very high magnifying power. (Seep. JT). 
 
 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 53). 
 
 Figs. 19 and 19a. An example of marked zonal aberration, being sections 
 of the cone of rays taken inside and outside of focus respectively. 
 (See page 38). 
 
 Figs. 20 and 20a. Another example of zonal aberration. (See page 38). 
 
 Figs. 21 and 21a. Example of the general figure of an objective being 
 tolerably good, but there is a region in the centre having a focus 
 somewhat beyond the main focus. (See page 38). 
 
 Figs. 22 and 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 pages 36 and 39). 
 
 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 39). 
 
 Figs. 24 and 24a. Examples of violent mechanical strain, due to im- 
 perfect mounting or bad annealing. (See page 49). 
 
 Fig. 24b. Example of the effects due to the presence of veins in the 
 material of the objective. 
 
CONTENTS. 
 
 PAGE 
 
 SQUARING ON ........ 8 
 
 ACHROMATISM ........ 22 
 
 FALSE CENTERING . . . . . . . 28 
 
 ASTIGMATISM- . . . . . . 30 
 
 SPHERICAL ABERRATION . . 35 
 
 ZONAL ABERRATION ....... 37 
 
 PERFECT FIGURE . . . . . 39 
 
 THE STAR IMAGE ....... 41 
 
 MECHANICAL STRAINS . - . . .'48 
 
 THE RELATION BETWEEN THE FORM OF THE CURVES 
 
 AND THE OPTICAL EFFECT OF FLEXURE . . 54 
 
 TERRESTRIAL TELESCOPES ...... 58 
 
 REFLECTORS . . . . . . . . 58 
 
 THE KNIFE-EDGE METHOD OF TESTING ... 59 
 
 THE COOKE PHOTO-VISUAL OBJECTIVE . . . 69 
 
 GENERAL TREATMENT OF OBJECTIVES . . 73 
 
 THE SECONDARY COLOUR ABERRATIONS OF THE 
 
 REFRACTING TELESCOPE IN RELATION TO VISION 79 
 
 DESCRIPTION OF A PERFECTLY ACHROMATIC REFRACTOR 101 
 
 AN EXPERIMENT WITH A 12^ INCH REFRACTOR . . 113 
 
The Adjustment and Testing of 
 Telescope Objectives. 
 
 T N these days all astronomers, whether amateur or other- 
 wise, 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 cerain 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 convenient and generally reliable form of tele- 
 scope, 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 objec- 
 tives which we have had submitted to us for our opinion, 
 we have found the greater proportion very indifferent in 
 their performance. Sometimes we have found second rate 
 objectives enjoying a reputation for high quality in some 
 quarters, chiefly for the reason, as it seems to us, that there 
 is a lack of published information of an authentic and com- 
 plete 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 of first rate quality are 
 nevertheless 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. Such facts as these, 
 among many others, as well as the wish to minister to the very 
 natural desire on the part of astronomers to understand the 
 
1^iO'*i' 
 
 instruments with which they work and upon whose good 
 performance they depend, prompt us to publish the following 
 remarks for the purpose of enabling them to gain the very 
 utmost advantage from the use of their instruments. 
 
 Squaring On. 
 
 In the first place it may be pointed out that portable and 
 pocket telescopes of under three inches aperture intended 
 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 render a perfect 
 adjustment of the objective unnecessary. Of course, good 
 objectives for astronomical purposes, if supplied with tube 
 and mounting, are alsp 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, 
 etc., 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 in some far distant country to which 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 the purchaser or a friend. It is therefore very important 
 that anyone who uses an astronomical telescope of any con- 
 siderable 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 diffi- 
 cult 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 a-0 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 a-0, 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 a-0 will now be transferred to the position a,-0, passing 
 centrally through the eye-piece. 
 
 Fig 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 ob- 
 jective 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; for there is no type 
 of object-glass in existence which will give a perfect image 
 of a point of light or star at a point situated even half a 
 degree only from the optic axis ; for, in the first place, the 
 oblique image is never a round star disc, but a sort of linear 
 formation caused by the astigmatism which mars the oblique 
 performance of refractors and reflectors in equal degrees. 
 But, besides this inevitable astigmatism, there is also an 
 oblique effect, known as " coma " or eccentric flare, whose 
 amount very essentially depends upon the form of the object- 
 glass. This defect is superimposed upon the inevitable 
 astigmatism, and in many cases is violent enough to com- 
 pletely disguise the latter. 
 
IO 
 
 In accordance, then, with the dependence of .coma upon 
 the forms of object-glasses, the latter may be divided, very 
 conveniently for our purpose, into the classes whose several 
 behaviours when tilted or thrown out of square are different, 
 thus requiring different treatment. 
 
 1st. There is the largest 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 optic 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. For all 
 the objectives (Figs. 2, 3, 4 and 5) included in this class give 
 a lateral image which is characterised by inward coma, or 
 coma in which the flare lies inwards towards the optic axis; 
 and they yield this effect in degrees depending directly upon 
 the degree of outward bulge of the curves of the object-glass. 
 
 Fig 2 Fig 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 i to 7|. This form has the 
 disadvantages of the last form, but in a minor degree. 
 
 Fig. 4 represents the form of double-objective which we 
 generally adopt ourselves. The radii of the crown lens are 
 to one another 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 D ray are 1-518 and 1-620 
 respectively), while the fourth surface is a concave of long 
 
II 
 
 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 correction 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 double 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 
 
 R g ,6. 
 
 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 considerable distance from the optical 
 axis. 
 
12 
 
 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. 
 In other words, such an object-glass as this is quite free from 
 coma in its oblique image. There is simply astigmatism 
 of a symmetrical character. 
 
 The mathematical conditions to be fulfilled in an object 
 glass that shall be free from coma will be found dealt with in 
 a paper by Mr. H. Dennis Taylor, entitled " Telescope 
 Objectives for Photographic Purposes," in Monthly Notices 
 of the Royal Astronomical Society, Vol. LIIL, No. 6. 
 
 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 ratio of curves it may be 
 noted that if a certain peculiar hard and colourless, yet low 
 refractive crown, made by Schott and Gen, of Jena, is used 
 for the crown lens, and ordinary 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 TbHTs. 
 If the radius of first surface is 5, then that of second ditto is 
 3, the third 3-05, and fourth 25. 
 
 3yd. 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 
 
 Rg.7 
 
 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. 
 
Such forms of objectives with their curves decidedly 
 bulged inwards towards the eye-piece yield outward coma in 
 their oblique images. The eccentric flare lies outwards or 
 away from the optic axis. 
 
 It may be said that the type of objective shown in Fig. 6 
 occupies a place by itself, dividing the first type, characterised 
 by outward bulge of the curves and inward coma, from the 
 third type, characterised by excessive inward bulge of the 
 curves and outward coma. 
 
 Of course, 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 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 9, which are respectively a cross section and a 
 plan of a cell and its attachments such as we generally supply. 
 In Fig. 8 the shaded section represents the cell which 
 holds 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 /-/ by 
 means of the three pairs of counter-screws shown at s t s t/ s tll - 
 On examination it will be found that, of each pair of counter- 
 screws, one (2) is threaded through the flange c and pushes 
 against the flange /, its office being to keep them separated 
 by a small interval, while the other' (i) passes loosely 
 through the flange c, but is threaded into the flange /, 
 its office thus being to draw the counter-cell nearer to the 
 flange/. 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 (2) the pushing 
 screw, for when tightened it tends to push the counter-cell 
 further out at that point, and the other (i) 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 tommy pin), 
 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 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 to the 
 proper tightness. 
 
 By referring to Fig. 9 it will be seen that supposing the 
 objective has to be pushed inwards (towards eye-end) near 
 the screws s t , 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 there, 
 then the same effect will be produced by pushing out the 
 objective at a point opposite to s /? and this may of course 
 be effected by separating the flanges by means of both 
 pairs of counter screws S H and s //y , the adjustment being 
 equally divided between them. 
 
 Or, again, if the counter-cell should require pushing off 
 at or near s //y , it may be done either by means of the screws s llt 
 or by pushing in by means of the two pairs of counter- 
 screws s, and s u , dividing the adjustment between them. 
 In order to get at these counter-screws, in the case of some 
 telescopes, it may be necessary to remove the dew-shade. 
 
 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 
 
Fig. 9 a. 
 
1 6 
 
 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. If the O.G. is 
 of the first type, then 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 eye-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, roughly speakin'g, the focus 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 delicate 
 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. 10 (), 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 
 
 * Figs. 10 to 24, inclusive, are represented by photography in the 
 frontispiece. 
 
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 some- 
 what to one side of the position of the focussed image, as in 
 Fig. u, 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 manipulation 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 eyepiece. The least disposition of the 
 rings to expand somewhat to one side of the focussed star 
 may then be detected and the corresponding minute adjust- 
 ment 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. 11). However, the 
 characteristic of eccentric expansion of the rings is the best 
 thing to observe in practice. 
 
i8 
 
 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 surrouud 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. 10 (b) 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 practice 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 practice 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 successfully 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 (a,), that is, if the objective is a really perfect 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 
 
19 
 
 adjusted, the observer may easily make a mistake and turn 
 the 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, etc. 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 whereabouts 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 and d t , 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 concentrically with regard to 
 the focussed image, nevertheless the disc of light is not round 
 but appears oval (d and d t ), or even of some irregular shape (c). 
 Such appearances are indicative of defects either 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 (p. 10) 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 
 
2O 
 
 considerably tilted, it will be found that the luminous disc 
 yielded by a star when out of focus, will still expand itself 
 symmetically round the position of the star when in focus, 
 but it will appear more or less oval in shape, as in Figs. 12, 
 d and d t (plate I.), 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 oblique pencils is, however, not so great as to prevent this 
 type of objective being the most eminently suited for photo- 
 graphic work where a relatively large field of view is 
 required. If an objective of this type is examined as to its 
 adjustment, 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 appearance inside focus were a horizontal 
 oval, then the tilting 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 (p. 12), 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 
 
21 
 
 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 thnem, and, if he takes pride 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 him to do so, at any rate with substantial 
 accuracy. All the following remarks apply impartially to all 
 the older types of double objectives without distinction, so 
 long as they are used for ordinary astronomical observations, 
 in which case 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 the camera, in which case the image is not submitted to 
 the scrutiny of such high powers. 
 
22 
 
 Achromatism. 
 
 Perhaps the first thing to direct the attention to in testing 
 an objective is the state of its achromatic correction. With 
 comparatively few exceptions the eye-pieces used with 
 astronomical telescopes are either Ramsden or Huyghenian 
 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 em'ergence 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 in the spectrum emerge strictly 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 lenses, emerge in such 
 a manner that the coloured pencils of which it is composed 
 seem, as before, to come from 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 
 
23 
 
 itself 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 Huyghenian 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 Huyghenian 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 Huyghenian or Ramsden eye-pieces, 
 as seen by the eye and measured by the standard of the spuri- 
 ous disc or real image, vary in direct ratio to the equivalent 
 focal lengths of such eye-pieces, and while the colour effects 
 of eye-pieces of high power may seem but slight, still the 
 coloured fringes produced by eye-pieces of lower powers 
 become decidedly aggressive to a fastidious eye. 
 
 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 Huyghenian 
 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 find there is 
 a very perceptible amount of red fringe round the image 
 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 
 
24 
 
 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. 
 The larger amount of colouration noticeable with the low 
 power is really due to both the eye and the eye-piece. 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 a considerable amount 
 of red fringe similar in character to but less in amount than 
 what he saw when using the low-power eye-piece ; this will 
 give some idea of the amount of colouration due to the eye 
 alone, for there can be no question concerning the perfect 
 achromatism of the star image yielded by a good reflector. 
 
 Then again, the relation existing between the aperture and 
 the focal length of the object-glass or mirror exerts a 
 most marked influence upon the amount of coloured fringes 
 produced by both eye and eye-piece, that is supposing 
 that the powers used are high enough in all cases to pass 
 the whole of the light from the objective or mirror through 
 the pupil of the eye. For it is evident that with a given 
 eye-piece in use giving a certain magnifying power with a 
 certain focal length, the doubling of the aperture of the 
 objective or mirror leads to the doubling of the breadth of 
 the pencil of light passing through the eye-piece and also 
 through the eye, and, therefore, the more colour producing 
 edges of the lenses of both eye-piece and eye are brought 
 more into operation. Indeed, the mischievous effects of the 
 coloured fringes produced by doubling the relative aperture 
 of the objective or mirror are quadrupled in amount when 
 
25 
 
 estimated in terms of the size of the spurious disc forming the 
 real star image, which, as has been hinted at above, should 
 always be taken as the standard of measurement of the 
 subjective amount of chromatic and other aberrations. For 
 while doubling the aperture or angle of the cone of rays has 
 led to doubling the circles of confusion in different colours 
 produced upon the retina, it has simultaneously led to the 
 halving of the size of the star disc, in accordance with the 
 well-known law of the size of the spurious disc forming 
 the image of a star varying in inverse ratio to the aperture. 
 Therefore, the coloured fringes are quadrupled in size as 
 compared with the star image. 
 
 In the case of the refracting telescope, besides the coloured 
 fringes produced by the eye-piece and the eye behind it, there 
 are the coloured fringes produced by the objective itself. No 
 one who has used even a three-inch objective upon celestial 
 objects can have failed to notice the blue fringe of light 
 surrounding all bright objects. The red fringe seen sur- 
 rounding the penumbra of light when the eye-piece is racked 
 inside the focus of a star, and the patch of concentrated red, 
 and of more diffuse blue light seen about the middle of the 
 penumbra of light when the eye-piece is racked outside of 
 focus, are effects well-known to all who critically examine 
 their instruments. Indeed, the ordinary double objective can 
 only be described as achromatic in an approximate sense. 
 The double objective has been used down to this day in 
 much the same condition that Dolland left it, and to those 
 who are aware of the very imperfect concentration of only a 
 portion of the coloured rays constituting white light into 
 one focus by double objectives, the great work done by such 
 instruments, and their very satisfactory although not perfect 
 defining power, must remain a matter of great surprise. 
 
 Such objectives made of ordinary crown and dense flint 
 glasses yield a focus, for white light coming from a distant 
 point of light such as a star, which may be described as the 
 spectrum doubled back upon itself. It is the aim of the 
 optician to so correct such an object-glass that those rays 
 to which the eye is most sensitive shall be refracted to the 
 minimum focus, or the focus nearest the objective. These 
 
26 
 
 are the yellow, citron and green rays. Then the bright red 
 rays about B or C are refracted to a focal point somewhat 
 further off the objective, along with the greenish blue rays 
 about F. The duller red light at the end of the visible 
 spectrum has a still longer focus, and the blue rays beyond 
 F a still more extended focus, and the violet rays are refracted 
 so far beyond focus as to be almost beyond recognition. 
 Hence, if a white object such as a planetary disc is focussed 
 in such a telescope, there is a fairly sharply defined image 
 thrown upon the retina painted by the most luminous rays 
 of the spectrum, the yellow green principally, while the green 
 and bright blue light, along with the full red and dull red, 
 form a halo of wasted light round the object, and this con- 
 stitutes the secondary spectrum as known to the observer. 
 For fuller information upon this very interesting subject, and 
 for an explanation of why it is that these colour aberrations 
 do not absolutely destroy definition in the case of large 
 refractors, we will refer the reader to Mr. Dennis Taylor's 
 paper on " The Secondary Colour Aberrations of the 
 Refracting Telescope in Relation to Vision " (see p. 79), and 
 also to his paper entitled "An Experiment with a twelve- 
 and-a-half-inch Refractor, whereby the light lost for defining 
 purposes owing to the secondary spectrum is separated out 
 and rendered approximately measurable" (see p. 113). In 
 this experiment, the loss of light theoretically arrived at in 
 the paper was practically confirmed. 
 
 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 
 
27 
 
 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 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 with 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. 
 
 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 greenish yellow 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 greenish yellow 
 disc, but without any trace of a red fringe, but, perhaps, 
 a trace of a green 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 greenish 
 yellow 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 blueish white star, like 
 Vega, still keeping the same eye-piece in use, it will be found 
 
28 
 
 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, 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 otherwise 
 conclude that the objective was under-corrected, when really 
 as achromatic as possible. 
 
 Turning to Polaris 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 the aperture be substituted. On racking inside focus 
 it will be found that not a trace of a red fringe is to be seen ; 
 indeed, 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. 
 
 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 
 
29 
 
 noticed that more red shows upon one side of the ring 
 system than another, and moreover, any bright object in 
 focus will show a red edging more 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 objective, 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. 
 
 Indeed, if the amount of eccentric colour is only small an 
 experienced observer knows how to neutralise it when using 
 high powers simply by altering the position of his eye. In 
 the normal course of things it is supposed that the centre of 
 the pupil of the eye is placed exactly in alignment with the 
 optic-axis of the eye-piece, and in such a case of course any 
 colour fringes due to the eye are distributed evenly around 
 the image. But if the pupil of the eye is displaced laterally 
 so that the pencil of light emerging axially from the eye-piece 
 is forced to traverse the pupil of the* eye eccentrically, then 
 eccentric colour effects are at once produced. For instance, 
 let a star or planet be focussed nicely in the middle of the 
 field of view of a rather high power eye-piece, then, if the eye 
 is properly placed behind the eye-piece, whatever coloured 
 fringes are seen are distributed impartially around the object, 
 but if the eye is moved towards the right hand, it will be 
 
30 
 
 noticed that the right hand margin of the object becomes 
 decidedly redder than before. This in itself exhibits the 
 imperfect nature of the colour corrections of the human eye. 
 An experienced observer may, however, turn this little fact to 
 account by acquiring the knack of so looking through his 
 telescope as to largely counteract small colour aberrations 
 due to atmospheric dispersion or imperfect centering. 
 
