si N9 LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class A SYSTEM OF APPLIED OPTICS OF APPLIED OPTICS BEING A {COMPLETE SYSTEM OF FOEMUL.E OF THE SECOND OEDEE, AND JHE FOUNDATION OF A COMPLETE SYSTEM OF THE THIED OEDEE, WITH EXAMPLES OF THEIE PEACTICAL APPLICATION BY H.- DENNIS TAYLOR v\ OPTICAL MANAGER TO T. COOKE AND SONS, LTD. OF LONDON, YORK, AND CAPE TOWN *>v or THF X UNIVERSITY ) or ILonUon MACMILLAN AND CO., LIMITED NEW YORK : THE MACMILLAN COMPANY 1906 All rights reserved CONTENTS WITH INDEX TO THE MOST IMPORTANT FORMULAE PAOB INTRODUCTION ....... 1 SECTION I FORMULA OF THE FIRST ORDER OF APPROXIMATION FORMULA I. Reflector. Conjugate Focal Distances . ... 6 II. Single Surface Refraction. Conjugate Focal Distances . . 7 III. Refraction by Thin Lens. Conjugate Focal Distances . . 10 IV. Position of First Principal Point of a Thick Lens . . .14 V. Position of Second Principal Point of a Thick Lens . . .14 VA. Back Focal Length of a Thick Lens . . . . .17 VI. Equivalent Focal Length of a Thick Lens . . , .-17 VII. .... 17 VIII. Percentage Variation in Power of a Lens due to Thickness . 18 SECTION II THE THEOREM OF ELEMENTS . . . " ; 20 SECTION III EQUIVALENT FOCAL LENGTHS AND PRINCIPAL POINTS OF THICK LENSES AND OF LENS SYSTEMS IXA. Position of First Principal Point of a Thick Lens or Pair of Elements, equivalent to Formula IV. . . . . .26 IXe. Position of Second Principal Point of a Thick Lens or Pair of Elements, equivalent to Formula V. . . . .26 156182 vi A SYSTEM OF APPLIED OPTICS IXc. Back Focal Length of a Thick Lens or Pair of Elements, equiva- lent to Formula VA. . . . . .-26 X. Equivalent Focal Length of a Thick Lens or Pair of Elements, equivalent to VI. and VII. . . . . .27 X!A. Three Separated Thin Lenses or Elements. Position of First Principal Point . . . . . .33 Xle. Three Separated Thin Lenses. Position of Second Principal Point 33 XII. Three Separated Thin Lenses. Equivalent Focal Length . 34 XIII. Four Separated Thin Lenses or Elements. Equivalent Focal Length . . _.,_ v~- 34 XIVA. Four Separated Thin Lenses or Elements. Position of First Principal Point . . . . ... 35 XIVs. Four Separated Thin Lenses or Elements. Position of Second Principal Point . . . . . .35 XV. Five Separated Thin Lenses. Equivalent Focal Length . 36 XV A. Five Separated Thin Lenses. Position of First Principal Point 36 XVB. Five Separated Thin Lenses. Position of Second Principal Point 37 XVI. Six Separated Thin Lenses or Elements. Equivalent Focal Length 37 XVlA. Six Separated Thin Lenses or Elements. Position of First Prin- cipal Point ...- ... .; -, ; - ^ - . . . ;. .:?. . 37 XVlB. Six Separated Thin Lenses or Elements. Position of Second Principal Point . /* . . . % . . . 37 XVII. Eight Separated Thin Lenses or Elements. Equivalent Focal Length ....... 38 XVIlA. Eight Separated Thin Lenses or Elements. Position of First Principal Point . . . .... 38 XVIlB. Eight Separated Thin Lenses or Elements. Position of Second Principal Point . . . . . 38 SECTION IV SPHERICAL ABERRATION. COMPLETE FORMULA OF THE SECOND AND OF THE THIRD ORDERS OF APPROXIMATION XVIII. (E.) Direct Refraction at Single Surface Spherical Aberration . 56 XVIII. (L.) Linear Value of above Aberration . . . * 55 XIX. (E.) Spherical Aberration of a Thin Lens or Element -.''" , 56 XIX. (L.) Linear Value of above Aberration - , c - i -% . , 57 XX. (R.) Refraction at a Single Surface Aberration to the Third Approximation . '".-- . . . . 63 XXI. (E.) ( Thin Lens or Element. Spherical Aberration of the Second and ) 64 XXII. (E.) [ of the Third Orders of Approximation . . ' , . J 65 CONTENTS vii FORMULA PAGE XXIII. (R.) Thin Lens or Element. Spherical Aberration of Second Order expressed in Coddington's Notation . . .67 XXIV. (R.) Spherical Aberration of Third Order expressed in Coddington's Notation . ... . . . .67 XXV. (R.> Aberration of a Parallel Plane Plate . ...... .79 XXVI. Aberration of First Element of a Thick Lens, including Versine Correction of the Order Y 4 / . . .88 XXVII. Aberration of Second Element of a Thick Lens, including Versine Correction of the Order Y 4 . . . .89 XXVIII. Value of 7/ 2 for Second Lens or Element of a Series, including Variation of Y 2 due to Aberration of L : . . .92 XXVIIlA. Aberration of Second Lens or Element of a Series, including that of the Order Y 2 4 due to Aberration of Lj . .92 XXVI I IB. Aberration of Third Lens or Element of a Series, including that of the Order Y 3 4 due to Aberrations of Lj and L 2 . 93 XXVI lie. Aberration of Fourth Lens or Element of a Series, including that of the Order Y 4 4 due to Aberrations of L p L 2 , an^L 3 93 XXIX. Variation in the Aberration of Second Lens or Element of a Series due to Variation in a 2 consequent upon Aberration of Lj 95 (27) to (31). Hybrid Spherical Aberrations . . .96 and 97 XXX. Differential of the Spherical Aberration with respect to a . 110 XXXI. Differential of the Spherical Aberration with respect to x . 110 XXXII. Linear Diameter of the Least Circle of Confusion . .110 XXXIII. Angular Diameter of above, subtended at Lens Centre . 110 XXXIV. (R.) Spherical Aberration of a Spherical Reflector . .113 SECTION V CENTRAL OBLIQUE REFRACTION OF PENCILS THROUGH THIN LENSES OR ELEMENTS. I. Normal Curvature Errors of a Thin Lens in Secondary Planes, in detail 120 II. (R.) Spherical Aberration of Direct or Oblique Pencils refracted through a Thin Lens . . "i . . . 120 III. (R.) Normal Curvature Error for a Thin Lens or Element in Secondary Planes. . ; . . . . 120 I V.(R.) Same as Formula II. (R.) . . . . . .124 V. (R.) Normal Curvature Error for a Thin Lens or Element in Primary Planes . , . . . .124 VI. Exact Formula for Curvature Errors of a Thin Lens or Element for Infinitely Narrow Pencils, Secondary Planes . . .127 viii A SYSTEM OF APPLIED OPTICS FORMULA PAGE VII. Same as above in Primary Planes . , ~ -. , . , . 127 VIII. Variation in Secondary Normal Curvature Error due to Variation of ft, when p is constant . .. . . . .129 IX. Variation in Primary Normal Curvature Error due to Variation of /x, when p is constant . . . . . . 129 X. Variation in Secondary Normal Curvature Error] due to Variation of ft, when / is constant ...... 129 XI. Variation in Primary Normal Curvature Error due to Variation of /x, when / is constant . . . . . .130 XII. Normal Curvature Error of Spherical Reflector in Primary Planes . 133 j-Exact Formula for above for Infinitely Narrow Pencils . .135 SECTION VI ECCENTRIC OBLIQUE EEFRACTION or PENCILS THROUGH THIN LENSES OR ELEMENTS I. The Spherical Aberration of all Pencils passing through a Stop of Semi- Aperture 8 . . . . . .142 II. The Spherical Aberration E.G. in Secondary Planes, in terms of D and U . . . . . . . . 142 III. The Comatic E.C. in Secondary Planes, in terms of D and U .142 IV. The Spherical Aberration E.C. in Primary Planes, in terms of D and U . . . . . .147 V. The Comatic E.C. in Primary Planes, in terms of D and U .148 VI. The Spherical Aberration E.C. in Secondary Planes, in Final Self- Interpreting Form . -. . . . .149 VII. The Comatic E.C. in Primary Planes, in Final Self-Interpreting Form . . . ";' . . . . 150 VIII. The Complete Formula, including both Spherical Aberration E.C. and Comatic E.C., in Primary Planes .... 150 VI HA. and B. Coddington's Formula for E.C.S in Secondary Planes . 151 VIIIc. Condition of Elimination of E.C.S . . . .163 VIIIo. Above in Quadratic Form . . . . 153 IX Differential of the E.C.s with respect to a . . . .154 X. Differential of the E.C.S with respect to ft . . .154 XI. Differential of the E.C.s with respect to x . . .. .154 CONTENTS ix SECTION VII ON SYSTEMS OF LENSES AND THE APPLICATION OF THE THEOREM OF ELEMENTS TO THICK LENSES FORMULA PAGE XII. Simple Lens. Eadius of Curvature of Secondary Image when Primary Image is Flat . . . . .155 XIlA. Simple Lens. Curvature Error of the Anastigmatic Image . 156 XIII. Condition necessary to the attainment of a Flat Anastigmatic Image . . . . . . .156 XII IA. The Petzval Theorem . . . . .157 XIV. The Petzval Formula for the Reciprocal of the Eadius of Curva- ture of the Anastigmatic Image . . . .158 66 and 67. Radii of Images formed by Central Oblique Refraction through Lens fulfilling Petzval Condition . . . .159 XV. Compound Lens. Reciprocal of the Radius of Curvature of the Secondary Image when the Primary Image is Flat . .168 XVI. Compound Lens. Reciprocal of the Radius of Curvature of either Secondary or Primary Image when Mean Image is Flat . 169 XVII. E.F.L. of a Triplet Lens showing Increment to Power due to Separations . . . . . .169 XVIIlA, Aberration and Curvature Error due to Oblique Refraction through a Parallel Plane Plate . . . .177 XVIII. (R.) Do., expressed in terms of 2 and w 2 . . . .177 XIX. (L.) Linear Value of Curvature Error due to Oblique Refraction through a Parallel Plane Plate . . . .180 XX. E.C. or Curvature Correction due to Eccentric Reflection from Spherical Mirror . . . . . " .189 SECTION VIII COMA AND THE SINE CONDITION VON SEIDEL'S SECOND CONDITION CENTRAL OBLIQUE REFRACTION I. The Length of the Comatic Flare in Primary Plane . . .195 II. The Angular Value of the Coma in Primary Plane, as subtended at Lens Centre ....... 197 III. The Angular Value of the Interval between Oblique Central Ray and the Focus for the Two Rays in Secondary Plane . . .198 IV. Linear Diameter of the Coma in the Secondary Plane . . 200 V. General Proof. Linear E.C. in the Secondary Plane ' . - . 213 VI. General Proof. Linear E.C. in the Primary Plane . . .214 VII. Condition of Elimination of Coma from a Double Objective . .216 x A SYSTEM OF APPLIED OPTICS FORMULA PAGE VIII. Relation between x and a for a Simple Lens free from Coma . 217 IX. Spherical Aberration of a Simple Lens free from Coma . .218 X. Condition of Least Spherical Aberration for a Simple Lens . 218 XI. Difference between Conditions of Least Spherical Aberration and of no Coma . . . . . . .218 XII Spherical Aberration of a Simple Lens differentiated with respect to a . . . . . . . . 218 XIII. Condition of Greatest Possible Constancy of Aberration when the Vergency varies . . . . . .218 XIV. Difference between Condition of Vergency Insensitiveness and of no Coma . . . . . . .218 XV. Angular Value of the Coma in Primary Plane by Central Oblique Reflection from Spherical Mirror . . . .219 SECTION VIIlA COMA AT THE Foci OF ECCENTRIC OBLIQUE PENCILS XVI. The Linear Value of the Spherical Aberration Coma in Primary Plane . . . . . . .225 XVII. Universal Expression for the Angular Value of the above, sub- tended at Lens Centre . . . . .225 XVIII. The Linear Value of the Coma Proper in Primary Plane . . 227 XIX. Universal Expression for the Angular Value of the above, subtended at Lens Centre . . . . .228 XX. Universal Expression for the Angular Value in Primary Plane of both sorts of Coma, subtended at Lens Centre . .228 XXA. Simple Lens. General Condition of Elimination of E.C.s . 229 XXB. Simple Lens. General Condition of Elimination of Coma . 229 XXc. Simple Lens. General Expression for the Coma when E.C.S are Eliminated . . ' . . . . .230 XXo. Simple Lens. General Condition of the Elimination of Astig- matism ....... 230 XXE. Simple Lens. General Expression for Angular Value of the Coma when there is no Astigmatism . . . .230 XXp. Simple Lens. General Expression for the KC.s when Coma is Eliminated . . . .- . . 231 XXo. Simple Lens. General Expression for the Astigmatism when there is no Coma . . . . . . 232 XXI. Expression for Angular Coma for Two Lenses in Succession . 233 XXII. Expression for Angular Coma for Three Lenses in Succession . 233 XI XXIIlA. Expression for Angular Value of the Coma in the Primary Plane, produced by Oblique Eefraction through a Parallel Plane Plate . . . 7 . . . Jp 237 XXI I IB. The above when an Element occurs at the Second Surface ; . " . 237 XXIV. Angular Value of the Coma in Primary Plane produced by Eccentric Oblique Reflection from a Spherical Mirror .,-,,: v ' - :. rr . ? 243 SECTION IX DISTORTION AND THE EECTILINEARITY OF IMAGES VON SEIDEL'S FIFTH CONDITION I. The Expression for the Tangent Eatio apart from Spherical Aberration 250 II. Coddington's Formula expressing Ratio between the Tangents of Emerging and Entering Principal Rays, inclusive of the Spherical Aberration . . . . . . .251 HA. Coddington's Formula for above in case of Two Lenses in Succession 254 III. New Universal Formula for Distortion of Image . . . 262 IV. Same, applied to a Series of n Lenses or Elements . . .264 (23). The Linear Distortion produced by a Parallel Plane Plate . .267 V. Formula for the Fractional Distortion produced by a Parallel Plane Plate . . . . . -.- . 268 Formula for the Magnification produced by a Telescope . .278 VI. Formula for the Magnification produced by a Simple Microscope or Lens ........ 279 VII. Formula for the Magnification produced by a Compound Microscope . 280 SECTION X ACHROMATISM I. Variation of the Power of a Lens due to Changes in Refractive Index consequent upon Colour . . . . .281 I A. Diameter of Least Circle of Chromatic Confusion . . .282 IB. Angular Value of above, subtended at Lens Centre . . .282 II. Condition of Axial Achromatism for Two Lenses in Contact . 283 III. Condition of Axial Achromatism for Two Separated Lenses . . 284 IV. Condition of Axial Achromatism for a Series of Separated Lenses . 285 Chromatic Variation of after Direct Refraction through a Parallel Plane Plate 286 xii A SYSTEM OF APPLIED OPTICS FORMULA PAGE VI. The Linear Value of above, as a Variation of v . . .287 VIA. The Exact Expression for the above . . . .287 VII. Differential of the Spherical Aberration of a Lens with respect to p. . 288 VIII. Two Separated Lenses, Value of y 2 in terms of y l . . . 290 IXA. Separation between Lenses of Two -lens Eye -piece necessary for Oblique Achromatism . . . . . .293 IXa The same when the Two Lenses are of Different Dispersive Powers . 294 IXc. Condition of Oblique Achromatism for a Three-lens Eye-piece . 294 X. Condition of Oblique Achromatism for a Four-lens Eye-piece . 295 XI. Single Lens. New Universal Formula for Ratio between Radial Dimensions of Conjugate Images for Different Colours . .297 XII. Same, applied to a Series of n Lenses in Succession . . .298 XIII. Condition of Achromatism when all Lenses of a Series are of the same Dispersive Power . . . . . .298 XIV. Parallel Plane Plate. Ratio between Dimensions of Variously Coloured Images ...... 300 SECTION XI A BRIEF SKETCH OF THE NORMAL AND OTHER CURVATURE ABERRATIONS OF THE THIRD ORDER TAN 4 , ETC. CENTRAL OBLIQUE REFRACTION (2). Functions of the y's corrected for Obliquity, Primary Plane . .314 (3). y corrected for Obliquity, Secondary Plane . . . .315 (4). Above, plus Versine Corrections to the y's, Primary Plane . .316 (5). Above, plus Versine Correction to y, Secondary Plane . .316 ECCENTRIC OBLIQUE REFRACTION (6). Obliquity Corrections to the y's, Primary Plane . . .318 (7). Obliquity Corrections to y, Secondary Plane . . . .318 (8). Versine Corrections to the y's, Primary Plane . . .319 (9). Versine Corrections to y, Secondary Plane .... 320 CENTRAL OBLIQUE REFRACTION (13). Corrections for Separation between the y's, Primary Plane . .324 ECCENTRIC OBLIQUE REFRACTION (14). Corrections for Separation between the y's, Primary Plane . . 325 CONTENTS xiii THE INTRINSIC SPHERICAL ABERRATION or THE THIRD ORDER CENTRAL OBLIQUE REFRACTION FORMULA PAGE (16). The Corrections of Second and Third Orders in Primary Plane . 327 (IV). The same Corrections in Secondary Plane .... 327 ECCENTRIC OBLIQUE REFRACTION (19). The Corrections of the Third Order in Primary Plane . . 328 (20). The same Corrections in Secondary Plane . . . .329 Summary of all above Results . . . . .329 THE PLATES NOTE AS TO THE REFERENCE LETTERS WHILE ordinary Latin capitals and minuscules were selected as being the more suitable for the diagrams, owing to their upright character, yet the corresponding small italic letters have been employed in the text because of their greater conspicuousness. In case this should cause inconvenience to any reader of other than English nationality, who is not quite familiar with both these forms of small letters, the following list of the two sets of equivalent characters may be found useful : LETTERS IN DIAGRAMS a b c d e f g h i j k 1 EQUIVALENTS IN TEXT LETTERS IN DIAGRAMS EQUIVALENTS IN TEXT a n n b c d P q P 2 e r r f 9 h s t u s t u i V V 3 k w X w X I y y m z z LIST OF PLATES PLATE I NO. OF DIAGRAMS BETWEEN PAGES Figs. 1 to 4h. Conjugate Focal Distances for Plane and Spherical Reflecting and Refracting Surfaces . . 6 and 7 PLATE II Figs. 4g to 7b. Conjugate Focal Distances for Second Surfaces and Lenses 10 and 11 THE PLATES xv PLATE III NO. OF DIAGRAMS BETWEEN PAGES Figs. 8a to 14b. Thick Lenses and Principal Points . . 12 and 13 PLATE IV Figs. 15a to 19b. Illustrating the Theorem of Elements 22 and 23 PLATE V Figs. 20 to 24. Principal Points of Systems of Separated Lenses . 36 and 37 PLATE VI Figs. 25 to 30a. Illustrating Principal Points of various Eye- pieces, etc, . - - , >- - . . 42 and 43 PLATE VII Figs. 31 to 34. Spherical Aberration and Illustrations of Shape and Vergency Characteristics . * . . 54 and 55 PLATE VIII Figs. 35a to 41. Spherical Aberrations of Second and Third Orders, etc. . . . . . . . . 70 and 7 1 PLATE IX Figs. 42 to 45a. Central Oblique Refraction through Thin Lenses or Elements . . . . .116 and 117 PLATE X Figs. 46 to 47b. Illustrating the True Curvature of Images formed by Central Oblique Refraction . .130 and 131 PLATE XI Figa 48 to 5 la. Illustrating Eccentric Oblique Refraction . .138 and 139 PLATE XII Figs. 52 to 60. Eccentric Oblique Refraction through Thick Lenses and Separated Lens Systems . .174 and 175 PLATE XIII Figs. 6 la to 6 8c. Elucidating Coma and its Structure . . .194 and 195 PLATE XIV Figs. 69 to 74. Various Phases of Coma and Astigmatism . . 208 and 209 xvi A SYSTEM OF APPLIED OPTICS PLATE XV NO. OF DIAGRAMS BETWEEN PAGES Figs. 75P to 76S. Further Phases of Coma blended with Astig- matism . . . . . . . 210 and 211 PLATE XVI Figs. 77 to 79f. Professor Silvanus Thompson's Drawings of Coma, etc. 212 and 213 PLATE XVII Figs. 80 to 83c. Spherical Aberration Coma at the Foci of Eccentric Oblique Pencils . . . .222 and 223 PLATE XVIII Figs. 84b to 86b. Coma Proper at the Foci of Eccentric Oblique Pencils . . . . . . . 226 and 227 PLATE XIX Figs. 87 to 93a. Distortion of Images 246 and 247 PLATE XX Figs. 94 to 98a. Distortion of Images formed by Systems of Separated Lenses . . . . .252 and 253 PLATE XXI Figs. 99a to 103. Distortion considered from a new standpoint . 266 and 267 PLATE XXII Figs. 104 to 11 Ob. Achromatism of Images, Axial and Oblique . 286 and 287 PLATE XXIII Figs. Ill to 115a. Curvature Errors and Eccentricity Corrections of the Third Order tan 4 . . . .312 and 313 PLATE XXIV Figs. 116 to 123. Dr. von Eohr's Graphs of the Deviations from a Plane Image for some Actual Lenses . 1332 and 333 or THf ( UNIVERSITY INTRODUCTION I TKUST that the student of Optics who casually scans the pages of this work for the first time, will not be alarmed by the complicated appearance of some of the formulae employed in the course of working out the conclusions, and therefore infer that it is necessary to be highly trained in mathematics in order to follow the lines of reasoning employed. For such is not the case ; all that is really necessary in the mathematical equipment of the student being an easy acquaintance with the ordinary manipulations of Algebra, together with a clear grasp of the Binomial Theorem, the chief propositions of Euclid, and the rudiments of the Differential Calculus. That granted, and given some instinct for the practical application of what he knows, then he will have no insuperable difficulty in following this work from cover to cover. The greater part is easy compared to the numerous problems and theorems which the average university student is called upon to solve, and which in so many cases are treated as of purely theoretical interest. After all, is not that the truest and most fruitful teaching of mathematics which fully recognises the mutual support between theory and practice ? Otherwise it is but natural if the student cleaves to the one and despises the other. I do not wish to imply that there is no scope for the employment of the highest mathematical skill in optical science; for, on the contrary, there are numerous problems in connection with the corrections of the third order of approximation, merely glanced at in Section XL of this work, which pre-eminently call for the elucidating and marshalling influence of some clear-headed mathematician who shall be thoroughly familiar with the properties of lenses from practical acquaintance, and not only from the theoretical point of view. The closer approach to perfection in the optical combinations of the future will lie in the more thorough elimination of the corrections of the third order, and in some cases of the fourth order, and the most highly trained mathematical skill, if it should ever deign to busy itself in this B 2 A SYSTEM OF APPLIED OPTICS country with the higher practical requirements of optical science, would doubtless be able to evolve corollaries of the greatest importance bearing upon this question. My chief object in working out the scheme of Applied Optics herein explained, has been to arrive at a complete system of algebraic formulae of the second order which can be applied to any optical system likely to occur in practice with results which in general very closely approach to accuracy. I have therefore confined myself for the most part to the attainment of those practical conditions which have to be fulfilled by the best optical constructions conditions which include, and run closely parallel to, Von Seidel's five well-recognised conditions. As far as I know, there is only one work in the English language professing to give a sketch of Von Seidel's methods, and that is Professor Silvanus Thompson's Contributions to Photographic Optics? after Otto Lummer, while there are numerous accounts of his methods published in German works, and several treatises built upon them, such as Steinheil and Voit's Handbuch der Angewandten Optik? 1891, and Von Rohr's Theorie und Geschichte des photo- graphischen Objective, 3 the latter a most instructive and valuable work ; and last, but not least, Dr. Siegfried Czapski's new edition of Der Theorie des optischen Instrumenten, 4 1904. This last work is a philo- sophical, broad, and general survey of the various problems which have to be faced, and if possible solved, by the optical designer who would rise superior to mere rule of thumb. But its perusal requires in many respects a higher level of mathematical training than is necessary for the understanding of this treatise. In the German language there exists quite a mine of optical literature written by men who are practical opticians as well as mathematical experts, while we have scarcely anything of a corre- sponding nature in the English tongue. The fact that such works as I have just mentioned have been published in Germany (as first editions, at any rate) for so many years, and yet no demand has ever arisen for English translations, is only too painful evidence of the apathy with which the Science of Optics has been regarded in this country. There are, of course, various works on geometrical optics which have more or less recently emanated from our universities, such as Heath's Geometrical Optics, 5 Parkinson's Optics, 6 Pendlebury's Lenses and 1 Macraillan and Co., 1900. 2 Teubner, Leipzig. 3 Julius Springer, Berlin, 1899. 4 Earth, Leipzig. 5 Cambridge University Press, 1895. 6 Macmillan and Co., 1900. INTKODUCTION 3 Systems of Lenses? Perceval's Optics, etc., which are excellent as furnishing material for purely mathematical students working up for examinations ; but the manner in which the various problems are dealt with is in many cases ill adapted for application in practice, while certain matters of the highest importance are ignored altogether. As a matter of fact there is not an English work on geometrical optics extant by whose guidance an ordinary photographic lens could be worked out in all particulars. Professor Silvanus Thompson's account of Von Seidel's system does not, however, give the impression that the latter's methods and notation are at all easy to comprehend, but certain it is that his system has been successfully employed for very many years by numerous mathematicians and opticians of the highest rank on the Continent, while the foundation-stone of English optical science has been left unbuilt upon. I here allude to the all-important work which was done about thirty years before that of Von Seidel by Sir George Airy, and still more by Henry Coddington. Sir G. Airy published some highly important papers in 1827 in the Cambridge Philosophical Transactions on " The Spherical Aberration of Eye-pieces of Telescopes," and another paper on the Achromatism of the same. Then Henry Coddington took up the work, and by the aid of some very ingenious devices of his own contrivance greatly added to the simplicity and universality of the formulae arrived at by Airy. In 1829 he published his labours under the title, A Treatise on the Reflection and Refraction of Light, which, although still the best work on geometrical optics from the practical optician's point of view, nevertheless contains many shortcomings, which I attribute chiefly to the fact that he had not had very much practical acquaintance with lenses and their properties. It is therefore with much diffidence that I venture to criticise and to supplement many of his methods and formulae, especially when I feel sure that had it not been for his labours this treatise would never have been undertaken. Another very important work on geometrical optics, now very little known, was Eichard Potter's Elementary Treatise on Optics, Part II. of which, published in 1851, contains certain formulas for spherical aberration of the third approximation. I may here state that the invention of the " Cooke " lenses for photography was not of a haphazard nature, but occurred in this way. I had been studying Coddington's work very carefully and did not feel quite satisfied with his method of working out the curvature 1 Deighton, Bell and Co., Cambridge, 1884. 4 A SYSTEM OF APPLIED OPTICS of the image formed by a lens, in the cases of both central and eccentric oblique refractions. He assumed the aperture of the pencil of rays in question to be infinitely narrow, and got at his results by the employment of the differential calculus. I saw that while this would be quite valid for such infinitely narrow pencils, still, as con- siderably broad pencils generally occur in practice, it struck me it might 'be worth while trying to devise a method not dependent upon the calculus, whereby the foci of broad oblique and eccentric pencils could be elucidated, when possibly some new results of practical importance might be forthcoming. About the year 1890 I undertook that task, and after meeting with many difficulties which almost com- pelled me to give up the investigation as hopeless, I at last succeeded in arriving at the results embodied in Sections V., VI., and VII. of this volume, and in so doing was fortunate enough to bring to light the formula relating to coma, a phenomenon that appears, strange as the fact may seem, never to have been noticed by Coddington. I then saw that the formula I thus arrived at implied corollaries of the greatest practical importance, and I was led almost directly to the conception of the Cooke lens, that is, of the older complex Cooke lens built up of two achromatic positive lenses and one achromatic negative lens. The simple Cooke lens was of later conception. Thus the theory preceded the practice, although I should say that there are certain other features of the Cooke lens, such as distortion and oblique achromatism more especially, whose theory I did not arrive at until a few years later, so that in that respect the practice preceded the theory. Having subsequently worked out a complete system of formula?, which I have proved and tested and found reliable in all manner of ways, and recognising the great importance of theory and practice working loyally together for future improvements, I thought that as soon as I had time enough at my disposal I would gather together and arrange what has been the interrupted labour of many years, with a view to publication, if by so doing I could, even in a humble degree, forward the development of optical science in this country, wherein it has lain so long neglected, or perhaps furnish some raw material on which some far abler heads than mine should at some future time found important corollaries not yet dreamed of. Considerations of space have compelled me to confine myself to theorems and formulae that I consider to be of the greatest practical value, and to leave out many corollaries of minor importance that might be dealt with in a future edition, were it ever called for. There are also many problems and theorems untouched upon, INTBODUCTION 5 which are only of theoretical importance or of interest from a mathematical point of view, and of little value to the practical optician, such as, for instance, the theory of caustics, planes of unit magnification, etc., about which the more mathematical student can obtain full information from various contemporary works of well- known repute, such as those mentioned above, as well as Coddington's work, which, however, is now out of print and often difficult to procure. It will be observed that I have not given the lines of reasoning by which the formulae of the first approximation are arrived at ; for I have assumed that the student will bring with him to the study of this work a knowledge of such elementary optical formulae. For those who wish to enter upon it without that knowledge I do not know a better book to recommend as a clearly written first guide to the formulae of the first approximation than Todhunter's Optics (in Part IT. of his Natural Philosophy for Beginners, 1877, which I believe is also out of print) or Lardner's Optics, and the series of articles on " Applied Optics " by Dr. Drysdale in the British Optical Journal, I think it must be conceded that, while the method of in- vestigating the foci of oblique and eccentric pencils of finite or large aperture explained in this work leads to novel and highly important formulae of the second approximation, and some others which are novel in many respects, it also opens out possibilities of working out formulae of the third and in some cases the fourth approximations, which in the hands of a skilful mathematician may lead to new and useful results of great importance ; while the application of the differential method of Coddington and other workers to infinitely narrow pencils is exceed- ingly limited in its scope and results, as I shall show. At first sight it seems a remarkable thing that a system of surfaces bound by the simplest of all known curves, namely, the circle, with their centres on a common axis, should give rise to problems which, if solved to a high degree of exactitude, are of such extraordinary complexity. I gladly take the present opportunity of expressing my thanks to Sir W. de W. Abney and Professor Silvanus P. Thompson for much kind encouragement and valuable help; and also to Dr. Moritz von Eohr for allowing me to reproduce some of his diagrams on Plate XXIV. In conclusion, I shall be only too glad if any technical errors or obscurities, which must, in spite of all care, exist in a work of this kind, are pointed out to me. H. DENNIS TAYLOR SECTION I A KECAP1TULATION WE will first of all recapitulate those well-known formulae of the first approximation relating to ultimate axial rays constituting direct or axial pencils, or, in other words, extremely narrow pencils whose central or principal ray coincides with the axis or straight line joining the origin or apex of the pencil to the centre of curvature of the spherical surface. Spherical aberration is in such cases a vanishing quantity and is therefore not regarded. Throughout this work it is assumed that all reflecting and refracting surfaces are either plane or spherical. Law connecting con- jugate focal dis- tances for plane reflector. Formula connect- ing conjugate focal distances for spherical reflec- tor. Case of a Plane or Curved Reflector Throughout the diagrams in this book light is supposed to be travelling from left to right. Plane reflector. Here if Q (Plate I.) be the origin and Q . . A, the principal ray, be perpendicular to the reflecting surface B . . R, then after reflection the rays will proceed backwards as if originating from a virtual point q situated on Q . . A projected and at a distance A . . q from the surface equal to A . . Q. On the contrary, if the incident pencil is of rays converging to the apex q, then they will be reflected back to a real point Q such that A . . Q = A . . q and Q . . . q is normal to K . . R If the reflecting surface be curved spherically as r . . r, Figs. 2a, 2b, 2c, and 2d, c being the centre of curvature and Q the origin or apex of the incident pencil, then the formula 1 1 A..g A..C A..Q ~~ I universally applies and interprets itself in all cases if the following conventions are strictly adhered to, viz. 6 PLATE. I PLATE. I SECT. I A EECAPITULATION The radius of curvature A..C is to be considered as an intrinsically positive quantity whether the surface be convex or concave; and then For concave reflector If rays of the incident pencil are divergent, then Q . . A is positive. Reflector Conven- If rays of incident pencil are convergent, then A . . Q is negative. If rays of reflected pencil are convergent, then A . . q is positive. If rays of reflected pencil are divergent, then q . . A is negative. And for convex reflector If rays of incident pencil are convergent, then A . . Q is positive. If rays of incident pencil are divergent, then Q . . A is negative. If rays of reflected pencil are divergent, then q . . A is positive. If rays of reflected pencil are convergent, then A . . q is negative. For instance, in the case of Fig. 2d we have T = - -r ^r, but Instances of appii- A ? A . . (^ cations of signs to by convention A . . Q is a negative quantity, therefore the formula is reflected pencils. T = - 7\ or ^ + T ~, therefore A . . Q comes out diver - A q " A . . 1^5 -C A . . v^j gent and positive. Should Q . . A or A . . Q be infinite or the rays of the incident pencil be parallel, then of course becomes zero, and be- 2 i A. . Q A. .q comes -r p or , and the rays converge to or diverge from the prin- cipal focus of the mirror. The dotted lines in the figures indicate negative distances, and the full lines the positive distances. Plane Refracting Surfaces In the case of normal or perpendicular incidence of small pencils at a plane refracting surface bounding a transparent substance whosf refractive index = //,, while that of the left-hand medium =^, the /* = refractive index, simple relationship A . . q = fi(A . . Q) holds good. See Figs. 3a and 3b. Spherical Refracting Surfaces In the case of direct refraction of normal pencils by spherical surfaces, as in Figs. 4a, b, c, d, e,f, g, and li, the formula or fJL _ [L- Aj = A77C"A..Q Formula connect- ing focal distances in case of refrac- tion at single surface. 8 A SYSTEM OF APPLIED OPTICS SECT. holds good if we put u for A . . Q, r for the radius A . . 0, and u for A . . q, and this formula interprets itself for all cases, provided the following conventions are strictly adhered to, viz. : Convention as to The radii of all surfaces, whether convex or concave, to be con- tances f fOCal ^ sidered intrinsically positive with respect to the conjugate distances whose signs are to be assessed. Then for convex surfaces Eays of incident pencil diverging, then Q . . A or u is positive. Figs. 4a and 4e. Eays of incident pencil converging, then A . . Q or u is negative. Figs. 4c and 4#. Eays of refracted pencil converging, then A . . q or it is positive. Figs. 4 a, 4c, and 4g. Eays of refracted pencil diverging, then q . . A or u is negative. Fig. 4e. And for concave surfaces Eays of incident pencil converging, then A . . Q or u is positive. Figs. 4& and 4/. Eays of incident pencil diverging, then Q . . A or u is negative. Figs. 4:d and 47i. Eays of refracted pencil diverging, then q . . A or u is positive. Figs. 4&, 4d, and 4A. Eays of refracted pencil converging, then A . . q or u is negative. Thus, in the case of Fig. 4c, A . . Q is convergent and therefore u is negative, and becomes gent. it r u Rays entering con- u - 1 vex surface conver- -- r And, again, in a case where Q. . A in Fig. 4 becomes less than - , then Rays leaving convex of course " = - gives a negative result, and the refracted pencil surface divergent. is shown to be divergent, as in Fig. 4e. If the rays of the incident pencil are parallel and therefore ;=- T- = zero, Q. .A therefore A KECAPITULATION and Focal distance when entering rays are parallel. If, on the other hand, QA = r,thenT = 0, and the rays of the fJL ~~~ 1 il refracted pencil are parallel. We are now in a position to consider the cases of two spherical surfaces in succession enclosing glass between them and forming a lens. We will assume the axial thicknesses of such lenses to be negligible, the two spherical surfaces being brought to a sharp edge in the case of collective lenses and the diameter or aperture being very small compared to the principal focal length, while in the case of dispersive lenses the two spherical surfaces may be supposed to touch one another on the lens axis, the axial thickness being zero. Let us take a case like Fig. 4 a, wherein the rays after refraction at the first surface are convergent and it, is positive. Let these convergent rays proceed through a second convex surface, as shown in Fig. 5 a. We saw that in the case of Fig. 4a the distance A . . q or u was /* . -, from which we get - = u u given by the equation = u r u u r u We can apply this equation to the refraction, taken in the reverse direction, at the second surface, as shown in Fig. 5a, Plate II., wherein A 2 . . Q 2 corresponds to u, and A z ..q = ii; only in this case A 2 . . Q 2 may be better expressed as = v, and the radius of curvature as s, so that we get /*_/*-! 1 U S V and !_/*-! /* v s it' But as the rays of the pencil are converging (left to right) into the second surface, and the distance it becomes, relatively to the second surface, negative, therefore the above equation becomes !_/*- 1 v But ~ by the refraction at the first surface was shown to be u Substituting this value in the above equation we get s it Refracted parallel. rays Two closely follow- ing surfaces consti- tute a lens. So far, lenses assumed to have no central thickness. Course of rays at second surface con- sidered reversed. 10 A SYSTEM OF APPLIED OPTICS Formula connect- or ing conjugate focal distances in the case of a lens. / i -= (fj.- !)(- + -) - - III. which well-known formula applies to all thin lenses whatsoever under the following conventions. Conventions as to signs of radii. Conventions as to signs of conjugate focal distances. Conventions as to signs of radii. Conventions as to signs of conjugate focal distances. Collective Lenses The focal length of a collective lens must be considered a positive quantity with respect to the conjugate focal distances. The radii of all convex surfaces are considered intrinsically positive, while the radii of all concave surfaces are considered intrinsically negative, their radii, of course, being always numerically greater than the radii of the convex surfaces in the same lenses, so that the deeper curved surface determines the character of the lens. If rays of incident pencil are diverging, u is real and + Figs. Qa and Qe. If rays of incident pencil are converging, u is virtual and . Fig. 6c. If rays of emergent pencil are converging, v is real and + . Figs. 6a and Qc. If rays of emergent pencil are diverging, v is virtual and . Fig. 6e. Dispersive Lenses The focal length of a dispersive lens is also to be considered a posi- tive quantity with respect to the conjugate focal distances. The radii of all concave surfaces are considered intrinsically positive, while the radii of all convex surfaces are considered intrinsically negative, their radii, of course, being always numerically greater than the radii of the concave surfaces in the same lenses, the deeper curved surface again determining the character of the lens. If rays of incident pencil are converging, u is virtual and + . Figs. 6& and 6/. If rays of incident pencil are diverging, u is real and . Fig. Qd. If rays of emergent pencil are diverging, v is virtual and + . Figs. 66 and Qd. If rays of emergent pencil are converging, v is real and . Illustrations. Mean- Figs. Qa, b, c, d, e, and / are illustrations of these conventions. dotted lines. 11 and As in Fi g- 4 > and generally throughout this book, all intrinsically positive distances are drawn in full lines, drawn thinner where PLATE .1 I u + v+ q q Fi^.G.a w* u- Fi^.G.d. u -t- V- -Q Fi^.G.e. V i. 6.f. PLATE .11. Fi^.4.h v+ q q i. 6.a u- _._S5Sr~q-^ Fi^.S.c. Fi^.T.a. Fi^.T.b A EECAPITULATION 11 virtual ; and all intrinsically negative distances are drawn in dotted lines, with their virtual extensions drawn lighter. Theorem of Central Projection Having now the formulae relating to axial pencils of rays, we may next consider the case like that shown in Figs. 7 a and b. Besides the conjugate axial pencils Q^.^, let another point of origin ' Q 2 , in the case of the collective lens, or another apex of convergence Q 2 , in the case of the dispersive lens, be taken at some small but appreci- able distance away from the axis, such that Q x and Q 2 are on a plane perpendicular to the axis. It is evident that a ray drawn from Q 2 through the centre of the lens will pass straight on, as it is crossing two elements of surfaces which are parallel and practically touching. If a straight line from Q is therefore drawn through the centre of the lens and produced until it cuts the other so-called conjugate focal plane q . . q (which is perpendicular to the axis and passes through q l} the conjugate focus to Q x ), then the point of intersection q z is where the conjugate image of the point Q 2 is formed. That is, the centre of the lens is always in a straight line between any point Q 2 or Q 3 of a plane object and its conjugate image q 2 or q^. This theorem is capable of a further extension, as shown in Figs. 8a and b, Plate III. Here are two cases in which the pencil of rays from Q 2 (here drawn in solid lines) is eccentric ; that is, none of the rays of the eccentric pencil actually pass through the centre of the lens owing to the stop s being interposed. But it is assumed that the rays constitut- ing such an eccentric pencil are but a part of a larger pencil of rays filling the whole lens ; and since the lens is assumed so small that all the rays refracted through it from any one point are caused to converge to or diverge from one and the same image point, therefore these eccentric rays may be regarded as coming under the same law, and the conjugate points Q, and q 2 may be considered to be strictly on a straight line of projection drawn through the centre of the lens. Thus the pencils of rays are assumed to be homocentric that is, all the rays constituting each pencil are assumed to diverge from or con- verge to one point. From this it follows that the distance q l . . q 2 v = (Q a Q)-J an( l the scale of any conjugate image formed of the v plane Q 1 . . Q 2 is - times the scale of the original. The scales of image and object are in direct ratio to their axial distances from the lens centre. The optic axis de- parted from. Oblique conjugate focal distances. When oblique pencils are also eccentric Definition of homo- centric pencils. Relative scales of object and its image. 12 A SYSTEM OF APPLIED OPTICS SECT. Limitations. Although this theorem, which is a part of the larger Gauss theory, is in its nature only true for minute angles of obliquity and for exceedingly narrow pencils, which never have more than a very small degree of eccentricity, yet it is of the highest importance when we proceed to ascertain that very important function of a more or less complex combination of lenses, known as the equivalent focal length. Corrected lens While the theorem is of little practical worth when applied to untrue for the parts 1 s i m pl e uncorrected lenses of substantial aperture, yet, for a combination but true for the of lenses yielding a flat and rectilinear image, it becomes absolutely true in the sum for the series, since the departures from its truth in any one lens are in that case neutralised by contrary departures from its truth in the other lenses. Gauss and Listing. ... .. Principal points or nodal points. Thick Lenses We may now proceed to deal with the case of lenses of consider- able thickness as measured along the axis. This subject was long ago worked out by Gauss (about 1838) and Listing (about 1868), and it will suffice to recapitulate here the most important results, although perhaps arriving at them by methods differing from theirs, but more convenient for our purpose. Let Figs. 9a, I, c, d, e,f, and g represent various forms of lenses, of central thicknesses A I , . A 2 , and radius c 1 . . ^ for first spherical surface, and c 2 . . r 2 for second surface. It is obvious that if any two radii Cj . . r x and c 2 . . r 2 are drawn parallel to one another and joined by the straight line r : . . r^, then the latter will cut the axis at the point C, so that we have two similar triangles c^Cr l and e 2 O 2 , and two similar mixtilinear triangles GA l r l and CA 9 r 2 , and the distance . . A. : C . . A_ r 2 , and moreover the straight line i i "2 'j . . r 9 cuts the first surface or its tangent at r v at exactly the same angle as it cuts the second surface or its tangent at r^. If, therefore, r . . r 2 represents a ray of light, it will obviously, if refracted out of the surface at r^ be deviated from the direction r 2 . . r l by exactly the same angle as it would be deviated from the direction T I . . r if refracted out- wards at the point r 2 , only the deviation will be in opposite directions. Hence the ray after refraction at ^ will pursue a course r . . t v and after refraction at ?' 2 will pursue a course r 2 . . t^, and these refracted rays are parallel to one another. If, then, r l . . t l and r 2 . . t. } are produced back- wards (if necessary) to cut the axis at two points p l and p z , we then get again two similar mixtilinear triangles r l A } p 1 and r A 2 p , and again have A I . . p l : A . . p z : : c^ . . r^ : c 2 . . r f These two points p l and p 2 are the two principal points of the lens or nodal points (sometimes PLATE.MI P P r i r z 12. PLATE.III 12. A EECAPITULATION 13 also called Gauss points), and, as we have seen, have this important property, that any ray which, while outside the lens, passes through the first principal or nodal point, will, after passage through the lens, emerge on the other side in a direction parallel to its first direction, and radiating from the second principal point ; moreover, the same ray, while traversing the interior substance of the lens, passes ex hypothesi through the geometric centre of the lens or the point C. As a corollary from the above principle, it follows that if we wish to know the relative sizes or scales of conjugate images formed by thick lenses, we must then measure the focal distances of such images from the principal points of the lens. The focal distance of the first image or object, virtual or otherwise, formed by the entering rays must be measured from the first principal point p l} and the distance of the second image formed by the emergent rays must be measured from the second prin- cipal point p 2 , when the sizes of the images will be in direct ratio to those focal distances. Our theorem of central projection still holds good, with this modification, viz. that the centre of the lens presents two aspects, or two different positions, according to whether the lens is viewed from one side or the other. Eegarded from the left hand the centre of the lens is practically the first principal point p l} but regarded from the right hand the centre of the lens is practically the second principal point p^, and these two points are but the refracted images of the geometric centre C of the lens. That is, p l is the conjugate image of C by refraction at the first surface, and p^ is the conjugate image of C by refraction at the second surface. Therefore the distances A l p l and A 2 p 2 may be derived from the Formula II., p. //. 1 1 it r u' in its more special application to Figs. 4/ and 4#. At the first surface we have u = A l ..p l (Fig. 9), which by convention is a minus quantity, while A 1 . . C = it, and is a plus quantity, and A l c l = r. Let r and s = first and second radii of curvature respectively, and let the thick- ness be denoted by t, therefore /A [A 1 1 A r .C i-Pi -1 but Aj.,0 r r Conjugate focal distances to be measured from the principal points. A thick lens exists virtually in two positions. Method of locating the principal points. r + s Distance of first principal point from first vertex. 14 therefore and A SYSTEM OF APPLIED OPTICS tr tr Thickness of a col- lective lens positive, and that of a disper- sive lens negative. Case of collective meniscus, first p. p. Case of dispersive meniscus, first p. p. Case of collective meniscus, second p. p. Case of dispersive meniscus, second p. p. S) - t(fJL - 1 )' Similarly, at the second refraction we have 1 yU, fJi 1 A 2 . . p. 2 = A777C s~ ' in which IV. A Q . . C = t- r + s therefore Distance of second and principal point from second ver- tex. ts V.PS--T- t ts s) - t(p - I)' These two formulas thus give the distances from the vertices A : and A 2 of the two principal points of a lens. They obviously give a positive result in the case of any double convex lens, which is as it should be, since these distances are really additions to the conjugate focal distances when both, as in Fig. 6 a, are positive. But in order to make the formulse apply to the case of the double concave lens whose normal object and image distances are virtual, 'but positive, we must consider t, the thickness, to be intrinsically a negative .quantity, thus making A l . .p 1 and A 2 . .p 2 negative quantities. For they are obviously deductions from the conjugate focal distances when both are positive, as in Fig. 66. That having been settled, then the formulse will interpret themselves correctly in all cases. In the case of the collective meniscus (Fig. Qe~) s must be entered as a negative quantity in the Formula IV., and being necessarily greater than r, then r + s comes out negative, and we get a negative denominator in the formula. Obviously in this case Aj . . p l is measured outside the lens and is a deduction from the value of u, if plus. In the corresponding case of a dispersive meniscus (Fig. 9/) Aj . .p l comes out positive, both numerator and denominator being negative. At the second surface in Fig. 9e the Formula V. gives both numerator and denominator negative and the result is positive, for A 2 ..^? 2 is an addition to the back focal distance v, if plus. In the corresponding case of the dispersive meniscus (Fig. 9/) Formula V. 15 yields a product of two negatives for numerator and a negative denominator, and A 2 . . p 9 comes out negative, being a deduction from a positive focal distance v. Fig. 9# represents a special case worthy of note, a case in which the two radii of curvature are equal, but of opposite signs. Here the distance of c, the centre of the lens, / T T\ from either A X or A 2 comes out infinity (or t-~- = t-j. The straight line joining the two points r l and r z , where the two parallel radii cut the surfaces, is parallel to the axis, and obviously after refraction by either surface will intersect the axis at a distance from the vertex of T S either surface equal to , and r or A, . . p. and A_ . . p, in the first fj. i [A i case negative and in the second positive. We shall also see later on that such a lens, of watch-glass form, really possesses collective power and can form a real image. But it is easy to see that if a real object is placed at the first principal point p v then, after passage through the lens, a virtual image will be formed at p^ of the same size as the original. In such case both u and v = o. We have, then, here an actual and realisable example of the theorem dwelt upon by various writers on optics, Dr. Drysdale for instance, in the British Optical Journal, to the effect that the two planes passing through the two principal points are planes of unit magnification, or, in other words, if an original object or an image lies in the first principal plane, then an equal-sized image of it, real or virtual, will be formed in the second principal plane. We shall have occasion to refer again to this theorem in the next section. Figs. 10 and 11 are peculiarly interesting cases, since we have the radii and thickness so related that the ray r^ . . ?* 2 within the glass is, after refraction outwards, either way parallel to the axis. This con- dition is seen to be fulfilled when t = i ~, + s _ ,, r in Fig. 10 being a negative quantity and in Fig. 11 a positive quantity. Such thick lenses as these may be said to have no principal points at all, and therefore no focal length, and their analogy to the Galilean and astronomical telescope respectively will be more fully realised later on. In Fig. 12 we have the simplest case of all, that is, the sphere, wherein the two principal points merge in the geometric centre. In Fig. 13 the case is extended to one in which the two radii of curvature are different, yet struck from a common centre. Here again the two principal points merge in the geometric centre. Figs. 14a and 14& show, for a collective lens and for a dispersive Two radii equal, but of opposite signs. Theorem as to prin- cipal planes. Lenses without any principal points and without focal length. A sphere has only one principal point. Other lenses with only one principal point. 16 A SYSTEM OF APPLIED OPTICS Course of oblique pencils with respect to the principal planes. Influence of thick- ness upon the prin- cipal focal length of a lens. Back focal length to be ascertained. lens respectively, the course of a complete oblique pencil of rays through the lens with respect to the principal points p l and p^ and the principal planes passing through the latter. Having now settled the positions of the two principal points of any lens, we may proceed to ascertain what influence the axial thick- ness t of a lens exercises upon its equivalent principal focal length. Fig. 14a represents a double convex lens forming a real image/, in its principal focal plane F, of an object situated at an infinite distance away on the left; Fig. 14& the corresponding case of a dispersive lens. Here the principal focal length required is the distance p^ . . F measured from the second principal point p 9 to the principal focal plane F. It consists of two parts : first, the back focal distance A 2 F measured from the vertex of the second surface ; and second, the distance A 2 . ,p 2 from the same vertex to the second principal point. The latter we have already got an expression for ; the former, or the back focal length, we must now proceed to formulate. After the first refraction, in the case of a collective lens, the axial pencil of parallel rays is converged to /, ; let A. . ./ = it. Then v 1 " 1 v 1 _zl_! u r u ' of which therefore Next Ult lilt 1 tU' , T-= -, and u=~ . u r u p.r ji - 1 r-0*-l)l /* - 1 At the second refraction we have yu. M 1 A therefore = s therefore and -(/A- IX} VA. Formula for the back focal length. Add A 2 . . p 2 to this from V., and we get A 2 ..F + A 2 ..^ 2 = -- s{/Ar-(/A- IX} + ) - (A* - s{/>tr - (/A - 1 X} + (/A - _ . . T ts VI. This, then, is the formula for the equivalent principal focal length E of a thick positive lens. If a small infinitely thin positive lens of principal focal length p . . F were placed at p z , it would form an image at F of distant objects of the same dimensions as that formed by the thick lens, of which latter it is the equivalent. In the case of the double concave lens, Fig. 106, we have Formula for the equivalent focal length of a thick positive lens. .p 2 also {/*r - (/A - 1 )/} ts (/A - + s) - (/A - i X) - i)' in which case, of course, the latter expression for A %2 . . p 9 comes out minus and as a deduction from the back focal length A 9 F, since t in this case must be entered as a negative quantity ; so that, just as in the case of the collective lens, we get the same expression for the equivalent principal focal length, viz. fJLTS .=E. VII. Formula for the equivalent focal length of thick dispersive lens. 18 A SYSTEM OF APPLIED OPTICS SECT. Formula giving modification due to thickness, as a percentage. Effect of thickness upon power in various cases. Now if the lens were infinitely thin, the reciprocal of its principal focal length would be simply (//, 1)(_ + _Y Calling this - and sub- tracting it from , we get rs _ E F' therefore so that therefore 1 _ 1. E~F r s F /*rs therefore F (/*-!)( E F VIII. This is perhaps the most convenient and significant mode of ex- pressing the modification of the power of any lens whatsoever which is due to thickness ; it expresses it in the form of a percentage of gain or loss as compared with the power which the lens would have if it were infinitely thin. It shows a loss of power in the case of double convex lenses, a gain in power in the case of double concave lenses, no alteration in power in the case of plano-convex, plano- concave, con vexo- plane, concavo- plane lenses, for in all four cases r + s becomes infinity ; while in the case of a collective meniscus, when r + s becomes negative, a greater and greater relative gain in power, consequent in thickness, is attained as the radius of the concave surface approaches to equality with the radius of curvature of the convex surface ; while, lastly, in the case of the dispersive meniscus a loss of power ensues on an increase of thickness, since both numerator and denominator of the function of t become negative. We have now arrived at the formula for the equivalent principal focal length E of any lens whatsoever, and also have located the geometric centre C and the two principal points p-^ and p^, from the latter of which the equivalent principal focal length is- measured. i A RECAPITULATION 19 We have next to inquire whether, in cases wherein the entering inquiry, is the rays are more or less divergent or convergent that is, when the entering rays are either diverging from a near object on the left of the lens or converging towards a real image to the right of the lens the thick lens still maintains the same principal focal length, or departs from it. Fig. 14a or 14& illustrates such a case. It is of the highest practical importance to know whether the law of conjugate foci for a thin lens - = ^ or = = - + - still holds good. In short, does v F u F u v - = :p, - pr ; that is, is - -^= + ^ - a constant. We shall be in a better position to answer this question when we have dealt with the problem by means of a device or theorem which is more general in its applications than any method which has been hitherto devised. This we will deal with in the next section. SECTION II Power of a single surface is a highly inconstant entity. The power of a thin lens is a constant quantity. Thick lenses com- pounded of infinitely thin elements and a parallel plane plate. THE THEOREM OF ELEMENTS WE have seen in the last section that the Formula II. relating to refraction of an axial pencil of rays at a single surface is by no means such a simple formula as the Formula III., which applies to the corresponding case of refraction of an axial pencil of rays by a lens bounded by two surfaces. In the case of the single surface, Fig. 4 a, for instance, if the rays are strongly divergent, then a large amount of positive refraction takes place ; but supposing the entering rays are converging to the centre of curvature C, then no refraction takes place ; while if the rays are converging still more to any point between C and A, then there ensues refraction of a negative character. Thus, from the practical point of view of refractive effect, we may disregard the so-called " optical invariant " of the late Professor Abbe as applied to a single refracting surface. Clearly a single surface is a somewhat puzzling and inconstant entity, which varies in its effects enormously according to circumstances. But not so the lens bounded by two refracting surfaces ; for whatever conditions of divergence or convergence may characterise the entering pencil of rays, the lens always adds or subtracts a constant refractive effect of its own which i\ is expressed by (ft !)[- + -) or =. Let us see, then, whether we cannot express any thick lens in terms of two complete lenses. Let Figs. 15 a, b, c, and d be four various thick lenses. Each one of these may be considered to be built up of plano-convex, plano-concave, convexo-plane, or concavo- plane lenses of infinite thinness, each lens consisting of any two of the above and containing between them a piece of plane parallel glass of a thickness equal to the axial thickness of the whole lens. For instance, the collective meniscus, Fig. 15a, may be considered to be built up of a convexo-plane infinitely thin lens e l at the left-hand 20 THE THEOEEM OF ELEMENTS 21 vertex of the whole lens, and a plano-concave lens - 1\ " t = t(- I, their distances from the plane surfaces being -. And we \' ft / /i have already seen that the principal equivalent focal length of all lenses having one surface plane is in no way altered by thickness, however great. Therefore in treating such lenses we may take any focal distances u Focal J distances or v that may be measured from the central or axial point of the plane J neasured from J er - tex of plane surface surface, add + - in the case of collective lenses, and add - in the I'- P case of dispersive lenses. Then u or v, as the case may be, will be referred to the principal point. Of course the addition of - is algebraical, for if the rays of a pencil emerging from the plane surface of Fig. 18 are converging, then v is positive, and -f - is an extension of that distance ; but if the f* rays of the pencil after emerging from the plane surface are diverging, then v is negative, and - becomes a deduction from its numerical value. ^ 24 A SYSTEM OF APPLIED OPTICS Case where collec- tive and dispersive lenses are ranged on a common axis. An apparent (incon- sistency explained. Conventions to be observed in case of mixed lenses having plane surfaces. But the same rule of adding - will not quite apply to the P" measurements of axial distances between neighbouring lenses of collec- tive and dispersive types mixed. For instance, the distance between the two lenses IS a and I8b, which have their plane sides towards one another, is indicated by the line s l ; that is, to the original air-space e n . . e, the distances - and - have to be added at each end. /* /* The distance between 18& and 18c is simply e., . . ^ as the two lenses are presenting their curved sides and two elements to one another, while the distance between 18c and I8d is e^..e. with - and - added H p. on at each end. It might here be urged that these latter are two dispersive lenses, and therefore - and - are negative quantities, so that they become deductions from the numerical value of the distance e 2 . . e l if the latter is positive. Here is a seeming inconsistency. But we must remember that if we are dealing with the two dispersive lenses 18c and ISd alone, then we treat them as positive entities, in which case both their thicknesses and any separation between them would be treated as negative quantities, so that s 3 would be the sum of -(P. 2 -- e J> - ( 2 . . gj), and -(e^.pj. But if we are tracing pencils through a series of collective and dispersive lenses ranged on a common axis, we can then treat all axial distances between such lenses as positive, provided the principal focal lengths of all dispersive lenses are considered negative relatively to the said distances and to the principal focal lengths of the collective lenses. Therefore in Figs. I8a, b, c, and d, if, as usual, we consider the distances s l and s positive, then s, would also be positive besides - and -, while the f- /" powers of 18c and I8d would be negative, and the powers of 18a and 18& positive. And this is the most reasonable convention to follow in the case of a series of mixed lenses. Such matters constantly demand the exercise of careful discrimination. SECTION III THEORY OF EQUIVALENT FOCAL LENGTHS AND PRINCIPAL POINTS OF LENS COMBINATIONS IN Section I. we have already worked out formulae for the equivalent principal focal lengths of thick lenses and for the distances of the two principal points from the two vertices. We will now prove the Previous formulae to identity of the formulae obtained from the theory of elements with 6 the above formulae already worked out for single thick lenses, and then prove that the sum of the reciprocals of the conjugate foci, as Constancy of equiva- measured from the principal points, is invariably constant and equal tTproved length to to the equivalent principal focal length. Fig. 19 represents two elements, exaggerated in diameter for clearness, separated by an axial air-space or distance s r Let -- be 1 f\ the reciprocal of the principal focal length of e , and j that of c.,, or /a the respective powers of the two elements. Let C be the geometric centre, such that l ..C:v.Cii/ l :/ r It is then plain that any slightly oblique ray passing through C will impinge upon e l and e 2 under exactly similar conditions, and will meet with exactly equal deviations when refracted through the elements, and therefore the rays after refraction both ways, Q . . p and p . . q will be parallel to one another, and if produced backwards will cut the axis at p 1 and p z , which two points are the principal points of the combination. Let the distance e 1 to p l be P p and e^ to p 2 be P 2 . We then have C f - - - , /1+/2 and if we are to make P I positive we have 25 26 A SYSTEM OF APPLIED OPTICS SECT. I i i = /i+/_j _/!+/-* P l C..^ /, sf, f, sj\ and Distance of first s f principal point from P l = -, ' l . first element. ./i + /2 "" s Also we have and if we are to make P 2 positive we have and Distance of second s f principal point from P 2 = ^ - ^r -- -. IX?.. second element. /] + Jz~ g Now if r be the radius of the curved surface of e v and s that of e 9 , then = = - and , =^ Also, if e. and e, were the elements of /! r /a f lid thick lens of thickness t, we have by our theorem s = - and u /iS. Substituting these values in Formula IXA. we get Identity of Formulae p " " IV. and IXA. r s t p.(r + a) - t(/j. - 1 ) p.