The D. Van Nolrand Company 
 
 intend this book to be sold to the Public 
 at the advertised price, and supply it to 
 the Trade on terms which will not allow 
 of reduction. 
 
THICK-LENS OPTICS 
 
 AN ELEMENTARY TREATISE 
 
 FOR THE STUDENT AND 
 
 THE AMATEUR 
 
 BY 
 
 ARTHUR LATHAM BAKER, PH.D. 
 
 MANUAL TRAINING HIGH SCHOOL, BROOKLYN, N.Y. 
 
 Author of 
 "QUATERNIONS AS THE RESULT OF ALGEBRAIC OPERATIONS' 
 
 ILLUSTRATED 
 
 
 NEW YORK 
 
 D. VAN NOSTRAND COMPANY 
 
 25 PARK PLACE 
 1912 
 

 Copyright, 1912 
 
 BY 
 
 D. VAN NOSTRAND COMPANY 
 
 THE-PLIMPTON-PRESS-NORWOOD-MASS-U-S-A 
 
PREFACE 
 
 THIS volume is the outcome of an attempt to answer 
 certain questions regarding the optics of the microscope 
 and telescope; questions to which no thoroughly satisfac- 
 tory answers could be found in any literature accessible 
 to the author. 
 
 Many answers were found, but they were discordant 
 and unusable for practical work, mainly by reason of their 
 complexity and seeming contradictoriness and lack of 
 co-ordination. 
 
 The following pages seek to answer these questions in a 
 manner so plain and simple that the average amateur can 
 find out for himself what is going on optically in his camera, 
 microscope, or telescope. 
 
 To this end the mathematics is of the simplest kind, so 
 that the busy man who has forgotten all or most of his 
 mathematics can nevertheless work his way through, pro- 
 vided he can use the simplest kind of algebra, two theorems 
 in elementary geometry and one in trigonometry. For 
 the reader who has not had trigonometry, the few simple 
 principles required are given in the text. So far as mathe- 
 matical difficulties go, any high-school student is sufficiently 
 equipped. 
 
 As an aid to concreteness and clearness the investigations 
 are based upon graphic principles as much as possible and 
 along intuitive lines. 
 
 For the more inquisitive reader who desires a more 
 
 iii 
 
 271628 
 
iv PREFACE 
 
 rigidly logical basis one investigation is given in analytical 
 form, as a supplement to the preceding intuitive ones. 
 This can safely be omitted by those not interested, without 
 destroying the continuity of the text. 
 
 The text is a working one, intended to give the reader 
 practical and intelligible rules of procedure, with full and 
 thorough explanations, so that the most cursory reader can 
 utilize them. Many practical examples are fully worked 
 out and many more given for practice. 
 
 Particular pains has been taken to reconcile seemingly 
 contradictory formulae for the same result, which, unrecon- 
 ciled, leave the reader in the deepest uncertainty, the fault 
 of most of the literature on the subject. 
 
 This volume, for the first time apparently, assembles 
 these rules, answers, and formulae in one consistent whole, 
 in a practical form intelligible to the non-technical reader. 
 
 The formulation of the methods of procedure is so stand- 
 ardized and simplified ( 106) that it is expected that the 
 reader can readily utilize the necessary calculations, con- 
 cretely visualized and checked by the graphic constructions 
 ( 107). 
 
 His ability to do so ought to render his use of optical 
 instruments that much more intelligent and interesting, 
 and enable him to know roughly how his instrument is 
 doing its work, what effect a change of lens or of its position 
 would have, how to decide in a rough way what form of 
 lens he wants for certain effects, and how he could modify 
 those effects. 
 
 The investigation is for a single monochromatic ray, and 
 therefore the questions of achromatic and spherical aber- 
 ration are not touched upon, as not being within the scope 
 of the simple treatment used. 
 
 The goal of the work is, of course, the practical calcula- 
 
PREFACE V 
 
 tions of 106, 107 and Chapter V, though many other 
 important calculations are gathered on the way. 
 
 To render the work less an isolated monograph and make 
 it more useful to the general reader, a number of sections 
 have been added, to round out the subject somewhat 
 toward the nature of a handbook and to increase its 
 practical value to the owner of an optical instrument; 
 not the least valuable of which will be the chapter on 
 Experimental Observations. This chapter will enable the 
 reader to get an experimental acquaintance with the optical 
 constants of his lenses. 
 
 The author makes little claim to novelty, except in the 
 simplification and workability of the rules of procedure. 
 
 A. L. B. 
 
 BROOKLYN, N. Y. 
 October 1, 1912. 
 
CONTENTS 
 
 PAGE 
 
 CHAPTER I 
 
 Surface Refraction 1 
 
 Construction for 3 
 
 Equation for 4 
 
 CHAPTER II 
 
 Thin Lenses, Equation 6 
 
 Direction of Light 8 
 
 Optical Center 9 
 
 Diagrammatic Investigations 
 
 Positive Lens 10 
 
 Negative Lens 12 
 
 Oblique Rays 15 
 
 Parallel Rays 17 
 
 Use of Formulae 18 
 
 Graphic Check 22 
 
 Diopters 25 
 
 Spectacles 26 
 
 Magnification 28 
 
 Copying 31 
 
 Exposure 33 
 
 Hyperfocal Distance 34 
 
 Magnifying Power 36 
 
 CHAPTER III 
 
 Thick Lenses 39 
 
 Principal Points 40 
 
 Nodal Points 42 
 
 Optical Center 43 
 
 Construction for Nodal Points 44 
 
 Image in Nodal Plane 47 
 
 vii 
 
viii CONTENTS 
 
 PAGE 
 
 Focal Length 49 
 
 Construction for Image 53 
 
 Use of Formulae 54 
 
 Graphic Tracing of Ray 54 
 
 Analytical Investigation 55 
 
 General Equation 57 
 
 CHAPTER IV 
 
 Combinations of Lenses 64 
 
 Thin Lenses in Contact 64 
 
 Thin Lenses not in Contact 65 
 
 Back Focal Distance: Thin Lenses 67 
 
 Equivalent Focus: Thin Lenses 67 
 
 Back Focal Distance: Light from Right. ...... 70 
 
 Back Focal Distance: Thick Lenses . 71 
 
 Equivalent Focal Distance: Thick Lenses 71 
 
 Nodal Distance 72 
 
 Resume 73 
 
 Use of Formulae, etc 74 
 
 Graphic Construction 76 
 
 Magnifying Power, Microscope 87 
 
 Telescope 88 
 
 Opera Glass 89 
 
 CHAPTER V 
 
 Telephoto Lens 90 
 
 Focal Length / 91 
 
 Telephoto Magnification 91 
 
 Focal Distances 92 
 
 Image Distance 94 
 
 Focal Radius 94 
 
 Object Distance for given Magnification 96 
 
 Reduction Factor . . ... . . ' * . - . . . . ' 97 
 
 CHAPTER VI 
 
 Reflection at Surfaces 98 
 
 Graphic Construction 99 
 
 CHAPTER VII 
 
 Experimental Observations . 102 
 
CONTENTS ix 
 
 PAGE 
 
 Radius of Curvature of Surface 102 
 
 Radius of Curvature of Surface: Small Radius .... 104 
 
 Focal Length of Thin Positive Lens 
 
 With Sun 104 
 
 Lens Distances 105 
 
 With a Telescope 105 
 
 Different Lens Positions 105 
 
 Equality of Object and Image 105 
 
 Comparison of Images 105 
 
 Focal Length of Thick Positive Lens 
 
 Highly Magnified Image 107 
 
 Swing of Camera 107 
 
 Movement of Screen 108 
 
 Angle of Vision 108 
 
 Unit-Screen Movement 108 
 
 Measurement of Image 109 
 
 Comparison with Standard Lens 109 
 
 Double Focus 110 
 
 Lens Displacement Ill 
 
 Focal Radius of Negative Lens 
 
 With Sun Ill 
 
 With Stronger Positive Lens 113 
 
 With Positive Lens and Comparison of Images . . . 114 
 
 Location of Nodals 114 
 
 Magnifying Power 
 
 Telescope 115 
 
 Visual Comparison of Images 115 
 
 Microscope 
 
 Visual Comparison of Images 116 
 
 Distribution between Objective and Ocular . . . . 116 
 
 Index of Refraction 118 
 
 Practical Suggestions 121 
 
THICK-LENS OPTICS 
 
 CHAPTER I 
 
 SURFACE REFRACTION 
 
 1. As this is intended as a working manual, and not a 
 treatise, it will be assumed at the outset that the reader 
 is acquainted with the fundamental principles of optical 
 refraction, the commonplaces of the elementary text-books, 
 viz.: 
 
 (a) Light rays are propagated in straight lines. 
 
 (6) In passing from one medium to another, a ray of 
 light is deflected towards the normal to the surface in 
 passing into the denser medium; vice versa in passing out. 
 
 2. Definition. The ratio of rise to slant of a line is 
 called the sine of the angle of inclination; of rise to run, 
 the tangent of the angle of inclination. 
 
 For example: In a roof whose vertical height (rise) is 
 3 feet, whose half width (run) is 4, the length of the rafters 
 (slant) will be 5, and f is the sine of the angle of inclination 
 of the roof to the horizon; f is the tangent (generally written 
 tan) of the angle of inclination, ratio of rise to run. 
 
 3. Definition. The ratio of the sine of the angle of 
 incidence to the sine of the angle of refraction is called the 
 index of refraction for the denser medium, the ray passing 
 into the denser medium from air. 
 
 1 
 
2 THICK-LENS OPTICS 
 
 4. Surface Refraction. 
 
 8 Jingle of incidence 
 Emergent Tay 
 
 Diagram showing angles of incidence, refraction, etc. 
 
 The index of refraction is generally indicated by the 
 Greek letter p, thus: 
 
 sin (3 
 
 A* = 
 
 sn 
 
 sn a 
 
 Unless otherwise specified, we will consider only two 
 media: air and glass. 
 
 5. Law of Refraction of Light. The index of refraction 
 for the same two media is constant, whatever the angle of 
 incidence. 
 