 If the lateral displacement of the eye were carried still 
 further so as to cause the emergent pencil of light to be roughly 
 bisected by the edge of the opaque iris, thus only allowing to 
 pass into the eye the pencil of light transmitted by about 
 half of the objective, then the secondary spectrum of the 
 objective would at once obtrude itself in an aggressive 
 manner, in the shape of yellow green^ fringes at one side 
 and purple fringes at the other side of objects. This is 
 especially noticeable in the case of refractors of relatively 
 short focal lengths. Altogether there is little doubt that 
 the clearness of vision through a telescope depends in no 
 small degree upon the way in which the observer applies 
 his eye to it. The observer who applies his eye eccentrically 
 and capriciously to a perfectly adjusted refractor may easily 
 delude himself into believing that -one of the lenses is out of 
 centre, whereas the observer who has acquired the proper 
 knack of looking can always get the best out of his instru- 
 ment. 
 
 Astigmatism. 
 
 The perfect fulfilment of the condition that all rays 
 striking the objective at equal distances from its centre shall 
 be equally refracted, is of 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 
 
3 1 
 
 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 then 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 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 practice be the point 
 actually focussed upon. As an example of the mischievous 
 effect of astigmatism, let an objective six inches aperture 
 and 90 inches focal length be taken, 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, 
 and the 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' 1 represents 
 the star image which would be seen if one of these focal lines 
 were focussed upon, its larger diameter being about three 
 
32 
 
 times the lesser. With the same amount of astigmatism the 
 least circle of confusion would show somewhat like a star 
 disc of about twice the proper size. Therefore such an 
 amount of astigmatism would render the instrument useless 
 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 therefore, 
 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 be not 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 as 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. Suppos- 
 ,ing 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 telescope, then the fault lies 
 in the latter. As an extra precaution, the eye-piece had 
 better be rotated to see whether the fault goes round with it 
 
33 
 
 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 superimposed and the astigmatic 
 effect is doubled. Or astigmatism 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 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. Therefore 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. 
 
 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 out- 
 
34 
 
 side 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 
 direction 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 that 
 is noticeable will be found to exist in the objective. 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 about to 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 astig- 
 matism. Many objectives are marked upon their edges, 
 indicating that they will perform at their best when corre- 
 sponding 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 dis- 
 appointed 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 
 
35 
 
 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 counter- 
 acted 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 (see page 50), after the objective has thoroughly 
 cooled down 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 
 sufficiently 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 complimentary 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 
 
 * See page 83 for maximum aberration permissible. 
 
36 
 
 appearance within focus in this case, and Fig. i$a the 
 complementary appearance to be seen outside focus. 
 
 On the other hand, if the central rings, when inside focus, 
 look about as luminous or even more so than the outer ring, 
 which is thin and weak (like Fig. 15^), 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 visibie. If there is residual aberration it may be 
 detected by the same symptoms as before, but they will be 
 less marked. For instance, Fig. 16 may represent the 
 appearance on one side of focus, and i6a 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 22^. As regards colour, it 
 is as well to remember that the diffusion of a purple 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 inside 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 
 purple 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 from 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. 
 
37 
 
 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 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 abetration, 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 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 such zonal aberration, which is caused by 
 imperfect figuring of one or more of the surfaces, it is best to 
 direct the telescope to a very bright star, using 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, 
 
38 
 
 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. iqa 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 2oa, which 
 represent the appearances to be seen inside of focus and out- 
 side of focus respectively. Another common fault is indicated 
 by the appearances shown in Figs. 21 and 21^, which 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 oj: 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 gradation of the 
 error from centre to edge, may or may not be due to imperfect 
 figuring. For, on the one hand, the radii of the spherical 
 curves may have been calculated and worked to with the 
 utmost nicety, with a view to 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. 
 
39 
 
 On the other hand, it is possible that all the surfaces may 
 be accurately figured and spherical, but, owing to a mis- 
 calculation 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 residues of 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 proportion 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 enhances the 
 brightness of and disguises the more central rings when out- 
 side focus. Figs. 22 and 220, represent the appearances, 
 inside and outside focus respectfully, 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. 
 
 Fig. 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 condi- 
 tions that any irregularity in the formations, of the rings will 
 be most easily detected. Fig. 23 scarcely does full justice to 
 that exquisite softness and regular gradation of the inter- 
 ference rings which should be visible when viewing a large 
 star (out of focus) through a perfect objective when th*e 
 atmosphere is clear and steady. 
 
4 o 
 
 As an observer gains experience in the use of an instrument, 
 and especially if he often directs his attention to the essential 
 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 
 eyesight 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 in and out of focus, 
 and examining the appearance of the interference rings, etc., 
 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 yield 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 examina- 
 tion 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 a 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 in- 
 strument, provided 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 say what 
 the fault is, or where it lies, without making an examination 
 of the out of Tocus appearances, unless the fault lies in 
 astigmatism, maladjustment, or want of true centering of the 
 lenses, for, on a steady night, these three faults may be 
 identified at the actual focus. 
 
 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 unnecessarily 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 light, 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. 
 
4 2 
 
 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. 25^) 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 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 perpendicular 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 
 
43 
 
 of a circle, it must appear a straight line in Fig. 25 (), 
 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 i 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 r b (Fig. 25), or the rectangle l-l (Fig. 250) 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 
 li-h t will exactly neutralize the light from h-h, also the 
 rectangle g-g t 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 t is considered, 
 d-b t will still exceed d-a i by one whole wave length (b,c), 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 is at /, 
 and there is a gradual falling off of brightness before d is 
 reached. The angular diameter of the interval / to rf, 
 subtended at the centre of the objective, is evidently equal to 
 ^|, or to ~^ h , for we supposed a-b to be equal to the 
 aperture, and i-f to be equal to the focal length. 
 
 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 
 
44 
 
 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 inter- 
 ference are very much modified, and the difficulty of finding 
 exactly the first point of interference d is immensely increased. 
 For now only a portion of the rectangular strip of wave 
 surface g-g is available for interfering with g-g t , and only a 
 portion of /-/ for interfering with i-i, although h-h t will almost 
 neutralise 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 exces of the area of g-i, g-h, i-h, 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 d-f must be increased. 
 
 Fig. 25 
 
 I K 
 
 g m 
 
 
 t 
 
 I 
 
 '' 
 
 
 + 
 
 ^ x 
 
 , 
 \ 
 
 * 
 
 y 
 
 
 
 \ 
 
 
 
 
 * 
 
 f / 
 / 
 
 
 
 
 
 
 
 ,' 
 
 
 
 
 
 ; 
 
 ^. 
 
 _- 
 
 
 
 
 
 1 
 
 1 
 
 L \ 
 
 1 1 f 
 
 ( ' 
 
 \ 
 
 \ 1 
 
 
 
 
 
 F 
 
 ic/. 
 
 25 
 
 a. 
 
 
 
 Fig. 25 b. 
 
 Sir George Airy, as the result of careful mathematical 
 investigation, 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 1-2197 wave lengths, or about 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 imaginary 
 
45 
 
 spherical wave surfaces is equal to one wave length will fall 
 at about h lt Fig. 25, or the thick line to the left in Fig. 256. 
 
 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, d is supposed to be 
 so placed that b-c 1-22 wave lengths, then, if the figure is 
 rotated about the axis i-f the point d will trace out a circle 
 about /, 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 from any wave surface 
 (a-b t ) again leaves a balance of luminosity. 
 
 The linear diameter of this first dark ring will then be 
 equal to 2FX ^' 22 *, 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 2XI A 22 \ 
 
 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 1-22 x, and if we take the wave length 
 of the most luminous light (citron green) as about ^-5-^0 inch, 
 then this amounts to -00081 inches, or T ^Vo^ n 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 off 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 ring. In all objectives 
 having their focal lengths equal to 15 times the aperture, 
 
4 6 
 
 then the linear diameter of the spurious disc may be said 
 to average -0004 inches or about ^Vo inch. With 6 inches 
 aperture this corresponds to an angular diameter of 0-9 
 seconds, and in a 12-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 rings 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 condi- 
 tions, 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 r22 V. it should therefore vary in inverse ratio to the 
 
 aperture 
 
 aperture. This is fully borne out by facts. 
 
 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 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 1-5 inches diameter. The mean of four 
 
47 
 
 measurements gave the diameter of the first dark ring (in 
 this case square in shape) as -0027 inch, while the formula 
 - (where * = T ^oir inch) gives -00266 as the theoretical 
 value. 
 
 A circular aperture, diameter 1-22 inches, was then placed 
 in front of the objective, when the mean of four measure- 
 ments gave a diameter of '0039 for the first dark ring, while 
 the formula - - gives 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 spectro- 
 scopic examination, this glass allowed to pass with freedom 
 only those rays lying between D and ", the maximum 
 transparency being rather nearer to E than to Z), and having 
 a wave length of about -j^^ 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 -0008 (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 telescope, 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 Encyclopedia 
 Metropolitana. 
 
 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 
 
4 8 
 
 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 ^V^ f an inch 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 very detrimental to 
 good definition, 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 : 
 
 ist, 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 
 
49 
 
 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 cheeks 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 bed itself anyhow and at random upon a single flange, but 
 the latter should be provided with three slightly raised faces 
 P, P and 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 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 upon 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. 24^. 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 equi- 
 distant fixed points for the lenses to rest upon and also for 
 
50 
 
 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. 
 
 (2) It is of the greatest importance to know that when an 
 objective is cooling down, as in the case of a telescope being 
 brought out of a house into the cold night air, the best 
 performance 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 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 tem- 
 perature, it tends to cool most rapidly upon the outside surface 
 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 deepened, 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. i$a, 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 large 
 
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 shows in good form. Besides, the effect 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. 
 
 Very often the penumbra, as seen inside focus, will show a 
 pronounced Q shape. That is it will have the shape of a 
 capital D with its back uppermost. This is caused by a 
 warmer stratum of air lying in the tube just behind the 
 object-glass. All these effects seem to be more noticeable in 
 the case of tubes which are only just large enough to pass all 
 the light from the objective ; a tube with plenty of room in it 
 certainly tends to better images. 
 
 (3) There are very few large glass 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 likely 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. 120. 
 Defects due to bad annealing 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 
 
52 
 
 objective, he would first examine them by polarized light, 
 and if not deemed satisfactory, either reject them or have 
 them re-annealed. 
 
 (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. I2&, would be 
 noticed, the projecting corners corresponding to the unsup- 
 ported parts of the crown. If, on the other hand, the crown 
 is imagined to 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 
 
53 
 
 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 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. i2b, and when in focus the spurious disc 
 will show somewhat like Fig. izb', or Fig. 18. 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 i2-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. 
 
54 
 
 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 perceptible alteration in the amount of the 
 deviation. This broad fact 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 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 distortions 
 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 outwards 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. If two tangents be 
 
55 
 
 drawn, one to each curve at the point where the ray passes it, 
 then we have the prism b-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 
 
 Fig. 26. 
 
 would then take up the position of the dotted lines, and the 
 prism b-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 
 i '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 b-a and a-c to rotate by 30 
 minutes of arc. This would represent an impossibly large 
 amount of distortion of the lens ; nevertheless, the only effect 
 upon the ray r-r-r would be to increase its deviation by only 
 one second of arc. 
 
 Fig26ci 
 
 And now let the same prism, refracting angle 9 -30', be 
 placed in the position shown in Fig. 260, so that the angle of 
 
56 
 
 incidence upon the first surface is only about a third of the 
 angle of emergence at the second surface. The prism then 
 corresponds 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 tangents b-a 
 and a-c, forming the prism b-a-c, to rotate by thirty minutes of 
 arc, as in the last case, then the deviation of the ray r-r will 
 be altered by no less than eighteen 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 dis- 
 tortion expressed by thirty minutes rotation, the more likely 
 amount of one minute had been supposed. It should here be 
 pointed out that since the star-disc is barely half - a - second 
 in diameter in the case of a 1 2-inch objective, therefore a 
 deviation of any rays from their true path (by distortion 
 of the objective) by any amount exceeding a quarter of a 
 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 be no reason for fearing the 
 ill effects of distortion, whether brought about by the great 
 weight of the lenses, 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, it only in the crown, for the flint is perhaps 
 
57 
 
 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 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 remem- 
 bered 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 practice. 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 distor- 
 tion is only about -g^o^ P art f ^ e e ff ect 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. 
 
 ,p, alteration of ray by refraction __ t s x ,, 
 
 alteration of ray by reflection TTJTo ^oTo 111 ' 
 
Terrestrial Telescopes. 
 
 In the case of refracting telescopes made for viewing 
 terrestrial objects, the quality 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 erect- 
 ing eye-piece. Therefore it is useless to expect good results 
 by testing such an objective with a Huyghenian or Ramsden 
 eye-piece, or good definition on the stars if high power 
 astronomical eye-pieces are applied to them. If the defini- 
 tion 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 sun- 
 shine, 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 
 either unusually large or viewed from too near a point. 
 
 Reflectors. 
 
 The reflecting telescope may be tested for quality in pre- 
 cisely the same manner as a refractor. The same spurious 
 disc should be seen at the focus, and the same systems of 
 rings be visible when the eye-piece is racked inside and 
 outside of the focus. If faults in the figuring or any 
 strains exist they should shew themselves in the same manner 
 as in the case of the refractor. But the difficulties of figuring 
 and perfect mounting are so great in the case of apertures 
 over 24 inches that it is donbtful whether any large reflector 
 in existence is capable of showing a distinct spurious disc 
 when directed to a star. 
 
59 
 
 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 Astro- 
 nomical 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 com- 
 parison 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 under- 
 stood by referring to Figs. 27 and 28, where o-o 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 at a 
 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 at a, provided the eye-piece 
 and eye are aplanatic. 
 
 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 
 
6o 
 
 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 o-o focus with accuracy at I it 
 is evident that sections taken through the cone of rays at 
 opposite sides of & and at equal distances from it will be simi- 
 lar in all respects excepting coloration, or, 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 scrutinize 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. If there is a slight amount of 
 positive spherical aberration then the section inside focus 
 (plate I., fig. 1 6) will show the luminosity growing more dense 
 towards the edge of the section than is the case with the 
 corresponding section taken outside focus (Fig. i6a), which 
 latter will appear softened off towards 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 of a star, will show variations or irregu- 
 larities 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 -01 inch short of the main focus, and 
 a cross-section be taken at '2 inches within the main focus, then the 
 
6i 
 
 In practice, however, the Huyghenian or Ramsden eye- 
 pieces used for such testing always possess a certain amount 
 of positive spherical aberration, which varies in proportion to 
 the square of the ratio between aperture and focal length of 
 the objective. Supposing the objective to refract all the rays 
 absolutely to one point, nevertheless the effect of the eye- 
 piece is to cause an appearance of positive spherical aberra- 
 tion ; the cross section of the cone of rays inside of the focus 
 will appear strong round its edge, while the cross section 
 outside of focus will appear softened off around its edge. 
 However, with ordinary objectives, having a focal length 
 equal to fifteen times the aperture, the spherical aberration 
 of eye-pieces of moderately high powers is only just per- 
 ceptible, and shows itself in a more marked degree with the 
 
 distance of the section within focus for the rays from the faulty zone 
 will be -19 inches, and the brightness of the section, where it cuts 
 through the zone, compared with the brightness of the rest, will be 
 represented by 2 2 = -" = ii Almost needless to say, such a zone. 
 
 19 2 
 
 distinguished by n / Q extra brightness, could not escape notice, especially 
 when its existence would be confirmed by a corresponding zone of 10 / 
 less 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 ,\ 2 where 
 
 r ' 
 
 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 over- 
 lapping 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 almost as un- 
 mistakably as they would do were the telescope directed to a star and a 
 critical examination made of the configuration of the interference rings. 
 
62 
 
 lowest powers. Therefore, a first-class objective must have 
 its spherical aberration very slightly over-corrected in order 
 to suit an average high-power eye-piece. 
 
 In Fig. 29 is shown a case of very strong aberration such 
 as would be yielded by a truly spherical mirror o-o 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 b, one-quarter of a-f from a, while c, one-quarter of a-f 
 from/, 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 rt, 
 then of course the edge 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 between 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 tocus 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 pjane a. Then the first rays to be stopped 
 will be those from about d t 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 
 
63 
 
 half disappeared, at the same time that the rays from about n 
 will escape the knife-edge altogether. Therefore the disc 
 
 B 
 
 will appear as in A 2. It should be borne in mind that 
 all these figures are in negative, the shading standing for 
 illuminated portions. 
 
6 4 
 
 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 o simultane- 
 ously, as in B 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 fths 
 of the full aperture. Rays from d will be cut off entirely, while 
 those from 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 o t , 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 simultane- 
 ously, leaving a half dark ring, while those coming from points 
 between o t and n, 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. 
 