(r + s) - t(p. - 1 )' M -i + /t-i ,, x/-i) which is identical with Formula IV. arrived at in Section I. Next we will work out the back focal length e^ , . . q, supposing the rays entering e l to be parallel, in which case q is on the principal focal plane. The image formed by e l in this case is at a distance f l from e l and a negative distance equal to /j s behind e , therefore we have 11/1 Calling 2 . . ^ = B, we have Formula for back and focal length for y 2 (y i _ ,<,) two separated B = . _ . elements. /] /2 THEOEY OF EQUIVALENT FOCAL LENGTHS 27 Then to get the equivalent principal focal length, or E, we have T? - Ti j- P - fjhfl ~ s / , s /2 JCi Jo + r ,7 7 1- -7 ? , and A A Formula for the equivalent focal X. length of two separated ele- ments. Putting ; I for / x , - ^j for / 2 , and - for s, as before, we then have rs E = - 1 - 1 _ /*(/* - 1 [) 1)}' which is identical with the Formula VI. for the equivalent focal length of a solid lens of thickness fj,s or t which we obtained in Section I. It will be found that Formula X. is universally true for couples of elements, provided our former conventions as to lenses are adhered to. If one of the elements is collective and the other dispersive, the stronger element should give the character to the lens, while the / for the weaker element should be entered in the formula as a negative quantity. For instance, if e^ is the stronger, having f = 9 and dispersive, while / x is 10 and collective, then, as in the case of the lens, we should consider the character of the combination, "by first intention as it were, to be dispersive, and the separation s, which say = 2, to be relatively a minus quantity, just as t was in the case of a dispersive lens ; so that E becomes, since f l is negative, (-10)( + 9)_ -10 + 9 + 2" Now as we assume the lens, by first intention, to be a dispersive lens, but with a positive sign, it is clear that E, being minus relatively, indicates that a real image is formed, and the combination, owing to the separation, acts as a collective lens, although by first intention it was dispersive. Thus the separation has reversed the character of the couple. If, on the other hand, we insert f l as a positive quantity in the Formula X., and / 2 as a negative quantity, and therefore s as a positive quantity, we shall then in the same case get E= +90, which comes to the same thing, E being + or of the same sign as f lt which is collective, and so indicating a collective resultant lens or combination. Concrete example of the use of signs and of the highly im- portant influence of separation on the character of a pair of elements or lenses. 28 A SYSTEM OF APPLIED OPTICS Inquiry. Is the ^constant* 1 This variance between the character of a lens or combination of elements by first intention and in actual result should be clearly borne in mind. Having now got the equivalent principal focal length E of the combination on the supposition that the entering pencils consist of parallel rays, we may now profitably return to our inquiry, whether, supposing the entering pencils to consist of either divergent or con- vergent rays, the sum of the reciprocals of the conjugate focal distances Q . . p and > 9 . . q will always be equal to the reciprocal of the equiva- lent principal focal length, and therefore constant. It is of the highest importance to know this. The first thing we want is the formula for the back focal length e 2 . . q, supposing the rays entering e l are not parallel. Let Q . . e l = H V and the conjugate focal distance after refraction through e be v v and the amended distance from the focus so formed by e l to the element e 2 be 2 , and the conjugate focal distance after refraction through e. 2 be v z . Then v 2 is the back focal distance required, and we have Back conjugate focal distance for first element. Front conjugate focal distance for second element. and Then :- = V l fl U l f u - 71 * v = = v*-S= , x -/j - S = . u^ -/j Then, since the rays are converging into e z , w 2 becomes a minus quantity, therefore l = i_ J_ -l + _ _!izA_ 2 /2 ~ U 2 /2 Al-Xl-/l) i -A) M-(g- Back conjugate focal and distance for second element. 2 {i - s ( u i -A)} ' /// ?{ /V = /2(/i M i ~ s ( lt i ~/i// . Add e 2 . . p 2 or P 9 to this in order to obtain the distance of the focus q from the second principal point p. 2 , and we have , 2 2 i +/. - i -/i)' in THEOEY OF EQUIVALENT FOCAL LENGTHS and 29 1 (A 11 V S' V 1 J \' t \v\ 2 // 2 1 1/ 1 1 \1 /!/-' After multiplying out denominator and cancelling we get i (fi u i ~~ su i + S A + A M i ~AAXA + A "" s ) P 2 + ^ 2 " ~7i A( A M i + S A + M - *i) Then the other conjugate focal distance = Q . . 2\ or w x + 1^ S A w i(A + A ~ s ) + S A Therefore Reciprocal of back conjugate focal length measured from second princi- pal point. therefore 1 /. ) A A/ 2 ( A which / 2 - v +A g XA A s ) /g ~A/ 2 )(A A/ 2 (%A + w i/2 - v + S A) - 1 ' = A/2 or = = constant. Thus the mutually dependent variables u l and t? 2 , the front and back focal distances respectively, have eliminated themselves, and we find that the sum of the reciprocals of the conjugate focal distances as measured from their respective principal points p l and p 2 is constantly equal to . If the reader will apply the same processes to a com- bination of three separated elements, he will arrive at just the same result, although the process is much more lengthy. Therefore the combination of two thin lenses or elements, however widely separated they may be, behaves like a simple thin lens of principal focal length E, such that = - if we put U for ?^ + P x and V for P 2 -f v y It only differs from a simple thin lens in that the two principal points are widely separated instead of both merging in the lens centre. Fig. 19 presents the case of two dispersive elements. It is commonly remarked that a thing cannot be in two places at once, but here we have an optical combination of equivalent focal Reciprocal of front conjugate focal length measured from first principal point. Sum of above two reciprocals = . 30 A SYSTEM OF APPLIED OPTICS SECT. A compound lens exists practically in two positions at once. Above curious fea- ture illustrated. A real pupil at the geometric centre implies two equal virtual pupils at the two principal planes. The principal planes are planes of unit magnification. length E (that is, it forms an image of infinitely distant objects on exactly the same scale as would be formed by a simple thin lens of principal focal length E) ; but from the point of view of Q, or the left hand, this equivalent simple lens is supposed to be placed at p , while from the point of view of q, or the right hand, it is supposed to be placed at p f It thus presents a dual aspect. Fig. 196 illustrates this curious feature. It represents the essentials of Fig. 19, p 1 and p^ being the first and second principal points, and Q . . Q x and q . . q l the two conjugate focal planes in which lie either an object or its image. The planes drawn through the two principal points perpendicular to the optic axis Q . . q are generally known as the principal planes, and can be shown to have the curious property that if any direct or oblique pencils of rays, such as Q . . p l and Q l . . p v strike centrally upon the first principal plane p l at certain points at certain distances from the optic axis, then the same rays will start from the second principal plane at similar points at the same distances from the axis (and on the same side of it). For instance, the principal ray Q . . p v together with two outer rays;Q . . c x and Q . . b r constitute the axial pencil striking the first principal plane at c^ p v and b v Also let the principal ray Q x . . p v together with Q t . . c^ and Q x . . &j, constitute an oblique pencil also striking the first principal plane at c r p v and b r Draw straight lines from these points parallel to the optic axis to intersect the second principal plane at c z , p 9 , and & 2 ; then these become the starting-points for the rays of the corresponding conjugate pencils p . . q and p . . q in such manner that the principal emergent ray p 2 . . q l is parallel to the principal entering ray Q l . .p r The proof of this theorem is really a simple one, for we have already seen that if we take two infinitely thin lenses L x and L of focal lengths /j and / 2 separated by a distance s, then the first principal point is the image of the geometrie centre C as formed by L X , and the second principal point is the image of the geometric centre as formed by L 2 . But the geometric centre is symmetrically disposed to the two lenses, and if / a = 3/ 2 , then the geometric centre is three times as far from L X as from L 9 . Therefore the image of C formed by L x is magnified or diminished in exactly the same degree as the image of C formed by L . Consequently, if we imagine a circular aperture or pupil to be placed at C, then the image of it formed in the first principal plane by L^ will be exactly equal to the other image of it formed in the second principal plane by L 2 . The two principal planes are in this way shown to be planes of unit magnification relatively to one another. Therefore if the bounding rays of any pencil whatever strike the first Ill THEORY OF EQUIVALENT "FOCAL LENGTHS 31 principal plane at distances d and d f from the axis, they will start from the second principal plane also at points distant by d and d f from the axis, although when actually passing through the plane of the geometric centre the distances d and d' may be more or less reduced or increased. Moreover, these distances d and d' invariably keep to the same side of the axis ; for since the images of the imaginary aperture placed at the geometric centre are formed by L x and L 2 in the two principal planes under similar conditions, therefore if the image of our pupil at C formed by L t at p l is the same way up as the original, then The two principal the other image of C formed by L 2 at p_ 2 is also the same way up, or P u P lls the same wa y if one image at p is reversed, then so is the other image at p 2 . It is interesting to note how the pencils of rays are set back in their course, as it were, by the distance p l p, 2 between the focal centres, which therefore constitutes in this case an overlapping of the conjugate focal distances, and corresponding shortening of the distance Q . . q. This theorem, that any two separated lenses on a common axis act as a simple thin lens of equivalent principal focal length E, is highly significant, and the important corollary follows from it, that all All optical systems, however complex, exhibit two final principal points, and that the sum of the reciprocals of the conjugate focal distances measured from those points is constant. For, supposing we have three thin lenses e^ e 2 , e^, of principal focal Proof of above lengths f^fyfy arranged on a common axis, as in Fig. 20, Plate V. theorem - Then the couple e l and e 2 have their geometric centre at c , and their two principal points at p l and ^? 9 , and have an equivalent principal focal length = E. Then, from the point of view of e , the combination e l + e 2 is tantamount to a simple lens of principal focal length = E placed at p^. It therefore follows that we have a new geometric centre C such that" (p 2 . . C) : (C . . e s ) : : E :/ 3 . Then the point C refracted by the equivalent lens at p z will be apparently transferred to P r p l . . P l being in this case conjugate to p 2 . . C, and C is also transferred to P 2 by the refraction of e. A , and we have two new principal points P X and P 2 for the whole combination of three lenses, which latter possesses a new equivalent principal focal length which we may call E g , which is also a constant with respect to the three lenses (so long as the separations are constant). It will be seen that lens systems tw i < n f nal prin " and ! P 2 . . e 3 ~ C. . e 3 32 A SYSTEM OF APPLIED OPTICS SECT. Therefore, from the left hand, or from the point of view of rays entering the combination, P a is the first principal point ; while from the right hand, or from the point of view of rays leaving the combination, P 2 is the second principal point. Principal points of a We now require formulae giving the distances e l . . P I and P 2 . . e y three-lens combina- or p ail( j p respectively. tion investigated. Let e, . ,e a = s., and e a . .e a = s a . 1 i 1 O Z First of all, from Formula IXA. we have by analogy SE ^^-E+yj-S' in which 2 a .. 22 Jl ' /2 ~ S l and fl+f Z " S l Therefore after substituting these values we get /f I -f {' - 1 + JZ - V/l + /2 ~ 5 1 Pi ' A 1 ~ f //. */ 3 1+/2~ S 1 x Jl^JZ therefore _ ^' l (A+/S - But we must add the distance ^..^ to this in order to obtain the required distance e l . . P, : and (see Formula IXA.); and on adding this to the above formula for p l . . P X we get W/ X +/ 8 - i) + *iftfJ 9 i/i{/i/ +/ 8 (/l + /2 - S l) - S 2(/l +/2 - S l) - S l/ 2 }, (A +/2 - i){/l/ in THEOEY OF EQUIVALENT FOCAL LENGTHS which reduces down to 33 MA - s i) + (A - S 2 )(A XlA. Next we require a formula for the distance P 2 . . e y measured from the second principal point P 2 of the triple combination to the element e . From Formula IXfi. we have by analogy S , in which S, as above, = s, + - ip and S l/2 / /. ~f" / o ~~ I S n ~f" / / / * !+/*-*! 3 V 2 A+A-Sj/ which reduces down to +/l g 2 ~ g l S 2 ) XlB. It is plainly evident that the Formula Xle. is the symmetrical com- plement of Formula XlA. For, if we trace the light backwards through the combination, then / 3 becomes f v s. 2 becomes s r while / remains f z , and thus the one formula may be turned into the other. Next, we require a formula for the equivalent principal focal length E 3 of such a combination of three lenses or elements. By analogy from Formula X. we derive E Q = in which, as in last two cases, .. .e 9 , or s 9 + -, -*r and E- ~ A/2/3 therefore A/2 A Formula locating first principal point for three elements. Formula locating second principal point for three ele- ments. 34 A SYSTEM OF APPLIED OPTICS Formula for equivalent focal length for three elements. which reduces down to /1/2/3 ~~2 \ 7 2 - s i) XII. We may now pass on to the consideration of a combination of four separated lenses or elements. Let Fig. 21 represent four elements e r e z , e a , and e 4 , separated by the three distances s^ s 2 , and s 3 . Our line of procedure is to consider this as a combination of two couples, viz. e l and 2 , having their two principal points at p l and p^ ; and e 3 and 4 , having their principal points at p s and p^. Then the distance p 2 . . p 3 or S is obviously the real separation between these two couples, whose respective equivalent principal focal lengths we will denote by E and E 2 . Then the separation between them is obviously equal to p 2 . . e 2 + s. 2 + e 3 . . p 3 = S, so that generally the equivalent principal focal length E 4 of the whole combination E E *V3 which Formula for equivalent focal length for four elements. f f f f J\JZ J3J4 /1 + /2~ S 1 /3+/4~ S 3 /1/2 /3/4 f *l/2 S 3/3 \ ' ''"'-i f -f i -^ /* 2 /l/2v/3/4 /3+/4-V and finally /I /2/3/4 /l+/2-*l XITT 4 ~(/l+/ 2 - S l)(/3 ^4 -/3 5 3 ) + (/s +/4 - S 3 ){/ 2 (/l ~ S l )-s 2 (/ 1 +/ 2 - 1 ; It is obvious that the two couples have between them a new centre of symmetry, C, or the optical centre of the whole combination, so located that it divides the distance p 2 . ,p 3 between the second and third principal points into two parts such that p 2 . . C : C . .p s : : E I : E 2 . Then the point C, refracted by E I} is transferred to P I such that p l . . l* l is conjugate to p 2 . . C. In the same way the point C, refracted by E 9 , is transferred to P 2 such that p 4 . . P 2 is conjugate to C . . p s . We now want formulse for the distances of the two principal points P X and P 2 from the outside elements e l and e 4 . In working out the principal points for two collective elements we found that if the first principal point fell to the right hand of e l then the distance e l . . p l was positive, and if it fell to the left hand, it was negative ; while if the second principal point fell to the left of e, then the distance p 2 . , e 2 was positive, in THEOKY OF EQUIVALENT FOCAL LENGTHS 35 and if it fell to the right hand it was negative. Bearing this in mind, it will be seen that in this case the required distance e l . . P x = (e 1 ..p l )-\-(p l ..'P l ) > the latter being negative in Fig. 21, and also that e^ . . P 2 = (> 4 . . e 4 ) + (p 4 . . P 2 ), the latter also being negative in Fig. 21. So we have e l . . P I = (e l . . pj + (p 1 -- Pj) (algebraically) A+/2-i E 1 + E 2 -S in which S or p, 2 . . e s = (p 2 . . 2 ) + s 2 + ( 8 . . ^ 3 ), which -F -P 2 / -f / 1 + /2 ~ S l /3 + /4 ~ S 3 On substituting this value of S in the above, we get P 1 A +/-! A/2 " , /3/ /l+/2- S l /3+/4 which reduces down to -.? or P _ _ _XIVA (/i +/ 2 ~ s i)(/3/4 -/3 s s) + (fa + /4 - S 3 )(/2(A - s i) - S 2 (/1 +/2 ~ s i)) first principal point. The distance of the second principal point P 9 from e^ is obviously 4 . . p 4 +PI . - P 2 , analogously to the last case, and is expressed by / 3+ / 4 - S3 E 1 + E 2 -S' in which S\ as before, is the distance p 2 . . p 3 , and E 2 is /8 + /4 - S 3 Therefore f g l/l , , Vl +/2 - ^l 2 "T i o T 33 S 3/3 1/3/4 f~l ! / ~ r 9 ~*~ f \tA-1- f ^ T -L e, . . P 2 or P 2 = , , -xx -T / / ^ 4 i T4,T O TT T T for O T f Q ' / A "Q /1/*> / Q I A ^1/1 WR-f I v o / 4 o * I*' _ i */ ,w 4 I*/ 1 in i o*/ ^i + A - s i which reduces down to 4 ..P 2 orP 2 ////.,/ \/ / / \-flf f \\ Four elements. For- /4l\/3 s 3 + /3 S 2 ~ Va/V/i +/2 ~ g i^ + /2(/3 s i + /1 S 3 ~ S 1 S 3^I YTV-o mula for position of (A +/2 - a/3/4 '/A) + (/3 +/4 - %){/ 2 (/! - l) - %(A +/2 ~ l) 36 A SYSTEM OF APPLIED OPTICS SECT. Symmetry of For- mulae XIV A. and XIVB. Formulae relating to more than four ele- ments undesirably complex. Case of five elements. On comparing Formulae XIVA. and XIVB. it will again be noticed that they are symmetrical to one another : f l in the first corresponds to / 4 in the second, / 2 corresponds to / 3 , / 3 to / 2 , and / 4 to /j, while s l corresponds to s 3 and s 2 to s f Hence one formula may be converted into the other by supposing the light to traverse the system in the reverse direction. We have now got general formulae for the equivalent focal lengths, and the positions of the two principal points for any combinations of two to four separated elements or thin lenses, stated in terms of the principal focal lengths of the several elements or lenses concerned and the separations between them ; and we have found these formulae relat- ing to a four-lens system to be sufficiently complex to deter us from proceeding any further on the same lines ; that is, were we to work out formulae for a five-lens, six-lens, and eight-lens systems, all likewise expressed in terms of the principal focal lengths of the several elements or lenses involved and their respective separations, we should arrive at undesirably bulky formulae. In such cases the results are perhaps best arrived at by the building up or cumulative process, yielding- formulae in which equivalent principal focal lengths of two or four lenses together constitute the terms. In this way we may deal with the case of five lenses as follows : Let e , e^, e a , e,, and e,. Fig. 22, be the five elements involved. Let E, 1' 2' 3 4' 5' _ 4 be the equivalent principal focal length of the first four elements, P T be the distance from first element e^ to first principal point P x of the same, and P 2 be the distance from second principal point P, to the fourth element e 4 . Then we may treat the whole as a combination of a simple lens of E.F.L. = E 4 placed at P 2 with another simple lens of E.F.L. =/ 5 placed at *^3 s, 22 i i P n , -i ' Pa' , Pz C Pj ( < Fi<> 23. 1 1 1 C 1 ', P C i "a * f > ' I ! 1 p 1 it / t - p. Fi^. 24. PLATE. V. Si 1 P ** i. 20 1 ' s . F? P c, p ' Ki r ' ( !~ ^ , p c, J* * < p. P, Fi<>. 2 I s r -S - r- 22 P, c p p' Sr 1 I V 1 i i P / P C P e* Fi^.23. i 1 t ', Pa C p > ' < P' p 1 Fi<5. 24. in THEOEY OF EQUIVALENT FOCAL LENGTHS 37 = , (s 4 + P,)/ 5 VAr Five elements. Posi- A V B. tion of second prin- E 4 + / 5 - (S 4 + P 2 ) cipal point. Formulae for Six Thin Lenses or Elements We may treat this as a combination of four lenses of E.F.L. = E 4 with another combination of two lenses of E.F.L. = E 2 . (See Fig. 23.) Then if P : = first principal point of the four-lens combination, and P a its distance e l . . Pj from e lt and P = the second principal point of the four-lens combination, and P., = distance 4 . . P 2 from e 4 , P 3 = the first principal point of the two-lens combination, and P = distance P . . e., then we have it O U P 4 = the second principal point of the two-lens combination, and P, its distance from e.. 4 o n T? Equivalent focal E 6 = - 4 2 _ - =-. XVI. length for six E 4 + E 2 - (s 4 + P 2 + P 3 ) elements. Then if P^ is the new first principal point and P 2 ' the second one for the whole combination, then putting P^ for the distance e l . . P ', and P/ for P 2 ; . . e fi , we have ( s + R, + P )E Six elements. Posi- P/ = - * * 2 *! *- + P, X VIA. tion of first principal E 4 + E 8 -(* 4 + P 8 + P 8 ) point. ( s + P -f P )E Six elements. Posi- and P ' = - - + P 4 . XVlB. tion of second prin- E 4 + E 2 - (s 4 + P 2 + P 3 ) cipal point. Another way is to treat a six-lens combination as a combination of three couples of E.F.L.s respectively = E p E 2 , and E 3 , then apply Formula XlA., XIs., and XII. Formulae for Eight Thin Lenses or Elements This consists of two four-lens combinations, whose respective E.F.L.S we may call E/ and E/. (See Fig. 24.) Let P X be the first principal point of first four-lens combination, and P X its distance from e^ Let P 2 be the second principal point of first four-lens combination, and P its distance from e 38 A SYSTEM OF APPLIED OPTICS SECT. Let P 3 be the first principal point of second four-lens combination, and P. its distance from e,. o o Let P 4 be the second principal point of the second four-lens combina- tion, and P. its distance from e a . i Let PI = the first principal point of the eight-lens combination, and Pj' = the distance e 1 . . P^. Let P ' = the second principal point of the eight-lens combination, and P 2 ' = the distance P/ . . e s . Equivalent focal length for eight Then elements. Eight elements. Position of first prin- cipal point. P' = ' + p Eight elements. Position of second principal point. and 34 - XVII. XVIlA. XVIlB. Various methods of treatment. Cemented etc. While combinations of eight separate lenses may seldom occur, yet combinations of four thick lenses are frequently employed, and we have seen that such cases may be treated as cases of eight elements, the elements .appertaining to each solid lens being considered to be t lenses, separated by a distance s equal to - ; while if any lenses are cemented together or in contact, then, in the above formulae, the separation s , s 4 , or s 6 (or whichever it may be, in its natural order) should be entered as equal to 0, while the principal focal lengths of the elements in contact may be entered as usual, even when of equal refractive indices, iii which case they exactly neutralise one another and may be treated as non-existent. Or if of different refractive indices and in contact or cemented, then, as one is necessarily a collective element and the other a dispersive element, the difference of their powers may be taken as the power of one resultant element, and thus the calculations, which are inevitably tedious in complicated cases, be considerably simplified. Or the E.F.L. and principal points of each thick lens may be worked out separately, resulting in a combination of four equivalent lenses, whose effective separations of course depend upon the relative positions of their principal points ; then any of the above formulae suitable to the case may be employed. Having once obtained the principal equivalent focal length of any one more or less complicated combination of lenses, and the position of the two principal points (sometimes called nodal points) with reference 39 to the first element or first vertex of the combination, and the last element or last vertex of the combination respectively, then, as all conjugate focal lengths are to be measured from those principal points, the positions of all images or original plane objects and their images can always be correctly assigned with reference to the first and last vertices of the combination, if so desired, provided that the optical corrections of the system are at least approximately well carried out. For it must be borne in mind that the above lines of reasoning and the consequent formulas are based upon the theorems of Gauss, which are abstractions in the sense that they would be of no practical value whatever if applied to lens combinations thrown haphazard together in such manner that no approach to flat, distinct, and rectilinear images were made at all. The more perfect the images formed by complex lens systems, so much the nearer to absolute accuracy become the deduc- tions from the Gauss theory as embodied in the formulae which we have arrived at in this section. A few illustrative examples of the application of the formulae to Examples, known combinations may now be given. Let a sphere of glass of refractive index = 1/5 and of radius r be treated by the method of elements. Then Good optical correc- tions assumed. The above theorems acknowledge no op- tical aberrations. Case of refracting sphere. and and /i=/ 2 =2r / 9r A. I At rr and the E.F.L. by Formula X. ) ^ 3 3 '4r~12r-4r~ 8r~2 ' E.F.L. of sphere. while either the first or second positions of principal points are given by Formula IXA. or IXs., so that 4 P, = Po = - 2r+2r- -r -r o o Centre and principal points coincide. Thus if the lens is a solid sphere, then the distances of the two 40 A SYSTEM OF APPLIED OPTICS SECT. principal points are both + and coincide with the centre of the sphere, but if the combination is of two elements separated by an air- space equal to -, then the two principal points would overlap by a dis- P tance equal to t^ = -; but the performance of the spherical solid /x 3 lens and its equivalent combination of elements are exactly identical from the exterior point of view, so far as the E.F.L. and conjugate focal distances and relative scale of images are concerned. Huygenian Eye -pieces The usual and older form of the Huygeniau eye-piece (Fig. 25, Plate VI.) consisted of two lenses of principal focal lengths 3 and 1, placed at a distance apart equal to half the sum of their focal lengths, that being the necessary condition for the variously coloured images Two cases of Huy- being of equal size. In many cases, however, the ratio of 2 to 1 for geman eye-pieces. ^ Q principal focal lengths is adopted, the same rule for separation of course prevailing. Treating the case generally we have E.F.L., or ff _ .'1-'2 - J_\ J _2 = O ( ft-fl \ f .f h */2 /I + /2 Vl+/2'' -'i " t "-'2 2 2 --M - / +f /I + /2 and so that the power of the combination is the mean of the powers of the two lenses. For instance if /j = 3 and/ 2 = 1> we et E.F.L. = 1|, and = = |(J + 1) = |, and if f^ = 2 and/ 2 = 1, we get E.F.L. = 1|, and ^ = |(| + 1) = f . The position of the first principal point p^ is given by f -fi + -'z /-A + ^2 Principal points of - 2 /ju_ 2 / Huygenian eye- "i = / / = + / p and r 2 7~T~f ~ + Jy pieces. f + f. f + f - -^ Thus Figs. 25 and 26 represent the essential features of any such combination having s =' l 2 - 2t THEOEY OF EQUIVALENT FOCAL LENGTHS 41 It is clear that since, as in all previous cases, the distance from the first vertex to the principal focal plane / = E P 1 , and P X is in this case the larger, therefore E P is a minus quantity and indicates that the image formed by the eye-piece at / of distant objects on the right is a virtual one. On the other hand, E P 2 is a positive quantity, as P is the lesser, and indicates that a real image is formed at F 2 of a distant object on the left. It is well known that a real image of the relatively distant object glass to the left is formed at F 2 . Thus either of the distances p l . . F I or p 2 . . F 2 represents the principal equivalent focal length of the combination. It is clear also that when used with a telescope whose objective is to the left hand, the eye-piece must be so placed that the primary image formed by the objective must be made to fall upon the first principal focal plane /, in order that the rays emerging from the eye- piece may be parallel and fit for normal vision. For it is clear that the rays converging to the image in the first principal focal plane /will, after refraction by the first lens, be con- verged to a real image in F I . . F I( in which plane also is the second principal point p 9 , where it is also in the principal focal plane of the second lens. This coincidence of the position of the real image formed between the lenses with the position of the second principal point is characteristic of combinations wherein s = 1 Q 2 , but it is a matter concerning the internal economy of the combination as it were ; and we must remember that the formulae we have worked out for equivalent focal lengths and positions of the principal points, in themselves deal with resultants and take no explicit notice of what goes on between the lenses, but only deal with the positions of objects or images from or to which the rays are proceeding before they enter the system and after they emerge from it. Thus in Fig. 26, with regard to the rays entering the combination, a simple thin lens (having a principal focal length equal to the E.F.L. of the combination) may be imagined to be placed at the first principal point p v so that the entering rays converging to a real image at / and f..p 1 are about equal to the E.F.L. of the system; while, after emergence from the eye - piece, the rays of pencils are either parallel, as if coming from a distant virtual image on the left hand, slightly divergent from a nearer virtual image, or else slightly con- vergent to a real image on the right hand ; but in all cases the focal distance of such image, which is conjugate to the distance /. ,p r is Image in first prin cipal focal plane is a virtual one. Image formed in second principal focal plane is a real one. Condition of use with a telescope. Formulae of this section deal with re- sultant effects only. An elementary lens equivalent to the eye-piece. 42 A SYSTEM OF APPLIED OPTICS Another aspect of the question. Principal rays do not pass through geometric centre. The exit pupil of an eye-piece. Definition of pupil; in the case of an image of a real stop. The entrance pupil. Case where stop and pupil are one. Pupil not necessarily placed at geometric centre of a combina- tion. measured from the second principal point p^. Therefore, supposing I is a particular point in the first image at /, and we join I by a straight line I . ,p l to the first principal point p , then if we draw another straight line through p 2 parallel to I . .p^, it will cut the plane of the second conjugate image at the point where the image of the point I is formed therein (assuming distortion to be eliminated). While we are dwelling on the case of the Huygenian eye-piece, Fig. .26, we may, with much advantage, discuss an aspect of this question of equivalent focal lengths of lens combination which may well appear puzzling to those studying the question for the first time. In our treatment of thick lenses and combinations of two thin lenses or elements we have assumed the centre or principal rays of oblique pencils of rays to pass through the geometric centre of the said lens or pair of elements, but in Figs. 2 5 and 2 6 this does not take place at all, and, in fact, the principal rays of oblique pencils are shown to cross the optic axis, not at the geometric centre C, but at or near F 2 , the second principal focal plane. Now it is the size of the distant object glass to the left that defines the sizes of the pencils of light entering the eye-piece, and we have seen that an image of the object glass is formed very near to F ?J through which image pass all the more 'or less oblique pencils of light emerging from the eye-piece. This image is the exit pupil of the eye-piece, and its centre or the point on the axis where it occurs is the exit pupil point of the eye-piece. The pupil point or points of an optical combination may then be defined as the point or points where the principal rays traversing the combination, or their projections, cross the axis. In this case the pupil point is where an image of the object glass would be formed by L 1% If the object glass on the left is brought nearer to the eye-piece, then the pupil point will, of course, move towards the right. The aperture of the object glass may then be regarded as the entrance pupil of the eye-piece, the pupil being an image of it formed by L I} and the exit pupil is an image of that image formed by L 2 . But cases of other optical combinations may be imagined, such as photographic lenses, wherein the stop or diaphragm may be somewhere in the middle of the combination, and be an actual stop and not merely an image of another stop. In some cases the diaphragm or stop forming the pupil may be placed exactly at the geometric centre of the combination, as for simplicity has been assumed in working out the formuke in this section, but in very many cases it is not so placed. In fact, the position of the pupil point of any combination is totally independent of the position of the geometric centre, and therefore of PLATE VI Fi<5. 27. / f * f\ / \ Li A F, t * 1 p / L p T; i n \ 1 \ * j " \ F V f F . / Fi<|. 28. j6. 3O.a. PLATE. VI i. 27. . 28 29. ifHRJK 'id. 3O.a. Ill THEOEY OF EQUIVALENT FOCAL LENGTHS 43 the two principal points. But it might be thought that if the pupil point is widely removed from the geometric centre, as is the case in the Huygenian eye -piece, then the equivalent focal length of the combination might be quite different, and that our formula for the same would no longer hold good. This matter is certainly worth inquiring into. In the first place, the theorem of homogeneous pencils as explained on page 11, may be applied here. For although we are considering the oblique pencils traversing the Huygenian eye-piece as avoiding the geometric centre (and therefore the principal points) of the combination, yet if we imagine such pencils to be homogeneous, but very much enlarged in angular aperture, then we arrive at a state of things in which, although the principal or central rays of all such pencils still avoid the geometric centre, yet there is sure to be some one ray in each pencil which does actually pass through the geometric centre and the two principal points, and since such centre-traversing rays are proceeding to or from the same image points as the principal rays of the same pencils (ex hypothesi\ we therefore clearly see that the relative sizes and positions of the conjugate images should not be disturbed by the fact that the real pupil point in an optical system does not coincide with the geometric centre, or that the apparent pupil points do not coincide with the principal points, if we assume that the final image approximates to perfection in all respects. Assuming that the theorem of the homogeneous pencil holds good we may prove the case algebraically thus : Let e 1 and e y Fig. 27, be two thin lenses or elements, and let P be the position of the stop where the principal rays of oblique pencils are constrained to cross the optic axis Q . . q, and let P be placed any- where not necessarily coincident with the geometric centre, which may be at C for instance. Let Q x . . b l . . P . . . . q l be one of the oblique principal rays proceeding from an infinitely distant point Q t on the left hand to the image point of it at q 1 in the principal focal plane q . . q v Before entering e l this principal ray is proceeding to p , the first pupil point, which is the apparent position of P as refracted by e v and on emerging from e 2 it proceeds apparently from p , the second pupil point, which is the apparent position of P as refracted by 2 . Let the separa- tion x . . 2 = S, and let e 1 ..p 1 = e 1 . . P = P . . e z = S - D a = C 2 , and p 2 . . e z = D 9 ; so that we have C 1 and D I conjugates as well as C- 2 and D 9 . Then we have The geometric centre traversed by one ray of each oblique pencil. Proof that the E.F.L. is independent of position of the pupil. A 44 A SYSTEM OF APPLIED OPTICS SECT. and c = VJi_ and Dg = /.(S-D,) ^ Let angle Q x . . p l . . Q = -v/r p angle & 2 . . P . . 2 = ^r 2 , and angle ?i ^9 ? = ^3 ' ^ nen C, tan ^ 2 = tan ^ =r* and C 9 tan ^ 3 = tan p a ^=r; 2 therefore and tan Now the back focal distance e g . . 57 '" = - -0259 - ( - -0906) = + -0647. Final principal points of the process lens. (Here the sign of p" for the dispersive lens has to be reversed.) This particular combination will be seen to afford a capital illustration of the application of our formula?, as it embodies certain features characteristic of meniscus lenses, which may easily lead astray a student taking up investigations of this sort for the first time. There cannot be too much Pitfalls as to signs. care bestowed upon the matter of signs ; for in prolonged and intricate optical calculations errors in signs are more likely to occur, and are often more difficult to detect, than errors in mere arithmetic. There is a very common term used in connection with the focal Back focal length, lengths of lens combinations, and that is the Back Focal Length, or the distance from the outer vertex of the last lens to the image formed by the lens of infinitely distant objects. It is obvious that the back focal length is simply the algebraic difference between the equivalent focal length and the distance of the second principal point from the outer apex of the last lens, or B.F.L. = E-P 2 . The principal points of lens combinations are also often termed Nodal points and nodal points and focal centres. These terms more fully emphasise focal centres - the fact that a straight line drawn from a certain point Q : in the first conjugate image or object to the first nodal point is always parallel to a straight line drawn from the second nodal point to the point q l in the final conjugate image where the image of the aforesaid point Q : is formed. Certain defects in lens systems which may more or less disguise this normal law of projection, will be dealt with in subsequent .Sections. 52 A SYSTEM OF APPLIED OPTICS SECT. Ill What constitutes a telescope. E.F.L. of telescope = infinity. A telescope has no principal points. Judged by the formulae we have been dealing with in the present inquiry, the combination of lenses forming a telescope is of peculiar theoretical interest. For the condition of clear vision through a telescope for normal eyesight demands that the primary image of distant objects formed in the principal focal plane of the object glass shall also be in the principal focal plane of the eye-piece. Therefore the separation s between the object glass and the eye-piece = F +/ or the sum of their principal equivalent focal lengths. Then in the formula for the E.F.L. of the combination F+/- we have F+/ s=0 and E = infinity P x and P 2 , or = : and ^ 4 Also our formulae for severally = infinity. Thus the image is formed at the geometric centre of the combination forming the telescope, but it has no focal power and no principal points, although it may possess immense magnifying power. The subject of magnifying power will be best dealt with in a subsequent Section (IX.) relating to distortion. SECTION IV SPHERICAL ABERRATION OF SIMPLE AND COMBINED LENSES AND CON- DITIONS OF ITS ELIMINATION VON SEIDEL'S FIRST CONDITION Spherical Aberration of Direct or Axial Pencils Aberration not hitherto considered. Method pursued by Coddington. So far we have assumed that, in all cases of refraction of axial pencils of rays by a spherical surface or their reflection from any spherical surface, the rays so refracted or reflected will still diverge from or converge to definite points situated in the conjugate focal planes. It requires, however, a very slight practical or theoretical acquaint- ance with optics to convince one of the existence of what is known as Spherical Aberration, or the aberration or wandering of the outer rays of direct pencil from the theoretical conjugate focal point which we have hitherto assumed. In our investigation of this phenomenon we shall find it most convenient to deal with the case of spherical refracting surfaces and lenses first, and with the case of spherical reflecting surfaces afterwards. We will first follow the method pursued by Henry Coddington in his Treatise on the Reflection and Refraction of Light, Part I., pp. 56 et seq., also 90 et seq. Let Fig. 31, Plate VII., be a typical case of a convex refracting Diagrams explained surface EAR' of radius r, on which is impinging a cone or pencil of rays diverging from the point Q I} the axis of the pencil or the principal ray passing through the centre of curvature 0. After refraction the rays converge again, the rays ultimately near the axis focusing at Q^ and the marginal rays Q . . R and Q . . R' at the point Q 2 r . From R drop R . . P perpendicular to Q x . . Q^. It must of course be Construction, understood that in the diagrams the distance R . . P or y, which measures the semi-aperture of the pencil, is much exaggerated relatively to the radius of curvature, in order to make it easier to follow the diagram. Let A be the vertex of the surface and let be the centre of its curva- ture. Then it is evident that ,/QjRO is the supplement to the angle 53 54 A SYSTEM OF APPLIED OPTICS SECT. of incidence, while ,,/Q/RO is the angle of refraction. Hence sin QjRO = p sin Q/RO, also we have Q a . . _ sin Q X RO _ sin /_ Incidence -\ Q! . . R sin Q X OR sin Q X OR Q 2 ' . . O _ sin Q 2 'RO _ sin /_ Refraction I Qa'TTR = sin Q 2 'OR = sin Q X OR J in which p = the refractive index ; therefore The fundamental ^2 ^ _ ^i *-* / i \ equation. ^Q 2 ' . . R Qj . . R' Let Q 1 . . A = u, . . R = r, A . . Q^ = u v and A . . Q 2 ' = u. 2 ; and let R . . P = y, Q 2 ' . . = w 2 r, and Q l . . = u + r. Then we have Q 2 ' . . R = Q 2 ' . . A - vers (A . . P) + vers (a . . P) ?/ 2 , f therefore 2V' - _-~---. Q 2 ' . . R u 2 2w 2 2 \r M 2 / % 2 \ Mj.Vr u 2 J 2 / We have also Q x . . R = Q x . . A + vers (A . . P) + vers (P . . b) 1 therefore 1 1 y 2 - f\ Q...B u 2u" 2 \r uJ u{^ u\r Therefore Equation (1) expands to u 2 \r w 2 /2 w I tt\r it/2 _ 1(1 + By dividing both sides by r and reducing we get /I 1\ M/I l\ 2 y 2 /I 1\ l/l therefore I 4. _( I i_ = l | I I 1 I - -- ; -- v -- - ' 2 \r u/ u\r uJ 2 ' u 2 r u r u\r u/ 2 u 2 \r u 2 / 2' or __ ,_ ^_ ^ _. PLATE. VII p U Q; A=U, A Q = U,, AvQi-* A z -Q r u t Ai-Q^U, Q, Fi<>s 34. PLATE. VII Q ; A=U,A-Q;=U,, A--Q=U,. Fi^s.34. 55 We may now insert approximate values of u in the above coefficient First approximate ,,..., ,. values to be inserted of /, treating it as equal to u v so that in the corrections we may in formulaj assume that fj. fj. p, 1 M 2 ~ %! ~ r u and - (see Formula II., Section I.), /I 1\ 2 1/1 1\ 2 and therefore (- -J becomes -*(- + -), and the above equation \F Ibf)' tt \f w/ becomes = zi - 1 + r M O r u \ VI + IV + (V- 1 _ IV! u\r u/ \ r u/\r which further reduces to the more convenient form First refraction. Re- _ r i _ - + " i i _ + A ) ( i + ' ) ,,2 XVIII. (R.) ciprocal of corrected U 9 r u 2/j. 2 \r uJ \r u J second focal dis- tance. As before, we will number all important formula, such as the above, with Roman numerals, and all of minor importance, but useful as steps in the investigation, with ordinary numerals. The function of y z in XVIII. is the correction to be applied to the reciprocal value -^ or ___! expressing the reciprocal of the length 1 ' 1 u, of the ultimate or paraxial rays, in order to convert it into - ^r-, ; A. * . v^o and the distance Q^ . . Q 2 ', or the longitudinal aberration within the glass, is therefore XVIII. (L.) . . Lmear value of 2//, 2 \r u/ \r u J //, above aberration. It is desirable to call all corrections to the reciprocal values of distances R corrections, and all corrections to the linear values of such distances L corrections. Formula XVIII. will be found to interpret itself in all cases if due regard is paid to the conventions which we laid down on page 10. If the entering rays are converging, and u therefore minus, there is u obviously no aberration if either ? = u or = -- ^ . Let it now be supposed that the pencil of light is refracted a The second refrac- second time by a second spherical surface closely following the first, as tion- shown in Fig. 32, wherein Q 2 ' is the point on the axis to which the 56 A SYSTEM OF APPLIED OPTICS SECT. ray R..Q 2 'is converging in Fig. 31. Supposing the collective lens which is formed by these two spherical surfaces to be very thin, and of a sharp edge at R or R\ then we have R . . P in Fig. 3 1 = R . . P 2 in Fig. 32, or y^ = y f Supposing in Fig. 32 that the ray Q 2 .". R is travelling right to left, originating from Q 2 , and entering the convex surface R . . A 2 , then putting Q 2 . . A 2 = v, and A 2 . . Q 2 ' = v v and radius 2 . . R = s, we have by application of Formula XVIII. (R.) therefore Second refraction. 1 _ 1 u-l/l 1W1 u+lN Reciprocal of last - = - --- " + ~r-f[- + - ) ( - + " - W 2 . (2) focal distance. v v i */a? \s 9/ \t v J y But v l in Fig. 32 is identical with 2 in Fig. 31, if the axial thickness of the lens is zero, only we must remember that the ray R . . Q 2 ' is converging into the second surface ; and while the distance A . . Q 2 ' or A 2 . . Q 2 ' is positive relatively to the first surface, it is nega- tive relatively to the second surface, by convention. So that in the last Formula (2) we have and _ = + t of Formula XVIII. (R.). i u z We may now insert the full expression for from Formula XVIII. (R.) in Formula (2), and thus obtain u * u-1/1 1N 2 /1 + c --^ - + - (- 2/x 2 \s v) \s v In the last function of - and - we must of course assume v to be s v its first approximate value as the focal distance conjugate to u. On adding together we then get Spherical aberration of complete lens. IN 2 / 1 + I- + -) l-T i fy ^.S V/ \S V /) J ^XIX. (R.) iv SPHEEICAL ABEEEATION 57 Thus we arrive at the formula of the second approximation, which contains also the old formula of the first approximation, viz. 1 , VI 1\ 1 111 - = (/*- 1)1- + -) -- or - = ---, v \r s/ u v F u which states the relationship between the conjugate focal distances u and v, which we previously obtained as Formula III., but we have gone further than in that case and arrived at a formula for the deviation from the strict conjugate relationship, a correction which has to be applied to the value of v obtained from Formula III. This correction is the spherical aberration, and is seen to vary as y z or the square of the distance from the axis of the point in the lens where the particular ray dealt with traverses the lens. If, in Fig. 32, / is the point where rays ultimately close to the axis come to focus after refraction at both surfaces, such that 1 111 A../ F u v' then the distance / Q 2 will be the longitudinal value of the spherical aberration, which will be expressed by the formula ii-lr/1 1\ 2 /1 iA+l\ /I 1\ 2 /1 u + 1\ 1 Linear value of ~ - -- - - - 22 above sherical ab- 1\ 2 /1 iA+l\ /I 1\ 2 /1 u + 1\ 1 near vaue o -)(- + ) + (- + -) ( - + ) . \ . O'l 2(A..Q 1 X It is evident that as a second step in accuracy, though not a final one, we may write Versines according to third approxima- vers. A. .P = tion. Jr -p. , vers. a . . P = --^r, ~.~ -. prr p , if- .and in which expressions we may enter approximate values of the terms in the denominators. In the statement of vers. a . . P, the distance P . . Q 2 ' occurs, which differs from P . . Q/ by the longitudinal aberration Q/ . . Q 2 ', which is a function of the quantity x which we want to arrive at. In stating a value for the versine a . . P we cannot afford to neglect this aberration Q/ . . Q 2 ' as a deduction from the radius of curvature of the arc R . . a. Let Q l ..A = u and A . . Q^ = u/ (the first approximate value for paraxial rays) and A . . Q ' = u as before. Then let 1 1 A V f\ f T 5 * 7T~> ~ - , A . . Q 2 A . . Qj M/ so that the longitudinal aberration Q/..Q 2 /= xuf. As the basis of our inquiry we still have the strictly true relationship iv SPHEEICAL ABERRATION TO THE THIRD OKDER 59 Then The fundamental equation. therefore also we have Then Q 2 ' . . = u/ - xu/ 2 - r ; Q! . . O = u + r. /o\ Formula for Q 2 '..o. (4) Formula for = (u/ - xu 2 } - = (u/ - xu/ 2 ) - = (u, - xu 2 } 2r- 2r or 2(P . . Q 2 ') + (a . . P) 2 \r therefore y z fl y 2 y 2 f \ ^( + x + -/ - -f. - -f x ) ; 2\u/ 2ru/ 2 4-w/ 3 4M/ 2 / ! 1 r u/ 4 rw / / We now want the reciprocal value of Q 2 '. . R, and as we wish to pre- serve all functions of 7/ 4 , the term (a) must be developed to two terms . . 1 I a a 2 in the sense that - - - = - + -, + -. u - a u U* w Therefore we get 1 1 Q 2 '..R f_ 2u, 2 ' rf _L / /I 1\ y 4 /I 1\ 2 \ ^ I ___ 1 J. ^ ( ___ I V 2uf\r V *4i^\f u/J f 60 A SYSTEM OF APPLIED OPTICS and 1 1 / f f \ y z /I 1 L rf _ I y _ y I rf 4. ^ I T t/ . I ^ T Formula f o r Q 2 ' R **/ \ 2w/ 2 8w/ 4 / 2w/ 2 V r * f J x 3 3 \ -- 1 ---- + _ I / 2 \2?' 3 T Z U/ ftty 8 2U/ 3 /' - 4M 2 Next we have yZ / y2 ^.2\ ^2 -i (**)- 2r therefore f(\ 1\ y 4 / 1 1 1 Q 1 ..R = M + ^-(- + -) + -r(-^- -5-5- 2 \r / 4 \2r 3 rw 2 2w Here again we want the reciprocal value of Q r .R, which, analogously to our procedure in arriving at Formulae (5), may be stated thus Q! . . R u (2u 2 \r uJ 4w 3 \r u/ I 4^x2^ ru 2 2u 3 1 ?/ 2 / 1 1 \ 2/ 4 / 1 2 1 \ 1 y* i 1 1 1 and Formula for i ru The insertion of On substituting Formulae (3), (4), (5), and (6) in our basis equation Formula (3), (4), (6), and (6) in the funda- 7 1 1 mental equation. ^(^2 0/rT 7 r? = (^i ' ' / 'Q T?' I . . XV m^ . . JA> we then get 1 / w 2 ? + _^fJ_ + ! .A.^M ^^i 2 \ O-j*3 't''n yii >" tytt ^ / I rr t* / \ ^ / / t* / / w- 1 & (Jb i / _t A --- - - V* ( 1 _ 3 _ _ + r ' 2 + 23 2 3 ' SPHEEICAL ABERRATION TO THE THIRD ORDER 61 In expanding this equation we may legitimately omit functions of or X s ,, as x is a relatively small quantity, and also omit functions of ^. Then we get, after cancelling out a few terms, f f ^ (\ \\ {u,- Msf + Mi ( --- ) \T 2ti/ r 2ui\r u// ~9 9 "R r 2 u/ ruf 2M/ 3 / u/ for the first side of the equation, which then becomes y' 2 f f / - ^ + p - p-x + which ?/ 2 f r ( f f -& - ^~-x + /-#*+ [ /*- - ^r- 2w/ 2r y V 2u/ %uf/\r u/ 3 So that the whole equation now takes the form _ _ * 2r 3 + r 2 Tl f (\ 1\ w 4 / 1 1 3 3 _ /. i A.\I ___ _ j __ i __ \ _ y ( _ __ __ >Lu 2u?\r u) 4&\W r*u ru 2 2u By dividing both sides by pr, and keeping functions of x on the left- Both sides divided hand side we then get by ^r. 2 \uf r 2 __ f &l 2u/\r u/ 4:U/\r u/ \2r 3 3 3 62 A SYSTEM OF APPLIED OPTICS SECT. from which ell f ( l l V 2\^~^ A A r u/ ' 2u/\r u// ' 4w/\r M / /\2r 3 r 2 u/ ru/ 2 u + r u + r (/I l\y 2 f/ 4 / 1 1 3 3 \} fj.ru fj.ru \\r u/2u 4:U\2r^ r 2 u ru 2 2u s J j u/ to be expressed it i s now desirable to express u. or A . . Q/ in terms of u and r, for in terms of u and r. 1 also 1 1 _ yu/w - (/z - 1 )M + r u + r r u/ pru /j.ru ' After substituting these values in the equation and cancelling we then get - r ( yVtt(l-2,0-2w<,*-l) + f*n I 2V pW )} _ y 2 /(/A- l)M-r\ /M + r\ 2 y 4 /^" I)M-^ \ /_+ A / 1 2\ yurM J\ fj.ru J 4\ /mt A/rtt/\2r 8 r\ fj.ru J 2\ from which 'X,{\ - -( 2~s~2 ~ ) f fj, 2 r z u* /) f //* - 1 ivi i\ 2 y 2 /i i ***(- K-+-) + o (-+- V . .O \ n / / \ ^ /}/ / V /*fl* \ ** // ^j/x \ / U'/ \ i Ui/ ALLU/ \i a ^/-i.iYiTlv.1 _ r u/\r w/\2f 8 _ __ A A A - 3 2 2 3 ' iv SPHERICAL ABERRATION TO THE THIRD ORDER 63 from which we derive f f(u*(l-2p) '" On multiplying both sides of the equation by Elimination of func- tion of x. (the function of 7/ 2 being supposed to amount to less than T ^th), we then get, if we neglect functions of y 6 , M -i/i r\Vi x = r ~ - T (- + -) (- + - jr LL \T It/ \? It 1\ 2 /1 /* + 1 V 2 (l - 2u) - 2w(u - 1) + r 2 if i '_ \ I ^ \r / /Jl / \ fl /!/ W/ \l IV (a) We may now add together all the functions of y and (c), (b) being expressed in the form contained in (6) Functions of sorted out. After multiplying out the factors contained in the large brackets and adding them to the terms in (a), we get, after much reducing and cancelling out, r _ /i_lV! + IVf 1 + /* + 1N \ 2 = /aberration by second approximation \ 2/* 3 \r uJ\r u ) y ~ \ as per Formula XVIII. (R.) -^ ) y*fl 1\/ 1 1 4 6 4 w,V 2 M 2 6 /A 4 1 1 /xru 6 [jiru 2 6 ____ _ __ 2 w 2 rw. 3 XX. First refraction. Fonnula for tne spherical aberration of the third approxi- mation. 64 A SYSTEM OF APPLIED OPTICS Aberrations of as- cending orders theoretically inter- minable. Formulae of the fourth approxima- tion generally un- desirable. Distinction between air value and glass value of the aberra- tion. Thus we again arrive at Formula XVIII. (E.) of the second approxi- mation (but divided by /A), while in Formula XX. (E.) we have the correc- tive formula of the third approximation. This generally is a correction of very much smaller value than the correction of the second approxima- tion. If we were to pursue the investigation still further, that is, were we to develop the fundamental equation given on page 59 to higher and higher degrees of accuracy, then we should obtain a series of formulae for the spherical aberration, first to the second and third approximations, being the above functions of y 2 - and y 4 , and the following approximations, being functions of y & , y 8 , etc., or rising even powers of y, and also increasing in complexity. The Formula XX. is not too complex, especially after it has been transformed into a more convenient and general form, to be sometimes useful in the higher problems which have frequently to be dealt with ; but approximations of still higher orders are for practical purposes undesirable. We have now got in Formulae XVI II. (E.) -f/i and XX. (E.) taken together a fairly exact corrective x to the reciprocal value of the distance Awhile - = - --- . It must A..Q/ or ui, such that - +x = Lens. Formula of the second approxima- tion again emerges. A . . y x ' MI fJiT be borne in mind that we are dealing with the distance u f as measured within the substance of the glass or other refracting medium. It is easily seen, therefore, that if the pencil of rays we are dealing with is refracted into air again at a second surface closely following the first, then, quite apart from any further spherical aberration imparted at the second surface, the spherical aberration imparted at the first surface may be looked upon as an angular deviation from the true direction, which will be multiplied by the refractive index on being refracted through the second surface. An aberration correction of value a inside of the glass becomes fj,a on being refracted out of the glass. Therefore our value of x, the aberration correction, must be multiplied by /j, to bring it outside the lens, when we may add the formula to the analogous formula appertaining to the refraction at the second surface, just as we did before when we took Formula XVIII. of the second approximation for the first surface, and then added to it the corresponding formula for the second surface, thus obtaining the Formula XIX. for the complete lens. Adapting that method to our present case, our formula for the spherical aberration to the third approximation for the whole lens is expressed thus v /*-lr/l 1\V1 /B+1N /I IN 2 / 1 /* +1 Ua VYT/-RN x = o 2 il - + - ) I - + - ) + ( - + - ) (-+ ) \y XXI - ( R -> 2/i- I \r u/ \r u / \s v/ \s v / ) iv SPHERICAL ABERRATION TO THE THIRD ORDER 65 + 8 1 + -: I s v /jiTir fjiT 6 + T- -. + /*V I 1 -r + -5- //2s 4 ^ 1 1 1 pS* S 3 ? .sr' Formula of the third XXII. (R.) approximation com- plete. These two corrections are to be added to the value of -, when - = 11 v v simply. So that if in any given case we work out the value of - + X, then we may take its reciprocal for the longitudinal value of the corrected conjugate focal distance of the two rays which are refracted through the lens at the height y from the optic axis. Or if the aberration is small relatively to -, then we may take the linear or longitudinal value of the aberration as v 2 X, so that, since for a collective lens X is nearly always positive, the longitudinal aberration is a deduction from v when v is positive, and an increase to v when v is negative or the emergent rays diverging. It is clear that the formulae we have now arrived at for the Present spherical aberration of a thin lens do not easily lend themselves to analytical problems, such as finding the form of a lens requisite to give or to counteract a certain known amount of spherical aberration, and the next desirable step is to put the formulae into a shape that is better adapted to manipulation, as well as more elegant and simple. formulae Introduction of a more Scientific Notation Here we cannot conceivably do better than adopt the beautiful Coddington's device device apparently invented by Coddingtou and explained on page 110 ex P lained - of his work before referred to. He shows how the reciprocal values of the radii r and s, and of the conjugate focal distances n and v, may be expressed by the use of two terms x and a. It may shortly be explained thus. Since - + - for the ultimate axial pencils = ^ and 11 _ i _ - r s 111 66 A SYSTEM OF APPLIED OPTICS SECT. Let tt 2F and let so that 1 -fa 1 -a._l 2F 2F ~F' let 1 1 + g and let K^IJF-H". , < 10) so that 1 +x 1 -x 1 1_1 I a. The characteris- Therefore a becomes the characteristic of the state of convergence or tic of the conditions Ji ver gence of the rays constituting the axial pencil traversing the lens, of V6r'611CV in relation to the power of the lens. For instance, if the rays of the entering axial pencil are parallel, or - = 0, then a 1 ; while if the 1 i conjugate foci are equal, or - = -, then a = ; while if the rays of the u V 1 1 1 emergent pencil are parallel, and - = -s and - = 0, then a= + 1. In short, we may style the term a the characteristic of the vergency of the pencil traversing the lens. x. The~characteris- Also x becomes the characteristic of the shape of the lens. If the tic of the shape of a 11 lens lens is equiconvex and - = -, then x = ; if convexo-plane, then r s x= + 1 ; if plano-convex, x= 1. If meniscus, such that r= 1 and s = 3, then is + 2 ; and if the same meniscus is reversed, then # is 2. Fig. 34, Plate VII., gives numerous self-explanatory illus- trations of the application of the two terms x and a to different cases. This device of numerical characteristics represented by a and x is invaluable in practical analytical calculations. After substituting the above expressions for -, -, -, and - in the Formulae XXI. and XXII., and arranging the terms in descending powers of x and ascending powers of a, we get iv SPHERICAL ABERRATION TO THE THIRD ORDER 67 Lens. Codding- 8/ 3 . ?^ + 4(u + l)ox + (3/a + 2)0* - 1) 2 + ^- JXXIII. (R.) ton ' s formula for -l tt-lJ spherical aberra- + 3(/* - 1) 2 ( - 8/* 3 - 13/* 2 - 2/4 - 10)a% 2 + (^ _ 1 )4( _ 1 5^3 _ 9/A 2 _ 3/4 _ 5 y ') XXIV. (R.) + 20* - 1)( 18/* 5 + 4r/* 4 - p? - 5pJ 2 )ax + 0* - 1) 2 ( - 30/* 5 + 6/* 4 - 4/* 3 - 5/* 2 )o 2 + ( 3/*' + 3/* 6 /, In Formula XXIII. we again have in a more convenient form Ooddington's Formula XIX. for the spherical aberration to the second degree of approximation, while. XXIV. is a further correction to it worked out on the same lines to the third degree of approximation ; both of them being corrections to -, the latter being ascertained by the simple law of conjugate focal lengths, - = =, . The first is a function J ' v F u spherical tion. Formula for the spherical aberra- tion by third approximation, in terms of a and x. of , the second is a function of j- . J J We shall find, on further investi- other corrections of gation of cases of axial pencils traversing combinations of lenses, and especially separated lenses, that many other corrections arise which are also functions of y 4 , and which it will be desirable to work out, where possible, and add to the same category of corrections as XXIV. It is easily seen that these formulae will interpret themselves correctly in all conceivable cases. It will be as well to call the Formula XXIV. the Intrinsic Aberration Function of the order y*. For we shall find that although certain other aberration functions of the same order y* will have to be considered, yet they will turn out to be functions of Formula XXIII. that is, they will be products of the latter formula into another function of y 2 , and are therefore functions of y^ in that sense only. It will be found that corrections involving higher powers of y than 2/ 4 involve degrees of cumbrousness and complexity which are out of all proportion to their importance or utility. If the reader will apply the reasoning of this Section to the corre- sponding case of a dispersive lens, in which preferably u, u t , and v, as well as r and s, are all positive for convenience in reasoning, he will the order if. 68 A SYSTEM OF APPLIED OPTICS arrive at precisely the same formulae. That is, it is best to assume the rays to be converging into the first concave surface of the dispersive lens, to be diverging after first refraction, and more strongly diverging after the refraction at the second surface, which is also concave. Conditions under It is clear that there is only one term in Coddington's formula which the aberration 2 may be o or nega- (which may be conveniently referred to as l^sA.', while the Formula tive. 4 * 8/ 3 XXIV. may be termed ^ A' x ) which can ever be negative, and that 1 Joy is the second term, involving ax, so that the only possible way of approaching to freedom from aberration in a simple lens is to make a and x of opposite signs ; therefore if rays are strongly diverging into a positive lens and a is positive, then x must be negative, and vice versd. For instance, if p = 1/5, then we have O % ^x 2 + 40* + l)ax + (3 p. + 2)0* - l)a 2 + - T = 7a; 2 + Wax + 3-25a 2 + 675, Formula XXIII. differentiated with respect to x. Condition of mini- mum aberration. and this will equate to if a = at least + 4 '4 5, when x will be about 3'15, implying a strong meniscus form with its hollow side turned to receive the divergent rays. With a still higher plus value for a, a value for x may be found to give a certain amount of negative aberra- tion. This fact is utilised in many systems of condenser lenses whereof the member nearest the source of light is made of a pronounced meniscus type. On differentiating the Formula XXIII. with respect to x we have which will equate to when ^a: (12) so that if p = 1/5, then x, for minimum possible spherical aberration, 5 must be -a. 5 If the entering rays are parallel and a= 1, then x +-, so that the radii of curvature will be as 2 : 1 2 or 1:6; while if //, is about 1/67, then x= + 1, or the lens of minimum aberration is convexo-plane. It will be meniscus if the refractive index is still higher. If we suppose p. = 1/5, then the Formulae XXIII. and XXIV. work out to iv SPHERICAL ABERRATION TO THE THIRD ORDER 69 ? ,2 f ^ Values of the two -r -j^jj-j 7x 2 + lOca 1 + 3'25or + 6 '75 [ orders of aberration when fj. - 1-5. - 4-625z 4 - 33-625az 3 - 60'1875a: 2 - 51'94aV - 55'55a 2 \ -28-19a 3 z- 131-06aa;-5a 4 -24-7j < From this it appears that the corrections of the order ?/ 4 must be always of a negative character when a and x are of the same sign, as when parallel rays fall upon a plano-convex lens, i.e. when a = 1 and x = 1 ; but it will be found that if parallel rays fall upon a convexo-plane lens, in which case a = 1 and x= + t, then the functions of ax?, a?x, and ax come out positive and nearly neutralise the negative terms. For instance, if /= 1, y = '25, and //, = l - 5, a = 1, x = + 1, then y 2 ^sA' gives llr 1 71 Convexo-plane lens 6 Tsl ' ~ 10 + 3 ' 25 + 6 ' 75 I = + OK 7s' refracting parallel yo / ' yo / rays. and ^A" gives / - 4-625 + 33-625 - 60-1875 - 51'94 - 55-55 + 28-19 ) + 131-06-5-24-7/ (16)(16)(27)/ 5 \ 768/ 5 ' or only g^th part of the correction to the second approximation. But if x also = 1, then the first formula gives 1 I/ \ 27 1 Plano-convex lens Qfi TsV / F + Qfi /3' refracting parallel rays. and the second formula gives 1 If M 5\- -li?I. ^?1 (16)(16)(27)/H j 256 / 5 ~ 96 / 5 ' or nearly a quarter of the aberration of the order y 1 . But if / is 2 doubled while y keeps constant, then the aberration ^A' is reduced y 4 J to th part, while the aberration - ^A" is reduced to ^nd part. These conclusions apply with equal truth to the corresponding concavo-plane and plano-concave dispersive lenses when refracting parallel rays. A SYSTEM OF APPLIED OPTICS A thick lens requires special treatment. However, we must not treat tins formula as if it represented the only aberration correction of the order y* which has to be dealt with. For in the case of thick lenses of large relative apertures, or a system of separated lenses, the formulae, before alluded to, which are functions of y*- into the aberrations of the second approximation, may often exceed in importance the intrinsic formulae of the third approximation. We have hitherto assumed that the thickness of the lens to which this Formula XXIII. refers is too small to sensibly affect its accuracy, but in general practice cases very often occur in which the thicknesses of the lenses concerned are so considerable that no approach to accuracy could be made without making proper allowance for it. Here we shall again find that the Theorem of Elements will enable us to effectually get over the difficulty. Application of the Theorem of Elements to Thick Lenses Let Figs. 35a and 35b, Plate VIII., represent two thick lenses, one a collective lens and one a dispersive lens, the conjugate focal distances Q l . . A I and A 9 . . Q 9 being also positive in each of the two cases. Let tangents to the two vertices A 1 and A 2 of the lenses be drawn. These then represent planes perpendicular to the optic axis, and as we Element planes. imagine two elements to be located at the two vertices, these planes may appropriately be called Element Planes. Moreover, if we are treating these thick lenses in accordance with the Theorem of Elements, it is obvious that the two element planes are also the bounding planes or surfaces of the imaginary plate of parallel glass which is supposed to lie between the two elements. Let b l ..A l and & 2 ..A 2 = Y 1 and Y., respectively, and let &/'-, and & 2 ' . . c g = y l and y 2 respectively. Now so far, in working out the formula for spherical aberration for a curved surface like A 1 . . b^, we have assumed y (or b^ . . c^ to express the perpendicular distance of b^ (the point on the curved surface where the ray in question is refracted) from the optic axis. But we might have assumed y to mean not b^ . . c^, but A I . . b^ that is the height Y X of the point where the same ray cuts the element plane, instead of the height where the ray cuts the curved Simplicity gained by surface; and it is obvious that the plan of measuring our y's along assuming the y's to the two element planes of any lens presents the advantage of great lie in che element ,. ., , , ,, ,. ,, planes simplicity, and renders it perfectly easy to assign the values of the successive Y's for a ray traversing a series of thick or widely separated lenses. PLATE. VIII. R, i. 36.b. Q. Q 2 q; qq, PLATE. VIII. iv SPHEEICAL ABERRATION OF THICK LENSES, ETC. 71 That once granted, then of what nature will be the corrections to the spherical aberration following upon the nonconformity between the Y's measured in the element planes and the y's measured up to the points where the same rays strike the curved surfaces ? We shall soon see that these corrections are comprised under the order of functions of Y 4 , and of higher even powers of Y. Of course, if the entering rays are parallel, there is ' then no disparity between Y X and y v and no disparity between Y 2 and y^ if the emergent rays are parallel. If we treat the whole lens as a self-contained entity, then if Q x . . A I = u, and A., . . Q., = v, as before, and = = (//, !)(- + -), we find that -Y -i-i'7i J- 1 + \"i v l - V J- 1 2 \ Y Y 3 / Y 2 \ 1 1_1 Y 4- ! - V (1 4- l \ ^. ; *i ~*~ ^ * i\ * + ~n ;' 2rJ u l 2ru l n ; 2ruJ so that Similarly Y 2 \ ^ 2 - . 2sv/ The above two formula 1 serve to indicate the general nature of the corrections involved, and we will return to a more exact investigation of this matter at a later stage. After this we will assume our ?/'s to lie in the element planes except where otherwise stated ; therefore we will retain the symbol y in place of the symbol Y which we employed in the above inquiry. We will first consider the thick lenses in Figs. 35a and 355, in terms of the aberrations of the two surfaces. The rays radiating from Q 1 are supposed, after refraction at the first surface, to converge to a point q situated at a distance u ( = A I . . q) from A r that distance being an intra-glass measurement. In the case of Fig. 355 they are supposed to be diverging from q after first refraction, a condition analogous to that of Fig. 35a. Now, the spherical aberration of the first surface as yielded by Formula XVIII. (R.) is a correction to the first approximate value of or v, and the longitudinal aberration is obtained by multiplying XVIII. by u 2 , as in Formula XVIII. (L.). We will call the longitudinal aberration so obtained y*ajP. Now, we wish to transfer the value of the aberration of the first surface to a new reference point A , so that we can add it to the aberration of the second surface. Therefore we have an aberration Thick lens. Form of the aberrations of the two surfaces. How aberration of first surface is trans- ferred to second vertex. 