 6. Trigonometric Law of Sines. In any triangle, the 
 sides are proportional to the sines of the opposite angles. 
 
 Proof. By definition of sine, 
 
 sin A = r 
 
 r * 
 
 sin B = - 
 a 
 
 Whence, by division, 
 
 sin A a 
 
 = = r Q.E.D. 
 
 sin B b 
 
SURFACE REFRACTION 3 
 
 7. Note. In all construction diagrams, the order of 
 construction is indicated by the alphabetical order of the 
 letters; e.g. in the diagram below, draw the circle a with 
 the radius indicated, then draw the circle 6, then locate 
 the point C, then locate the point Z), and then, since this 
 ends the series of letters, draw the refracted ray as shown. 
 
 8. Construction for Surface Refraction. 
 
 'a,with radius k 
 b,witk radius m 
 
 surface at 
 point of in- 
 cidence 
 
 Diagram showing construction for surface refraction. Given the 
 incident ray, to find the refracted ray. 
 
 For refraction out, interchange the letters C and D. 
 
 9. Definition. Any arbitrary line through the center 
 of curvature is called the axis of the surface. The point 
 where it pierces the surface is called the vertex. 
 
 10. Convention as to Signs. Distances measured to 
 the right from the vertex are considered as positive; those 
 to the left negative. 
 
 Note. In the following investigations, the diagrams 
 will usually be so taken as to make all the elements con- 
 
4 THICK-LENS OPTICS 
 
 sidered positive. This will give normal equations which 
 may be considered typical for all cases. 
 
 11. Equation for Surface Refraction (incident onto 
 denser medium). 
 
 By the Law of Sines ( 6), 
 
 sin QOR = sin <* 
 
 = n sin < 
 
 sin <f> _ sine of incident angle 
 sin <' ~ sine of refracted angle 
 
 P F-limit ofP Q Source 
 ofroA) 
 
 Jrri 
 
 sin FOR = sin 
 
 . sin QOR = /* sin FOR 
 
 QO-PR = n-PO-QR 
 
 [sin QOR = sin FOR 
 
 As R approaches S, P approaches some point F as a 
 limit. 
 
 .'. (u r) w = /. (w r) u 
 
 or, in the more usual form, 
 
SURFACE REFRACTION 
 
 w u r 
 
 ~~u = dist. from vertex to source, + to right 
 w = dist. to image, + to right, to left 
 r = radius of curvature, + to right 
 fi. = index of refraction 1 
 
 12. When the source is very distant, i.e. u = oo, w takes 
 the special value /, and 
 
 / = distance to point through which rays parallel to the 
 axis in the rarer medium meet the axis after 
 refraction 
 = focal radius of the surface for parallel entering rays 
 
 (r meas. from vertex to right is pos. and vice 
 
 versa). 
 
 N. B. This formula holds whether light comes from 
 right or left, or surface convex or concave. 
 
 T 
 
 13. For emergent rays / = ^r = focus for rays par- 
 
 allel in the denser medium. Note how it differs from that 
 of 12. 
 
 14. Graphic Check. Check the calculation of / by 
 similar triangles drawn to scale, 
 
 in which the sides are repre- 
 sented as shown. This will 
 readily detect large errors of cal- 
 culation in time to prevent their vitiating later calculations. 
 
 1 The /* represents the ratio of the sine of the angle of incidence to 
 the sine of the angle of refraction. In the case of incidence on a denser 
 medium it will be the index of refraction and an improper fraction, 
 but in the case of incidence on a rarer medium it will be the reciprocal 
 of the index of refraction and a proper fraction. 
 
CHAPTER II 
 
 THIN LENSES 
 
 15. By thin lenses is meant lenses whose thickness can 
 be practically neglected in comparison with the other 
 elements under consideration. 
 
 By 11, for the first surface, 
 
 d - = P ~~ 1 = 
 w u r f 
 
 Similarly, for the second surface, the image of the first 
 surface being the object of the second surface, and the 
 index for the second surface being inverted, because the 
 ray is emergent instead of incident (see 11, footnote), 
 
 1 1 
 
 /* _1_ _ /M. FFor significance of the letters 
 
 v w' s see the diagram 
 
 or ,_ = = _ 
 
 w' v s f 
 
 The negative sign is used merely to make the formula 
 below conform in looks to that of 91 and similar ones. 
 This convention has no effect either on the numerical 
 value of F ( 16) or on the sign of F. 
 
 6 
 
THIN LENSES 7 
 
 If the lens is thin, so that practically w = w', then, by 
 subtraction, 
 
 ./ ' r. 
 
 r = radius of first surface 
 
 s = radius of second surface 
 
 / = focal length of first surface for incident rays 
 
 /' = negative focal radius of the second surface (see 
 
 preceding paragraph) = - 
 m fji = index of refraction 
 16. For a very distant source, i.e. u = oo, we get the 
 
 special value for 
 
 J"-l*g+)-fr-l)g 
 
 1 1 JL 
 
 v u F 
 
 F = principal focal 
 length of the 
 lens 
 
 = value of v for 
 (u = oo) 
 
 That is, for a very distant object (horizontal rays) all 
 the horizontal rays pass through F, the focal point. 
 
 17. Since distances from the vertex to the left are nega- 
 tive, we get 
 For a double convex lens 
 
 11 / 1 IV 1 11 
 
 1 Heavy black face type indicates numerical values without regard 
 to direction; light letters indicate true values, taking account of direc- 
 tion where this is necessary. The black face type will be used when 
 
8 THICK-LENS OPTICS 
 
 For a double concave lens 
 
 11 n (\ ,l\ 1 1 1 
 
 --^=(^-1)^ + -^ = - = --- 
 
 18. Starting with w + and large, if F is +, - must be 
 
 greater than -, in order to make positive. There- 
 in v u 
 
 fore v must be smaller than u, and the image is nearer the 
 lens than the object is. 
 
 If r > s, thus making the lens thicker in the middle, 
 or if r < 0, i.e. negative, thus making the lens a double 
 
 convex lens, then -= is negative, and therefore F is negative 
 and must lie on the left. 
 
 Keeping u + (or on the right) and large, which makes - 
 
 vi 
 
 small, the only way to make < (i.e. neg.) is to 
 
 1} \Ji 
 
 make v negative. 
 
 In other words, for a thinner-in-the-middle lens, later 
 called a negative lens, u, F, and v have the same sign 
 when the object is real. 
 
 19. For a thicker-in-the-middle lens, later called a posi- 
 tive lens, and a real object, /''and v must have different 
 signs from u. 
 
 Stated in another way, 
 
 For a + lens, the further focus (from the object) is the 
 
 active focus. 
 
 For a lens, the nearer focus is the active focus. 
 
 20. Light from the Right. In formulae used hereafter, 
 the positive lens (thicker in the middle) will be considered 
 
 the absence of direction is to be specially emphasized. In other cases 
 the context will indicate whether the quantities have direction or not. 
 
THIN LENSES 9 
 
 as having a negative focal length, and negative lenses 
 (thinner in the middle) as having a positive focal length, 
 because the effective focus lies in these respective directions. 
 
 21. Light from the Left. In future formulae, the posi- 
 tive lens will be considered as having a + focal length, 
 and the negative lens as having a focal length, because 
 the effective focus lies in these directions respectively. 
 
 Since this makes the sign of the lens and the sign of the 
 focal length concordant (a great gain in uniformity), the 
 light will hereafter be assumed to come from the left unless 
 otherwise specified. 
 
 Note. The diagram in 15 was so taken because all 
 the quantities are positive, thus giving a normal formula 
 (see 10, note) applicable to any diagram when we take 
 account of the changes in sign of the various quantities. 
 So also in diagrams of 67, 72. 
 
 22. Notice that if the media on the two sides of the 
 lens are not the same, the nearer and farther focal-point 
 distances will be different (Conf. 71). This will easily 
 be seen by, in the preceding investigation, taking /*/ for 
 the second surface instead of n, and finding F for u = oo. 
 Reversing this by taking p! for the first surface and p for 
 the second surface, we get a different value for F. 
 
 Unless specifically mentioned, we assume that we have 
 air on both sides of the lens, the usual condition, and there- 
 fore the two focal-point distances the same. 
 
 23. Optical Center. The investigation of 65, which 
 may be read here, shows that for any lens there is a point, 
 rays passing through which are parallel before and after 
 refraction by the lens. For a thin lens this point must 
 be where the axis of the lens pierces the lens. Rays through 
 this point are not changed in direction. 
 
 24. Diagrammatic Investigations. In the diagram- 
 
10 
 
 THICK-LENS OPTICS 
 
 matic investigations which follow, the determination of 
 image from object will be made by means of two definite 
 kinds of rays. 
 
 (a) One which after refraction passes through the proper 
 focus point, i.e. rays from the object parallel to the axis 
 of the lens, as if from a distant object, 
 
 (6) One unchanged in direction before and after refrac- 
 tion, i.e. a ray through the center of the lens (or in the 
 case of a thick lens, the nodal points, see 63). 
 
 To emphasize the characteristics, say, 
 
 Horizontal rays always refract to the focus. 
 
 Central (or in the case of a thick lens, nodal, see 63) 
 rays pass through without angular deviation. 
 
 Note. Each point in the object sends rays in all direc- 
 tions, and of course we choose those which serve us best. 
 
 25. Diagrammatic Derivation of Image (object outside 
 of focal distance). 
 
 f-lens 
 
 JL Horizontal ray 
 
 Object 
 
 Jxis of lens 
 
 Note the order of the letters (see figure} ; this order, formulated, becomes 
 a rule of procedure. 
 
 Convention. The object is represented by a heavy' 
 arrow, the image by a light arrow, the focus by F, the 
 thin lens by a vertical heavy straight line. 
 
 AB represents one of the first kind of rays, which, we 
 know, must go through F; AC represents the second 
 
THIN LENSES 
 
 11 
 
 kind, which goes through without refraction. Hence D, 
 the point where the two rays meet, must correspond to A, 
 or be the image of A. 
 
 The diagrammatic procedure of finding the image D, 
 of an object A, may be symbolized by the letters (a great 
 help in future diagrams) 
 
 hi > / 2 to c 
 meaning : 
 
 From some point in the object pass horizontally (h) to 
 the lens (I), then through ( >) the right-hand focus (fa) 
 to the center line (c). The intersection will be the image 
 point, (/i would mean left-hand focus.) 
 
 Observe that F and v are +, while u is . Compare 
 with 19. 
 
 26. In a similar way we get the following diagrams. 
 When u > F numerically, we get a real aerial image 
 
 which can be made visible by interposing at the image 
 point a piece of ground glass. The rays from the object 
 to the lens are divergent raySj those from the lens to the 
 image are convergent rays. 
 