 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 
 
65 
 
 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 fifteen times the aperture, we find 
 it possible to find the focus with no greater error than T o tn 
 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 parabolic 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 zorie, 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, 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 along the axis 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 ToVoth 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 Fig. 29, the faults showing them- 
 selves in the form of the irregular appearances shown in the 
 Figs. A , B and C, although perhaps in less pronounced forms. 
 
66 
 
 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 con- 
 siderable experience. The knife-edge test has its uses, but we 
 can confidently state, after comparing both methods together, 
 that for the purpose of detecting errors of workmanship, etc., 
 the direct focussing method has the decided advantage in 
 practice. 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 simultaneously, 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, because the objective may be absolutely free from 
 aberration in itself, whereas there should be sufficient 
 negative aberration to counteract the influence of the eye- 
 piece. The knife-edge test makes no allowance for the 
 requirements of the eye-piece. It is interesting to know that 
 the keenest of observers, the Rev. 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 
 focus with a certain eye-piece, the cardboard is removed, 
 a disc equal to the aperture in the latter is placed over the 
 centre of the objective, thus leaving exposed the 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 
 
6 7 
 
 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 corresponding vacuity 
 on the other side of focus, and such feature ougfit not to 
 escape the notice of the most casual observer. 
 
 We have emphasised the words more distinct just 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 microscopic, 
 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 tests 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, 
 
68 
 
 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. 
 
 The theory of interference of light would lead one to 
 expect that any cone of rays, converging to a point, must 
 be broken up into a series 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 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. Now, 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 something like a series of fine 
 Newton's* rings ; but there seems to be no marked indication 
 of such an effect in the case of ordinary double objectives, of 
 the larger sizes at any rate. But this may doubtless be 
 accounted for by the fact that only a limited range of the 
 spectral colours constituting white light is refracted to any- 
 thing like one and the same focus. It is significant that in 
 the case of our Photo-Visual Objective, whose achromatism 
 is practically perfect, the interference rings seen when at a 
 considerable distance either inside of or more especially 
 beyond focus, certainly show themselves fringed with colours 
 to a much more marked degree than is the case with 
 ordinary double objectives. 
 
 Whether the shells of light and darkness are all strictly 
 conical right up to the spurious disc, or whether a longitudinal 
 section through the axis and near the focus exhibits some sort 
 of curve, is an interesting question which will be found dealt 
 with on page 89 following. For, as may be gathered from 
 the appendix (pages 87 to 89), there is the strongest possible 
 
6 9 
 
 evidence of the cone of rays tapering off into a cylindrical 
 form as the focus is approached. 
 
 But, by way of 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 i : 29, and JL-th 
 of *2i = '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 '^- 7 -^ = '0013, or approximately 
 g^th of an inch. The intervals between the more interior 
 rings always grow successively smaller, but the quickest 
 gradation is from the outside one to the third or fourth. 
 
 The Cooke Photo- Visual Objective. 
 
 This objective, although composed of three lenses, may 
 yet be included in Class 2, before dealt with, since it is 
 perfectly free from coma. Thus, no amount of tilting within 
 reason will make it show any side flare, but only astigmatism 
 of the types illustrated in the Frontispiece, Figs. 12, d, d' t and 
 d" ' . So that the same method of squaring on as has been 
 recommended for Class 2 might be used in this case, were it 
 not that we have a very much simpler and better method to 
 recommend which has the advantage of being very delicate, 
 
 Fig. 30. 
 
 and yet can just as well be carried out without opening the 
 roof of the observatory. It is a method whereby the sixth, 
 or back surface of the objective can be made to face the 
 
7o 
 
 eye-piece accurately and squarely by a reflection test. Of 
 course, if the back surface is accurately squared on, then all 
 the other surfaces follow suit. We will explain the method 
 by means of the accompanying diagram, Fig. 30, where O 
 is the objective and E is the eye-end, with the eye-piece 
 taken out. 
 
 First, a piece of dark material V such as velvet or dark 
 brown paper is held up in front of the objective, and at some 
 distance in front of it. The object of this is to furnish a dull 
 background against which the eye, placed at the eye-end, 
 may see the aperture of the objective as a circular grey 
 patch of light. Then a piece of white card C with a hole H, 
 about one-fifth inch diameter is held diagonally behind the 
 eye-piece and in front of the eye, so that the eye may view 
 the objective through the hole. This card must be brightly 
 illuminated, when a reflected circular image of the eye-end 
 aperture, showing fairly white and with a black spot in 
 its centre corresponding to the hole, should be visible in 
 the back surface of the objective. This image will have 
 an apparent diameter equal generally to about one-eighth 
 part of the apparent size of the objective. It will not be 
 easily seen if the background V is too pale or too brightly 
 illuminated. Now, it the objective is correctly squared on, 
 then the bright circular image of the eye-end aperture should 
 appear exactly in the centre of the objective. If it is not 
 but is displaced towards the right hand, then the objective 
 must be tilted outwards on the same side by means of the 
 antagonistic screws of the counter-cell and vice versa. When 
 giving the final adjustments it is important that the observer 
 should keep the hole in the eye-screen exactly in alignment 
 with the axis of the tube. He can easily see whether he is 
 right in this respect by watching the reflected image, which 
 should show like a circular white patch enclosing a dark 
 spot which will appear exactly in its centre if the eye-hole is 
 upon the axis of the tube. Should one of these objectives 
 show any coma or side fiare under any circumstances, the 
 only conclusion is that one of the separate lenses has got 
 slightly misplaced by a shock or something of that sort, 
 and the observer should preferably return it to us tor re- 
 
adjustment, since no amount of tilting the objective as a whole 
 would ever cure it. It would indicate that one of the com- 
 ponent lenses had got slightly out of square with respect to 
 the other two. The lenses composing this objective require 
 more perfect centering and more careful adjustment than 
 those of ordinary double objectives, and therefore we recom- 
 mend even more careful handling if possible. It is so 
 important that the lenses shall be allowed no side play 
 whatever, that we have adopted a form of cell in which 
 the lenses are confined by a perfect fit at all temperatures, 
 but without nipping, by means of a simple and reliable zinc 
 and brass compensation, which has been tested carefully 
 between the range of 20 to 120 Fahrenheit with no difference 
 of fit manifesting itself. Compensated cells are used for all 
 sizes of 4 inches aperture and upwards, while in sizes below 
 4 inches aperture a spring is introduced which always gently 
 presses the three lenses up home against two out of three 
 equidistant cylindrical cheeks. The flint lens of this objective 
 is so shaped that flexure, supposing it to exist, can have 
 scarcely any optical effect, moreover, the borosilicate flint is 
 the hardest glass with which we are acquainted, and there- 
 fore very rigid. The two positive lenses are turned opposite 
 ways, and therefore any flexure due to their weight results in 
 contrary optical effects in the two cases. But here again the 
 substantial central thickness of these lenses adds very much 
 to their rigidity. Of course, the same optical tests of per- 
 formance as have been recommended for ordinary objectives 
 may just as appropriately.be applied to our Photo-Visual 
 Objectives, although the nature of the colour corrections is 
 quite altered. In the ordinary objectives the observer looks 
 for a ring-system, when inside or outside of focus, of a 
 greenish yellow colour fringed with red inside focus, while 
 outside focus he expects to see a bright blue flare and a 
 bright red point superimposed upon it. But in the case of 
 the Photo-Visual O.G. the system of rings will be found 
 generally of a pearly white colour on both sides of focus, 
 while the closest scrutiny will fail to show either the red 
 spot and blue flare outside focus, or the purple and blue 
 fringes of colour surrounding bright objects which he had 
 
almost learned to look upon as necessary evils in the case of 
 ordinary double objectives. It will be found that under 
 high powers the definition of fine lunar details, etc., will 
 show a clearer and more black and white appearance than 
 when viewed through ordinary objectives, while coloured 
 details are rendered with that fidelity well known to be one 
 of the great merits of the reflecting telescope. 
 
 Since the blue and violet rays are refracted to the same 
 focus as the most luminous rays, of course a perfect 
 photograph can always be taken by exposing the plate in 
 the visual focus, and it will be found that a circular plate 
 of at least the same size as the object glass will be very 
 well covered, any stars imprinted at the extreme margin 
 showing but slight diffusion and astigmatism. With the 
 usual focal length of eighteen times the aperture given to 
 these objectives, this corresponds to a circular field of at 
 least 3 10' diameter. 
 
 But it is of little use photographing planets, excepting in 
 the case of a very large instrument, without first enlarging 
 the primary image several diameters, by means of a Barlow 
 lens of some sort, for the direct image is so small and fine 
 that its details are inevitably obliterated by the relatively 
 coarse grain of the sensitive plate. 
 
 We have before pointed out that when ordinary objectives 
 are in a condition of cooling down, the image always shows 
 more or less under-corrected for spherical aberration, the 
 edge rays falling to the shortest focus. 
 
 The same transitory fault, but existing happily in a less 
 noticeable degree, will be found in the case of the Photo- 
 Visual Objective, and should accordingly be allowed for. 
 
 For observations upon faint objects, where light is of 
 the greatest importance, we would strongly recommend 
 our single achromatic eye-pieces. These are also specially 
 adapted for transit instruments, or for use with micrometers. 
 Consisting only of a single cemented combination, there are 
 two less surfaces to reflect back light, which make up for the 
 two extra surfaces in the objective. (This represents a gain 
 of about eight per cent, in light.) The field is rather smaller 
 than that given by a Ramsden eye-piece, but is very flat and 
 
73 ' 
 
 free from linear distortion, while the lens is placed at the safe 
 distance of its own focal length behind the delicate spider 
 lines. As, however, Huyghenian eye-pieces are generally 
 used with the Photo- Visual Objective, we generally make a 
 practice, unless otherwise required, of so adjusting the colour 
 correction of the objective as to yield a perfectly achromatic 
 image when a Huyghenian eye-piece, magnifying about fifty 
 times the aperture, is in use. This would represent a power 
 of 300 upon a 6" O.G. With lower powers than these the 
 objective necessarily appears a trace under-corrected owing 
 to the greater influence of the eye-piece, while with still 
 higher powers the same objective appears a mere trace 
 over-corrected, as it really is. And if a single achromatic 
 eye-piece is used with the same objective, it necessarily 
 follows that the over-correction for colour asserts itself still 
 more, as such an eye-piece in itself exerts no positive colour 
 aberration, like the Huyghenian or Ramsden constructions. 
 Nevertheless, these slight variations in colour correction, 
 dependent upon the type and the powers of the eye-pieces in 
 use, are too slight to appreciably impair the clearness of 
 vision, and would escape all but an experienced eye, and 
 thus may be regarded as of little importance compared to an 
 appreciable gain in brightness in the case of faint objects. 
 
 General Treatment of Objectives. 
 
 In conclusion, we will make a few remarks upon the care 
 of objectives in general. In a dry, pure air all objectives will 
 keep clean and beyond the need of internal wiping for a 
 considerable period, many years in many cases. All that is 
 necessary is to keep the outer surface clean by occasional 
 dusting with a fine brush, followed by a careful wiping with 
 a wash-leather or soft cambric, which must be free from all 
 trace of greasiness. Finger marks must on no account be 
 allowed to remain upon the surfaces, or curious cometary 
 appendages to stars will often result. The visibility- of 
 exceedingly minute stars depends very essentially upon 
 the perfect cleanliness and freedom from greasiness of the 
 surfaces. But in more humid climates the interior surfaces 
 
74 
 
 will occasionally require thoroughly cleaning. Many glasses 
 have the curious property of condensing a fine film of 
 moisture upon their surfaces when the air is humid ; they are 
 more or less hygroscopic. Ordinary crown glass, such as 
 is almost universally used for the positive lenses of double 
 objectives, has a perceptible tendency in this direction, while 
 the flint glasses, generally used for the negative lens, show 
 no such tendency. It is curious that flint glasses are very 
 much more easily cleaned that crown glasses; the dirt seems 
 to cling to the latter in many cases, when a preliminary 
 wipe with pure alcohol is strongly recommended. We have 
 often had ordinary objectives sent to us to be examined 
 and cleaned which have been many years either in use 
 or perhaps standing idle without internal cleaning, and 
 which have shown an extraordinary network of fine white 
 threads all over one of the internal surfaces. On taking the 
 lenses apart we have always found that this fine network was 
 upon the crown lens, and consisted of a minute thread-like 
 fungus which had more or less invaded the whole surface 
 and nourished itself mostly upon a fine film of sweat or 
 moisture which can be distinctly recognised upon the surface, 
 and also upon salts rendered available by a slight decomposi- 
 tion of the surface of the glass. For when this fungus has 
 been long established, it is impossible to wipe off all traces of 
 it, it leaves behind it a decided deterioration or etching of 
 the surface closely following the threads of the fungus. At 
 the same time the closest examination of the flint lens fails 
 to show any trace either of moisture or of fungoid growth. 
 All that can be noticed upon the flint is a film of tarnish. 
 Recognising then the indirect mischievous effect of damp 
 when allowed to accumulate on the interior surfaces, is it not 
 better to resort to prevention rather than cure? W T e are 
 afraid that the general practice is to leave objectives to take 
 care of themselves. For those, however, who are disposed 
 to take a little trouble to keep their objectives dry, we 
 would recommend the simple device which has been so 
 often applied to reflecting telescopes for absorbing the film of 
 dew deposited upon their mirrors after use on damp nights. 
 While using the instrument a thick circular pad of many 
 
75 
 
 thicknesses of flannel was kept toasting before a fire, and on 
 the cessation of observations this bone-dry pad was laid on 
 the top of the mirror and the cover of the tube put on. It 
 quickly absorbed all traces of moisture, and exactly the 
 same policy might be adopted with refractors, only in this 
 case the flannel pad should be introduced into the dew 
 cap and then the cover put on. It cannot be too well 
 remembered that where there is no moisture there is generally 
 no chemical re-action, and the presence of a film of moisture 
 is not only likely to interfere with the perfect and regular 
 refraction of the figured surfaces, but also to promote -and 
 foster fungoid growth and consequent decomposition of the 
 surfaces. Not only will a thoroughly toasted flannel pad 
 keep dry the front of the objective, but it will inevitably dry 
 the interior surfaces, provided the pad is used fairly often ; 
 for the air contained inside the cell and between the lenses 
 is continuously but slowly interchanging with the outer air 
 by means of diffusion, also more quickly by means of the 
 alternate expansions and contractions following upon every 
 change of temperature. Therefore, if the air outside an 
 object glass is kept dry, then the air inside the object glass 
 will sooner or later partake of the same dryness. And we 
 would recommend precisely the same treatment of our Photo- 
 Visual Objective, the back lens of which closely resembles 
 ordinary crown glass in all its properties. In the case where 
 an objective is not likely to be used for a long period, by far 
 the best way is to take it out of the telescope tube and place 
 it in a round tin box just large enough to hold it. This box 
 should have a closely fitting lid, and if a thoroughly dried 
 flannel pad is put in on top of the objective, so much the 
 better. 
 
 As regards the tarnish which we have above alluded to as 
 being noticeable upon the flint lens of an ordinary objective 
 after a few years of use, we are very glad to be able to 
 reassure the owner of such a flint that this film of tarnish, 
 generally looked upon with suspicion, is really a very good 
 friend to the observer, inasmuch as it increases the trans- 
 parency of his objective. We have seen so many proofs, 
 some quite unsought and some the result of intentional 
 
76 
 
 experiments, ot the unmistakably increased transparency of 
 tarnished surfaces as compared with freshly polished surfaces 
 as to remove all doubts on this point, so that we can con- 
 fidently assure observers that whereas a thin plate of dense 
 flint glass of the type usually used for objectives will, when 
 freshly polished, reflect back from its two surfaces about 
 ii per cent, of the light falling upon it, and transmit 89 per 
 cent., the same plate, when tarnished to a sort of dull grey 
 brown or blue (as viewed by reflection), will reflect back only 
 about 5 per cent., and transmit 95 per cent., a perceptible 
 gain of 5 to 6 per cent, in transparency. 
 
 If, however, the decomposition of a surface took the form, 
 not of a transparent film of tarnish, but of a dull milky film, 
 visible by scattered light, then of course the effect would be 
 entirely different, and would utterly spoil the lens. But we 
 have never yet come across a case of milky decomposition of 
 any glasses that have come under our direct inspection, 
 although we have heard of it taking place with certain 
 phosphate glasses in tropical countries. 
 
 In conclusion, should any of our Photo- Visual Objectives 
 show unmistakable signs of requiring cleaning on any in- 
 ternal surfaces, we would strongly recommend the owner 
 to forward the objective to us, since taking them apart and 
 putting them together again is not quite such a simple 
 operation as in the case of ordinary objectives, and requires 
 considerable care and experience. 
 
Supplementary Note to Third Edition. 
 