72 A SYSTEM OF APPLIED OPTICS SECT. from the first surface denoted by y z a^, implying a linear aberration equal to y~a^, which, regarded from the new point A 9 , or the vertex c -11 i f " ? AM 2 \ 2 ? * 2 of the second surface, will be equal to . / /. 9 = 2 -- ^ in V (u- t) 2 J " l l (ti- ty- & the case of the collective lens, and y l x /,- -- . 2 in the case of the dis- \Jlv T" v/ persive lens, as an R correction. If we now add in the aberration of the second surface we have the joint aberration, referred to the point A 2 , expressed by \ \77>. Tw" Sum of the aberra- tion of the two for the collective lens, and surfaces. , for the dispersive lens, as R corrections. Furthermore, if the ?/s are so small that the versines of the curves are small and negligible quantities, we then have (u t\ "IT*/ and Relationship be- for co u ective l ens an d tween the two y's. for dispersive lens, which is a very simple relationship. Let us now treat the same lens by the method of elements. We may then denote the conjugate focal distances for the first element by u l and v r and those for the second element by w. 2 and v z , so 111 .111 that + = TT, and + --. U l i /l W 2 ^2 /2 Same thick lens Then at A^ we must imagine a convexo-plane element, and at A 2 a of e eiements metbod plano-convex element in the case of the collective lens ; and a concavo- plane and plano-concave element at A X and A 2 respectively in the case of the dispersive lens. The rays which converge to or diverge from q after refraction by the first surface of the first element will, after refraction at the second or plane surface of the element, converge to a5 or diverge from a new point distant from A l by - ( = vj. Then, since the separation between the two elements is -, we shall have a spherical aberration for the first element, which may be called #i 2 ( a i + PI)> PI being the aberrative function for the second 'or plane surface. This aberration becomes iv SPHEKICAL ABERRATION OF THICK LENSES, ETC. 73 when referred to the second element at A., for the collective lens, and for the dispersive lens ; while y 2 2 will be 11 t 2 /---\ / r 2/ A 1 !" \ 81 V l M Vv-*v J - / v % > for the collective lens, and y 2 2 will be ?/ / 2 /-A / r ,=*=- -4=- /* /* /* ?/ is obviously equal to -7 -, which we got before for the solid lens, It T f and the same applies to their reciprocals ; only, in the case of the imaginary elements separated by - we have supposed to exist the inner f* plane surfaces of the said elements, which do not exist in the solid lens. But it is clear that we can legitimately imagine the two inner interpretation of a plane surfaces of the two elements to exist in the solid lens, provided thick lens by theorem .'.,', * . _ Cf elements, that we also imagine to exist a solid parallel plate of glass of thickness t lying between and touching the said two elements. There would then be four plane surfaces to be imagined, two bounding the elements and two bounding the parallel plate. At each one of such plane surfaces, provided that the rays traversing the 74 A SYSTEM OF APPLIED OPTICS SECT. The six constituents of the whole aberra tion. interior of the lens are not parallel, a certain amount of aberration takes place. We may call the spherical aberration for the first element, as before, y^(a^ +p l ), x being the spherical aberration of the curved surface and p l the aberration of the second or plane surface of the element. Then the spherical aberration of the first surface of the parallel plate may be written y*(p^) ; the spherical aberration of the second surface of the parallel plate written 2/ 2 2 (/>./) ; and the spherical aberration of the second element may be y^(p^ + a^), in which p is the spherical aberration of the first or plane surface of the second element, and 2 that of the curved surface. So that the whole series of aberrations, referred to the point A 2 , may be expressed by Now it is plain that if a pencil of rays passes, however obliquely, from one piece of glass bounded by a plane surface into another piece of glass of the same refractive index and bounded by another plane surface in close contact with the plane surface of the first piece of glass, then no refraction and therefore no aberration whatsoever can take place. In other words, the refraction or aberration which takes place when the pencil of rays emerges from the first piece of glass into air is exactly neutralised by the opposite refraction or aberration ensuing on the same pencil being refracted again immediately into the second piece of glass, so that the two plane surfaces might be absent and the glass be solid and homogeneous so far as any optical effect upon the pencil of rays is concerned. Therefore in our series of aberrations it is clear that y*p l + y*p^ = and yp r\ u \ K, .. cos c/> therefore The exact form- x l ^ cos <' lllfl j ' P" 7 ^^^ *^i == P" "^l COS cp COS (b COS cp exactly. This can be reduced into an approximately accurate algebraic form, thus Since ,/ q-,..A, , 1 QT . . R, cos (h = -^ and q l . . Rj cos (f> therefore above equation becomes which iv ABEEEATION OF PARALLEL PLANE PLATE 77 in which we may insert the approximate values of q^ . . A 1; = fjn^, and for Q! . . A a write u lt making n 2 ( ^ \ ( uM, + jr* - I W-i \ 2uU-,/ /* L'M, , 2 1 = ~ l 1 2 therefore we get and therefore and 2, 1 1 2W 1 9 1 9 t- - 1 a^ {j? ' 2uf' First plane surface. (15) Formula of second approximation. It is clear that this formula applies to both cases, 36a and 36&, and that the aberration is of a minus character, implying an extension of the first approximate distance A } . . q. We can also derive Formula (15) from the Formula XVIII. expressing the spherical aberration of a single spherical surface. For the plane surface is but a spherical surface of infinite radius, so that - in XVIII. becomes zero, and the result is r Formula (15) (with a conventional difference of sign), which confirms our result. Further, it will be readily seen that the case of the con- vergent rays entering left to right into the plane surface is but the reversal, as it were, of the case of divergent rays passing out of the glass from right to left, and the same formula can be applied. Therefore the same formula which applies to the converging rays entering in Fig. 366 will apply also to the diverging rays leaving the glass in Fig. 36a. Turning our attention to this case, then let A 2 . . Q 2 = - = v 2 and A 2 . . Q 2 ' = Course of rays con- sidered reversed. 78 A SYSTEM OF APPLIED OPTICS SECI-. A 2 . . q = u 2 and A 2 . . It, = a,. Then we also have the relations and therefore the following identities hold good Cl-t dn = Qjf\ ~ Cln , t t u l = v 2 and v> 2 = u^ + - . Then we have at the second surface therefore * _ P ,P ~ 1 < y /ig\ I ^9 Q V A V / v 2 u 2 2fi- v 2 A This expresses the aberration of the pencil of divergent rays emerg- ing from the second surface, on the condition, of course, that the ' rays are diverging from a fixed point at a distance = u t) within the substance of the glass. After being refracted outwards they are subject to the aberration given in above Formula (16); and this aberration is of the opposite tendency to that which the rays met with on entering the glass, and implies a shortening of the first approximate value -2. But we have now to add the aberration produced at the first surface to that produced at the second. Aberration of first Iii order to transfer the aberration produced at the first surface to surface transferred / utt \ to second surface. the new reference point A , we must multiply (15) by ( L ) 2 , thus VuM, + tJ getting P 1 P 1 1 t ~n i /i e\\ I 1 " f^~ ~ ^1~ I /^l 1 i = (of Formula (16))= - ~" 9~i~ * ~8\ // independently of the second refraction. On adding the aberration of the second surface from (16) to the above, we then get 1 _ p +/ / 2 + 1 2fj? v/ 2fji iv ABERRATION OF PAEALLEL PLANE PLATE 79 This should, for the sake of practical convenience, be expressed in terms of v 9 and a y so that /* > i (do 2 therefore i i //' 2 - l n - A-j 1 ^^ _ Z2_/ YYV m ^ parallel plane plate. 1 _ 1 , A 1 " * *2~j YYV /R ^ Aberration of a ~ ~ ~ + oa ~i l - ^VAv.^rv.; If the same line of reasoning is applied to Fig. 366 the same result will be obtained, provided that u and v 9 are considered negative ; but if they are also considered positive then the spherical aberration will work out with a minus sign before it. In fact, we find that the aberration given by a parallel plate of glass is always of a negative character, if we compare its influence with that of a collective lens under normal conditions. If a pencil of divergent rays traverses a parallel plate, then the outer rays of the pencil on emergence are diverging from a point nearer to the second surface than the point indicated by the first approximation ; while in the case of a convergent pencil of rays the outer rays after emergence are converging to a point farther from the second surface than the point indicated by the first approximation. In short, the aberration is of the character of that yielded by a dispersive lens, and we shall afterwards find that this analogy holds good in other respects also. We also find from XXV. that the amount of the aberration increases inversely as the fourth power of the distance of that point from the second surface from which or to which the emergent rays are diverging or converging, and therefore there is no aberration in the case of Wj or v 2 being infinite or the rays parallel. We also find from our formula that 80 A SYSTEM OF APPLIED OPTICS Linear value of the above aberration. p - and therefore the latter term is the linear aberration, which thus varies inversely as v, directly as a 2 2 , and directly as t. Therefore it is plain that when the pencils of rays traversing the interior of thick lenses are strongly convergent or divergent, and the pencils are of wide aperture, the parallel plate aberration may be very considerable. The notation. A Detailed Confirmation of the Theorem of Elements Having worked out the Formula XXV. for the aberration pro- duced by a parallel plate, we are now in a position to give the general confirmation of the theorem of elements as applied to thick lenses. This proof can best be presented in the form of a balance- sheet (see p. 81), on one side of which we insert the successive aberra- tions of the six surfaces in their order, two belonging to the first element, two to the parallel plate, and two to the second element ; while on the other side we gather together the aberrations of the first pair of surfaces and express them as the aberration for the first element, the aberrations of the third and fourth plane surfaces and express them as the aberration of the parallel plate, and the aberrations of the two last surfaces and express them as the aberration of the second element. Then in comparing the one side with the other the identity of the two sums is clearly established, while at the same time it is also clearly seen on looking down the left-hand side that the whole sum for the six surfaces is identical with the sum of the aberrations of the first and sixth surfaces only, the intervening aberrations neutralising one another. The notation is as follows : y l is the height of the ray where it cuts the first element plane, y z is the height of the ray where it cuts the second element plane, % and v 1 are the conjugate focal distances for the first element, w x is the distance from first vertex to the point to which the rays are converging after refraction by the first surface, and w 2 and v z are the first and second conjugate focal distances for the second element, so that % and ti 2 are within glass measurements, so that ? 2 = il l t (t being the thickness), and therefore SPHERICAL ABERRATION OF A THICK LENS 81 o H g p[J |~ T- S b + g to ^ -u- ^ rO S 05 ^ ' ^ O ' rH ! ;T ^ + r- 4 -i 1 sT - + ^ *" -* ^> 3. - a., + If a i a. + ~ U ^ I < aT + ^ ^Ta. 4"! ^ >, ^ ^' + -' ^ tf 1 w ^T~ -^ ^ i J ^ jT C. *s T ~v ~^ ' C) ^ V Hi* "* r-iiT -Ij? " ^ ^\t ^ s^ ~> -~^. ' ?- T-H s 1 - 1 : ^* + g > 1 * ^s ^ * I 1- ^ I 1 i S "a. 04 f- ^i *V, cS f- '^ i i q i i 1 4. B 1 n 7Q O (M a. J tn a. lr ^_ !N ^ J w =*- * , n oq v^ I a. 0?" ' ^ 1 ill"" i II + > ( y e ." C_ a - + O ,j i .2 II 2 s * = * ^ ^ 1 ^ 1 " ^ T *S OJ 02 ^ zn a? tc -g a ! 82 A SYSTEM OF APPLIED OPTICS SECT. On both sides of this balance-sheet all the aberrations are referred to the second vertex of the lens or to the second element, and y. 2 is (u \ t j ; also the aberrations of %2 + /*/ / V \ the parallel plate are similarly treated, so that ?/ 2 becomes y l ( t j , X A^*^ while ?' x and r 2 are radii of the first element, and s x and s 2 those of the second element. In the above formulas it has been the more convenient for our purpose to consider u 2 as a positive quantity, and the sign prefixed to the formula for each surface shows whether the aberration is + or with respect to the final results. But after gathering together the two last formulae into one formula for the second element, the con- vention of u 2 being is resumed. A Practical Illustration As a further confirmation of the above theorem, and as an arithmetical illustration of the practical application of Coddington's Formula XXIII. and the above Formula XXV. to a thick lens, Treatment by ele- treated by the method of elements, we will take the case of a merits. f /I 1\ 1 lens of principal focal length =1 '5, such that (^- + ~)(f J -~ 1) = [Tg> yu, = 1-50, r = 1, and r 2 or s = 3, while the central thickness t = '75. We will suppose u^ to be infinite and the entering rays parallel. The power of the first element = = -, therefore / x = 2. Powers of the two r i elements. u - 1 '5 The power of the second element = 1 = , therefore/ 2 = 6. Let us suppose y l to be '40 ; then since 2 = v 1 and _t_ Relation between Uc> V ~*-~~n. 2 - '50 3 thetwov's. y9 = y\-^-y\ ~-y\ * = ^i' 1 V 1 ^ therefore Values of the two Then we have a = 1 and x = 4-1, while at the second element a ' B - we have nf ~ ~ i .K > ' ' a 2 ~ 1 .K ~ iv SPHEKICAL ABERRATION OF A THICK LENS 83 and 2 ' of these two lenses. It may then be assumed that the distance t becomes an air-space of thickness -, so far as our present purposes are concerned. ^ It is clear that the vertical difference between y^ and Y x is the Y Y horizontal distance ^ . . &/ multiplied by 1 . or *. But b b f is the versine of the curve of radius r l for the semi- chord & / . . c r It is sufficiently accurate to suppose that iv OTHER ABERRATIONS OF THE THIRD ORDER 87 Then we have V 2 V V 3 w Y + x = Y + * (18} Expression for yi\in. 2^ u^ l 2r 1 w 1 * terms of YI, etc. We next proceed to find the value of y 2 in terms of y r It is plain that t X i . i i ' 1 I m X 1 ^ 9 l y y i jjj winch I L * I so that _ _ 1/rp _ Y i 2 _ Y 2 2 \ Xl Expression for y 2 in 2 ~ * u\ 2r.. 2r 2 / v l ' terms of y,, etc. in which we may insert for y l the value given above in (18), so that Y 3 1 / Y 2 Y 2 \ Y y = Y + l - -IT - ) or ^Y Y 3 Y 3 Y 2 Y Expression for t/ 2 in 1/2 = Yj - - T ^-^- + ^ ^ - + ^ ; terms of Yj and Y 2 , i l XI i 11 *21 CuC. in which, as we have already seen, Y 3 Y 3 Y 2 Y Y _l = Y 2 , so that y 2 = Y 2 /A v t As the last three terms are small quantities compared to Y 2 we may say that V 3V V 3V V 3V ,,2_V2, 1 1 2, ^JL_2 , X 2 X l . y 9 x etc - 88 A SYSTEM OF APPLIED OPTICS SECT. If the rays are converging into the second element, as in the diagram, then, as u 2 in this case would be negative, all the above terms would arithmetically work out positive. We saw from Formula (18) that Expression for y? in terms of Yj, etc. Aberration of first element corrected for versine. Aberration of second element corrected for versine. therefore V 4 / V 2 a =21+ So that, having now obtained expressions for y 2 and y* in terms of Y x 2 and Y 2 2 , we may state the aberration of the first element to be (20) (21) and the aberration of the second element to be V 2 -.A.') 1 + Y 2 - / 11 2 l\~v ( - JV-i - -S ----- ) I V ru u ruJ j These formulae, however, are open to objection in their present form. In the application of (20), for instance, to the first element of a thick positive lens in which the first surface is concave and therefore r l is negative, and still supposing that the entering rays are diverging into the first element, as in Fig. 35, it is plain that y l will Y 2 be less than Y., instead of greater, so that - 1 - should turn out negative fjttj if the formula is quite self-interpreting. But obviously r l should be entered as a negative quantity ; moreover, by our conventions previ- ously laid down, u should also be entered as a negative quantity, and Y 2 therefore would remain positive, which is obviously wrong. T-.U In order to render Formulae (20) and (21) quite self-interpreting, we may leave u and for j 2 intact, while putting 1 1 for , for etc. Then becomes 1+04 , and therefore Formula (20) becomes Formula (20) in self- interpreting form. Obviously if r and / a become negative, then by convention becomes tt, iv SPHERICAL ABERRATIONS OF THE THIRD ORDER negative with respect to /^ and (1 + a ; ) is therefore negative, like manner Formula (21) becomes In 2^(/t- XXVII. Formula (21) in self- interpreting form. Since / in the denominators of the first two functions in the inside brackets may be expressed as n/ 2 , it is evident that the corrections in the inside brackets in both Formulas XXVI. and XXVII. are aberrations Y 4 of the order -= similarly to the intrinsic aberration functions of the third approximation. It is clear that these formulas may be applied to any pair of elements constituting a thick lens. Thus the corrections that have to be added to the first values of the aberration to the order Y 2 , as ascertained from Y I and Y 2 in the element planes, are functions of Y 4 and of the -aberration of the second approximation as expressed in Formula XXIII. Precisely the same formula will be obtained by the same course of reasoning in the case of the negative lens, Fig. 35&, although in the intermediate processes the signs of T and t are different. As these corrections are consequent upon the curved surfaces Above versinecorrec- retreating from the element planes, we may fitly call them the versine corrections of the order Y 4 , in distinction from the intrinsic aberrative corrections of the order Y 4 as expressed in Formula XXIV. tions distinguished from intrinsic cor- rections of the same order. Practical Application of the Intrinsic Aberration of the Order Y 4 to the same Lens as before As an instance of the arithmetical application of these aberration formulas of the order Y 4 we will take the same lens of radii 1 and 3, thickness '75, Y x = -40, and Y 2 = -30, with entering rays parallel, for which we worked out an aberration of the order Y 2 equal to + '0645. Applying the Intrinsic Aberration Formula XXIV. we get for the first element, since x l = +1, and a x = 1, Q40) 4 f - 4-625 + 33-625 - 60-1875 - 51'94 - 55'55 + 28-19 + 131'06-j 27(2)H _5_24-7/ (27)(32) l 32 Intrinsic aberration of the third order for first element. or about of the aberration of the order Y 2 , which was +'02 33 3. 90 A SYSTEM OF APPLIED OPTICS SECT. Above aberration 111 order to transfer this to the second element, we must, as before, transfei vertex. transferredtosecond multiply by ^ = \, thus getting - "00048. For the second element, with # 2 = 1, and a 2 = 9, as before, we get (-30) 4 } - 4-625 - 302-625 - 60-1875 - 4206'9 - 4499'3 - 20548'7-\ 27(6)~H - 1179-6 -32958-7 -24-7 / Intrinsic aberration '0081 of the third order = / 97 w fl N 5 (~ 63785} = '00246, for second element. or about j^th of the aberration of the order Y 2 , which was + '0255. So that we have - -00048 for first element and - -00246 for second element. Total of above. Total. . -'00294 for the intrinsic aberration corrections of the order Y 4 . Aberration of the To work out a formula for the aberration of the parallel glass ailei plate no^im- plate also to the order Y 4 would scarcely be of any importance, for, as portant. a ru i 6) eve n the parallel plate aberrations of the order Y 2 are small compared to the aberrations of the elements. Application of the Versine Corrections to the same Lens We will now turn to the versine corrections of the order Y 4 for the above lens. At the first element we have Y 2 /Y 2N Versine correction wn ich = 0, since u. = infinity, for first element = 0. * At the second element we have, as applying to this case, 8/2 2 in which, since u^ = infinity, the first term vanishes. In the remain - / T\ ing two terms v l = 2, yu-=l"5, ?' = 1, r 9 = 3, and u 9 = ( *, ) / -75\ . ' P = ( 2 - ) = 1'5, so that the formula becomes \ 1 *o/ <- aM )t- (1-5)(1)(- 3-375) ( Versine corrections / 4 1 \ / 1 \ for second element. = '0255 - n ^ + i element. This formula is open to the objection that if L 2 were dispersive, then V - would be positive instead of negative, and the correction to Y would u 2 come out as an increment instead of the decrement, which it so obviously is. But we can make the formula universally self- interpreting by adopting the same device as in the case of the versine corrections, thus arriving at Above formula in /,. \ 2 c y 2 / i , \ / ^ self-interpreting y 2 m Y p) { 1 + fr^\(r^)&l \ XXVIII. form. \V 4/ a 3 1 \l-.a 1 //j Now, if / is dispersive, it is negative relatively to / 1? so that j is /2 negative, while 1 + a, 2 and 1 a t are both positive, therefore the correction to Y 1 comes out negative. The spherical aberration of L 2 may now be written in the form or, if we express Y 2 in terms of Y ls in the form Whole expression for the aberration of Lo, including that of the third order. V 2/M 1 /A'N/I^^I A' /ii?iVla \ . Y i (-) c/ 3 ( A 2 )i l + 773 A 1 1 iT ~ ) 7 S i p U i 7 8/ 2 Vi - a i//2 so that the aberration of the order Y x 4 , when separated out, is berration o the / \ 2 1 1 /l-t-nX/" third order for L 2 Y (Sl) ^.(A / f V-i i >%. (23) isolated. \V 8/ 2 3V ''i/^ M-V/f Aberration of the The Y's modified by In this case we may say that the modification of Y at the second ceding lenses. ^ ens an( ^ ^ ne consequent modification of its aberration is due to borrowed aberration. Let it now be supposed that another lens is added to the right hand of L 2 and at a distance = s 2 from it. Then it is evident that the aberration of L t will not only affect Y 2 , but will generally affect Y 3 in still greater degree, since L 3 is further removed from L I . The aberration of L X will be transferred right through L 2 on to L g . Not only so, but L 2 will add (if it is a collective lens) its own aberration to the aberration of L : passing through it, and therefore Y will be affected by the two aberrations borrowed from L X and L 2 . OTHEE ABEEEATIONS OF THE THIED OEDEE 93 We will here refer forward to Fig. 96, Plate XX., which represents a case of four collective lenses or elements in succession, so arranged that all the w's and v's are equal and positive. The first lens only is supposed to give an aberration whose linear amount is Q t . . q lt while the other three lenses are supposed to be free from aberration and to simply copy through from focus to focus the aberration given by L^ ; yet the cumulative effect upon the successive Y's is most marked, and The cumulative they grow larger and larger as we proceed from left to right. Of course, if L 2 , for instance, is a dispersive lens, then the effect of Y*'s. its aberration on Y 3 will more or less neutralise the effect of the aberration of L r The formulae giving the modifications of the aberrations of the third and fourth lenses due to aberrations borrowed from the preceding lenses are naturally more complex and unwieldy than XXVIII., and it will suffice to give the complete expressions for the spherical aberrations of the third and fourth lenses of a series of four widely separated elements or thin lenses, without detailing their working out. The student may easily verify the formulae for himself. We have already obtained the expression for the second lens or element in Formula XXVIIlA., and we will adhere to the highly convenient All the Y's to be ex- expedient of expressing all the Y's of the succeeding lenses in terms ^ es of Y,. Then the formula for the corrected spherical aberration of the third lens is, in self-interpreting form, ^ 1 ~1~ CCo / n / JL -i . t %n I Whole expression for 1 ' XXVIIlB. tne aberration of L 3 , including that of the third order. 4/, 15 0l z* 2 V J J and the formula for the fourth lens is 1 2 ,u* Y/ . ., ., ?','' V 2 / v v \2> *1 A' WM I xxvnic. Whole expression for the aberration of L 4 , including that of the third order. The formula for the fifth lens would evidently contain ten terms, and that for the sixth lens fifteen terms. In the case of large 94 A SYSTEM OF APPLIED OPTICS Vergency variation for L. 2 due to aberra- tionofLj. Vergency variation for L 3 due to aberra- turns of 1.,+Ls. Vergency variation for L 4 due to aberra- tion80fL 1 ,L 8 ,andL 3 . apertures and separations, the corrections of the order Y 4 may form a large percentage of the spherical aberrations of the order Y 2 . Vergency Variations consequent upon the Aberrations of one or more preceding Lenses There is now a further modification of the aberration of L in 2 Fig. 36 to be considered, which, strictly speaking, applies even when L 2 is in contact with L I} but applies with much greater force if S is large compared with v^. Hitherto in assessing the value of the vergency characteristic a for any lens or element, we have assumed that there is a fixed point Q from which or to which the rays are diverging or converging before entering. But in Fig. 36 it is clear that in the case of L 2 the entering rays are converging to a varying point Q 2 ', which recedes farther and farther from Q 2 in proportion to Y x 2 , the recession being a function of the spherical aberration of L^ We may regard Q 2 . . Q 2 ' as a variation of either v } or % 2 , and since in Fig. 36 u 2 is minus, we have therefore so that we have in which therefore and JL. -XiVV. A - 3 ! 11 2 2 a 2 + Aa 2 1 2/2 1+a V2 _ 2/ 2 Y/ 2 , ^ In the same way we find that and 2 V2 ' 2 + AI A' 4/; 3 l \^ 4/ 2 3 , V2 /wvw.N 2 Y, 2 A/ /W 2 2 i; s \ 2 Y/ 2 A/ / 2 8 t' 8 \ 2 ) - . / J -J.A ( J JO.] + -} A' I- 2 - 2 -^) +--LA 3 (- -*) [ (26) \W X*W 4 A "VWV ' >^%V ' iv SPHEKICAL ABERRATIONS OF THE THIRD ORDER 95 Now if we differentiate the formula for spherical aberration of the Differentiation of second lens with respect to 2 we get th "P^ricai aber- ration formula with respect to a. in which we may substitute Formula (26) for da, 2 , and and then get for Y 2 2 , XXIX - / 2 \ 2 In this formula Y 2 has been expressed as Y, 2 ! -^ ), which has \V/ V cancelled out the -\ of (24), and as / x can be expressed as nf y we see ^2 V 4 that the correction is of the order -^ and is the expression for the /2 variation in the spherical aberration of L 2 consequent upon the varia- tion in a due to the aberration of L r In the same way the complete expressions for the functions of da 3 and da^ can be worked out. In these two cases of the effects of the aberration of one lens upon another we have assumed that the rays entering the first or left-hand lens are either diverging from or converging to a definite point on the axis: But if we have to look upon these rays as principal rays, each such ray being the central ray of a pencil, then it often happens that such principal rays are constrained to pass through a definite point on the axis after passage through one, two, or perhaps all of the lenses of a series, owing to a diaphragm with a circular aperture being placed at the desired crossing point. In such a case, of course, it is the more simple and convenient to regard the rays as travelling from right to left, and the formulae ex- pressing the corrections to the aberrations consequent on borrowed aberrations may then be worked in inverse order. However, these considerations do not strictly apply in the present section, but only when we come to deal with the optical characteristics of lenses other than spherical aberration, and especially distortion. Complete formula for variation in a 2 consequent from aberration of Lj. Summary of the Spherical Aberrations of the Order Y 4 On summing up these spherical aberrations of the order Y , we have for each element or thin lens First, as applying to all single lenses, and in the case of all 96 A SYSTEM OF APPLIED OPTICS First, the intrinsic aberration functions. Second, the versine corrections to the aberration. Third, the correc- tions to the Y's due to aberrations of preceding lenses. Fourth, the correc- tions to the n's due to aberrations of preceding lenses. Y 4 elements, the intrinsic aberration function of the order -^ as expressed by Formula XXIY. Second, as applying to all single lenses, and in all cases, the versine Y 4 corrections to the aberration of the order -^ as expressed in Formula / y 4 XXVI. for the first element of a thick lens, and also of the order * / as in Formula XXVII. for the second lens element. Thus in a series of lenses, Formula XXVI. applies to the first, third, fifth, seventh elements, etc., and Formula XXVII. to the second, fourth, sixth elements, etc. Third, but only where separations exist between lenses or elements, the corrections to the aberration of a lens or element due to the Y 4 variation in its Y caused by borrowed aberration of the order -^ as expressed in Formulae XXVIIlA., B, and c. Fourth, but only in the case of one lens being preceded by others, and especially if widely separated, the corrections to the aberration of a lens or element due to the variation of its vergency characteristic, Y 4 and caused by borrowed aberration of the order -j- & as expressed in Formula XXIX. Hybrid Spherical Aberrations Let it now be supposed that in a system of lenses the above aberrations of the order Y 4 do not neutralise one another, but that there is a perceptible balance left over ; then the question arises, can they be neutralised by a contrary overplus of .aberration of the order Y 2 ? We shall soon see that they cannot. Let it be supposed that Y represents the extreme semi-aperture of second order cannot system of lenses iii which we are seeking to eradicate all the be properly neutral- * .... ised by a contrary spherical aberration, and that there is a residue of minus aberration of f *** the order Y * Then > of course > ifc is q uite possible and practicable to counteract this residue by leaving in the system a residue of plus aberration of the order Y 2 , so that we have An aberration of the /,* +//,* = 0, (27) in which // represents a certain coefficient of Y 2 , and /// represents a certain coefficient of Y 4 . Then it is obvious that the relationship of these two coefficients is given by ///- "//a" (28) HYBKID SPHEEICAL ABEEKATIONS 97 Let us now take another measure of the semi-aperture, smaller than Y, and call it y. Then since the coefficients and their relation- ship are constant, tho only variable being y, then we have fjif +f//y* to express the aberration for the smaller semi-aperture y, and if we differentiate this expression with respect to y we get (2/ /y+ 4/ //y )Jy. (29) Then it is plain that we can equate this differential coefficient to 0, thus : 2f/y + 4/^y 3 = 0, in which (from 28) f// = //^; so that we then have and (30) tion. Evidently, then, at a distance from the axis- such that y=-j=> mum hybrid aberra- ** " there is a maximum deviation from a true balance of the two orders of aberration, and the amount of this maximum deviation may be easily determined as follows : Since Y 2 Y 4 v '-= , therefore at the height j= from the axis the state of the aberration is v 2 given by an expression exactly analogous to (27), viz. //2T+///y 4 becomes Y 2 Y 4 1 // +///, in which //y^ mav be substituted for f n (from (28), so that we then have ,Y 2 .Y 2 , . , /,Y 2 //-r--//^-, which = +- // , (31) or exactly one-fourth part of the + aberration of the order Y 2 to which the ray passing through at the extreme semi-aperture Y is subject. This theorem is illustrated in a striking and convincing manner by the diagram, Fig. 37. Let L . . D be the optic axis of a system of lenses of semi-aperture = D . . P, placed somewhere towards the left hand, and let A 2 . . P represent the longitudinal value of a residual amount of negative spherical aberration of the order Y" 4 to which the edge ray is subject. Then let there be introduced such an amount of positive spherical Maximum hybrid aberration is one- fourth of the aber- ration of the second order and of the same sign. 98 A SYSTEM OF APPLIED OPTICS SECT. Zone of aberration explained. aberration of the order Y 2 as will neutralise the negative aberration of the order Y 4 . That is, A l . . P = P . . A 2 , and represents the longitudinal value of the positive spherical aberration of the order Y 2 . Then, as these two aberrations for the edge ray are equal and opposite, the said ray will, of course, focus at P in the same plane as D, the focus for ultimate centre rays as given by formulae of first approximation. But if the abscissae of the curve D A I are made to vary, as y 2 " or the square of the height from L . . D of any point in the curve, and the abscissas of the curve D . . A 2 are made to vary, as y* or the fourth power of the height from L . . D, then it is easy to see that the resultant curve joining loci of actual focal points for rays traversing the system at different heights from the axis will be the curve D . . m . . P, Y 2 having its maximum abscissa at m, where y 2 = , and that m . . b will 2i be exactly a quarter of P . . A x or P . . A 2 . Here we have the explanation of a phenomenon familiar to many opticians who have attempted optical systems of large relative aperture, and found it impossible to obtain a well-defined axial image of a point owing to the presence of what we may fitly call " a zone of aberration," which exhibits itself in the form of a bright diffuse zone or annulus within the cone of rays, which is visible through an eye-piece placed either inside of the focus or beyond it. While the edge rays at the height Y from the axis and ultimate centre rays may be brought to the same focus, yet the rays traversing Y the system at a height equal to r=- intersect the optic axis at perhaps a considerable distance either short of or beyond the focal point for axial and edge rays. The reason why, when the eye-piece is placed well Phenomena at the within or beyond the focus, the phenomenon gives rise to a bright zone, is rendered plain by means of the diagram, Fig. 38, which accurately represents the rays coming to focus in a case where there is hybrid aberration, brought about as in Fig 37. If the eye-piece is made to focus upon a plane somewhere about a . . a, it is evident that a condensation of rays occurs about half-way between centre and periphery of the circular penumbra or section of the cone of rays. On approaching the focus, as at position b . . b, the condensation of rays is still more marked, but it occurs now relatively nearer to the centre, while at b f . . b f the zone of aberration is at its most distinct phase and has a radius of about one-fourth of the radius of the whole penumbra. The extreme edge ray focuses or cuts the optic axis at P, focus. HYBRID SPHERICAL ABERRATIONS 99 which is supposed to be the focal point also for the rays ultimately close to the axis, as given by the formulae of first approximation. The whole distance m . . P along the axis over which the hybrid aberration spreads itself of course corresponds to the maximum distance m . . b in Fig. 3 7. If the eye-piece is made to focus upon planes beyond the focus in this case, then a ring of rarefaction or a comparatively dark ring will show itself, corresponding to the bright ring visible inside focus. In the plane c . . c the central bright nucleus is very marked. It is clear from Fig. 37 that the bright zone of aberration will always show itself on the same side of the focus as the aberration of the order Y 2 , while a corresponding dark zone will show itself on the same side of the focus as the opposing aberration of the order Y 4 . It is the existence of outstanding aberration of the third approxi- mation or of the order Y 4 , as represented by P . . A., in Fig. 37, which is supposed to have necessitated our having in the system an equal and opposite aberration of the second approximation or of the order Y 2 , as represented by A I . . P ; and we have seen that the incongruity between the two orders of aberration gives rise to a maximum amount of hybrid aberration whose amount m . . b is always one-fourth of the amount of the aberration A I . . P of the order y 2 to which this extreme ray is subject. We have also seen that all the aberrations of the order Y 4 which Y 4 arise in a lens or system of lenses are functions of --. From this it follows that if in place of each lens of a combination we substitute two lenses, each being of half the power or double the focal length Y 4 of the original, then, instead of an aberration represented by -j- 5 , we / Y 4 \ 1 Y 4 * have an aberration represented by 2 ( . r-g 1 or 75 Tims, supposing we are troubled with a zone of aberration at the focus of any given system, and it cannot be eliminated by opposing plus aberrations of the order Y 4 against minus aberrations of the same order, then we can at once reduce the zone to one-sixteenth part (as a general proposition) by the expedient of splitting up the lenses, or at any rate the most violently curved one, into two lenses each of half the power of the original. It is also evident that the linear amount of hybrid aberration in any given case and the consequent intensity of the zone will be multiplied 16 times on doubling the aperture. It is also worth while to glance at the case of the hybrid aberration which arises when we correct a certain amount of aberration Opposite effects at the two sides of the focus. Favourable effect of dividing up powers of lenses upon a zone of aberration. The next higher order of a zone of aberration. 100 A SYSTEM OF APPLIED OPTICS SECT. of the fourth approximation, or of the order Y 6 for the extreme ray by an equal and opposite amount of aberration of the second approxima- tion, or of the order Y' 2 . Eig. 39 illustrates this case. We then have //Y 2 +////Y 6 = 0, from which Y 2 1 Where the hybrid aberration is at its maximum. Aberration of the order Y H generally small compared to that of the order Y 4 . therefore, substituting, we have //y 2 -// yi^ 8 to represent the hybrid aberration for any other height of ray = y. On differentiatin this we have = 0, /. l - Sy*rfy = 0; and on equating this expression to we get 3|J = 1 and y* = -*, .% y = -~ = '7598Y. Then it is for this height of ray y that the maximum amount of hybrid aberration occurs, and its amount will be given by =/ 1 Y 2 (-577 - -192) =/ 1 Y 2 (-385). Hence the maximum amount of the hybrid aberration occurs for a ray which traverses the system at a distance from the axis equal to about three-fourths of the extreme semi-aperture, and the amount of it is about three-eighths of the outstanding aberration of the order Y 2 to which the extreme ray is subject. But of course the amount of aberrations of the order Y 6 will, generally speaking, be but a small fraction of the aberrations of the order Y 4 . Hence we may regard the hybrid aberration curve as a combination of the curve of Fig. 37 with a much flatter curve of the character of Fig. 39. The latter will have the effect of raising an elevation or wave on the curve of Fig. 3V at about h. An Important Corollary One very obvious corollary from all the preceding investigation is That if for any optical system the aberrations of the two higher IV SPHERICAL ABERRATION 101 orders Y 4 and Y 6 are eliminated or of an imperceptible and negligible amount, then our formulae of the order Y 2 , as applied to elements, etc., will be strictly accurate. The best possible test case for this proposition is provided by an optical system whose curves are strictly spherical, which is known not to show any perceptible zone of aberration at the focus, and whose focal distance for the ray traversing the extreme edge of the aperture has been proved by the most rigorous possible trigonometrical calculation to be exactly equal to the focal distance for rays ultimately close to the axis, as determined by the formulae of the first approximation. Conditions under which formulae of the second approxi- mation are accurate A suitable test case. Application of the Method of Elements to a large Telescope Object Glass The following astronomical objective of 12-inches aperture and focal length of 176'13 inches measured from the vertex of the fourth surface serves as a capital example of the application of the formulae for spherical aberration of the order Y 2 which we have worked out. Radii of Curves, etc. Collective Lens t\= + 59'8" r z = + 90-15". Centre thickness = 1". Refractive index of the crown glass for C ray= 1'5146 = /* Dispersive Lens r 4 = -410". Centre thickness = 1". r.= -84-7" Specification of 12- Refractive index of the flint glass for P C J 6B aperture ob- IOPT.IVA the C ray = 1-6121 = M. jective. The focal length for parallel rays measured from the vertex of the fourth surface, as trigonometrically calculated for the C rays, is for the ultimate centre rays = 176*1306" and for the ray 6 inches from the axis = 176 '127 2 Aberration under-corrected by - -0034" We will now apply the algebraic formulae of the second approxima- tion to this objective, by the method of elements. We have 7 = 59 T 8 ' " ^ = 1 16 ' 2068 = v v from which subtract -*, which = '66024 ./i /i 66024 M 2 = -115-54656' 102 A SYSTEM OF APPLIED OPTICS SECT. First element. Second element. r^p'-v/.- 1 "-' 8 * 8 ' _i_ i_i V 2 JZ U '< 1 + 1 175-1846 115-54656 . , .'. v,= +69-6244". The axial separation between vertices of second and third surfaces is '013" ; and subtracting this from v 2 we get u s = +69-6114", 1 = -6121 ft*u-r i_i_i i _^ ^'3~/3 138-3761 69-6114 and/ 3 = 138-3761 ; , .'. ? 8 = - 140-08, the rays being convergent. From ?; subtract t. 2 M and we get then M 4 = +139-4597". -, .-./,= 669-835, 1 i _! _ _! _ J ^~/4 w 4 ~ 669-825 139-4597 1 1764806' Therefore v t = 176-1306. as stated above, and the distance is 4 minus only with respect to. the dispersive lens, since the rays are convergent. So we now have 1= 116-2068 ( + ) = -115-54656 / 2 = 175-1846 ( + ) u 3 = +69-6114 /= 138-3761 (-) 4 = 669-825 (- != +116-2068 ,= +69-6244 v 3 = -140-08 v 4 = -176-1306 M 4 =+ 139-4597 We may now assess the values of the characteristics a and x. 1 + a, --^=0, .-. a x = - 1; x^ +1. 1 + a 2/ 2 so that ' 350-3692 1 115-54656 , from which 1 + a 2 = - 3'03228, o-= -4-03228; SPHEEICAL ABERRATION 103 !+, 1 1 +a 3 1 2/3 V ' 276-7522 69-6114' so that 3 = + 2-97567; g = +1. 1 + a. 1 1 , 1 frv->Tn -HrVd'^Tl 1 j. - a. Q'fiftfiflR 2/4 < ' ' 1339-650 1 '^Q'd^QT* AiV1 " " "*^" A I V* 4 1 / IFWVVSV, so that Third element. a 4 = + 8'606 ; 4 = - 1. Fourth element. We have next to express the y's or heights of the ray from the axis where it cuts each element plane in terms of the corresponding y l at the first element plane. We have . 2_2/ M 2\ 2 . The y's expressed in ~ terms of y r Next we must transfer the spherical aberrations of all four elements to one common reference point, which is, of course, the vertex of the fourth surface or the locus of the fourth element. Calling the aberration function by the symbol fff^-Jfi f r the first element, R -j- s A'^y* for the second y /2 8 /i element, etc., then the aberration of the first element transferred to the fourth will be expressed by 1 /\ 2 /\2/ai\2 i / a. \ 2 Aberration of first _L_A' ( -M I- 2 -) f^l v 2 = A x I ^ 2 3 ) V 2 element transferred S/! 3 I \l 4 / \M 3 / W 4 / l 8/ t 3 A^WgM/ 3 to fourth. The aberration of the second element transferred to the fourth is 2 /, \ 2 Aberration of second 1 /w\ 2 /tf\ 2 1 /vr\ 2 /u\* erraon o secon LA'( 2 ) l-*l V 2 = -AM-^- 1 ) I- 2 ) w 2 element transferred W 2 W \V 8 /2 3 2 V% 4 / V Vl / yi ' to fourth. The aberration of the third element transferred to the fourth is 1 / \ 2 1 /?; \ 2 A/ w \2 Aberration of third A // t '!J\ O ^ * I \ / "'O *9 \ 9 -yr. A' ( -2- ) y 2 = Z-T-.A , ( -^ - J - J ? /i 2 5 element transferred 8 /3 W ' 8 /3 3 Vw 4 / ^ v i ?! 2^ to fourth. 104 A SYSTEM OF APPLIED OPTICS SECT. and the aberration of the fourth element is Aberration of fourth 1 A ' 2 _ JL A ' ( u * u s u *\ ~ v 2 element. s/ 4 3 ^ 4 ~8/ 4 * At^vfe/ s It is interesting to note in the above functions of the u's and the vs that, after we have corrected the aberrations to one reference point, and also expressed the ?/'s in terms of y l , we then always get a function containing n 1 terms in both numerator and denominator when the number of elements = n, and that as we pass from one element to the next the first term in the numerator disappears, and appears again as the last term in the new denominator ; and the first term of the denominator disappears, and appears again as the last term of the new numerator. The full statement of the aberration of the first element is Aberration of first 1 / v,v z v s \ 1 f/x + 2 . v . 9 p? element fully stated. ^(^^^{^T^ 1) A + (3,* + 2)0*- ) 't , which is a correction to - - or . The second parallel plate correction is already a correction to or 1 -, viz. ^ M 2 -l v 2 /% 2M 3 'if*' 2 ' m whlch y * Vl \v so that we have Aberration of second , M 2 - 1 y^ / U 2 U 3 parallel plate. 2 ^2M^ u^ 2 \v~v^v which works out to Value of above. - ("0000000004 9),^ 2 , which added to the previous amount gives - (00000000464)t/ 1 2 , and since 3^ = 6, this = - -000000 167, which must be deducted from the total we found for the spherical aberration = + -000000289 - -000000167 Final total aberra- +'000000122 tions of objective. which is the final correction to - , so that the final longitudinal error at v the focus is obtained by multiplying the above final result by v* or (176'13) 2 , giving a final spherical aberration at the focus of 0038 inches, which scarcely perceptibly differs from the '0034 which was arrived at by a rigorous trigonometrical calculation of the course of the same edge ray through the objective. Aberration of the It is theoretically true that for this objective there exists an imperceptible. q aberration of the order Y 4 , but it is an imperceptibly small amount of about +'0004 longitudinally, resulting in a zone of rays focusing '0001 short of the focus for edge and centre rays. The aperture of such an objective would have to be at least 24 inches, giving a SPHERICAL ABERRATION 107 zone of aberration of -'0001 x(2) 4 = -'0016 in., before it would become perceptible at the focus by the most refined optical tests. The chief value of the above example is illustrative, arid there is no necessity in practice for being accurate to so many decimal places or for adopting the device of elements in a case of an ordinary double objective whose aperture is only y^th of its focal length ; for were it the case that there existed at the focus a longitudinal aberration of -|- or YQ-th of an inch, it would be possible to correct it by departing from true spherical curves, either by parabolising the figures of the surfaces or the reverse, thus bringing about a slight deviation for the rays which increases as y 2 approximately. There- fore it by no means follows that, because a given optical combina- tion yields an axial image of a point of light which shows no trace of outstanding spherical aberration, therefore a calculation of the course of an edge ray, either algebraic or trigonometric, will also show no aberration. Hence the desirability of comparing the results of an algebraic calculation with the results of a rigid trigonometric calculation if we wish to thoroughly test the accuracy of the former. Many optical designers would prefer to employ trigonometric calculations of spherical aberration rather than any other, even in the case we have just dealt with. Indeed, it is doubtful whether in the case of some of the highly complex constructions of five or more thick lenses forming modern microscope objectives, any method can be as easily applied as the trigonometrical one, provided that not only the focus for the extreme edge rays relatively to the ultimate centre rays is calculated, but also the focus for the rays passing the aperture at a height y equal to about |-ths of the full semi-aperture. Thus any discrepancy between the focus for the intermediate zone of rays and the joint focus for the central and edge rays would at once indicate the presence of an aberration of the order y* and perhaps y 6 . Or, suppos- ing the focus for the edge rays not to coincide with the focus for ultimate centre rays as calculated by the formulae of the first approxi- mation, then the calculated relative position of the focus for the zone of radius ^ths would at once show any departure from the law of the aberration varying as y- simply, and thus reveal the presence of an aberration of the next higher order. It is certainly true that the trigonometrical method is very much more applicable to broad axial pencils than to any other case of refraction that can arise. Although trigonometrical calculations of the course of a ray through an optical system are often highly desirable, yet these are merely How a small aber- ration may be neu- tralised by depart- ing from spherical Trigonometrical methods often pre- ferred for axial pencils. 108 A SYSTEM OF APPLIED OPTICS mechanical processes which, more especially when applied to oblique and eccentric pencils, do not lend themselves at all to analysis. They Empirical nature are empirical and uuinstructive, or at any rate barren of enlighten- of tri & onometrical rnent unless a larc;e number of calculations are carried out in which methods. certain factors, such as radii or separations, are varied, and the results of such variations carefully noted. All tin's involves much empirical work ; whereas by the aid of algebraic formulae, although they may be not quite so exact, leading principles can be established, and the tendencies of the corrections consequent upon the variation of any one term can always be worked out with very little trouble, and it is by the intelligent grasp of the general tendencies that an optical con- struction may be varied in its parts until the utmost possible perfection is realised. The designing of a An Example of the Practical Analytical Application of Formula XXIII. Before dealing with the spherical reflector, we will give another useful example of the practical analytical application of Formula XXIII., or Coddington's formula for spherical aberration. While we have seen that if we wish to arrive at a correct estimate of the total aberrations of the second approximation for thick lenses, we must treat them by the method of elements, still we must not lose sight of the fact that for analytical purposes, when planning out new combinations of lenses whose thicknesses are not great compared with, their focal lengths, we may with approximate accuracy treat such lenses as wholes, and then, if we desire greater accuracy, check the aberrations by the application of the method of elements. For instance, we may wish to design an object glass for telescopes with the interior surfaces of the two lenses of equal but opposite radii of curvatures, so that the two lenses will touch all over, and can be cemented together by Canada balsam. Let the crown glass lens be outermost and have a refractive index = ^ = 1'5, and the flint glass have a refractive index = /i. 2 = 1'6, and let the ratio of focal lengths for crown and flint be 3 : 5, so that F I = + 3, and F., = 5. Then, since the rays entering the first or crown glass lens are parallel, we have a l = 1 ; then u_ 2 for the second lens = F l = 3 ; and we have 1 + a. I _ + 3' 1 + a = 3i and a, = + 2-L Now we can express x for the second or negative lens in terms of iv SPHERICAL ABERRATION 109 x^ ; for, as the two contiguous radii of curvature have to be equal and will be of the same sign (as the lenses are of opposite sign), we have 2(^-1)5 20*!- 1)3' ~ =(1 -* t )2 = 2 - 2x v and ^ = 1 - 2x v so that the spherical aberration for the combination is pressed m terms of 1 - 2.',) 2 + 10'4(2J)(1 - 2*,) + 4'08(2i) + 6'83 , which we must then equate to 0, getting i)(l - 2^)-+ (5A)(4-08) + 6'83 = 0, 12o (6 ' ^ 4Tl + 24a ^ S) + (24 ' 266 - 4 + 22-21 + 6-83 j -=0, 345^ - -493^ + -493 - ('05 - -20^ + ^Oa-; 2 + '2022 - -4044^ + -185 + -057) = 0, 14. f ia: 1 2 + '111^ -'001 = 0, a.- x 2 + -765^ = -007, ^ 2 + -76^ + (-3S) 2 - -007 + -145 = -152 ; .-. Xl + '38 = * v x T52 - * '39, aj 1= --38 4= '39= 4 -01, or -'77. Hence the crown lens, if placed outermost, must be practically TWO solutions of the equiconvex, or else have its radii in the ratio, 177 to 23, or nearly ec i uatlon - 8 : 1. It can be shown that if we have the two lenses with principal focal lengths in the ratio 1 : 1'875, and the refractive indices 1'5 and 1*62 respectively, then in the same manner we get the equation in final form xf + -486*! = - '025, x* + '486^ + (-243) 2 = - -025 + -059, ^ + 243= * V 7 034= *-18, and x l = -'243 4= '18; 110 A SYSTEM OF APPLIED OPTICS SECT. therefore finally != -'063 or --423. A very slight increase in the focal ratio over the above 1'875 : 1 will render the equation insoluble, the nearest approach to freedom from spherical aberration being made when x = about "2. Limits of focal ratio The ratio 1'9 : 1 for the principal focal lengths with the refractive cemented 60868 t0 be ibices l'^ 2 and 1'62 is just about the limit, a higher ratio of focal lengths producing undercorrected spherical aberration. Two often useful formulas are the differentials of the spherical aberration with respect to the two characteristics a and x, which we will here give. First, the differential with respect to a : Differential of XXX. Second, the differential with respect to x is Differential of . . 2 g / 9 ,, ./ , i\, A'J/- TiritV va dv \ > A. V I = n;. ~ ~rs^ H ; r cCLx. XA.X1. s/ 3 y V8/ 3 ' y J S/'IX/*-!) 8 ^-1)) spect to~K. gy means O f these formulae the effect of any contemplated change in a or x for any lens is easily ascertained ; or, on the other hand, the value of dx or da required to effect a given small change in the spherical aberration is soon arrived at. It will be as well to repeat here the formula for the least circle of confusion that is, the smallest section or circular aperture through which the rays of a pencil subject to spherical aberration will pass. It is practically the best possible approach to a focus that the pencil is capable of, and its linear diameter is worked out by Coddiugton on page 1 2 of his work. Thus the linear diameter of the least circle of confusion is Linear diameter of av / ^z \ least circle of con- ( . A ' ), XXXII. fusion. 2V 8/ 3 /' and its angular diameter subtended at the lens centre is therefore o Angular diameter of a/ a 2 least circle of con- _( A' ) XXXIII. fusion. 2V8/ 3 wherein a is the semi-aperture of the pencil at the lens, v is the second conjugate focal distance, and Y^A'J represents the spherical aberra- 1 tion, as a correction to -, as usual. Thus the angular value of the v least circle of confusion varies inversely as the cube of the focal length SPHEKICAL ABERRATION OF REFLECTOR 111 when a is constant, and as the cube of the aperture when v is constant. For simple lenses of relatively small aperture, however, the circle of confusion consequent upon the differently coloured rays being refracted to different foci far exceeds the least circle of confusion consequent upon the spherical aberration, a matter which we may have occasion to refer to again in Section X., on Achromatism. The Aberration of a Spherical Reflector We will conclude this Section by working out the formula for the spherical aberration for an axial pencil of rays directly reflected from a spherical reflector, either of concave or convex form. In this case we cannot do better than follow Coddington's method as explained on page 1 8 of his work. Let Fig. 40 represent a divergent pencil impinging on a concave mirror, and Fig. 41 a convergent pencil impinging on a convex mirror. Let the radius r in both cases be considered intrinsically positive, in which case the distance Q . . a or u will be positive by the conventions laid down on page 7. Let Q' be the focal point by first approximation. Let the circular curve a R have its centre at 0, so that r = . . a = . . R. Then it is clear that the ray Q . . R or R . . Q makes an angle QRO with the radius or perpendicular . . R, which is equal to the angle ORg* made with it by the reflected ray ; therefore sin QRO = sin OP\.q, and we also have sin ROq = sin ROQ, so that we have the strict relationship O..j O..Q q . . R Q . . R' (32) About q as a centre draw through R the arc R . . b cutting the axis at I ; about Q as a centre draw through R the arc R . . c cutting the axis at c, and from R drop R . . d perpendicular to the axis; and let R . . d = y, let a . . q^ the required corrected focal distance = v', and let a . . Q' the focal distance by first approximation = v as usual. Now in the above equation (32) the distance O..q evidently = r (v xv 2 ) if we denote the linear aberration Q' . . q by xv 2 ; also, if the angle RQ', is not large, we may say that q. . R = Q' . . R xv 2 . But it will be found that the introduction of xv 2 into both the numerator and denominator of the ratio q.. R will not affect the Coddington s pro- cedure followed. The fundamental equation. result as regards the formula of the second order of approximation, 112 A SYSTEM OF APPLIED OPTICS 1 unnecessary. which we are proceeding to work out, and therefore its introduction is only required if a formula of the third order, involving y*, is wanted. The introduction of This was clearly shown in the course of working out the aberration of the aberration itself the third order for a spherical refracting surface on page 54, wherein the introduction of the required aberration x into the more exact statement of the fundamental equation did not lead to any modification of the formula of the second approximation itself, but only to modifica- tions of the formula of the third approximation. Since, however, the aberration of a spherical reflector is already much smaller than in the case of a lens of the same relative aperture, even in the most favour- Formula of the third able case, it is scarcely worth while working out a formula of the third UQ " order of approximation. Therefore we may assume that v r Then we have then Q . . R = (Q . . a) - (a . . c) = u - {(a . . d) - (c . . d)} if f\_ ?/YI i\ H _j _i i!/!_ i x \ \2r~2/ = 2W J' a d Q..R~ tt + 2tt 2 \r u)' Therefore, on putting the whole equation together, we get or r-vf f(l l\} = v \ 2v\v r// 2u\r On dividing both sides by r we then get _. 2u\r ) \v r 2v\v r r u 2u\r u Now by first approximation 111 21 1111 at or = , .'. = v J u r u v r r u iv SPHERICAL ABERRATION OF REFLECTOR 113 therefore the equation becomes 1111 f(\ 1\/1 IV 2 . ,112 1 - - - = ?r \ - + - ) ( ) i m which - + - = - or -p : v r r u 2 \u vJ \r u/ u v r / therefore finally we get 111 1/1 1\ 2 , Spherical aberra- J = f ~ u + r \7 ~ / r ' tion of reflector. Hence if u is infinite and the impinging rays are parallel, the 9 o y y aberration becomes ^ simply or ^ ; whereas in the case of a lens of principal focus/, of glass of refractive index = 1'5, and of the shape to give the minimum possible aberration for parallel rays (when x would 5 ? /2 / 4\ be +- and a be 1), the aberration would be ',- 3 ( 8^1. So that the Aberration of spheri- t of \ i / ca l reflector much reflector shows to very great advantage compared to a lens of the same smaller than that of aperture and focal length, even when most favourably shaped. It will be remembered that the Formula XVIII. that we arrived at for the aberration in the case of a single spherical surface of radius r was 1W1 4 Now in the case of reflection it is legitimate to consider the refractive Reflection assumes or reflective index to be - 1 ; that is, the sine of the angle of SV^Jf** 8 index incidence = 1 (the sine of the angle of reflection). If, then, we put /i = 1 in the above formula for the refracting surface, we then get -2 /I 1W1 ,1(1.1) r \r u/ which is identical with Formula XXXIV. This analogy will be found in later Sections to apply in all corresponding cases between a reflecting and a refracting surface of the same radius, so that we have only to stipulate p, = 1 in order to convert the refraction formula into the corresponding reflection formula. SECTION V Angle of obliquity Flat images required of optical systems. Conjugate focal planes assumed. CENTRAL OBLIQUE KEFRACTION OF PENCILS THROUGH THIN LENSES OR ELEMENTS WE have now investigated the spherical aberrations to which a direct pencil of rays is subject whose central or principal ray coincides with the optic axis of the lens or lens system, and our next task is to trace out what happens to those pencils of rays which are refracted centrally but more or less obliquely through a thin lens or element that is, in such manner that -the principal ray of each pencil traverses the centre of the lens or element. It is obvious that we here have to do with a new variable in the shape of the angle formed by the principal ray of each pencil with the optic axis. The extended images which it is sought to obtain by means of optical systems such as the telescope, microscope, and the photographic or lantern projection lens, are always flat images of plane objects. In the case of the telescope or the photographic lens when used on distant objects, the oblique pencils of rays entering them consist of practically parallel rays, which may be considered as originating from points in an infinitely distant plane. The image in the case of the telescope has to be presented to the eye in that state best adapted to simultaneously distinct vision over a considerable angular extent of field ; that is, the image presented to the eye must be approximately flat. This condition of flatness of image applies with still greater force to the camera and lantern projection lens ; and as often as not they have to form flat and well-defined images of strictly plane objects. Therefore, throughout our investigations of oblique pencils we shall treat all such pencils of rays as diverging ' from points which lie in a plane normal to the optic axis, or else as converging to points in a plane normal to the optic axis, and all such planes that pass through points on the optic axis which are conjugate to one another, we will 114 115 call conjugate focal planes. We will also assume the existence of The element planes planes tangent to the vertices of curvature of any lens, or, in other a s am assumed, words, the same element planes which we assumed in the last Section, reserving the consideration of any corrections to our formulae depending upon the versines or departure of the spherical surfaces from such .element planes for Section XI. We shall then find that the position of the focus or mutual crossing point for the two extreme rays of an oblique pencil, as defined by its distance from the lens centre, measured parallel to the optic axis, is essentially a matter of the spherical aberrations which take place at each surface of the lens as well as of other corrections of a some- what different character. Let Figs. 42 and 42a represent the case of oblique refraction of a pencil through the first surface of a double convex and double concave lens whose optic axis is P . . p. Let r f be the centre of curvature, x the vertex of the surface, and Notation, etc., ex- r' . . a^ the radius of curvature, or shortly r, and let P . . Q be the p a original plane object, and Q a radiant point in it. Let the angle of obliquity P . . a a . . Q be called 2 . . # 2 represent y, and /TT r + r-^5 + -r \ " n the aberration. v r r 2(U + r) ii- \ + r/\*-f i;+5/2 Now to above we must add the spherical aberration due to the first surface, taking y^ from (2), so that we then get the complete M value of as follows : 9 , tan 2 < ^ X TT ' """ V^/TT \ ' 'Xfl ' ^9I^ / ul * li Y* TT [ \ \ r r 2(U + r) u 2 u*\ J + r\ \ii-r Value of - N including V the aberration of first surface. _ } _! v v + s |2 )1 /*-lfl 1 ) 2 (1 /x+l|f/ T _ r \ 2 ,J te + ^rr\- + TTt i- + ( -fr-n( u tan ^fr +4*1 )2 2{j? [r UJ \r U J [V ^U + r/ J (7)R. Turning now to the refraction at the second surface we have v' negative as the rays are converging ; therefore, including its spherical aberration, we have Value of - including -i n / x i (-, 1^2/1 , -n ( / \2 surface. Length v to be re- Then, after having got the value of T; ( = d 2 . ./), we have to reduce duced to the axis. ^^ distance to the axis. Drop the perpendicular /. .x' to the lens axis, then evidently tan ") 1 1 ' . . a, = 1 1 1 " = ~ + \n ~^rv F or ~ = ~ + ten ^7w TT \ (9) K. when a; is measured * V 2 2(V + s) x v ^ 2(V + s) which last expression is symmetrical to the other end correction in Formula (6). After adding (9) to Formula (8), while substituting Formula (7) for -^ therein, we then get the complete formula CENTRAL OBLIQUE REFRACTION SEC. PLANE 119 1 ./I 1\ 1 u 1 - = (/- 1) I - + -J - = + #rj + tan 2 - ff - - a* \r s / U w 2 2(U + ?) = first end correction, U tan = first surface spherical aberration, 2 f 1 1 11 U + r| (u-r tf + *J2 = correction for converting it into v\ _ J) 2 J^O s V V tan $ = V + s + tan 2 < = second surface spherical aberration, 1 2(V + s) = second end correction. Complete formula (or i. X As in general the middle correction of the above is small relatively Approximate values J to be inserted in the to ~, ^> -, and -, we may again insert approximate values of u and tf ; ( and since Li U. 1 1 V - ff y st a PP roximatlon ) u ]/U(/*-l)-r\ 2 Vs reduces to -I - r^ , a /x,\ rU r(U + r) 1 U(/*-l)-r ^ - r reduces to fr^ -,T J ^ , and -^ to - r= . ; U(/x - 1 ) - r u-r r(U + r) also since ulu-1 ulu-1 1 S- V(w, - I ) ; = ~ or , ; = cr - - - , . '. -7 reduces to p,,^ . -. - v V s v V s ?/ + s s(V + s) After separating out from Formula (10) the products of the two spherical aberrations into A* and also substituting the above values of - 5 it - r, -, and . we then get it? it-r' v + s' 1 U /A .T = F~U ^~ 2+ V _ j - 4- r + UAr + U - ~ ,S V Includes the aberra- tion of all pencils of semi-aperture A. = spherical aberration of all pencils of semi-aperature A, 120 A SYSTEM OF APPLIED OPTICS Aberration of first surface. Corrections convert- ing it into v". Aberration of second surface. The two end correc- tions. .1 fl - , X V n - i> i+ IV Ur \ 2 2/, 2 , VUG vr U/ \r ' )> 1 ! ** U / \ TJ + f/ r(U + r) 1 ) 2 fU(/*-l)-n A i ^"^ rU a i\ 2 / ^U(u-l)-r U + rJ I r(U + ?) [>.