 27. Virtual Image (object inside the focal distance). 
 
 + lens 
 
 Object 
 
 Object inside the focal distance: the dotted lines show rays made 
 less divergent after refraction. 
 
 Diagram showing how an object within the focal point gives a virtual 
 image, an image erect instead of inverted as in the previous diagram, 
 and which cannot be made visible by the interposition of a piece of 
 ground glass. Unlike the previous case, it renders the divergent rays 
 less divergent, but not convergent. 
 
12 
 
 THICK-LENS OPTICS 
 
 28. Convergent Rays. 
 
 Notice that the converging rays are rendered more con- 
 verging. 
 
 lens 
 
 Convergent rays""- 
 
 derial 
 some 
 
 object, real 
 previous lens 
 
 C F 
 
 Aerial object outside or inside the focal distance. 
 Diagram showing the effect of interposing a + lens in the path of 
 converging rays, thus producing a real image. 
 
 N. B. In tracing images, notice what a different result 
 we get for the same position of the object, influenced by 
 its being a real, or an aerial object with converging rays. 
 This is of great importance in tracing images. 
 
 29. Negative Lens. In the same manner we get the 
 
 -lens 
 
 Divergent rays: virtual image. 
 
 following progressive diagrams, showing the result of mov- 
 ing the object outward at the left until it disappears at oo, 
 coming in again at the right from oo and moving down to 
 within the focal distance. 
 
THIN LENSES 
 
 13 
 
 -lens 
 
 Jl 
 
 Object 
 at* 
 
 D 
 
 YlTtUdl 
 
 image a point 
 Parallel rays: virtual image a point 
 
 -lens 
 B / 
 
 Jirt 
 
 inal\object 
 <z convergent 
 ^'~^ rays* 
 
 Convergent rays: aerial object outside the focal distance: virtual 
 image. 
 
 derail jieal image 
 -.object at oo 5^ 
 
 Fiftual' -IT^C^- x 
 
 .Convergent rays: aerial object at focal distance: image virtual or 
 real, at infinity. 
 
 Note. The letters A BCD have the same progressive 
 signification as in 25. 
 
 Notice that the formula has now become hl>fi to c. 
 
14 
 
 THICK-LENS OPTICS 
 
 30. Collecting these results, we have the following formu- 
 lation of procedure. 
 
 Formula for diagram tracing of rays from object to 
 image : 
 
 Positive lens hi > / 2 to c 
 Negative lens hi > /i to c 
 
 For explanation of symbols, see 25. 
 
 Convergent rays: aerial object inside focal distance: real image. 
 
 Notice, what has been elsewhere spoken of, the effective 
 focus for a -f lens is the right-hand one; for a lens, 
 the left-hand one. (Light from the left.) 
 
 31. In some cases it becomes necessary to trace back 
 the rays from the image to the object (see" 37, Ex. 13), in 
 which case we have: 
 
 Formulation of procedure for diagram tracing of rays 
 backward from image to object: 
 
 Positive lens rf 2 l \ \ c/ 2 to c 
 Negative lens rf\l \\ cf\ to c 
 
 meaning, draw a ray (r) through / 2 to the lens (I), and then 
 along a parallel to a center line through / 2 (|| c/ 2 ) to the 
 center line (c). f\ is the left-hand, / 2 the right-hand focus. 
 The use of this formulation will be found to be of great 
 help in tracing graphically the conjugate points of a lens. 
 In fact, without the mechanical aid of the formulation, 
 
THIN LENSES 15 
 
 it is exceedingly difficult at times for the novice to do so 
 without error, especially in tracing from image to object. 
 
 32. Diagram for Oblique Rays. 
 
 By 25 A' is the image of A. A second ray from A, 
 AB, must go to the 
 same point, "5 A'. But 
 by 3^ it must also 
 go through the point 
 C, where the parallel 
 center line DC pierces 
 the focal plane. 
 
 But we can con- 
 sider the ray A B as an oblique ray, and the formula for 
 oblique rays is evidently 
 
 rl ><#> 2 1 1 to c 
 
 meaning, draw a ray (r) to the lens (I), and then through 
 ( 0, the secondary focus determined by a parallel through 
 the center intersecting with the focal perpendicular (< 2 ||), 
 to the center line from the object (c). Omission of "to c" 
 gives the direction of the refracted ray, independent of the 
 origin. 
 
 This formula evidently includes that of 25 as a par- 
 ticular case. 
 
 The reverse formula for tracing from image to object is 
 rfal || c<f> 2 to c, the interpretation of which is similar to 
 that of 31, of which this is the general case. 
 
 For a negative lens, the corresponding formulae are 
 
 rl <i || to c and r<i I \\ c$\ to c 
 
 This is one of the most important sections in the book for 
 enabling the investigator to get a quick and graphic idea 
 of the location of the images due to a succession of lenses, 
 
16 
 
 THICK-LENS OPTICS 
 
 allowing, as it does, any ray to be traced, whatever the effect 
 of the lens upon it. 
 
 Start the ray from the intersection of the object with the axis. 
 Each new intersection with the axis will locate an image. 
 
 The same principle applies to the refraction through a 
 surface and reflection from a surface, the surfaces being 
 typified by vertical straight lines, as in the case of lens 
 surfaces. 
 
 Since the point A' lies on the line A A' through the 
 optical center, its position will not be changed by twisting 
 the lens about a vertical axis through the optical center. 
 This will have an important bearing in subsequent sections. 
 
 The reverse formulation (see 31) is r<f> 2 l \\ c<f> 2 to c, and 
 r<f>il || c<i to c, for + and lens respectively. 
 
 33. As an example of the application of the section 
 
 Diagram illustrating the tracing of oblique rays from the object A, 
 through the three lenses, 7, //, ///, to the final image, ///. The foci 
 and images of the lenses are marked correspondingly /, II, III. The 
 rays in their different courses are marked by encircled numerals. This 
 is, in a very distorted form, the course through a compound microscope, 
 / being the objective and //, /// the lenses of the eyepiece, produ- 
 cing the virtual image, ///. See also 95, Ex. 2. 
 
THIN LENSES 
 
 17 
 
 above and 25, we have the adjacent diagram of rays 
 passing through three lenses: I, II, III. 
 
 Surface refraction can be traced in a manner similar to 
 that of 32 by the formulae 
 
 Incident rays rs -> <fo 1 1 
 Emergent rays rs > < 2 1 1 
 
 where <fo || means the point determined by a parallel 
 through the center of curvature and a perpendicular (to 
 the axis) through the focus of the surface, 3 radii (assuming 
 /A = |, 12) from the vertex, measured through the center. 
 < 2 || similarly, but 2 radii ( 13) from the vertex, measured 
 away from the center. (See Appendix, Figs. 1-4.) 
 
 Example 1. Try this method in checking the results 
 of 72, Exs. 7, 8, 10, 19. 
 
 Example 2. By 32, drawing a ray from a point in 
 the axis (see (D in preceding diagram), show that in order 
 to get a virtual image (the case of a positive lens used as 
 a microscope) the object must be within the focal distance. 
 (Conf. 93, Ex. 4.) 
 
 34. Parallel Rays meet in the Focal Plane. 
 
 By Elementary Geometry, three rays through a point 
 cut off proportional parts on any two parallels; hence 
 
18 THICK-LENS OPTICS 
 
 AD = OB A'D' OB OG 
 
 AE = OG ~ A'E' l 7 ^' ~ ~A r E' 
 
 But by similar triangles, etc. 
 
 OC_ OB OG OF^ 
 CD' ~ A'D' ~ A'E' ~ FE' 
 
 Therefore the triangles OCF and OD'E' being similar, C 
 and F are equally distant from the line OG 
 
 That is, parallel rays focus in the focal plane (the plane 
 through the focus perpendicular to the axis) at a point 
 determined by the center line; and, conversely, rays from 
 a point in the focal plane emerge parallel, parallel to a 
 line from the point through the center. 
 
 35. Standard Formula. One formula (viz. = - 
 
 V v u / 
 
 is used throughout the book, the proper sign (+ or ) 
 being given to the numerical values when used. The use 
 of the two formulae of 17, one for the positive lens and 
 one for the negative lens, as is the practice of some writers, 
 is apt to lead to confusion, since both formulae apply to 
 both lenses under some conditions. It is the difficulty of 
 distinguishing these conditions that makes the trouble for 
 the non-expert. Hence the decision at the head of this 
 section, since then the only difficulty arises from the 
 selection of the + and signs. This selection is guided 
 by the rules of the next two sections. (See note to 17.) 
 
 USE OF THE FORMULA - = 
 
 (See diagrams of 25-29.) 
 
 36. Positive Lens. Real object, diverging rays, real 
 image (object outside of F): 
 
THIN LENSES 19 
 
 / and v must have a different sign from u. 
 Real object, virtual image (object inside of F): 
 
 u and v must have a different sign from /. 
 Converging rays, aerial object: 
 
 u, v, and / have the same sign. 
 
 For light coming from the left, / is positive (for light from 
 the right, / is negative). (See 76.) 
 
 37. Negative Lens. Real object, diverging rays: 
 
 u, v, and / have the same sign. 
 Converging rays (converging outside of F), virtual image: 
 
 u different sign from v and /. 
 Converging rays (converging inside of F) real image: 
 
 u and v have different sign from /. 
 
 For light coming from the left / is negative (for light from 
 the right, / is positive). (See 76.) 
 
 EXAMPLES 
 
 Check each calculation by an actual drawing, to scale 
 ( 38), to avoid large errors; guide the drawing by the 
 formulation of 30, 32, and see 38-41. 
 
 Decide on the direction of the ray, thus fixing the sign 
 of u and / (say from the left). If from the left, / will be 
 positive for a positive lens and negative for a negative lens; 
 u will be negative for a real object or an aerial object with 
 diverging rays therefrom, and positive for an aerial object 
 and converging rays. (See 76.) 
 
 From the data given find the other elements. 
 
 1. Positive lens with F = 1 ft. 
 
 (a) u = - 11 in. /. v = - 11 ft. 
 
 (6) u = -- 10 in. /. v = - 5 ft. 
 
 (c) u = - 1 in. /. v = - T V ft. 
 
 (d) u = - 20 ft. /. v = f $ ft. 
 
20 THICK-LENS OPTICS 
 
 (e) u = - 2 ft. /. v = 2 ft. 
 