 In a little hand-book entitled " The Amateur's Telescope," published 
 in 1920, there is raised a discussion, on page 71, concerning the first 
 known application of the method of testing telescope objectives by auto- 
 collimation. We take this opportunity of making known the fact that we 
 introduced the method of auto-collimation into regular use in our work- 
 shops as far back as 1890, our first flat mirror being of 16 inches diameter 
 and unsilvered. A small metal bulb reflecting an electric arc lamp was 
 placed near the principal focus. This forms an artificial star, the light 
 from which radiates out through the objective, after passing through 
 which it forms a parallel beam which is reflected by the mirror back 
 again through the objective which brings the light to a focus close along- 
 side the original star, where it is scrutinized by the eye-piece. The test 
 is carried out in a dark passage quite independent of weather conditions, 
 and owing to the double passage of the light through the objective all 
 faults and errors in figuring are doubled in effect so that the test is of 
 twice the delicacy of the usual test on a real star, and has the additional 
 advantage of generally being carried out under superior atmospheric 
 conditions. All of our objectives of over 4 inches aperture, made since 
 1890, have been tested by this auto-collimation method. 
 
79 
 
 The Secondary Colour Aberrations of 
 
 the Refracting Telescope in 
 
 Relation to Vision. 
 
 By H. DENNIS TAYLOR. 
 
 Reprinted from tJie 
 
 MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY. 
 Vol. LIV., No. 2. 
 
 T T is notoriously easy to grow so accustomed to a supposed 
 necessary evil as to become practically oblivious of its 
 presence. The astronomer who diligently uses his refracting 
 telescope ceases to notice the ever-present halo of purplish 
 light which surrounds the objects of his scrutiny, and is 
 somewhat startled, or even offended, when some brother 
 astronomer, accustomed to use his reflecting telescope, takes 
 a look through the refractor, perhaps for the first time, at a 
 planet or a star, and recoils with an expression of surprise 
 at the unsightly amount of coloured fringe which he sees. 
 And, certainly, to one who sees it for the first time and has 
 always heard the refractor spoken of as achromatic, the 
 secondary spectrum seems far too aggressive a fault to be 
 ignored. 
 
 Much has already been written about the secondary 
 spectrum of the refracting telescope, but, I venture to think, 
 to relatively little practical purpose. 
 
 The theory of vision through the refractor has, so far as I 
 know, not yet received adequate attention. At the same 
 time, the production of an objective practically as achromatic 
 as the reflector has in many quarters been looked upon as a 
 mere refinement, because the very serious nature of the 
 secondary colour aberrations of ordinary objectives, especially 
 when their apertures are measured by feet, has not been 
 sufficiently realised. I will now proceed to give what seems 
 to me sufficient evidence of the detriment and loss of utility 
 suffered by the larger refractors owing to the imperfect 
 nature of their achromatic corrections. 
 
8o 
 
 Plate II. illustrates the secondary colour aberrations of the 
 refracting telescope in the form of graphic curves. 
 
 First of all, in Fig. i, is shown the colour curve of the 
 great Lick refractor as plotted out from the actual measure- 
 ments of the foci for the various colours, executed with great 
 care by Mr. James Keeler by means of a large spectroscope."" 
 
 The great objective is so corrected that the two rays, B 
 and F come to exactly the same focus, which is taken as the 
 zero in the following table, where the colour aberrations are 
 preceded by a + sign when the respective colours focus 
 beyond the focus for B and F, and by a - sign when they 
 
 focus short. 
 
 TABLE I. 
 
 Lick Refractor. Focal Length = 57 feet. 
 
 Distance of foci from 
 
 Wave length. Fraunhofer line. focus for B and F. 
 
 Inches. 
 
 6867 B o'oo 
 
 6562 C 0-24 
 
 5889 D 0-45 
 
 About '5650 Minimum focus 0-47 
 
 4861 F ooo 
 
 4341 G'orHy +i'45 
 
 4101 h +276 
 
 Visual focus o 20 
 
 In Fig. i, as in all the other colour curves which I give, 
 the wave lengths are set off in arithmetical order along the 
 perpendicular on the left hand : that is, the wave lengths 
 (expressed in thousandths of a millimetre) are measured by 
 the ordinates. A scale, divided to inches and tenths, is set 
 off along the base line ; that is, the variations in focal length 
 for the rays of various wave lengths are measured by the 
 abscissae. 
 
 Now, it seems to be a fact that large refractors are rather 
 over-corrected compared to smaller ones. Otherwise they 
 would show an aggressive amount of secondary red. We want 
 a colour-curve more truly representative of medium-sized 
 refractors of average colour correction. Smaller objectives 
 are generally calculated to refract the rays C and F to the 
 
 ^Publications 0} the Astronomical Society of the Pacific, Nov. 9, 1890. 
 
8i 
 
 same focus, but in practice I have good reasons for believing 
 that the average state of correction may be said to be one in 
 which the ray half-way between B and C is united with F. 
 The best state of correction is largely a matter of taste or 
 individual eyesight ; nevertheless, I have assumed the latter 
 state of correction as the best to form the basis of a series of 
 calculations, whereby the colour-curve shown in Fig. 2 has 
 been arrived at. The Lick measurements are too few and 
 too far apart to enable one to draw in the colour-curve 
 with a sufficient degree of reliability from one end of the 
 spectrum to the other. Therefore I calculated out the colour- 
 curve (Fig. 2) for a 2-foot object-glass of 30 feet focal length. 
 The data on which I have proceeded are as follows : 
 
 1. The very careful series of measurements of Refractive 
 Indices for Ordinary Crown Glass and Ordinary Dense Flint 
 Glass, executed by Dr. Hopkinson, and published in the 
 Proceedings of the Royal Society, 1877 (" Refractive Indices of 
 Glass"). These are perhaps the most accurate and reliable 
 measurements of this sort ever executed. 
 
 2. It is well known that the refractive index for any ray of 
 known wave length can be expressed in terms of that wave 
 length with reasonable accuracy by means of Cauchy's 
 formula 
 
 where /x, = the refractive index, A = the wave length, and 
 a, b, c, and d t etc., are constants depending upon the 
 physical properties of the particular glass in question. As 
 Dr. Hopkinson points out, if we wish to calculate //, from x 
 with as great a degree of accuracy as may be obtained by 
 actual observation, then it is necessary to work by the 
 formula extended to four terms, as shown above. This I 
 have done first working out the values of a, b, c, and d for 
 crown glass and flint glass respectively from the observed 
 refractive indices for the four spectral lines as follows : 
 
 TABLE II. 
 
 Ray. Wave length A. Ray. Wave length X. 
 
 B "6867 F -4861 
 
 D 2 -5889 H! -3968 
 
82 
 
 Next, I calculated the necessary values of for the crown 
 lens and for the flint lens, which are necessary for refracting 
 
 the ray half-way between B and C (wave length = -6715) to 
 the same focus as the ray F (A = -4861) at a focal distance of 
 
 30 feet. Here, as usual, stands for the sum of the recipro- 
 
 PI 
 
 cals of the radii of the crown lens, and for the sum of the 
 
 Pz 
 
 reciprocals of the radii of the flint lens. Then, if F represents 
 the principal focal length of the combination, we have 
 
 Let 
 
 Pi Pz Pi Pz 
 
 c-A^Md-D.i-i 
 
 Then 
 
 ^=A + B.i + ci + D;i ....... (2) 
 
 A 2 X. t X G 
 
 or 
 
 which is the formula expressing the reciprocal of the focal 
 
 length for any ray in terms of the wave length of that ray. 
 
 Differentiating (2) with respect to A, it will be found that 
 
 *(l?) ^- 2 *- 3 ( B + 2C *- 2 +3Dx- 4 )rfA ..... (3) 
 On equating (B + 2O- 2 + 3DA- 4 ) to o, it will be found that 
 
 That is, the ray whose wave length will satisfy equation (4) 
 will be refracted to the minimum focal length. (The sign of 
 the root in (4) has been assessed from actual trial.*) 
 
 * It is interesting to note that the use of Cauchy's dispersion formula 
 carried to four terms (as above) only gives rise to errors in the value of ^ 
 not exceeding and generally less than -ooooio ; which error, in the case 
 of the crown lens, would give rise to an error of only -018 inch in a focal 
 length of 360 inches, and an error of '009 inch for the same error in p in 
 
I will here give Dr. Hopkinson's Refractive Indices for the 
 
 Crown and Flint glasses respectively for the four selected 
 rays or Fraunhofer lines : 
 
 TABLE III. 
 
 Ray. Crown. Flint. 
 
 B 1-513624 1-615704 
 
 D 2 1-517116 1-622411 
 
 F 1-523145 1634748 
 
 H! 1532789 1-656229 
 
 From these the following constants were obtained : 
 
 TABLE IV. 
 
 Constant. Crown. Constant. Flint. 
 
 a 1 = +1-5031566 a z =+1-5973804 
 
 b l = + -0053061 b z + -0086972 
 
 c^ = - -00021147 c z - -000084923 
 
 d l = + -00001742 d z + -000027465 
 
 For a focal length of 360 inches for the F ray, it will be 
 found that 
 
 Pi 74H292 Pz 149-318 
 
 on the supposition that the ray half way between B and C 
 (A -6715) is refracted to the same focus as F. 
 
 From these data the objective constants, A, B, C, and D 
 see equation (2) are found to be as follows: 
 
 TABLE V. 
 
 A = + -00275948 
 B = + -000013045 
 C - -0000022725 
 D = + "000000050085 
 
 From the above data the colour curve shown in Fig. 2 was 
 derived. The focal lengths of the rays of the spectrum A, B, 
 C, D 2 , E, b, F, G, h, and H 15 as well as six other rays of 
 intermediate wave lengths (where spaces seemed to require 
 it), were calculated and gave the abscissae of the curve. It 
 
 the case of the flint lens. An error of -ooooio in observing p. from a crown 
 prism of 60 angle would imply an error in reading with the spectrometer 
 equal to about three seconds of arc only. Therefore the errors likely to 
 arise from the above application of Cauchy's formula are no greater 
 than the errors likely to arise in taking observations of ^ with the greatest 
 care by means of the best of spectrometers, 
 
8 4 
 
 will be noticed that its character is much the same as the 
 actually measured curve of the Lick refractor, if we allow 
 for the great difference of focal length. Some years ago 
 Professor Piazzi Smyth was good enough to send to Messrs. 
 Cooke a series of measurements of the focal lengths of rays 
 of various colours executed with his great spectroscope, 
 which consisted of a large diffraction grating, a collimating 
 telescope of 4-in. aperture and 36-in. focal length, and an 
 observing telescope of 4-in. aperture and 68-in. focal length. 
 Allowing for there being two objectives in use, one being 
 nearly twice as powerful as the other, and the nature of the 
 colour correction, which brought the D and the G rays to 
 the same focus, the colour aberrations as measured by him 
 (if we allow for the difficulty of attaining great exactness) 
 were found to be on a scale proportionate to the focal powers 
 concerned. The colour aberrations of the 27-in. Vienna 
 telescope have also been measured by Professor Vogel and 
 found to be substantially of the same character as those of 
 the Lick refractor, only less in amount in proportion to the 
 smaller focal length. So far, then, calculation and actual 
 observation are substanti-ally in agreement. 
 
 We have now to consider an aspect of the question which 
 is of very great practical importance, and also extremely 
 interesting. Large refractors are looked upon as valuable 
 because (i) of their supposed great light-grasping or space- 
 penetrating power, and (2) because of their greater separating 
 or dividing power, which enables them to deal easily with 
 very difficult double, binary, and multiple star systems 
 which are totally inaccessible to smaller instruments. Their 
 greater separating or dividing power upon close double stars, 
 etc., is of course fully proved, and is well known to increase 
 directly as the aperture ; it is chiefly the light-grasping power 
 of refractors with which I intend to deal. 
 
 First of all, I will point out what seems to me to be a 
 most inexplicable fact (?). It has been asserted with regard 
 to the Lick refractor (in Table I.) that the eye does not select 
 a focus at that point of the optic axis where the greatest 
 concentration of light occurs, viz., at the minimum focus. 
 That the eye is able to select any distinct focus whatever 
 
85 
 
 but of all the innumerable various coloured foci which are 
 continuously distributed over several inches of length in the 
 case of very large refractors is in itself a surprising fact. 
 But there is found to be no ambiguity in practice about the 
 exact whereabouts of the visual focus of large refractors. It 
 is asserted by Mr. Keeler that the eye places the visual focal 
 plane at a distance of -27 inch, or about a quarter of an inch 
 behind the position of minimum focus. 
 
 In Fig. i, f l - -f l is the visual focal plane as placed by 
 Mr. Keeler. It is seen to cut the colour curve in two places 
 in the red near C, and in the bluish green beween E and F. 
 These two colours, being about complementar}^, together 
 would form an apparently white focus. In fact the quality of 
 whiteness and not luminosity would seem to be the eye's 
 main requirement, if this visual focal plane is rightly placed. 
 Now, when an optician designs an object glass for visual 
 purposes, he reasons thus : This objective has to yield an 
 image which shall appeal as much as possible to our sense of 
 vision ; therefore it is of the utmost importance that those 
 rays of the spectrum which exert the greatest effect upon 
 vision should be refracted to the most compact focus possible; 
 which amounts to this, that the visually most active rays 
 must be refracted to the minimum focal length, so that the 
 brightest part of the diagrammatic colour curve may be 
 embraced tangentially by the focal plane." He pursues this 
 sound line of reasoning, embodies it in practice, and gets the 
 brightest yellow-green rays condensed to the most compact 
 focus. And then (if Mr. Keeler's estimate of the position of 
 the visual focal plane is the true one), when he comes to 
 actually use. the telescope, his eye most persistently ignores 
 the very focus which has been specially designed to suit it, 
 and selects a focus of its own, far inferior in brightness, at 
 quite a different position on the axis. 
 
 This seems the very height of illogical perversity. As an 
 example of discordance between design and use it compares 
 very well with the well-known case of the planter who 
 
 * Of course the focal plane is here supposed to have a substantial 
 thickness, as will be explained further on. 
 
86 
 
 supplied his black labourers with wheelbarrows to lighten 
 their toil, and found them next day carrying them fully 
 loaded, but on their heads ! 
 
 Professor Harkness, in an article on the Secondary Spec- 
 trum in the " Amer. Journal of Science and Arts," 1879, 
 September, puts forward the following ingenious method of 
 finding the position of the visual focal plane when the amount 
 and character of the secondary spectrum are known. 
 
 In Fig. 9 is shown the same colour curve as that of Fig. 2 
 (for a 30-foot refractor). Across this colour curve there are 
 drawn horizontally, so as to be bisected by the colour curve, 
 a series of short lines whose length is made proportional to 
 the visual brightness of the light at each point of the colour 
 scale. After the ends of these lines have been connected we 
 have the boomerang-shaped figure shown in Fig. 9. (The 
 figure as drawn does not pretend to any accuracy). Let 
 this diagram be drawn on stiff paper and then cut out and 
 balanced upon a knife-edge (which, of course, must corre- 
 spond to an ordinate). Professor Harkness maintains that 
 the position of the knife-edge necessary for a balance should 
 indicate the position of the visual focal plane. 
 
 Now, however applicable this solution of the problem may 
 be in the case of small objectives where the secondary colour 
 aberrations are not too great to be embraced between the 
 limits of the focal plane, yet it is wholly inapplicable to the 
 case of large objectives. For the image of a star is a spurious 
 disc, of a well-ascertained diameter, and what we are really 
 concerned with is the quantity or luminous value of the light 
 which actually goes to the formation of that star disc. For, 
 as I shall presently show, the light of certain other vagabond 
 colours focusses at points so far beyond (if not short of) 
 the visual focal plane as to form penumbra or circles of 
 aberration on the latter, which are so very much larger than 
 the star disc that the light forming these said penumbra is 
 diluted tens of times, hundreds of times, and even thousands 
 of times, according to the colour. Therefore, such light 
 certainly does not sensibly contribute to the brightness of the 
 star disc, and Professor Hackness's method is no longer 
 applicable, unless applied only to that brightest part of the 
 
87 
 
 curve which can be comprised between the limits of the focal 
 layer; a term which I shall now proceed to explain. 
 