-.,, y/ -l)-sU f^ ^ s(V + s) J 2 11* JLI /1'^\T? O 2 \ J + vA f i S 1 > V / V V + s/ \ 1 H4.M? , I. (R.) vU + r V + s/2 = the two end corrections from ( 1 0). The expressions (11), (12), (13), and (14) together constitute what we will call the normal curvature errors, as corrections to the reciprocal of the conjugate focal distance of the axial pencil of rays of semi- aperture A. Complex as these expressions are, they nevertheless simplify down, without any further compromise, to the simple expression tan 2 (i u+ 1 Includes the aberra- tion common to all pencils of semi-aper- ture k. The normal curvature error in secondary plane. so that our complete formula becomes 1_1 1 ja P~ l j( I 1W1 p+l\ fl 1W1 /A+l x r u & 2+ 2^ 2 lv ^. r + U/ \r ' U / ' Vs ' VJ \s ' V tan 2 fj, + I ~2F w ' II, (B.) III. (R.) The reader is strongly recommended to verify these reductions for himself. Thus in III. we arrive at the same result as did Coddington by a considerably different method, in which he neglected the spherical aberration of the pencil, as expressed in II. The Rays in the Primary Plane We will now trace through the lens the two rays which are refracted at the extreme ends of that diameter of the lens lying in the plane of the paper in other words, symmetrical pairs of rays in the primary plane. v CENTEAL OBLIQUE EEFEACTION PEIMAEY PLANE 121 Let the two rays Q . . e l and Q . . h l impinge upon the element plane at e l and h l at equal perpendicular distances = A (the semi- aperture) from the lens axis P . . r' (Figs. 44 and 44a). Then, correctly, the distances or ys of these two rays from the The two y'a to be normal ray Q . . r' . . q are respectively m 1 . . o l and n 1 ..t l ; but for our present purposes we will assume y l to be e l . . q in the element plane, and y to be h^ . . c v also in the element plane. Then, approximately, if a^ . . Cj = B l as before, = (A+r tan U U + r U = (A (\ tt A -r tan < y- J = (A - B^f. It is evident that the ray Q . . e l meets with more spherical aberration than the ray Q . . h , so that while the former is refracted to f^ the latter is refracted to / 2 on the normal or axial ray Q . . r' . ./ 2 , and therefore the point q f where they intersect will be slightly to one side of the oblique axial ray Q . . / . ./ 2 . Let x l denote the required distance d l . . q'. Let f l denote the distance d l . .f and let/ 2 denote the distance d^. ./ 2 . Draw q'. . p' perpendicular to the oblique axis Q . . / . ./ 2 . Then we have the equation x i~fi _ ii >\ _ /2 ~ x z 2 / 2 or from which But f l and / 2 involve y* and y* respectively, since they are affected by the spherical aberration. Now that part of the expressions for - and j which is common to a l 2 both of them is the term ", and denoting the spherical aberration by the term o^y 2 , we then have fJ- p 1 1 w, 9 1 1 w, 7- = - + ? /9 and = - + -i A ^ v- 2 / 2 ^ A* Expression Expression for y 2 2 The fundamental equation. 122 then becomes i-{j and A SYSTEM OF APPLIED OPTICS l corrected for the *i compounded aberra- tion of first surface. (15) R. 2 j. ,, 2 u ( U \ 2 A* + 2Ar tan <6= + I r tan $ ) U + r \ U + r/ + ^4 2 - 2^4r tan d>== h ( r tan d> ^ ) U -t- r V U 4 r/ -^ - u v2 which in skeleton form is equivalent to + (A' 2 + 2AB y? and Value of the function of t/j and y 2 . = ^ 2 + 3 tan2 therefore in full p p-l 1 2 1 ^ = ^^"U + ' ^2(U-fr) Value of the com- pounded aberration of first surface. + **o (~ + ff ) (- + ^TT" ) f ^ 2 + 3 tan 2 ^>( f 2u 2 \r U/ \r U / 1 U + r = w i = (!/i 2 + ?/ 2 2 - Turning now to the refraction of the same two rays at the second surface, Figs. 45 and 45a, we have the upper ray Q . . ^ . . q after both refractions cutting the normal or axial ray s f . . q' at the point f t while the lower ray Q . . h . . q meets with more spherical aberration and cuts the oblique axis s' . . q f at / 2 '. Therefore the two rays intersect or come to a focus at q" a little v CENTEAL OBLIQUE REFRACTION PEIMARY PLANE 123 to one aide of s'. . and = -. -- ,. Then as in the previous /I <* 2 '-/l /2 d 2--/2 X -2 d 2--h case, supposing e. 2 . . c g = Y x and c 2 . . h. 2 = Y 2 , we have - | - If /" Vl 7l and, as before, and 1 1 ., , r T7 . x 2 2 - Y 1 Y 2 ) corrected for com- a- 2 pounded aberration of second surface = A 2 - / sV \ 2 + tan' 2 &{==- - } ^\V + s/ sV V / sV \ 2 As tan TT + tan 2 d> ( =- - ) V + s ^\V + s/ .-. Y, 2 x a v 4 2 + 3 tan 2 < 3 tan 2 or, more fully, 1 u-1 -1/1 1\2/i i : /< _ -(- + -} (- ^ ' _ -- - - -v' 2/z 2 \s V/ \s X/ g y x2\ Value of the com- A 2 + 3 tan 2 ( ) / (17) E. pounded aberration W + s/ J of second surface. Drop ^''..X perpendicular from q" to the axis s.. 1 _ 1 1 ^V 2 tan 2 \ _ 1 ~~ + ~~ On summing up all corrections in their order we then get 1 1 1 , ii 1 _!_/' _L_ton^*-/v _ XT 7 - T^ - TT 1 * T"0 Udll X F U tf Involves expressed )/ U in terms of U and r = first end correction, = A 2 + 3 /x ?- --TT f - f U / I r \U -r ?7 j = spherical aberration of first surface, spheri- cal aberration of first surface. Corrections convert- ing u into v. 124 A SYSTEM OF APPLIED OPTICS SECT. u/ TT ii-r\ 2 / 1 1 \1 , + -fsl U tan < == ( ^ - - (conversion of u into v ) u z \ \J +rJ \u- r v + /2 Compounded spheri- ^ cal aberration of + ~n~z \ v second surface. . P xs v = spherical aberration of second surface, Correction convert- j 1.1 + tan 2 ^ me - into ^ . V = second end correction. Then after selecting out the product of A* into the sum of the two aberrations and substituting approximate values of T2> u r, -, u' ' u - r and -^ , as we did in the case of the analogous formulee for rays in the secondary plane, we then get Includes the aberra- 111 u, u + 1 f/1 1 \ 2 /l u+l\ /I 1\ 2 /1 M + tion of all pencils of =^ = ^-^+7-9+ n 9 \ ( ~~ + TT ) ( ~ ~* TT~ ) + (~ + TT : )(~ + ir semi-aperture^. u 2p? {\r U/ \r Vs V/ Vs V Aberration of first + ^- (1 + _L WI + ^1) ( 3 tan 2 *f -^L-V [ (18) R. surface. 2/x 2 \r U/ Vr U /I ^VU + r/ j Aberration of second , r~ -i * ~ l I " j t" ' ' 17 Q .-^,2 JL( ' " 1 t /i o\ T? surface. h l^Va + V/ Vs + "T"/\ l ^VVTs/ / ! U^, KU + r) 1 ; )T Corrections convert- ^ ^ ^U(^-l)-r U + r J I r(U + r) ing M into v\ VC,/ - n - *1 1 ' ' (21) R. s(V + s) J 2 The two end correc- T mu T tions. r VU + r V + s/2 The above expressions (18), (19), (20), and (21) therefore together constitute the normal curvature errors to which the rays in the primary plane are subjected when refracted centrally as well as obliquely by the lens. Analogously to the last case, all these expressions simplify tan 2 3u + 1 down to the simple expression ~v so tna t the complete formula becomes Includes the aberra- 1 1 1 ^ /*-lf/I 1\ 2 /1 tion of all pencils of v = tr ~ TT + ^ "^ + ~K~ semi-aperture A. ^ "P IV. (R.) The normal tan2< ^ g j curvature error in + . primary plane. ^ CENTKAL OBLIQUE REFRACTION 125 As regards the correction for obliquity we have again arrived at the same result as did Coddington, only we have in Formula IV. added the spherical aberration which is common to all the pencils, whether direct or oblique. We have recapitulated these processes chiefly in order to form an introduction to more important results yet to be arrived at, also bearing in mind the principle that complex investiga- tions of this sort are understood in less time and with less effort when all processes (except perhaps reductions) are given in full. The differential process as applied to infinitely narrow oblique pencils by Coddington and other writers, resulting in Formulae VI. and VII., also leads to Formulae III. and V. with less trouble, it is true ; but the developments dealt with in subsequent Sections of this work and the corrections of the third order of Section XI. could not be derived from them. If the reader takes the trouble to pursue the same lines of reasoning in the case of a negative lens with the entering rays con- verging and the emergent rays diverging, or the cases of Result is the same as for infinitely thin pencil. The formulae versally true. uni- or Entering rays converging into a positive lens Entering rays diverging into a negative lens, he will again arrive at the same formulae, if due regard is paid to the conventions already laid down. The further convention with regard to meniscus lenses must be also observed, viz. that the radius of the deeper curve shall be considered positive and characteristic of the lens and the radius of the shallower curve negative relatively, so that the spherical aberration corrections and curvature errors for the shallower surface will come out negative with respect to the same corrections for the deeper surface, and the result for the whole lens be the algebraic difference. Then the final formulas emerge just as before. ii ., As to the expression t^, it will be found to be but another way The term t-^ does of expressing the correction, due to thickness, to be applied to the not affect th e pre- l sent formulae. reciprocal value of - (by first approximation), and it has no further significance in the present investigations. Having now got the corrections for curvature of image formed by pencils traversing the lens obliquely but centrally, tan 2 e 2F 126 A SYSTEM OF APPLIED OPTICS SECT. in secondary planes and primary planes respectively, and these being small corrections relatively to the values of ^ or y if the angle of obliquity is not more than a few degrees, therefore the linear or longitudinal (L.) corrections are expressed by Secondary plane. tan 2 p, + I tan 2 + 1 . Linear value. ~2F T ~ ~*>F" m secondary planes, and Primary plane.. tan 2 (f> 3//, + 1 2 tan 2 < 3/x, + 1 . Linear value. ~2F~ ft r " ~2F~ ^~ m P nmar y P lanes > and we may therefore treat these quantities as the versines of the curved images formed by rays in the two planes, and calling the required radii of curvature of the two images E and E v we have and 2R = F tan 2 fji+l ^ V2 tan 2 p+l 2 A* (Ftan<) 2 2F p. (Vtan) 2 = 2F F tan 2 3/x + 1 v<> tan 2 (f> B/J, + 1 2 /, 2F /* 3^ + 1 ' 2 p 2F p. therefore Radius of curva- ture of image, E = F ^ (22) secondary plane. t j - + ^ and Radius of curva- ture of image, R =F ^ (23) primary plane. / + 1 Curvature of image whether V = F or whatever its value may be. Thus the curvature of consta^i PrOXimately " na e ^ or some distance from the optic axis is independent of the distance V of the image from the lens, and depends solely upon F and upon the refractive index /j, of the glass, and is independent of the shape of the lens. Supposing p = 1*5, then the radii of curvatures 3 3 are respectively -F and F. o 11 If we take the difference between the E corrections tan 2 (f> 3/x + 1 tan 2 //, + 1 Expression for the we then get astigmatism of a tan 2 /3/>i + 1 /*+l\ tan 2 central oblique 2F \ u~ ~M~"/ r F pencil. \ CENTEAL OBLIQUE REFRACTION 127 as the E correction expressing the astigmatism at the oblique focus for any degree of obliquity <. The same simple expression also applies in the case of a spherical reflecting surface. Clearly no variations in the refractive index can affect the astigmatism, nor do they in any substantial sense affect the curvature errors. For, supposing the refractive index is 1'6 instead of 1'5, we then get radii of curvatures of F - =F( - 6154) instead of Small effect of A/z i .f- upon normal curva- F(-6), when /*= 1-5; and F^ = F('276) instead of F(-2727), when ture errors. So that it would require a refractive index of a very abnormal character to much affect the results ; for even if /j, were oc , But when we F then F and would become the radii of curvatures. o , come to deal with combinations of collective and dispersive lenses, we shall find variations in refractive indices of one unit of the first decimal place of the highest importance. We' may here with advantage compare our results with the exact formulae for oblique central pencils worked out by Coddington, and given on page 120 of his work. He adopted the course of supposing the pencil of rays to be an infinitely narrow one, and therefore the effective aperture and thickness of the lens to be vanishing quantities ; he then worked out the oblique focal distances by a strictly differential- method, arriving at the formula 1 / cos - z l/ cos in the secondary plane, and COS (f) COS --~ cos 1 1 - + - r s COS u VI. VII. Secondary plane. Exact formula for thin pencil. Primary plane. Exact formula for thin pencil. in the primary plane, in which u is the oblique distance from the radiant point Q to the lens centre. v is the oblique distance from the lens centre to the corresponding conjugate focal point. , tan 6 , etc. We shall also find that when the aperture of the oblique pencil becomes large enough to show perceptible spherical aberration, then among the corrections of such higher orders we find corrections involving the square and higher powers of the aperture, so that the curve traced out by the foci of the two extreme rays of a pencil of large aperture will not be exactly of the same character as the curve traced out by the foci of two rays infinitely close to the principal ray. This means that the amount of the spherical aberration of a very oblique pencil of semi-aperture A will not be the same as the spherical aberration of the axial pencil of semi-aperture A. It is, however, obvious that while in any system of separated lenses or elements the principal rays of the pencils may cross the axis just where one lens or element occurs, and thus be refracted obliquely but centrally through the same, yet such principal rays must traverse most of the lenses eccentrically as well as obliquely. In the next Section we will deal with such cases' of eccentric oblique refraction ; but before proceeding to that it will be as well to deal with a few very useful formulae in connection with the curvature errors which we have arrived at in the shape of Formulae III. and V., or tan 2 u,+ I " _ secondary planes, and NORMAL CURVATURE OF IMAGES in primary planes. 129 + tan 2 - - - It is often very desirable to know the effect of a change in the refractive index upon these curvature corrections. We "will first deal with the case of the curvature being constant ; that is, - + - or - is constant, so that -. or ^ is variable as LL r s P f P varies. In secondary planes we have X. so that for a constant focal length the higher refractive index, imply- ing shallower curves for the lens, yields a flatter image. Primary plane. Variation in curva- ture error due to dp, when - is constant. Secondary plane. Variation in curva- ture error due to dp, when - is constant. K 130 SECT. Primary plane. Variation.! in curva- ture error due to dp. when - is constant. / In primary planes we have tan 2 c 3^+1 tan 2 ' 2/ tan 2 < ~w XL So we get the same differential as in the case of the secondary plane. This we should, of course, expect, since the astigmatism as measured by tan 2 3/A + 1 tan 2 //,+ !_ tan 2 = constant, whatever may be the value of /A, and therefore the changes in curva- ture consequent upon d/j, must be identical in the two planes. The Spherical Reflector We have yet to consider the case of a spherical reflecting surface and its effect upon pencils of rays reflected obliquely but centrally. Let Fig. 47 a, Plate X., represent a spherical reflector of semi-aperture C. . Ej or C . . E 9 = A. Let Q . . Q' be a finitely distant flat object per- pendicular to the axis C . . Q. Let be the centre of curvature, the radius being . . C = r. Primary Plane We will deal with rays in the primary plane first. Draw a straight line Q' . . . . S from Q' through the centre of curvature ; this then becomes the theoretical axis of the oblique pencil, so that S . . E and S . . E 2 are the two heights for the two extreme rays, which heights we will call y l and ?/ 2 . It is clear that if/* is the ulti- mate focal point for rays close to the oblique axis Q' . . S, then the ray Q' . . E 2 , after reflection, will cut Q' . . S at a point / 2 , the ray Q' . . E 1 from the upper edge will, after reflection, cut Q' . . S at/ x , and/. ./ 2 and / . . /! will be the linear spherical aberrations proportional to y 2 and y^, and these two rays reflected from the extreme edges of the mirror will cut one another at a point q slightly outside of the oblique normal ray Q' . . S. Draw q . . p perpendicular to C . . Q. Then, as in the case of oblique refraction at a spherical surface, we may put Q' . . S = u, f..S = v\ and q . . S = x, and let the angle of obliquity Q'CQ = 111 1 and - v = = h tan- - --^7 -\ - v = --- r, u u 2(M -- r) v F u 2( - r) 132 A SYSTEM OF APPLIED OPTICS SECT. so that Equation (24B) becomes Abbreviated value of 1 - Expression for y v - = =3 + tan 2 tan 2 <^> 57-^-r + A' A* + 3 tan 2 # . (24c) r 2( - r) [ \w - r/ J Distance x to be re- . We must next reduce the oblique distance q . . S or x to the duced to the axis. ax ^ s Q . . Q O f the mirror. From q draw q . . X perpendicular to the mirror axis, so that C . . X, or X for short, becomes the required corrected distance. It is clear that X=#+-<^-- v-, in which v \ / may be put as its first approximate value such that - = .^ - -. Then we have ( _ wl 2 ( r - -'/ u-r and - rj 2(r - v) Value of 1 1 ( r - v] 2 1 1 = 1 u tan d> - V ; . 9 , X a; 1 ^u-r\ 2(r-v) x 2 in which we may put v CENTRAL OBLIQUE REFLECTION FROM MIRROR 133 1 so that 1 1 = X x r-v\ z tan - 1 F /' 1 x i -r, / u-r/ 2(r - v)\F u) (24D) Value of = ampli- fied. On inserting the previously worked out value of - from 24c we then get 1 1 1 ,= = = -- X F u 2(u - r) 3 t - u tan ur i vT^^ ~ -) 2 - (2*E) u - rJ 2(r - v)\F u/ Value of ^. with all A corrections. Now A',4 2 is obviously the spherical aberration for the direct or axial pencil originating from Q on the axis and of semi-aperture a (which was y in our investigation of the direct spheriqal aberration in Section IV.), and should be kept separate ; so that after inserting 1/1 1\ 2 .. /. I I u-r ur \ -( ) for A . we then get since = - - and w = 5 ~ 1 r \r u/ \ r u ru 2% - r) 1 111/1 1\ 2 v = ^-~+ (- -) A I 1 u r\r u/ + 3 tan 2 d>- r -r\ z . (24F) ur / Full value of = after JK separating out the common aberration. Then the last line of the above simplifies down thus = 3 tan 2 d> - + tan 2 rf> . r - tan 2 d> -^ f 2(u - r) 2r(u - r) 1 3 2u-r } ,{r+Q(u-r)-(2u-r) r + - - - -7 'r = tan 2 64 ^-^^ \ = tan 2(w - r) r 2r(u - r)j 2r(u - r) = tan 2 4> ^A -r) so that finally we get 2 1 - = tan 2 d> ; r F _ X ~ F tan 2 }* + A* \u r/ J \ u - r) so that the spherical aberration to which the two extreme rays in the secondary plane are subject is expressed by The spherical aber- , ration of the second- A' -I A 2 + tan 2 ary rays. I while the other corrections for reducing the distances concerned to the axis are the same as before, so that following the analogy of Formula (24E) we have - X F u r 2(-r) \ I 1 _ i-i ; (24o) u-rj 2 (r -v)\F u so that after insertion of the term A' in full we get j X F u r \r u Full value of = after ^ / \ a X 1,9. 2 , 1 2 , 9 ./r\l /2u - r\- /n . . separating out the + - tan- < + tan 2 ---- - % 2 tan 2 I - . )- -- -( - - ) , (24H) common aberration. r 2 ( u ~ r ) \2n - rJ 2(u - r)\ ur the last line of which 1 1 2% T = tan 2 - + tan 2 ~ r ^r ^2u-r ^2ru-r 2 1 fl econary pane. ,/2(-r) + r - (2w- r)) n The normal curva- = tan 2 d> { - > = 0. + Maw r._n I 2ru-r ) So that there is no curvature error for rays in secondary planes and the image is flat. CENTRAL OBLIQUE REFLECTION FROM MIRROR 135 From these results it follows that the curvature error tan 2 = in primary planes represents also the astigmatism of oblique pencils, and it is thus seen that it is exactly the same as for a lens of the same principal focal length. For a lens we have the formulae for curvature of oblique pencils tan 2 /A + I . 2F in secondary planes, and tan 2 - 3 + 1 = in secondary planes, 2F - 1 = tan 2 < = in primary planes, which agree with the curvature errors which we have already worked out. This last formula, however, can be shown to be inexact, for there are corrections of higher orders, functions of tan 4 0, tan 6 . , = - - -- ^ in primary planes, and cos< F in secondary planes, u (241) (24J) Curvature error in primary plane and astigmatism identi- cal. Normal curvature errors for lenses. Result of assuming refractive index in above = -1. Exact formulae for normal curvature errors for spherical mirror. in which formulae u is the oblique distance Q'.. C of our Fig. 47a, and v is the oblique distance ( = C . . q) of q, the focus, from C, the centre of the mirror surface. These formulas are exact for infinitely narrow pencils, and practically accurate for cases in which the aperture of the mirror does not amount to one-tenth part of the principal focal length. If in the above two formulae we suppose - to vanish, we then get impinging rays par- u allel. 136 A SYSTEM OF APPLIED OPTICS SECT. V in primary planes y = Fcos<. Fig. 47& shows a spherical mirror of principal focal length = C . . F, and of radius of curvature = 2 (C . . F). A pencil of parallel rays whose principal ray is Q l . . C is incident upon the mirror at an angle Q : CF = , and of course reflected off F at the same angle. Let a circle of radius -- be drawn touching the a mirror centre at C, and the principal focal point or plane at F. Here, then, C . . ^ is Coddington's v. But in order to compare his formulae with those we have worked out we must first reduce his oblique distance v to the axis by drawing q l . ,p l perpendicular to the axis C . . F. Let C . . p 1 = V. Then it is clear that V = (C . . qj cos $ = v cos <, v = F cos , therefore Primary image is It is clear that the formula worked out by Coddington differentially * i m pli es tna ^ the locus of curvature for the oblique foci is a circle of radius -, for the triangle CF# is always a right-angled triangle having always flat. its right angle at q r so that v or C . . q l invariably = F cos <. Secondary image is In secondary planes we have by Coddington's formula F v = - F sec (t>, cos which, of course, requires a plane image to satisfy that condition, the focus for secondary rays falling at q^ when the focus for primary rays falls at +S TJ The eccentricity C. Value of the Y's in detail, U-D and this is the coefficient of the aberration at the second refraction. Reverting to Formula I., page 120, Section V., giving the complete All the corrections statement of corrections applicable to the secondary rays of the central oblique pencil, it will easily be seen that the R correction (see Formula present case. (14)) expressing the differences between the oblique - and - and the axial Y? and ^ respectively will apply just the same in our present 140 A SYSTEM OF APPLIED OPTICS SECT. case of the eccentric oblique pencil. Also the expression (12), which gives the E correction necessary for converting the - of the first refraction into the - of the second refraction, will equally be required, while expressions (11) and (13) for the two spherical aberrations will be replaced by corresponding expressions with those values of y and Y given above in (25c) and (26s) substituted therein. Hence we get the following formula, after selecting out the joint spherical aberrations for the semi-aperture S T y which constitute a correction common to all the pencils, whether axial or otherwise i_i i j* /A-im iy/i X~F~U iY 2 + 2/, 2 lVr + U/ \r + (27) LW_iL_VI V7 Vs ' V from (25c) (28) VU(/.-l)-rW Utan ^ _KT -MO V/UO^ " -' ' - - - - - - ,gg\ from (26fi) UP Vs_ J-D~ n * V + ; (31) 8/'Z' in which formula X = the horizontal distance a 2 . . X'. Then (28) becomes /t-1/1 iy/1 p+l\(( UD \ 2 UD Ur 2a \r + UAr + U 7\\U-D/ + " U - D ' U + r (28A) and (30) becomes ^-1/1 iy/i U + _I\J/UD y ^2"V 5 + VA5 V /IVU-D/ (30A) vi ECCENTKIC OBLIQUE KEFBACTION 141 from which we can again select out from (2 8 A) and (30A) the function of the two aberrations so we then get for the whole formula, after somewhat simplifying down Formuk (29), 1 1 1 + ^ + ' 9 2 u 7. + uA 7- + '~irV (27) ur> TTr /from (28)\ UD ^tan 2 ^ (2 9 A) 1 1 \1 + VuT7- + TTJ2 tan * ( 31 > Before simplifying down the above complex formula it is expedient Adoption of the to adopt Coddington's device which was explained on pages 65 and 66. sha P e and ver & eQ cy TI 1 characteristics r and Eecapitulating, we have since ^r r + ^r for the ultimate axial pencils 1,11 11 r-ir and V = "2F~' so that lj-a 1 -a ~2F~ ^F~ then ^ +x ~20*-l)F 142 and so that A SYSTEM OF APPLIED OPTICS 1 1 - x 1 +x l-x 1 1 After substituting the above values of -, -. - -. and = in the above r s U v Formulae (27), (32), (33), (34), (29A), and (31), excepting in those expressions involving D, which for the present it is desirable to keep Secondary Plane, intact, the above formulae simplify down to the following : 1 - 1 _ i _ !/ \* * x F u~zr a > IP Reciprocal of back focal distance cor- rected for thickness. Eccentricity correc- tion dependent on coma. Formulae (39) and I. apply to all pencils. Formula (40) applies to all oblique pencils, whether central or not. Formula II. applies only to eccentric pencils. Spherical aberration for all pencils. Normal curvature error. + Eccentricity correc- tion dependent on spherical aberration. 4 0* 2 )0* - iX 2 . f rom (29A) and (31) u 3 ^| / DU \ _ f H"[J - D/ tan 2 2(u+l) 1 DU 4/m + - (x - a) }~ =r j from (33) and (34). . i. from (27) (40) III. ^7 - - 4r (fj, l)\. ^ n (39) and I. we have the complete formulae for the reciprocal o f the distance from the back apex a of the focus of the axial pencil whose semi-aperture at the element plane is S^ f: or A, correspond- U JL/ ing to Formula XXIII., page 67, for a pencil of semi-aperture = y. These Formulae (39) and I. apply to all pencils, whether axial, oblique, or eccentric. Formula (40) is the same again as Formula III., page 120, which we De f ore worked out for the full lens aperture and for oblique but . ... ,,. central pencils. It is now seen that it applies to all oblique pencils whether central or eccentric. Formula II. is a further function of the spherical aberration of ^he lens applying onlii to eccentric pencils. Since the spherical rr J i , aberration is almost invariably positive or or the same sign as the power of the lens, this correction is also almost invariably positive. ECCENTRIC OBLIQUE KEFKACTION 143 We may call II. the diaphragm correction or stop correction dependent upon the spherical aberration of the lens and the degree of eccentricity. In III. we have a further correction applying only to eccentric Formula III. applies pencils. It is a second stop correction due to the presence of coma in the lens or eccentric oblique refraction. It is as well, before entering more closely into the nature of these stop corrections II. and III. and their causes, to first investigate the case of the rays of the same eccentric oblique pencils contained in the primary plane. Rays of Eccentric Oblique Pencils contained in the Primary Plane In this case we may follow much the same lines of construction as we did in tracing rays in the primary planes of central oblique pencils, Figs. 44 and 45. In Figs. 50 and 50a let % and w l be the two points where the two extreme rays in the primary plane passed by the stop D\ . D ( strike the first element plane of the lens. Join the radiant point Q to ?', the centre of curvature, and produce it to the ultimate focus of the pencil at q. Obviously ray Q . . 7i x meets with more spherical aberration than does ray Q . . w lt and therefore intersects the normal oblique ray Q . . / . . q at / x nearer to the lens than the point / where ray Q . . w l intersects Q . . r 1 . . q. Let c l . ,n l = y l and c x . . w l = T/ O . Let O x be the point where the principal or central ray of the pencil cuts the element plane. Let aperture of stop = 2$ as before. Let = =- , . c^ = r tan 6 or tan == , and we shall then find that 2/i 2 + V Z 2 + 9M (of Fig. 50) = A* since in Fig. 50 y l = A + (B l + C) and y z = -A while in Fig. 44 we may consider that y^A + (B + C) and y 2 = A - (B l + C) (of Fig. 44), C), Identity of the final wherein (7=0 as the refraction is central, so that in all cases we arrive at the same result when y* + y + yjj z or y^ + y^ y^y^ are expressed in terms of A (the semi-aperture of the pencil in the element plane) and B l and C. vi ECCENTEIC OBLIQUE EEFEACTION Now approximately or 145 and (and r tan Q \ 9 = tan< ^lJ + r/ tC -u r tan # Q ' 1 ft 4. ^"' /=/\ 2 ' UD Ur \ U ) 2 and yo 2 = - tan oS ( ( r\ and ^1^2 = \ tan 2 o> tan2< Ku U -V ' UD Ur \ U ) 2 / UD Ur \ 2 / U \ 2 ) . Ur V 2 ,/ UD Ur W U \ ^ U \* } U + r ) n< ^Vu-D ' U + rA^U-D/ ' V^U-Dy 1 Ur V 2 / UD Ur \/o U \ /_ U \ 2 I< PVU-D' 2jY UD U + r) n< %-D ' U + rA^U-Dy 1 ' V^U-Dy 1 Ur V 2 / U \ 2 U + r/ ~ V U - D/ J _/^ u y ( 3tan2 /-/ UD y 2 V U - D/ r i. \U - Dy 1 UD U,- , / U,- V, <) U - D U + r VU + r, Full value of the function of ^ and y z . The Refraction at the Second Surface At the second refraction, illustrated in Figs. 51 and 5 la, after adopting the same construction and putting x >2 for the required distance of the focus - tan U = c 2 . . n Y 2 2 = -(Utan ^"~MS)' UD Vs / Vs (45) therefore the sum of the compounded aberrations at the two surfaces is w l(/l + 2/2 + ^1%) + W 2( Y 1 2 + Y 2 2 - Y 1 Y 2) VI ECCENTKIC OBLIQUE KEFKACTION 147 We have then to add to the above the two end corrections for obliquity (31), and also the correction (29) or (29A) for converting - 1 U into v ; and then after gathering together all corrections and putting ^ for the corrected reciprocal of the final axial or horizontal distance j\. a. 2 . . X' of the final focus q" fr m the back vertex of the lens, we get, after cancelling out in (49) and (50), the following complete formulae: X F U 2 A * u-u/i iwi ,fti\.'/i , ivi h fcl\\/r^Y v* u ; (47) (49) (50) (51) (52) spending i reduces Primary Plane. Reciprocal of back (53) focal distance, cor- rected for thickness. (54) or Spherical aberration j for all pencils. , p. v Normal curvature error. Eccentricity correc- IV. tion dependent on spherical aberration. 1 2^ iVr ' IjAr ' U ) ' Vs ' VAs ' V M* U -DJ ^' l( ( l , ^ , /i+1 ),( 1 , l }\ 1 ,/* +1 W UD Y 2 3t _^-i^/i ivi M+i\H/ UD \ 6tan2 . .-. 9 i I ' r V T 1 1 + TT / / 1 \ TT Tk ^ J 2/x 2 l\s V/\s V /JJVU D/ /*-!(/! /*+l\ /I /* +1 \l3tan 2 6 2/i 2 1 \r ' U / + \.s + V /] 2/il r(U + r) s(V + s) / After again adopting the same device as in the last corre case of rays in the secondary plane, the above complex formuL down to 1111 v t X ~ F ~ U ~ f ' 4F 2 1 / + 2 ., "] yu, 3 \f U \ 2 1 Un* ,V + 1 Vl/ ^ 2F /. a 3 \f DU VJ 4- ' ' 1 II ^-UVU-D/'J Eccentricity correc- tion dependent on Ratio between the Eccentricity Correc- tions in the two planes. Conventions under which the formulae are universally true. 148 A SYSTEM OF APPLIED OPTICS 3tan 2 < r , 2(,*+l), ^ DU V. The Formula (54) is the spherical aberration common to all pencils of light passing through the stop. Formula (55) is the normal curva- ture error for all oblique pencils, central or eccentric. Formula IV. gives the stop correction for all eccentric oblique pencils due to the spherical aberration of the lens ; while Formula V. gives the stop correction for the same pencils due to coma in the lens. All these are R corrections to be applied to the first approximate . 1 I ,/l 1\ 1 1 1 value of -ry, as obtained from ^ = u - 11 - + - - TT , or ^-^. V v \t s/ U Thus the R corrections due to the presence of the stop, viz. IV. and V., for rays in primary planes come out just three times the corresponding stop corrections for rays in secondary planes, viz. II. and III. The student may with advantage pursue the same processes in the case of positive and negative lenses and meniscus lenses with the entering rays both divergent and convergent, the stop being real, and either in front of or behind the lens, or else virtual only, adhering always to the following conventions, consistently with those already laid down on page 10. Collective Lenses. Bays constituting the pencils. Principal rays. COLLECTIVE LENSES OR MENISCI Entering rays diverging, U is + intrinsically. converging, U is - Emergent rays converging, V is + ,, diverging, V is - Stop in front of lens and real, or entering principal rays),.,. . . . diverging y JD 1S + mtnnsically. Stop behind lens and virtual, or entering principal rays) , . converging J " Stop behind lens and real, or emergent principal rays) . converging / " Stop in front of lens and virtual, or emergent principal) . rays diverging / " Thus we may write D' for the distance from lens to where the principal rays cross the optic axis before entering the lens, and D" for the refracted distance, conjugate to the former, between- the lens and the point where the principal rays cross the optic axis after refraction. vi ECCENTEIC OBLIQUE REFRACTION 149 DISPERSIVE LENSES AND MENISCI Dispersive Lenses. Entering rays converging, U is + intrinsically Rays constituting diverging, U is - the pencils. Emergent rays diverging, V is + converging, V is - Stop behind lens and virtual, or entering principal rays \ , . . . . ,. * >D is + intrinsically. Principal rays, converging J Stop in front of lens and real, or entering principal rays') _, . i (^ IS ,, diverging J Stop in front of lens and virtual, or emergent principal \ . i * t -LJ IS T i* rays diverging J Stop behind lens and real, or emergent principal rays } . / J-' IS 11 converging J Seeing that such principal rays are compelled to cross the axis of the lens at the centre of the stop, or at any image of such stop, there- fore that centre has to be regarded as an axial point from which such principal rays are diverging or to which they are converging, and since these principal rays are refracted by the lens in precisely the same manner a=5 any other rays, therefore it is universally true that D' and D", in relation to any one lens in any particular case, are conjugate focal distances, such that I- 1 1 (56) D"~F" D" Therefore we can carry Coddington's device one step further and let introduction of the B stand as the characteristic of the state of divergence or convergence new ver & en y cnar - actenstic p for the of the principal rays with respect to the lens, so that principal rays. 1+/3 1 1-0 1 ,, 7 , ^F =v and TF = D- ft is thus closely analogous to a, and may be called the vergency characteristic for the, principal rays. Then (^ ^J converts into 4F 2 DU 2F 77; ^ and = ^r into ^ , since the D we have so far been dealing (p - a.y U - D p - a with was the front conjugate distance D', relating to the entering principal rays. Therefore Formula IV. becomes 3tan 2 < 1 |>+2 , / 8 1 ITT The s P herical 2Fu(u - 1) (a - fl) 2 U - \ X ^ + ^ aX + ^ * + ^ ~ ^ t^l I ' aberration Eccen- j tricity Correction. and Formula V. becomes, after multiplying by -(//,), 150 A SYSTEM OF APPLIED OPTICS SECT. " a The comatic Ec- centricity Correc- tion. The above two corrections com- bined. Abbreviated form for the Eccentricity Corrections. Comparison of above results with Cod dington's formulae. or more conveniently l)a + (/* + l)z[. VII. So that these two stop corrections may be bracketed together thus 1 FM + 2 a' ~ /2 i //... i \ ..~ , /o ,n\/ i \ '? . r* - 2 2 VIII. We may often have occasion to write this formula in the abbreviated form (58) For rays in secondary planes the 3 tan 2 is replaced by tan 2 <. Subject to the conventions as to the intrinsic signs of U, V, D', and D", these formulae are universally true of all lenses, provided their axial thicknesses are very small. Calling the inevitable curvature errors tan 2 d> u, + 1 , tan 2 d> 3 a + 1 =- . - -- ana --.,-- , which are incidental to central oblique 2f P 2r fj, pencils, the normal curvature corrections, then Formula VI. expresses what is nearly always a plus stop correction due to the joint effect of the spherical aberration and the selective action of the stop upon eccentric pencils, while Formula VII. expresses what is a very variable stop correction, sometimes plus and sometimes minus, due to the joint effect of coma, or eccentric oblique refraction, and the selective action of the stop. Thus diaphragm or stop corrections may be defined broadly as corrections applicable to oblique pencils refracted eccentrically through a lens, causing more or less serious departures from its normal curvature corrections. It is more convenient to call these diaphragm corrections eccentricity corrections, or E.C.s for brevity. Turning now to the comparison of these results with those worked out by Coddington, more especially in his Prop. 123, p. 132, it might be thought on first inspection that they are quite at variance. In secondary planes he arrived at the formula for an infinitely thin pencil refracted eccentrically through a lens = - k ~ F h 2F ' (59) vi ECCENTKIC OBLIQUE KEFRACTION 151 wherein h'is A v > 7 = OUI> TT' an " 72 A. /i U A while his term V; -r - ; , ^a;2 + 2(u + l)(a - B)x + 2(u + !)(/* - l)a/? u(u- 1) (-/3) 2 V-l V ^ IvTTTA Coddmgton's For 3 v j-viiiA. m ula. (1 \ 1 " 2 V + - )f2 4 in secondary p./ K~ planes and (3V + -fe 5^ in primary planes are inclusive formulae, embracing not only the corrections due to eccentric refraction of oblique pencils, but also the corrections due to their central refraction. If, however, we take the normal curvature corrections 7 in the form 1 -- 2F I and add them to our corresponding Formula VIII. we get 1 or 3 (//x+2 , , w . 2 + ( " + }ax + ( * + }( " ' } - l)a(a - VIIlB. This will be found to reduce exactly to Coddington's ,- / i\ / i\~v Formula VIII. con- J ( V + - or 3 V + - ) \ . firmed by Codding- T-. \ 1 W / ^ A \ *-* * > It* J fj./ (J./J ton's results. Hence it is evident that in his formula he got the normal curvature Mixed-up nature of corrections, the E.C.S due to spherical aberration and the E.C.s due to ^^ instons coma all mixed up together in a manner unfortunately most incon- venient for practical purposes. It is a most curious fact that throughout Coddington's work there Coddington appar- is no allusion to such a well-recognised thing as " coma " ; indeed it is coma, doubtful whether he could have been aware of its existence without at least attempting to work out a formula for it and its effects. On 152 A SYSTEM OF APPLIED OPTICS SECT. page 159, in the course of discussing aplanatic combinations of lenses in contact, he says : " The next question that offers itself is the advan- tage to be derived from a combination of lenses when a pencil passes through it centrically but obliquely. It will, however, easily be seen that as the effects of obliquity in this case are totally independent of the form of a single lens, so they cannot be removed or diminished by any combination." While this statement is quite true in regard to the normal curvature of image, yet the possibility of coma being either Coma met with in present or absent is entirely overlooked. Every practical optician is practice 1 * 7 ptical aware that some objectives for telescopes are extremely sensitive to being thrown out of square, while others are not ; the former show strong coma at the foci of even slightly oblique pencils, while the latter show little or none, but only pure astigmatism, while simple lenses show the same differences, only there is spherical aberration superadded. Such objectives without coma give better definition for a considerable angular distance from the axis than do those whose oblique images are marred by coma or eccentric oblique refraction ; although the normal curvature of image and astigmatism can be shown to vaiy only slightly in different cases. We will revert to this subject again with greater advantage at the end of Section VIII. The phenomenon of coma is not only deeply interesting, but of great practical importance, and we will reserve a more thorough investigation into its properties for Section VIII. Before concluding this Section, we may with advantage consider a question that may already have occurred to the reader with regard to Formula VIII. for the Eccentricity Corrections. incongruous nature Since the E.C.s consequent upon the spherical aberration of the of the two Eccen- } tricity Corrections, lens vary as -. -^, and the E.C.s consequent upon coma in the lens vary as , and since the value of ( } increases more rapidly a - p \u- p/ than does ~x when the stop is removed farther from the lens, a p therefore the plus E.C.s consequent upon the spherical aberration must rapidly overtake in value the comatic E.C.s, therefore we should expect that there should be a limit to the distance of the stop, beyond which it will be impossible to obtain an excess of minus comatic E.C.s. or even a neutral balance of minus comatic E.C.s against plus aberration E.C.S. Limits to the useful In other words, if we want to modify the normal curvature of position of the stop. j ma g es i n the direction of flattening them, we must take care that our stop is not placed too far from the lens, or else the plus aberration VI ECCENTRIC OBLIQUE REFRACTION 153 E.C.s will inevitably prevail and the images be more curved than before. Now, if we have eccentric refraction of oblique pencils through a simple thin lens, and we wish to preserve the normal curvature of the images, then we must equate the E.C.s to ; that is, we must have - 1 40* + l)ax + (3ft + 2)0* - l)a 2 + - VIIIc. - 2( - /3){(2/x + !)(,* - 1) + 0* + 1M = 0, This formula yields the following quadratic equation : VIIlD. In order that E.C.s may be js possibly eliminated, it is obvious that we must have the right-hand side of the equation equal to 0, from which, since a is a known quantity, we may derive the limiting value of /3, and then obtain the necessary correlative value of x from the left-hand side. In this way we may derive the following limiting values for /3 and x : If p = 1*5, and a = 1, then f +1-45 p \or-3-93 and x= { , - lor + 1'7G If = + 1-45, then the stop is -817F in front of the lens. If y8= 3'93, then the stop is at a distance = '40 5F behind the lens. In either case x indicates the meniscus form of collective lens with the concave side facing the stop. If /A = 1*6 and a again = 1, then i +1-523 \or- 2-757 and .r = I or +1-6 2 If = +1-523, then the stop is 79F in front of the lens. If j3= - 2-757, then the stop is -53F behind the lens. So that the above stop distances are the maximum permissible if we wish to get our images natter than the normal "by means of E.C.s. Condition for equat- ing E.C.s to 0. Quadratic equation derived from above. When A =1-5. When /* = 1-6. 154 A SYSTEM OF APPLIED OPTICS SECT, vi Hence we cannot expect to obtain a flat final image from such a combination as a Cooke portrait or astro -photographic lens if the separations between the simple lenses composing it exceed the limits implied in the above Formula VIIlD. It is often useful to know the effect upon the Eccentricity Corrections of a lens (as expressed in Formula VIII. of this Section) of slight alterations in the value of a, /3, or x, and we will here give the differentials of the E.G. formula with respect to these three characteristics for rays in primary planes. 1st, with respect to a Differential of the Eccentricity Correc- tions when a varies. from which we see that the effect of a change in the divergency of the entering rays is somewhat complex. 2nd, with respect to /3 3 tan 2 1 f ., . Differential of the 3 tan^ T __ \ i \ 1 ( r E.C.s when (3 varies. f L(a - #) 3 l .! (a -~B)' 2 \ (a - #) 3 l .! (a -~B)' 2 which is necessarily an expression of a much simpler nature than the last. 3rd, with respect to x Differential of the = F/ (p. + 2) 2fr + 1) | _ _ /. + 1 1 j ^W - 1-T _ _ E.C.s when x varies. / ( - P? l^ ~ I) 2 ^ - 1) which is perhaps the most useful of the above three differentials. SECTION VII ON SYSTEMS OF LENSES AND THE APPLICATION OF THE THEOREM OF ELEMENTS TO THICK LENSES SOME consequences of the greatest practical importance follow from the various formulae arrived at in the last Section. First of all, since and - represent the relative normal u, u, curvature corrections of any simple lens, and as these functions stand generally in the ratio of 1 to 2 '2, while the Eccentricity Corrections in primary planes are always three times the corresponding E.C.s in secondary planes, it follows that the two normal curvature errors of a simple lens cannot possibly be simultaneously neutralised by E.C.s, due to the presence of a stop placed anywhere on the optic axis. If the normal curvature errors in primary planes are neutralised by E.C.s, so that the image formed by rays in primary planes is got quite flat, in which case . tan 2 ^ S/J.-T 1 KC.s (in pr. plane) = ^ - L tan 2 ei/u + 1 1 3a+ IN M1 then =M ~ - - ) will represent the remaining curvature -p. 3 fj. / tan 2 (f> / be the R correction to the reciprocal value of the Confirmation of the combined focal length F of two lenses in contact ; then, if the final Petzval Tneorem - image is free from astigmatism, F 2 (# tan 2 <) is the versine of such anastigmatic curved image. Therefore we have the equation fly N J J M_IP which condition follows from the fact that the primary E.C.s (due to the presence of an axial stop) required for throwing back the curved image formed by central oblique rays in primary planes on to the curve of the anastigmatic image are always three times the secondary E.C.s required for throwing the image formed by central oblique rays in secondary planes on to the same anastigmatic image. From this equation we get 11 1/11 or x J\>P Value of the curva- ture correction for anastigmatic image. 158 A SYSTEM OF APPLIED OPTICS SECT. and the versine of the curve of the anastimatic image (62) Then if r = the required radius of curvature, then (Ftan<) 2 1/1 1 \_ = s ( ~ -- F 2 tan 2 i--n A tan 2 (/> "~ 2P L If If respectively. Hence the radius of curvature of the secondary image , | _ p ft> ^ an( j O f t ], ie p r i mar y image = f^^ v Then, if - is the M) -T power of the combination when there is no separation between the lenses, it follows that I - 1 _ f^ 1 1 - 1 PP ~ ^ F"P ^ P P"P' ~fr from which l = -^ and P = F^^^ 1 . (65) ^ * > ~ - * VII COMBINED LENSES IN CONTACT 159 On substituting this value of P in the previous two formulae for the radii of curvatures of the two images we get = F simply. (66) =r - simply. (67) Radius of secondary image - ( F ' Radius of primary image = ( F ' Radii of the two normal images when Formula XIII. is ful- filled. It is interesting to observe, then, that the normal curvatures of the two images yielded by a compound lens fulfilling the condition XIII. are the same as if the lens were a simple lens of the same focal length, but made of glass having an infinitely high refractive index. So that we may regard the particular case of the Petzval Formula XIIlA. being equated to 0, as in Formula XIII., as a device for making a lens whose refractive index is virtually infinity, with regard to its influence on the compound normal curvature errors. Therefore it is quite clear, from what has preceded, that E.C.s must perform a part in this compound lens, if the two images are to be simultaneously thrown back into a plane image. That is, eccentric oblique refraction is absolutely necessary to the attainment of the desired flat and anastigmatic image, in the case of contact combinations fulfilling Formula XIII. It is plain that the Petzval condition XIII. demands that if the combination is to have a positive focus, the collective lens must, in order to possess the preponderating power, be made of a glass of higher refractive index than that of which the dispersive lens is made (so that Pfi p = N/i^, or the principal focal lengths are in inverse ratio to their refractive indices), a condition which was impossible to fulfil consistently with achromatism until the era of the new optical glasses was inaugur- ated at Jena. The new dense barium crown glasses combining a refractive index as high as 1*61 with a dispersive power as low as -^ for rays C to F, and the new crown or very light flint glasses having a refractive index of 1'52 to 1*54 with a dispersive power as high as ^, were the crea- tions of the celebrated firm of Herren Schott & Gen., of Jena, who thus rendered it possible to embody the Petzval condition in combina- tions of two or more lenses in contact. Dr. Hugo Schroeder's con- centric lens was apparently the first photographic lens in which Petzval's sum 2j was equated to with any degree of success ; but not only does the far too small difference of refractive indices yet available render it impossible to get much focal power from such combinations, Above combination equal to a simple lens of infinite re- fractive index. Further necessity for E.C.s to get a flat image. The new Jena glasses. The Concentric Lens. The first contact combination equat- ing the Petzval Formula to 0. The small balance of power available. 160 A SYSTEM OF APPLIED OPTICS Imperfect correction against spherical aberration. Dr. Rudolph's ana- stigmat. Dr. Rudolph's and Dr. Eniile von Hoegh's improved anastigmat. Petzval condition not quite fulfilled. but the fact that Schroeder made it a condition in his concentric lens that the plano-convex collective lens of high refractive index should be cemented to the plano-concave dispersive lens of low refractive index, precluded him from the advantage of freedom from spherical aberration. A reference to Fig. 52 renders it evident that any ray entering the dispersive lens parallel to the axis is refracted away from the axis, so that its distance from the axis when traversing the collective lens is greater than its distance from the axis when traversing the dispersive lens. This variation in ?/ 2 would have little significance if the glass of the collective lens were of lower refractive index than that of the dispersive lens, but in the case of this abnormal pair of glasses the variation in y introduces an aberration of the third order which is fatal to the elimination of spherical aberration, so that, as a matter of fact, sharp definition, even on the axis, could not be secured with any F F larger aperture than about , or in larger-sized lenses. After- wards Dr. Rudolph of Jena, in Germany, got over this difficulty with considerable success by adopting the expedient of opposing two cemented combinations A and B, A comprising an abnormal pair of a collective and a dispersive lens, of which the collective lens had the higher refractive index, while B was a normal pair in which the collective lens had the lower refractive index. Combination A was undercorrected for spherical aberration, but this defect was counteracted by the opposite fault in B ; also a rough approximation to the Petzval condition was secured by a suitable division of the powers of the lenses relatively to their refractive indices. In this way much larger relative apertures were obtained. Later Dr. Rudolph, closely followed by Emile von Hoegh, devised a still better symmetrical construction for each half of the lens, which was made to consist of a double concave dispersive lens cemented between an inner meniscus collective lens and an outer double convex collective lens, the refractive index of the dispersive lens being approximately a mean between the high refractive index of the double convex collective lens on the one side and the low refractive index of the meniscus collective lens on the other side. Dr. von Hoegh's lens is generally known as the Goerz lens. In each half lens the so-called Petzval condition, (68) was almost but not quite fulfilled. In order to fulfil it exactly, either VII SEPAKATED LENS SYSTEMS 161 the power of the dispersive lens would have to be increased, or its refractive index decreased, but the exigencies of cemented combinations preclude the simultaneous fulfilment of other conditions, consistently with sufficient power being obtained. As the extreme differences of refractive indices between the new abnormal pairs of glasses are as 1'6 to To, it is evident that any contact combination of thin lenses fulfilling the Petzval condition must have the power of the collective lens or lenses equal to 16, as against 15 for the power of the dispersive lens or lenses, the resulting power of the combination being 1- or only -j^th of the power of the collective lens or lenses. This is a limitation implying the use of very powerful or strongly curved lenses in order to gain a comparatively long focused combination, whose normal curvature errors in primary planes are three times the normal curvature errors in secondary planes, and therefore in the proper relation for being simultaneously neutralised by E.C.s left in the system for that purpose. Powerful constituent lenses result in rela- tively small power. The Case of Separated Lenses or Elements So far, then, we have considered the application of the formulse arrived at to combinations of very thin lenses in contact. We have yet to consider their application to either thin lenses more or less widely separated, or to thick lenses considered either singly or in combination. Some twelve years ago, in the course of thinking over the general results arrived at in the last two Sections, especially in relation to the normal curvatures of image characteristic of simple or achromatic lenses, it suddenly occurred to the author that since the normal curvatures of image due to any lens, whether simple or com- pound, are fixed by its refractive indices and power alone, and are independent of the state of the rays entering the lens, whether con- vergent, divergent, or parallel, then it should follow that the normal curvature errors of an achromatic and aberration-free collective lens should be neutralised by the normal curvature errors of an achromatic and aberration-free dispersive lens of the same power (and made of the same glasses) placed at a considerable distance behind the collective lens ; while the combination would, as a result of the separation, have considerable power or yield a positive focus, so long as the rays from the collective lens are convergent to a distance behind the dispersive lens less than the principal focal length of the latter, or more especially when the rays entering the first or collective lens are parallel. But such complete neutralisation of normal curvature errors could obviously M How collective and dispersive lenses of equal powers may neutralise each other's normal cur- vature errors even when separated. 162 A SYSTEM OF APPLIED OPTICS SECT. The above two lenses must be free from coma. Effect of separations on the formulae. The , power gained by separation be- tween collective and dispersive lenses is an unqualified net gain. not ensue if any E.C.s were allowed to interfere, therefore both these achromatic and aberration-free lenses must be free from coma or give symmetrical oblique refraction ; otherwise pencils of rays traversing one of the lenses centrally, but the other necessarily eccentrically, would be subject to E.C.s, and their final foci be either shortened or extended, and thus the desired result be prevented. This idea led to further .experiments and calculations, which we will now deal with. We must first ascertain how the formulae which have been arrived at, are to be applied to combinations of thin lenses on a common axis, but having considerable separations between them. In Fig. 53 let Lj represent such a compound collective lens free from coma and aberration, of principal focal length =/ 1 , and L 2 a compound dispersive lens also free from coma and aberration, and made of the same glasses, and having the same principal focal length / ( = / 1 ). Let the rays entering L I be parallel. Then at the distance /j behind L x is formed the curved image s . . s due to rays in secondary sections of oblique pencils, and the still more curved image p . . p due to rays in primary sections of the same oblique pencils. The dispersive lens L 2 will project an enlarged image of these to a distance b behind it, such that T = -r, where a plane anastigmatic image will be I) / S f Jl J% formed. Or treating the said plane as an origin for the pencils in the reverse direction, it is evident that after such direct and oblique divergent pencils (such as that from q) have been refracted by L 9 , they will then virtually radiate from points in the curved surfaces, s . . s in secondary planes and p..p in primary planes, which are exactly the same curved images as are yielded by the positive lens L 1; so that all the pencils will emerge strictly parallel leftwards from L : . The theorem that the normal curvature errors of two equal collective and dispersive lenses will neutralise one another, even when the lenses are widely separated, is thus almost self-evident when once pointed out; but the more general theorem that the curvature errors and E.C.s of a system of separated lenses are the simple sum of the curvature errors and E.C.S of the individual lenses, and that the power gained by separation is a net gain and carries with it no curvature corrections whatever, requires further demonstration. It might at first be thought that the fact that the centre of each lens of a separated system views the same point of the original object or its image under different angles of obliquity, and views the same curvature error from different distances, would lead to unavoidable complications, but this is not so. VII SEPAKATED LENS SYSTEMS 163 In Fig. 53 let $ = the original angle of obliquity of a central Demonstration of the or eccentric pencil impinging on L a . As throughout the foregoing above theorem. processes, the angle is always the angle contained between the optic axis and that ray to or from the real or virtual radiant or focal point Q which passes through the centre of the lens. The corresponding oblique focal point about Q, to which the rays converge after refraction by Lj, subtends a new angle 6 at the centre of L 2 . Let us assume that the linear aberrations of Q' from the focal plane P..P..P do not exceed Y(jth part of yi, as is the case if the angle < does not exceed 14 degrees. Let ^ any E corrections, including normal curvature errors ,and E.C.s, for the first lens ; let S = the similar R corrections for 2 L 2 in neither case amounting to more than 10 per cent of j or respectively. l 1 Then j + A the reciprocal value of the corrected focal length of the oblique pencil we are dealing with, and if the same pencil traversed L 2 under the same angle of obliquity <, then the corrected reciprocal value of the back focus would be Sol, * I ' (69) .supposing 4- 2 The two angles and 6 assumed to be equal. ^ or the first lens is for the moment neglected. i But the second lens L 2 views Q under the angle 6, and it is evident that Tan tancj in terms of Also the R corrections for the oblique pencil tra\trsing L a , expressed by B l will from the point of view of the second lens The same R correc- 4/1 tion as viewed from become . , . ,. ,1 L i and i, or increased in inverse proportion to the tiveiy. respec- square of the distance ; for generally if v = the linear amount of the General argument, curvature error in question (referred to the axis) and is a small quantity compared to / a or / x s, then 1 1 v and then if / x becomes / x s, then 1 1 164 A SYSTEM OF APPLIED OPTICS SECT. so that the R correction from the point of view of L is 7 ^- 5 , as v (fi ~ s r against ^ for the same R correction from the point of view of L^ ; but (A-*) 2 and moreover is only another way of expressing an -$S lt therefore Ji 2 /i with reference to L, the R correction from L, is (-~ L -] 5-ftj,. as Vi ~ / a/; above. Next, the R correction to which the same pencil is subjected on traversing L 2 under the new angle of obliquity 6 is evidently tan* 0-, which = *-- tan' 2 Therefore the sum of the R corrections for both lenses from the point of view of L 2 becomes /, tan" * / /, or Sum of the R cor- rections for the two '] ) """ ri i* , Lx \ (7ft\ lenses. V/j - sJ 2 Vj l / t V And if this last expression is multiplied by B 2 , or the back focal length squared, we shall then get the linear value, reduced to the axis, of the sum of the R corrections of the two lenses. As we have seen before, = = - +-, and B = / 2 '{ 1 ~^ so that (70) x B 2 becomes /i ~~ s h /2 + (/i " *) Linear value of the ^ f / 2 (/i ~ ) Y( /i \ 2 tan 2 ^l !J 1, above. Next, in order to reduce this to an R correction of the reciprocal of the equivalent focal length of the whole combination, we must divide (71) by (E.F.L.) 2 or the square of the equivalent focal length of the whole combination. Now the E.F.L. is the axial distance of the back principal point from the final image plane, at which point a pin-hole would have to be placed in order to throw an image of the same dimensions as that yielded by the combined lenses ; on which supposition the E.F.L. is equal to B 7 1 , which ~ s A/2 /79\ vii SEPARATED LENS SYSTEMS from Formula X., Section III. Therefore (71) H- (E.F.L.) 2 165 which * /, tan 2 at each lens. We need only take the sum of the R corrections appertaining to the several lenses and multiply them by (E.F.L.) 2 tan 2 should be referred. Then it is clear that, if the original object is infinitely distant and the rays of pencils parallel, it be- comes quite a matter of indifference whether the angle is referred to the outer vertex of the first lens or to the first principal point. Clearly there is no difference in such a case. With regard to the second conjugate focal distance, it is obvious that tan must also be measured from the second principal point. The Gain in Power due to Separation Now the reciprocal of the E.F.L., or = for brevity, or the equiva- lent power of the combination (73), = 1 E.F.L. A/ ^ , as we have 1 1 seen above, and this is made up of : two parts, viz. 7 H- 7, or the /I /2 c simple difference of the powers of the two lenses, and 7-7-, which is the '1/2 gain of power due entirely to separation, so that while 7 + 7 may be A _ / 2 zero if the powers of the two lenses are equal, one collective and the other dispersive, yet there remains a considerable surplus power, repre- ~ s sented by j-?, in the case of the same two lenses separated. /1/2 Sum of the R correc- tions assessed upon the E.F.L. Same important theorem applies to three or more separ- ate lenses. Tan -/> should always be referred to the two principal points. 166 A SYSTEM OF APPLIED OPTICS SECT. The gain in power due to separation is a net gain. A practical illustra- tion. The Petzval condi- tion also applies to separated lenses. The radius of the anastigmatic image independent of the separation. When the Petzval condition may be largely ignored. An instance. But as we have seen from Formula (73) the curvature errors or E.C.s appertain solely to j + j, therefore the great gain in power repre- - s A / 2 sented by ^r is an unqualified net gain and carries with it no normal A/2 curvature aberrations whatsoever. - s 1 Supposing we have/! - l,/ 2 = - 1, and s = '25, then -7* = + '25 = -, A/2 or the equivalent power of the combination, entirely due to separation, is one-quarter of the power of the collective lens a very considerable amount, especially if we compare it with the case of two lenses in contact fulfilling the so-called Petzval condition ; the collective lens being of power 16 and the dispersive lens of power 15, and the resulting power of the combination being only 1, or T ^th part of the power of the collective lens. Now it obviously remains perfectly true, that even in the case of a separated pair of a collective and dispersive lens such as we have been dealing with, the condition XIII. must be fulfilled if a flat final image, free from astigmatism, is to be secured ; and it still remains true that ' ^ = - (see XIV.) if that P/KP N^ r condition is not fulfilled ; but since the radius of curvature r of the anastigmatic image is the same whether the two lenses be in contact or separated, it is obvious that the shortening of the E.F.L. due to separation means virtually a flattening of the anastigmatic f image, for becomes much greater than if there were no separa- tion. Therefore it follows that a departure from the so-called Petzval condition, which would lead to serious astigmatism in the final image of mean flatness thrown by a contact combination, would lead to a much less serious astigmatism in the final image of mean flatness thrown by the same two lenses when separated. For instance, let us take two lenses, one collective, of focal length 1 5 and refractive index 1'5, and the other dispersive, of focal length 16 and refractive index also 1'5, thus not fulfilling the Petzval condition at all. The radius of curvature of the anastigmatic image thrown by these two lenses in contact is given by Reciprocal of the radius of the ana- stigmatic image. Power when in con- tact. 1 1 1 while r 15(1-5) 16(1-5) 22-5 24 360' 1_J_ JL _L F~l5~T6~240' SEPARATED LENS SYSTEMS 167 so that r = (l'5)F. tauce = 4, we find 1 Then if the two lenses be separated by a dis- E.F.L. 15-16 -4 -16 xTsT 1 240 _! 5 60~240 _ 48' Thus we have ?= 360 as before, while the E.F.L. is reduced to 48, so that r is now (7'5)F instead of (1'5)F. Thus the effect of a departure from the Petzval condition is reduced to a vanishing quantity, so that if we construct photographic lenses on the principle of gaining a considerable proportion or all of their power by separation, then we need no longer be restricted to carrying out the Petzval condition ; we can ignore it to some extent in favour of a more general and elastic rule, viz. that the power of the dispersive lens must be approximately equal to the power of the collective lens, or the sum of the powers of the collective lenses if there are more than one. This is one of the two supplementary principles which underlie the Cooke photographic lenses, and many others which have been introduced since they were first made public. And now it will be easily seen that a true anastigrnat might have been made long before the advent of the new Jena glasses. For instance, we will take a crown glass collective lens of refractive index = 1'5, and whose 7=Y, and a dense flint glass dispersive lens of Ji i i refractive index = 1'6 whose 7 = ^, the two being separated by s= 7. /2 15 Here the Petzval condition is fulfilled, but if the lenses are in contact the power is - ^-^ and the system is dispersive, but the power when = + . When put into the separated by 7 is 16-15-7 -6 ( _ 15)(16) -3^ triplet form, like a Cooke lens, a very fair rectilinear ariastigmat lens could be and has been produced, but not so good as when the newer F Jena glasses are employed. The Cooke lens of aperture , known as Series la, for astronomical photography, is practically an ana- stigrnat in which the refractive index of the dispersive lens is consider- ably higher than that of the two collective lenses, and the Petzval condition is very considerably departed from, yet the final image is quite flat and shows only a trace of astigmatism within an angle of 20 degrees. Before proceeding to the question of thick lenses it is desirable to arrive at two more very useful formulae relating to contact or separated combinations. If the final image yielded by a photographic lens has Power when separ- ated. Anas tigmat s could have been produced by the aid of the old crown and flint glasses only. A practical instance. The Cooke lens for astronomical photo- graphy. 168 A SYSTEM OF APPLIED OPTICS SECT. When ^the primary image js made flat. a little residual astigmatism away from the axis, it yet remains desirable to attain an approximately flat image, and two useful compromises suggest themselves. 