 (/) u = - li ft. /. v = 3 ft. 
 
 2. u = - 2f .'. v = 2f 
 
 3. u = - 6, v = 1 .'. F = f 
 
 4. w = 3 in., i; = 18 in. .'. F = - 3f in. 
 
 5. u = 12, v = 1 /. F = H 
 
 6. r = 5, s = 7, negative lens, /* = $, w = 60. 
 Ans. F = \ 5 , v = -V/, double concave. 
 
 7. Positive lens, r = 7, s = 5, /* = |, w = 60. 
 
 ^.ns. Light from right, F = 35, v = 84, concavo- 
 convex. 
 
 8. Positive lens, r = -- 7, s = 5, /* = f , w = 60. 
 
 ^.ns. F = -> v = f f, double convex, light from 
 right. 
 
 9. Negative lens, r = 5, s = 7, n = |. 
 Ans. / = 15, /' = - 21, F = 35. 
 
 10. Negative lens, r = 7, s = 5, /* == |. 
 
 Ans. Light from left, / = 21, /' = -- 15, F = - 35. 
 
 11. Negative lens, r = 7, s = 5, /A = |. 
 
 Ans. Light from right, / = - 21, /' = 15, F = 35. 
 
 12. Convex lens, light from right, F = 5.813, object 
 30.56 in front. Where is the image? 
 
 Ans - - = 57T^+ AM o=0-03273 -0.1720= -0.1392 
 v 60. ob o.olo 
 
 - /. = - 7.183 to left. 
 
THIN LENSES 21 
 
 13. A telescope has a field glass of 23f inches focus and 
 an erecting eyepiece composed of 4 lenses as follows, 
 reading towards the eyepiece, 2, If, If, If inch focus, 
 with the separations 2J, 4, 2 inches. To trace the con- 
 jugate foci. 
 
 Since the last four lenses are fixed and the focussing is done 
 by adjusting the combination relative to the field glass, 
 we take as the starting point the virtual image seen by 
 the eye. This will be seen at a distance determined by 
 the "set of the eye" of the observer. (See 109.) 
 
 We assume the "far set" eye and the rays to the virtual 
 image parallel. This makes the object for the fourth lens 
 (counting from the left) at the focus of that lens; and 
 indicating by v\, u\ the conjugate distances for the first 
 lens, etc., we have the following series of conjugate distances. 
 
 11 
 
 V.i = 00 , U 4 = - -- 
 
 ,3 = 2 - = .625. -- .-,-. 987 
 
 = 4 + .937 - 4.937. ^ - I = A. ,. % = - 3.012 
 
 *=- 3.012 + 2.25 =- .762. -^ - 1 - J 
 
 .'. ui = - .552 
 
 This shows that the focus of the field glass should be 
 .552 inches in front of the first lens. This is approximate 
 only, since the real lenses must be treated as thick lenses, 
 as in Chapter III. 
 
22 
 
 THICK-LENS OPTICS 
 
 The lenses, with their corresponding images and foci, are designated 
 by Roman numerals. The dotted lines show the course of a ray from 
 the foot of the aerial object, with the construction lines .( 35). Notice 
 how the inverted object (aerial) is converted into an erect one by the 
 second lens. The virtual image being at infinity, the last course of 
 the ray is horizontal, as shown. 
 
 GRAPHIC CHECK ON CALCULATIONS 
 
 38. Inspection of the diagrammatic constructions will 
 show that they fall under one or the other of the following 
 diagrams, or modifications of these. 
 
 Where all the quan- 
 tities are on one side of 
 the zero line (lens line), 
 as in the left-hand dia- 
 gram, we have (by sim- 
 ilar triangles) : 
 
 a v b v , v . v a + b 1 
 
 7 = -i r~r = whence r - f = . r = * 
 a + b u' a+b f u f a + b 
 
 1 ,1 1 
 
 or r 7 = - 
 
 u f v 
 
 which is the same as in 16. 
 
 Worded, this becomes, The reciprocal of the mid line = 
 the sum of the reciprocals of the end lines (when all the 
 quantities are on the same side of the zero line). 
 
THIN LENSES 23 
 
 39. Similarly for the right-hand diagram, unless we take 
 into account the signs of the quantities, in which case: 
 
 Where two of the quantities are on different sides of 
 the zero (lens) line (right-hand diagram) The reciprocal 
 of the mid line = the reciprocal of the end line on the same 
 side of the zero line as the mid line the reciprocal of the 
 end line on the other side of the zero line. 
 
 USE OF THE DIAGRAMS. To insure accuracy in signs and 
 to detect material (large) errors, plot these diagrams to scale. 
 Lay off the end lines any distance apart, draw the diagonals 
 and see if the mid line fits in size and sign. Or lay off f and 
 u (any distance apart) and then by means of the two diagonals 
 determine v. 
 
 EXAMPLES 
 
 1 1 J_ 1 1 1 
 
 -11" =TTl2 12' -10" -5-12~l2' 
 
 1=*- J- Ex 6 -i- = U l 
 
 61-6 18 3 "* 18 
 
 7 ' 5 
 
 GRAPHIC CHECK ON CALCULATIONS 
 
 This method is given in some detail because so many books use 
 one or the other of the diagrams. 
 
 40. Pos. lens with + f. By similar triangles 
 u u + v' 111 
 
24 
 
 THICK-LENS OPTICS 
 
 which is the same as the equation of 17 for a positive 
 lens. 
 
 USE OF THE DIAGRAM. Set off / and u in proper size 
 and direction, draw the line through 
 their ends: its intersection with the 
 45 line will give v both in size and 
 sense, u on the right indicates an 
 aerial image made by some preceding 
 lens. 
 
 Variations of this are, as the object 
 
 moves from the left, being aerial when on the right of 
 the lens, the rays of light coming from the left, 
 
 / 
 
 45' 'line 
 
 
 
 Diagrams showing the relative sizes and positions 
 of u and v. 
 
 Real object beyond the focal distance, real 
 image, inverted. (Light from the left.) 
 
 (D Real object within the focal distance, virtual 
 image, erect. 
 
 (3) Aerial object within the focal distance, real 
 image, erect. 
 
 @ Aerial object beyond the focal distance, real 
 image, erect. 
 
 41. Negative lens with f. In th6 same manner, we 
 get the following diagrams: 
 
 USE OF THE DIAGRAMS. (See 40.) 
 
THIN LENSES 
 
 25 
 
 Notice. When u and v have the same sign, the image is 
 erect. When u and v have a different sign, the image is 
 inverted. (Both lenses.) 
 
 Real object beyond the local distance, virtual image, erect. 
 
 Real object within the focal distance, virtual image, erect. 
 
 Aerial object within the focal distance, real erect image. 
 
 Aerial object beyond the focal distance, virtual inverted image. 
 
 POWERS OF LENSES: DIOPTERS 
 
 42. In the expression --- 
 v u 
 
 -.> - f is called the power 
 f f 
 
 of the lens (to alter a light- wave front), and when v and u 
 are expressed in meters (or its practical equivalent, 40 
 inches), the power units are called diopters. 
 
 Similarly, - = p and - = p' are called, for convenience 
 
 v u 
 
 of reference, powers of the distances. 
 
 If we call the power of the lens Z>, then 
 
 D = P - 
 
 - 1) (q - q') 
 
 Dioptric units are generally used by opticians in con- 
 nection with (thin) spectacle lenses. The power of a 
 combination of lenses equals the sum of the powers. (See 
 75.) 
 
THICK-LENS OPTICS 
 
 EXAMPLES 
 
 43. Ex. 1. u = - 80 cm., / = 20 cm., v =? (Light 
 from left.) 
 
 Ans. p = = 
 
 100 
 ' * " 376 " 26f 
 
 Ex. 2. u = - 20 in., ti = 5 in., / =? 
 
 Ex. 3. / = 10, u = - 8, v = ? (Positive lens.) 
 
 n 40 40 
 
 Ans. D = JQ = p - -g = 4 = p + 5. .'. p = - 1. 
 
 Ex. 4. Four lenses in contact: (a) a plane concave of 
 4 diopters; (b) a positive meniscus of r = 2 in., s = 5 in.; 
 (c) a biconvex of 50 cm. focus; (d) a biconcave of 33 J cm. 
 focus. What is the focus of the combination? (Light 
 from left.) 
 
 Ans. (a) = - 4 D. (b) = ~ - = 6 D. 
 
 W-E-.ix W-'-g ift 
 
 Therefore, combination = -4D + 6D + 2D-3D = Z>, 
 and the result is a positive lens of 100 cm. focus, projecting 
 a real inverted image. (See 25.) 
 
 44. The focal length expressed in inches gives the num- 
 ber of the lens. (Obsolete.) 
 
 45. Spectacles for Farsighted. Positive lens, virtual 
 image. 
 
THIN LENSES 27 
 
 The formula of 16 becomes: 
 
 1 1 
 
 -v -10 
 
 the 10 and v being taken negative, because we want a 
 virtual image. See diagram of 27. v must be the nearest 
 'distance at which the wearer can conveniently see without 
 spectacles, 10 being the distance at which he holds the 
 book. The image is virtual. 
 
 46. Spectacles for Nearsighted. Negative lens. 
 
 In this case the formula of 16 becomes: 
 
 1 1 1 
 
 - v u - f 
 
 u, v, and / have the same sign, 37. v must be the greatest 
 distance at which the wearer can see clearly without spec- 
 tacles, 10 inches being the distance at which the book is 
 held. The image is virtual. If / is less than v, he can 
 see objects at all distances over 10 inches, since the virtual 
 image is always within his visual distance. See first dia- 
 gram of 29. 
 
 EXAMPLES 
 
 1. If longest distance for distinct vision is 15 cm., what 
 lens will enable the wearer to see all distant objects? 
 Ans. 15 cm. or under. 
 
 2. Book is held at 1 ft. with concave 6 in. focal lens. 
 Where is the image? Ans. 4 in. 
 
 3. A man can read distinctly at 15 cm. What lens must 
 he use if he wants to read easily at 60cm.? Ans. f = 20. 
 
 4. If the nearest distance for distinct vision is 15 inches, 
 what focal length of spectacle is required if the book is 
 held at 10 inches? Ans. 30 in. 
 