 It is of the very greatest importance in this enquiry to 
 know what are the limits between which the image of a point 
 of light or a star may be said to be approximately in focus in 
 the case of well-figured objectives. Fig. 5 embodies the 
 generally, and I believe universally, accepted idea of the 
 longitudinal section of the cone of rays from an objective or 
 mirror converging to form a spurious disc at D, and then 
 diverging again in a similar manner on the other side. It is 
 supposed as a deduction from geometrical optics that the 
 figure is that of two similar cones on a common axis, with 
 
 their points overlapping by a distance equal to 2 -r times the 
 
 diameter D of the spurious disc. Here F represents the focal 
 length, and A the aperture of the O.G. In Fig. 5 I have 
 represented the average case of the focal length being fifteen 
 times the aperture. Let it be supposed that the star disc is 
 viewed through an eye-piece powerful enough to magnify the 
 spurious disc to a very appreciable size, as in the case of 
 ordinary double-star work. Then, if Fig. 5 is true to fact, it 
 is evident that if the eye-piece is pushed within or beyond 
 focus by an amount equal to the diameter of the spurious disc 
 
 F F 
 
 multiplied by-r-, or shortly D, then it will focus upon a 
 A A 
 
 section of the cone of rays whose diameter d is equal to 2 D, 
 
 F 
 and if pushed in or out by the amount 2 D , then it will 
 
 focus upon a section of the cone of rays where d ~ 3 D, and 
 
 so oru 
 
 p 
 Now, in the case of any objective where -^ = 15, it is well 
 
 known that the spurious disc is about -0004 inch in diameter 
 (or rather less). Therefore, according to Fig. 5, if the eye- 
 piece were pushed in or out of focus by a distance equal to 
 0004 x 15, or -006 inch, then the appearance should be a 
 penumbra of light equal in diameter to twice that of the 
 spurious disc. NOW it happened to occur to me that I would 
 determine the limits of good focus not only theoretically, by 
 application of the usually accepted rule, but I would also 
 
88 
 
 confirm it by experiment. What, then, was my surprise when 
 I found that, so far from there being any agreement between 
 the tacitly accepted theory and actual fact, I actually had to 
 push in the eye-piece or draw it out in order to expand the 
 spurious disc into a penumbra of twice its size (see Fig. 50 
 and b), by an amount equal to -03 inch on each side of focus, 
 or five times as much as I had theoretically expected, while 
 I could push the eye-piece in or out by '015, or nearly -02 
 inch, without sensibly increasing the size or destroying the 
 character of the spurious disc. This astonished me, and led 
 me to make a series of experiments with different eye-pieces, 
 apertures and focal lengths, and varying relations between 
 apertures and focal lengths. The results turned out to be 
 independent of the focal lengths of the objective and the 
 power of the eye- piece, provided the magnifying power was 
 sufficient to show the star disc. Focal lengths of 9 inches (in 
 which case the secondary spectrum is practically nothing) up 
 
 to ii feet were tried, with the results shown in Table VI. 
 
 p 
 The first column gives the relation -j- ; the second gives the 
 
 whole distance of travel of the eye-piece between the point 
 where d 2D on one side of focus and the point where rf 2D 
 
 on the opposite side of focus. Of course D increases as -^ 
 and theoretically the taper or angle of the cone of rays also 
 
 varies inversely as -r, so the theoretical distances of travel, 
 J\ 
 
 including that on both sides of focus (for producing the 
 condition d=2D) given in the third column, are calculated 
 
 as varying as \ij~J The fourth column gives the ratio of 
 actual to theoretical distances. 
 
 TABLE VI. 
 ^ Actual. Theoretical. "Ratios. 
 
 8-4 -02 less -004 5 + 
 
 15 -06 -012 5 
 
 30 -18 -048 375 
 
 60 '54 about "192 2-8 
 
 These measurements do not admit of great accuracy ; but 
 so far as they go they clearly prove that the taper of the 
 cone of rays becomes much more gradual as the focus is 
 
89 
 
 approached, or as the diameter of the cone grows small in 
 comparison with the wave length. 
 
 Any observer can verify these results for himself on his 
 own telescope. The discrepancy between calculation and 
 fact is far too great to be explained away, and the conclusion 
 is that Fig. 6 pretty accurately represents the longitudinal 
 
 section of the cone of rays near the focus when ^ = 15. 
 
 Figs, ja and jb show the differences between the theoretical 
 and actual cones for a large relative aperture. I can scarcely 
 suppose that I am the only observer who has noted this 
 curious law, but I am not aware that the modification of the 
 taper of a cone of rays near its focal point has ever been 
 studied as a deduction from the undulatory theory of light. * 
 But here we have a fact which renders possible what before 
 seemed impossible, viz., the production of a tolerably definite 
 focus out of the confusion of many foci ; for since the bundle 
 
 of rays is nearly cylindrical for a distance of nearly '02 on 
 
 F 
 each side of focus (when -r- = 15), evidently colour aberra- 
 
 r\. 
 
 tions to that amount on each side of focus will not give rise to 
 loss of light owing to any part of it falling outside the limits 
 of the star disc. In all the colour curves shown in Figs, i, 2, 
 3, and 4 I have drawn what I call the focal layer,/- -/, of a 
 width in accordance with the above experiments. The focal 
 plane may be indicated by a line drawn down the centre of 
 the focal layer, while the outer dotted lines mark the limits of 
 the focal layer. Supposing an eye-piece is focussed upon the 
 focal plane, then the two rays of other wave length which 
 
 * It can be shown that no amount of residual spherical aberration can 
 possibly account for this apparent drawing out of the cone of rays. In 
 order to guard myself from being misled by the well-known variability of 
 the focus of the eye of an observer, I made use of very high power eye- 
 pieces for the greater number of the above experiments. The most 
 powerful eye-piece used had an equivalent focal length of -15 inch, and in 
 this case the utmost error in reading that could arise from the extremes 
 of focal adjustment likely to take place involuntarily in my eye would not 
 amount to more than -g^jth of an inch. For the reasons indicated above, 
 the use of eye-pieces of equivalent focal lengths greater than about -inch 
 would be likely to give rise to very large absolute errors in the readings. 
 
go 
 
 focus respectively upon the inner limit and the outer limit of 
 of the focal layer will produce upon the focal plane two 
 penumbra of light whose diameters are equal to twice the 
 diameter of the focussed disc (see Figs, ja and yb) ; that is, 
 about one-quarter* of the light which focusses upon the two 
 limits of the focal layer will fall within, and contribute to, the 
 brightness of the spurious disc. Any light focussing sensibly 
 beyond the outer limit or sensibly within the inner limit of 
 the focal layer will contribute a still less fraction of its 
 illuminating power to the spurious disc or star image. 
 
 On careful consideration it will be conceded, I think, that 
 the thickness of the focal layer, when denned as that thick- 
 ness necessary for the condition that a ray focussing upon one 
 of the limiting planes shall produce a penumbra upon the 
 focal plane of a diameter equal to twice that of the star disc, 
 errs if anything on the side of liberality ; for the integrated 
 values of the very small fractions of the light contributed to 
 the star disc from the colours (if any) focussing at various 
 distances outside the limits of the focal layer can scarcely be 
 supposed to balance the loss (rising upwards to 75 per cent.) 
 of light incurred by the star disc as regards those colours 
 which focus at points just inside the limits of the focal layer, 
 and yet just outside the limits of sharp focus, or between 
 the outer dotted and the inner solid lines in the diagrams. 
 Hence the luminous value of the light of that range of wave 
 length A* which is embraced between the limits of the focal 
 layer may be taken as a fair and very liberal measure of 
 the light-grasping power of the objective when used for star 
 work. 
 
 Although the observations recorded in Table VI. do not 
 
 F 
 admit of great exactness, especially when -^- becomes great, 
 
 yet they seemed to me reliable enough to prove that the 
 
 / F\ 2 
 thickness of the visual focal layer does not vary as \~jr-) > but 
 
 (p\i6 P 
 
 -JT ) , since doubling the ratio ~j- was found 
 
 * This fraction should really be considerably less than one-quarter, 
 since it is a well-known fact that the greater portion of light in the out of 
 focus penumbra is concentrated in the outer annulus. 
 
to be about treble the thickness of the focal layer (or rather 
 more than that). Turning again to Fig. i, it will be seen 
 that I have indicated one focal layer / x --/ x just where the 
 published measurements of the Lick refractor state it to be 
 situated, its centre or the focal plane being at -27 inch behind 
 the minimum focus. Now the curve L--L--L in Fig. i 
 (as in Figs. 2, 3 and 4) is the curve of luminous intensity for 
 the normal solar spectrum, the wave length scale being 
 exactly the same as that at the left-hand end of each diagram. 
 For this luminosity curve I am indebted to the very valuable 
 researches and experiments of Captain Abney and Major- 
 General Festing, as published in the Bakerian Lecture for 
 1886.* 
 
 Since Fraunhofer's first attempt to measure the relative 
 luminous values of the various colours of the solar spectrum, 
 several systematic photometric investigations with the same 
 object have been entered upon, especially those of Vierordt 
 in 1869; but I venture to think there are no photometric 
 measurements of the relative luminous values of the coloured 
 constituents of white light which can be more safely relied 
 upon than those embodied in the luminosity curves L - L - L 
 here presented. Captain Abney and Major-General Festing 
 found their own eyes to be very fairly typical of the average 
 results obtained by them wken testing the colour-sensitive- 
 ness of other persons experimented upon. 
 
 They also well established the law that the visual photo- 
 metric value of any mixture of colours is always equal to the 
 simple sum of the separate visual photometric values of the 
 separate coloured constituents, whether the constituents are 
 complementary or not, and this supplies additional reason for 
 doubting the existence of the supposed visual focal layer 
 /!- -/! (Fig. i) in the case of the Lick refractor. 
 
 I should remark that Captain Abney's luminosity curve is 
 the curve for the prismatic spectrum, and gives the maximum 
 intensity about wave length -576 (citron-yellow). On con- 
 verting Captain Abney's curve into the curve for the normal 
 spectrum with great care, I obtained the curve L - L - L with 
 
 * " Colour Photometry," Phil. Trans. Part ii., 1886. 
 
92 
 
 the maximum intensity transferred to wave length '565, citron- 
 green in colour. It will be noticed that the Lick refractor is 
 very exactly corrected to refract this most luminous colour to 
 minimum focus. 
 
 Now in Fig. i the outer limits of the focal layer / x - -f t 
 (placed in accordance with Mr. James Keeler's estimate of 
 the position of the visual focal plane) cut the colour curve at 
 two points near C in the red. and at two points near F in the 
 greenish blue. If four horizontal lines are drawn through 
 these points right across the luminosity curve L - L -L, then 
 we get the two approximately rectangular strips / - - / and 
 / 1 - -/ 15 and obviously the amount of light embraced by the 
 focal layer will bear the same ratio to the whole light passed 
 by the objective as the ratio which the combined area of the 
 two strips / - - / and / t - - / x bears to the whole area of the 
 luminosity curve L-L-L. Hence it can be seen at a glance 
 that if the visual focal layer is really placed by the eye at 
 /!--/!, then the amount of light concentrated in the star 
 image must be only about one-tenth part, at most, of the 
 whole light transmitted by the objective ! This is far too 
 improbable to be true, and therefore, as well as for other 
 reasons already partly stated, I feel very sceptical about the 
 position of the visual focal plane of the Lick refractor being 
 at /,_ - -f 1 , as it has been stated to be. Let it be supposed, on 
 the other hand, that the focal layer is really situated at/- -/, 
 nearly tangential to the colour curve. Then, on drawing 
 horizontal lines from the two points where the colour curve 
 is intersected by the outer limit of the focal layer, across the 
 luminosity curve, we get a strip which, besides being very 
 much broader, also includes the very brightest part of the 
 spectrum ; and it can be seen from a glance at the diagram 
 that the luminous value of the light concentrated into the 
 star image will be at least one-half of the value of the whole 
 light transmitted. Thus theory most certainly demands that 
 the visual focal plane shall be so placed as to be nearly 
 tangential to the bend of the colour curve. 
 
 It so happens that ordinary bright green glass, if of suf- 
 ficient thickness, only transmits green light, whose maximum 
 intensity is about wave length 5400, very close to the point of 
 
93 
 
 minimum focal length for ordinary objectives. Any observer 
 possessing a large refractor of at least 8 inches aperture will 
 find, as I have done, that if the focus is most carefully 
 adjusted with a high power on a white star, and then the 
 image is viewed through the green glass and a careful 
 readjustment of focus made, that the eye places the focal 
 plane for the unimpeded white light at about '02 inch only 
 behind the green or minimum focus. I have also tried this 
 method in the following case. 
 
 A i2*5-inch objective of 15 feet 5 inches focal length was 
 arranged to be its own collimator; that is, it was placed close 
 to, and with its optic axis perpendicular to, a plane optically- 
 worked mirror. A small speculum metal bulb was placed in 
 the eye end, just a little to one side of the principal focal 
 point, which bulb produced an artificial star by reflecting the 
 light from a very small electric lamp placed at one side. The 
 light from this star diverged to the objective, passed out 
 parallel, was reflected back by the mirror, and again con- 
 verged by the objective towards a focus situated in the eye 
 end close to the original star. This image, formed after two 
 passages through the objective, could then be examined by 
 ordinary eye-pieces. Now it scarcely needs explaining that, 
 provided the speculum bulb is kept at an unvarying distance 
 from the objective, then the colour aberrations (as well as 
 any other faults in the objective) will be just double the 
 colour aberrations of the same objective used in the ordinary 
 way. So that by this method I practically had to deal with 
 the secondary spectrum of a 25-inch objective of 30 feet 10 
 inches focal length, and its colour curve would be the same 
 practically as that of Fig. 2. 
 
 A knife-edge placed parallel to the focal plane, but worked 
 backwards and forwards towards or from the objective by a 
 micrometer screw of 100 threads per inch, was used for 
 finding the exact positions of foci, which could be done with 
 no greater liability to error than about '007 inch if the mean 
 of several trials was taken. By this Foucault method I found 
 that the focal plane, when the green glass was used, was 
 about '02 inch within the focal plane selected by the eye 
 when the white image was viewed without any coloured 
 
94 
 
 glass. About the same result was obtained when using 
 the ordinary eye-piece method with and without the green 
 glass. 
 
 These results seem to agree very well with what theory 
 would indicate, and therefore I feel justified in placing the 
 focal planes where I have done in all the diagrams. I have 
 placed them a trifle (-01) too much towards the right in Figs. 
 2, 3, and 4, but I have compensated for this by drawing the 
 dotted lines going across to the luminosity curves from the 
 proper points on the colour curves. 
 
 Sir George Stokes, some years ago, devised a very simple 
 and yet efficacious method of showing up in rather a sur- 
 prising way the presence of the secondary spectrum in any 
 objective. Fig. 8 serves to explain the principle of this test. 
 An objective, a a, is focussed very carefully upon a vertical 
 white line drawn on a black ground. At the focus a small 
 circle is shown, which serves as a cross -section, as it were, of 
 the focussed line of light, which is, of course, supposed to be 
 perpendicular to the plane of the paper. An opaque screen 
 with its edge straight (and perpendicular to the paper) is now 
 placed in front of the objective at 5 s so as to practically 
 reduce its aperture to a semi-circle. A little consideration 
 will now show that any edge-rays focussing short of the 
 previously-found focal plane will now strike the focal plane 
 on the right-hand side, r, of the image, and that any edge- 
 rays focussing beyond the same plane will strike the latter at 
 the left-hand side, /, of the image. 
 
 It is found, in fact, that the right side, r, is conspicuously 
 margined with yellow or citron-green, and the left side, /, 
 with reddish purple shading off into blue. Doubtless, the 
 reason why this stray light is not noticed when the full 
 aperture is in use is because in that case we get a combination 
 of yellow-green and purple on each side of the image, and 
 these colours, being roughly complementary (excepting per- 
 haps in the case of very large objectives), form a greyish 
 halo on each side, which naturally does not obtrude itself 
 upon the attention so forcibly as the unsymmetrical com- 
 bination of a citron:green halo on one side contrasted with 
 a purplish and blue halo on the other. 
 
95 
 
 Even if this test is applied to a 2-inch objective, these 
 secondary colours may be easily detected. A black line seen 
 against a white ground, such as the rod of a distant weather- 
 cock, will answer equally well as a test object. 
 
 The above appearances, as exhibited by achromatics with 
 the average degree of colour correction, clearly prove that the 
 focal plane is not quite at the minimum focus, but are quite 
 consistent with the position of the focal plane as I have 
 shown it in the diagrams, where the inner limit of the focal 
 layer is placed about tangential. If the visual focal plane 
 were placed still further back, we should expect to see, under 
 Sir George Stokes's test, a much greater amount of secondary 
 green (as well as orange and bluish-green) in the case of 
 large objectives than is actually seen. But the application 
 of this test to an 8-inch objective did not seem to me to yield 
 much more secondary green than the same test applied to a 
 3^-inch. Thus the visual focal layer seems to cling to the 
 minimum focus, and if in the case of ver}' large telescopes it 
 should really be proved to be placed further back by normal 
 eyes, as at f^-f^ in Fig. i , then our present theories of visual 
 luminosities would have to be very radically overhauled in 
 order to harmonise theory with fact, and to show why the eye 
 should prefer in such an overwhelming degree the focus at 
 / 1 - /! instead of the concentrated focus at/- -/. 
 
 Table VII. gives roughly the estimated loss of light, repre- 
 sented by the shaded portion of the luminosity curves, incurred 
 by objectives of the sizes named through the presence of the 
 
 secondary spectrum. 
 
 TABLE VII. 
 
 Luminous value of whole light transmitted in each case = 100 
 
 36 in. Lick refractor 57 feet, F. L., loss = 27 
 
 24 in. refractor 30 feet, F. L., loss = 42 
 
 12 in. ,, 15 F. L., ,, = 21 
 
 6 in. ,, 7-6 ,, F. L.. ,, = . 9 
 28 in. Greenwich refractor 28 feet, F. L., loss = about 50.* 
 
 * For some kinds of astronomical work a relatively short focal length is 
 highly desirable, but the above figures indicate that such a condition 
 cannot be obtained without some sacrifice of that light-grasping power 
 which should be appropriate to the aperture. 
 
9 6 
 
 Thus the Lick refractor has a great advantage owing 'to 
 its relatively great focal length. 
 