1. The image formed by rays in primary planes may be got flat, leaving the image formed by rays in secondary planes still somewhat curved concave to the lens. In such case what will be the radius of curvature (r) of such secondary image ? It is evident that the primary E.C.S which throw the primary image back on to the focal plane must be equal to If p = the power of the collective and ^ that of the dispersive lens, the simultaneous secondary EC.s will then be tan 2 + 1 J_ V rt + -tan $2\2P u~ 2N ^ and in primary planes the E.C.s will be equal to Value of the E.C.s in secondary planes. Value of the E.C.s in primary planes. Therefore the final curvature R correction in secondary planes will be 1 2 ^ + 2N u 2N 1 ( 1 9 / -t- 2\2P~7^ Curvature errors > (75) minus E.C.s in secondary planes. which reduces to 2 ,|J_ J_ so that the versine of the image curve and 21P ,/,, XVI. The three Formulae XIV., XV., and XVI. give at a glance, as it were, the degree of approximation to "an anastigmatic focal plane attainable in any suggested combination of lenses of known powers and refractive indices, whose combined equivalent focal length, if separations exist, can also be derived from Formula (72) if only two lenses are employed, or from the more complex formulae given in Section II. if there are more than two. No photographic lens of separated lenses can be made to give achromatic and rectilinear images with less than three constituent lenses, and if P a is the P.F.L. of the first collective lens L x , N the P.F.L. of the dispersive middle lens L 2 , and P 2 the P.F.L. of the back collective lens L 8 , s x the separation between L x and L 2 , and s 2 the separation between L and L 3 , then the E.F.L. of the combination for parallel rays is given by the formula Reciprocal of the radius of either secondary or prim- ary image. The minimum num- ber of lenses required for a photographic lens. 1 1 1 XVII. Triplet lens. The increment to power due to separations. The first part of the above formula is the simple sum of the powers, 170 A SYSTEM OF APPLIED OPTICS or the E.F.L. of the three lenses if thin and placed in contact, while the latter part of the formula gives the further increment to power due entirely to the separations. Cases where separa- tions lead to loss of power. Huygenian pieces. A Significant Corollary relating to Eye-pieces We have just alluded to the increment to power accruing to a combination of two collective and one dispersive lens, consequent upon separation. Eeferring back to Section III., p. 46, we found that a certain four-lens erecting eye-piece whose lenses were of focal lengths 1, 1'25, 1'25, and '80, only gave an equivalent focal length of '31. Here is a case wherein the separations have led to a noticeable decrement to power ; for while the sum of the powers of the four lenses is -^ we have ^ ^, T = Therefore while the normal curvature 26 errors will E.F.L. -31' be proportional to -- , and consequently the radius of curvature of the anastigmatic image be proportional to '26, yet the E.F.L. is "3 1 only. That is, the radius of curvature of the 26\ anastigmatic image, if formed, will be smaller (in the ratio ^y] than the radius of the same image if a simple equivalent lens of E.F.L. = '31 were used. The above eye-piece is a comparatively favourable case, having s. 2 or the second separation greater than usual, which leads to increment to power. In most cases shortness is aimed at, when the ^ vrr- of E.F.L. course grows smaller compared to the sum of the powers of the four lenses, and therefore the curvature errors of the final image are bound to increase. A flat or nearly flat image for rays in primary planes is generally aimed at, and therefore it will be seen that the less is the power realised in the combination, the more relatively violent will be the curvature of the same final image as formed by rays in secondary planes. The shorter is such an eye-piece, the more difficult it becomes to attain a satisfactorily flat field of view. eye- We also saw that in the case of the Huygenian eye-piece with lenses of focal lengths 3 and 1, we got 12 114 OH = 3*' e A 4 / 2 = 3- Hence the curvature of image will be twice as strong as that for the equivalent lens. VII EFFECTS OF SEPARATIONS ON IMAGES m In the case where the two lenses were of the focal lengths 2 and \, then E.F.L. 3 ,.,11 3. = -, while -j + T = - 4 /I /2 2 Ramsden eye-piece. The ideal eye-piece. Erection of the in- verted image by reflecting prisms. The eye-piece used in prismatic tele- scopes. and here again we have the same disadvantage. In the case of the Ramsden eye-piece of lenses of focal lengths 1 and 1 and separation 1, the same argument again applies, but as the separation is generally about % or , leading to a gain in power, the construction comes out about on a par with an ordinary four-lens erecting eye-piece. The practical conclusion of these arguments is. that the ideal eye- piece is one which consists of a single lens, self corrected for spherical and chromatic aberration by being built up of a dispersive lens, and one, or better still, two collective lenses. If it consists of a dispersive lens between two collective lenses, then any effective separation between the components (in the form of thickness perhaps) counts for gain in power and not loss as in the eye-pieces just considered. Then if the image has to be erected, crossed doubly reflecting prisms can be resorted to. The modified form of Kellner eye -piece now so commonly employed in prismatic telescopes does not fall far short of this ideal, and it must be conceded that the images that it yields are not only superior to those yielded by four-lens erecting eye-pieces in regard to angular extent and flatness of field and freedom from astigmatism, but also as regards freedom from certain other curvature errors and E.C.s of a higher order which we shall glance at in Section XL It will also be seen that the use of a pair of double total reflecting prisms between the eye-piece and the objective rather helps to flatten the image formed by the latter. For they are equivalent to placing a pair of thick plane parallel plates in the path of the pencils of con- verging rays whose principal rays radiate from the centre of the objective, so that the oblique foci are subject to parallel plate correc- tions tending to throw them back relatively to the axial focus. This relieves the eye-piece of a certain amount of eccentricity corrections. It will, however, be seen that the position of the prisms relatively to the primary image will make no difference to their flattening effect upon the same. Application of the Theorem of Elements So far as we have yet proceeded, it has been assumed that the Thicknesses cannot 'axial thicknesses of the lenses we have been dealing with have been aways The favourable effect of the reflecting prisms. 172 A SYSTEM OF APPLIED OPTICS How the theorem of elements is to be applied. quite negligible quantities, very small compared with the radii or focal lengths of the lenses in question. While excessive axial thick- nesses in the lenses building up optical systems are objectionable for obvious reasons, and as much as possible to be avoided, yet thicknesses far too great to be neglected in our computations arise in most cases. Now the formuke of the order tan 2 arrived at are in their very nature and origin more and more exact in their results in inverse proportion to the fourth power of the angles of obliquity (0) dealt with ; and, if a pencil of rays crossing the axis of a lens system at a given diaphragm point is traced through all the other lenses at a small enough degree of obliquity, it may obviously traverse all the lenses very closely to their centres, even if the lens system is of considerable length. In Sections II. and III., etc., we have already dealt with the theorem of elements as applied to thick lenses, and we will now see how the same theorem may be applied in the computation of normal curvature errors and E.C.s. Let Fig. 54 be a double convex lens and Fig. 55 a meniscus collective lens, Fig. 54 a double concave lens and Fig. 55a a meniscus dispersive lens. Eecapitulating, it is obvious that close to the axis the double convex lens may be considered to be built up of two infinitely thin elementary lenses e l and e^ e 1 being convexo-plane, and e 2 plano- convex, the two enclosing between them a parallel plate of glass of a thickness equal to t, the axial thickness of the lens. It is not quite so obvious, but nevertheless is demonstrable, that any departures from exactness in the formulae of this Section, due to the refraction of the pencils through outer parts of the lenses where the thicknesses are widely different to the central thicknesses, take the The corrections of form of corrections of the higher orders tan 4 and tan 6 , etc. These the third order, etc. ... , , ... , . -, \. VT higher developments will be dealt with in Section XI. In the same way the collective meniscus lens may be considered built up of a convexo-plane elementary lens e v and a plano-concave elementary lens 2 , enclosing between them a parallel plate of glass of a thickness equal to t, the axial thickness of the lens. If r and s are, as before, the two radii of curvatures, then the power of e is simply - 1 and the power of e simply + or while x, the characteristic of the shape of each elementary lens, is + 1 simply for e v and 1 simply for e f Then in assessing the consecutive values u^ and o,j with respect to e v and u 2 and 2 with respect to e 2 , or the respective axial distances from which or to which the axial pencils diverge before VII 173 refraction, we must look upon e l and e 2 as two distinct lenses separated by an air-space equal to -. Also in the case of slightly oblique and eccentric pencils, the prin- cipal rays of which cross the optic axis at any known diaphragm point at a known distance D/ or D" in front of or behind e l (according to which /3j is assessed), we can always assess the value of D 2 ' and /3 2 with respect to 2 consistently with the same supposition, viz. that ^ and 2 are two separate lenses separated by an air-space equal to -. In this way the values of a and ft for each element may be arrived at in a very simple way. The Effects of a Parallel Plane Plate upon Obliquely Refracted Pencils We have next to consider whether, besides the influence exerted by the parallel plate on the spherical aberration of the axial pencil, it has any influence upon the corrections of the oblique pencils that should be taken into account. It is obvious enough that if the rays constituting pencils emerge in a condition of parallelism No effect uponpencils from e l} and consequently traverse the parallel glass plate in a condition para e of parallelism, then the plate cannot possibly exert any influence upon them, and they will emerge from the plate and enter e still in a parallel condition. But if the rays of pencils are converging to or diverging from points at a distance from the plate, not very large com- pared with t, then the plate exerts an influence on oblique pencils which it is necessary to investigate before we are in a position to properly bring the theorem of elements into practical application. We already have the complete Formula XXV., Section IV., for the spherical aberration (to use an expression which is here rather a misnomer) of a direct pencil refracted through a flat parallel plate, but for our present purpose we shall first require Formula (15), Section IV., which gives the spherical aberration occurring at the first flat surface, which formula was of the form V 1 _ * A* ~ 1 a 1 _ n 2 /7fi\ Aberration at first I ~ u ~ it 1u A u l ' Plane surface. in which a is the semi-aperture of the pencil at the first surface. We can now bring this formula into requisition when investigating the case of oblique pencils. Let Fig. 56 represent the case of a divergent oblique pencil Notation. ?&! . . Q . . W Y Let Q..A 1 = w, and let /j.u = u. Then let Fig. 56 be 174 SECT. The fundamental equation. the corresponding case of a convergent oblique pencil, both entering into a plane glass surface. In the first case % and & : should be con- sidered positive, and in the second case negative. Let semi-aperture of pencil B a . . n^ or B X . . w 1 = a, as before. Let A 1 ..w 1 = y 1 and A 1 . . w a = y 2 , and A l . . B x = H! ; and let the angle between the principal ray B x . Q and the perpendicular A l . . Q be called %. Since ray Q . . % is the most oblique, it therefore meets with more aberration than ray Q . . .w lf and after refraction cuts the perpendicular Q . . A a at / 2 farther away from the surface than / x for the refracted ray q 1 . . w v Let q l be the desired point where the two extreme rays Q . . n-^ and Q . . w l intersect in the primary plane after refraction. Draw ^i Pi at right angles to the axis or perpendicular Q..AJ. Then if we put x l for q l . . A l or the corrected value of &, /i for L and / 2 for / 2 . . then (A, . . = (A, . . from which Adopting our device used on former occasions, let then A / 2 Talue of the com- pounded aberration. Primary plane. Primary plane. Ob- liquity correction + aberration. Now and = (H x - ftj) 2 = (u tan x - f^) 2 = 2 tan 2 x - 2aw tan = (Hj^ + dj) 2 = (u tan x + i) 2 = w 2 tan 2 x + 2ait tan y 2 = (H, 2 - a, 2 ) = ( W 2 tan 2 x - a, 2 ) = u tan 2 x l = J__/* 2 -l x l fM 2/x 3 ii 3 l 21 _ _ ^Z. 1 tan 2 x ,.2 (77) PLATE. XII. Fi^.59. PLATE. XII. Fi^.GO. vii OBLIQUE EEFKACTION THKOUGH PLATE 175 In the secondary plane we have y 2 = H 2 + a x 2 = w 2 tan 2 x + a i 2 > 1 1 u' 2 - tan 2 v + a 2 ), l and u. 1 u 2 -! u' 2 -l ,_ s Secondary plane. - '- 2 tan 4 x - 2 3 a i ' / Obliquity correction #1 w ^ w -^ M plus aberration. Hence, as in the case of eccentricity corrections, the correction for Ratio between the , ,. , , P . o . ., ,. i ,1 corrections in the obliquity or the function of tan ^ is three times as much in the two p i anes . primary plane as in the secondary plane. Second Surface We will pursue the investigation in the primary plane. At second surface of Fig. 56 we have the same state of things as is represented in Fig. 56a at the first surface, only that in the latter figure we must imagine the light to be passing from right to left, instead of from left to right, and under either supposition the Formula (77) equally applies, so that we have i 2 _ i u 2 - 1 f* * f* oj.9 r^ 9 = g 3 tarr \ - 2 3 a% , V V Alb V _//,"/' and therefore 1 luu 2 1 M 2 ~l Second surface. - corrected or = ^ + , 3 tan 2 v + ~- t a leaving the plate, whereas the angle is the angle included between the optic axis and a ray drawn from the oblique image point Q to the principal point of any lens system. Therefore in computing our parallel plate corrections we must always insert the actual angle of obliquity under which the principal ray of the pencil enters or leaves the plate, and this angle ^ may be considerably different from the original angle , which is always assessed in relation to the first principal point of the system ; but ^ is easily calculated from 0. Let Fig. 5 7 represent the essentials of Fig. 5 8 that is, a collective lens Lj, a dispersive lens L 2 , and a collective lens L 3 in succession ; and let P be the point on the axis where the principal rays of all pencils traversing the system are made to cross that is, P is the pupil point, where a stop of variable aperture is placed. N 178 A SYSTEM OF APPLIED OPTICS SECT. Let it be carefully noted that the axial glass thicknesses in this diagram (57) are supposed to be drawn equal to -th of their real P amounts, as shown in Fig. 58, and also that the refractions, shown apparently as" surface refractions, are really the refractions due to the passage of the principal ray through the successive infinitely thin elements e lt e 2 , e 3 , etc., e l being con vexo- plane, e z plano-convex, e^ concavo-plane, etc. The principal ray is traced through the system as a solid line. Every refraction of the principal ray at an element plane leads to an apparent shifting of the diaphragm point P. For Successive pupil rays first entering the system the apparent diaphragm point is at p lt P mts - and that is what is known as the entrance pupil point of the system ; while the axial point p 6 f , from which the principal rays apparently diverge on emerging from the system, is the exit pupil point of the lens. For e l the front pupil or diaphragm point, or the point to which the principal rays are converging before entering, is p l} and the corresponding diaphragm point to which the principal ray converges after refraction by e l is p^ ; that is, p l and pi are conjugate foci with respect to the element e t . We will denote the distance e 1 ..p l by D/, and the distance e l . .p^ by D/', so that D/ and D/' are conjugate focal distances. Either of these distances D/ or D/' determines the characteristic quantity 13 1 for the element e l} and we can either write or 2 fl D/' 2/i -D/' (wherein / x stands for the principal focal length of e^), and so deter- mine $! For e^ the front diaphragm point is p^ or ^> 2 , and the back diaphragm point is p^ or p a . Distance 2 . . p, 2 = D 2 ', and c^ . . p.' = D ", either giving /3 2 , and so on. As it is scarcely possible to exhibit clearly all the various refractions to which the principal ray is subject in Fig. 57, unless it were on a much larger scale, therefore the minor refractions exerted by e s and e^ are not represented therein. Now it is evident that if ty is the first angle between the incident principal ray and the axis before refraction by e 1} then fa, representing the angle between the principal ray and the axis after refraction by e- t will also be the required angle of either incidence or emergence under which the principal ray enters or leaves the parallel glass plate of thickness t lt and it will be greater than ty, since e l is collective. If the lens L t had been drawn in its actual thickness, it is evident that the principal ray would have had to be shown vii OBLIQUE REFRACTION THKOUGH PLATE 179 traversing it at a smaller angle = , or the angle of obliquity in the The expression of ^ tan ij/ l , tan \L, etc. , interior of the glass plate. But by considering e 1 and e^ not as mere m terms of tan ik surfaces, but as complete though infinitely thin lenses, and also sub- stituting an air-space equal to in place of t lt then i/r a becomes the f*i .angle of incidence or emergence into and out of the first glass plate, which is what we really want. Moreover, all the diaphragm distances D/, D 2 ', and D/', D 2 ", etc., etc., for the principal rays, and the image distances u v u 2 , and ^ and v 2 , etc., etc., for rays constituting the pencils, .all come out to their proper values by means of this simple device. Now it is evident that tan ^ - tan ^ f , tan ^ = tan ^ \ 7n tan ^ 3 = tan ^ V, 1 1 2i . D/D/D/D; , D/D/D.'D/D; tan ^ = tan f J , tan ^ = tan ^ .and finally tan ,A - i2, tan n - D "D "D "D "D "D * l/j J^ 2 U< A U^ U^ J_7g Hence in applying the oblique correction of Formula XVIII. (E.) the original tan ^r must be multiplied by the corresponding factor * ? or D 'D 'D ' l T\ I "T\ Z "T)"> as ^ e case ma y ^ e - 1^ i g clear that if the rays of pencils D l L>2 i^ 3 entering L x are parallel as if coming from an infinitely distant object, then angle ty is the same as 0. All these diaphragm stop or pupil distances have, in the first place, to be worked out in any given lens system, as a necessary step to deriving the characteristics ft v /3 2 , etc., for each element. The Transference of the Parallel Plate Corrections to the Final Focus But we have yet to carry these parallel plate corrections through to the final focus and convert them into corrections to ^,-^f of the L.r .L. system. Eeferring to Formula XVIII. we have two corrections to the reciprocal value of the perpendicular distance v from the second plate surface of the point from which or to which the pencil diverges or converges. The second formula is a function of the aperture of the pencil, and is of the same nature as spherical aberration, and we have 180 A SYSTEM OF APPLIED OPTICS already dealt with it in Section IV. It applies to all pencils, whether axial or oblique, and may for our present purposes be left out of consideration, leaving us only the oblique correction to - expressed as Parallel plate. Linear value of oblique correc- tion. (81) For our purposes we must now convert this into a correction to the linear value of v by multiplying it by v 2 , and then we get V -i 3 tan 2 x . XIX. (L.) This is the absolute linear value of the oblique correction due to a parallel plate of thickness t v It is thus seen to be independent of the amount of u or of v, and is merely a function of the thickness, angle of obliquity , and refractive index /A. Referring to Fig. 57, it will be readily seen that after we have got the linear oblique correction due to passage through the parallel plate t l from Formula XIX. (L.), we can then express it as a correction to by multiplying it / 1 \ 2 by ( / ^ we transform it back aain to its linear value at the con- jugate focal distance v 2 by multiplying by v*, so that the linear correction to v z after refraction through e, 2 is expressed by - 1 t - 3 tan* (82) It must be borne in mind that all parallel plate corrections, reduced to linear value, are essentially of positive value with respect to the final focal distance of a collective system ; there is, therefore, no question of signs to trouble us. They all take the form of linear transferences of oblique foci from left to right, or in the direction in The oblique plate which the light travels through the system. For the same reasons same sign ultimately, these corrections considered as reciprocal corrections, as in Formula (81), are all of negative import with respect to the final focal power, if the latter is positive ; and since their value in the primary plane is three times their value in the secondary plane, they amount for all practical purposes to the same thing as minus eccentricity corrections. Having now got Formula (82) expressing the linear correction to %, we then express it as a correction to by multiplying by |, and 181 then reduce to its value as a linear correction to # 3 by multiplying by v^, when we get " 1 3 tan 2 ^-2^4, (83) Pl V% 2 and so on, until after refraction through e 6 we get, as the linear correction to the oblique final conjugate focal distance v & , the amount (84) Then to convert this into a correction to the reciprocal of the equivalent focal length of the combination we must multiply (84) by Also tan 2 ^ = tan 2 fa = tan 2 i|rf ^J , as we have seen before. After inserting these values we therefore get, for the case before us, i / /nv-nvn \ 2 /T) ' 13 (?&& 3 tan 2 ^(^i / / 1 \yi^tt 4 f^t%/ T VDj r VE.F.L. (85) R. In the same way the final oblique plate correction due to the second parallel plate of thickness t 2 is expressed as / / r r \ 2 / D'T) - ? ( -***- fji 2 \u^ 5 u & ' 3 tan2 1 ^2 ^3 and, finally, the third glass plate of thickness t^ gives us M, 2 -l J < 86 ) R - ' v As the quantities D' and D" and u and v have always to be worked out for each element at the outset for the purpose of arriving at the characteristics a and /3 for each element, the application of the above formulae entails very little extra work. There is another way of working in these parallel plate corrections, but the above method is the simplest and most straightforward. Having now explained the nature of the method of calculating the normal curvature errors and eccentricity corrections, etc., of any ptical system, so as to define the state of the final image with regard to flat- ness, curvature, or astigmatism, we will conclude with three series of carefully checked calculations as applied to three different optical constructions of which the curves, thicknesses, separations, and refractive indices were all known with reasonable accuracy, and whose final images were also carefully observed and accurately measured. First oblique plate correction trans- ferred to final focus. Second oblique plate Correction t?an 8 - ferred to final focus. Third oblique plate correction trans- ferred to final focus. 182 A SYSTEM OF APPLIED OPTICS Instances of the Practical Application of the Formulae of this Section to actual Lens Constructions 1st. A Series Ic Cooke Lens for Stellar Photography of 6'5 inches aperture and 43'05 inches measured equivalent focal length (Fig. 58). As the foci for the D ray lend themselves best to visual measurement, we will take the heads of the calculations for that ray LI L 2 L 3 Refractive indices. ^ = 1-5180 /*= 1-6035 /*= 1-5180 Radii Radii Radii Radii of surfaces. r 1= +10-64 r 8 = - 14-54 ?- 5 = + 67-35 Thicknesses and equivalent air- spaces. r 2 = +72-45 ^ = -83 tl = -547 r 4 = - 10-35 r fl = + 13 L=-325 -^ = -203 L = '75 -^ = '494 Pa Ps Separations. Axial air-space A l = 4 - 39, and A 2 = 6'85. Diaphragm or pupil point where principal rays cross the axis is taken as being "40 inches behind vertex of fourth surface. The powers of the six elements are therefore Powers of the six elements. 1 -518 1 1 -518 1 1 -6035 1 /! 10-64 20-54 f z 72-45 139-865 / 3 14-54 24-092 1 -6035 1 1 -518 1 1 -518 1 / 4 10-35 17-15 / 5 67-35 130-019 / 6 13 25-096 The first entering rays are supposed to be parallel and % = oc, starting from which we get the following data for the six successive elements (each element being styled by E x ) Values of u, V, a, and .'' for the suc- cessive elements. 1 1 /! ~ 2~0-54 MI ~ ! = + 20- E i from which a x = - 1 54 (convergent and plus) x l - + 1 HI <)(\-KA i -i o y ODD _6 v z = + 17-492 (convergent and plus) from which a 2 = - 14'992 11 17-4Q 2-4'39= +13-102 (convergent and plus) /s 24-092 v s - - 28'723 (convergent and minus) = + 2*677 x s = + 1 PEACTICAL EXAMPLES 183 E 4 } M 4 = 28-723 - -203 = + 28'52 (convergent and plus) 1 1 ' 1 5 t; 4 = 43-018 (divergent and plus) a 4 = + '203 * 4 = -1 E rj 5 u 5 = 43-018 -f 6-85 = 49'868 (divergent and plus) v. = 80-895 (divergent and minus) a 5 = + 4'214 f i o o - o i y u, = 80-895 + -494 = 81 '389 (divergent and plus) /e 25 ' 096 # 6 = +36-285 (convergent and plus and = back focal length) a 6 = "383 6 = -1 We have now to assess the value of /3 for each element. Starting from the pupil point or the intercrossing point of the principal rays placed at "40 inch behind the fourth element, we have for E 4 D/' = - -40 behind, and D 4 ' = + '391 (conjugate to D 4 ") Values of D', D", and R = + 8 6 "7 5 /3 for the successive for E 3 D 8 "= - (-391 4 -203)= - -594, and D s ' = + -58 .'. /3*= + 82'12 elements - for E 2 D/ = -58 + 4'39 = + 4'97, and D 2 ' = - 5-153 .-. /3 9 = - 55'284 forEj D x " = 5-153 + -547 = + 5'70, and D/ = - 7'89 .-.^--6'207 for E 5 D 5 ' = 6'85--40 =+ 6'45, and D 6 " =- 6'786 .-. /3 5 =+ 39'316 for E D 6 '= 6-786 + -494 = + 7"28, and D 6 " = - 10-25 .;, /3 6 = + 5'894 Then the E.C.s in secondary planes, as ascertained from Formula VIIL, Section VI. (substituting tan 2 (/> for 3 tan 2 ^>), may be expressed shortly as tan 2 < 1 i , -w-^wr f 1 and come out as follows : for E x E.C.s = + -00276 tan 2 ^> Eccentricity Correc- for E 2 = +-01 02389 AA-QIQ^ (These two being dispersive elements, the * or 8 " J signs of the E.C.s have to be reversed forE 4 - -0014101 ^ before summing up. for E 5 = + -0036 181 for E 6 = - -015281 2 E.C.s for E 3 and E 6 = - '020594 tan 2 $ EC.s for Ep E 2 , E 4 , and E 5 = + '018027 Total of above, Total for system - '002567 tan 2 (f> secondary plane. 184 A SYSTEM OF APPLIED OPTICS Total normal curva- ture errors, second- ary plane. Total normal curva- ture errors, primary plane. First plate oblique corrections, second- ary plane. Second plate oblique corrections, second- ary plane. Third plate oblique corrections, second- ary plane. Total of same. Final total, second- ary plane. Final linear error. Observed error. The normal curvature errors in secondary planes of the four collective elements as ascertained by **.ti?(I 1 ' >) = + -085734 un* / 1 */ 2 *^5 ^ 6 and the same for the two dispersive elements - '081032 ,, therefore the total normal curvature errors in secondary planes = + '004702 tan 2 < The normal curvature errors in primary planes of the four collective elements as ascertained by tan 2 3p+ 1/1 I I 9 VT + 7~ + 7~^ & LI \r. r 9 /, I J 1 J A J o and the same for the two dispersive elements = - '180847 ,, therefore the total normal curvature errors in primary planes = + '008258 tan 2 Parallel plate corrections in secondary planes for Lj^ as ascertained by = + '189105 tan 2 tan 2 *J "' ' 1 vD/7 \u 2 u 3 u 4 u 5 u & / VE.F.L. and for L 2 as ascertained by 2 _ i\*2 ^"2 ' ,, / r /- n '"n ' \ 2 / , \ 2 tan 2 4, - 456 UMMJ VE.F.L. = - -00070028 tan 2 1 \ 2 r-1 = - -000078 and for L 3 as ascertained by . . 1 FL. = -'00002537 Total = - -00080366 tan 2 < and three times that quantity in primary planes. Summary. On summing up in secondary planes we have + "004702 tan 2 < for normal curvature errors, - '002567 tan 2 < for eccentricity corrections (E.C.s), - '000804 tan 2 < for parallel plate corrections, + '001331 tan 2 < being the final error, which it is now desirable to express as a linear deviation from the focal plane. To that end it must be multiplied by - (E.F.L.) 2 . Let < be 7 degrees, for which the tangent = '132. Then the linear deviation, in the secondary plane, from the focal plane at that angle is + '00133 x - (-132 x 43'05) 2 - - '043 inch, while the actually measured deviation was - '040 inch. VII PRACTICAL EXAMPLES 185 Primary Plane On summing up in the primary plane we have + '008258 tan 2 < for normal curvature errors, - '007701 tan 2 < for eccentricity corrections (E.C.s), - '002412 tan 2 (f> for parallel plate corrections, - "001855 tan 2 being the final error, from which the linear error at 7| Final total, primary degrees from the axis plane. = ( - -001858) x - (-132 x 43-05) 2 = + '059 inch, Final linear error, while the actually measured deviation was + "030 inch. Observed error. Thus the measured deviation in the secondary plane agrees more exactly with the calculated result than the deviation in the primary plane. The whole field of this lens did not extend to much more than 10 degrees from the axis. We shall have occasion to refer to these residual discrepancies in Section XI. Process Lens The next example is shown in section in Eig. 59. It is a leos specially designed for copying or process work, also composed of only three lenses. The following curves, etc., are for an E.F.L. of 8"55 inches. /* 1D = 1-6103 1^ = 1-6103 f t 3D = 1-5240 Refractive indices. r 1= + 1-264 f t = -2-09 - 6 = --5325 Radii. r 2 = - 1 -48 r 4 = + -553 r 6 = +2-8 ^ = 105 1 = '0652 * 2 =-358 2 ='222 < 3 =-110 -^='0722 Thicknesses. = -232 A 2 = -0053 /!= +2-0711 /,- -2-425 /,. -3-4246 a,= -l &=- 16-796 SB.= Separations. Focal lengths and characteristics. a 2 = + 1-4179 j8 2 = +27"945 E 3 a 3 = --3978 &= - 132-711 E 4 / 4 =+ -90611 a 4 =--6462 /? 4 = +5-626 / 6 = -1-01622 a s =+-8552 /8 5 =- 6-1186 / 6 -+5-3432 o e = -'2423 ^ 6 =+28'877 186 A SYSTEM OF APPLIED OPTICS E.C.s, Secondary Plane Eccentricity Correc- tions, secondary plane. Total. Total normal curva- ture errors, second- ary plane. Oblique plate cor- rections, secondary plane. Final totals. Calculated error. Observed error. Calculated error. Observed error. Discrepancies. for E 1 = + "0053 128 tan 2 for E 3 = + -0034821 for E 5 = + -4544450 for E = - -0183698 tan 2 forE^ = --462 1330 for E r = - -0222200 + "4^32399 tan 2 -5027228 tan 2 + 4632399 Total E.C.s = - -0394829 tan 2 < Normal Curvature Errors tan 2 < ~2~ tan 2 < + 1 / 1 1 = --659896 + -055024 tan 2 Normal curvature errors in primary plane + "14020 tan 2 < Parallel plate corrections for L t = - '0060907 tan 2 L 2 = - -0029855 L 3 = - -0001 118 Total - -009 1880 tan 2 Summary Secondary Plane. Total normal curvature errors + "055024 tan 2 < Total KC.s . . . - -039483 Total parallel plate corrections - '009188 Primary Plane. + -140200 taii 2 - -118449 - -027564 Final error + "006353 tan 2 - "005813 tan 2 < Taking the angle of obliquity < to be 14 2 ; , whose tangent is "25, and multiplying above results by (E.F.L.) 2 , we get - ( + "006353)("25) 2 (8"55) 2 = - "029 inch in secondary plane, while actual measurement gave - "005 inch in secondary plane, - ( - "005813)("25) 2 (8"55) 2 = + "0265 inch in primary plane, while actual measurement gave + "05 inch in primary plane. Owing to the difficulty in accurately measuring the radii in such deep curved combinations, such discrepancies as the above may be partly due to statements of radii being inexact. But the principal cause of the discrepancy is due to the unmistak- able presence of minus corrections of the order tan 4 , which will be better understood after reading Section XI. VII PRACTICAL EXAMPLES 187 Series III. Cooke Lens This is composed of four lenses, the dispersive lens being compound ; see Fig. 60. E.F.L. = 10 inches. L l L 2 L 3 L 4 Z D = 1-5101 /*D=l-53 r 1= +2-158 r 8 = -3 r 2 = -r 4-655 r 4 = - 1 ^=603 * 2 =-044 65 /*= 1-6110 ^=1-5101 472 r 5 = +1-150 r 7 = + 12-65 150 r 6 = -1-910 r Q = + 5-843 3 =-218 / 4 =-393 Refractive indices. Radii. Thicknesses. A x = '08 A 2 = A 3 =-90 Separations. Diaphragm or pupil point '25 behind vertex of sixth surface. /!= +4-2305 a x = - 1 Pi= ~ 8 ' 566 i = + 1 Focal lengths and characteristics. / 2 = +9-1256 a 2 = -5-764 ^8 2 = -38-634 2 = - 1 /,= - 6-4716 3 = + 3-943 ^ 3 = +33-012 z 3 = + 1 / 4 =- 2-1435 a 4 - -019 /3 4 = + 10410 4 = - 1 / 6 =+ 1-8822 5 = -105 6 = -9-262 a%= +1 / 6 =+ 3-126 + 912 )8 6 = +26-008 a: 6 = - 1 / 7 =+24'8 a 7 = -31 p 7 = +75-307 Zy= + 1 / 8 = +11 '45 a 8 = - 1-607 j8 ft = +23-624 X R = - 1 ' o O E.C.S, Secondary Plane E 2 + -07401 3 tan 2 E 4 + -104650 E 6 + -001617 E 7 + -001 347 Ej - -000077 2 tan 2 < E 3 - -0839500 E 5 - -1002700 E 8 - -0226700 rf Eccentricity Correc- tions, secondary plane. + -181627 tan 2 - -2069672 tan 2 < + -181627 Total = - -025 34 tan 2 < Total. 188 A SYSTEM OF APPLIED OPTICS SECT. Normal curvature errors, secondary plane. Total oblique plate corrections, second- ary plane. Final totals. Normal Curvature Errors i 1 1 1 1 i tan 2 < +_1/_1 _ ^ u^ \f ~ f ro V5 J 6 + -393624 tan 2 < - -512616 = + -171320 Total = +-05 2 3 20 tan 2 < Normal curvature errors in primary plane = + '11631 tan 2 <. Parallel plate corrections for L : -- - -0108010 tan 2 < L 2 = - -0005574 L 3 - - -0048564 L 4 = - -0000536 Total - '0162684 tan' 2 < in secondary plane. Nor. curv. errors E.C.S Par. plate corr. . Summary Secondary Plane. + 05232 tan 2 < - -02534 - -01627 + 01071 tan 2 < Primary Plane. + 11 631 tan 2 <^ - -07602 - -04881 - -00852 tan 2 < Supposing the angle of obliquity to be 14 2' as before, then after multiplying above final errors by - (tan 2 <)(E.F.L.) 2 or by-('25) 2 (10) 2 we get linear deviations from the plane image of '067 in secondary planes and +'053 in primary planes. The actually observed errors were '04 in secondary planes and no perceptible error in primary F planes, with the lens stopped down to . In the three concrete instances given it will be observed that the thickness of a lens exerts influence in two ways upon the oblique pencils refracted through it : first and most important, the separation between the two elements very largely alters the relationship between the several D's, and consequently the /3's, for the two elements ; and secondly, by introducing a parallel glass plate. This last generally gives rise to much smaller effects than the first, and yet in the three instances given it is too large to be neglected. There is no manage- able formula whereby a thick lens can be treated as a whole. vii CASE OF SPHEEICAL MIRKOK 189 Eccentric Oblique Reflection from a Spherical Reflector It would scarcely be worth the necessary space to work out fully and independently the formulae applying to eccentric oblique reflec- tion for a spherical mirror, as their practical applications do not compare in importance with the corresponding formulae relating to lenses. However, there is a short cut to the formulae relating to a spherical reflector which may be followed with advantage. We have already noted, in connection with the formulae for spherical aberration and central oblique refraction, that the refraction formulae may be transformed into the corresponding reflection formulae by the simple device of substituting the value 1 for //,. Let us take the formula for E.C.s in the secondary plane, which is 1 M n 2/ (a-pJVG"'- l)Ll/*- 1 - 2(a - [j? ~\ ~T I - J and make the substitution therein of 1 for /JL, and we then get If the power of a lens is concentrated into one surface only, then the other surface is plane and x is + or 1 . In the case of a spherical reflecting surface the power is also concentrated into one surface, and x = 4- or 1 ; it does not matter which. Therefore the term con- taining x 2 cancels out and there remains simply while in the primary plane tan 2 becomes 3 tan 2 <, and the correction . j, is of course extra to the normal curvature error =- r Here, just as in the case of the lens, 1 +a 1 , 1 + B I ir^ ~^T = i>" D' being the distance of the stop from the mirror vertex. Thus a is the vergency characteristic for the rays constituting pencils, and /? the vergency characteristic for the principal rays. If the reader will pursue the investigation in detail and ab initio for a mirror with a 190 A SYSTEM OF APPLIED OPTICS SECT, vn stop placed in front, he will arrive at precisely the same formula as that which we have just derived by substituting 1 for /* and 1 for x. We have in the Gregorian and Cassegrain forms of reflecting telescope two cases to which the above formula applies, for it is clear that while there is central oblique reflection from the large concave mirror, yet there is eccentric oblique reflection from the small concave or convex mirror as the case may be. But, as the angular extent of field taken in by even the lowest power eye-piece rarely exceeds a degree, the question as to which form of reflecting telescope gives the flattest final image is of little practical consequence. Such telescopes are essentially very ill adapted, owing to their construction, for taking photographic views covering an angle of view at all comparable to what can be embraced by refracting instruments. SECTION VIII COMA AND THE SINE CONDITION -VON SEIDEL/S SECOND CONDITION CENTRAL OBLIQUE EEFRACTION IT is now our object to investigate much more closely than we have yet done the nature of that phenomenon known to practical opticians as coma, and sometimes as side-flare. We shall find that many of its manifestations are of an exceedingly interesting nature, of great theoretical interest as well as of great practical importance. For a small amount of coma at the oblique focus of a point in a distant object Great importance of formed by a lens system may cause much more mischief to the defini- coma " tion than either astigmatism or spherical aberration, or both combined, so that it is eminently desirable to arrive at reliable formula of the second approximation by the employment of which it shall be possible to eliminate coma from any desired lens system. In Section VI. we arrived at Formulae VI. and VII., which to- gether give the Eccentricity Correction or modification to the normal curvature of image due to the presence of an axial stop or diaphragm causing the pencils to traverse the lens eccentrically instead of centrally. Formulae VI. will be seen at once to be a function of the spherical aberration of the lens. Now it is obvious that if we have two thin lenses in contact so arranged as to give equal and opposite spherical aberrations, as is the case in the object glass of a telescope, then as the compound lens gives no axial spherical aberration, and Formula I., Section VI., proves that the spherical aberration for the oblique eccentric pencil is the same as for the axial pencil of the same aperture, therefore there should not ensue any eccentricity correction due to pencils traversing the compound lens eccentrically. This is certainly the case, and Formula VI., if applied to the two lenses, will be found to be zero. For the formula for the spherical aberration for the axial pencil is, written shortly, 191 I 192 A SYSTEM OF APPLIED OPTICS SECT. Spherical aberration 1_ , , / v 2 1 , . , v 2 _ ft ( , . for a pair of lenses 0/3^1^1 +a7ik A 9m ~ u > W in contact. 8/ i 8/ 2 and the formula for E.C.s in the primary plane, also abbreviated, is as follows Spherical aberration 3 tan 2 $ 1 3 tail 2 ^ 1 , . E.C.s for same pair ^7 T~ ~'~ a \ 2 A l + of T n R~\2 A 2 (*) of lenses. "l \ 1 F*V *J* ( a ^"Pz) which should also be expected to = 0. That this is really the case is evident from the following relations, which obviously exist in the case of two thin lenses in contact. For supposing both to be collective we have Relations between , the characteristics u v D ' D " for a pair of lenses , in contact. tnat IS, L?s = i-i , - 2/ 2 and /i **-&= -(1 -i)/ /I Relations between n \ /2 ~ ' - 3. and O - /Q\ / From the above Equation (1) obviously A' 2 = A'/ y V, so that \/i/ if we take Equation (2) and substitute therein this value for A 2 and the value of (a 2 /3 ) from Formula (3) we then get No axial spherical 2 , aberration implies wVnVVi - *** f ' A' - i "it- W 111L/11 " "X ro ~"~" i* i r\ \c* *"**- I " no sphencal aberra- 2 v / (a - B ) 2 / (a p ) J tion E.C.s. Hence in the case of a combination of thin lenses in contact from which the spherical aberration is eliminated for an axial pencil, there are therefore no E.C.s consequent on spherical aberration. But it by no means follows that the combination is free from coma or side-flare for pencils refracted through it obliquely. That is, if we imagine a diaphragm to be placed in front of or behind such a compound lens, via INVESTIGATION OF COMA 193 then the application of Formula VII. to the two lenses will not necessarily give a zero result ; in other words, coma may be strongly in evidence. For this formula gives us the modification to the normal curvature of image consequent upon the selective action of the stop upon the rays of oblique pencils which are characterised by corna, so that we may call VII. the formula for comatic E.C.s, just as we may conveniently call VI. the formula for aberration E.C.s. The Formulation of Coma The question now arises, whether from the comatic E.C., Formula VII., we can derive other formulae which will give us not only the actual size of the comatic flare at the focus when the whole aperture is in use and the refraction oblique and central, but also the size of the comatic flare when the pencils are not only oblique but eccentric, owing to the presence of a stop. These formulae are of such vital importance as to justify a thorough investigation for central oblique pencils, while we may leave the case of the coma at the foci of eccentric pencils to the neTt Section. In the course of working out such formulae we are also helped to a much clearer understanding of the phenomenon, and the course of the rays which produce it. Let L..L 1( Fig. 61, represent a lens, Q the oblique radiant point in the plane P . . Q, p the conjugate focal point or image of P, and q the conjugate focal point or image of Q as formed by the ultimate oblique centre rays close to Q . . C ; and let it be supposed that the lens is free from every defect excepting coma, which in this case is inward coma, that is, having the flare eccentric towards the optic axis P . . C . .p, the brightest and most condensed end being at q on the oblique axis Q . . C . . q, and the most diffused end at e. Then as our Formula VII. Line of argument, for comatic E.C.s is absolutely independent of the aperture of the lens, and obviously equates to when oblique pencils are centrally refracted (since in that case ft becomes infinity), and as we have seen that the normal curvature errors are also independent of aperture, therefore, since spherical aberration is supposed absent, the conclusion is that any pairs of rays refracted through the lens at equal distances from and on opposite sides of the oblique axis Q . . C . . q come to a focus in the same plane as q, the focus for the ultimate rays close to Q . . C . . q. But if such pairs of oblique rays focussed at the same point as the ultimate central oblique rays, that is, if the oblique pencils were homocentric, then evidently there could be no comatic E.C.s 194 A SYSTEM OF APPLIED OPTICS SECT. under Formula VII. Therefore, since they focus or intersect in the same plane as do the ultimate central oblique rays, but not at the same point as the latter, the only other possible explanation is that they focus in the same plane, but at a different distance from the lens axis. For instance, in the case of Fig. 61, if the ultimate central oblique rays focus at q, then the extreme pair of rays Q . . L and Q . . L : focus at e, and other pairs of rays refracted by the lens at points nearer to its centre will focus at intermediate points in the line q . . e. We have now to find how these focal points are distributed along the line q . . e in the focal plane. It is obvious from the fore- going that the primary section of the cone of rays is at a minimum at q . . e, in the plane wherein symmetrical pairs of rays such as Q . . L and Q . . Lj intersect after refraction. If now we can find the point / where the ray Q . . L after refraction crosses the centre ray Q . . C . . q, C L then clearly the distance (/. . g) ' ' will give q . . e, the length of the O . ./ A device for obtain- comatic flare. In order to get at this we must imagine a stop S : . . Sj flu- e enS omatlc to be so placed centrally on the lens axis as to just let pass simul- taneously the centre ray Q . . C . . q and the extreme ray Q . . L . ./; then it is obvious that/. . e will be the longitudinal value of the stop correction or E.G. as a variation of V or C . . p, the back conjugate focal distance, which is due to that particular position of the stop and degree of obliquity . Let S = semi-aperture of stop, and A semi-aperture of lens. Let d . . C as usual = D, P . . C = U, and C . .p = V. Then 2*+l . DU by comatic E.G. Formula V. (This form of the formula is the most convenient for our present purpose.) We then have the relations ./ - ~ - - r..C r..C U ^"^, wherein P.. Q = U tan <; . . DU tan = A\J - DA - DU tan <, so that our condition that the stop just allows the extreme ray Q . . L and centre ray Q . . C to pass demands that PI b ' i Fi^.67.a. P- ,d -A B- -r=3 r-i. ^- 1.5. A. - -.S X. * +.+ U = -2 Fi^.68.a. INVESTIGATION OF COMA AU 195 2 TJ tan + AU - 2 tan 2U tan ( + A so that Formula V. becomes or, more conveniently, tan which is the correction to ~= required to convert it into V There- fore the required linear stop correction 7t . . > or / . . e is obtained by multiplying (4) by V 2 , unless V is very large compared to F, and then q . . e or the length of the comatic flare will be obtained by A A multiplying /. . e or h . .p by , or approximately by ; so that V tan in which, resorting to our former device, we may substitute V, thus arriving at , l(2/*+ !)(/*- IJa + Oi+l)*, -, 2F for T^ - 1) This, then, is the formula for the length of the comatic flare, sup- posing that the other aberrations are absent. It is evident that it is not affected by the stop S x . . S : , which we have used as a stepping-stone in the line of reasoning, being taken away, thus bringing the full aperture into use. The formula therefore applies to the full aperture 2A of the lens. It is now seen that the length of the coma increases The correction to . Formula for the length of the comatic flare. 196 A SYSTEM OF APPLIED OPTICS Distribution of the rays in the primary plane. A corollary. The distribution of the brightness. Variation in focal lengths for different lens zones. Zonal aberration. ' as the square of the aperture, other factors being constant, and there- fore the lateral displacement (like q..e) of the foci for symmetrical pairs of rays increases as the square of the distance from the oblique axis or ray Q . . C . . q of the points where they impinge on the lens. We are now in a position to construct a diagram of the course of the rays in the primary plane, which gives rise to coma, in more detail. Fig. 62 illustrates the same case as Fig. 61, only with the coma farther exaggerated for the sake of clearness, and with more rays filled in. Here the pair of rays Q . . b t and Q . . b refracted at distance = 1 from the oblique axis Q . . A come to focus at point b l + & 2 at 1 unit from q, the focal point for central rays ; the rays Q . . c x and Q . . c. 2 refracted at distance = 2 from the oblique axis Q . . A come to focus at point c l + c 2 at 4 units from q ; while the pair of rays Q . .d l and Q . . d refracted at distance = 3 from the oblique axis Q . . A come to focus at point d 1 + d 9 situated 9 units from q, and so on as the square of the aperture. It follows, as an obvious corollary from the law of the length of the coma increasing as the square of the aperture, that, provided the length of the coma is very small compared to its distance from the lens, as is usually the case, then the distances q . . b, q . . c, and q . . d from the focus to the points where the rays Q . . b lt Q, l ..c l , and Q . . d l intersect the central oblique ray Q . . A . . q must vary as the aperture, or as the respective distances A . . b l} A . . c l} and A . . d^ The coma in Fig. 62 is too much exaggerated to permit of this relationship being properly shown. In the primary plane it is clear that the rays are most crowded together at the end q of the coma, and most diffused at the other end e where d l -f- rf 2 intersect. Hence the former is the bright end, and the latter the diffused end of the flare. Supposing that the lens were divided into concentric rings or zones, and each zone in turn allowed to throw an image of the point Q on to the plane p . . q, it is very evident that as the image of Q formed by the two extreme rays in primary planes falls at d l + d, 2 nearer to the optic axis than the foci for smaller zones of the lens, therefore the equivalent focal length may be said to vary for different zones ; the larger the zones of the lens the smaller the equivalent focus of such zones. In other words, the equivalent focal length differs from that of the ultimate central portion by amounts varying as tan (f> and as the square of the aperture. This property of a lens subject to coma has been well emphasised by Professor Silvanus Thompson, who has applied the term " zonal aberration " to the phenomenon. INVESTIGATION OF COMA 197 It is clearly of the greatest practical importance, when estimating or eliminating coma in a combination of lenses, to have an expression for the angular value of the comatic flare, that is ?-^-- Of course this is obtained by multiplying the Formula I. by ~ or -^FT, by which we get q..e II. The angular value of the coma, as subtended at lens centre. It is clear that in the case of Fig. 61 we have both a and x positive, while at the same time the coma q . . e is inwards or towards the optic axis P . . p. It is very important to adopt a convention with regard to conventions as to the sign of coma. In Formula II. the angular cOma comes out negative, signs of coma. We will consider any such comatic flare to be negative which is inward, or whose diffused end lies towards the optic axis (or whose bright end C (Fig. 66) lies away from the optic axis); and this rule must apply whether the coma is real or whether it is merely virtual, and irrespective of whether the lens in question is collective or dispersive. For instance, Fig. 61 gives, on a smaller scale, the case of a dispersive lens corresponding exactly to the case of the collective lens in Fig. 61. Here, also, it will be easily seen that the coma e . . q is likewise inward or towards the optic axis. Also both a and x are positive. Therefore it is clear that the minus sign must still prefix Formulae I. and II. with respect to the dispersive lens ; and then, as we shall see farther on, the comatic functions of a series of lenses can all be simply added together, and there will be no need for reversing the signs of the functions for dispersive lenses before summing up. The case is intrinsically quite different to that of the eccentricity corrections. In short, the fact that the formula for coma is a function of ^ shows that the sign of f may be ignored, lens is implied in the sign of a. Moreover, the sign of the The Part Played by the Secondary Rays in Coma Formation We may now turn our attention to the consideration of sym- metrical pairs of rays contained in the secondary plane, any two rays refracted through the lens at equal distances above and below A. Since we are assuming the existence of coma without astigmatism (a condition which is hypothetical in the case of a simple lens except under very special cases of eccentric refraction, but quite possible and quite common in the case of certain compound lenses), we have, of 198 A SYSTEM OF APPLIED OPTICS SECT. Difficult nature the inquiry. of course, to assume that a pair of rays in the secondary plane intercross or focus in the same focal plane q..p as do the pair of rays in the primary plane, and it is obvious that they will focus somewhere in the straight line p . . q lying in the primary plane and passing through the optic axis. The line of reasoning whereby the position of this focal point for two rays refracted at the distance A from the lens centre in the secondary plane is determined is long and difficult, and perhaps it is unnecessary for our purpose to do more than give a brief sketch of it by the help of Fig. 63. This method consists in assuming the two rays Q . . T' and Q . . T" in the secondary plane to be refracted through the sharp edge of the lens immediately above and below the point T, aud finding by spherical trigonometry how much the vertical plane containing these two rays after refraction is angularly deviated (in the primary plane) from the plane containing the same two rays before refraction ; for it can be shown that such a deviation always takes place. In Fig. 63 the two incident rays Q . . T 7 and Q . . T" respectively have to be repre- sented by one straight line Q . . T, and the two emergent rays T r . . q f and T" . . q f by another straight line T . . q' ; but those two straight lines are not one ; they form a small angle with one another at T, and the angular displacement of T . . q' witli respect to Q . . T is outwards or away from the optic axis. Having got a general expression for this deviation (which depends upon the shape of the lens, etc.), we next compare it with the lateral parallel displacement which occurs to the central my which passes through the two principal points, 2h an d p 2 , of the lens and its geometric centre, as shown by the solid lines. We then arrive at the formula, III., for the angular displacement of the focus q l for two rays in the secondary plane from the focus q for the ray passing through the geometric centre of the lens that is, the angular value of q . . q f subtended at T Angular value of interval between oblique central ray and secondary focus. tan III. Thus we obtain a value which is just one-third of Formula II. So that if, in Fig. 63, q is the point where the central ray strikes the focal plane, and q f is the point where the two rays Q . . T' and Q . . T" in the secondary plane come to focus, then if we make q . . q" = 3(^ . . q'), q" will be the point where the two rays Q . . E' and Q . . E" in the primary plane come to focus, the two sets of rays belonging to the vin INVESTIGATION OF COMA 199 same zone or circle of the lens, which we have assumed to coincide with its sharp edge. The Diameter of the Coma in the Secondary Plane The following line of reasoning for obtaining the diameter of the comatic Hare in the secondary plane may be pursued consistently with the theorem of coma which we have just explained. We have supposed that the four rays which, two by two, impinge upon the two extremities of the secondary axis of the comatic circle and define its size in the secondary plane, are refracted through the lens zone at points 45 and 135 degrees in both directions from the neutral point p r (Fig. 64), that is, rays from /, /", j\, and j 2 . Confining our attention to the pair j" and j 2 immediately above and below the point n, as shown in dotted lines in Fig. 61, we have C . . n = (C . . L) cos 45 = A= (A being the semi-aperture of the lens). The dotted circle in Fig. 64 then represents the eccentric zone limited by the stop S' . . S', and its radius is obviously . We have already found the crossing point / for the two rays C . ./ and L . ./, which gave us the linear E.G. in primary plane (=f..e), from which we got q . . e. We now want the corresponding E.G. for the two rays n . . s in the secondary plane passing above and below n ; and in order to find it we must imagine the diaphragm moved back from d to d', such that Q . . d f produced passes through n ; then, calling the diaphragm distance (d f . . C) I)', for short, we have, if angle P . . d f . . Q = 6, ^4 1 U-D' 1. -L' - */2 -4 ^ x /2 U tan< tan 9 and dividing by D' we get J_ 1 U-D' D'U A 1 tan UD' ' U - D v ^2 tan . Hence the required E.G. is expressed by Formula III., Section VI., with the above value of ^ vc, inserted ; that is, U-D /*+/ x) 4ua + r - -- '(x - a) \ * / A A = Diameter of the coma in the sec- ondary plane. 200 A SYSTEM OF APPLIED OPTICS which is more conveniently expressed as Then the linear E.C. obtained by multiplying by V 2 is (6) Then the secondary diameter of the comatic flare is obtained by multi- 2A aperture in secondary plane 2 A I - a plying (6) by - - ^ , that is, by v/2 or ~ ^r- ~y~ v* So that we get 1 l-o IV. for the secondary axis of the comatic flare, which is just two-thirds of the value given by our previous Formula I. for the primary axis of the flare. To trace out mathematically what happens to the rays from Q other than those we have dealt with, and which are retracted through the sharp edge or belong to the same lens zone, is a much more difficult task. It has, however, been undertaken by Professor Finster- walder and others, and the results may be shortly explained by Fig. 64. The comatic circle. Structure of Pure Coma We will now give a brief explanation of the comatic flare, while reserving until later the general proof that this theorem of coma necessarily implies the ratio of 3 to 1 for the E.C.s in primary and secondary planes respectively. Let the circle s'. .p'. . s". .p" of Fig. 64 represent one of the concentric zones of a lens, the optic axis of such lens being perpendicular to the paper. Let C be the point in the distant focal plane where the ray passing through the geometric centre of the lens strikes ; let P be the point where the two rays in the primary plane, p' , . P and p" . . P, come to focus ; and S be the point where the two rays in the secondary plane, / . . S and s" . . S, come to focus, C . . S being ^ of C . . P. About a point half-way between S and P draw the circle S . . K' . . P of diameter = S . . P. This circle we will call a comatic circle, on which strike all the rays refracted through the zone s f . .p" . . s" . .p' of the lens, only the way in which the striking points are distributed around the comatic circle is 201 a peculiar one. Starting from p' in the primary plane, the point towards which P (the point of the comatic circle most remote from the centre ray C) lies, we may reckon our rays by their angular distance from p' measured along the zone. The ray from j', a point 45 from^', will strike the comatic circle at K 7 at a point 90 from P ; the ray from s', 90 from p', strikes the comatic circle at S, 180 'from P ; the ray j" , 135 from jo 7 , strikes the comatic circle at K", 270" from P, and so on. That is, every ray passing through the lens zone at an angle from the neutral point p' strikes the comatic circle at a point situated by 20 from the corresponding neutral point P. Thus all the striking- points of rays are subjected to what may be termed a degree of torsional displacement equal to 0. In Fig. 64 the comatic circle, for clearness, is shown too large in proportion to the size of the lens zone. Fig. 65 shows the structure of the comatic circle far more truly, for it is constructed on the supposition that the lens zone is infinitely large compared to the comatic circle, so that the inclination of all the rays shown therein to the primtry plane P . . P ' is the true measure of their angular distribution round the lens zone. Also it is supposed that the diagram 65 represents a view of the comatic circle as if looking along the oblique central ray, so that the lens zone would, strictly speaking, appear as an ellipse. But the angle of obliquity is assumed to be small enough to allow us to treat the lens zone as a circle, of immense size compared to the diagram. As a corollary from this torsional effect on all rays (except the neutral pair striking the comatic circle at P t ), it follows that every point in the comatic circle is the mutual striking point of two rays originating from two points in the corre- sponding lens zone which are 180 apart or diametrically opposite So that each straight line drawn across Fig. 65 represents two rays, one from one point in the lens zone, and the other from the opposite point. A marked feature of the case is that all the rays cut the straight line drawn from the lens centre to the intersection S' of the two rays S . . S' and S x . . S' in the secondary plane ; but let it be noted that these intersections are at different distances from the plane of the diagram or comatic circle, so that the seeming intersection of all the rays at S' is apparent only. Fig. 65a is designed to elucidate these points further. It is a perspective view of the comatic circle and the same rays coming from the lens zone as those shown in Fig. 65, wherein the rays are numbered I, -2, +1, +2, etc. The + sign means that the ray in question, after intersecting the comatic circle, proceeds to cut Distribution of the rays round the comatic circle. The torsion im- parted to the rays. Every point in the comatic circle re- ceives two opposite rays. A common inter- section axis for all rays from each lens zone. 202 A SYSTEM OF APPLIED OPTICS SECT. Distribution of the commonlnt^fsection axis - the ray projected through S' from the centre of the lens, at a point beyond the plane of the comatic circle; while the -- sign means that the ray in question cuts the projected central line before it intersects the comatic circle. Thus rays of the same sign and number cut the ax ^ s ^ intersection at the same point, and those of equal numerical values, but opposite signs, cut the axis of intersection at points equidistant from, but on opposite sides of, the plane of the comatic circle. The two rays marked s and s l in the secondary plane cut the comatic circle at one point S', also shown in Fig. 65a. For the sake of clearness, each ray is drawn as a solid line up to its intersection with the comatic circle, and as a dotted line after its intersection. Also each ray is marked with the same numbers and signs as in Fig. 65, so that each ray may be identified in both diagrams. The relative aperture of the lens zone is assumed to be very large. Outline of the com- atic flare denned. The Distribution of the Gomatic Circles formed by Different Lens Zones + The next Figure, 66, shows a series of comatic circles and their relative distribution for a series of lens zones of semi-apertures = 1, 2, and 3, from which it will be easily seen that the two tangents to the series of comatic circles embrace an angle of 60, and intersect at the point C where the .central ray cuts the focal plane. For we have seen from Formula III. that the distance C . . B, from the central ray C to the point B where the two rays in the secondary plane intersect, is -^ of C . . D. Therefore, assuming the comatic circle t' . . B . . t" . . D. with its centre at e, to exist, we have - = i- C.. = sin (t' . . C . . e) = sin 30, therefore the angle between the two tangents is 60. Such an expanding series of comatic circles makes the well-known balloon-shaped side-Hare or coma instead of a point of light at C. Then C is the end of the coma at which the greatest intensity of light concentration occurs, while D, the opposite extremity, is marked by the greatest diffusion of light. We will call C the root of the coma, and D its extremity. If the extremity of a comatic flare lies towards the optic axis of a lens, then the coma is negative or ; if it lies away from the optic axis, then the coma is positive or + . The signs preceding Formulae I., II., and III. are arranged to always give results in accordance with the above convention, bearing in mind that no difference of sign is required to be made in applying these formula? to dispersive lenses, of which instances will be given later. VIII THE SINE CONDITION 203 The student wishing to study the formation of coma corresponding HOW pure coma may to any particular lens zone cannot do better than take one-half of a be exhibited - Goerz Double Anastigmat, with the stop to the front to receive nearly parallel rays from a distant bright point of light. The lens may be rendered opaque except for a narrow zone near the edge of its aperture, and then, on examining the focus with an eye-piece, while tilting the lens to a certain degree of obliquity, a very fine example of pure coma without much admixture of astigmatism may be obtained, and the duplex circle of Fig. 70 may be watched as it closes up to focus. It A fcaif lens zone is particularly instructive to cover up half the zone, when, at the focus, jJjJJJJJJJJ app c * r m atic a complete ring of light will still be obtained. circle. The Sine Condition By many optical authorities, especially on the Continent, it has been asserted that if a lens fulfils what is called " the Sine Condition," it will then show no coma. The late and much lamented Professor Abbe, of Jena, was the first to prove that if a lens L . . L X (see Fig. 67) is so shaped relatively to the conjugate axial foci P and p that LPS - = constant for all values of L . . S or y, then pencils refracted sin Lj?S obliquely but centrally through the lens, such as pencils LQLj and L^I^, will be free from coma. It can be proved that if the lens fulfils the sine condition, then, if we take a new point of origin Q to one side of the axis, but in the same focal plane as P, the length of path Q . . L -f L . . q = the length of path Q . . Lj + L : . . q, and therefore two elements of a wave of light starting together from Q meet again at q simultaneously upon a common point situated on the central oblique ray, there being, therefore, no lateral displacement. But to plan a lens that will fulfil the sine condition in any particular case by trigonometric methods is far more laborious than arriving at a direct result by a simple algebraic formula, and it may easily be proved that our formula for eliminating coma, (2/i + !)(//, l)a + (//,+ 1) = 0, can be deduced directly from Professor Abbe's sine condition, and is the algebraic expression of that condition. Let us consider any pair of conjugate rays such as P . . n and n . .p (Fig. 67), and suppose they are each produced into the lens until they meet at n, then the per- pendicular n . . S is common to the two triangles ftPS and npS, and sine npS n . . P . , - KFT = simply, sine nrb n. .p Then let us consider a pair of conjugate rays refracted extremely The sine condition implies equal " opti- cal lengths " for ex- treme rays of an oblique pencil. The sine condition made the basis for our formula for no coma. 204 A SYSTEM OF APPLIED OPTICS SECT. closely to the lens axis (see enlarged diagram of the centre of the lens, Fig. 67a). If the two conjugate rays P . . b and p. . d are produced in- wards and meet at h, then in the extremely narrow triangle libd the base b . . d is the course of the ray within the lens, the angle hbd is the angle of deviation at the first surface, and the angle hdb is the angle of deviation at the second surface, but at such extremely small angles, the angles of incidence or emergence and angles of deviation bear the constant relation [j,:fj, l, and we may say that the angle of incidence of the ray P . . b is to the angle of emergence of the ray d . . p as h . . d is to h . . b ; so that ultimately when h is brought down to the axis it will be so placed as to divide the thickness t of the lens into two parts A, corresponding to b . . h, and B corresponding to h . . d. Then A angle of emergence of d .. p B angle of incidence of P. . b Let P..L and L..p in Fig. 67 be another pair of conjugate rays refracted by the extreme thin edge of the lens ; then it is obvious that the sine condition demands that P . . L = P . . n Qr P..h P. L /*_(P..&) + (&../0 = U + A L . . p n . . p h . . p ' h . . p (d . . p) + (d ' . . h) V + B ' therefore r> T TT A P..L_U + A ( } L..^> V+B' Now let perpendicular L . . S = y, then Ke verting to Formula (7), giving the ratio between A and B, it is obvious that the ultimate angle of emergence of ray d . .p is expressed by (- + v)> and the ultimate angle of incidence of ray P. .b is similarly /I i\ expressed by ( - + ~}. Therefore putting t for the central thickness of the lens we have 7- TT and B = '1 J\ /I 1\ /I 1 therefore Formula (8) becomes THE SINE CONDITION 205 u 2 V U 7 / \ 7 \J ' \ O * ' / 1 1 \ W 2/1 f = ' -~5 " ~ ' V + (-J + - 2 W s, y + U in which we may put f f _?//! 1 ~ 2r + 2s~"2\r + s so that (11) becomes U- V- >flM) u y 2 /! 1\ 1 s h v 4) * 7+ 6 H 4)- lG ^ \ V 5/ -TT W 2 /l 1\ j. ' 1 .1. 1 i r h u (12) 2 \r s/ 1 l \^.i l l \ r + u) + Vl + vJ 1 + a; 1 On resorting to the former device of making .. _ n = - Reductions. 1 a; 1 1 + a 1 1-a 1 , Vi-.i- = f^, and ->- =^, and substituting these in the 2/0*- 1) 8 1 2/ U' 2/ smaller terms, then (12) becomes From (13) we derive from which we get, on reducing to a common denominator and leaving out the latter, 16UV/VO* - I) 2 + 4y*V/G* - !){(! - x) + 0* - 1)0 - )} - !){(! - X) + 0* - 1)(1 - )} + ?/ 4 {(l - X) + (/, - 1)(1 - a)} 2 - I) 2 - WVMt* - !){(! + *) + (A* - 1)(1 + )) - !){(! + x) + (p. - 1)(1 + a)} - y 4 {(l + X) + 0* - 1)(1 + a)} 2 = 0. Neglecting functions of ?/ 4 , which belong to a higher order of approxi- mation and are small compared to the other terms, we get 206 A SYSTEM OF APPLIED OPTICS SECT. Conclusion from ful- filment of the sine condition. Reciprocal of the radius of the sine surface. Two corollaries. 