28 
 
 THICK-LENS OPTICS 
 
 5. If the shortest distance for distinct vision is 1 m., 
 what length spectacle is wanted for object at 25 cm.? 
 
 Ans. -7 = -7^ 1=4 1=3 diopters, or/ = 33 J cm. 
 
 .25 
 
 6. Best vision at 3 ft. With 1 ft. spectacles the book 
 should be held at . . .? Ans. 9 in. 
 
 7. A longsighted person with + glasses of 40 cm. length 
 finds he must hold the book no nearer than 30 cm. for 
 comfort. What is his nearest point of distinct vision? 
 Ans. 120 cm. 
 
 8. A longsighted person can only see distinctly at 48 
 cm. or more. By how much will he increase his range of 
 vision with convex spectacles of 32 cm. focus? Ans. 
 48 - 19.2 = 28.8. 
 
 9. A person whose distance of most distinct vision is 
 20 cm., uses a reading glass of 5 cm. focus. How far from 
 the book must it be held? Ans. 4 cm. 
 
 47. Magnification for Convex Lens, Real Image (Cam- 
 era, etc.). 
 
 f-lens 
 
 a . Ill 1,11. ... . 
 
 Since =7, or - H = j for positive lenses, 
 
 v u f v u f 
 
 y 
 
 - = M = magnification factor = ratio of image to object 
 
THIN LENSES 29 
 
 = ratio of image distance to object distance 
 v f 
 
 = f " = iTH 
 
 T. _ . ,. size of image image b v f 
 
 Magnification = ~ r^ - = - -- = - = 7 
 
 size ot object c a 
 
 u = ^ -{- f = distance to object 
 
 v = f M + f = distance to image 
 If M = 1, then u = 2/, v = 2/. 
 
 If M = 1, which is equivalent to saying, image same 
 size but not inverted, then u = 0, v = 0. 
 
 Note. Ordinarily (when we are taking account of direc- 
 tion) -f- M indicates erect image; M, inverted image. 
 Do not get the two cases confused. 
 
 EXAMPLES 
 1. In Ex. 9, 46, what is the magnifying power? 
 
 Ans. 
 
 - T 
 5 4 
 
 2. An engraver uses a 4 in. focus magnifying glass, close 
 to the eye. Where must he hold his work to get a magnifi- 
 
 4 
 
 cation of 4? Ans. - - = 4. .*. u = 3. 
 4 u 
 
 3. An object is 3 ft. in front of a 6 in. lens. What is 
 the magnification? Ans. J/(3 J) = i- 
 
 4. v = 8, 4-inch lens. Ans. M = 1, u = 8. 
 
 5. v = 12, 4-inch lens. Ans. M = 2, u = 6. 
 
 6. v = 16, 4-inch lens. Ans. M = 3, u = 5J. 
 
30 THICK-LENS OPTICS 
 
 7. v = 20, 4-inch lens. Ans. M = 4, u = 5. 
 
 8. An object 8 cm. high is 1 m. from an equiconvex 
 lens of index of refraction 1.5 and radius of curv. 0.4 m. 
 Where is the image and what is its size? Ans. Image 
 f m. on other side and of height 5J cm. 
 
 9. In Ex. 1, 37, what are the magnifications? Ans. 
 12, 6, T V, A, 1, 2. 
 
 10. Converging lens, with object 5 in. from lens. Image 
 = 6 times the object. Where is the image and what is 
 the focal length? Ans. v = 30 in. / = -\-. 
 
 11. Required an image of 3 mag. by lens of F focal 
 length. How far must the screen be from the object? 
 
 Ans. \F. 1 + -L = 1. .-.x = \F. 
 
 6 x 6 x r 6 
 
 12. An object is a distance d from a screen, and a thin 
 pos. lens is placed to form an image. If the lens be moved 
 a distance p = Vd 2 4 df, another image will be formed 
 whose linear dimensions are to those of the former as 
 (d - p) 2 to (d + p)\ 
 
 13. A disc 1 inch in diameter, 8 inches from the eye, is 
 seen through a convex lens of 8 inches focus, placed half^ 
 way between. What should be the diameter of the lens 
 to see the whole of the disc at once? What is the distance 
 of the image from the eye? Ans. Diam. of lens = f in. 
 Dist. of image from eye = 12 in. 
 
 14. A concave lens is blackened except a circle of 4 cm. 
 diameter at the center. A beam of sunlight through this 
 gave an illuminated circle of 20 cm. diameter on a screen 
 64 cm. from the lens. Show that the focus of the lens is 
 16 cm. Use the first diagram of 29. 
 
 15. If u = 2/, then v = 2/, M = 1. 
 
THIN LENSES 31 
 
 j|/f f u f' f 
 
 16. If u is very large, show that -=-^7 = - ; , = J , 
 
 M u j j j 
 
 practically, and therefore that for distant objects the sizes 
 of the image are proportional to the focal radii of the lenses. 
 48. Copying, Enlarging, etc., with the Camera. 
 To find distance to plate, etc., for given magnification 
 or reduction. 
 
 v = distance from lens to screen (plate) 
 = camera extension. 
 
 = F (M + 1) = ^ + F (47) 
 M = magnification factor 
 N = reduction factor; M = -~ 
 F = focal length of the lens 
 u = distance from lens to object 
 
 = F + ^ = F + NF 
 
 Strictly these distances should be measured from the 
 nodal points (see 63), but approximate values and meas- 
 urements are sufficient for a first adjustment, the final 
 being made by trial. 
 
 EXAMPLES 
 
 1. 6 in. lens, 12 in. drawing, 4 in. copy, whence N = 3. 
 
 2. 6 in. lens, 4 in. plate, 12 in. copy, whence M = 3. 
 Ans. v = 24, u = 8. 
 
 3. A candle stands a yard from the screen. What lens 
 and where must be used to get an image 5 times as large? 
 
 Ans. 5 in. lens, 30 in. from the screen. 
 
32 THICK-LENS OPTICS 
 
 49. Best Position for Condenser. 
 
 Condenser of joe al length f 
 
 Jfounting of 
 the objective 
 
 JTejative 
 
 \_ 
 V 
 
 1 
 
 v 
 
 - = M = Magnification 
 
 v = (1 + M) f 
 1 + M 
 
 u = 
 
 M 
 
 f 
 
 [16 
 
 [48 
 
 To get all the picture within the cone of rays, 
 a V 
 
 g~~ V -x 
 
 whence x can be calculated. 
 
 To get the cone of light to just fill the mounting, 
 
 a V 
 
 c =: V - x - u + b 
 
 whence u can be found. 
 
 For sunlight, V = F, (U = oo), whence 
 a 
 
 F = 
 
 g - 
 
THIN LENSES 33 
 
 If x = ' = - whence u 
 
 can be found. 
 
 Example. F = 3, / = 6, M = 5, a = f , c = }, b = f . 
 
 t^i. J- - C7-317 
 
 50. Exposure. 
 
 T = time of exposure for distance v of image, with aper- 
 ture d 
 
 (the subscripts indicating the corresponding quantities for 
 some known exposure with a satisfactory result) 
 
 tZY (^ . P\. Tl = (M__T) 2 
 
 + 1\ 2 U.S. 
 
 /Y 
 
 /J ' 
 
 + 1J U.S., 
 
 U.S. = U.S. numbers 
 /, /i = the / numbers; see 51 
 M, MI = magnification 
 
 In use, disregard the letters in which there has been no 
 change of conditions, and see note at the end of Example 1. 
 
 EXAMPLES 
 
 1. With an 8 in. lens, with //20 enlarging 5 times, 40 
 seconds exposure was required. What exposure is required 
 for a 9 in. lens //30 and enlarging 6 times? 
 
 sec. 
 
34 THICK-LENS OPTICS 
 
 Notice that the / number determines the exposure with- 
 out regard to the focal length. E.g. //20 requires the same 
 time whatever the lens. 
 
 _., focal length 
 
 ol. slowness tactor = rr- 2 = n, written = 
 
 aperture diameter 
 
 f/n and called the / number. 
 
 Time of exposure varies as the square of the slowness 
 
 factor. For example, a ~- i ens requires -^j- the exposure 
 
 / . 169 13 2 
 
 of a - lens. ^ = ^ 
 
 The U.S. numbers give the relative time of exposure, 
 + -UJ1.U 
 
 whence we can find corresponding / and U.S. numbers by 
 the formulae 
 
 / number = 4 VU.S. number 
 (j numberV 
 
 U.S. number = , 
 
 52. Hyperfocal Distance. When the object is too near 
 to the lens, what produces confusion in the picture is the 
 overlapping of images. It is found that a slight overlap- 
 ping is not distinguishable. This occurs when the two 
 images of a given point are not more than T ^ inch apart. 
 Hence up to the point in front of the lens where the separa- 
 tion of images is not more than T <j inch, all objects will 
 apparently be in focus. 
 
 The distance of this point from the lens is called the 
 
THIN LENSES 
 
 35 
 
 hyperfocal distance, the distance to the nearest distinct 
 
 object beyond 
 
 which all objects 
 
 are apparently in 
 
 focus. 
 
 The different P. 
 images of a given 
 object areevidently 
 scattered over a circle. The largest circle permissible with- 
 out confusion is called the circle of confusion. 
 
 F = focal length in inches 
 P = nearest distinct object 
 Q = image of P 
 
 a = radius of circle of confusion 
 d = diam. of stop in inches 
 
 f = -7 =/ number 
 
 1 
 200 
 
 m. 
 
 By sim. triangles 
 But 17 
 
 d/2 
 
 l_ _ __ _ 
 
 F + x~*~ u ~ F 
 
 1 
 
 u = F 
 
 F + x F (F + x) 
 
 Fd 
 2a 
 
 focal length X diam. of stop 
 . diam. of circle of confusion 
 
 /Till 
 
 100 F 2 . 
 
 100 F 2 
 
36 
 
 THICK-LENS OPTICS 
 
 = distance to nearest distinct object, 
 beyond which all objects are 
 apparently in focus 
 
 Note. The diagram is drawn for a thick lens. If the 
 two vertical dotted lines should be brought together, it 
 would make the proper diagram for a thin lens, with no 
 change in the mathematics. 
 
 53. Magnification and Reduction for Negative Lens. 
 "Lens 
 
 Objec\ 
 
 u 
 
 T-> i ,- 1 image f v v 
 
 Reduction = = -T^T = 7 = 1 - ? 
 