 (1) If the aperture of a large objective of, say, 15 inches 
 aperture is doubled while the focal length remains the same, 
 it is evident that the colour curve remains the same while the 
 light distributed along it is quadrupled in intensity at every 
 point ; but the focal layer is reduced to about ^rd, and, 
 therefore, the range of wave length AA embraced by the 
 smaller focal layer is reduced, but not reduced to quite so 
 little as ^rd (to know exactly to what fraction would require 
 a very precise knowledge of the colour curve near its 
 minimum), but roughly to about one-half, and this half in- 
 cludes rather more than half the light, and the result comes 
 out that doubling the aperture of large objectives while 
 keeping the focal length constant does not quadruple the 
 light-gathering power for star work, but only a little more 
 than doubles it. 
 
 (2) If both aperture and focal length are simultaneously 
 doubled, then it is evident that the abscissae of the colour 
 curve will be doubled, while the focal layer will remain of the 
 same thickness ; and it will now be found that the range of 
 wave length AA embraced by the focal layer will be reduced 
 to about f rds ; but the light in this portion of the curve is 
 quadrupled in intensity, and the part actually embraced is of 
 higher luminous value, and therefore the light-gathering 
 power may be expressed as a little more than f rds of 4, or, 
 say, about 3. 'Thus doubling all of the dimensions of the 
 telescope will only increase the light-gathering power about 
 three times instead of four times. 
 
 (3) If the aperture of a large objective is kept the same 
 while the focal length is doubled, it is evident that the 
 abscissae of the colour curve are all doubled ; but the thick- 
 ness of the focal layer being about trebled, it will be found to 
 embrace considerably more than the former range of wave 
 length AA, and the light-gathering power for star work will 
 be accordingly increased (roughly, nearly i times). * 
 
 * It would be unsafe to venture upon more accurate statements or 
 formulas relating to light-gathering power before this important subject 
 has received yet further elucidation. 
 
97 
 
 But it may be asked, " Does not the light which accurately 
 focusses in the star disc become quadrupled when the aperture 
 is doubled as in case i ? " Certainly it does, but it should be 
 borne in mind that, owing to the spurious disc shrinking to 
 half its former size, light from the outer zones of the objective, 
 which focusses a little within or beyond the disc, and which 
 before contributed considerably to its brightness, now escapes 
 it altogether, so that although the total light within the disc 
 gains in one way yet it loses in another. (See Fig. 8). 
 
 When all the facts relating to this complicated question 
 are taken into consideration, we can scarcely escape the 
 conclusion that customary notions of the space-penetrating 
 power of large refractors as compared with smaller ones 
 need serious revision, and that all estimates of the relative 
 magnitudes of stars based upon the customarily conceived 
 relations between aperture and visibility are more or less 
 misleading, especially if we take into consideration the large 
 amount of variety in the colours of stars. For it is evident 
 that the relative light- gathering power of the same telescope 
 on a star in which yellow-green rays preponderate will be 
 very much greater than its light-gathering power for a star 
 in which blue rays or extreme red rays preponderate, since 
 the blue-green and blue and violet rays are so hopelessly 
 wasted, while the red rays about C and beyond are also 
 much wasted. 
 
 Another important fact should not be overlooked. It is 
 just in the case of small stars approaching the verge of 
 visibility that any gain or loss of light is of the most vital 
 importance. And there is a considerable weight of evidence 
 to show that the relative luminous value of the blue and 
 violet rays rises as the total illumination grows less* (Captain 
 Abney's experiments do not, however, confirm this). Hence 
 there is probably a greater loss of light in the case of small 
 stars than in the case of bright ones, owing to the absence 
 of the blue and violet, and therefore the reclaiming of the 
 latter into the visual focus rises to greater importance in 
 these cases. 
 
 *Modcrn Chromatics, chap, xii.; Helmholtz's Popular Scientific Lectures, p. no. 
 
 G 
 
98 
 
 Supposing it established that the visual focal plane is 
 really situated close to the minimum focus in large telescopes 
 it might be urged that the images of stars should appear 
 decidedly green or greenish-yellow, and not white. This 
 is partly the fact, at any rate in the case of small stars ; but 
 in the case of bright stars, the principle that the colours 
 of the spectrum, and especially the brightest colours, tend 
 to appear yellowish-white when raised to great intensity,* 
 holds good. 
 
 The concentration of light into the star image is, of course, 
 very great in the case of bright stars viewed through large 
 instruments. And the fact that the halo of wasted light 
 surrounding the star image is not more aggressive than it is 
 is easily explained by the fact that such light is for the 
 greater part so enormously diluted, and also by the fact 
 that the light forming the star image has the advantage of 
 comparatively great concentration, and stands out boldly by 
 force of contrast. The purest white paper only reflects fifty 
 times as much light as the deepest black paper under the 
 same illumination ; and as a matter of fact it can be shown 
 that a photometric contrast of only twenty to one is to the 
 eye as good as a black-and-white contrast. 
 
 It scarcely needs explaining that the effect of the secondary 
 spectrum upon vision of objects having considerable exten- 
 sion, such as the planets or the moon, is to throw a haze of 
 light over the object ; the straying colours corresponding to 
 each point of the image are spread into a halo which overlies 
 the images of surrounding points, and while no light can be 
 said to be lost (in the case of the moon), in a luminous sense, 
 yet it is lost for defining purposes as certainly as in the case of 
 stars. The image is a tolerably well-defined one, but having 
 th.e appearance of being viewed through more or less mist. 
 If the object shows various colours, as the belts of Jupiter, 
 then the image may be likened to many coloured prints which 
 one sees, in which the outline and details are rendered sharply 
 in black, while the colours are sponged on anyhow ; in the 
 same way the image of Jupiter may be defined very well as 
 
 * Modern Chromatics, chap. xii. 
 
99 
 
 regards outline and details by means of the most luminous 
 green, yellow-green, and yellow rays; while the colours, such 
 as the blue rays and the more extreme red, are spread on 
 where required in a sort of diffused wholesale fashion. There 
 is little to wonder at, then, in the fact that reflecting tele- 
 scopes have got the reputation for defining coloured details 
 more clearly than the larger refractors. It is the blue which 
 is principally missing, and a number of experiments have 
 been made which tend to show that the colouring power and 
 the denning power for delicate details, of blue rays, is of a 
 higher value than their small visual luminosity would lead 
 us to expect.* 
 
 I thought it would be of great interest to show the graphic 
 colour curves (calculated in exactly the same way) for the 
 33-inch Lick Photographic Objective (about 47 feet F. L.), 
 and also for the standard 13-inch Photographic Objective of 
 135 inches F. L. for charting the heavens. The focal layers 
 are shown in both cases at the minimum bend. The large 
 relative aperture of the 13-inch reduces the focal layer 
 considerably. It will be seen at once that the focal layers 
 embrace only a comparatively small portion of the range of 
 wave length to which the photographic plates are sensitive, 
 and there is little doubt that the enlargement of the star 
 images which takes place when either the stars are brighter 
 or the exposures are increased may easily be explained by 
 supposing that the aberrant rays begin necessarily to affect 
 the plate at greater and greater distances from the true star 
 disc. If this is the case, then, we should expect a marked 
 difference between stellar photographs taken with reflectors 
 and those taken with refractors. Of the former I have not 
 sufficient information to enable me to confirm this point. If 
 a curve of photographic sensitiveness were supplied in Figs, i 
 and 3 it would certainly be found that there is a considerable 
 quantity of actinic light lost for defining purposes through 
 the presence of the secondary spectrum, especially if we allow 
 for the fact that the minimum part of the colour curve bends 
 back and returns upon itself much more abruptly than in the 
 
 ^Modern Chromatics, p. 185; Helmholtz's Popular Scientific Lectures, p. no. 
 
100 
 
 case of the visual colour curve. This should be expected 
 from the fact that the irrationality of dispersion between 
 ordinary crown and dense flint glasses is greater as the blue 
 end of the spectrum is approached. 
 
 It is abundantly evident that an objective which would 
 refract all the colours as well as the blue actinic rays to 
 exactly the same focus will possess more practical advantages 
 than would appear at first sight. Hoping to introduce to 
 this Society on another occasion an objective which will do 
 this, and which is quite practicable up to reasonably large 
 sizes at any rate, I must now bring to a conclusion a paper 
 which has considerably surpassed the limits which I had at 
 first intended for it. 
 
101 
 
 Description of a Perfectly Achromatic 
 Refractor. 
 
 By H. DENNIS TAYLOR. 
 
 Reprinted from the 
 
 MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY. 
 Vol. LIV., No. 5. 
 
 OW that photography is becoming such an indispensable 
 supplement to eye observation in astronomical work, 
 the need for a form of refracting telescope which is perfectly 
 corrected for photographic as well as for visual purposes be- 
 comes more and more apparent. And if, simultaneously with 
 that condition, a great improvement in the character of the 
 visual image can also be obtained, then we have the elements 
 of a very substantial improvement in the refractor. In a 
 paper which I had the honour of reading to this Society in 
 November last," I pointed out the considerable loss of light for 
 defining purposes which must take place owing to the usual 
 colour aberrations ever present in the case of double object 
 glasses made of ordinary crown and flint glasses. I there 
 gave the losses of light, as determined by a theoretical 
 method, for certain objectives of various sizes. Since then I 
 have been able to carry out a delicate experiment with a 
 i2|--inch object glass, whereby the amount of light lost 
 owing to the colour aberrations was separated from the real 
 image, and rendered approximately measurable. The result 
 certainly confirmed the figures giving percentage of light 
 
 * See page 79. 
 
102 
 
 lost which I had previously arrived at by an a priori line of 
 reasoning. I hope to have the pleasure of describing this 
 experiment in a future paper. Thus I feel justified in saying 
 that the principal improvement in the visual image to be 
 expected from the almost perfect achromatism attained in 
 this new objective is a more brilliantly defined image, the 
 fine details being rendered in more black and white contrast 
 than we have been accustomed to see, thus standing out, 
 in an artistic sense, with greater sharpness. For this new 
 objective is to all intents and purposes as achromatic as a 
 reflecting telescope. My justification for this statement is as 
 follows : 
 
 It must be remembered- that in practice the reflecting 
 telescope, mirror and Huyghenian eye-piece combined, is 
 not absolutely achromatic. However absolute may be the 
 achromatism of the primary image, yet the eye-piece, if of 
 the Huyghenian or Ramsden type, always introduces its own 
 colour aberrations. Taking the case of a Huyghenian eye- 
 piece of i -inch equivalent focal length, the virtual longi- 
 tudinal colour aberration from B to H* (G ; ) which it would 
 introduce would amount to about three-hundredths of an 
 inch. This amount in the case of a reflector of a focal length 
 equal to seven-and-a-half times its aperture is about equal, 
 as regards detriment to vision, to a longitudinal colour 
 aberration of six-hundredths of an inch in the case of either 
 a reflector or a refractor of a focal length equal to fifteen 
 times its aperture. I shall shortly point out that in the case 
 of even an objective of 2 feet aperture, and of 36 feet focal 
 length, of the new construction, the residual longitudinal 
 colour aberrations need not be expected to amount to as 
 much as six-hundredths of an inch ; while at the same time 
 an achromatic single lens eye-piece may be employed, 
 presenting no spectrum and having only two reflecting 
 surfaces. Or, supposing a i-inch Huyghenian eye-piece to be 
 generally employed on such a telescope, the objective may 
 be sufficiently over-corrected to counteract the longitudinal 
 colour aberrations of that eye-piece. 
 
 The problem of devising a perfectly achromatic object 
 glass has occupied the attention of many scientific men and 
 
io 3 
 
 opticians for a considerable time. The Rev. Vernon Har- 
 court spent twenty-five years or so of his life in melting and 
 trying new forms of glass with a view to producing two glasses 
 of similar rationality. But none of his experiments resulted 
 in practical success. The problem has also deeply occupied 
 the attention of Sir G. G. Stokes and the late Professor 
 Pritchard. In short, it is a problem which has exerted a 
 great fascination over many minds. Some, notably Pro- 
 fessor Hastings, in America, and Professor Abbe, of Jena, 
 fully realised that practical success need not by any means 
 depend upon the employment of only two lenses in the objec- 
 tive. Professor Hastings published an article in the American 
 Journal of Science and Art (for December, 1879) i which he 
 gave the results of calculations of several forms of objectives 
 made of three kinds of glass, in some of which a remarkably 
 near approach to perfect achromatism seemed to be possible. 
 However, none of these combinations seem to have been 
 worked into practical shape, and, so far as I know, principally 
 because one or other of the glasses employed - all, I believe, 
 the production of Mons. Feil, of Paris could not be made 
 hard enough or durable enough for practical use. 
 
 In 1884 the justly celebrated firm of Herren Schott und 
 Gen. started their optical glass manufactory at Jena for the 
 principal purpose of making both old and new varieties of 
 optical glass on a commercial scale, at the same time address- 
 ing themselves in a most thorough and scientific manner to 
 the problem of producing two glasses whose rationality of 
 dispersion should be identical, at the same time that the 
 difference between their dispersive powers should be sufficient 
 to enable a perfectly achromatic double object glass to be 
 constructed, without having to resort to impracticably deep 
 curves. 
 
 'But in spite of the skill and scientific method brought to 
 bear upon this problem, and exhaustive trials of all known 
 substances capable of being fused into glasses, it cannot be 
 said that that aim has been realised ; and I really believe that 
 a practicable double object glass, even reasonably free from 
 secondary colour aberration, is as far from being realised as 
 it was ten years ago. For it should be borne in mind, when 
 
104 
 
 aiming at an object glass of two glasses which shall be 
 perfectly achromatic, that although it may be possible to 
 considerably reduce the discordance between their respective 
 rationalities of dispersion, yet the reduction of secondary 
 spectrum at the focus accruing from that improvement may 
 be largely counteracted or even wholly neutralised by the 
 deeper curvatures or higher powers of the two component 
 lenses, which are rendered necessary by that small difference 
 of dispersive powers of the two glasses which has hitherto 
 inevitably accompanied the closer agreement in rationality of 
 dispersion. In short, whether using two or three lenses, the 
 deeper the curves, or the more powerful the component lenses 
 required for obtaining a given focal length, the more close 
 and strict must be the concordance of rationality of dispersion 
 between the glass (or two glasses combined) forming the 
 positive element and the glass forming the negative element 
 of the objective. To be sure, certain double combinations 
 have been made in which the secondary spectrum has been 
 reduced to about half the usual amount. For instance, Dr. 
 Czapski, of Jena,* describes a form of objective of which 
 two or three were made by Bamberg, of Berlin, and also, I 
 believe, by Dr. Czapski himself, in which the positive lens 
 was made of Schott's dense barium phosphate crown S -30, 
 and the negative lens of Schott's borate flint S '8 or S '7. This 
 is somewhat similar to a combination of the same flint with a 
 nearly similar crown which was carried out by Messrs. Cooke 
 about seven years ago, and found to make a more than usually 
 achromatic object glass. But the borate flints are too soft for 
 practical purposes, and, moreover, Dr. Schott has informed 
 us that it is almost impossible to get good discs of this glass 
 of over 5 inches aperture. 
 
 A few years ago Professor Hastings patented in America 
 another form of double object glass in which the secondary 
 
 * In Dr. Czapski's work, Theorie der optischen Instnimenten (nach Abbe), 
 on page 131, will be found graphic diagrams showing the somewhat 
 reduced amounts of secondary spectrum exhibited by these combinations 
 as compared with that exhibited by an ordinary object glass of crown 
 and dense flint. 
 
105 
 
 spectrum was reduced to about half. For his positive lens 
 he made use of potassium silicate crown (Schott's O'i3), and 
 for his negative lens Schott's boro-silicate flint glass O'i6i. 
 This combination turned out to be impracticable : the crown 
 is too hygroscopic, and the flint, Dr. Schott has informed us, 
 cannot be made good enough. 
 
 Having considered all the facts of the case, I was inevitably 
 led, as a few other inquirers have been led, to the conclusion 
 that there was little prospect of making an object glass of two 
 lenses whose achromatism would be so much superior to that 
 of the ordinary objective as to fully compensate for the other 
 practical disadvantages which would ensue, either in the 
 shape of curves difficult to work or a confined field of view, 
 not to mention several other serious drawbacks. 
 
 Some two years ago I entered into an investigation as to 
 the possibility of producing a triple object glass, free from 
 secondary spectrum, whose curves should be such as to 
 permit of all the other necessary or desirable optical qualities 
 and certain practical advantages being attained. After a very 
 searching and prolonged trial of all the likely combinations 
 of glass picked out of the numerous new and old types of 
 glass, put at the disposal of the optical world by the genius 
 of Dr. Schott and Professor Abbe, and by help of every 
 opportunity for investigation and experiment afforded me in 
 the workshops of Messrs. Cooke, I at last realised a triple 
 combination of three different sorts of glass, which, while 
 yielding a degree of achromatism almost greater than any- 
 thing which I had at first ventured to hope for, at the same 
 time rendered possible the achievement of certain other 
 important optical qualities and practical conveniences which 
 yet remain to be specified. 
 
 Taking the optical qualities first 
 
 i. This objective can be made free from spherical aberra- 
 tion for all colours simultaneously. This is of the utmost 
 importance in the case of an objective having any pretensions 
 to perfect achromatism. In a double objective of ordinary 
 construction having a double concave or concavo-plane flint 
 lens, it is well known that when the brightest yellow-green 
 rays are perfectly corrected for spherical aberration, then the 
 
io6 
 
 red rays are under-corrected, and the blue rays are over- 
 corrected for spherical aberration ; a condition of things, 
 however, which escapes notice in the presence of the greater 
 fault of the secondary spectrum. 
 