0. - a)}l a/ 2f -2f Then on writin - lor U, and - for V, and multiplying all terms by (1 a)(l + a), we get, - x) - a)} + x) and this simplifies down to (14) which, as we have already seen in Formulas I., II., and III., etc., is the condition of no coma,, which we previously worked out from quite different premises. It can also be shown that if, when the sine condition is fulfilled, the incident and emergent rays are produced to intersect within the lens, then the radius E of the circular curve L . . S. . Lj along which the pairs of conjugate rays thus intersect is given by the formula 1 R 1 1 Thus, when U is infinite E = V ; when U = V, E is infinite, and the surface L . . S . . Lj is flat ; but when V > U, then the curve of radius E is reversed in sign and faces convex to the longer conjugate focus. ' We may call this spherical surface of radius E the sine surface. When a lens is free from coma, or fulfils the sine condition, two important corollaries can be deduced from the conditions prevailing and these are, firstly, that the point S, Fig. 67, where the sine surface cuts the optic axis, is always exactly in a straight line between any original radiant point Q and its image q ; and secondly, this point S is so situated with respect to the two principal points, p l and >, of the lens as to divide p l . . p 2 into two parts, such that ( ?1 ..S):(S..^ 2 )::U:V. Therefore S falls between the two principal points if both U and V are positive, as in Fig. 68 ; but if U and V are of different signs and the conjugate foci on the same side of the lens, as in Fig. Q8a, then the point S falls outside the principal points, and in this case behind them. SOME MANIFESTATIONS OF COMA 207 Some Manifestations of Coma Eeturning now to the consideration of the structure of coma, we have seen that, in the absence of other aberrations, a lens manifesting coma forms for each zone of the objective or lens a duplex circle in the focal plane, whose actual diameter is given by Formula IV., and its angular diameter, as viewed from the lens centre, by two-thirds of Formula II. Thus for any given lens zone the diameter of the comatic circle varies as the tangent of the angle of obliquity of the incident pencil ; and for any given angle of obliquity the diameters of the comatic circles and the distances of their centres from the oblique central or principal ray alike vary as the square of the diameters of the corresponding lens zones. It now becomes interesting to inquire what sort of figures will be When the focal plane traced out by the rays going to form such comatic circles first, is ^P 8 ^^ when the focal plane is departed from either towards or away from the lens ; and. second, when that usual accompaniment of coma, viz. astigmatism, is also present. We will first of all deal with pure coma as projected upon planes In the case of pure nearer to or farther from the lens than the focal plane in which the ' duplex comatic circle is formed. Here Fig. 65 will at once help us to form an idea of the figure traced out by the rays on a plane somewhat nearer to the lens. This figure represents what would be seen by the eye placed in and looking in a direction parallel to the straight line joining the centre of the lens to the centre of the comatic circle. Therefore, since the inclinations of all the converging rays to the plane of the diagram are equal, if we mark off on each ray a point such as w^, w , etc., such that the distances from all such points to the points where the same rays cut the comatic circle are equal, then the curve w l . . w >2 and w{ . . w 2 , etc., through all these points will be one of the out -of -focus comatic curves. The resemblance to a hypocycloid is Hypocycloidal at once apparent. In fact, it has been proved by Finsterwalder (what nature of the curves, is in entire conformity with the formula we have worked out) that the comatic curve traced out by the rays from any one lens zone is such a curve as would be traced out by a point in a uniformly rotating circle whose centre is simultaneously travelling at half the rate and in the same direction around another fixed circle. Fig. 69, Plate XIV., illustrates this. In all the figures r..r is the rotating circle, and /../ the fixed circle that the centre of the former travels round. While the centre of r . . r travels once uniformly round /../ the circle r . . r has rotated 208 A SYSTEM OF APPLIED OPTICS Out - of - focus coma for five concentric lens zones. The effect of adding astigmatism. uniformly on itself twice. Now r . . r is the same size as the comatic circle in the focal plane, and thus represents the amount of torsion to which the rays are subjected ; while the fixed circle/. ./ may be zero or of any size, for it simply represents the circle traced by the hollow coned surface of rays upon the selected plane of projection (supposing that the rays were all refracted accurately to a point at the centre of the comatic circle). Thus the size of /. ./ simply depends upon the distance of our plane of projection from the focal plane. If the plane of projection coincides with the focal plane, then /. ./ vanishes to a point, and in that case we have to imagine the rotating circle r . . r rotating on itself twice while its centre remains stationary, which hypo- thetical case explains the duplex comatic ring. It is really a double loop in its ultimate closed-up form. Fig. 70 a and d show two phases of the comatic curve at equal distances on each side of the focal plane in which the comatic circle is formed, followed by three more out-of- focus phases b, c, d. All these and the following figures have been traced out by the employment of a geometric machine in accordance with the above law of coma formation. Next, let us take a lens giving pure coma, and consider the tracings made near the focal plane by each of five concentric zones of the lens of radii, 1, 2, 3, 4, and 5. Then at the focus we shall have a figure like Fig. 66, a series of duplex comatic circles, but at a little distance on either side of the focus we shall get Fig. 71. Next we may consider the effect of the usual astigmatism being added to the coma. The effect of astigmatism is, at the focus for rays in the primary plane, to substitute a short and nearly straight focal line for the point, and at the focus lor rays in the secondary plane to substitute another straight focal line of the same length as the former for the point, these two focal lines being at right angles to one another. Consequently, the figure to be expected in the plane of each focal line is the figure that will be traced by a point in the comatic circle rotating on itself twice, while its centre travels with a harmonic motion up and down the whole length of the focal line. Fig. 69a illustrates this action, at within the primary focus, at P the primary focus, at L the least circle, at S the secondary focus, and at 0' beyond the latter; while in Fig. 72, P is the figure thus traced at the focus for the two rays in the primary plane which mutually intersect at the point p. Then, if the plane of projection is transferred to a position half-way between the two focal lines or at the circle of least confusion, we get the tracing L ; and then, on transferring the plane of projection to the secondary focal line where the two rays in the secondary plane PLATE. XIV 69. a O a' 7O. L 72 C S Fid 73. PLATE. XIV 69 Fi<> 69 a. P L a O a' 7O. L 72 'id 73. Fi^.74. VIII SOME MANIFESTATIONS OF COMA 209 intersect, we get the tracing S, s being the intersection point for the two rays from the zone which lie in the secondary plane. Tracing a is taken within the primary focus. Fig. 73 is a series of phases of astigmatic coma, all for the same lens zone, in a case where the degree of astigmatism bears a still greater proportion to the comatic circle. a is within the primary focus, P is at the primary focus, b half-way between the primary focus and the least circle, L is at the least circle, c is half-way between the latter and the secondary focus, s is at the secondary focus, and d beyond it. Fig. 74 is the complete series of tracings for five-lens zones in a case of coma combined with very moderate astigmatism, taken in the focus for primary rays for all zones, as the lens is supposed to be free from spherical aberration. Fig. 75 P, Plate XV., is the complete comatic formation for five- lens zones at the primary focus, in a case where the astigmatism is more pronounced than in Fig. 74. Fig. 75 L is the phase of the same which occurs at the least circle, and Fig. 75 S the phase of the same which occurs at the secondary focus. Figs. 76 P, L, and S show the phases, corresponding to the last, of astigmatic coma in a case where the astigmatism is relatively still more violent. Throughout all cases of astigmatic coma it will be noticed that the form of the loop is different for each lens zone. For it is obvious that the length of the focal line increases as the diameter of the corresponding lens zone, whereas the comatic circle, whose rotation and travel produce the loop, increases as the square of the corresponding lens zone. Hence for the smaller lens zones the straight line formation predominates, and for the larger lens zones the circular element or loop- like effect predominates. Figs. 7 5 P and 7 6 P both show this feature. The phase of coma indicated in Fig. 76 S, when all the infinite series of zones are filled in, as in the actual case of real coma formed by an aberration-free object glass, is perhaps the most beautiful, being a shell-like formation which at first sight looks complicated and puzzling. The comatic formations yielded at the oblique foci produced by uucorrected lenses are still further complicated by the fact that the foci for each lens zone vary by spherical aberration, but by the kind permission of Professor Silvanus Thompson * we are enabled to here reproduce some actual sketches taken by him at the oblique foci of a * And also by permission of the Royal Photographic Societj T . Phases of astigmatic coma from the same lens zone. Astigmatic coma for five concentric lens zones, primary focus. Astigmatism more pronounced, primary focus. Same at least circle, and at secondary focus. Same with astigma- tism still stronger. Form of the astig- matic loop varies for each lens zone. Beautiful nature of the effects. Spherical aberration adds a further com- plication. Prof. S. Thompson's experiments. 210 A SYSTEM OF APPLIED OPTICS SECT. simple plano-convex lens whose face was divided up by annuli of black varnish into a series of concentric transparent zones of finite width. Of course a good deal of colour fringe which was actually present does not show in these reproductions, which will be seen to exhibit practically the same character as the curves we have just dealt with. A full account of his experiments was given in a most interesting arid instructive paper printed in the Photographic Journal for December 1901 ; which should be carefully studied by all interested in this branch of optics. Some of the paradoxical con- sequences of coma therein described are exceedingly interesting. If Fig. 78 E be carefully observed, it will be noticed that the tracing of light for the outermost zone is at the focus for the rays in the primary plane, and the curve is in the same phase as any one of Varying phases for the curves in Fig. 76 P. But the curves in Fig. 78 E for the smaller is zones. j eng zoneg are more open loops, for, owing to the spherical aberration, the two primary rays of such zones focus beyond the plane in which the comatic curves were taken. In short, the effect of spherical aberration upon the comatic curves is to cause the latter to assume more or less different phases for the different lens zones. The great broadening out of the outermost zone tracing so marked in Fig. 77 F is of course due to the outer lens zone having a finite and appreciable width, the loops for the outer edge and inner edge of the zone being widely different, owing in large part to the spherical aberration, while the zones between these two all contribute their light to intermediate loops. Our comatic loops Fig. 79a illustrates the figures obtained by Dr. Adolph Steinheil neii'^^r^onometri ^y elaborate trigonometrical calculations applied to the case of the cal calculations. 6 -inch refracting telescope at Konigsberg made by the celebrated Frauenhofer. He selected four zones of the objective, as in Fig. K, and calculated the oblique foci for eight rays equally distributed round each of the said zones, and found where they impinged on the plane passing through the axial focus (see G and H) on a second plane '35 of a millimetre nearer the objective (see I and J), and on a third plane 70 of a millimetre nearer the objective (see K and L). He thus arrived at the comatic formations H, J, and L, whose identity with our previous results is plainly evident. He then, after a few alterations in the curves of the objective, got it to give symmetrical oblique refraction, the sine condition being fulfilled, and the resulting oblique foci shown in Fig. 796, N, P, and E, then showed pure astigmatism only. PLATE. XV. i. 75. S i. 76. S 76 L. PLATE. XV. i. 75. S 76 S vni GENERAL PROOF OF THEOREM OF COMA 211 General Proof of the Theorem of Coma Having now given a certain explanation of the formation of coma and shown many figures synthetically formed by way of illustration, and others either drawn from actual experiment or trigonometrical calculation, all of which confirm one another, it will now be as well to give a general proof that our theorem of coma will necessarily lead to all comatic eccentricity corrections in the primary- plane being three times as much as the simultaneous eccentricity corrections in the secondary plane. In Fig. 79c let C be the centre of an aberration-free objective yielding coma, and let the eccentric circle c x . . G . . c 2 . . H represent the outline of a pencil of rays where it impinges upon the plane of the lens. Then . ./ is the eccentricity. Let the radius or semi- aperture of the eccentric pencil /. . c x or /. . c 2 be r. About C describe the circle R 3 . . R 3 , touching circle c x . . G . . H at G, another circle R 2 . . R 2 passing through c x and c 2 at the upper and lower extremities of the secondary diameter of the pencil, and another circle R x . . R x touching the circle c x . . G . . c 2 . . II at H. Then G and H are the points where the two extreme rays in the primary plane are refracted through the lens, while c x and c 2 are the points where the two extreme rays in the secondary plane are refracted. Turning our attention to the oblique focus (Fig. 79d) formed by light filling the whole aperture R! . . R I} we have the lens zone R : . . R x forming the duplex ring R^, the lens zone R 2 . . R 2 forming the duplex ring R^, and the lens zone R 3 . . R 3 forming the duplex ring R 3 '. Here let it be borne in mind that Fig. 79d is really very small compared with the lens aperture R X . . R r We will assume that the distance, such as C . . h, between the central ray C and the outermost point of any duplex ring is N" times the radius of the duplex ring. We have so far assumed this ratio to be 3:1, but as it is desirable to make this proof quite general in its bearing and be applicable also to comatic formations of a higher order, we will assume the outermost point of each comatic circle to be displaced from the central ray by a distance equal to N times the radius of each comatic circle. Secondary Plane Here we may proceed as follows : First we may express the radii R 2 and R 3 of the two-lens zones 212 A SYSTEM OF APPLIED OPTICS Radius of second lens zone. Radius of third lens zone. Ratio between radius of second comatic circle and path of secondary rays pro- jected on it. E 2 . . E 2 and E g . . E 3 in terms of R p the radius of the outermost zone, and of r, the radius /. . c { of the eccentric pencil ; thus = (C . . (C . - 2Er + 2r 2 ; Along the lens zone E 9 (15) (16) . b equal to d . . c , ^2 mark off the arc c l and join d to b by the chord d..b.. Also join a to b by straight line a . . b, and then from the centre c draw c . . e perpendicular to a . . b, and bisecting the latter at e. Then for .the moment we will assume the circle E 2 . . E 2 to represent the comatic circle formed by lens zone E 2 . . E 2 ; in which case we have the ray refracted through the lens zone at c : striking the comatic circle at 5, c : . . b being the torsion imparted to the ray in the comatic circle. Then since c l . .b = c l . .d, therefore the chord b . . d is bisected at n, and angle JCcj = c-fid. But angle bad = one-half of angle bed, therefore angle bad = angle bCc^. But angle bad is also equal to Cba. Therefore angle Cba angle bCc^ Therefore a . . b is parallel to C . . c x and e . . b is equal to C . . n, which latter obviously = C. . . /, so that we have a . . b = 2(e . . b) = 2(C . . n) = 2(C . ./) = 2(E, - r), from which we then derive a . . b 2(E 1 -r) EQ (17) - 2R/ + 2r 2 Turning now to the real comatic circle E/ in Fig. 79d, which is formed by lens zone E 2 . . E 2) we have & x as the point where the ray from G! strikes the comatic circle, and A 2 . . 7^ is obviously parallel to a. . b of Fig. 79c. Now we have already seen, from Figs. 65 and 65a, that the perpendicular to the diagram drawn through A 2 is a sort of axis through which pass all rays from E 2 . . E 2 which intersect the comatic circle ^ . . A 2 . . & 2 . Therefore the two rays in the secondary plane from c 1 and c 2 which strike the comatic circle at & : and k z respectively, will intersect one another at a point somewhere on the perpendicular through A 2 , whose distance from the plane of the diagram can be expressed in terms of A 2 . . & x or A 2 . . Jc 2 . Now clearly a . . b 2r 2 PLATE. XVI Fi<>.78. Screen at Moved in I Moved in Focal Plane. 0-35 millimetre. 1 07 millimetre. 79. a. Screen at I Moved in I Moved in Focal Plane. 0-35 millimetre. 0-7 millimetre. M PLATE. XVI. Fi^.79. Fi^.77 Fi^.78. Screen at I Moved in I Moved in Focal Plane. 0-35 millimetre. 07 millimetre. 79. a. Screen at I Moved in I Moved in Focal Plane. I O'J5 millimetre. I 0-7 millimetre. M VIII GENERAL PKOOF OF THEOREM OF COMA 213 R/ 2(R 1 -r) 2r 2 (18) Length of secondary ray as projected on second comatic circle. If now A 2 . . & 2 is multiplied by - , # 2 being the angle made with tan 6*2 the central ray by any of the rays refracted through the lens zone R 2 . . R 2 , we shall then arrive at the distance within or beyond the plane of the diagram at which the two rays & x . . A 2 and k 2 . . A 2 intersect, and this is the required linear E.G. in the secondary plane. Now if we write l for the angle made with the central ray by the rays refracted through the outer lens zone R x . . R a , then we have, if F = the focal length, C..H tan 6 l = F and if we take tan # x as the unit we have and G..d C..d R -pr- = tan 0^ TT = tan ^ J , u..n i$j C G- = tan "R = tan (19) (20) Therefore the linear E.G. in the secondary plane (from 18) 1 Ri 2 N / R 2 -2R 1 r+2r 2 ', 7^2 tan 6 l ~ _ K , R/ 2(R 1 - r) R x 1 in which, as we have seen, R 2 tan B l ' 2r 2 , so that finally our linear E.C. tan V. with which we have yet to compare the linear E.G. in the primary plane, which we will now proceed to formulate. Primary Plane Here we have to deal with the two rays refracted through the lens at G and H. The ray from G strikes the comatic circle R 3 r at Semi-angle of cone of rays from outer lens zone. Semi-angle of cone of rays from second lens zone. Semi-angle of cone of rays from third lens zone. Linear E.G. in the secondary plane. 214 A SYSTEM OF APPLIED OPTICS Distance between points where the two primary rays strike the plane of the coma. Distance behind comatic plane where the two primary rays intersect. Linear E.G. in the primary plane. the point g, while the ray from H strikes the comatic circle and we first require the linear distance g . . h. Now g..h = NE/ - NE/ = N(E/ - E./) SECT. j at h, !,/, E, 2 - 4E,r + 4r 2 \ | . . 0. . h = N{ E, 1 - - p * - ) -. (21) BI Iii Fig. 79 let j?../i be the plane of the diagram Fig. 79d, and g . . F and h . . F the two rays we are dealing with which intersect at F beyond the plane of the coma p . . h. We have just obtained a formula for the distance g . . h, and now what we want is the linear E.G. correction F . . p measured parallel to the ray through the centre c of the lens and perpendicular to the plane p . . h of the comatic rings. Let x represent this required distance F . . p. First we have the ray H . . h . . F making the angle O l with the centre ray or with F . .p ; the other ray G . . g . . F makes the angle S with F . . p (while each secondary ray c x . . d . . F makes the angle # 2 with F . . p). Thus we have x tan 6 1 -x tan 3 = (g . . h) ; . . #(tan $ l - tan # 3 ) (g . . h) ; _ / ,\ / 9 rtv x ~ (9 ' n ) tan a _ tan ^""T in less than j^-ths of the principal focal length, and that for the image formed by rays in secondary planes somewhat less than ^ths of the principal focal length. But since the extent of image utilised in such cases seldom amounts to more than two degrees from the axis, these curvature errors do not seriously matter, so we have the fact that the principal factor which determines the superiority of one form of objective over another as regards its definition away from the optic But coma at oblique axis is simply the presence or absence of coma. For instance, a double achromatic objective with the collective lens placed first and of a meniscus or convexo- plane form will yield a very considerable amount of inward coma at its oblique foci which, at even five minutes of arc from the axis, is considerable enough to spoil definition; while if. 216 A SYSTEM OF APPLIED OPT [OS Coma causes sensi- trveness to squaring Coma quite avoid- Condition for elimi- nation of coma from a two-lens combina- tion. the collective lens is plano-convex and still placed first, the opposite sort of coma will prevail, although it will not be quite so bad as in the former case. It is also obvious that forms of objectives characterised by strong coma W QJ ^ verv sens jtive to being slightly thrown out of square, a highly undesirable condition, for the mischief caused to definition by such corna may far exceed the mischief caused by the inevitable astigmatism. We cannot get rid of the normal curvature of the images nor the astigmatism in thin contact combinations, but we can get rid of the coma, and therefore it is of the highest importance in the case of telescope objectives, especially when designed for photographic purposes, that they should be designed free from coma, and to that end we may proceed as follows : Formula II. of this Section gives us the angular value of the coma yielded by any lens, so that in the case of the two lenses constituting a telescope objective that is to be free from coma, we have Stan < . % / _ l \ 1 /*lv/*l ~ l / > 9 , -A- J tan G) l AA*1 ~ l ) a i l > x l U. VII. Let F! = +1 and F 2 = -, a : = I, the collective lens being placed first. Then . . 2 = 1, so that a 2 = + 2J. Relation between the x't for no coma. Let fj, 1 =l'5 and /u, 2 =l'6. Then, leaving out common factors, we have from which finally we derive _ q-iq ^ ~ ^ + We may now insert this value of # 2 in our formula for spherical aberration for the two lenses and equate them to 0, thus 8(' 3-25 + 6 -75J ELIMINATION OF COMA 217 ~ 3 ' 43a i + ' 474 ) 2 + 10 ' 4 ( - 3 ' 43 *i + - from which we get ((70-592V 2 - 19-5^ + 1-35)] 1 *V - IK + If - -028^ ( - 83-23^ + 11-502) I = 0, + 22-21 + 6-83 j which reduces to -8Ix*+ 1-21^ + -493 = 0, + x 1 *-l-5x = +'608, x* - 1-5* + (-75) 2 = -608 + -5625, Xl - -75 = N/l-1705 = 1-082 ; .-. a 1 = -75 1-082 = --332 or + 1-832. The first result is the most convenient, as it implies radii in the ratio 4 2 to -= or 2 to 1, in which case we have 3 o = - 3-43( - J) + -474 = + 1-617, \ O/ or radii in about the ratio of ^ : ^7. or + 1 : 4^, which implies a concavo-convex dispersive lens. Among useful formulae is one for the spherical aberration of a single lens free from coma. In order to be free from coma we must have + DO- ) + 0- from which (2/i+ !)(/*- 1) Relation between x x = - - -- r - a- VIII. and a in single lens v* "*" * / free from coma. Then, on substituting this value of x in the formula for spherical aberration, we get j^ 1 ^+2(2^+1)^-1)2 (2^ 8F J5^W^1 " (/, + l) 2 ' a "^ + 1} p. 1 After adding together the three functions of a 2 and reducing, we get 218 A SYSTEM OF APPLIED OPTICS SECT. Spherical aberration V 2 , 2 -, of simple lens free -^ FTT^ 1 ~ a *> ' IX " from coma. olt I (/A - I ) 4 which is a simple expression for the spherical aberration of a lens free from coma. We have seen before that a simple lens gives the least possible spherical aberration when Condition of least \x _ _ (p + 1)0"- _ _ spherical aberration. u + 2 Then X -VIII ..-*, _ . Difference between / 1 > conditions of least . " ._ ^ _ M/* ~ * / VT spherical aberration in + !)( + 2) and no coma. If a = 1 and yu, = 1'5, then the above 75 -75 1 -(2'5)(3-5) ( " -^75 = "lif B6 ' so that the difference between the two values of x required to fulfil the conditions of freedom from coma and minimum aberration is only a small one. Let us now consider the lens from another point of view. Suppose we wish the lens to satisfy the condition that if a varies or the vergency of the entering rays alters, then the spherical aberration shall remain constant, or, at any rate, vary in the least possible degree. We must then differentiate the spherical aberration formula with respect to a, and we have Differential of spheri- -> 2 i c cal aberration with d a -(A')y^ = ^- - -Ufa + l)x + 2(3,* + 2)fc. - l)a }da, XII. respect to a. of (J-(p- - 1 ) ^ which equates to when Condition of con- stancy of aberra- w _ (-V + 2)(/x - 1) tion when vergency 2(u + 1) "' AJ.11. varies. ^ Here again it is instructive to compare this formula with VIII. and X. For instance, we find that xni -vm - -(V 20* +1) 2(/ +1) ' Difference between condition of vergency . \\\ _ \ _ P-\P- ~ 1 ) YTV insensitiveness and ~2(u+l) a -A.lv. of no coma. '75 If a = 1 and //,= 1'5, the above = - = '15; and again we VIII 219 find there is not a very great difference between the values of x for fulfilling the two conditions of constancy of aberration when a varies, and freedom from coma. Of course, the same methods may be ex- Discrepancy between tended to compound lenses such as telescope objectives, and it will be very 'great; 1 ' found that the form of objective which we worked out as free from coma with x l = '332 will also not differ very seriously from the form of objective necessary to give the least possible change in the spherical aberration when a varies, as, for instance, when the entering rays become slightly divergent instead of parallel. To fulfil this condition x l would have to be about '40. Thus there is not such a large discrep- ancy between the two conditions as has been asserted by some writers. Spherical and Parabolic Reflectors at Open Aperture We have already had several instances before us of the conversion of any formula relating to refraction into the corresponding one relating to reflection by simply inserting the value 1 for p. In this case, also, it will be found that the formula for coma at the oblique focus of a spherical reflector at open aperture may be obtained from the Formula II. for the angular value of the coma for a lens of open aperture. The latter formula was 3 tan (f) 1 ( . -> , _v ~ Trrio / Tvl \"I J ' + 'A/ jt ~ i)o. + lfJ,+ 1 )X (A . \"") 4r ' fji^p, 1) v. i Here there need be no ambiguity about the meaning of x in the case of the above formula, since (//,+ 1) becomes = 0, while a is 1, as in the case of the lens when the entering rays are parallel, while it is if the rays are diverging from the centre of curvature, and + 1 if they are diverging from the principal focus. Our formula therefore becomes 3 tan XV. Let it be supposed that the semi-aperture A is I foot, and the principal focal length 20 feet, and entering rays parallel as usual, so that a = 1 ; then the coma will be + and outward, and its angular 3 1 amount tan 1flnn . If tan = - , then at 2*4 inches from the axis 1 oUU 100 we shall have coma whose angular value at the mirror centre will be 160TOOO' and ltS Hnear Value Wil1 be 160-000 = 261,6 th f a f 0t r ^ part of an inch, a very small quantity. Angular coma in case of central ob- lique reflection. SECTION VIIlA Central oblique refraction excep- tional. Two sorts of coma. COMA AT THE FOCI OF ECCENTRIC OBLIQUE PENCILS So far we have got the universal Formula IT., giving the angular diameter of the longer axis of the comatic flare (as subtended at the centre of the lens) on the assumption that the principal ray of the oblique pencil passes through the centre of the lens. But in the numerous cases of systems of more or less separated lenses it is the exception rather than the rule for central oblique refraction to take place ; in most cases the principal rays of such pencils are refracted through the lenses at considerable distances from their centres, and as it is highly important to be in a position to eliminate coma at the oblique foci of such lens systems, we must therefore work out the formulae appropriate to the eccentric oblique pencils refracted through them. In the first place, a very little consideration will show that there are two sorts of coma, or rather coma caused in two different ways, to be dealt with in the case under consideration. First, there is coma which is simply part of the general coma already dealt with, which may be present in the lens and show at full symmetrical aperture. Second, there is coma resulting from the presence of spherical aberration Direct axial pencil in the central oblique pencil. Indeed, this sort of coma may manifest i tse lf m the case of a direct axial pencil limited by an eccentrically placed stop. For instance, let Figs. 80 and SOa represent an uncor- rected lens with an axial pencil, refracted eccentrically through it, owing to the presence of the circular but eccentrically placed stop. Then let Figs. 81 and 8 la represent cases in which the pencil is obliquely refracted by the lens, but the stop is central and of an aperture allowing of the same aperture of the pencil where traversing the lens, as in Figs. 80 and 80a. Then such oblique pencil is subject to the same spherical aberration as the axial pencil of the same aperture ; but we will suppose that there is no coma of the sort that we have yet 220 * ' COMA OF ECCENTETC OBLIQUE PENCILS 221 dealt with ; in other words, we will assume that the lens gives symmetrical oblique refraction. Of course, it will also give con- siderable astigmatism, but for the sake of simplicity we will assume the astigmatism to be absent and the focus to be exactly the same as for the axial pencil. It is at once obvious from the Diagrams 80 and 81 that there will ensue an eccentric formation at the focus whose structure in the primary plane is perhaps more clearly shown in Figs. 805 and 816. Suppose we arrange our stop s . . s so as to pass the central ray at one extreme of its aperture, and the outer ray at the other extreme of its aperture, as shown in Fig. 81, and that we place a ground glass screen perpendicular to the optic axis at the point/ where the extreme outer ray passed by the stop intersects the centre ray. Let Fig. 82 represent a view of this screen when looking towards the centre of the lens, a . . & . . c the periphery of the lens, and d . . e . ./ the outline of the eccentric pencil where it traverses the lens. We can then plot out the figure thrown on the screen or plane of the diagram by the rays which are refracted through the lens at points in the zone d . . e . ./ of the eccentric pencil, in the following manner. From /, which is the point where both the centre ray Q . ./ and the ray from g (the other extremity of the eccentric pencil) strike the screen, radial lines may be drawn to as many points in the circumference or zone d . . e . ./ as may be desired, say points every ten degrees apart as measured from /. Then the lengths of these lines from / to the points where they cut the eccentric zone d . . e . ./. . g will give the values of the y's or the distances from the lens centre of the points in the lens where ea-ch ray is refracted, from which the relative longitudinal spherical aberrations of such rays may be calculated, and from those the distances from the central ray / to the points where each ray cuts the screen or the plane of the diagram. It is obvious that all such displacements on the screen must take place along the radial lines drawn from / ; all rays, except the extreme one, cut the central ray through / at points on the latter situated farther from the lens in calculable degrees, that is, at points nearer to the observer. Having worked out the point on each radial line where the corresponding ray from the zone d..e..f..g cuts the plane of the diagram, and joining all such points together, we obtain the curve shown, which is exactly the same sort of curve as in Fig. 76 P, resulting from coma combined with astigmatism. For it is evident that while we are at the focus for the two extreme rays from the zone contained in the primary plane, yet we should have to retreat farther from the lens before we arrived at the focus for the two rays from w l and w^ on the Symmetrical oblique refraction assumed. How the comatic loop is derived. The result is an comatic 222 A SYSTEM OF APPLIED OPTICS The same loop de- rivable from the axial pencil with eccentric stop. zone which are contained in the secondary plane and strike the comatic loop at w-l and iv 2 f . Hence there is astigmatism introduced by the selective action of the stop. We have already seen from Formula VI., Section VI., for E.C.s, that if we place a diaphragm in front of a collective lens having positive spherical aberration, so as to cause a pencil to traverse the lens eccentrically, then the E.C. consequent on spherical aberration will always be positive ; that is, the intersection point for rays both in primary and secondary planes will be brought much nearer to the lens, and by three times as much in primary planes as in secondary planes, which last condition implies the existence of the astigmatism which we have independently arrived at in Fig. 82. It is obvious, also, that the comatic curve obtained in Fig. 82 may be derived also from the case of Figs. 80 and 80a; but of course the combination of an axial pencil limited by an eccentric stop does not occur in practice. Now let be the point on the screen where the principal ray Q . . h, or the ray through the centre of the stop or of the eccentric zone or circle d . . e . ./, cuts the plane of the diagram ; then the line . ./ will be the length of the whole comatic formation in the primary plane, for any comatic curves traced out by rays from smaller zones than d..e..f..y will all be found to lie between and /, as in Fig. 76 P. Investigation of the Coma due to General Spherical Aberration Construction. We may now proceed to work out a formula for the length of such a comatic formation in the primary plane in the following manner. Let Figs. 83 and 83a represent a case of an oblique and eccentric pencil, limited by the stop s . . s, refracted through a lens at a. The origin or focus of the oblique pencil is Q, and its focus for rays ultimately close to the oblique axis Q . . a . . j is at a v Let the ray Q . . k grazing the lower edge of the stop focus at b on the oblique axis, the principal ray Q . . c passing through the centre of the stop focus at c a , and the other extreme ray Q . . t focus at d, so that % . .b, % . . c l} and ! . . d are the longitudinal spherical aberrations, being therefore proportional to (a . . &) 2 , (a . . cf, and (a . . t) 2 respectively. Let the angle of obliquity P . . a . . Q or < be measured at the lens or element centre as usual. Let / be the point where the two extreme rays Q . . t and Q . . k passing the stop intersect, and through / draw e . ./ . . g perpendicular to the optic axis P . . a. Then the size of the comatic formation is evidently at a minimum in e.,f..g, where the two extreme rays in 83. c. 83. c. VI HA 223 the primary plane focus, and the total length of the coma is obviously given by e . ./. Let P . . a = U, and a . . a-^ referred to the optic axis be V as usual. Let the semi-aperture of the stop be S and the semi-aperture of the pencil where it traverses the lens be A. Let the vertical distance from a to c, where the principal ray cuts the lens, be L,* and let the distance of the stop from the lens = D. Let the formula for spherical 2 aberration be stated shortly as |^ (A ), in which / is the principal focal length of the lens. For y we shall have in turn to substitute various other values. Then we have the following expressions for the longitudinal spherical aberrations : Then we have the following relations : (24) in which (d . . g) = (b . . e) - (b .. . d) = (b . . e) - {(^ ..d}- (% . . 6)}. V2 V 2 (25) V Also from (24) It is clear that L is the same thing as the eccentricity C of Section VI. 224 A SYSTEM OF APPLIED OPTICS (26) Also (e.. y) = ..e)- (b . . cj= (/>.. e)-(a l ..c 1 ) + (a l .. 9) = (A' (27) Now Formula for the length of the aberra- tion coma. 1 ^ . '. (e . ./) = ^(A')(3^ 2 )LV. (28) Its length varies as Hence e . ./, or the length of the coma, varies directly as the eccentricity L. Formula (28) may be put into more general and 9F 1 T) convenient form by substituting - for V, and U tan T ^ or ~ a Generalisation of the formula. tan - for L, and then we get (../) = 8f* *) tan f)Tji ' Now since the diaphragm is nearer the lens than Q, then /3 in the above Formula (29) will be of positive value and numerically greater than a ; therefore /3 a will be positive. Also, since V is positive and real, therefore 1 a will also be positive. Also A' is positive, there- fore e . ./ will be positive also. But we shall find it convenient to treat e . .f as a negative quantity, for the corna is obviously inward coma, a flare lying towards the optic axis ; e is the position of the centre or principal ray of the eccentric pencil, and therefore e. ./is a diminution of the distance from the optic axis. We must therefore reverse the sign of e . ./ by writing COMA DUE TO SPHEEICAL ABERRATION 225 or, in full, (e . ./) = tan . ./) = _ (A/) tan Zf a -i)Wr (3 / ,+ 2)(/.-l)a 2 1 1 Full formula for XVI. length of the aber- ration coma. But the most convenient formula of all is one expressing the angular value of e . ./ as viewed from the lens centre, which is of course obtained 1 1 - a by multiplying the above formula by y or by ^r- t by which we then get e..f 1 s ^ i i f 5- j, - U a - YVTT XV il. Universal formula for the angular va i ue O f the ab- erration coma. Fig. S'3a and c illustrates the analogous case of a dispersive lens in which also ft is + and numerically greater than a, so that a ft is again negative and therefore gives a minus value to Formula XVII. This is as it should be, for it is plain from the diagram that the coma produced is again inward or towards the optic axis. Since the formula is a function of 7^, it is evident that the sign of /has no influence on the sign of the result ; in fact, the sign of the lens is really implied in the value of a /3. Thus it will be found that Formula XVII. is universally true of all cases. We may now turn our attention to the case of the coma of eccentric and oblique pencils consequent upon coma proper. Investigation of the Coma Proper at the Foci of Eccentric Oblique Pencils Fig. 84 represents a case of a collective lens giving pure inward coma at the focus of a central oblique pencil, spherical aberration and astigmatism being ,supposed to be absent, while Fig. 84a represents the corresponding case of a dispersive lens. Fig. 846 shows on a larger scale the structure of the focus for the collective lens. As in the last case, A is the semi-aperture of the eccentric pencil where Construction. it strikes the lens. /=the principal focal length of the lens; L = the eccentricity or the height A . . C from the lens axis at which the Q 226 A SYSTEM OF APPLIED OPTICS SECT. principal ray strikes the lens. Q . . A is the central oblique ray passing through the lens centre at an angle of obliquity = ; & is the point where the extreme ray Q . . k . . b passing the stop s, and nearest the lens centre, intersects or focuses on the central oblique ray ; c is the point where the principal ray Q . . C . . c focuses on the central oblique ray ; and d is the point where the extreme ray Q . . t, passed by the stop s . . s and most remote from the lens centre, intersects the central oblique ray. Then the two extreme rays passed by the stop, Q . . t and Q . . k, intersect one another at the point /. Through / draw e..f..g perpendicular to the optic axis; then e..f is the length of the coma at the focus of the eccentric oblique pencil as limited by the stop s . . s. Referring back to our method of finding the length of the coma yielded by the open lens (not shielded by any stop), we obtained a formula (4) having its application to Fig. 61. This formula expressed the eccentricity correction to be applied to ^ in order to convert it 1 v into j- for any given semi-aperture A of the lens, on the supposition C . . it that the hypothetical stop was always so placed as to just pass the central oblique ray and the other ray cutting the lens at the semi- aperture A from the lens centre. We may apply that formula again in the present case of Fig. 83 or 84. It was ... ., , ., 3 tan < which we may write shortly as -p, 2 In the present case it is obvious that the linear distance a . . b is the above eccentricity correction - ~ 2 (C f )A, with the semi-aperture A . . k or ~LA substituted for the former A, and the whole multiplied by V 2 , so that ( . . b) = 3 ^^(C'XL - ^)V 2 . (30) Likewise and (32) We may now proceed in a manner analogous to the last case. We have PLATE. XVIII t o. PLATE. XVIII A' vnu COMA PKOPEK OF ECCENTKIC PENCILS 227 /, \ J-J -^i / / \ i -i \ J-J "- (b..e)-^-=(f..g) = (d..g)- T -, (33) in which (d. .g} = (6 . . e) - (b . . d) = (b . . e) - {(a ..d)-(a.. &)}; .-. (d..g) = (b..e)-{(L + A)-(L-A)}^j(V)V*-, (33a) . -. from (33) and (33), (b . . e^A V =- 2A (i.. 6 ) = (L + ^)|(C')V 2 . (34) Also, from (33) V y/ v y y // /\ " tail

Thus the interior functions in the formula are closely analogous to those in the formula for E.C.s, VIII., Section VI., only 4F 2 replaces 2F, and _ replaces . 7 ^ > , while the comatic function is reduced to a-fi (a - p) z a half. If the same processes are followed in the similar case of the dispersive lens, exactly the same formula will be arrived at, provided VIIlA COMA OF ECCENTKIC PENCILS 229 that our convention is adhered to which makes inward coma or flare towards the optic axis negative, and outward coma positive, irrespective of whether the lens in question be collective or dispersive, for, as we have seen, that matter really tells in the sign of a ft for the lens in question. A good test case for the correctness of signs in Formula XX. in their application to collective and dispersive lenses is one in which a plano-convex collective lens is placed in contact with a concavo-plane dispersive lens of the same radius of curvature and of the same index of refraction. Thus it is clear that, especially if cemented together, the two lenses will merely form a parallel plate of glass, and act as such. Then the Formulae XX. for the two lenses will in this case be found to equate to in all circumstances, since a 2 = a i > A = & , and x 2 = x v and therefore (a., /3. 2 ) = (a l &). Coma in Relation to E.C.s and Normal Curvature Errors, etc. Some Interesting Corollaries. Many important deductions may be drawn from the formulas arrived at in this and previous Sections. 1. Supposing that in the case of eccentric oblique refraction through a simple lens the E.C.S are eliminated, leaving the normal curvature errors of the lens intact, then what will be the result as to the presence or absence of coma at the foci of oblique pencils ? Such a condition has often to be fulfilled or closely approached in Cooke lenses. First of all we have for the elimination of E.C.S from a lens the condition if -2 a-f3) C'\ = from which we derive XXA. On the other hand we have for the elimination of coma from a lens under the same circumstances the condition tan from which 4/ 2 */_ 1 A' XXA. XXB. Condition of elimi- nation of E.C.s. Condition of elimi- nation of coma. 230 A SYSTEM OF APPLIED OPTICS SECT. Formula for coma when E.C.s are eli- minated. Condition of elimina- tion of astigmatism. Formula for angular coma when there is no astigmatism. Case of a lens giving Hence it is clear that when E.C.s are eliminated there will be a preponderance of spherical aberration coma at the foci of eccentric oblique pencils, for which the formula will be f an A i 3 A 2 - --?- 0/ . .A.'. 4/ 2 2(o - 0) XXc. 2. Let it be supposed that the E.C.s are so arranged as to neutralise the normal oblique astigmatism of the lens, then what will be the condition of the oblique foci as to coma ? For the elimination of astimatism we have from which we derive and A'-2(-0)C' = -<-# p, _ A' + (a - /3) 2 2( a - #) AAD. as the condition of no astigmatism. If now we insert this value of C' into the above formula for coma, XXA., we get ' / \ a /3^' > ^ P> \ 4/ 2 t 2 (a - ft) ) XXE which expresses the angular value of the coma when there is no astigmatism. Then it is clear that if A' = (a /3) 2 there will be no coma at the foci of eccentric oblique pencils. Fig. 85 illustrates an example of this case which will be alrea( ty familiar to many readers. It is the case of a plano-convex lens of crown glass receiving parallel rays passed through a stop F fixed at a distance D' = - in front of it. Here, the refractive index O being 1'5, it is clear that after refraction by the first plane surface (P*\ = -), which z/ is then the centre of curvature of the second surface, and therefore the principal rays will all impinge upon the second surface as if viii, COMA IN KELATION TO E.C.s 231 diverging from the centre of curvature, and will consequently meet with perfectly symmetrical refraction, and there will be neither astig- matism nor coma at the oblique focus/. Here we have x= I, a = I, $=4-5, and (a /3) = 6. A' works out to ~(l + 10 + 3-25 + 6-75J = -^(27) = 36. '75 v ) t o C' works out to = -6. Therefore if we insert these values of A' and C' in above Formula XXE., we then have tan 0J-36 -( - 6) 2 | Formula for coma 4P I 2(-6) "/ = But it is evident that other conditions may be found, leading to no astigmatism, which will yet permit of the presence of coma, especially when u is less than F and v negative, and therefore a greater than -f 1. We may now inquire what will be the formula for E.C.s when coma is eliminated. The formula for E.C.s in the primary plane is 3tan 2 4>( 1 1 and if for C f we substitute its value from XXB., which holds good when coma is eliminated, we then have 2/ and therefore the E.C.s 3tan 2 r 1 ,\ Formula for E.C.s . ( A > . X X T^_ 117 It a Tl //\YM Q Id AllTYlin _ .. TT -/ - 75v>- ( AXF. when coma is elimin- / ( - PY ated. Lastly, we have the formula for astigmatism, in which we may substitute the value of C' which holds good when there is no coma, and we then have f JL_ \a- _ 1 (a-/?) 2 (a -fi 232 A SYSTEM OF APPLIED OPTICS SECT. Formula for astig- wn j cn fi na H y tan 2 , ,/ _ oy _ A /, matism when there = _L_L J v "' " I XXG is no coma. / t (a - B)' 2 ) ' In the course of the preliminary planning out of optical systems, such generalisations as the above are often useful. Application of the Formulas to a Series of Separated Lenses We saw that the formulae for eccentricity corrections were functions of tan 2 fa and had to be multiplied by V 2 or F 2 in order to reduce them to their longitudinal value as corrections to the focal length, and that in adding together the functions for a series of separated lenses no notice need be taken of the successive modifications of the angle for the different lenses, all that was required being the simple- algebraic sum of the corrections for all the lenses ; so, in the case of a series of separated lenses we may in the same way apply the Formula XIX. for coma directly to each lens in turn, for the formula is a function of tan < simply, and the linear amount of coma yielded by each lens is obtained by multiplying by V. Fig. 85 shows a lens L giving a certain length of coma e . ./. It obviously makes no difference to the linear value of e . . f whether we assume it to be referred to the point C at the centre of the lens and in terms of tan fa, or to the point D and in terms of tan 9 . For supposing Formula XIX. gives us a certain value M tan fa for the angular value of the coma as viewed from C ; then, supposing C . . F = V, the linear value of the coma is simply MV tan <^ . If, on the other hand, we assume that D is the position of the back lens of the combination and that V or y C . . F = n(D . . F), or D..F=-, then obviously tan fa = n tan fa, and therefore the length L of the coma referred to the point D is given by y L = (M tan <^)V = M(n tan ^)- = (M tan 2 )(D . . F), IV which is the same result. But it is clear that the semi-aperture A of the oblique eccentric pencil where, it traverses each lens in turn must be carefully inserted. For brevity let us write Formula XX. as simply 3 tan d> 1 then for two lenses or elements in succession, whether separated or not, the formula will take the form VIIlA 3 tan 3 tan 233 Formulae for coma XXI ^ or two l enses * n and for three elements or lenses in succession, whether separated or not, 3 tan d) 1 , . xn ,} . , l - (a, - ft)^ K 2 and so on up to any number of lenses or elements in succession ; the semi-aperture of the pencil where it traverses each lens or element plane being expressed in terms of the semi-aperture of the pencil at the first lens or element plane of the series. successon. Formulae for coma for three lenses in succession. Coma produced by Oblique Refraction through a Parallel Plane Plate However, our formula for coma is not yet quite complete, for in the case of thick lenses we have to deal with two elements and a parallel plate, and we must now work out a formula for the coma produced when a pencil of converging or diverging rays is refracted obliquely through a parallel plane plate. That spherical aberration coma is produced in such a case is evident from the inspection of Figs. 86a and 8Qb, and also from experiment. Let A. . h be the second surface of a piece of parallel plane glass of Construction, thickuess = t and refractive index = /i. Let b . . K and d . . H be the two extreme rays of the oblique pencil, and c . . E the middle or principal ray of the same. Let a be the focal point for the rays ultimately close to the normal Q . . A, which, if the pencil were in- definitely extended, would be a ray perpendicular to the plane surfaces. Then we must imagine that the origin of the pencil or the point from which all the rays originally start is at a point Q on A . .a produced backwards and at a distance to the left of a equal to t- , and the P diagrams chiefly represent the course of the rays after emergence from the second surface. Then, as we have seen in Section IV., page 79, 234 A SYSTEM OF APPLIED OPTICS SECT. the rays are subjected to a negative aberration which, as a correction to - or T - , was found to be v, .' a?, in which a 9 was the distance v A. . a 2p s tr of each ray, where it cut the second surface, from the normal ray A..Q. On multiplying the above formula by v 2 we then get the longi- tudinal aberration for any ray, so that we have (A..&)= ( -- is negative and the coma is inward. But if the optic axis is at 3 . . O 3 , then both v and D" are negative ; but D" is greater than v, so that v- D" is positive, and the coma has become positive. This device covers the case of the parallel glass plate, supposing it is either a detached and independent unit in a lens system with an air- space on either side of it, or if it forms part of a convexo-plane or concavo-plane lens, in which case no element occurs at the second surface. Case wherein an But if, as is usual, an element does occur at the second surface, tlien we nave onl y to refer to those data wnich nave had to ^ e worked out for the various elements in order to find a simple sign determinant in the form of (u D')/ for that element which occurs at the second surface. If the element is a collective one, then/ is entered as positive, but if a dispersive one, then / must be entered negative, while the u arid the D' must be entered with those signs prefixed which have been vim COMA BY OBLIQUE KEFEACTION THROUGH PLATE 237 already assigned in accordance with the conventions laid down on pages 148 and 149, Section VI. Thus, then, when no element occurs at the second plane surface we have the formula = 3 tan x with (r-D xv ) as sign determinant : XXIIlA. or if there is an element at the second plane surface, then = 3 tan x 9 3 ./ A' 2 , with (u - D')/ as sign determinant. XXIIlB. ' The aberrations in the diagram are much exaggerated, for clearness, and the crossing points for principal rays are of course determined by formula? of the first approximation only, all aberrations being ignored. In this way the point where the principal rays of pencils entering a lens system cross the axis (generally the stop centre or its image) is determined in the first instance; and supposing, as usual, that the angle made by the principal ray with the optic axis at the first element is i/r, then the angle ^ which the same principal ray makes with any particular parallel plane plate may be obtained in the way described on page 179, Section VII., where it was shown that if n elements precede any given parallel glass plate, then tan x = tan ^ f^k^V ' ^ , etc. ; * J \ * J y, - L 'n while the semi-aperture A of the pencil where it cuts the second surface of such parallel glass plate may be obtained in the manner described on page 103, for it is the same thing as the semi-aperture y for the axial pencil. Supposing there is an element at the second surface of any given parallel glass plate, and it is the ?ith element of the series, then (A^=( U ^--^A*. (48) \VjV 2 . . #_./ If there is no element at the second surface, then the focal distance v of the emergent pencil may be specially assessed with respect to the second surface of the parallel plate, as also the focal distance (or D") for the principal rays in accordance with Formula XXIIlA. Application of the Formulae for Coma to two Actual Lens Systems We will now conclude this Section with two examples of the actual application of the formulas for coma to two of the photographic lenses that we dealt with in Section VII. Parallel Plane Plate. Formula for angular coma with no element at second surface. Same when element occurs at second surface. 238 A SYSTEM OF APPLIED OPTICS Stellar Cooke lens First we will take the Series Ic Stellar Cooke Lens, whose curves of 43-in. focus. an( j a n other data were given on page 182, Section VII. Kecapitu- lating, we have the formula for E.C.s in the abridged form (in secondary planes) while we have seen that the formula for coma is (50) from which it is seen that if we take the function 20' already worked out for the E.O.s, and divide by two, and multiply the whole formula by *, and substitute tan $ for tan' 2 <, then we shall arrive at the angular comatic corrections for each lens or element in turn. Proceeding in this way we get, taking each element in succes- sion (the actual sign of a /3 being indicated over each) First element. -00784 - tan <. Second element. Third element. E ( + -002597 + -003820) 3 ^ 2 P** A *(*-*} = + -02775 tan 2 /2 i' -0002383 E, ,- 8 = ~ ' 07709 tan 2 /8 ^^l Fourth element. Fifth element. Sixth element. ( + 0000437- -0007269) -00046152 -011511 - 3(a * _ E 5 2/ 5 EL = + '02863 tan <. = - -006145 tan = + . 00 533 tan vim APPLICATIONS OF COMA FOEMUL^E 239 E 1 + -02129 tan (/> E 3 -'07709 tan E 2 + -02775 E 5 - -00614 * " - '08323 tan ,, . + '08300 tan < - '00023 tan = total for all elements. Total angular coma for six elements. This result implies a minute amount of inward coma at the oblique focus ; but we have yet to work out and add in the parallel plate comatic corrections. First Plate For the first parallel plate we have the angular coma (/A 2 - l)t, g = 3 tan Xi with (w 2 D./)/ 2 as sign determinant, which " wherein tan -v/r = tan , as the original object plane is infinitely distant. This formula gives - '0000761 75^* tan 4>, (u s - D/)/ 9 being (-)( + ) Second Plate For the second parallel plate we have the angular coma 0* 2 -lXa^a = 3 tan x 2 ^ o s ^4 2/**t^ with (w 4 U/)/ 4 as sign determinant, which - 3 tan cA D l /D 2 /D 3' (^ I !)^ o tan ""-" ~-" (which gives the result - '00 0005 878^ tan <, O 4 -D/)/ 4 being Third Plate For the third parallel plate we have the angular coma with (u 6 D 6 ')/ 6 as sign determinant, which D/D'D'D/D/ (u 2 -P = o tan ~->* - ^, r ^i~~77-^r v D 1 / 'D a ' / D 8 "D 4 // D 8 " 2/,% ( 240 A SYSTEM OF APPLIED OPTICS , SECT. which gives the result + -000000 72 15A 1 2 , (M O - D 6 ')/ being ( + )( + ). So .we have - -000076175^^ for L x - 000005878^ 1 2 for L 2 2 for L + L 2 + -000000722^^ for L co^ecSons'for three Total - - '000081331^ tan c Add previous total ) , . , , i \~ - "00023^- tan d> from the six elements J l r Final total. - '00031 1^ 2 tan Total angular value of the coma at final focus. On multiplying this result by (E.F.L.) tan A? we shall get the linear value of the inward coma at any angle from the axis. Let tan < = for about 5 degrees, A l = 3 inches (the full aperture was 6^ inches), while the E.F.L. is 43 inches, then our multiplier is Length of the coma (43)^(9) = 32J, and ( - -00031)(32) - - '01 inch. This is more at 5 degrees from the axis. than the coma which was sensibly inward actually measured ; indeed, at about 7 degrees from the axis there was no coma at all. The existence of just perceptible inward coma at from 1 to 6 degrees from the axis, its absence at about 7 degrees, to be superseded by more and more out- ward coma as 10 to 12 degrees was approached, was a characteristic which manifests itself in the final image of many such combinations, and is explained in exactly the same way as we explained the Positive coma of a existence of zones of aberration. For besides the comatic corrections of the order tan , for which we have worked out the formulae, there exist comatic corrections of higher orders whose formulae will be more complex in inverse ratio to their relative numerical importance. Hence, if we refer back to Fig. 39 and let the curves represent two orders of comatic corrections which are left over at the final focal plane and are equal and opposite at any given distance from the axis, so as to bring- about absence of coma at that point, then at a point somewhere between that neutral point and the axis there will occur a maximum of coma of the same character as the lower and most important order of coma for which we have worked out the formulae, while outside of the neutral point the coma of the higher order will more and more prevail. In this case we have slight residual negative coma of the order tan pitted against residual positive coma of the higher order tan 3 , so that inside the neutral point slight inward coma prevails, and vim 241 outside of it outward coma prevails, and would show up much more strongly were not the effective aperture of the combination for oblique pencils largely reduced by the obliquity. For our second illustration we will fall back upon the process The Cooke Pro- lens, Fig. 59, whose radii, etc., and KC.s are all given on pp. 185 cess Lens. and 186. Here again, in order to convert the eccentricity corrections for each element into comatic corrections, we must first halve the inside comatic E.C.s, and then multiply the whole aberration E.C.s plus half 3A 2 (a B) the comatic E.C.s by - , and substitute tan for tan 2 < ; we *J then obtain the following comatic corrections for each element in turn : ( + '0063848 - 000536Q) 3 ( ai T^ 1 ^ 1 2 tan = + -051 125^ tan 0. First element. ( + -0018885 + 2/ 2 1 2 tan = - '1559QA* tan . Second element. l ( + -0000786 - -001 7803) " A*( 2 3 tan < = - -088849^* tan = + "81219^2 tan d>. Fourth element. ( + -3120633 - -383254) 3(as " 2 /5 \ViV = - '71453 J, 2 tan V l V Z V ^ V E x + -051125 E 2 - -15590 E 4 + -81219 E 3 - -088849 E 6 + -080489 E 5 - -71453 tan <> = + -080489^^ tan ^>. Sixth element. --95928 +'94380 -'95928 Total =- '01548^^ tan . Total angular coma for six elements. We have yet to add the three parallel plate corrections. In this 242 A SYSTEM OF APPLIED OPTICS Total parallel plate corrections for three lenses. Final total. case we supposed the rays constituting the pencils entering the first lens to be parallel ; therefore tan i/r = tan <. First Plate The formula for the first parallel plate is therefore (54) with (u. 2 D 2 ')/ 2 as sign determinant, which gives us *007877-4. a tan <>. Second Plate f n tan (55) ^) 1 "D 2 "D 3 " 2/A 3 M 4 2 with ( 4 D 4 ')/ 4 as sign determinant, which gives us + '0023 2 9^4^ tan . for, Uall Third Plate D/D/Dg'D/D/ 0**-! /x 6 (56) (with u 6 D 6 ')/e as si g n determinant = +'0000327^f tan 0. Summing up we have for second plate + '002329 for first plate . . - '007877 for third plate + '000033 for second and third plate + "002362 + '002362 Total plate corrections - '00551 5^ x 2 tan Total from elements . - '01548 Final total . - '02100^ 1 2 tan Supposing A 1 the semi-aperture = ^th inch, and tan < = '25 (for about 14 degrees), and the E.F.L. = 8'5 inches, then the linear amount of the inward coma at 14 degrees from the optic axis will be given by Length of the coma ( '021)(-)( )(8'5) = '00178, a very minute amount of inward at 14 degrees from the axis. coma. As a matter of fact, there was a just perceptible inward coma at that angle of obliquity visible with a high-power eye-piece, while the lens showed unusually free from comatic aberration of higher orders. It will be noticed that the parallel plate in the first lens gives a comatic effect about 3 '5 times as strong as the much thicker second plate, owing to the fact that the rays are converging more strongly through the first plate than they are through the second plate. APPLICATIONS OF COMA FOEMUL^ 243 These particular instances do not show relatively very strong Coma for parallel comatic corrections for the parallel plates, and these might legitimately be neglected ; but cases of much thicker lenses may sometimes occur, or cases in which the divergence or convergence of the rays through the plates is relatively very much stronger, leading to very serious comatic corrections which cannot be neglected. Such cases are, perhaps, the most likely to happen in microscope objectives ; so that our formulae for coma of the order tan would be incomplete without those applying to parallel plates. Coma at the Foci of Eccentric Oblique Pencils Reflected from a Spherical Mirror In the case of the spherical reflector, we may occasionally have to deal with the eccentric oblique refraction of pencils, as occurs off the small concave or convex spherical mirror of the Gregorian or Cassegrain reflecting telescopes. Here again if we take Formula XX. and substitute 1 for , we then arrive at the formula e..f 3 tan XXIV. which is the universal expression for the angular value of the comatic flare subtended at the vertex of the mirror ; the vergency characteristics a and /3 being assessed in accordance with the usual conventions, and A being, as usual, the semi-aperture of the pencil where it impinges upon the mirror. Angular coma at foci of eccentric oblique reflected pencils. SECTION IX DISTORTION AND RECTILINEAKITY OF IMAGES VON SEIDEL'S FIFTH CONDITION The simplest case of image projection by a pinhole. The pinhole replaced by a thin lens. WE now have to consider another very important condition which has to be fulfilled by any optical combinations that are designed to project on to a plane surface images of exterior objects which extend to many degrees from the optic axis, and are at the same time required to resemble the original in the sense that the linear distances of image points from the axis point shall be strictly pro- portional to the tangents of the angles that the corresponding points in the original subtend at the front apex of the lens or at any other point on the axis of the same. Fig. 87, Plate XIX., illustrates the ideally simple case of the projection of an image of the original flat object B . . C on to a flat screen b . . c by means of a pinhole P. Let A . . a be the straight line drawn through the pinhole P perpendicular to both planes B . . C and b . . c, and we may regard it as the optic axis ; let C, D, and B be three points in the original whose images are projected, in straight lines, to c, d, and b, their respective image points ; then we have a..b A..B a..d A..D b ~ Jt> .d and c A..C -^-p , and also the tangents a..P A..Fa..p-A..P,..p-A..P' SOthatWen0t nlyhaVe a constant ratio between all radial distances in the image and all radial distances in the original, but also a constant equality between the tangents of angles subtended by points in the original and the tangents of angles subtended by the corresponding image points. And it is clear that these relations will continue to hold good whatever may be the ratio between what we may term the focal distances A . . P and P . . a. Next we may suppose the pinhole to be enlarged, and a small and very thin collective lens to be inserted in it, after which we shall have 244 SECT. IX GENERAL THEORY OF DISTORTION 245 the relationship ~ = ^ - T ~ if the most distinct image is to be X . . (I _C J\. . . S? projected on to b . . c. But we are passing our narrow pencils of rays through the centre of the lens in this case, and, as we have seen in Section L, rectilinear projection ensues with reasonable accuracy throughout a very large angle of view. But let us go further and suppose that we have two separated lenses, as in Fig. 88, which we will suppose to be plano-convex with their convexities turned towards one another, and of equal powers. Let there be a screen or stop placed half-way between them to compel the effective pencils to cross the axis at S, the geometric centre of the system, and let the two conjugate focal distances A . . L t and L 2 . . a be equal, so that the image is equal to the original. Let B..C..S..D..E be the course of an oblique principal ray from B. Let p l be the first principal point, being the image of the stop centre S as formed by the lens L : , and presented to outward view ; and let p 2 be the second principal point or the image of the stop centre, similarly formed by the lens L 2 . We are now going further than we did in Section I., and must therefore take notice of the spherical aberration of the two A. B lenses, for we are supposing the angle of obliquity -r- 1 to be consider- A . . p l able, so that the ray B . . C traverses L x and L 2 at a substantial distance from their centres. Under these circumstances it is clear that the image of S formed at p l or p 2 is subject to spherical aberration ; the ray S . . C (tracing it backwards) after refraction at C seems to proceed from anc ^ ^ an e = ^ v~ 5 tan 77 K..N M..X K..N M..X The tangent ratio. ' tan e N . . Y H . . M H . . M N . . Y ' yZ y2 vers (A. . M) = vers N . . B = 2r 2s Also -i/2 2s /,, , f\fl , f H . . M x . . M , _ y*_ 2r ~b' + 2sb'' + 2rb 7 in which we may neglect functions of the thickness t (which is The thickness may independent of y), especially as we shall eventually apply our formulae to elements of no thickness and parallel plates in the case of having to deal with very thick lenses. Therefore we may write H..M 2\r sJb" next , y* 2brJ ' N..Y = 250 A SYSTEM OF APPLIED OPTICS SECT. K -- N M..X_ f y/l I\ } bi y*(I_l\ } ' H..M N..Y \ 2b\r sJjc( 2\br csJ ) ' tan bi f 1/1 1\ ft\ \ '___M _ _( __ 1 __ I 4. *L I _ ^ 'tone cl 2 &'\r sJ 2\br cs K..N M..X MV */ 2 ri/i i\ 1/1 m~i The tangent ratio. ,. H ^ . s ^ = -|_l ^-{-(^H- -)--(-__) }J. (5) Here it is desirable to express &' in terms of yu,, r, and Z>. We have _/*-! _1. l_/i-l l_^-l)-r ft' r b ' ' b' pjr pb p,rb ---- -_/*-l/l , 1\ \ ' T I \/ i \ ' T p/ii) p.rb p \r b Relatively to c and the second surface we also have cs COC p.CS fJ.CS //, \C S therefore on substituting we get The tangent ratio in ^ ! fl /! jx 1 ^ ^ } n terms of u, r, s, b, ' = - M + * _(_+-__ + _ ) U2 I and /. tan e cL 2u, lr\r &/ s\c s/J Remembering that the so-called rays that we are dealing with are principal rays, each of which is supposed to be the central ray of a pencil or cone of rays which is limited by an aperture of which X is the centre, we may now adopt the device described on page 149, The characteristics Section VI., and use the characteristic /3, substituting 9 . for ^-, and /? and x introduced. I - R 1 . for -, assessing the signs of I and c according to the conventions there laid down, and we will also adopt the characteristic x for the shape of the lens, so that 1+a; 1 I -x 1 ;=- and 20.