 N object / / 
 
 u f 
 
 f+u f+u 
 
 As the object is moved nearer the lens, the image grows 
 
 larger, until with the object at 
 the lens we get unit reduction. 
 54. If converging rays (due 
 to convex lens) are coming from 
 the left, we have the following 
 diagram, the real image of the 
 convex lens being the aerial 
 object of the negative lens. 
 
 ens 
 
 Image 
 
 Serial 
 object 
 
 Image 
 Object 
 
 magnification of image of the + lens, due to 
 the lens. 
 
 aerial object f u 
 
THIN LENSES 
 
 37 
 
 This is the telephoto combination spoken of in Chapter V. 
 v is there called the bellows extension and denoted by E, 
 
 p< 
 whence M = 1 + and E = f (M - 1) 
 
 55. Magnifying Power of a Positive Lens used as a 
 Microscope. 
 
 +lens 
 
 In this case the image seen by the eye is virtual and 
 subtends the angle shown in the diagram. Since this 
 angle is very small, it is measured by its tangent, and 
 
 = tan 9 = **- 
 
 The angle subtended by the object is 
 
 c 
 
 u + y 
 
 and the ratio of these is the 
 
 Apparent magnification = 
 
 9 
 
 a 
 
 f u cd 
 
 = f (v + y) + vy 
 fd 
 
 cd 
 
 ? / 
 f-u 
 
 fv 
 
 t + 7 
 
 i_ i 
 
 v u 
 
38 THICK-LENS OPTICS 
 
 , (<*-y)y 
 id 
 
 This is a maximum = 1 + j when y = - d, which ex- 
 plains why some readers like to push their spectacles down 
 towards the end of the nose. 
 
 The distance d, the distance for distinct vision, is gen- 
 erally taken as 10 inches (25 cm.), but should be taken a 
 specific value for each observer. 
 
 56. Ordinarily the conventional magnification of the 
 lens is stated as the ratio of the actual size of the image to 
 that of the object, viz. 
 
 a v 
 Conventional magnification = - = - 
 
 v (v + f ) 
 
 vf 
 
 EXAMPLES 
 
 1. A convex lens, of focal length inch, is used by a 
 14 inch (nearest distance for distinct vision) eye. What is 
 the magnification? 
 
 Ans. Mag. = 14/t + 1 =71. 
 
 2. A 2 yard eye uses a 2 ft. lens. How far from the 
 glass should the object be placed? 
 
 Ill - 3,, 
 
 Ans. -z -- = o5 therefore u = = ft. 
 
 D U A & 
 
 3. Check Ex. 2 by comparison of images, graphic con- 
 struction, 32. 
 
CHAPTER III 
 
 THICK LENSES 
 
 57. One method of finding the equivalent focus of a 
 thick lens is to select a thin lens (spectacle lens) which 
 will give on the ground glass an image of exactly the same 
 size as the thick lens gives, the object being very distant. 
 The focal length of this thin lens will be the focal length 
 of the lens under consideration. On the mounting of the 
 thick lens mark off this distance from the ground glass 
 when in focus. This point is called the principal point 
 of emergence. 
 
 Turning the lens around, we get a similar point for the 
 other end of the lens. These two separated points mark 
 the points from which evidently measurements for focal 
 radii are to be made, and correspond to the optical center 
 of a thin lens. 
 
 Like optical centers, these principal points will be found 
 to be points around which the lens can be twisted (about 
 a vertical axis) without affecting the image on the ground 
 glass. 
 
 58. Suppose an object to take the successive positions 
 a, 6, c, and then the aerial objects at d and e, with the 
 resulting images a', 6' ... 
 
 39 
 
40 
 
 THICK-LENS OPTICS 
 
 At the optical center of the thin lens, and only there 
 (Conf. 47), will the object and image have the same size 
 and sense (image not inverted). 
 
 Diagram showing the images resulting from successive positions 
 of the object, and the resulting changes in size of the image; 
 i.e. its magnification with the corresponding images a', &', c'. . . 
 
 59. In 32 we found that revolution about an axis 
 through the optical center, the point from which the focal 
 radius is measured, did not disturb the image. Experiment 
 shows that revolution about an axis through the principal 
 point of emergence, the point from which we measure the 
 focal distance, does not disturb the image of a distant 
 object. This is the point around which panoramic cameras 
 are revolved. 
 
 60. Principal Points. Just as image and object have 
 the same size and relation at the optical center of a thin 
 lens, so we might anticipate that for a thick lens the 
 image and object would have the same size and relation at 
 the principal points, the points from which the radii are 
 measured. (Conf. 69.) 
 
 61. We can find these principal points as follows: 
 
Horizon tal ray 
 
 / 
 
 THICK LENSES 
 
 /*~**\ 
 
 / ^Principal plane 
 
 41 
 
 Principal pl 
 
 Diagram showing construction for the determination of the principal 
 points. The order of the letters indicates the order of construction. 
 6, c, found by 7. The upper half gives the construction for one 
 principal point, the lower half for the other. 
 
 Caution. This construction applies strictly only to 
 points near the axis, but it serves to illustrate the principle 
 for future use. 
 
 Theoretically we could, by picturing the surfaces as 
 straight lines, get a correct graphic construction, but the 
 disparity between the thickness of the lens anji the radii 
 is generally so great that the graphic construction is of 
 little value by reason of its acute intersections. 
 
 62. Since there are two points around which we can 
 
 Diagram illustrating the apparent horizontal transference between 
 the principal planes. 
 
42 
 
 THICK-LENS OPTICS 
 
 revolve the lens without effect on the image (the lens 
 being reversed so as to make each one a point of emergence), 
 i.e. two points like the optical center of the thin lens, two 
 points where the object and image have the same size and 
 relation, as shown by the diagram (Conf. the diagram 
 of 61), the effect is as if the rays from the object passed 
 to the first principal plane and then were transferred hori- 
 zontally to the other principal plane so as to keep the 
 object and image the same size from plane to plane. (Conf. 
 105 after reading 64.) 
 
 This equivalent pair of parallel surfaces is called the 
 equivalent thin-split. 
 
 63. Nodal Points. The following construction gives 
 two new points of importance called nodal points. 
 
 Font* 
 
 ~Foc 
 
 Diagram illustrating the location of the nodal 
 points. 
 
 From A, a point in one focal plane, draw the horizontal 
 ray A B, which is of course refracted to the focus F r . Draw 
 AC parallel to BF'. By the property of principal planes 
 C is carried to Z), and by the property of rays from a point 
 in a focal plane (see 35) DN' is parallel to BF'. 
 
 N and N' are two points, nodal points, which have the 
 property that incident rays through one of them (e.g. N) 
 emerge parallel through the other (e.g. N'). In this they 
 resemble the optical center of thin lenses. (Conf. 70.) 
 
THICK LENSES 
 
 43 
 
 64. Evidently HN = H' N' = F'H' - FH. 
 
 [Equal triangles, etc. 
 
 Therefore if FH = F'H', i.e. the focal distances the 
 same, due to same media on both sides of the lens, the usual 
 case (but Conf. 71), then H and N coincide, likewise H' 
 and N'. 
 
 Every incident ray through the first nodal point emerges 
 as a parallel ray through the second nodal point. There- 
 fore the angle subtended by an object at the first nodal 
 point equals the angle subtended by the image at the 
 second nodal point, just as in the thin lens the angles 
 subtended at the center by object and image are the same 
 size. 
 
 In the human eye, the second nodal point is within the 
 crystalline lens about .4 mm. from the back. (Conf. 71.) 
 
 65. Optical Center. 
 
 2 d surf. 
 
 C Center of Center O 
 1* surf ace of 2* surf ace 
 
 Construction. From the centers of the two surfaces 
 draw parallel rays and find the point C as shown. 
 
 By sim. triangles 
 
 = - (left-hand diag.) 
 
 - CO - r . 
 
 , -- (right-hand diag.) 
 
 CO = 
 
 s r 
 
 7* 
 
 00' = 
 
 er 
 
 r 
 
 (s - r - e) 
 
 s r 
 
44 
 
 THICK-LENS OPTICS 
 
 EXAMPLES 
 1. Neg. lens, r = 2, s = - 3, e = 1. 
 
 Ans. AC = 
 
 , A'C = 
 5 5 
 
 2. Pos. lens, r = - 3, s = 2, e = 1. 
 
 Ans. AC = 
 
 66. Construction for Nodal Points. 
 
 Caution. These constructions apply strictly only to 
 points near the axis, but they serve to illustrate the prin- 
 ciple. (Conf. remark, 61.) 
 
 Draw a and a' parallel through the centers 0, 0', giving 
 the points B, B' . From the ray BB' construct the re- 
 fracted rays c, c'. Where c, c' prolonged cut the axis will 
 
 be the nodal points N, N f . Where BB' cuts the axis is 
 the optical center, since c is parallel to c', being equally 
 inclined to the parallel radii, a, a'. 
 
 AC 
 A'C 
 
 AO - 
 
 CO 
 
 er - ef 
 
 ^^ 
 
 ftr 
 
 = AA' 
 es 
 
 s 
 + AC 
 e/' 
 
 -r /'+/ 
 - Seediag.65 
 
 f 
 f 
 
 /x- 1 
 
 - /tAS 
 
 J - 
 c 
 
 ft- 1 
 AA' 
 
 ir-i 
 
 ; f'+J 
 
THICK LENSES 45 
 
 Hence the position of the optical center is fixed for two 
 given surfaces a distance e apart, since only constants 
 enter into its value. 
 
 If the light comes from the left, A and A' interchange 
 
 places, also r and s, and A'C = , > AC = , ]_ 
 
 67. Calculation for Nodal Points. 
 
 Evidently any ray passing (after refraction) through G 
 will enter and emerge in parallel lines, since the surfaces 
 at the points of incidence and emergence are parallel (being 
 perpendicular to the parallel radii from 0, 0'). 
 
 Evidently a ray pointing to N before refraction will 
 after refraction emerge in a parallel direction as if coming 
 from N'. 
 
 N and C are conjugate foci for the first surface, there- 
 fore, 11, 
 
 [Note. Use the diagram on the right (for reasons given 
 in 21, note) until second reading.] 
 