 2. The objective is so arranged as to give the largest 
 possible field of good definition. The image of a star formed 
 at, say, 2 from the optic axis, is free from coma or side flare, 
 and also from colour. If a star is focussed into the centre 
 of the eye-piece and the objective is tilted, even very con- 
 siderably, then the image of the star remains symmetrical 
 and without side flare, and only showing the inevitable 
 astigmatism. This condition is of the utmost importance in 
 the case of an objective when used for photographing celestial 
 objects having considerable angular extension, or for transit 
 instruments which are generally provided with eye-pieces 
 having a certain amount of lateral travel. 
 
 3. The curves are also such that a ray parallel to the optic 
 axis traced through the margin of the objective enters and 
 leaves the flint lens at approximately equal angles. This is 
 of considerable importance in the case of deep curves, and 
 especially in large-sized objectives, because the flint lens is 
 by a long way the weakest, in a mechanical sense, of the 
 three lenses, and therefore most liable to flexure across its 
 diameter, and such flexure would in the large sizes, and 
 under ordinary circumstances, give rise to considerable 
 
ID; 
 
 aberrations of the marginal rays from true focus. But if 
 refraction of such marginal rays takes place about equally at 
 both surfaces, then the optical results of such flexure or 
 sagging of this lens are reduced to practically nothing. The 
 action of the edge of the flint lens upon the marginal rays is 
 in this case the same as that of a prism when set at minimum 
 deviation, which may be rotated through a relatively con- 
 siderable angle without bringing about a very perceptible 
 increase in deviation. 
 
 The accompanying diagram is a section of the objective 
 taken along a diameter. The focal length is eighteen times 
 the aperture. If specially required, the aperture can be made 
 equal to one-fifteenth of the focal length in the case of sizes 
 up to 8 or 10 inches aperture. A relatively large focal length 
 is, however, most to be recommended. 
 
 The lens L x which is placed first to receive the parallel 
 rays is made of Schott's baryta light flint glass 0-543, whose 
 refractive index for D ray is about 1-564. 
 
 The negative lens L 2 is made of Schott's new boro-silicate 
 flint glass, which is a variety of their 0-164. Its refractive 
 index for D is 1-547. 
 
 The glass of which the third lens L/ 3 is made is a sort 
 of light silicate crown (Schott's 0-374), which differs from 
 ordinary hard crown in having a lower dispersive power and 
 a somewhat different rationality. Its index of refraction for 
 D is about i'5ii. 
 
 The separation between the negative lens L 2 and the 
 positive lens L 3 is designed for the purpose of correcting the 
 spherical aberration more perfectly for all colours. 
 
 The contiguous second and third surfaces r z and r z are 
 curved to exactly the same radius ; also the fourth and fifth 
 surfaces r and r s are curved to the same radius. The last 
 surface r Q is concave, and curved to a radius equal to nearly 
 twice the focal length. There are thus three hollow surfaces 
 whose figuring can be directly tested by reflected light, while 
 each of the other three convex surfaces can be tested for 
 figuring in an indirect manner. For instance, if all three 
 concave surfaces are approved, and faults at the focus are 
 found to exist which must be causejd by defective figuring in 
 
io8 
 
 one or more of the convex surfaces, then they can be tested 
 in the following manner : 
 
 If the faults at the focus are due to defective figuring in 
 the second surface, those faults will more or less totally 
 disappear on introducing some liquid having a refractive 
 index approximately equal to that of glass between the two 
 surfaces r z and r s . If it is desired to test the fifth surface, 
 then the brass ring separating L 2 from L 3 is taken out and 
 put between the second and third surfaces, while L 2 and L 3 
 are brought close together. The objective being first tested 
 in this state with its surfaces dry and the faults at the focus 
 noted, then the refracting liquid may be introduced between 
 the fourth and fifth surfaces, when, if the fifth surface is 
 faulty in figure, the faults at the focus will more or less 
 disappear. But if both the second and fifth surfaces are 
 not found to be faulty by this method, then the conclusion is 
 that the first surface r^ is really at fault. Thus, all surfaces 
 may be easily tested either directly or indirectly, and the 
 whereabouts of small errors of figuring revealed, which are 
 too fine to be appreciated by the use of spherometers or other 
 mechanical tests. Thus no time need be lost in refiguring 
 the wrong surfaces. 
 
 The baryta light flint and light silicate crown glasses used 
 for the two exterior lenses are as hard and durable as 
 ordinary crown glass and if anything more colourless and 
 transparent. 
 
 The boro-silicate flint glass used for the negative lens is of 
 quite a different type from that used by Professor Hastings 
 for his patented doublet. Indeed, this flint is the only avail- 
 able glass which can be used for apochromatic objectives, 
 and can be turned out in large-sized discs. Its mechanical 
 hardness is very great greater indeed than that of hard 
 crown. It is beautifully transparent to all the brighter rays 
 of the spectrum, but begins to absorb the violet rays after 
 the G line is passed, This characteristic imparts a pale 
 lemon yellow colour to a lens when of two inches thick or 
 over, but it is scarcely noticeable in the case of lenses of 
 moderate sizes. 
 
 In a town atmosphere, where sulphretted hydrogen or 
 
log 
 
 sulphuric acid abounds, it would not be safe to use this flint 
 for an exterior lens ; but when enclosed between the two 
 positive lenses, as in this objective, we have no reason for 
 expecting it to be any less permanent in its polish than the 
 dense flint glass which has hitherto been used for double 
 object glasses of the usual construction. Or even supposing 
 a slight amount of tarnishing of its surfaces did take place 
 after many years, as is the case with the flint glasses hitherto 
 used for telescope object glasses and prisms, it may yet be 
 asked whether the transparency or light-transmitting power 
 of the objective would be in any degree impaired. We know 
 well enough that a tarnished optical surface does not look 
 so brilliant by reflected light. It may seem a somewhat 
 startling statement to make, but nevertheless it is a fact, that 
 certain flint glasses which we have experimented with by 
 local tarnishing have been found to transmit actually more 
 light where tarnished than where the surfaces had had their 
 original polish preserved, and in no case has it ever been 
 found that tarnish in any perceptible degree interfered 
 with transparency." We have yet to make some further 
 experiments in this direction, and I hope at some future 
 time to throw the proofs of this statement on the screen 
 for your inspection. Those who use ordinary refractors 
 may congratulate themselves on the fact that the light- 
 gathering power of their objectives slightly increases with 
 age, provided the surfaces are kept clean. However, leaving 
 that interesting point for the present, I may say that enough 
 experiments, extending over about a year and a half, have 
 been made with this new glass to fully satisfy Messrs. T. 
 Cooke and Sons of its lasting properties, and to warrant them 
 in guaranteeing the permanenc}' of this objective over many 
 years. 
 
 I will here give a table setting forth the optical properties 
 of the three glasses, and also showing how very closely all 
 the rays of the spectrum can be refracted to one focus. 
 
 *Of course this statement is not intended to apply to the case of a 
 surface which is actually corroded, so as to appear at all grey or milky by 
 reflected light. 
 
no 
 
 12 3 4 5 6 
 
 Combined Proportional 
 
 Partial Partial Partial Dis- Dispersions Proportional 
 
 Region of Dispersions Dispersions persions. or C to F Dispersions for 
 
 Spectrum. or Au., for or Au, for Au..-\-Au z being Unity Boro-silicate 
 
 Glass 0-543. Glass 0-374. for Glasses for Glasses Flint o'6 5 8. 
 
 0-543 and 0-374. '543 + 0-374 . 
 
 C to F "01115 '00844 
 
 (roooo) ( i -oooo) 
 
 A to C '00374 '00296 
 
 (o-3354) (0-3507) 
 
 D to F -00790 "00593 
 
 (0-7085) (0-7026) 
 
 E to F* -00369 -00274 
 
 (0-3309) (0-3247) 
 
 FtoG(Hx) -00650 '00479 
 
 (0-5830) (0-5675) 
 
 F to HJ* -01322 -00976 
 
 (1-1857) ( I-I 564) 
 
 This table is calculated on the supposition that 
 
 01959 
 
 I' OOOO 
 
 roooo 
 
 00670 
 
 3420 
 
 3425 
 
 01383 
 
 759 
 
 7052 
 
 00643 
 
 3282 
 
 3278 
 
 01129 
 
 5763 
 
 5767 
 
 02298 
 
 1-1730 
 
 i'i745 
 
 Pi Pz 
 
 where = the sum of the reciprocals of the radii of the first 
 PI 
 
 lens and = the sum of the reciprocals of the radii of the 
 
 third lens. This pretty well represents an average case, and 
 with the present meltings of glass gives the most perfect 
 colour correction. The curves may of course be varied 
 within certain limits in order to compensate for variations in 
 the optical qualities of the glasses. 
 
 But under the conditions above defined it is evident to 
 those conversant with optics that the partial dispersions for 
 these two positive lenses combined, for any given region of 
 the spectrum, is obtained by simply adding together their 
 respective dispersions for that region of the spectrum. 
 
 In column 2 are given the partial dispersions of baryta 
 light flint glass 0-543 for the different regions of the spectrum 
 indicated in column i. 
 
 In column 3 are given the corresponding partial disper- 
 sions for the light silicate crown glass 0-374. The figures in 
 brackets beneath these in columns 2 and 3 indicate the relative 
 
 *Figures relating to spectral regions E to F and F to H x ar- calculated 
 from Messrs. Schott's measurements by means of Cauchy's dispersion 
 formula. 
 
Ill 
 
 partial dispersions for each region of the spectrum when the 
 dispersion from C to F for each glass is taken as unity. 
 
 In column 4 are given the sums of the partial dispersions 
 for each region of the spectrum, obtained by simply adding 
 together the partial dispersions for 0-543 an< ^ '374 respec- 
 tively for each region of the spectrum indicated in column i. 
 
 In column 5 the combined dispersion for the two positive 
 lenses for the region C to F is taken as unity, and the relative 
 proportions of the combined dispersions for the other regions 
 of the spectrum are expressed in fractional parts of the com- 
 bined dispersion from C to F. Finally, in column 6 are given 
 the corresponding proportional dispersions in the same sense 
 as in column 5 for the boro-silicate flint-glass 0-658. I need 
 scarcely explain that the perfection of the achromatism of the 
 triple combination is made manifest by the remarkably close 
 agreement between the proportional dispersions for the two 
 positive lenses combined, on the one hand, and the propor- 
 tional dispersions for the negative lens on the other hand. 
 
 The rationality of dispersion, as it is generally termed, is in 
 remarkably close agreement. 
 
 These figures do not pretend to accuracy in the fourth 
 decimal place, but I would point out that an error of -ooio 
 in any one of these figures would, in the case of a 6-inch O.G. 
 of 108 inches focal length, correspond to a longitudinal aber- 
 ration of only -02 inch ; an amount which, I believe, the most 
 delicate test could scarcely reveal, owing to the cone of rays 
 close to focus merging in such a remarkable manner into the 
 cylindrical form. 
 
 Owing to the expensive nature of experiments of this sort, 
 the first objectives actually made on this principle and finished 
 were of the moderate aperture of 3^ inches ; but these were 
 found so successful as to leave no doubt about much larger 
 sizes being practicable should the glass be forthcoming. 
 However, a recent visit to Dr. Schott and Professor Abbe, at 
 Jena, has quite set at rest any doubts as to the possibility of 
 turning out discs of considerable sizes, many discs of moderate 
 dimensions, up to 9 inches and possibly 12 inches diameter, 
 being already available. Dr. Schott is of opinion that 
 only actual experience with future meltings will determine 
 
112 
 
 up to what sizes it will be possible to manufacture perfect 
 discs. 
 
 It is easily seen from the diagram that the total thickness 
 of glass in this new objective is a trifle over i times the 
 thickness of glass used in an ordinary double objective of the 
 same aperture, and it might be thought that a serious loss of 
 light might result therefrom in large sizes. I have recently 
 made some experiments upon the transparency of some blocks 
 from 4 to 5 inches thick of the three glasses used in the 
 objective, and have been pleased to find that a thickness of 
 5 inches of the baryta light flint absorbed only 40 per cent, 
 of candle light ; a thickness of 4^ inches of boro-silicate flint 
 absorbed 27 per cent, of candle light (though doubtless more 
 of daylight) ; while a thickness of 5-^ inches of the light 
 silicate crown absorbs only 20 per cent, of candle light. 
 
 These figures must be regarded as preliminary. But as far 
 as they go these results are surprising as showing what an 
 improvement has taken place in the transparency of glasses 
 (a few blocks of ordinary crown and flint glasses were also 
 tried and found nearly as transparent as the block of 0-374) 
 within the last few years, when thicknesses of 4 inches only 
 were generally found to absorb as much as one-half of the 
 light. 
 
 Thus this triple objective might be made in very large sizes, 
 2 feet aperture or so, before the loss of light due to thickness 
 would become so great as to put it on an equality as regards 
 light-gathering power with a Newtonian reflector of average 
 condition and similar aperture. 
 
An Experiment with a 12! in. Refractor, 
 
 whereby the Light lost through the Secondary Spectrum is 
 separated out and rendered approximately measurable. 
 
 By H. DENNIS TAYLOR. 
 
 Reprinted from 
 
 THE MEMOIRS OF THE ROYAL ASTRONOMICAL SOCIETY. 
 Vol. LI. 
 
 T N a former paper which I had the honour to read before 
 -* you last November I showed theoretical reasons, founded 
 upon the spectrometrically-observed refractive indices of 
 crown and flint glasses, for supposing that a very serious 
 amount of light must be lost for defining purposes in the 
 case of large-sized objectives ; or, in other words, only a 
 portion of the light transmitted by the objective really con- 
 tributes to the formation of the spurious disc or star image, 
 while the other aberrant rays only form a useless halo of 
 wasted light, which only fails to be more bright and con- 
 spicuous than it actually appears because of its diffusion 
 over comparatively large areas of the focal plane. And 
 I pointed out that if the laws of geometric optics applied 
 strictly to the cone of rays condensing to and expanding 
 from the star image at the focus, then the proportion of light 
 actually contributing to the formation of the star disc in the 
 case of a large refractor (say 2 feet aperture and 30 feet focal 
 length) would be only a very small fraction of the light 
 transmitted that, in fact, anything like distinct vision would 
 be impossible. But owing to the great discrepancy between 
 deduction from the laws of geometric optics in this case and 
 the actually observed state of things at the focus, caused, 
 
H4 
 
 no doubt, by the operation of interference in a manner 
 not yet worked out, the colour aberrations of much larger 
 objectives are rendered innocuous in effect compared with what 
 they might be. For owing to the remarkable manner in which 
 the cone of rays tapers off into an approximately cylindrical 
 form for some distance on either side of the focus, the depth 
 of focus (to borrow a term from photographic optics) is in- 
 creased about fivefold, or the " focal layer " as defined in my 
 previous paper is increased fivefold. 
 
 These focal phenomena very much complicate the problem 
 of estimating the proportion of light transmitted which goes 
 to the formation of, and contributes to, the brightness of the 
 star disc. However, by means of a graphic method I arrived 
 at certain approximate results. At the same time I had in 
 my mind's eye an experiment by which I hoped to check or 
 confirm the data arrived at by the theoretical method. The 
 object of this experiment is to separate out the amount of 
 light which, owing to the secondary colour aberrations, falls 
 outside the limits of the spurious disc or star image, from 
 the amount of light which actually contributes to the forma- 
 tion of the star disc. The theorem on which this experiment 
 depends is tolerably simple. In* the first place the image of 
 a star formed by an ordinary refracting telescope is a spurious 
 disc, formed by that light which falls within certain limits of 
 accurate focus. Round this disc are an infinite series of 
 circles of aberration, formed by those other colours which 
 fall more or less within or beyond the true focus. The 
 brightest aberrant colours form the smallest circles of aberra- 
 tion, of sizes comparable to the size of the spurious disc, 
 while the less bright colours (approaching the two ends of the 
 spectrum) form circles of aberration of relatively enormous 
 size. In the case of a 2-foot o.-g. of 30 feet focal length, the 
 circle of aberration for the red B ray is *oi66 inch diameter, 
 or about 40 times the diameter of the spurious disc, which is 
 0004. The G (H y ) ray forms a circle of aberration -0666 
 inch in diameter, or 166 times the diameter of the spurious 
 disc. Representing the spurious disc or star image by a circle 
 o of an inch diameter, as in fig. 8, then the circle of aberra- 
 tion for the B ray will be 2-1 inches diameter, and the same 
 
for the G ray will be no less than 83 inches in diameter. * 
 In plate III., fig. 8, I have filled in certain circles of aberra- 
 tion for rays of other colours, as indicated.! 
 
 Now let it be supposed that such a 2-foot o.-g. is focussed 
 upon a distant absolutely black line (fig. 5) drawn upon a 
 uniformly white background, it being supposed that the 
 angular width of the black line is about twice the angular 
 diameter (o"'4) of the spurious disc. Fig. 6 will then repre- 
 sent the structure of the telescopic image of the black line. 
 