-1)/ r 2(/x-l)/ , On substituting these values we find that Formula (8) works out to The tangent ratio in tan 77 Z>T f _L _ ( ^ 1 final form. ^ = c [l + |J2 ^ZT)\^ + 1)a; + ^ ~ J ^ j J' which we may briefly write -1 1 + ,y. 2 T' /. IX DISTORTION 251 When (/4-f !)#+ (fi l)a = 0, then the intersection points of rays entering the lens and the same rays leaving the lens all lie on a plane passing through the sharp edge of the lens, and the tangent condition is fulfilled. The Effect of Spherical Aberration upon the Distortion We may now consider the further addition to our formulae consequent upon the introduction of the spherical aberration of the lens. Figs. 94 and 94a, Plate XX., illustrate the case. The principal rays from or to X, instead of converging to or diverging from Z, as supposed before, really converge to or diverge from z owing to spherical aberration. Thus z . . Z is the linear spherical aberration whose value is expressed shortly as ^ (A^e 2 , if y = H . . M as before, and if, in the full value of A', /3 is substituted for a. Therefore the true value of B . . z or c f is c ~ 3 (B')c 2 , writing B' instead of A', because we are dealing with the question of the spherical aberration of principal rays ; and so the true value of -7 or - --is - + ,oB' which may be written in the form c B . . z c 8/ 3 c f ^A , , c 2/11 + --. f-fsB k in which - , may be expressed as ,- - . = ; 2/ 4/ z ) 2j 1 - p 2/ 1 - p On substituting this value of -, corrected in accordance with the c spherical aberration in Formula L, we then get tan 7j tane cL 4/ 2 which in full is tan 77 &f, f 1 ft, 1 (n+2 , ^ Coddington's if-- > i\~ /.. i\o\ . formula expressing 6f, f lit, 1 fM+2 , = - 1 + T/2 -7 - r\( (0* + ! )* + (/* - l)/8 \ + -, o\ ^4*? CL 4/V(/^-l)^ J l-\k-l /2 \ -, o tane CL 4/V(/^-l)^ J l-ft\fk-l ratio between tan- 3 _ rli gents of emerging + 4(^ + 1)^ + (^ + 2)0* - I)/? 2 in which b and c are the conjugate focal distances by first approximation, so that - + - = - simply. b c f This is Coddington's formula for the relationships of tan rj and tan e for one lens. We shall, however, soon see that it is not an sal. 252 A SYSTEM OF APPLIED OPTICS SECT. When crossing point of principal rays is denned after pass- age. Formula varies ac- cording to position of the stop. Two or more lenses in succession. Spherical aberration must be carried for- ward to following lenses. universal formula, and will not interpret itself in all circumstances. In the case of Figs. 94 and 94a we have supposed X to be the point where the principal rays cross the optic axis, and the spherical aber- ration only affects the value of c by reducing it, therefore - is increased in value by the aberration. But let us suppose that the point where the principal rays cross the optic axis is defined after passage through the lens ; let there be a stop at Z in the case of the collective lens instead of at X ; then it will be A . . X or b that will be reduced by spherical aberration, and - should obviously suffer a decrease from the normal - But it is tan e c clear that the value of ft, if the stop were at Z, might be anything between 1 and -f 1, so that ^ would still be of positive value, while we want a negative value in order to make - less by the tan e spherical aberration. Coddington showed that in any case in which the crossing point of the principal rays is defined after passage, then must be 1 l+p substituted for -- - in Formula II., and this works out quite correctly. He then proceeded to adapt the above Formula II. to the cases of two or more lenses in succession. In Fig. 95 let L x the first lens be receiving principal rays diverging from a point X x on the axis to the left, then after refraction they are subject to spherical aberration, and the ray figured above crosses the axis at z 1 instead of at Z l the ultimate focus, and passes on to the second lens L g . It is clear that while Z x . . z l is a decrement to c x it is an increment to A 2 . . Z l or b 2 . Therefore the statement of - for the second lens needs modification tane 2 in order to cover the variation of & 2 consequent on the variation of c : . Coddington made the necessary correction, and thereby obtained the Formula HA. which is applicable to two lenses in succession, such as a Huygenian or Eamsden eye-piece ; but in extending the application to the case of a four-lens or erecting eye-piece, which was one of the main objects in view throughout his investigation of distortion, he made a strange omission. For in his series of formulae, while carrying the spherical aberration of L! through to L 2 , that of L 2 through to L 3 , and that of L g through to L 4 , he omitted to carry the aberration of L x through L g on to L 3 and L 4 ; nor did he carry the aberration of L 2 through L 3 on to L 4 . PLATE. XX H PLATE. XX. IX DISTORTION BY SPHERICAL ABERRATION 253 But the omitted operations can be shown to be as important and sometimes much more important than the processes which he retained. Fig. 96 represents a case whicli furnishes a capital illustration of the necessity for carrying the aberration of any one lens right through to the following lenses. Let there be four lenses, L 1} L 2 , L g , and L 4 , all of equal focal lengths and equal separations, the latter being four times the focal length of any one of the lenses. Let it be supposed that a set of principal rays is radiating from a fixed point at a distance in front of L! equal to twice its focal length. Let . . P l be one of these principal rays forming an angle e x with the axis. Let it be supposed that all four lenses are quite free from aberration, and also that the tangent condition is fulfilled, so that the refractions all take place in one plane perpendicular to the optic axis and passing through the lens centres (equiconvex lenses are here implied). Then it is obvious that the course of the principal ray through the series is . . P : . . Q a . . P 2 . . Q 2 . . P 3 . . Q 3 . . P 4 . . Q 4 , and what takes place at one lens is a repetition of what takes place at any other, and the emergent ray makes an angle y 4 with the axis equal to e^ Next, let it be supposed that a very slight spherical aberration is introduced in L 1? so that the principal ray . . P 1} instead of being refracted accurately to Q 1? is refracted to q lt so that Q l . . q l is the linear aberration. Supposing this to be a small quantity, say 1 per cent of L x . . Q ls then we have the ray striking the second lens plane at a height L 2 . . ^> 2 which will be 2 per cent greater than L 2 . . P 2 . Then the image point of q^ thrown by L 2 will obviously be q 2 , and Q 2 . . q 2 will be very nearly equal to q l . . Q ls as the conjugate focal distances are equal and the variation very small. Let . . L x = u lt Lj . . Qj = v lt L 1 ..q l = v\, q l ..L% = ii 2 , ~L. 2 ..q 2 = v\, L, . . P 2 = ?/ 2 , and L 2 . ._p 2 = y\. Then the increment to L 3 . . P g will be 4 per cent, and that of L 4 . . P 4 will be 6 per cent. But it is not our purpose to take notice of the variations in the y's in our functions of T' and B', because they involve corrections of a higher order, namely, of the order yf. What we are chiefly concerned with are the new functions of B r and y 2 which have to be introduced in order to express the cumulative increment to tan 77, for evidently tan r? 4 Ul 4j & it = -^.-^.-f.-^= - 'Git?/ \i> 2 - -Qlv z / \v s - 'Oli Four lenses in suc- cession. All four lenses first supposed free from aberration. Effect of introducing spherical aberration in first lens. Cumulative effect of the aberration of first lens. and on writing u = v=l, the above becomes -01)}, 254 A SYSTEM OF APPLIED OPTICS tan = (1 + 07), or we may say that Hence Coddington's omission to transfer all the aberrations through the series is fatal to the accuracy of his formulae for more than two lenses in succession. It will be as well, however, to repeat here his formula for two separated lenses in succession, which is quite correct although very unwieldy tan Coddington's dis- tortion formula for ] C 1 C 2 two lenses in suc- cession. 1 +2 Formulae for three or more lenses highly complicated. Application of Formula IIA. to a Huygenian eye-piece. Emergent rays par- allel. The student will find his formulae for lenses in series dealt with on pages 162 to 172 of his work, and, after perusing the same, will be obliged to concede that, even as they stand, they are very complex and ill adapted for practical purposes, especially when any variations in the position of the limiting stop always render certain modifications necessary. If, however, the omitted functions for the transferred aberrations were also taken into account, then Coddington's formulae for three or four lenses, when completed, would become unmanageably complex, or at any rate full of pitfalls for the unwary. This is essentially the case in a method which seeks to interpret distortion only in terms of the relationship between the tangents for finally emergent principal rays and the tangents for the same rays before entering. Let Fig. 97 represent a Huygenian eye -piece, for which Coddington's two-lens formula is quite correct. Let it be supposed that an objective away to the left is projecting a truly rectilinear image on to the plane P . . P (if L x were not interposed). Let two principal rays from the centre of the objective be considered, one r 1 . . 1\ aiming for a point in the outskirts of the image, and one r . . r aiming for a point in the image very near the optic axis. After these two rays are refracted by Lj they proceed, through a new and imperfect image formed at p . . p in the principal focal plane of L 2 , on to L 2 , by which they are again refracted, r x . . r a to cross the axis at / 2 , and ?* 2 . . r 2 at/i ; F . ./ 2 being the linear spherical aberration. But the rays constituting the emergent pencils represented CONSTANCY OF TANGENT KATIOS 255 by these principal rays emerge in a very nearly parallel state, as if coming from an infinitely distant object, that being the state of the rays best adapted for distinct vision by the normal human eye. Therefore so long as tan ^ bears the same ratio to tan e l as tangent 7/ 2 bears to tan e , the eye will notice no distortion, and straight lines in the distant object will appear to the eye through the telescope as straight lines wherever they may occur in the field of view. That is what takes place when the functions of y 1 in Coddington's Formula HA. equate to 0. But let us consider what will happen, supposing we no longer confine ourselves to receiving the emergent rays into the eye, but draw out the eye-piece with a view to throwing a real image of the object (the sun for instance) onto a white screen S . . S at a little distance behind the eye-piece. It is clear that such an image will no longer be free from distortion. For the principal rays, although emerging in the Lateral displace . ment of emergent right direction, as implied in the constancy of , will be subject to principal rays. a lateral displacement consequent on the aberration F . ./ 2 . If they all radiated from F there would be no distortion on the screen S . . S, and the ray i\ . . i\ would strike the screen at Q ; but instead of that it strikes the screen at q, and Q . . q is the linear distortion or displace- ment of the image point q from the correct position Q. The linear amount of this distortion Q . . q varies as the cube of the distance from the axis. On an infinitely big image, either virtual or real, the absolute displacement Q . . q is relatively a vanishing quantity ; but relatively to the image formed on S . . S it may be a very large quantity. Now the amount of linear spherical aberration of principal rays taking place in the case of a four-lens eye-piece is very much greater than in the case before us, and the student will find, what is well known to many opticians, that if he takes an erecting telescope free from An experiment with distortion and directs it to an object containing straight lines, and a then pulls out the eye-piece until it throws an image onto a ground glass screen a few inches behind the eye lens, he will then see that the positive distortion of the straight lines, or pincushion distortion as it is often called, is very marked. On the other hand, let an extremely short-sighted person use the same telescope on the same object. He requires a virtual image a few inches from his eye to be formed, and therefore pushes the eye- piece nearer to the objective than its normal position ; when he will see all the straight lines distorted in the opposite sense, for there will be strongly marked negative or barrel-shaped distortion. 256 A SYSTEM OF APPLIED OPTICS SECT. Formulae of greater It is quite plain, then, that Coddington's formulae are quite inade- quate to deal with cases in which real or virtual images are formed at finite distances, instead of at infinite distances. We therefore require formulae of perfectly general application, and the following Hues of reasoning will guide us to what we want, as well as lead to much greater simplicity. So far, all that has been taken into account is, first, that the rays constituting pencils finally emerging shall be parallel as though emanating from an infinitely distant image, and, second, the constancy or otherwise of ~ - for principal rays traversing the system at varying heights from the optic axis, and therefore traversing the several lenses at varying degrees of obliquity. Extension of the Inquiry As yet the positions of the planes where the various real or virtual images are formed have not been properly considered. Let Fig. 98 represent a collective lens L, placed behind a real image . . 0', such real image being projected without any distortion from X, which point may perhaps mark the centre of a telescope objective, and is thus the point on the optic axis from which the principal rays of the pencils going to form the image . . 0' radiate. Let the distance from to the lens be greater than the P.F.L. of the lens, so that it projects another real image of . . 0' at I . . i'. Then as X . . N is greater than . . N, therefore the focal point Z conjugate to X will be nearer to the lens than I . . i'. Supposing Z is the ultimate point by first approximation, then z is the real point where the principal ray XMz crosses the axis before proceeding to i', and Z . . z is the linear aberration. Let Fig. 98a represent the corresponding case of a dispersive lens, exactly the same notation applying. It is best always to choose for our typical examples cases in which all the quantities are conventionally Relationship be- positive. What we now want is a formula expressing the relationship tween the sizes of between the size of the image I . . i' and the size of the original image the images. . . 0' presented to the lens. That is, we want to find out by how much the ratio between the radial dimensions of the two images as painted by the eccentric principal ray X . . M . . i' departs from con- v stancy or from the ideal or normal relationship expressed by -. u Here we are assuming that the conjugate focal distances b and c for principal rays are measured from the point 1ST, the axial point of DISTOKTION. PKELIMLNAKY LNQUIKIES 257 the tangent surface M . . N. We must next inquire, from what point must the conjugate focal distances u and v be measured, if aberration- Must u and v be free refraction of the principal rays at M in the tangent surface is to measured from N ? lead to rectilinear projection or an image of . . 0' that is free from distortion ? Is N the required centre of projection ? The theorem that a lens through which are refracted a system of If Formulae I. and II. eccentric pencils, which fulfils the tangent condition and is free from central projectJn spherical aberration, also fulfils the condition of central projection implied? through the point N, may be proved algebraically thus In Fig. 995 let JST . . M be a lens fulfilling the tangent condition for a system of principal rays radiating from Q. That being the case, then all refractions of such principal rays will virtually take place in the plane M . . N". Let the lens also be aberration free for all distances, so that the law of conjugate focal distances by first approxi- mation will strictly hold good. Let F = the principal focal length of the lens L. Let Q . . N = 6, and let q be the focus conjugate to Q, so that i !_!_! N.. q c ~F b ' Let p . . d be a plane image or object placed anywhere between Q Construction, and L, and perpendicular to the axis. Then let p be a point in such plane image which also lies upon the principal ray Q . . M. From p draw the straight line p . . N through the centre N of the tangent surface, and produce it onwards until it intersects the refracted principal ray M.. .q. .g at g. From g draw g . ./ perpendicular to the axis. Let the distance N . ./ be v, and d . . N be u. Assuming N to be the centre of projection, then the question is, what must be the relationship between v and u ? Since the point g is on the line of projection from the original p through the centre N of the tangent surface, /. f..g=(p..dF- or 0-, Xp..d=0; Therefore we get f..g also = (v - c)-, if Y = M . . N. 0- = (v - c) , in which Y = 5 u ^ c b-u ... o%(.-)-Z2 u ^ ' c (10) 258 A SYSTEM OF APPLIED OPTICS v v- c b v - c b - , and v = : u : u b u c v = vbu b -u c cbu v= - c(b - u) c(b - u) ' .-. v{c(b - u) - bu} = -cbu, cbu (U) c(b -u) -bu in which expression we may put . 1 c(b - u) - bu , and - = - - ~- v cbu 1 and then we have bF 1_I~ 5 ~ F F b bF . bF N is proved to be the common reference point for conjugate distances " and v as well as b and c. Effect of the separa- tion between the principal points. 1-1-1 v ~F u ' (12) So that the simple law of conjugate focal distances, connecting Q . . N and N . . q (or b and c) for the principal rays, co-operating with rectilinear central projection through N for the corresponding image points p and g, also satisfies the same simple law of conjugate focal distances for the two image distances u and v ; that is, a distortion- free lens forms its image in strict conformity with the condition of rectilinear projection through the centre of the tangent surface. We have now to inquire whether the above line of reasoning will apply to a lens having appreciable central thickness, that is, will the above theorem apply when the lens thickness is such as to lead to very appreciable separation between the two principal points ? In order to answer this question we must know how the point N is situated with respect to the principal points. Fig. 99c represents a thick collective lens. The tangent surface is of course the plane containing the sharp edge M of the lens, and N is the axial point of the same. C is the geometric centre of the lens, and p l and p 2 are the two principal points, while % and a 2 are the two vertices, the radii being r and s as usual. ix DISTORTION. PRELIMINARY INQUIRIES 259 Formulas IV. and V., page 14, fix the positions p l and p^, or the distances a : . .p 1 and , . .p 2 , as respectively tr ts n TiH . ; * r . ~'^\ CILHJ. / v ., - \ . Now as t(/j, 1) is generally a very small quantity compared to p(r 4. s) } representing as it does the very small effect upon the positions of p 1 and p 2 exercised by the refraction of the two curved surfaces as compared to two plane surfaces, we may legitimately omit it and write t r t s r + s A* T Then we have . . N obviously = tf , -therefore r + s i TkT\ / \ -.-r ' 99 or /u, r + s and N.. ^ r - s - 1 r + s p r + s (13) which expresses the proportion borne by N . . p 2 to the separation Pi- -Pz between the principal points. This formula can be written in a more convenient form, in terms of x, N (u,-l)-(u.+ l)x Position of N with ___ _ __. (15) reference to the ^i -Pz ^0* ~ -0 principal points. Now let it be supposed that the two conjugate focal distances for principal rays b and c bear the same ratio to one another as p l . . N to N . .p 2 , and therefore that so that from this we get fl _(/*-!) -(/*+!) 7 A-* ~ ; Point N distant from the principal points in proportion to b and . Above theorems therefore apply to the two principal planes. The thickness only alters value of F. Formula for tangent condition fairly ac- curate for thick lenses. 260 A SYSTEM OF APPLIED OPTICS and therefore u + 1 P = ?* SECT. (16) But we have seen that the tangent condition is fulfilled when (yu, I)y8 + (/x + 1)& = 0, which is the same thing. The conclusion is, then, that when the tangent condition is fulfilled the tangent surface cuts the optic axis so as to divide the distance between the principal points into two portions p 1 . . N and N . . p 2 respectively, proportional to b and c. Therefore if two principal planes are drawn through the two principal points (Fig. 996) parallel to M . . N they will obviously be cut by Q . . M and M . . g at equal heights. Also, by the law of principal points, the ray p 2 f . . g' through the second principal point is parallel to the ray p'. . p- through the first principal point. Therefore the conjugate distances b and u on the one hand and c and v on the other hand will be measured from the principal planes. So that if we suppose the gap between the two principal planes to be closed up by sliding the two halves of the diagram into one another, as it were, we then arrive at the state of things first assumed in our inquiry, for p and p 2 f will become merged in N, while &/ and k 2 f will be simultaneously merged in a common point M. The only difference made by the thickness, if not excessive, is in the value of =, but the equation T + - = - + ~ f course always J. C w C/ holds good, and we still have the equivalent of central projection of the image through the point K Thus in Fig. 996 the dotted lines and accented letters indicate the state of things when the separation between the two principal points is allowed for, and the full lines and unaccented letters the state of things when the gap between the principal planes is closed up. It will now be seen that, with regard to the fulfilment of the tangent condition or any departures from it, it is scarcely necessary to the attainment of accuracy to treat a thick lens by elements, although it becomes desirable to do so when the thickness becomes excessive, for the refractive effect of the curved surfaces (as compared with flat surfaces) upon the linear positions of the principal points grows as the square of the thickness, and leads to the above theorems becoming inapplicable. Let us now revert to Figs. 98 and 98a; and, as usual, let X . . N = b and N . . Z = c, M . . N = y, . . N = u, and N . . I = v, = e and /MzN = r, UNIVERSITY or ix DISTOKTION INVESTIGATED ANEW 261 and let v r\ j T j X . . (J = cfjj and z. . L = d 2 . Then we have I . . i' = d. 2 tan 77, and . . 0' = d^ tan c ; ' 0. .0' ^ wherein d 2 = fl - e + ^TgB'e 2 , and ^ = 6 - M ; ., r-c/l-lJB'tf) I . . ^ _ V 8/ 3 / tan 77 . . 0' b - u tan e b-u\ v\ l-(3 4/ 2 // tan e ' in which we may next insert the value of already worked out, and t 1 11 6 which was expressed shortly in Formula (9) as tan 71 6 r f /, , 1 so that Formula (18) amplifies to * bv c . 7T~~r\> ~7I - \" -- T - a 779 / I ir?9\ -- T . . c(6 - ) I i \ 1 - /? 4/vJ I 4/ 2 \ 1 - in which we may now, following Coddington's useful device, substitute 2f , 2/ 2/ , , 2/ . Jv for w, r-^ - for TJ, y for o, and . -- for c, on which l+o ' 1 - a ' 1 + l-p c(b-u) (l+a)(l-p) , C 1-a becomes ^ -- 'A - ~, and - becomes - =. (l-a)(a-/?)' V 1-0 On substituting these values in Equation (19) we then get 0..0' (l-a)(a-0)l\ I-/?/ (1-0) 2 4/ 2 /l 4/ 2 4/ 2 l-/3 which, if we neglect functions of ^, y 2 (l-0)^y 2 ) /3) 2 4/ 2 / 4 -0 y 2 262 in which A SYSTEM OF APPLIED OPTICS 1 +g_/l +a\/ 2/ \_v . l-o \ 2/ Al-a/ J so that the formula finally becomes which in full is Universal formula -I 1 + / _ 1 Jj 0* + 1) + (/* - l)/3 j + - ^| ~, + 4 (/A + l)/fo + (3fj. + 2)(/z - for distortion of image. Thus we find that the change required in the formula for - - in tan e order to convert it into a statement of the ratio between the radial dimensions of the two conjugate images is an unexpectedly simple one, involving the simple substitution of -- ~ for ^ in the spherical v b a ~P ~P aberration function, and - for -. If the reader will pursue the same u c process in the case of X being nearer the lens than . . 0', the case of the stop being placed behind the lens, or any other case he likes to choose, he will arrive at the same formula ; in fact, it is quite universal and interprets itself in all cases. Two Applications of Formula III. to Combinations of Lenses We will now show how this formula simplifies the problem of arriving at the distortion produced by a series of separated or non- separated lenses in succession, even when employed for projecting real images on to plane surfaces at finite distances. separated Let Fig. 99 represent two lenses in succession placed in alignment behind either a real plane object . . : or an image projected by another lens. Let it be supposed that the lenses are very thin, and that the principal rays cross the axis somewhere about z, and then proceed to intersect the conjugate focal plane I . . i' where an image (in this case inverted) of . . O l is formed. From Oj draw Oj . . Lj . . i straight through the lens centre, then i in the plane I..i r will be the correct place for the image of the point O l to be formed if there is no distortion ; but owing to the operation of distortion the image of the point 1 is really formed at i', and i . . i' is the linear distortion ; which, for example, may be 1 per cent of the correct ix DISTOBTION OF LENS SYSTEMS 263 V radial dimension I . . i, which latter, of course, = (0 . . 0^ . This Ul exaggerated radial dimension I . . i is then presented as an image to the lens L 2 . It is clear that if L 2 is so circumstanced as to form an image of I . . i' without in itself exercising any distorting effect, ther>. if we draw a straight line from the centre of L 2 through i' to cut the conjugate focal plane J ../' at j', then j f becomes the image point of the point i', whereas the image of the true point i would be thrown to j ; therefore j . . j' is the correct projection or image of the linear distortion i , . i' ; that is, the lens L 2 will simply form a correct image of what is presented to it if it is free from distortion, while if it does exercise any distortion itself, it is obvious that it will add its own distortion, j'. .j v for instance, to that which is already presented to it. If the two distortions are of opposite signs and equal, then the final image will, of course, be a true image of the original. Our Formula III. simply represents an increment or decrement to the ideal radial distance from the optic axis of any image point located or defined by a principal ray passing through the lens at a given height y from the axis, and is therefore quite independent of the sign of the lens ; in fact, 7 is always positive, and the sign of the lens is really ** 1 always implied in the term --- ~ in the spherical aberration function, and in /3 in the function of the tangent condition. Therefore the Simple summation distortion functions involving if for a series of lenses will be the formS simple sum of the distortion functions for the individual lenses. In of lenses. the case of two lenses, we have the image to object ratio for the first lens o..o'~ % i ' ^-jBfvvi 1 / 1 and the image to object ratio for the second lens is given by Vl + (T 2 ' + ! ] On multiplying these two formulae together we get T i' a v f / 1 \ 11 2 / 1 \ 11 2 [ 11 2 2 - Distortion 12/1 i / T ' _L "R ' \ "l i/ r T / , "DM "2 i I "l "2 l/f)l\ for two ! -/ - -{ I 4- I , 4- hS.. I -r I 1 -r- r>~ I ^ -r I / I A I 1 1UA uWU j f\ f\ \ D ~^l /J^itlfi O "^9 /JJT9 /1/J* O JT 9jJ\/ (J . . U M^l a i~Pi 'Wi \ a z~ Pz ' Va l^/i 7 2 * succession. from which the function of =nhr^Fi ma Y be left out, as it is a correction ISA 2 /*, 2 of the order ^/ 4 . Therefore the total distortion of the series is the sum of the distortions of the individual lenses. But it is obvious fonnul 2 264 A SYSTEM OF APPLIED OPTICS Distortion for- mulae for three or more lenses in succession. that y will have to be inserted at its proper value for each lens ; and all the y's may be expressed in terms of y l} for yn ~~ y-1 ) y q Vo Vl j cLC* So that the formulae for a series of n lenses or elements must be written in abbreviated form, 1 IV. An objection to the validity of the above formulae. First case. E.C.s of L.j assumed to neu- tralise the normal curvature errors. T> 2 /472Vc~ z ' V2 X( l In such cases y : for the first lens may be taken to be ^ tan , which connects the functions with the angle of obliquity of the pencil of rays in question. It will be as well to now consider an objection that may be raised to this series of formulae, and at first sight a very plausible objection. It may be urged against it that it does not allow for curvature of image. Let Lj, Fig. 99a, be a collective lens which by central oblique pencils forms an image ^i-Fi which for rays in primary planes is curved as usual to a radius equal to about/! OjJ* + 1 or If so, then will not the primary focal point at q 1} and not its projection O l on the focal plane, form an object, as it were, from the point of view of a second lens placed at L 2 ? Let L 2 be a dispersive lens of the same power and material as L 1? and let it project an enlarged image of O l . . F x or q l . . Fj on to another plane 2 . . F 2 , which image, if L 2 is free from E.C.s, will be a flat one. Now the primary focal line q 1 is formed on the oblique principal ray Lj . . O x (unless there is coma, but that is dealt with by separate formulas), and assuming L 2 free -from distortion and coma, and at the same time to have no curvature of image, in the sense that the E.C.s balance the normal curvature errors and therefore the lens projects a flat primary image of a flat object, then the image of the point q l will be projected to q on the refracted principal ray and on L 2 . . ^ produced, so that the versine or curvature error q^ . . s 2 will be 2 265 times q l . . s^ Also the focal plane q, 2 . ./ 2 will in this case be con- jugate to q 1 . ./! in exactly the same sense that 2 . . F 2 is conjugate to 1 . . Fj. The curvature of image would in this case be copied through from one image to the other, without the point O 2 being disturbed. For 2 would then be the centre of an out-of-focus oval, being a section of the eccentric peucil of rays whose axis is I' . . q 2 . But now it may be urged that supposing the E.C.s of L 2 are eliminated so that its normal curvature errors become equal and opposite to those of L 15 then it will throw upon F 2 . . 2 a flat image, and if we still assume the line of central projection L . . q l to be produced to cut the focal plane F 2 . . 2 at . IO2 . M Fi<5.99.c. i. IO3. i. lOO.a. i. 100. PLATE. XXI. M i. IO3. i. lOO.a. IX DISTORTION OF PARALLEL PLANE PLATE 267 the ultimate focus A after refraction, according to the first approxima- tion. Through A draw the focal plane A . . F parallel to the surfaces. It is obvious that if the focus were formed at A, as it would be by a thin perpendicular pencil, there would be no distortion ; but the oblique rays are subject to aberration, the ray K . . b intersects the normal ray P . . A at 6, the principal ray R . . c intersects it at c, and the ray H . . d at d, and the longitudinal aberrations A . . 6, A . . c, and A . . d are proportional respectively to (P . . K) 2 , (P . . R) 2 , and (P . . H) 2 . But the principal ray R . . c, when produced, cuts the focal plane A . . F at B to one side of the true point A. A . . B is then the linear or absolute value of the distortion, and our problem is to express it in terms of the radial dimensions of the image, which, of course, necessitates our knowing the whereabouts of the optic axis of the system, of which the parallel plate forms a part. In the first place, we are supposed to know the angle of obliquity PAR or ^ ; we required and ascertained it before for other parallel plate corrections. Then we also have the formula for the linear aberration c . . A from page 80, Section IV., which was wherein in this case a, the semi-aperture of the larger direct pencil, is P . . R, which we will call h, while v = P . . A. It is clear that h v tan \r ~~ L r/ z \9 ' ~ O 32 (* tan X) > also A . . B = (c . . A) tan x = lljfv* tan tan x . (22) (23) But so far there is nothing to determine the sign of the distortion. Let GJ..O!, 2 ..0 2 , and 3 . . 3 represent three possible and different positions of the optic axis. Then A . . O/, A . . 2 ', and A . . 3 ' are the respective radial dimensions of the image, in terms of which we want to express the displacement A . . B. Let D 1} D 2 , and D 3 be the points where the principal ray cuts the optic axes 1 . . 1; 2 . . 2 , and 3 ..0 3 . Then, in pursuance of the conventions previously adopted, the distance from T) 1 to the second surface is + in both cases, for the principal ray is diverging from D : on emergence. The distance from D 2 to the second surface is in Fig. 100, as the principal ray is Formula for the linear distortion yielded by parallel plane plate. 268 A SYSTEM OF APPLIED OPTICS SECT. converging to D 2 , but is shown to be + in Fig. 100ft, as the principal ray is diverging from D 2 after emergence. The distance from D 3 to the second surface is shown in both cases, as the principal ray is converging to D 3 after emergence. Let these distances be c l} c 2 , and c 3 respectively. In Fig. 100 the distance A . . P or v is , and in Fig. lOOa is + . Then, if the above conventions are adhered to, we have A . . O/ = (v - c x ) tan A . . 2 ' = ( - c 2 ) tan A . . 3 ' = (v - c 3 ) tan and is - in both cases ; and is - in Fig. 100 and + in Fig. lOOa ; and is + in both cases. A B Evidently, then, r ~- gives the distortion as a fraction of the (v - c) tan x radial dimension of the image. Then A . . B in the numerator, having no sign, may always be considered + , but (v c) in the denominator acts as a sign determinant. In full, then, the fractional distortion is tan 2 x- (24) Formula for the fractional distor- tion yielded by parallel plane plate. Normally the ratio between the sizes of the two conjugate images in the case of a parallel plate is simply unity, therefore we find the corrected ratio to be V. In the case of an optical combination containing thick lenses the quantities from which we can pick out v and c have to be assessed at the outset, as we have seen before. But we must remember in this case that while v and c may be known quantitatively, yet their signs must not necessarily be taken in connection with or with respect to the element following the parallel plate, but must be assessed with respect to the parallel plate itself in strict conformity with the above convention. Should any parallel plate not be followed by an element, still the quantities v and c are easily inferred from the values v and c or v and D" of the preceding element. Under these circumstances Formula V. will be found to interpret itself in all cases, and give a positive result when the displacement A . . B is from the optic axis, and a minus result when it is towards the optic axis. DISTOKTION ILLUSTRATED 269 Some Concrete Examples of the Application of the Distortion Formulae We will now take the process lens whose curves and other data Process lens, were given on page 185, Section VII., and work out its total distortion by the Formula IV. we have arrived at, taking the quantities a and ft, x and /, etc., as before arrived at. Then we get the following quantities for each element, the values of the function of T' and those of B' being stated separately, or shortly as /T 7 and /B', and assuming y l to be 05 inch : /T/= - -001 13275 /B/= + -0095 122 Total - + -0083795 /T./= + -0008095 /B 2 '= - -0062888 Total - -0054792 /T 3 ' = - -00018385 /B 3 ' = + -00127723 Total + -00109338 /T 4 '= + -0007870 /B 4 ' = --01 309 17 Total --01 23047 /T 5 '= - -0006549 /B 5 ' = + -0111008 Total + -0104459 /T 6 '= + -0006872 Total for e l + -0083795 + -0010934 + -0104459 + 0199188 /B r ;= - -0048547 Total - -004 1675 Total for e - -0054792 - -0123047 - -0041675 - -0219514 + -0199188 Final total for six elements = - -0020326 But we have indicating a slight negative or barrel-shaped distortion. yet to add the parallel plate corrections. For the first plate P l we have tan = ~ which tan

2 = + 5'122 = M 4 , and the principal rays also are divergent, .-. v 2 -%= + 4-8485, Distortion of second and the distortion is therefore positive, and it works out to parallel plane plate. + '01596 tan 2 *. PS In this case tan x 3 = tan* 1,2 3 4 5^ an( j ^ an( j c ^ are quantitatively the C 1 C 2 C 3 C 4 C 5 same as c and & 6 of the sixth element. The rays of the emergent pencils are divergent, and v s = + 14*1 046 = U R , and the principal rays are divergent, .-. v 3 -c 2 = + 13-7469, Distortion of third and the distortion is therefore positive, and works out to parallel plane plate. +-Q15976 tan 2 *. We then get a distortion for P 2 = + "01596 tan 2 * P 3 +-01 598 tan 2 * + -03 194 tan 2 * P x --01 383 tan 2 * Total distortion of Total for parallel plates = + '01811 tan 2 * plates. >05 . 2 On multiplying by tan 2 *, which we saw was ( ) , we then get \ ^ ' _ _ o/ a total distortion for the three plates = + '00498 to which we have to add the distortion for six elements = - '00203 Final total for whole and our grand total is + "00295 lens. 271 The E.F.L. was 8'55" and (E.F.L.) tan =l'63 inches, so that at a distance from the optic axis =1*6 3 inches, corresponding to an angle of 10'47', the linear distortion is ( + -00295)(1'63) = about + '005 inches, an amount barely perceptible by any but very delicate tests. As a matter of fact, this lens was very carefully corrected for rectilinearity, and at much greater angles from the axis very slight negative distortion was just perceptible. Having now dealt with a case in which the relative separations are not large, it will be as well to apply the same formulas to the well-known cases of the Huygenian eye-piece, and the four-lens erecting eye-piece, in which the separations are very considerable. Huygenian Eye-piece Let this be the usual combination of two convexo-plane lenses of focal lengths 3 inches and 1 inch separated by a distance s = 2 inches. Then as the image is formed in the principal focal plane of L 2 , or 1 inch in front of it, it falls therefore half-way between the two lenses. The E.F.L. of the eye-piece = 1'5 inches. We may assume the principal rays entering L x to be parallel if the focal length of the object glass forming the image is relatively very long, so that /^ = - 1 . Also the rays are converging into L x as if to form an image 1 '5 inch behind L p therefore c^ = - 5. The principal rays are converging into L 2 to a point 1 inch behind it, therefore /3 2 - - 3, also a 2 = + 1, and a 2 - /3 2 = +4. Also 2/2 = ^-. The distortion for L x works out to and for L 2 works out to Let /,= 1'5. 108 V' 2 y\ t The characteristics and other data. Total or Final result. which, if y l = '2, gives a distortion of + 1 --. This is at an angular 1 OU distance from the centre of the apparent field of view, such that _ -2 _ 1 uctll (I) "1 ~ = -_~~~ I'D 7'5 The distortion Supposing we substituted a single convexo-plane lens of the same yafenttonvexo-pSe power for this eye-piece, it would have to be 1*5 in focal length, lens. 272 A SYSTEM OF APPLIED OPTICS while y would be the same as the y^ of the eye-piece = '2, and (3 would then become 1. In that case the distortion would work out to 88 eye-piece. or five times as much as the The Causes of the in- difference is partly due to the fact that in the eye-piece the principal ra y s are strongly convergent into the eye lens instead of parallel, which causes a much closer approach to the fulfilment of the tangent condition (which requires /3 2 to be 5) than in the case of the simple equivalent lens, but principally because of the relative reduction in y^ For supposing an equivalent simple lens is substituted for the eye-piece, then its y would necessarily be equal to the y l of the above eye-piece, and if y^ = -th of / x (and IV 1 2 2/0 then = -th of /A it is clear that ?A would be -ths of / the n n focal length of the equivalent lens. Thus the principal rays are caused to be refracted through the eye lens of a Huygenian eye- piece three times as close to the axis as in the case of the equivalent 3 lens, while the power of the eye lens is - of the equivalent lens, so 2> that the relative distortion of the eye lens, other things being equal, /l\ 2 /3\ 2 may be expected on that account alone to be reduced to (-5) (-5) i J.T ^*' \^' = lth. The formula for distortion for the Huygenian eye-piece will be found to work out to about a minimum, when x l = Q and x 2 + 1, in which case the field lens is equiconvex, and the eye lens convexo- 5 plane, when the total distortion is + -^$1- But such a combination has certain other disadvantages. Sometimes Huygenian eye-pieces are constructed with a ratio of focal lengths between the field lens and eye lens of 2 to 1, which enables a flatter field of view to be obtained than with the ratio 3 to 1 ; but with the ratio 2 to 1 the approach to freedom from distortion is not quite so good. The Four-Lens Erecting Eye-piece Its inventor. This well-known and useful optical device seems to have been arrived at quite empirically by the monk De Eheita, who evidently had been experimenting with various combinations of lenses in series in conjunction with a telescopic objective. But the theory of it was not worked out until very many years later, by Sir George Airy and Henry DISTOKTION OF FOUE-LENS EYE-PIECE 273 Coddington, and even then not in one sense completely. Fig. 101 The course of the shows the course of a couple of pencils of rays through such an eye- [^eye-piece * piece, from their points of origin in the first object or aerial image i . . i to their again concentrating into a second aerial inverted image i 2 . . i. 2 in the principal focal plane of the eye lens L 4 , so that after emergence from the latter the rays constituting the pencils are parallel and fit for vision by the normal eye placed behind it at P. Since the objective of the telescope is supposed to be placed at a considerable distance to the left hand, and the principal rays of the various pencils or cones of rays are supposed to radiate from the centre of the objective, therefore such principal rays are brought to a focus at at a distance behind L a equal to or a little more than its principal focal length ; not only so, but an image of the aperture of the objective is formed at that position, where it is usual to place a stop Position and func- with a circular aperture a little larger than such image of the objective, tion of first stop- whose office it is to screen off stray light reflected from the interior of the tubes. Then a second image of the objective or an inverted image of is again formed behind the eye lens at P ; that is, the principal rays again come to a focus or cross the axis at P, where the pupil of the eye is placed to receive them and the pencils of rays which they represent. But, as we shall see later, this second image of the objective, or exit pupil, is an exceedingly rough and imperfect one. Fig. 101 is a correct drawing to scale of a four-lens eye-piece which was specially adjusted with great care to show an apparently rectilinear image when used as a magnifier on a set of straight lines ruled on a flat surface placed at i . . i, the eyesight of the observer being normal. The object was to see whether the sum of the formulae for distortion for the four lenses would in that case work out to zero. The stop at was at a distance =f l behind L x . The data for this combination were as follows, the refractive index being T53 for all four lenses : Separation Sl = 2-24" x 2 =-2 s =5-24" 76 b. 2 = -34 = + 3 ' 51 and 04 - f3 l = +5 c 2 = - "40 /. /3 9 =+12'3 ^ 2 = +6-35 .-.03=+ -29 and a. 2 - /? 2 = - 1 2 T 274 / 3 =2-03" A SYSTEM OF APPLIED OPTICS =+5-64 c=+3-17 SECT. .'. /3 8 =- '28 .'. a 3 = -4-64 and a 3 -/3 3 = - 4'36 / 4 =1'41" / 3 =+l 6 4 =-l-04 c 4 = + -60 /. /3 4 =-3-7 U 4 = + 1-41 V= oc .'. a 4 = + 1 and a 4 - 4 = + 4'7 From which we get the following values of the distortion when ,= -1-115 i, 8 = + -72 - The personal equa- tion. Parallel straight lines viewed through a circular aperture may appear dis- torted. Distortion of the third order. =+ ' 00512 = -- 0190 = -- 0203 Total = + '0235 The total result is a positive distortion of about 2-^- per cent, which, although small in itself, is in excess of the distortion yielded by any one of the four lenses. But 2-^ per cent of distortion could scarcely go unperceived under a searching test. How is it that this apparent discrepancy between theory and practice arises ? It is partly due to the fact that a good deal of the personal equation arises in the case of a series of straight lines or chords viewed through a circular aperture. The real image formed in the principal focal plane of the eye lens is bounded or limited by the field diaphragm within the circular aperture of which it is formed. Now, it can be shown that a series of parallel straight lines viewed, without any lenses whatever, through a circular aperture do not appear to be straight to all observers ; to some, including the author, they invariably appear somewhat barrel-shaped, as if by the presence of negative distortion, while a square drawn with sides so curved inwards as to represent a case of 2 per cent of positive distortion at the corners (and therefore 1 per cent at the middle of the sides) appears to be perfectly rectilinear when viewed through a circular aperture just well clearing the corners. The reader should try this experiment for himself, and will then become convinced of the difficulty there is in saying whether an eye-piece is really free from distortion or not. Furthermore, in the four-lens eye-piece, consisting as it does of four widely separated lenses, the distortion corrections of the higher order ix DISTORTION OF FOUR-LENS EYE-PIECE 275 7/ 4 in some cases may form a very appreciable fraction of those which we have formulated of the order '^, and this is chiefly true of the corrections affecting the eye lens. To be sure Coddington, on pages 168 to 170 of his work, in dealing with the four-lens eye-piece, makes it appear that the distortion formula of the order ^~ for the four lenses may be reduced to zero ; but we have seen that he neglected in working out his formulae to allow for the spherical aberration of the first lens being carried through to the third and fourth lenses, and that of the second to the fourth, operations which, as we have already seen, are really as vitally important in his scheme as carrying forward the aberrations of each lens to the next following lens, which he did allow for. Hence his conclusions on page 170 were erroneous. We have seen that the formula for distortion which we have worked out is quite independent of such accumulated variations of b and c in each lens, that is, so far as the formulae of the order -^ are / concerned. But Fig. 102 will help us to see that the aberrations exerted by each lens upon the principal rays must necessarily have an effect upon the distortions of the following lenses which we cannot altogether neglect. In Fig. 102 the deviation of the principal ray Departure of the from its theoretical course is a little exaggerated for the sake of JjXj^JjL f r m clearness. The solid lines indicate the theoretical course of two the ideal path. principal rays through the lenses according to the formulae of the first approximation, by which the values of b and c, and therefore /3 for each lens are assessed. But .the dotted lines indicate the actual course of the same principal ray, which deviates largely from the theoretical course, especially at the eye lens. It is clear that our method of expressing the y for each lens in turn in terms of y v deviates more and more from the truth as we work towards the eye lens, and this fact is just as important whether we work out our distortion completely by Coddington's scheme or by our own. After allowing for these modifications of 7/ 2 , y^ and y^ and /3 2 , /3 3 , and /3 4 , it can be shown that our principal ray, striking L : at a height ^ = '20 from the axis, is subject to a distortion of the order y 2 4 , y 3 4 , and y*, equal to ith part of the distortion of 2 ^ per cent previously arrived at and of the opposite sign. On page 91, Section IV. (Fig. 36), we showed how, when two lenses are separated from one another on a common axis, the spherical aberration of the first lens gave rise to a spherical aberration in the 276 A SYSTEM OF APPLIED OPTICS Hybrid distortion. Above explained. Hybrid distortion in- creases as the fourth power of the angular field of view. second lens of the order y. 2 4 , and similarly for any subsequent lenses ; and the same influences operate in the case of the four -lens eye- piece. Moreover, there exists for each lens the intrinsic aberrations of the order y*, not only as regards the spherical aberration, but also the aberrations from the tangent condition. So that the distortion formulse for a four-lens erecting eye-piece, supposing we take all of the order y 4 into account, as well as those of the order y 2 , are of a highly complex nature. The fact that the corrections against distortion are generally of a hybrid nature, involving the opposition of these two orders of corrections, is made apparent by rigidly testing the rectilinearity of an eye-piece which has an extra large field of view. It will then be found that there exists a small amount of positive or pincushion distortion of straight lines in the inner zones of the field of view, while in the outermost zone there is quickly increasing negative or barrel - shaped distortion of straight lines. This is illustrated in exaggerated form in Fig. 103. The case is exactly illustrated by means of Fig. 37, in which the left-hand curve may be taken to represent + distortion of the order // and the right-hand curve distortion of the order y*. These neutralise each other at a certain distance D from the axis or centre of the field D of view ; but at a distance equal to -j~ from the axis there occurs a \'2 maximum of + distortion equal to ^th of the distortion that occurs at D, and outside that a rapidly increasing distortion. In the case of certain forms of four-lens erecting eye-pieces largely favoured by Continental opticians, and consisting of four compound and achromatic lenses, this compound curvature of straight lines, consequent upon a still greater degree of distortion of the order ?/ 4 opposed by distortion of the opposite sign of the order y 2 , is still more noticeable. It is clear that since the distortion of the order y* increases as tan 4 ?/ or the fourth power of the semi-diameter of the apparent field of view, therefore the size of the latter cannot be very much increased without the hybrid distortion showing itself in an aggressive manner. Doubling the size of the field of view will multiply the defect sixteen times. Cooke Photographic Lenses These lenses, which are composed of two simple collective lenses containing between them a simple dispersive lens, form good practical examples of the embodiment of the formula MAGNIFICATION 277 {1Y + - ' B B'} - 0, 4 I ~ for the two collective lenses of focal lengths /j and f s are separated from the dispersive lens by separations s r and s. 2 , which are proportional to/! and/ 3 ; and when the distances from the object to L x and from L 3 to the image are also proportional to / a and / 3 , and L a and L 3 are symmetrically shaped with respect to one another, then clearly the conditions of vergency as well as of shape of the lenses L : ' and L 3 are all symmetrical if the principal rays are supposed to cross the optic axis at the centre of the lens L, ; so that = = 0, also a L & = (a 3 /S s ), and a : = a. v /3 l = /S 3 , &\ # 3 , etc. Therefore the system is free from distortion, and practically remains so under all normal conditions. Magnification We have yet to consider the important question of the magnifying powers of lens systems. It is quite obvious that if the eye views a distant fiat object and fixes itself upon some central point C, then various other points in the object will seem to be distant from C by certain angles fa, fa, etc.; and their apparent distances from C as measured in the plane of the object will be proportional to tan^, tan< 2 , etc. On approaching to a distance equal to -th of the first distance, the apparent distances of the same points from C will be proportional to n tan fa , n tan fa , etc. If, instead of approaching n times nearer, an optical contrivance causes principal rays to make angles equal to n tan < x , n tan fa , etc., with the axial line through C, in place of tan fa and tan fa, etc., then clearly the magnifying power = n. So that if, in the case of the telescope, we write tan < for the tangent of the angle included between the optic axis and the principal ray from any point in the distant object, and tan <' for the angle made with the optic axis by the same principal ray after emerging from the instrument, then clearly - - will express the magnifying power. This is of course equivalent to the ratio in Airy's and tan* Coddington's Formulae II. for the distortion of eye-pieces ; in which 278 A SYSTEM OF APPLIED OPTICS Use of the dynamo- meter. tane = tan<, or the original visual angle subtended at the object glass, and tan 77 = tan telescope. form -, , in which F = the equivalent focal length of the object glass, and / the E.F.L. of the eye-piece. Supposing neither F nor / are exactly known, then the familiar device of measuring the diameter of the image of the aperture of the object glass formed just beyond the eye lens with a dynamometer, when the telescope is focused for distant objects, and dividing the same into the aperture of the object glass, may always be relied upon to give fairly exact results. Theoretically the method is quite exact, as the following reasoning will show. When set for normal eyesight the first principal point of the eye-piece is distant from the second principal point of the objective by a distance equal to F +/. Now let F = mf, so that m is the magnifying power. Then the two conjugate focal distances, with respect to the eye-piece, of the object glass and its image will Proof of the accuracy of the dynamometer. clearly be (m + I)/ and = (m + 1 )/ and 1 1 _ / ~~ (m+[)f 771+1 or (m+l)f and - -/; 1 m (m+1)/ and consequently the image of the objective will be -th of the original size ; and therefore the ratio m expresses the magnifying power of the telescope. The only thing which militates against the accuracy of this method is the violent spherical aberration to which the image of the object glass is subject in many cases. Also many cases arise in the case of three- or four-lens eye-pieces in which the image formed behind the eye-piece is not really an image of the objective at all, but is an image of the stop between the first and second lens of the eye-piece, which is, either intentionally or not, made too small to pass the full image of the object glass thrown into it by the first lens. In such cases the best plan is to place an artificial circular aperture of smaller size over the object glass and divide its aperture by the diameter of the image of the same formed by the eye-piece. IX MAGNIFICATION 279 The conventional standard of distance. The Simple Microscope Here we have to deal with a somewhat different state of things, for the apparent size of the original objects, which are close at hand in the first instance, is evidently quite arbitrary ; a short-sighted person may view an object with his naked eye 6 inches away, and see it magnified three times relatively to a person who can only see it clearly with the naked eye at 18 inches away. Therefore the convention has been adopted of accepting 1 inches as the standard distance at which the normal naked eye can comfortably view small objects, and therefore all microscope magnifying powers are estimated relatively to that conventional standard. First, it is clear that in the case of using lenses of low magnifying Advantage of being power the short-sighted person will clearly have an advantage, as he can place his magnifier nearer to the object and deal with more divergent rays than the long-sighted ; and, again, the question is further complicated by the variation occurring in the distance of the eye behind the lens. Let / be the E.F.L. of the lens, u its distance from the object, and D the distance of the eye from the lens, all in inches. Then the J_ = fu 1 short-sighted. conjugate focal distance v will be image from the eye will be and the distance of the u ~f u ~f If the eye were at the lens centre, then clearly the conventional magnifying power would be quite independently of the position of It the second conjugate image, but the eye is at a distance from the image which is reduced by D, therefore the magnifying power becomes 10 v u v- D 10 u fu- D(u -/) 10 fn-D(n-f) fa-D(u-f}' Formula for the mag \ I nification of a simple ' microscope. As a general rule v is a minus quantity, since the emergent rays constituting the pencils are diverging. If they are converging, then of course D gives a gain in magnifying power instead of a loss. 280 A SYSTEM OF APPLIED OPTICS SECT, ix The Compound Microscope Here there is a real image of the original formed behind the objective, and this image is viewed through an eye-piece, which yields a further magnifying power. Let F = the E.F.L. of the objective, and / that of the eye-piece, and U and V the conjugate focal distances of the object and image respectively, and let it be assumed that the rays emerge parallel from the eye-piece. If the eye were placed at the first principal point of the objective it would see the object under a magnification equal to -y ; and if it could turn to the second principal point and look the other way it would see the conjugate image under exactly the same visual angle, and the magnifying power would still be -==. If the eye then views the conjugate image through the eye-piece, the magnifying power will be obviously increased in the ratio -^ ; there- fore the whole magnifying power will be 10 V v'7 Now we may call V, or the distance from the second principal point of the objective to the enlarged image, the effective length of tube, which may also be written as nY, so that we have 1 - 1 -!- 1 *- 1 m) U F F~ nY ' so that our formula becomes Formula for the /is v magnification of a in / 7 ljl__ 1 - i ft ( % ~~ \ VTT compound micro- \ F / / \ f I scope. As in the compound microscope an image of the objective is formed just behind the eye-piece, therefore the eye cannot be far removed from the latter if the whole field of vision is to be seen ; nor, in the case of high-power eye-pieces at any rate, will the state of divergence of the emergent rays very appreciably affect the truth of the above simple formula, SECTION X ACHROMATISM So far as we have yet proceeded, we have generally treated the rays refracted by any particular lens, element, or parallel plate as if the refractive index p were a fixed quantity. Our next task is to consider what follows from the refractive index, varying, as it does, for the differently coloured rays usually constituting the pencils of light refracted through lenses. It may fairly be assumed that the reader will be quite familiar Elementary formulae. with the simpler formulae relating to achromatism, yet for the sake of completeness it is desirable to recapitulate the usual formulae, and then pass on to the new theorems and formulae contained in this Section. First of all from our familiar formula for a thin lens we deduce and since - + -( or - for brevity ) = r s\ p J / (p,-l)l 1 \ Au 1 Variation of the power of a lens due F/ fj, - 1 F to colour variation. So that the variation of the power of a lens consequent upon a variation A//, in the refractive index is equal to the power of the lens multiplied by ~, which is the well-known expression for the disper- fj, - 1 sive power of the glass for the range of rays dealt with. 281 282 A SYSTEM OF APPLIED OPTICS SECT. Then from the formula for conjugate foci ^ = ^ - == we derive, if U is constant arid F varies, as in Formula I., Linear value of chromatic aberra- tion. so that -IF (1) Thus the linear chromatic aberration, as measured along the optic axis, varies directly as the square of V, the distance to which it is projected by the lens, just as in the case of spherical aberration, only with this difference, that the linear chromatic aberration is quite independent (except in the higher orders) of the aperture or form of the lens and of the state of divergence or otherwise of the entering rays. Thus the characteristics a and x do not as yet enter into the case at all, and the chromatic aberration depends only upon the power of the lens and the dispersive power of its material. But it is quite clear that the aperture of the lens must exert a proportional effect upon the size of the least circle of chromatic aberration through which the range of coloured rays will pass. This least circle is obviously situated half-way between the focal points for the two extreme colours concerned, and its diameter is equal to half the linear chromatic aberration multiplied by the ratio of 2 a aperture to the conjugate focal distance V, or =r , wherein a is the semi-aperture of the pencil or lens. So that the diameter of the least circle of chromatic aberration is expressed by Diameter of least circle of chromatic confusion. Angular value of above subtended at objective. * V 1. y2 \2a 2V- 1 F /V A/* 1 - 1 F and its angular diameter as subtended at the lens centre is IA. IB. which shows that, supposing the aperture is constant, the angular diameter of the least circle of chromatic aberration varies inversely as F, a fact which was realised in a very practical manner by astronomers and opticians such as Huygens and Hevelius in the early days of the AXIAL ACHROMATISM 283 simple objective, for they made a great point of having the focal lengths of their telescopes as long as possible, 120 feet being nothing unusual. We have also seen in Section IV., page 110, that the least circle of confusion consequent upon spherical aberration has an angular diameter which varies inversely as the cube of the focal length when the aperture is constant. If we put two thin lenses in contact, with a view to producing an achromatic image in the conjugate focal plane of the compound lens, then we must fulfil the equation A .. 1 A .. 1 II. -1 F IL - I F r l P-2 i r 2 in which A/A I or A/u, 2 refer to the respective differences in refractive indices for any two coloured rays of the spectrum that may be fixed upon. These are generally the orange-red . ray known as the C ray, and the blue-green ray known as the F ray. Since in all known glasses the refractive index increases as we ascend the spectrum from red to violet, and A/i is always of the same sign for different lenses when it refers to the same spectrum interval, must be of opposite signs, and therefore it is clear that ^- and that /*2 ~ l F (2) Two lenses in con- tact. Condition of axial achromatism. that is, the dispersive powers of the glasses forming the lenses must be in inverse proportion to their powers or in direct proportion to their focal lengths. Also, since the resultant power of the contact combination is simply =- + , it is clear that the fulfilment of Equation II. *1 *2 demands that the lens of the greater power shall be made out of glass of the least dispersive power, and then its power will prevail over the other. So that if the combination is to have positive power, then the collective lens must be made of the glass of the lower dispersive power ; and if the combination is to have negative power, then the dispersive lens must be made out of the glass of the lower dispersive power. Thus if Dispersive powers in proportion to focal lengths. then =- will be 4ths =- , and the power of the combination will be 284 A SYSTEM OF APPLIED OPTICS SECT. _1 3 1 _2 _!_ "F! 5 "Fj " 5 F/ The glasses usually This is the ratio of dispersive powers generally prevailing in the g lasses used f r ordinary telescope objectives, the collective lens being generally made of a crown glass having a dispersive power of for the spectrum interval C to F, and the dispersive lens out of a dense flint glass having a dispersive power for the same spectrum interval equal to . It is clear that any contact combinations of ob a collective with a dispersive lens may be achromatic for all degrees of divergence or convergence of the entering rays. Thin Lenses Separated by an Interval Should an interval s exist between the two lenses, Formula II. will no longer apply. Since ^ ^- is the chromatic aberration /*i~ *i of the first lens, and j , V is the longitudinal chromatic ab- /i Ih ~ erration as measured along the axis, or the chromatic variation of v lt therefore from the centre of the second lens as a reference point the chromatic aberration of the first lens which must be neutralised by the chromatic aberration of the second lens. Therefore the formula for achromatism is Two separated 1 A/^ v^ 1 A/* 2 lenses. Condition of T ~ ~~1 2 "*" T ZT = axial achromatism. / 1 /*i ~~ % /2 /*2 ~ /0 \ 2 So that the greater is the separation multiplier (-M the greater is the \u 2 / chromatic aberration which the second lens has to counteract. /V \ 2 But, since ( M can be made practically equal to unity by assuming v l to be a very large quantity compared to s, as when the rays leaving L x are about parallel, the formula in such circumstances becomes practically the same as Formula II. Axial achromatism Hence it is clear that while Formula III. may be equated to for of two separated J lenses not constant, any given value of u^ , yet if u^ varies considerably and thus causes AXIAL ACHROMATISM 285 7) \ - 1 ) to vary, the condition of achromatism will no longer hold good. ittg/ Hence no separated combination of a collective with a dispersive lens can possibly be achromatic for all degrees of divergence or converg- ence of the entering rays. While the equivalent focal length of the combination is constant, as we saw in Section III., yet the chromatic aberration varies according to the radiant distance %. But it can be Axial achromatism shown that under certain circumstances a combination of three or more ^g*^ separated thin lenses may yield a practically constant chromatic tically constant, aberration under all circumstances likely to occur in practice. We may now extend Formula III. to a larger number of separated lenses. Supposing we have three lenses, then from the centre of the last lens as a reference point the chromatic aberration of the first lens as a variation of - is Mo A /*i - and that of the second lens is and that of the third lens is 1 simply. Proceeding in the same way for n number of lenses we get the general formula A _ 1 A/^ fv^) z 0> t - iV 1 A/* 2 (V z ..v n -i ' / i] - 1 \_ 11 / f n - 1 W /i A^i v U 2 U 3 "'n / J-2 ^2 k *8' * / u_ 1' / /* which is strictly applicable also to a series of elements. ,. Condition of axial achromatism for a series of separated lenses. The Linear Chromatic Aberration of a Parallel Glass Plate But in order to apply the formulae to thick lenses by the method of elements, we must next work out the formulae for the chromatic aberration of a parallel plate of glass. Let Fig. 104 represent a case of a divergent pencil of white light originating from Q and passing perpendicularly through the parallel plate of thickness A l . . A 2 or t, and Fig. 104a the corresponding case 286 A SYSTEM OF APPLIED OPTICS SECT. of a perpendicular pencil of rays converging to Q. After refraction at the first surface the less refrangible rays, such as the red, will be divergent from or convergent to r 1} such that r^ . . A l = yu, r (Q . . Aj), while the more refrangible rays, such as the blue, will be divergent from or convergent to b l} such that ^ . . A l =yu, b (Q. . Aj) ; so that the distance between & a and r-^ will be Q . . A l (//, 6 yu, r ), or, shortly, (A/z). Then as a correction to the reciprocal value of the distance i\ . . A 2 in the case of Fig. 104, the quantity w(A/^) becomes ^ ; that is, (r l . . A 2 ) 1_ 1 MA/A 1 ij . . A 2 r l ..A. 2 (fr.u + t) 2 n r u + t (fj, r u + t) 2 ' After refraction at the second surface ( becomes ^ or .u + t ,u + t - or -, which =- - , and 7 - U ** vo becomes - , v ?