 = n 
 
 e / 
 
 if light comes from the left 
 
46 THICK-LENS OPTICS 
 
 Therefore AN dist. from first vertex to corresponding 
 nodal point 
 
 -e/ 
 
 (f -L fi _ y ^ li^ht comes from left J 
 
 Similarly A' N f = distance from second vertex to corre- 
 sponding nodal point 
 
 e/' 
 
 ( = ( f -L f ' V ^ light comes from the left) 
 
 N N' = distance between the nodals 
 
 e/ + e/' = Q-l) (/ + /') + ( 
 
 E 
 E 
 
 if light is from the left 
 
 = e (r - s + e) Q - 1) 
 /A (r s + e) e 
 
 e (r - s - e) (A* - 1) . f Ught . g f rQm j f 
 /* (r - s - e) + e 
 
 e (neglecting very small terms) 
 
 = - e f or glass 
 
 o 
 
 68. In computation check the numerical value for N N f 
 by the separate values, e AN A'N'j and check by 
 graphic construction, as in 73. 
 
THICK LENSES 47 
 
 Ex. 1. Show that AN:A'N' has the ratio between 
 r and s, the radii of the surfaces. 
 
 69. Image in One Nodal Plane of Object in the Other. 
 
 If P is the object in one nodal plane (which may be 
 outside the prism altogether, see 72, Ex. 3), we can find 
 its image in the other nodal plane by tracing known rays. 
 
 The explanatory details of one diagram apply equally to the other. 
 
 The known rays are rays through the center of curvature, 
 which enter the corresponding surface without refraction. 
 
 The ray PO, which is unrefracted by the first surface, 
 gives an image R in the plane RC. The image R becomes 
 the object for a new image Q, made in the nodal plane N f 
 by the unrefracted ray O'R. 
 
 PN NO QN' = N'O' 
 RC == CO' RC 
 
 Therefore 
 
 PN 
 QN' 
 
 Therefore 
 PN = 
 
 PN 
 RC 
 
 RC 
 QN' 
 
 NO 
 CO 
 
 CO' 
 
 CO' 
 
 [Sim. triangles 
 
 N'O' 
 
 = 1 
 
 [CO r 
 \Cp' ~ s 
 
 ON 
 O'N' 
 
 NO CO' 
 N'O' ' CO 
 
 [Sim. triangles 
 
 = - Sim. triangles 
 s 
 
48 
 
 THICK-LENS OPTICS 
 
 Hence the object in one nodal plane has an equal and 
 erect image in the other nodal plane; i.e. all rays passing 
 through P in one plane will pass through Q in the other. 
 PQ is parallel to 00'. The image in one nodal plane is 
 transferred horizontally without change of size to the other 
 nodal plane. (Conf. 60.) 
 
 70. Lens separating Different Media. 
 
 Diagram showing the paths of two sets of rays 
 when the principal points and the nodal points do 
 not coincide. Compare this with the diagram of 
 74, where the principal points and the nodals 
 coincide. 
 
 71. The human eye illustrates this case, the aqueous 
 humor being on one side of the lens and the vitreous humor 
 on the other, under which circum- 
 stances the principal points and the 
 nodal points are separated and the 
 two foci are different. (Conf. 22.) 
 The principal points are very close 
 together at H, about 2 mm. behind 
 the cornea; the nodal points almost as close together at 
 N, about 7 mm. behind the cornea. 
 
 The anterior focus is at F, about 13.7 mm. in front of 
 the cornea, and the posterior focus at F f , about 22.8 mm. 
 behind the cornea. 
 
THICK LENSES 49 
 
 Note. Since the investigations of this book are gen- 
 erally for the case where the 
 principal planes and the nodal 
 planes coincide (lens in air), the 
 term nodals has been used indis- 
 criminately for the coincident "T 
 points H and N. 
 
 72. Focal Length of Thick Lens. 
 
 For parallel incident rays, the image by refraction from 
 the first surface will ( 12) be at a distance / from the 
 surface, and 
 
 fJL 1 /A 
 
 ~^~ = 7 
 
 Therefore the distance of the first image from the second 
 surface will be 
 
 (If the light comes from the left, this distance will be/ e.) 
 If v = the distance of second image from second surface, 
 then, 11 
 
 f + e v~ s f 
 
 Therefore - - ** + ^ - u. e + f 
 
 ~ + " 
 
 Whence v = f (f + e) 
 
 
 
 / = - - = focal rad. for 1st surf. 
 
 /' = - = neg. focal rad. of 2d surf. 
 
 e = thickness of lens 
 
 r = rad. of curv. of 1st surf. 
 
 s = rad. of curv. of 2d surf. 
 
 F = principal focus = focus for parallel rays 
 
50 THICK-LENS OPTICS 
 
 Therefore 
 
 N'F = A'F - A'N' = "" 
 
 = F 
 
 e) 
 
 which is called the focal length, for reasons indicated in 
 57. 
 
 / 
 
 [ If the light comes from the left, we have F = , 
 
 \ /* (J ~r J 
 
 EXAMPLES 
 1. Negative lens, r = 5, s = 7, /* = 1.5, e = .2. 
 
 Ans. f = ^ = 15, f = - l -^t - - 21,6 
 
 = .2 + 15 - 21 = - 5.8, AN = .34, A'AT' = .48, 
 F = 36.2. Notice that both nodal points are outside the 
 lens. Light from the right. 
 
 2. Negative lens, r = 5, s = 7, p = 1.5, e = .2. 
 
 ^ns. Light from right. / = 15, /' = 21, A N = -.055, 
 A'N' = .077, F = 5.80. Notice that the nodal points are 
 both inside the lens and close to the surfaces. 
 
 3. Light from right, r = 7, s = 5, ft = 1.5, e = .2. 
 
 Ans. Positive lens. / = 21, /' = - 15, A N = - .451, 
 A'N' = - .323, F = - 33.87. Nodal points are both 
 outside and behind the lens. 
 
 4. Light from right, r = 7, s = 5, ^ = 1.5, e = .2. 
 
 Ans. Double convex lens. / = 21, /' = -- 15, AN 
 = - .078, A'N' = .056, F = - 5.85. Nodal points are 
 inside and very near the surfaces. 
 
THICK LENSES 51 
 
 5. Light from left, r = 5, s = 7, i*> = f , e = .2. 
 
 Ans. f = 15, f = - 21, AN = - .32, A' N' = - .45. 
 Nodal points outside the lens. 
 
 6. Light from right, r = 7, s = 5, /* = f , 6 = .2. 
 Ans. / = - 21, f = 15, A TV = - .48, A' N' = - .34. 
 
 7. Double convex lens, r = f , s = 1, e = J, /u, = f . 
 Light from right. 
 
 . / = - f, /' = - 3, AN = - A, A'N' = A, 
 
 8. Double convex lens, r f , s = 1, e = , /* = f. 
 Light from left. 
 
 Ans. / = , /' = 3, A N = A = 1-58, A' N' = -- A = 
 
 - 0.21, F = 0.947. 
 
 9. Negative lens. Light from left, r = |, s = oo, 
 e = .1, ft - f. 
 
 Ans. /=-3, /' = oo, AN = 0, A'N' = - A = 
 
 - 0.0625, F = - V 5 = " L875. 
 
 10. Double convex lens, r = }, s = 1, e = }, /x = |. 
 Therefore light from right. 
 
 Ans. / = - f , /' = - 3, AN = - & = - 0.157, 
 A'N' = A = 0.210, F = - 0.947. 
 
 11. Double convex lens, r = f , s = 10, e = J, 
 /A = f . Therefore light from right. 
 
 Ans. / = - |, f = - 30, A N = - T f T = - 0.0236, 
 A'N' = AV = 0.315. 
 
 12. Double convex lens, r f, s = 100, e = J, 
 ft = |. Therefore light from right. 
 
 Ans. / = - |, /' = - 300, A N = - 0.00248, A' TV' = 
 .331. 
 
52 THICK-LENS OPTICS 
 
 (Examples 10, 11, 12 are to show how the flattening of the 
 lens causes the node to approach one face.) 
 
 13. Piano convex lens, r = 16, s = oo, e = 2, /u, = . 
 Therefore light from left. 
 
 Ans. f = 48, /' = oo, AN = 0, A' N' = |, F = 32. 
 
 Notice that in a piano convex the nodes are independent 
 of the finite radius. Ditto, piano concave. 
 
 14. Positive meniscus, r = 10, s = 16, e = 2, ^ = f . 
 Therefore light from left and lens convex towards the left. 
 
 Ans. f = 30, /' = - 48, AN = - 2, A' N' = - 3.2, 
 F = 48. Both nodes outside. 
 
 15. Non-curvature lens, r = 10, s = 10, e = 2, ^ = |. 
 Therefore as in Ex. 14. 
 
 Ans. f = 30, /' = - 30, AN = - 20, A 1 N' = - 20, 
 F = 300. Therefore as in Ex. 14 both nodes outside. 
 
 16. Double convex lens, r = 10, s = 16, e = 2 } 
 /u, = f . Therefore light from left. 
 
 Ans. f = 30, /' = -- 48, A N = fj, A' N' = - i, 
 F = - 2 r 4 9- Nodes inside. 
 
 17. Double convex lens, r = 10, s = 16, e = 2, p = 
 f . Therefore light from left. 
 
 Ans. f = -- 30, /' = -- 48, AN = i, A'tf' = - f, 
 F = 12. Nodes inside. 
 
 18. Piano convex, r = GO, s = 16, e = 2, /x = f. 
 Therefore light from left. 
 
 Ans. f = oo, /' = - 48, AN = f, A'N' = 0, F = 32. 
 Nodes inside, one tangent. 
 
 19. Concentric lens (Ross lens), r = 3, 5 = 1, e = 2, 
 
THICK LENSES 
 
 53 
 
 Ans. f = 9, /' = - 3, AN = 3, A 1 N' = 1, F = - 4J. 
 The nodes coincide at the center. 
 
 20. In Ex. 8, if u = 2.594, whence v = 1.492, show that 
 ( 75) xy = 0.947 2 . 
 
 73. Graphic Check. To de- _Z / r 
 
 tect large errors check the cal- S*/-; 
 
 culation by similar triangles, 
 
 drawn to scale, in which the sides \ /' 
 
 are as shown. 
 
 A large error will be quickly 
 
 detected in this way before it has time to vitiate the fol- 
 lowing calculations. 
 
 74. Construction for Image (Conf. 25). 
 
 Jl 
 
 C'D is parallel to AC ( 63). 
 By sim. triangles 
 
 u k m F , ,. 
 