 In the first place, every point of either edge of the original 
 black line is painted by the object-glass as a spurious disc c, e, e. 
 In fact, the edges may be considered as formed by an infinite 
 linear series of spurious discs, e e, &c., each of which, be it 
 remarked, is surrounded with its own series of circles of 
 aberration, as indicated. Also the image of the continuous 
 white surface is made up of an infinite series of spurious discs, 
 each surrounded with its own series of circles of aberration. 
 It is evident, therefore, since the radius of the largest circle 
 of aberration (for the G ray) is about -$ of an inch; that all 
 spurious discs forming the white surface whtch fall within 
 3*o of an inch from the black line will throw some portion of 
 their aberrant light into the area occupied by the image of 
 the black line. That is, the image of the black line cannot 
 be truly black, but is really filled in by the light contributed 
 by the infinite series of circles of aberration which overlap it. 
 Supposing the area of white surface to be sufficiently large 
 (i inch square, or more), then we may consider the image 
 of that white surface to be built up of two moieties or classes 
 of light. 
 
 ist. That light which is refracted truly to focus, and 
 forms a spurious disc for every point of the white surface. 
 
 *I have left out of consideration the violet rays between G and H, as 
 their luminous intensity is exceedingly small. Moreover, the circle of 
 aberration for the H ray would fall entirely outside the limits of the 
 diagram. 
 
 f To make this diagram approximate to the truth, I have deducted a 
 constant (equal to four times the diameter of the spurious disc) from the 
 diameter of the circles of aberration, as calculated by geometrical optics. 
 
n6 
 
 This is the light used strictly for defining purposes, and it 
 forms the defined image of the black line. 
 
 2nd. That light which is not refracted to correct focus, 
 but forms halos or circles of aberration around each point of 
 the white surface. This light, wasted for defining purposes, 
 yet paints an area of uniformly bright surface not appreci- 
 ably differing in relative size from the true image of the white 
 surface, but with a very blurred outline. 
 
 Now the image of the white surface, as we see it, is 
 essentially composed of the above two elements super- 
 imposed on one another, and it scarcely needs explaining 
 that the area of the image of the black line, being entirely 
 unoccupied by light of the ist class, leaves the light of the 
 2nd class, or the wasted light, to reveal itself. 
 
 In the white surface we have the whole light transmitted 
 by the object-glass represented. 
 
 Within the area of the image of the black line, on the other 
 hand, we have laid bare to our view a narrow section of that 
 uniformly bright area of wasted light which really overlies 
 the whole image of the surface. 
 
 Therefore, if we can measure the relative intensities of the 
 light falling within the "black" line as compared with the 
 surrounding white, then we shall obtain the ratio betweeli 
 the light which is wasted for defining purposes and the whole 
 light transmitted by the objective. 
 
 Here it should be borne in mind that a considerable 
 proportion of wasted light consists of light of those colours 
 which are so slightly aberrant from true focus as to form halos 
 or circles of aberration, of diameters equal to from i times to 
 3 times the diameter ot the spurious disc. It is obvious, 
 then, that if we require this moiety of wasted light (see small 
 circles round e, e, e, &c., fig. 6) to contribute to the brightness 
 of the more central parts of the image of the black line, then 
 we must have the black line as narrow as possible, consist- 
 ently with it presenting an appreciable width. Thus, in 
 order to obtain the most accurate result, the black line should 
 be very narrow indeed, say of a width equal to the diameter 
 of the spurious disc. But unfortunately the narrower is the 
 black line, the more hopelessly difficult does it become to 
 
'measure the relative intensity of the wasted light which falls 
 within it. 
 
 The wider the black line becomes the darker it grows down 
 the centre, owing to the failure of the smallest circles of 
 aberration to overlap it. Thus the most accurate practical 
 measurement likely to be made of the relative brightness of 
 the wasted light in the black line will be apt to under- 
 estimate the real proportion of wasted light, for the possibility 
 of making such a measurement depends upon the image of 
 the black line not being less in width than about i\ times 
 the diameter of the spurious disc at least, so I found in 
 actual practice. The way in which the measurement was 
 carried out was as follows : 
 
 I took a very perfect objective (O O, fig. i) of I2'6 inches 
 aperture and about 185 inches focal length (almost exactly 
 fifteen times its aperture), and fitted it up in a horizontal 
 tube in a dark passage, opposite a flat optically worked 
 mirror R R of 16" aperture (but not silvered). The line 
 A A represents both the optic axis and a perpendicular to 
 the mirror. S represents the original black line on a white 
 background, and Sj. represents the reflected image of the 
 same formed after two passages through the object-glass. 
 This method has great practical advantages over any other ; 
 it does away with the necessity for viewing the original 
 object through a great distance of unsteady air out of doors, 
 which would also be likely to throw atmospheric haze over 
 the image of the black line and vitiate the results ; it enables 
 the observer to be in darkness and isolated from currents of 
 disturbing air. Not only so, but there is the further great 
 advantage that the secondary longitudinal colour aberrations 
 of the objective are doubled in amount owing to the double 
 passage of light through the objective. We have the 
 secondary spectra of two such objectives added together. 
 The diameter of the spurious disc at S x is the same ('0004 
 inch) as that which would be yielded by a 25'2-inch objective 
 of 370 inches focal length, or by any other objective having 
 the same ratio of focal length to aperture. I have also 
 found that the drawing out of the cone of rays into the 
 cylindrical form near the focus is not disturbed in amount by 
 
n8 
 
 the fact of the reflection from the mirror and the double 
 passage of light through the objective/" 
 
 Thus, then, we have represented at S 2 precisely the same 
 state of things (except for the angular diameter of the spuri- 
 ous disc) as we should have at the focus of a 2$-2-inch 
 objective of 370 inches focal length supposing it were focussed 
 upon a. very distant black line (drawn on a white ground), 
 and of such a width that the objective would form an image 
 of it equal in width to either our original black line at S or 
 its image at S^ Fig 2 is a view as seen from the objective of the 
 arrangement of slits. Fig. 3 is a view of the same from the 
 eyepiece side. Fig. 4 is a longitudinal plan and section of 
 the apparatus taken along the optic axis. All three views 
 are drawn to double the original size. S S is the original 
 black line formed as follows : Three steel jaws or slips, D 1} 
 D 2 , ad D 3 , with carefully worked straight knife edges, were 
 clamped on to a common flat surface, the utmost care being 
 taken to insure exact parallelism between the two edges of 
 the second slip, D 2 . Close to and parallel to the knife edge 
 of D 1} were ruled with a dividing engine two fine lines -002 
 inch apart, the same being done close to the slit S x upon the 
 flat surface of D 2 . This was for guidance in adjusting the 
 widths of the slits formed between the contiguous knife edges. 
 The slit S S was adjusted to -002 or T -g. of an inch width. 
 The thick line K K represents a section of a thin metal 
 partition dividing the tube into two, and preventing any light 
 reaching the left-hand side of the tube and the slit S x S lf 
 which should be kept in darkness. The flat face of the slips 
 D! and D 2 , facing the objective and on the right hand of the 
 partition K K, was then smoked carefully over a piece of 
 burning magnesium wire. This operation covers the face 
 with a beautifully fine and pure white deposit of magnesia. 
 
 * This fact, that the degree of drawing out of the cone of rays, as the 
 focus is approached, is independent of whether fhe light traverses the 
 objective twice or only once, is fatal to any attempt to account for this 
 curious phenomenon by residual spherical aberration. It seems to be 
 dependent on nothing else but the ratio between aperture and focal length 
 or the angle of the cone of rays. 
 
rig 
 
 At the same time, if great care is not exercised, the slit S S 
 gets filled up. But after a few failures I was able to get the 
 flat faces of the two steel slips (D x and D 2 , fig. 2, to the left of 
 the partition K K) very uniformly smoked over to a pure 
 white, while at the same time the width of the slit S S was 
 reduced down as nearly as possible to a uniform width of 
 ooi or T o^o fan inch. A piece of black paper, P (figs. 3 and 4), 
 was afterwards fastened against the back face of the two 
 jaws, so that the triangular space behind S S formed a sort 
 of dark well behind it. Now, the white surface being illumin- 
 ated brightly by the small electric lamp, L (fig. 4), it is evident 
 that the slit S S forms an absolutely black line of T oVo f an 
 inch width. A line ruled with the blackest ink on a white 
 surface is not quite black, nor is it likely that a line of only a 
 thousandth of an inch in width could be ruled at all, at any 
 rate perfectly enough to bear much magnification. A small 
 rectangular slip of white paper, P, also smoked with magnesia, 
 was then carefully fastened over the upper part of the black 
 slit and facing the objective, for the purpose shortly to be 
 explained. The width of the slit Si S x was adjusted to a 
 trifle less than T ^oo- f an m ch. W in figs. 2 and 3 repre- 
 sents a sort of window cut out of the two steel slips, D 2 and 
 D 3 . A positive eyepiece, E, of a power of 450, which is 
 sufficiently high to magnify the slit Sx Sj to a very percep- 
 tible width (8 minutes of arc), was mounted behind the slits. 
 The apparatus being pushed into the tube of the telescope 
 (which has been previously adjusted) and the glowlamp L ex- 
 cited to its fullest power, then a very little manipulation of 
 the telescope will bring the reflected image of the black slit 
 S S vertically across the window W, where it can be viewed 
 by the eyepiece E and nicely focussed. Fine adjustments 
 are then required to bring the actual slit Si Si and the 
 image of the black line simultaneously into the very best 
 focus, so as to avoid any parallax. It is already evident 
 enough to the eye that the image of the black line, although 
 pretty sharply defined, is not quite black, but is filled in with 
 wasted light. We now wish to isolate this wasted light, and 
 directly compare its brightness with the brightness of the 
 image of the surrounding white surface. By means of a fine 
 
120 
 
 adjustment of the telescope the image of the black line as 
 seen in the window can be brought into exact alignment with 
 the slit Si SJL. Now, it is evident enough that if the image 
 were really black, then, on making the above adjustment, 
 the slit Sj S x should go suddenly quite black. But it cer- 
 tainly does not do so ; it goes very little darker indeed than 
 when the image of the white surface shines through it. As a 
 result of contrast, the black line in the window appears fairly 
 dark a dark grey but its continuation below, as isolated 
 by the jaws of the slit, appears a relatively bright line. Now, 
 let it be supposed that the optic axis of the objective as well 
 as a normal to the surface of the reflector cuts the surface of 
 the steel strip D 2 at the point A. Then we have the image 
 of the black line so thrown (upside down) as to fall between 
 the limits d and &, partly across the window and partly into 
 the slit S x S x from a to b. But between the limits b and c 
 we have the image of the white paper P (forming part of the 
 whole white surface) thrown across the slit Si S^ When 
 properly adjusted, then, we have from a to b an isolated strip 
 of the light which is lost for defining purposes owing to the 
 secondary colour aberrations ; while from b to c we have a 
 strip, of similar width, of the image of the white surface 
 representing the luminous value of the whole light transmitted 
 by the objective, for it is made up of both the aberrant light 
 and the light which is strictly available for defining purposes. 
 In carrying out this experiment any observer ignorant of the 
 secondary spectrum would be surprised at the comparatively 
 small difference between these two portions of the slit. I 
 proposed to measure the relative luminosities of these two 
 narrow strips of light, a to b and b to c, by placing small pieces 
 of neutral-tint glass, g, in front of the portion of slit b c and 
 with their upper edges at b (see figs. 2, 3, and 4) until it 
 appeared to be of a brightness equal to the part a b. I 
 first tried a single thickness of a pale neutral-tint glass, 
 which, by very careful and repeated experiments by means 
 of the Rumford photometer test, I had found to transmit 56 
 per cent, of light. On fastening this behind b c, its bright- 
 ness was reduced, but it still appeared somewhat brighter 
 than a b. I then put two such strips (which I had proved 
 
121 
 
 by similar tests to transmit 31 per cent, of the light, thus 
 agreeing with calculation) behind b c, and now the part of 
 the slit a b was rather the brighter of the two. 
 
 Thus I judged that the brightness of the wasted light, 
 represented by a b, was about intermediate in value between 
 31 per cent, and 56 per cent, of the whote light (represented 
 by b c), transmitted by the objective-- that is, the wasted 
 light amounted to about 40 per cent, of the whole light 
 transmitted by the objective. It scarcely needs pointing out 
 that the photometric estimation of the relative brightnesses 
 of two such narrow strips of light as these, placed end to end, 
 renders anything like exactness impossible; I could, however, 
 make sure of my limits. It should be remembered that the 
 width of the black line and its image was T -oVo of an inch, 
 which is about 2| times the diameter (-0004) of the spurious 
 disq corresponding to each point of the original. If I had 
 narrowed the black slit still more, say to "0004 or so, and at 
 the same time had been able to estimate the relative bright- 
 nesses of the parts a b and b c, I should have found the 
 percentage of wasted light in a b still greater compared 
 with b c. But tremor in the telescope might, even at the 
 best of times, vitiate any measurements, could they be taken, 
 with the black line and slit as narrow as that. The greatest 
 precautions were taken to guard against vibration and cur- 
 rents of air ; the objective had the usual colour correction, 
 was perfectly free from spherical aberration, and perfectly 
 squared on, and always allowed to arrive at a uniform 
 temperature before experimenting, for slight warming and 
 cooling give rise to negative and positive spherical aberra- 
 tions respectively. The experiments were carefully repeated 
 on several different occasions. I never saw, even for an 
 instant, the part of the slit a b become darker than. a bright 
 grey when made to exactly overlap the image of the black 
 line. I occasionally noticed tremor in the definition of the 
 black line as seen in the window, but it was rarely enough to 
 cause momentary encroachments of light into the slit, the 
 measurements of whose brightness were only taken when the 
 image of the slit in the window was seen to be perfectly 
 steady. The width of the slit S l Si was slightly but 
 
122 
 
 certainly narrower than the image of the black line, so as to 
 insure the edges of the white surfaces on either side being 
 thoroughly hidden. 
 
 But here it might be asked whether some, if not a large 
 part, of this wasted light could not be accounted for by those 
 diffraction rings Which are well known to surround the 
 spurious disc. The integrated brightness of innumerable 
 series of such rings overlapping one another over the area of 
 the black line might be expected to give a perceptibly 
 luminous effect. I scarcely knew what to expect theoretically 
 from this cause, but resolved to put it to the test of an 
 experiment with a 3^-inch triple objective (on the principle 
 which I had the pleasure of bringing to your notice in March). 
 This objective, of about 59 inches focal length, was mounted 
 in exactly the same way as the i2*6-inch in front of the fiat 
 mirror, and exactly the same system of slits (figs. 2, 3, and 4) 
 and eyepiece as had been used for the i2'6-inch o.-g. was 
 applied to the 3^ o.-g. in precisely the same way. Here a 
 very different state of things was at once apparent. After 
 making all the requisite adjustments, and getting the image 
 of the black line exactly focussed into the slit S A Si, the 
 latter became almost totally black. I say almost, because I 
 suspected a small trace of light to be present, but it was so 
 small that it could scarcely have amounted to as much as 
 10 per cent, of the brightness of the white surface. At the 
 same time I should expect to see a little residual light, owing 
 to the above-mentioned diffractive phenomena. However, 
 there was no mistaking the marked difference in the behaviour 
 of the two telescopes the first with the secondary spectrum 
 of an ordinary 25-inch objective, and the second with practi- 
 cally no secondary spectrum. The latter gave more black 
 and white definition than the former. In my paper on the 
 secondary spectrum of November last I estimated, on theo- 
 retical grounds, that the light lost for defining purposes, 
 owing to the secondary spectrum, in the case of a 24-inch 
 objective of 360 inches focal length should be not less than 
 42 per cent, of the whole light transmitted. My experiment 
 with the i2'6-inch objective used as its own collimator, so as 
 to be equivalent to a 25'2-inch objective of 370 inches focal 
 
I2 3 
 
 length, has left no doubt in my mind that that estimate is 
 not an exaggerated one. Other things being equal, if we can 
 win back to correct focus this aberrant light in the case of a 
 24-inch objective of 360 inches focal length, we shall then be 
 increasing the brightness of very small stars viewed through 
 it in the ratio of 60 to 100 ; in other words, we shall add 66 
 per cent, to their brightness, and increase space penetrating 
 power accordingly, as well as obtain a more perfect definition 
 upon objects presenting extension and details. 
 
 I have above described the image of the black line as seen 
 through the little window, showing the white surface at either 
 side, as appearing fairly dark, and I arrived at the conclusion 
 that the real luminosity was about f of that of the white sur- 
 face. It then occurred to me to try the effect of viewing a 
 narrow strip (J inch wide) of neutral-tint glass transmitting 
 56 per cent, of light, and placed in front of a uniformly 
 illuminated sheet of white paper placed at the end of a dark 
 passage. I then went to such a distance off as to cause the 
 strip of glass to present an angular diameter of from 5 to 9 
 minutes, and was surprised to see how much the apparent 
 dark line formed by it stood out. It appeared of a medium 
 grey tint. I then tried the effect of two such strips super- 
 imposed, together transmitting 31 per cent., and then at the 
 same distance the effect was a distinctly marked line of a 
 dark grey colour, the contrast seeming to me, if anything, 
 greater than that visible in the case of the image of the black 
 line seen in the window of the eye-piece in the i2'6-inch 
 telescope. 
 
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