- 2 . . A 2 (pr* + t) z (fi r u + 1)' 2 Now, supposing the other ray, or the blue ray, were also radiating from the same point r a as the red ray before refraction at the second surface, then after refraction we should have the blue rays apparently radiating from &/, such that the distance & 2 ' . . A 2 would be equal to r, . . A, r, . . A, a r u + t - 2 , which = ] -^ or - , so that t (3) so that ^. is the increment to - due to colour consequent upon the fj. r U + t V second refraction only. But we have seen that the chromatic aberration brought over from the first surface and referred to the point A 9 was . ~- , so that the chromatic aberrations of both OVK + if. surfaces are /j, r u + 1 /j, r u t ' 2 ^ 2 z "' Parallel plane plate. But ^ r u + 1 = fj, r v, so that the chromatic aberration becomes The chromatic vari- ation of - n s-. as a correction to - V. V A,.V V PLATE. XX II. b, r , i. IO4.a. PLATE. XX II. b, x AXIAL ACHKOMATISM and the linear value r 2 . . 6 2 of the aberration is simply yj The linear chromatic variation of v. If Ayu, refers to a large interval of spectrum and the glass is highly dispersive, it is more correct to write t^- or <^-. VIA. /M*6 Wb The same line of reasoning applied to Fig. 104a leads to the same result, provided we consider v negative, so that in the case of Fig. 104 we have 1 1 A/t ~ + * b 2 . . A 2 v . and in the case of Fig. 104 we have l>2 . . A 2 V {Jifv 2 Iii both cases we find the linear chromatic aberration i\ . . & 2 ranges The dispersion al- left to right ; a plus increment to p implies a transference of the focal ^^ s point in the same direction as the light is travelling, and in this sense the effect of a parallel plate is similar to that of a dispersive lens only with this difference, that while the chromatic aberration of a dispersive lens is -. ^-r and thus independent of u or v, in the case of the parallel plate the chromatic aberration - varies inversely as v 2 , and of course vanishes when v becomes infinite and the rays parallel. We might have arrived at the same result more shortly in this way. Since the linear transference of the focal point due to passage through a parallel plate is, as we have seen in Section L, t , then in differentiating with respect to //, we get y-~ ^ - = t ^, only we P- P" should have missed noting the effects taking place at each surface. Chromatic Variation of the Spherical Aberration So far we have studied the effects of //, being a variable upon the formulse of the first approximation, and it is now desirable to investigate the effect of p being a variable upon the spherical aberration of a lens. It is a subject of considerable importance in the 288 A SYSTEM OF APPLIED OPTICS SECT. theoretical designing of achromatic object glasses of larger aperture, and especially when of relatively short focal length. For it is some- what futile to take elaborate precautions to ensure any two colours being refracted exactly to the same focal point by formulae applying Importance of two to ultimate central rays, and also get the spherical aberration of the from^phericaf aber* ra 7 s of tne one colour perfectly corrected and then allow the rays of ration - the other colour to be subject to strong spherical aberration, thus to a large extent discounting the advantages of achromatism. If we take the formula for spherical aberration, , , , - and tor ^ put I - + - ) (p If, or simply ( - ) , since ^ is a variable / P ' J depending on //,, we may then write it in the form 1 /0*+2)fr- 1),. 2 + *(/* + !)(/- !) 2 ^ + (V+ 2)0*- D 3 a2 Q o < *t- T U.*v T ~~~~ fX 8/>l /* /* /* I , ^- l) }i-.\ P i } On differentiating with respect to //, we shall then find that Differential of the ~ 5~q i ( 1 5 )# 2 + 4( 2u - 1 9 }aX + ( 9u 2 - 1 4u + 3 + - op I \ it*/ uv us sphencal aberration with respect to p.. + /3 ^2 _ 2 , Supposing /A = 1 '5 this works out to O, /> *^ , O.1O *' i O. p r + o-ao; + o 1 ocr 4- o / VII. If - = 1 (for a focal length of 2), x = + 1, and a = + 1, and y = ~r t then P 13 the formula works out to + r-^-^/i ; and since '01 is a very liberal 1 ^O allowance for dp for the brighter part of the spectrum in the case of glasses of low refractive index, we then get -A = = about -001. But the spherical aberration in such a case would be AXIAL ACHROMATISM 289 1 1 , .. 1 27 1 l6 = 48< 27) 16 - 768 = 28 ' or 36 times the above variation due to dp. Such a small quantity as this might almost be neutralised by parabolising the curves of an object glass or the reverse if there were only rays of one colour to be dealt with ; but it is clear that if we have perfect correction for spherical aberration for one colour, whether it be by a perfect balance of curves or by figuring, then a very minute amount of spherical aberration for another colour will be perceptible under high magnifying powers, so that the correct balancing of the spherical aberration for all colours as far as possible assumes a great importance. This means that in the case of a double achromatic object glass it is desirable to fulfil the condition d {gjp A >l} - ^t{2 s A >l} = i ( 6 > or if it does not or cannot equate to 0, then we must introduce another influence to effect it. In the case of an ordinary achromatic objective with the collective lens at the front and double convex, and the dispersive lens double concave or concavo-convex, but in close contact with the collective lens, it will be found that the chromatic variation of the spherical aberration as expressed shortly in (6), and in detail for the collective lens in Formula VII., is negative ; that is, the dispersive lens exerts the greater influence, so that the more refrangible rays are over-corrected for spherical aberration when the less refrangible rays are accurately corrected. Apparently Gauss was the first to point out that a separation The separation de- between the two lenses could be made to neutralise this defect. ce ad P ted b y ,,. Gauss. Let Iig. 105 represent two lenses separated, Lj . . L 2 being half the collective lens and L' . . L" half the dispersive lens, and L 2 . . F the optic axis. Let F be the focal point by first approximation for the red ray (ray C) and / the focal point for the blue ray (ray F) for the collective lens, so that F . ./ or shortly 8, is the linear chromatic aberration which *i.lM-,i .A ft-i.v Let the semi-aperture of L : be Y and the semi-aperture of L 2 or the height L" ..r be ij v We will assume the red ray L, . . F to be the standard ray which gives the values j and j. fl /2 290 A SYSTEM OF APPLIED OPTICS SECT. Two separated lenses. Value of y, 2 in terms of // , . Oblique images formed by a separ- ated double objective cannot be achroma- tic. Let L 2 . . F be v 1 and L" . . F be u z , and the separation L 2 . . L" be s. Let the height L". . b to where the blue ray Lj . ./ cuts the second lens be called y z . Then our object is to express y z in terms of y^ allowing for the dispersive effect of the first lens. First we have yi = Y^, (7) y* = 1 "1 ^ ^ 1 ' VV 1 /1/*1~ /1/*1~ M 2 xr^o/V, Au, \ . , . Mo = Y - Y I - L - L -s ), in which Y y, ; V, V, \U a u,-, - I / V, (8) so that so that finally .U 2 ft - 1 2 _ - 2/*l U - VIII. So that for the second or dispersive lens the spherical aberration may be written shortly and since / 2 is negative, therefore the aberration is negative, and the variation -2- 1 W -r comes out a positive one, and may be made to neutralise the negative result of Formula (6) as applied to the same combination. Unfortunately, however, the separation between the two lenses of a double objective exercises a most prejudicial effect on the images a little removed from the axis ; it is impossible to have the different coloured oblique images of any given star depicted at the same point on the focal plane, the blue images falling farther from the axis than the red (if the collective lens is at the front), while there is also a large amount of corna, so that the available field of good definition is very much restricted. OBLIQUE ACHROMATISM 291 OBLIQUE ACHROMATISM AND CHROMATIC MAGNIFICATION The foregoing remarks about the double separated objective brings us to the question of the conditions which determine whether an optical system forming an image of a real object, distant or otherwise, shall paint the said image on a dimensional scale which shall be independent of the colour or refrangibility of the various rays making up the pencils of white or mixed light diverging from the original object. We have just noticed that an achromatic objective consisting of two separated lenses with the collective lens to the front is only achromatic for the axial image, and that the oblique image of a star is not a true image, that it is drawn out into a minute spectrum, the red end of which lies towards the optic axis. If the dispersive lens were at the front, then the opposite state of things would result, and the blue end of the spectrum would lie nearest to the optic axis. It will be as well in the first instance to recapitulate the inquiry made by Sir George Airy and Henry Coddington into the conditions for securing oblique achromatism or equal magnification for the different colours that have to be fulfilled in the case of two-lens Huygenian or Eamsden eye-pieces or three- or four-lens erecting eye- pieces. It is assumed in all such cases that the oblique pencils of rays emerging from such eye-pieces are made up of parallel rays, that is, that they are proceeding from an apparently very distant or infinitely distant virtual and magnified image. Such being the case, then it is clear that if the oblique image of Coloured constitu- any point of white light, such as a star, is to appear to the eye as ^ys em^rgiSgsepar- one white image, then the variously coloured rays constituting the ated but parallel, mixed oblique pencil must be emerging parallel to one another, and whether or not there happens to be any lateral separation of such variously coloured pencils of rays does not matter, provided that the virtual image is infinitely distant. We saw in Section IX., pages 247 to 254, that freedom from distor- tion in such a case depended upon the ratio of the tangent of the angle of emergence of the principal rays to the tangent of the angle of incidence being a constant throughout the field of view, and that Formula HA. was, for two lenses in succession ten, tane -l 292 A SYSTEM OF APPLIED OPTICS SECT.. Now if we differentiate the functions T' and E f for each lens with respect to //,, we shall find that the variation in the functions corre- sponding to d/j, comes out very small compared to the functions them- selves ; we worked out a case on pages 288 and 289, where the 1 chromatic variation in -^ 3 A.y was only g^tfo part of the latter, and in most cases likely to occur in practice it would amount to still less. Now the function A' is almost exactly similar to B'. And since the Chromatic variation distortion functions in eye-pieces rarely amount to more than 5 per of the distortion very cen j. Q f ^ Q ra( jj a } dimensions of the image, it is not to be expected that -^g-th part of that, or less, would be at all noticeable. So that we need not in ordinary practice trouble ourselves about the chromatic variation of the distortion functions. It is in the exterior magnification function -^ - (for n number of lenses) that we must look for the vastly more important chromatic variation ; for it is plain enough that all the terms with the exception of &! are variables; they depend upon focal lengths, and the focal lengths are different for the different colours. Conditions of oblique Let Fig. 106 represent two thin lenses Lj and L 2 in succession, of achromatism of eye: f oca i lengths /, and /, of the same glass, and separated by an interval pieces for normal j\ ji- . . .. vision. s (less than /i). Let principal rays be diverging from an axial point Q to the left, so that Q, . . L x = b^ If these two lenses are used as an eye-piece for a telescope or microscope then Q will represent the centre of the objective. Also, in order to suit normal vision, the rays constituting the pencils of any one colour emerging from L 2 must be considered parallel, so that v z = oc. In such case it is clear that if the variously coloured images are to appear all of the same size, then a multi-coloured principal ray, which enters the eye-piece all as one,, must, after being split up by the first lens into a fan of diversely coloured rays, emerge from the second lens with such variously coloured constituent rays parallel to one another, when they will all appear to originate from one and the same point in the infinitely distant image. Therefore for all eye-pieces the condition for achromatism for oblique pencils is that tan 77 = constant for different values of /A; that is, that dp tan 77 = 0. Therefore we first want to express tan 77 in terms of 6 1; y^f lf / 2) , s, and y^. We have Tan ?; not to vary with /A. tan T? = ^ ; C 2 1 "A fA OBLIQUE ACHKOMATISM 293 .'+. , = C 2 *2 /2 C 1~ S /2 /a( C -/ fcvsr* ta '-?-^'trW-,v& jy * o T f O , fl 1 1 /I l\sl /11X Two-lens eye-piece. tani ? = yi{7-r + 7-l/-r//r Valueoftan7,. VI l /2 Vl V/2- We now have to differentiate this expression with respect to /i. Leaving out . , which is a constant, we have mi TCP . i p 1 1 A/1' o 1 1 Au, , 1 . 2 Au, The differential of - is - ^-, , of T is -r- - , and of p is T - - A A /* ~ l /a /2 /* ~ * '1/2 A/2 /* ~ therefore we have (1+ L.. 2 * +-L.) A ^ =0 ; A/2 "1/2 /i /a i and , ,. Two-lens eye-piece. s _ J\ + ./ 2 IXA. Separation neces- / ' sary to oblique T achromatism. If &j is large relatively to f l} then we arrive at the well-known rule of the separation being half the sum of the focal lengths. If b is relatively small, then the lenses must be more widely separated. In order to secure better corrections for astigmatism, distortion, Condition of oblique achromatism for a two-lens eye-piece. 294 A SYSTEM OF APPLIED OPTICS or coma, it is sometimes desirable that the two lenses of such an eye-piece shall be made of glasses of different dispersive power. Let ,-j --_- -. = the dispersive power of the first lens or field lens, and _ = the dispersive power of the second or eye lens, and let the dis- persive ratio AM M-l then it can be shown that IXB. 6, from which it appears that a stronger dispersive power in the field lens leads to a smaller separation, and a stronger dispersive power in the eye lens to a greater separation. It will scarcely be necessary here to recapitulate the much more complex and lengthy processes of the same nature which have to be gone through in order to arrive at the condition for oblique achroma- tism for eye-pieces consisting of three and four separated lenses. Let it suffice to simply state the results. The reader will find the investigation in full in Coddington's work, Part I., pages 259 to 268. Condition of oblique achromatism for a three-lens eye-piece. Condition of Oblique Achromatism for a Three-Lens Eye-piece Let/u/2, an d /s be the principal focal lengths of the three lenses, all being made of the same sort of glass, and s^ and s 2 the first and second separations, and ^ for the first lens being assumed infinite or relatively large. Then the achromatic condition is A/2 + A/8 + A A - 2 /i s 2 - 2 /2 s i - 2 / 2 * 2 - 2M + 3 v 2 = 0. IXc. Condition of Oblique Achromatism for a Four-Lens Eye-piece x, / 2 , / 3 , and / 4 be the principal focal lengths of the four lenses, all of the same sort of glass, and s l} s 2 , and s 3 the three separations in order, and let \ be considered infinite or relatively very large. Then the achromatic condition is OBLIQUE ACHEOMATISM /1/2/3 +/1/2/4 +/1/3/4 +A/3/4 2/ 1 / 2S3 -2/ 1 / 3 6- 3 -2/ 2 / 3S3 S/gS.^ + S/^Sg + S/^Sg + 3/38^3 + S/ Sg + 3/ 2 s 2 s 3 295 x Condition of oblique achromatism for a four-lens eye-piece. The condition for a five-lens eye-piece works out to a very much more cumbersome formula. Fig. 107 will help us to realise the very restricted usefulness of all these formulae. It represents the last or eye lens of one of these eye-pieces, preferably that of a four-lens eye-piece. Since the lenses are all simple, therefore the chromatic aberrations all sum up together, so that at the position about P, where the principal rays cross the optic axis and where a rough image of the object glass is formed, the crossing point p for the blue rays is very much nearer the lens than the crossing point P for the red rays. We have two oblique principal rays one red, Q x . . . . P . . Q, and one blue, q l . . o . .p . . q which entered the eye-piece as one ray, finally emerging separately but parallel to one another, and to the normal eye with its pupil placed at P or p the two rays seem as one. But supposing we wish to use the eye-piece for projecting a real image of what is seen in the telescope or microscope on to a screen Gr . . Q at a short distance to the right, and for that purpose draw out the eye-piece. It is perfectly clear that such an image cannot be achromatic, for the red ray will strike the screen at Q and the blue ray at q ; so that the blue image of any extended object will be painted on a larger scale than the red image. On the other hand, let it be supposed that a very short-sighted person uses the eye-piece. He will have to push the eye-piece farther in towards the objective, in order that the emergent rays of pencils may be divergent as though proceeding from a virtual image Q a . . G : 8 or 10 inches to the left hand. It is again clear that such an image cannot appear achromatic, for the blue principal ray appears to be coming from a point q l nearer to the axis than the point Q l for the corresponding red ray ; the red image is now painted on a larger scale than the blue image. Supposing we want to project real or virtual images to or from finite distances, then what help or enlightenment can we possibly obtain from formulae of the nature Lateral displace- ment of coloured constituents of the principal ray. Real image larger for the more re- frangible rays. Virtual image larger for the less refran- gible rays. Constancy of tangent ratios useless where real images are formed. tan 296 A SYSTEM OF APPLIED OPTICS SECT. Such formulae, however useful they may be for eye-pieces, are absolutely useless for working out the oblique achromatism of combinations, such as photographic lenses, which are expected to form real images of real objects at finite distances. Formulae of Perfectly General Application We must therefore seek for a formula of perfectly general applica- tion, and in so doing may with advantage pursue the same method or line of reasoning that we followed in arriving at our general formula for distortion in the last Section. In Fig. 108 let a principal ray radiate from Q and take the eccentric course Q . . N . . P . . j through the lens M . . N. We are supposing the lens free from spherical aberration and the tangent condition fulfilled, since we are discussing solely the effects of variations in the refractive index. Let there be an image formed at o . , whose, radial dimension is o. From o draw o . . M through the centre of the lens, and produce it to cut the conjugate image plane I . . j at /. Let it be assumed that the principal ray Q . . N . .j is of the standard colour, for which the refractive index yu, applies, and that the conjugate images o . . and I . .j also apply to rays of the same standard colour. It is clear that another more refrangible coloured ray coincident with Q . . N before refraction will take a different course N . . p . . / T after refraction, and j..j l will be the linear dispersion between the two colours. Now what we want is an expression for j . . j l in terms of I . ./, or the radial dimension of the blue image in terms of the radial dimension of the corresponding red image, supposing we fix upon those two colours. Let I . .j be i, and I . ,j l be i l} and o . . be o ; Q . . M be b, O . . M be u, M . . P be c, M . . I be v, and M . . p be c v Then we have i = o - , u also o , = M . . N = i o -u v - c b v- c b v~c v b-u c c b -u u 1 and b v-c-. . 1 A//, i,=0i --- . wherein c, - c- - - -r-c ; b-u c x / //, - 1 so that ( 1 A /* b v -( c -J-~ } 1=0 x J * b-u i \,, OBLIQUE ACHROMATISM 297 A/1 1 A/* '- -^ - c 1 Au, . 1 Aw, -+7 ,C + (*'- 6)7 -- C ///-! '/'/*- i fv-c v A/i^ = -- f- _ -- _ V - M I C il) ffi b v - c( cv A/A 1 ^ -w c I v-c p-I f )' in -which the outside function =0-, from Formula (13); u cv 1 ^/*j On adopting Coddington's device we fiud that 2/ 2/ CV 1 - P 1 - a 2/ so that finally we get 2f_ 2/ -/T I O f. 2 A, (14) XI. Single lens. Uni- versal formula for ratio between object and coloured image of same. a very simple and convenient formula which can be applied with the greatest ease to any number of lenses or elements in series. The term /has disappeared, but its value is really implied in a /3, which terms are, of course, assessed with regard to the ray of standard colour. On applying the same line of reasoning to the corresponding case of a dispersive lens, or any other cases whatever, exactly the same formula will be arrived at. An objection may be raised to the above formula on the ground An objection, that a ft is in itself a variable, for it varies as /, which varies inversely as /A 1 ; but if we insert the variation in a /3, we then get for our formula - = -^ l + M t 1 = -n + (A/.) 2 n\ i / T \9 / I ' a. p \fi 1 (yu, - 1 ) / J The correction involved is thus seen to be of the order ( "^ ) or the 298 A SYSTEM OF APPLIED OPTICS SECT. Universal formula for ratio between object and final coloured image for a series of lenses. Series of lenses. Con- dition of oblique achromatism when all of same disper- sive power. square of what is already a very small quantity. Hence it may be legitimately neglected. In applying Formula XL to a series of lenses in succession, it is clear that a lens will copy or transfer forward any want of chromatic conformity in the radial dimensions of any image presented to it by the preceding lens or lenses, and at the same time add its own chromatic error, and so on. Therefore the expression for a series of n lenses is Four - lens erecting eye-piece. 1 + 2 A/* 2 XII. "1 r*i r"l 8 "2 r*2 x u n A-'n. /*n ~ " ' Then, if all the lenses are made of the same sort of glass, the condition of oblique achromatism is simply 1 1 1 = 0. XIII. f) O a l ~ Pi a 2 ~ P-2 a n ~ Hn Let us apply this formula to the ordinary Huygenian eye-piece wherein / a = 3, / 2 = 1, and which we have seen is achromatic when s = 2, provided that b l = ,/-00487 -01041\ , - F 2 -) for the interval F to G : \ / and it is clear that there will be prevailing dispersion of the crown lens along the axis from D to A', the A' ray focusing beyond the D ray ; from F to D the dispersion of the flint lens will predominate, and the D ray will focus inside of C and F ; while for the region Proportional sectional dispersions for crown and flint Condition for bring- ing C and F rays to one focus. The interval C to F, or Ayu,(Cto F), usually taken as unity. Formula for the chromatic error D to A'. Formula for the chromatic error F toD. Formula for the chromatic error F toG'. 308 A SYSTEM OF APPLIED OPTICS SECT. F to G' the flint glass dispersion will again prevail, and the G' ray will focus considerably beyond the C and F rays. As an example we will take the case of a double objective of 30 feet focal length composed of the crown and flint glasses whose main characteristics have been given above, only the values of - and 1 Pl - are so calculated as to cause a ray half-way between B and C of Pz the spectrum to focus to the same axial point as the ray F, which arrangement is likely to give the best colour correction for an objective of that size (upwards of 2 feet aperture). VARIATION or F FOR THE DIFFERENT COLOURS (IN INCHES) FOR A TELESCOPE OBJECTIVE 30 FEET E.F.L. Ray . A' B E F G' h H, Minimum Variation + 33 + 05 -04 -20 -24 -19 1 + 80 + 1-40 + 1-88 Table of chromatic errors of a 30-foot double objective. It will be noticed that the total is 2'02 inches, and that the largest minus variation occurs about half-way between D 2 and E, where it is '24. This is about the brightest part of the spectrum from a visual point of view, and since the maximum light concentration The minimum focus, obviously occurs at the minimum focus where a high value of A//, or range of spectrum may coincide with a very small variation from the minimum focal point, it is highly important that this light concen- tration should coincide with the position in the spectrum of the greatest visual intensity, unless the objective is specially designed for photographic purposes, when the greatest effectiveness and best definition is obtained by arranging for the minimum focal length and Chromatic correc- maximum light concentration to occur for a ray a little on the less refrangible side of the G' ray (the hydrogen blue ray), at which position in the spectrum the usual photographic plate is most sensitive. We will here give the variations in F for such a telescopic objective for photographic purposes of the same focal length of 30 feet. VARIATION OF F FOR THE DIFFERENT COLOURS (IN INCHES) FOR A PHOTOGRAPHIC TELESCOPE OBJECTIVE 30 FEET FOCAL LENGTH Table of chromatic errors of a 30-foot astro - photographic objective. Ray . , . Variation 1 A' + 2-16 B C + 2-20 + 1-38 1 D 2 + 88 E + -40 F + 16 G' h + 05 H, + 18 x SECOND AEY CHKOMATIC EEEOKS 309 Here it will be seen that the brightest visual rays are scattered along the axis for two inches or so beyond the photographic focus, and the image of a star formed by the G' ray is therefore surrounded by a large halo of wasted light, which imprints itself more and more on the photographic plate as the exposure is extended ; and that is why the photographs of the brighter stars come out so abnormally large when those of small magnitude have just imprinted themselves. Triple Telescope Objectives The only way known of getting rid of the secondary spectrum is by resorting, if possible, to a combination of one dispersive lens enclosed between two collective lenses, the two latter being made of two different sorts of glass, so chosen that the mean of their partial relative dispersion l + 3 , etc., for various regions of the spectrum shall Pi Ps correspond as closely as possible with the corresponding relative partial A / dispersions -*, etc., etc., for the same spectrum regions for the glass Pz used for the dispersive lens. In this way the glasses employed in the Cooke Photo -Visual Objective were chosen ; with the result that the linear secondary colour aberrations for such an objective of 30 -foot focus are reduced to less than one-tenth part of an inch for the whole range of spectrum A' to Secondary spectrum H T , which is only one-twentieth part of the 2'02 inches, the total twentieth, axial chromatic error given above for the ordinary double objective of the same focal length. Why the Secondary Spectrum of Large Double Objectives does not render Clear Vision impossible Eeturning to the case of the visually corrected objective, it can be shown that if the usually accepted theory of the formation of the image by rays of any one colour is correct, then anything like distinct vision through a 30-foot objective of 18-inch to 24-inch aperture would be impossible. Fig. llOa, Plate XXI L, shows a section of the usual conception of the cone of rays converging to form the well-known spurious disc or star image at the focus, and then diverging again, so that the beam of rays takes the form of two straight-sided cones with both their points cut away to the diameter of the spurious disc. If this really represented 310 A SYSTEM OF APPLIED OPTICS SECT. X the case, then only a very small fraction (about 15 per cent) of the light refracted through a 30 -foot objective would be utilised for defining purposes, all the rest being wasted. Happily, however, the real section near the focus of the converging and diverging beam of The tapering-off of rays is as in Fig. 1 1 Qb ; the angle between the two sides of each cone thefocus fraySliear decreas es as the spurious disc is approached, or tails off into the cylindrical shape. This can be proved by experiment, and it is a strange fact that while mathematicians have spent a good deal of work upon the conformation of the spurious disc and its surrounding diffraction rings as they are formed in the focal plane, yet none have entered upon an investigation of the conformation of the cone of rays along the axis as it approaches the spurious disc. Such an investiga- tion, based upon the wave theory of light, should be most instructive and of the highest importance. The tapering-off It can also be proved by experiment that the tailing off into the most marked with ! j i v i i i ' TTI P large relative aper- cylindrical shape takes place in a more marked degree in the case of tures - cones of rays of large angular aperture than in the case of cones of small angular aperture, which fact tells in favour of objectives of relatively large aperture, and discounts their other disadvantages in a substantial degree. However, we are here trenching on the borderland between geometrical and physical optics, with the latter of which this work does not profess to deal. For further information on this subject the reader is referred to a paper entitled " The Secondary Colour Aberrations of the Eefracting Telescope in relation to Vision," in the Monthly Notices of the Royal Astronomical Society, vol. liv. No. 2, also to " Description of a Perfectly Achromatic Eefractor," in the same publication, vol. liv. No. 5 ; both by the author. SECTION XI A BKIEF SKETCH OF THE NORMAL AND OTHER CURVATURE ABERRATIONS OF THE THIRD ORDER TAN 4 (/>, ETC. PERHAPS the most important corrections that the optical designer has importance of a to take into consideration in the course of working out photographic P lane lenses are those relating to the curvature of image or the deviations of the image from an ideal flatness. We found that the deviations from a plane image as calculated by the formulae of Sections V. and VI. applied to the three lenses given as examples in Section VII. differed appreciably from the actually measured results. These discrepancies are indeed scarcely too large to be accounted for by inexactness in the measurement of the curvatures, especially in the cases of the deep curves employed in the process lens and the four-lens Cooke lens. It can be shown, for instance, that an increment of plus value in the convex curvature of a lens of low refractive index, together with a rather smaller minus increment in the convex curvature of a lens of high refractive index, may have the effect of quite reversing the character of a small residual oblique astigmatism without affecting the principal focal length of the combination ; while the increments in question may easily escape all but the most exact methods of measurement. But in the cases worked out the character of the image curvatures at still greater distances from the optic axis proves that the discrepancies are chiefly due to the presence of curvature aberrations of a higher order than those we have yet dealt with. CENTRAL OBLIQUE REFRACTION t The Three Corrections to the ?/s First of all, for the purpose of calculating the y's, we have assumed the refractions to take place in a plane tangent to the vertex of each 311 312 A SYSTEM OF APPLIED OPTICS SECT. be departed from. surface or element, and we may first consider the nature of the The element plane to corrections which would have to be applied in order to allow for the 2/'s being reduced to perpendicularity to the normal oblique ray passing through the centre of curvature, since all the formulae for spherical aberration assume the T/'S to be measured at right angles to the aforesaid ray. Primary Planes Ee verting to Section V., page 121, dealing with the question of oblique rays passing centrally through a lens, we had at the first surface, Fig. 44a, the equation * The corrections for obliquity defined. The versine correc- tions denned. The corrections for positions denned. T7' leading to r = l + !/2 x i in which x l denoted the required oblique distance from d, the oblique vertex, to q, the crossing point of the two extreme rays in the primary plane. These y's were the distances c . . e and c . . e l reckoned in the element plane, as shown in Figs. Ill and 112. Now it is clear from these figures that the y's, so reckoned, become more and more incorrect as the aperture and the angle of obliquity $ increase. The y's are subject to three corrections: (1) the correction for obliquity; (2) the versine correction, and (3) the correction for the positions of the T/'S or the lateral separation generally existing between them. (1) The corrections for obliquity consist in converting the distances c . . e and c..e l in the element plane into the distances e . . g and e l . . a t measured perpendicularly to the normal oblique ray Q . . r. (2) The versine correction is due to the retreat of the spherical surface from the element plane. Let the extreme ray Q . . e be pro- duced to cut the spherical surface at k ; from k draw k . . I perpen- dicular to the normal ray Q . . r ; then through e draw e . . h parallel to Q . . r, and cutting k . . I at h ; then the distance k . . h is the versine correction applicable to e..g in order to convert it into k . . I, which latter is the real y upon which the spherical aberration should correctly be based. (3) Still another species of correction has yet to be applied a correction not of the values of the y's, but a correction for their posi- tions. Fig. Ill explains this. We must bear in mind that the Formula (1) gives the value of ^ or -, that is, the reciprocal value of the focal distance d . . q measured along the normal oblique ray Q. .c. .p, with absolute correctness, provided that PLATE. XXIII PLATE. XXIII. XI CUKVATUKE EEKOKS OF THE THIKD OEDEE 313 1st. The spherical aberration function w is correctly formulated. 2nd. The values of the two y"'s, k . . I and n . . t, are correctly given ; and 3rd. The two y"'s are at equal distances from the focus q, that is, that I . . q = t . . q, in which case the two y"'s would be in one straight line. But it is plain that this can only happen when either the radius r is infinite and the refracting surface is plane, or when it or d . . q is infinite, when, of course, the separation t . .1 becomes a relatively vanishing quantity. It is clear in Fig. Ill that the lateral translation of k . . I or y" towards the right hand, while assuming its length to remain constant, must cause the crossing point q to move to the left hand, nearer the lens ; that is, the correction due to the separation of the ?/"s is in this case of a plus nature, since it adds to the value of -. x It is also clear that this separation of the two y"'a gives rise to a Correction for posi- correction to - which operates in the primary plane only. We shall 16 also find that it works out as a function of tan 4 < and 2 tan 2 <, and therefore comes under the head of the forruulse of the third approximation. We will now treat these corrections more explicitly. tions only applies in primary planes. The Correction for Obliquity Primary Plane It will be better to deal with the question in general terms, first taking the obliquity correction. Let a = the semi-aperture A . . e or A . . e l of the pencil where it Notation, crosses the element plane. Let b = the distance A . . c from the lens vertex to the point in the element plane where the normal oblique ray Q . . r cuts it.* Then if the angle PAQ = <, and QrA 0, as before, and P . . A = u, as usual, then b or A . . c = r tan 6 = rtan< ^db' Let e --ff = yi and e i--ffi = 1/2- Then, as in our earlier inquiry, in Fig. Ill, yi 2 = ( a + j)2 and ys 2 = ( a _j)2 and yi y 2 = ( a 2_j2) t Then in the right-angled triangles c . . e . . g and c . . e^ . . g l it is clear that ' (e . . gY or y/ 2 = y* - y* sin 2 ceg and (e l . . gj* or y^ = y 2 2 - y 2 2 sin 2 ce$ ; In this Section the terms a, b, and c will supersede the corresponding terms A, B, and C of Sections V. to VIIlA., as they are more convenient for manipulation. 316 A SYSTEM OF APPLIED OPTICS SECT. similarly a 2 \ -J; " Primary plane. Value of the func- ( . , 9 1 ) tions of ///', / 2 ", and = \ ( a " + ^ ~ V ~ 35V + (* + 3 ^ 2 )~ / "i w i- the two last terms being the new terms consequent on the versine corrections. Secondary Plane Here we see from Fig. 113 that if the chord s . . s represents the circular aperture of the lens surface seen edgeways of semi-aperture = a, plus a correction shortly to be dealt with ; then the right-angled triangle, whose hypothenuse is the y required, consists of the side h . . n, and a vertical side above h, perpendicular to the diagram, so that y 2 = (h . . rif -f- (the vertical from A) 2 , and clearly ru and the vertical side over h a 2 a ( a 2 \ = vertical side over A ( = a) + - = a 1 + - ): 2r u \ 2ru/ /. (vertical side over h) 2 = a 2 ( 1 + - so that 2\ / 2\\ u>,w v2 = \ & 2 (1 e' 2 )( 1 H ) + a 2 ! 1 H ) (W-. \. \ ru/ \ ru/ J Secondary plane. = j/ a 2 + m _ tf e z + / a 4 + a 2m l ] . (5) Value of y ' 2 u r \ ru) the two last terms being consequent upon the versine corrections. The term a 4 --, which is independent of the angle of obliquity (f>, is thus seen to be common to primary and secondary planes, and is, in fact, a function of the spherical aberration consequent upon the axial xi CURVATURE ERRORS OF THE THIRD ORDER 317 or oblique pencil expanding in diameter as it traverses the distance between the element plane and the spherical surface. We also see that the functions of a 2 ^ 2 or a 2 tan 2 and of tan 4 < or 6V are three times as great in primary planes as in secondary planes. ECCENTRIC OBLIQUE REFRACTION We may now deal with the more complex y's involved in the case of eccentric pencils on the same lines, taking as our basis equation = (:r - r) > leading to = ? + w^V, 2 + yf + y,y 2 ) The basis equation. rr \f i J II _ II f ifi ** * 6 *> 1 -y=i , when, as usual, < A . . O Iff \-) is the angle PAQ and D = S . . A. So that b and c are both functions of tan <6. Let us then denote A. .N or tan by the r u- D J symbol c, A . . c or r tan < - - being b, and the semi-aperture of the pencil 1ST . . e or N . . e l where it cuts the element plane being a, as before. Obliquity Corrections to the y'& Primary Plane Here let e . . g = y and e l . . g l = y%. In the right-angled triangle e . . c . . g we have as before (e . . g) 2 or y^ 2 = (e . . c)' 2 - (c . . g)* ; w ^2 _ 7 , 2 _ w 2 y\ ~y\ y\ that is, . . ^2 _ 7/ 2 1 _ t an 2 .A ____,, 2/ y\ ~y\\ l tan 9.. , / -y\ \ W ~T I and similarly 318 A SYSTEM OF APPLIED OPTICS SECT. ft 2 = (b + c + of .-. y^ = (b + c + a) 2 (l - e 2 ) y = (b + c- a) 2 . -. y^- = (b + c- a) 2 (l - e 2 ) w* -< + c ) 2 - * 2 ' vM = {( b + c ) 2 - 8 }(i - 2 ) and = {(a 2 + 36 2 ) + Qbc + 3c 2 } Wl plus the new terms in the shape of functions of e 2 . which are Primary plane. Value of obliquity ( - a 2 e 2 - 3iV 2 - 3cV - 6ke 2 W, (6) functions in terms of y lt y z , and w r which are clearly functions of a 2 tan 2 <, 3 tan 4 , 3 tan 4 , and 6 tan 4 < respectively. Secondary Plane Turning to Fig, 115, it is clear that c. . N or b + c only is subject to the obliquity correction, so that (b + c) 2 modified for the obliquity ' 2 l-e 2 and y" 2(a i - {(* + c ) 2 (i ~ g2 ) + a ' 2 } ft) i = {(J 2 + 2ic + c 2 )(l - e 2 ) + a 2 }^ = (ft 2 + 2Jc + c 2 + o 2 ) Wl + the new terms in the shape of functions of e 2 , which are Secondary plane. Value of obliquity / 7 2 2 >> > OA/. 2\ /T\ functions in terms of ( ~ be ~ ~ ~ ^ce^ v (7) \A ^ Wr all minus functions of tan 4 <, and, as usual, one-third of the corre- sponding corrections in primary planes ; but the function of a z e 2 or a 2 tan 2 is again absent. Versine Corrections to the ?/s Primary Plane Eeverting to Fig. 114, let k . . I = y^ and n , . t = ?/ 2 ". Here the versines of the curved surface with respect to the element plane measured parallel to P . . A are obviously proportional to (c -t- a)' 2 1 v and (c of, and the increment to y' or e , . g = (c + a) 2 &T U approximately, and similarly xi CURVATURE ERRORS OF THE THIRD ORDER 319 and = (b + c + ) 2 (1 - e 2 ){ 1 + (c 2 + lac + a 2 ) \ + ( C 2 _ 2ac + a 2 ) and the new terms consequent upon the versine corrections are (J2 + C z + a 2 + ^ab + 2ac + 2&c)(c 2 + 2ac + a 2 ) 'n* + (6 2 + c 2 + a 2 - 2a6 - 2ac + 2&c)(c 2 - 2ac + a 2 ) - + 2k + c 2 - a 2 )(c 2 + a 2 ) - x ru which, after multiplying out and cancelling, gives us i Primary plane. { (3& 2 c 2 + 6k 3 + 3c 4 ) + 1 2a 2 c 2 + 1 4a?bc + (3a?b 2 + a 4 ) I a) ls (8) Value of functions of y", y^\ and w r in which the terms 3a 2 6 2 + a 4 appertain to the central oblique pencil also. So we have (35 2 c 2 + Qbc 3 + 3c 4 ) all functions of tan 4 , Lboth functions of a 2 tan 2 <, and the functions of a 4 and 3 2 6 2 before worked out for central oblique pencils. Secondary Plane Turning to Fig. 115, it will be seen that in the right-angled triangle, whose two sides including the right angle are p . . g' and a", the latter being perpendicular to the plane of the diagram and over the point p ; evidently p . . g' = N . . g subject to a double versine correction approximately equal to + ,2r 2r/ u xi is also subject to a double versine correction approximately equal to 320 A SYSTEM OF APPLIED OPTICS SECT. (^L ^!v V2r + 2vV so that we have and also 2ru and ru so that the hypothenuse squared, after correction, or and so that the new terms consequent upon the versine corrections are /jo m o o (b 2 + 2bc + c 2 + a 2 = (a >2 6 2 + 2a 2 ic + a 2 c 2 + a 4 + c 2 6 2 + 2&C 3 + c 4 + a 2 c 2 ) - w, ; ' ru J in which 2 6 2 + a 4 appertain to the central oblique pencil. Here it is instructive to notice that while the terms a 2 b 2 + b 2 c 2 4- 2bc 3 + c 4 are one-third of the corresponding terms in the primary plane, the term 2aV is only one-sixth part and 2a?bc only one-seventh part of the corresponding term in the primary plane, while the term a 4 is common to both planes. xi CURVATURE ERRORS OF THE THIRD ORDER 321 The Corrections for the Separation between the two y's Central Oblique Refraction So far we have been considering, in a qualitative sense, the nature of the small corrections which have to be applied to the two y's in order to convert them into the y"'s. We will now deal with those corrections which are due to the separation between the two y"'s to which we have previously alluded. We may again legitimately express the necessary corrections in terms of the uucorrected y's, since to express them in terms of the corrected y'"s would lead to functions of the order tan 6 which are beyond the scope of this inquiry. First we have our fundamental equation, with reference to Fig. Ill, y Zl = T) . . (7 = y & , The basis equation. /I ' /2 wherein /j so far has been held to mean the distance d . . q l} whereas it should be the distance I . . q (Fig. Ill), the versine d . . I being deducted. Similarly / has been held, so far, to mean the distance d . . q 2 , whereas it should be the distance t . . q y the versine d. .t being deducted. But as regards the numerator x f lt it is clear that since xfi is simply the distance q l . . p, therefore if we deducted the versine c . . I from /J we should also have to deduct it from x ; that is, the terms (x /i) and (/, #) are not affected by our corrections; but obviously/! and/ 2 in the denominators must be corrected for the versines, so that the above equation becomes 11 x ~f^ - y fi~ x . y\ 9 yv 9 ) - 1 - nt ** " ill & f 7l _ _ 2r 2 2r ^ 1 % 2 + f r & ' We may now express / : and / 2 in terms of d . . f or w and the spherical aberration, so that we may put 11 ; 2rV /* * / 2rU /// 2 /' B + - (y 3 + y 8) + (y 3 + y 3)__ + (y 5 + y 5) 1 A /A v ^ J 2nt 2 /A x r 7 r&J = 2/1 + 2/2 + 0V + ?/ 2 3 )T + (^ rt Since the ^'s are generally much smaller quantities than r and it, we may treat the second and third terms of the denominator as variants of (y l + 2/ 2 ), so that - becomes multiplied by + y 2 yi + ^ n /* and after multiplying out we get 1 H + ^i 3 + y 2 3 . *> + / 5 x \ii + _ _ etc _ etc< Here we may neglect the last two terms, and also the two terms (the seventh and tenth) involving -^, since they involve functions of the order y* and y, which we are not dealing with. The second and fifth terms cancel one another, while the third and ninth add together, so that we get finally CUEVATUKE EEKOES OF THE THIED OEDEE 323 1 = 1 + * . V* + y* + w 3/i 5 + y* J_ _ ^ (^i 3 + ^g 3 ) 2 l _ } Primary plane. P* Vi+Vz A* Vi + V* ' 2rh /* (h + y^ 2rit Value of -.corrected ( ' * for separation be- tween the if a. The last term in this formula need not be heeded, as it does not involve the spherical aberration at all ; for if, in our original equation, we suppose that there is no aberration whatsoever, and therefore that ./i = * =/.- and yet suppose that the two y's are at unequal distances from q\ and then correct it for the versines as before, we then get x - it which finally works out to x u (y\ + y^f 4r 2 ^ 3 ' which means that the distance x is to be measured from a point very slightly to the left of the vertex ^ by a minute amount varying inversely as u. This curious result doubtless follows upon our assuming the versines to vary exactly as f, which is not strictly true. Anyway this term has nothing to do with our present purposes and may be ignored, so that we have, after dividing out the functions of y l and y , y-2* - Here the first line is the result of the second approximation, which we have had to deal with before in Sections V. and VI. After adding together the second and third lines we get finally Primary plane. x = & + fa + ^ - w*> + fay* - a^v + y^-h (12) Reduced value of ' corrected for separa- We have in the process leading to this result dealt with Fig. Ill, tion between tfl e y's. 324 A SYSTEM OF APPLIED OPTICS Sensitiveness of x to the separation be- tween the y's in the case of narrow pen- cils. in which the two y's are at opposite sides of the normal oblique ray,, and we have treated both y's as positive quantities. Under this assumption, then, y 2 in the next Figure (112) would have to be considered negative, so that the above yfy.-, 2^/ 1 2 y 2 + y^j ^ would become three negative quantities. Now it is clear that if a, the aperture of the pencil, vanishes, then y l and i/ 9 . become numerically equal, and the above correction of the third order will not therefore vanish, and a little consideration and a reference to Fig. 115a will show that this correction to the oblique focal length x, due to a lateral separation between the two y's, should not vanish when the aperture of the pencil vanishes, for the smaller is the aperture the smaller is the angle between the two rays bounding the pencil in the primary plane ; and assuming their two focal points q and q l on the oblique normal ray r . . Q' to remain fixed, it is clear that the position of their intersection point q becomes highly sensitive to even a most minute lateral separation between the y's. For instance, if the two rays through k and n focus at fixed points q and q l} while n . . t is transferred laterally to n' . . t' without changing its length, then the point of intersection of the two rays will be transferred from f to f". But the formula of course vanishes when either one of the ?/s = 0, since then one ray becomes the normal oblique ray Q . . /. If the reader will carry out a similar investigation in the case of Fig. 112, treating both y's as positive quantities, he will arrive at the correction -(2/i 3 J/ 2 + %i 2 2/2 2 + 2/i2/2 3 XrT , which when worked out p. 2ru' and expressed in terms of a and b for central oblique refraction, or of a, b, and c for eccentric oblique refraction, will lead to exactly the same formulae as ISTos. (13) and (14) below. We may proceed to convert (12) of the third order as follows : First, in the case of Fig. Ill for central oblique refraction we have y = a + b and y z = a b, so that - = - + - (a 2 + 3b 2 x u p ^ dealt with before) plus the following new terms 1a 4 + 2a?b - 2ab z - 4> -(2a 4 - (13) been. Primary plane, and the function of the third order is finally Corrections to - for w x ^ the separation be- A 1 tween the ?/'s. f or cen t ra l oblique refraction. CUKVATUKE EEEOES OF THE THIED OEDEE 325 Eccentric Oblique Refraction Turning now to the case of eccentric oblique refraction, Fig. 114, wherein the two ijs are again on the same side of the central oblique ray Q . . r, we have y 1 = (I + c) + a and y 2 - (b + c) - a. This being the case, we shall find that the value of the consequent function expressed in terms of a, b, and c, works out to (14) The first two terms apply to the central oblique refraction which we have just worked out, while the last six terms, all involving c, follow from the eccentricity of the pencil. It is interesting to note how the terms of Fnrmulie (13) and (14) equate to 0, when in (13) a = &, or in (14) a = b + c, for, of course, when this is the case, y z becomes zero, or, in other words, the lower ray coincides with the normal oblique ray, so that the case is fully met by the usual spherical aberration formula -y-?. It might at first be thought that Formulae (13) and (14) should equate to when a vanishes, but this is not so, for we have seen that the narrower is the pencil the greater is the sensitive- ness of the position of the focus to a minute lateral separation between the two y's. These corrections of the third order consequent upon the relative lateral displacement of the two y's, obviously come into force in the primary plane only, and there is nothing corresponding to them in the secondary plane. It is clear that such corrections as (13) and (14) could not apply to a parallel glass plate or a plane surface, since r would become infinite and the value of the formulae vanish. It is also clear that when we come to add to the functions of the third order for the first surface the corresponding functions for the second surface, then - 1 , which is the " inside glass " value of the spherical /* aberration, will become fo lt or the outside glass value for the same aberration. Primary plane. Corrections to for X the separations be- tween the y's. The separation cor- rection only valid in the primary plane. 326 A SYSTEM OF APPLIED OPTICS SECT. We have now considered all the corrections of the third order which have to be applied in order to convert the ys of the second approximation into the y's of the third approximation, the correcting formulae being functions of 2 , or the spherical aberration formulae of the second approximation for the two surfaces which resulted in the formulae for curvature errors previously worked out in Sections V. and VI., which, as applied to the single surface that we have been considering, was fj? \r (see Formula XVIII. (K.), Section IV.), and the corrections that we have been dealing with in this Section are of course all products of the corrections to y l or y 2 into the part of the above formula included in the large brackets. THE INTRINSIC SPHERICAL ABERRATION OF THE THIRD ORDER But we have yet to consider the intrinsic spherical aberration of the third order in its application to oblique rays ; that is, we have to find what are the modifications to the curvature corrections consequent upon our taking into account Formula XX. (E.) of Section IV. (page 63), which is a function of y*, and therefore, in its present application, of tan 4 (f). We will first deal with the case of Central Oblique Refraction Primary Plane Here we must revert again to the fundamental equation dealt with on page 121, Section V., applying also to Fig. Ill in which we must now stipulate that T = l + "^i 2 + Mi and ^ = + Wl y 2 2 J\ /2 the last terms expressing the intrinsic spherical aberration of the order y 4 as given in Formula XX. (R). We are not now to consider any corrections to the y's involved in ^_y 4 , since such procedure would only result in corrections of the order y 6 , etc., but have to find what is the xi CUEVATUEE EEEOES OF THE THIED OEDEE 327 result of the introduction of this new term on the curvature corrections in the primary plane. We have then 4 1 ' so that the equation 7=T + -y 2 2 + / 2 U ft " //, 2 / 2/1 + 2/2 now becomes + + ) + y 2 (* + V + ***)! . . / 2 \w /A 2 n sJyi + 9t r(2/i + 2/2) + -(2/1 3 + 2/ 2 3 ) + ;;(yi 5 + 2/ 2 5 ) lb UL fA y\ + y have been already worked out, and we may confine our attention to the functions of %. We have, in Fig. Ill, y l = (a + b) gfy - - (a 4 - i V = a * a 4 + 10a 2 6 2 + 5i 4 therefore Primary plane. ^ = ^ + w (a 2 + 36 2 ) + v (a 4 + 1 Oa 2 6 2 + 5& 4 ). (16) Value of ^ from (15) . ^ '' after reduction. The functions coa and %a are, of course, the spherical aberrations of the two orders to which all pencils of semi-aperture a are subject, whether axial or oblique. Secondary Plane Here y 2 = simply a 2 + 5 2 , and .-. f - a 4 + 2a 2 6 2 + 6 4 , and we have therefore Secondary plane. ^ = + w(a 2 + ft 2 ) + X (a 4 + 2a 2 6' 2 + 6 4 ). (17) Value of ^ after re- *C I/- *^ duction. 328 A SYSTEM OF APPLIED OPTICS Here again aa 2 and p^a 4 are the spherical aberrations of the two orders common to all pencils, axial or oblique ; while it will be seen that the correction ^(2 2 J 2 -f & 4 ) is only one-fifth part of the correspond- ing correction in primary planes. Eccentric Oblique Refraction Primary Plane Here we have, in Fig. 114, 7/! 2 = (b + c + a)' 2 The conditions are here the same as dealt with on page 143, Section VI., Fig. 50, and we have 2 Value of functions of \. Primary plane. Functions of X , y v and I/, after reduc- tion. 9i * + ^ 2 ) + x (^ + y^ + y*y* + y*y z + y^). (18) The functions of w have already been dealt with. On working out the functions of y in terms of a, b, and c, we get x (a 4 + 10a 2 c 2 + 30& 2 c 2 + 20a 2 ic + 206 3 c + 20k 3 + 5+ 5c 4 - x [a 4 + {I0a 2 (b- + 2k + c 2 ) + 5(b* + 45 3 c + 6# order a 2 tan 2 < order tan 4 $ in which the underlined terms 4 + 10a 2 6 2 + 5Z> 4 relate, as we have seen, to central oblique pencils also. Secondary Plane Here y 1 = simply (b + c) 2 + a 2 , and y* = (b + c) 4 + 2a z (b + c) 2 + a 4 , and Secondary plane. Function of X after reduction. - 2bc + c 2 ) + a 2 + etc. > i + x/a 4 + 2a 2 (& 2 + 2k + c 2 ) + (6 4 + 46 3 c + 66 2 c 2 + 4k 3 + c 4 ) J, (20) UNIVERSITY or XI CUEVATUEE EEEOES OF THE THIED OEDEE 329 in which the terms a 4 +2 2 & 2 + & 4 apply to central oblique pencils. The functions of w have already been dealt with. Thus we find that the corrections involving the intrinsic spherical aberration of the order y* are five times as great in primary planes as in secondary planes, and that all the terms are represented in both planes. It is advisable to now gather together our results in the form of a table as follows : Functions of &>! Primary Plane Terms to be added for 3rd Approximation 2nd Approximation Terms tc Central Oblique Pencils. Add for Eccentric Pencils. Central Oblique Pencils. 2 + 36 2 + 6bc + 3o 2 - ft' 2 e 2 - 3&V 1 ru (4a' 2 & 2 46 4 ) - '2rii Add for Eccentric Pencils. - 3c 2 e 2 - Obliquity corrections. Versine iwoc j i Versine + 12 2 c 2 ) J corrections. r +(-246% 2 -166c 3 -16& 3 c -, ) Separation - 4c 4 + 8a 2 bc + 4a 2 c 2 ) ; - - f corrections. Z.TU ) + 2bc + c- Secondary Plane _ - We*- I - c-e 2 - 2&ce 2 / Obliquity \ corrections. + 2a 2 c 2 ) ru corrections. Functions of % l Primary Plane Central Oblique Pencils l: Add for Eccentric Pencils 4 ) il (306V- + 20&C 3 + 206 3 c + 5c 4 + 20a?bc + 10aV). Secondary Plane The Functions of ^ Leaving the functions of - , out of present consideration it will be 2ru seen that the functions of eoj under the head of second approximation have already been fully worked out for both surfaces of a lens in Sections V. and VI., wherein we found that the term 6bc or 2bc 330 A SYSTEM OF APPLIED OPTICS SECT. resulted in the formulae for comatic stop corrections or E.C.s, and 3c 2 and c 2 in the formulae for spherical aberration E.C.s. Therefore the terms (14a 2 6c and 2ct?bc) for primary and secondary planes in the third approximation are comatic functions involving the semi-aperture squared, and the high ratio of 7 : 1 between primary and secondary planes instead of the 3 : 1 for the second approximation is significant of much that requires working out. The two terms (12V and 2aV) imply spherical aberration stop corrections dependent upon the aperture of the pencil, whose influence is six times as powerful in primary planes as in secondary planes. All the other terms with one exception imply the usual ratio of 3 : 1 in primary and secondary planes. The exception alluded to is the term V in the obliquity corrections which does not appear at all in the secondary plane. This is also a highly significant term, and explains a phenomenon commonly observable at the foci of oblique pencils passing through certain Double side flare. photographic lenses, and that is a sort of double coma. For instance, when a little way inside of the focus the section of the oblique cone of rays shows over-correction for spherical aberration in the primary plane, and the primary plane only, while in the secondary plane the spherical aberration may be about correct. Thus there appears to be a side flare both towards the optic axis and away from it. The terms 12aV and 2aV may also tend either to aggravate or to mitigate the above effect. As regards the corrections, functions of =, , which follow from the 2ru lateral separation between the two y's, although they apply only in the primary plane, yet their quantitative value may usually be regarded as by no means unimportant compared to the obliquity and versine corrections. The Functions of % : Turning to the functions of % ly or the intrinsic spherical aberration of the third order, it is interesting to see that the corrections in the primary plane are exactly five times as much as in the secondary plane. The significance of this discrepancy between the ratios 3 : 1 and 5 : 1 for the functions of w l and ^ x respectively, together with the presence of the separation corrections in the primary plane only XI CURVATURE ERRORS OF THE THIRD ORDER 331 (supposing we leave all corrections involving a, or the aperture of the pencil, out of consideration), will shortly become apparent in studying the actually measured or calculated curvature of image corrections for certain photographic lenses, figured on Plate XXIV. The peculiar comatic formation which will satisfy the ratio of 5 : 1 between the E.C.s of the third order was shown on Plate XVI., Fig. 79/, as being formed of a series of duplex comatic circles distributed over a length equal to five times the radius of the largest one ; while the size of the formation will vary as tan 3 < instead of as tan ; which may often be too considerable to be ignored. Also the image of the stop centre thrown by the first lens may be subject to a considerable spherical aberration leading to variations in b 2 , again dependent upon the first angle of obliquity <. These aberrations of image curvature of the third approximation present an ample field for the exercise of a higher order of mathe- matical skill than has generally been called for in the present work. The complete reduc- tion of the formulae of the third order highly laborious. Corrections of one lens affected by pre- ceding lenses. 332 A SYSTEM OF APPLIED OPTICS SECT. Dr. von Rohr's graphs of curvature errors for narrow pencils. Errors involving the aperture not con- sidered. Dr. Steinheil's lenses. Some Practical Examples of Hybrid Curvature Errors Before concluding this Section it will be instructive to reproduce in Plate XXIV., by the kind permission of Dr. Moritz von Rohr, a few diagrams from his most valuable and painstaking work, Theoric und Geschichte des Photograpliisches Objectivs, which furnish illustrations of certain of these curvature aberrations of the third order which we have been dealing with. These graphic curves show the deviations from true flatness, in primary planes by the dotted line, and in secondary planes by a solid line, of the images of distant objects thrown by various types of photographic lenses. They were worked out by careful calculation, on the supposition that the stop of the lens was in its usual working position, but reduced almost to a point ; that is, the curves traverse the fjoci of infinitely narrow oblique and eccentric pencils. Thus all corrections of the third order involving a (the aperture), such as we have lately been dealing with, are eliminated. Therefore, if the stop of any of the lenses were opened out to considerable working aperture, as in practical use, it would by no means follow that the curves of aberrations from the flat image would remain like these diagrams ; indeed, in many cases the curves would become very substantially modified, in some cases favourably and in other cases unfavourably, a fact which somewhat discounts the value of these diagrams from the practical photographer's point of view. Each of the lenses here dealt with is supposed to be placed on the left hand, and to be 3 - 5 inches equivalent focal length on the scale of the plate ; the ordinates represent angular distances from the optic axis ; the abscissae represent the aberrations from the plane image, but for the sake of clearness these are four times exaggerated. Every 5 degrees are marked off along the vertical, and every millimetre of horizontal aberration along the horizontal base line, which represents the optic axis. Fig. 116 is the curve for Steinheil's Orthostigmat Lens, Fig. 118 for his Antiplanat, and Fig. 120 for his Rapid Antiplanat. These three curves are substantially of the same character. The broad features are the under -corrected field and over -corrected astigmatism within 20 degrees of the axis. The image formed by rays in primary planes (dotted) is more nearly flat than the image formed by rays in the secondary plane (solid). This failure to come up to a plane image simultaneously is due to the imperfect approach to the fulfilment of the Petzval condition. PLATE. XXIV. 30 M m lm 2O Ml IS IO i i i I I L. I I l l I i. 117. i. 118. . 119 4O i<. 120 12 I. i. I 2 2 i. I 23 PLATE. XXIV. 30 m 20 M :m 15 IO , i<. 117. Fi<>. 118. Fi<>. 119 4O . I 2O i. 12 I. i. I 2 2 Fi<5. I 23 XI HYBRID CURVATURE ERRORS Old Boss Lens. Doublet Here we have strongly marked plus curvature errors tending to round images (concave to the lens) of the second order or varying as tan 2 <, against which are working minus curvature errors of the third order varying as tan 4 . Hence the latter rapidly overtake and more than neutralise the former as we get away beyond 25 degrees, but the + curvature error for rays in primary planes is at a maximum at 20 to 22 degrees, but apparently beyond 30 degrees for rays in secondary planes. But the curve of errors is of the same general character in the two planes, although the maxima and points of crossing back over the focal plane do not coincide. Fig. 122 for Dr. Rudolph's Wide-Angle Anastigmat furnishes a Dr. Rudolph's Wide- capital example of the same general features as the last three, excepting ng e that the maxima much more nearly coincide, and the astigmatism is reversed. Fig. 123 for an old type Ross Doublet Lens is a case similar to the preceding for rays in primary planes, but it is doubtful whether the curve for rays in secondary planes shows any decided tendency to a maximum followed by a curve back again ; indeed, aberrations of the order tan 4 in both primary and secondary planes counteracted by curvature errors of the order tan 4 ; the latter at the outskirts of the field asserting themselves so much as to throw the images back behind the focal plane. The maximum for secondary rays is at about 22 degrees, and that for primary rays at about 18 degrees from the optic axis. Fig. 117 gives the curves of errors for an old form of Cooke Lens, Series Ilia (the lens figured in Fig. 60, Plate XII.), and Fig. 119 gives the curves of errors for the well-known Goerz Double Anastigmat (the older cemented doublet). These two cases are of the same general character, excepting that in Fig. 117 the primary image is by first intention curved back convex to the lens, and is slightly concave to the lens in Fie;. 119. But the most remarkable characteristic lies in Old Cooke Lens, Series Ilia. 334 A SYSTEM OF APPLIED OPTICS SECT. XI Independence of the curvature errors of the third order in the two planes. The disturbing effect of aperture upon the curvature errors of the third order. Future progress de- pends upon elimina- tion of curvature errors of third and fourth orders. the fact that the curvature errors of the order tan 4 < are decidedly negative for rays in secondary planes, but positive for rays in primary planes. Hence the manner in which the two curves cross one another at 27 and 30 degrees respectively, after which there follows a rapid mutual separation. Now it is clear that were the ratio between the aberrations of the third order invariably 3 : 1 or any other fixed ratio between the primary and secondary planes, then such graphs as these could never arise. But since (leaving all terms containing a out of consideration, as we are dealing here with pencils of infinitely small aperture) the third order functions of - are in the ratio 3:1 in the primary and secondary planes, while the third order functions of ^ are in the ratio 5 : 1 in the primary and secondary rays, then we can clearly see that in the case of the functions of - being of the opposite sign to the functions f* of ^ we may easily have the total aberrations of the third order plus in one plane while they are minus in the other. The separation corrections to the y's, existing, as we have seen, only in the primary plane, cause a still further degree of independence between the curvature errors in the two planes. And the scope for vagaries of this sort is still more enlarged when we come to deal with the images thrown by pencils of relatively large aperture, for we have seen that in the primary plane there are functions of a that are seven and six times the corresponding functions in the secondary plane. Therefore it is that, if we take the lenses we have dealt .with and open out their apertures and locate their oblique foci (by obtaining the best possible distinctness of image), we may find the curvature errors come out substantially different to those shown on Plate XXIV. It is clear, then, that it is not always practicable to determine the working character of a lens by calculating its curvature errors for infinitely narrow pencils only. It will easily be seen that the future progress of photographic lenses towards perfection depends chiefly upon the successful elimination of the curvature errors of the order tan 4 , and the doing of it with the simplestossible lens construction. Printtrily R. & R. CLARK, LIMITED, Edinburgh. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. AUTO. DISC. MAR 11 *87 AUG291955LU JAN 3 1967 6 3 MAY S5 LD 21-100n-9,'48iB399sl6)476 YE 00688 GENERAL LIBRARY - U.C. BERKELEY BoooaaMosa 156182 "C- ~ 4