 " - 1 - T ~ ^= -J (upper dia 
 
54 THICK-LENS OPTICS 
 
 u k m F ., ,. 
 
 * == "T : 7 = F^~v (1 Wer di 
 Whence 
 
 I - - = - 
 
 v u F 
 
 Hence the distances of object and image from the nodal 
 points obey the same law as the' distances from the lens in 
 the case of thin lenses, the focal length being the distance 
 from the nodal point of emergence to the principal focus. 
 
 The nodal planes take the place of the two coincident 
 faces of the thin lens, and the constructions and calcula- 
 tions are carried on as if the thin lens were split and then 
 the two edges of the split separated the distance between 
 the nodal planes. 
 
 75. Exercise. From the diagram show that 
 
 &* - F) (v - F) = F 2 
 
 or, as it is generally written, xy = F 2 , x and y being the 
 distances of the object and image from the focal points. 
 
 76. Use of Formulae. Decide upon the direction of 
 light and give the corresponding signs to/,/' (see 36, 37). 
 Then select the proper formulae corresponding to the direc- 
 tion of the light. (Light from the left makes the / of the 
 positive lens + , a seeming gain in concordance of signs.) 
 
 77. Graphic Tracing of any Ray Path. 
 
 This follows the formula of 32, rl $2 ||, exactly, 
 except that the lens line is split apart the distance between 
 the nodal planes, the points in one nodal plane being 
 
THICK LENSES 
 
 55 
 
 dragged horizontally to the other. The order of the letters 
 indicates the order of construction, x being the line sought. 
 
 78. Since the nodal planes are really plane surfaces, 
 their intersections with the paper will be straight lines, as 
 drawn in the diagrams. Therefore, having the nodals and 
 foci of two lenses given in position, we can find the nodals 
 of the combination by 61, by taking the initial rays 
 horizontal. 
 
 Jf JV F 
 
 Diagram showing the tracing of a ray through two successive lenses, 
 the principal planes of each lens being indicated by letters in horizon- 
 tal lines. 
 
 Example 1. Try this on the combinations of 107, 
 Exs. 4, 8. 
 
 Example 2. See 95, Ex. 2. 
 
 These two sections are an extension of the principles of 
 32, and are equally important in the application to nodal 
 planes. 
 
 ANALYTICAL INVESTIGATION 1 
 
 79. The previous investigation has assumed some facts 
 
 J The remaining sections of this chapter are for those inquisitive 
 readers who desire a somewhat more rigorous logic and less depend- 
 
56 
 
 THICK-LENS OPTICS 
 
 as self-evident. The investigation of this section is for 
 the purpose of putting these facts on a more strictly logical 
 basis, to meet the criticism to which the preceding sections 
 might be open to the casuist. 
 
 80. Before entering upon the discussion, we give some 
 preliminary principles. 
 
 y = dist. above (or below) the x axis, of an arbitrary 
 point on the line 
 
 x = distance of the point, to right or left of the y axis 
 a, b = corresponding distances for some fixed point 
 x and y have many values, one for each point 
 a and b are constant, fixing some definite point 
 
 By sim. triangles 
 
 x a 
 
 d 
 
 m 
 
 or, as it is generally written 
 
 y b = m (x a) 
 This is called the equation 
 of the line referred to the 
 axis, since x and y taken in 
 corresponding values fix any 
 point on the line. Their 
 values could be used to 
 plot points on the line; or 
 corresponding values 
 measured from a point on the line will satisfy the equation. 
 From the equation we can locate the line by assuming a 
 value for x and calculating the corresponding value of y, 
 and then plotting the two values, thus locating a point, 
 and so on. In other words, an equation of a line gives us 
 a clue as to where the line lies. 
 
 8 
 
 
 /^ d 
 
 y 
 
 !> 
 
 
 
 
 
 ' 
 
 
 \ 
 
 >X^ 
 
 
 b 
 
 i 
 
 s/6 
 
 a 
 
 
 i 
 
 x axis 
 
 
 
 
 
 
 
 
 ence upon intuition. They can safely be omitted by those not inter- 
 ested, without destroying the continuity of the text. 
 
THICK LENSES 
 
 57 
 
 The advantage of the equation is that we can operate 
 upon the equation algebraically and then interpret the 
 result geometrically, without going through all the pecul- 
 iarities of a geometrical diagram. 
 
 81. General Equation of a Refracted Ray. 
 
 , (2), and (D represent the ray before, during, and after refraction 
 
 For rays through points near the vertex A, so that the 
 point of incidence is practically over A, 
 
 1. The equation to line is ( 80): 
 
 y b = m (x OA) 
 
 2. For line (D 
 
 y - b = m' (x - OA) or (y - 6') = m' (x - OA') 
 
 3. For line (D y - b' = m" (x - OA') 
 
 4. From equation 2, b - b' = m' (OA' - OA} = m'e 
 By 4 sin r = ^ sin r(D 
 
 [r means the angle between r and , etc. 
 
 c _ sin r 
 r sin c 
 
 But by 6 
 
 Therefore - sin c = sin r = /* sin r 
 
 5. 
 
 d . 
 
 = P> - sm 
 r 
 
 [d si 
 r ~s 
 
 sin r(D 
 
58 
 
 THICK-LENS OPTICS 
 
 Therefore c = b + m (OC - OA) [Eq. 1 taking y = c 
 
 = b + mr 
 
 Similarly d = b + m'r 
 Therefore (b + mr) sin c = p (b + m'r) sin d(j) 
 
 or 6 + mr = p (6 + ra'r) 
 
 Since sin c(D = sin 
 
 prac- 
 
 tically, the two angles being 
 nearly 90 each 
 
 6. Whence pm f = m 6 = m bu 
 
 Making - = u 
 
 7. Similarly pim' = m" 6V 
 
 E = , pi = index of refraction for 2d surface 
 
 s = radius of 2d surface 
 
 XT */ -, . m bu , f eu\ . me rri 
 
 Now b' = 6 H e = 6 ( 1 ) H [Eqs. 4, 6 
 
 8. = gb + hm 
 
 r. 
 
 Lr> X.L- i eu e 7 
 
 Putting 1 = gr , - = h 
 
 , t , , , , (1 eu) . men' 
 
 m" = /*im' + bu' 
 
 m bu 
 
 [Eqs. 8, 7 
 
 9. 
 
 - 
 
 p. 
 
 meu 
 
 10. = kb + Im 
 
 ^ ... 7 , pi 
 Putting k = u -- 
 
 euu 
 
 = ug u - 
 
 Z = 
 
 eu 
 
THICK LENSES 
 
 59 
 
 If 6 = Y-m(X-OA) 
 
 X, Y being co-ordinates of the 
 point on the line which is 
 considered as the source of 
 the ray 
 
 then 
 
 6' = g Y + m (h - g X - OA) 
 
 m 
 
 From these 
 
 Whence 
 
 k Y + m (I - k X - OA) 
 
 m" - k Y 
 m l-k(X-OA) 
 
 h-g(X-OA) 
 
 m" (x - OA' 
 
 11. Or 
 
 h-g(X-OA)\ 
 + l-k(X-OA)) [Eq ' 3 
 
 * m n( x OA , , h-g(X-OA)\ 
 
 y ~l*[l-k(X-OA)}~ \ X ^l-k(X-OA)) 
 
 Since gl hk = ~ 
 
 the equation of the emerged ray in terms of X, Y, the 
 co-ordinates of the source. 
 
 82. If X be taken such that Z - k (X - OA) = ^ 
 
 that is 
 then when 
 x = OA' - 
 
 X = OA + 
 
 = OH, say 
 
 h-g(X -OA) 
 
60 
 
 THICK-LENS OPTICS 
 l-^ 
 
 = OA'-- \h- 
 
 Mi 
 
 I t 
 
 X- 
 
 [Si 
 
 ince gL hK = 
 
 = OA' 
 we will have 
 
 say 
 = F 
 
 and in the planes of these two points (H and H') the object 
 (F) and the image (y) are equally distant from the axis 
 of the lens; the rays are transferred horizontally. (Conf. 
 62, 69.) 
 
 These points are called principal points, and perpendic- 
 ular planes through them principal planes. 
 
 83. If X be taken such that I - k (X - OA) = 1, 
 
 whence 
 then when 
 
 X = OA + 
 
 l-l 
 
 ON, say 
 
 x = OA f - \h-g(X- OA)\ = OA' - (h - g l -jl 
 
 I- 1 
 
 X - OA = 
 
THICK LENSES 61 
 
 and when Y = 0, then also y = 0, and 
 
 m" - k Y 
 ~ I- k(X -OA) 
 
 since kY = and I - k (X - OA) = 1 
 
 84. Since m = m", the rays before refraction and after 
 refraction are parallel (Conf. 63), and the image is not 
 deflected so long as this point N' is not moved. (Conf. 
 59.) 
 
 The points N and N' are called the nodal points. 
 
 85. If /MI = /A (e.g. air on both sides), then H and H r 
 coincide respectively with N and N'. (Conf. 64, 71.) 
 
 86. If m" = 0, i.e. the ray is parallel to the axis after 
 refraction, then from eq. 10 
 
 6 = 7- m 
 k 
 
 and the equation of the incident ray is 
 
 y + -j- = m (x OA) or y = m ( x OA -- ^ J 
 
 87. If we also take y = 0, so as to find where the ray 
 crosses the axis, then 
 
 A*i erf 
 
 x = OA + \ = OA + - - -- ^ - 7 [Eq. 10 
 k , PI euu 
 
 u u --- 
 
 f- M- 
 = OF, say 
 
 88. If m = 0, i.e. the incident ray parallel to the axis, 
 then from eqs. 8, 10 
 
62 THICK-LENS OPTICS 
 
 and the equation of the refracted ray becomes, eq. 3 
 y - Q ^f = m" (x - OA') or y = m" (x - OA' + | 
 
 whence if also y = 0, in order to find where the ray crosses 
 the axis, then the distance to the crossing point is 
 
 /*i _ eu 
 
 x = OA' - \ = OA' --- ^ -- ^ -- T = OF', say. 
 k , /u-i euu' ' J 
 
 u' u -- 
 /* ^ 
 
 F and F' are called the focal points. 
 89. For /AI = /A, the usual case 
 
 OF = OA+ 
 
 . 
 I*, (u u) euu 
 
 OA- fU'~< 
 
 /^ (/ + /'-) 
 
 OF'=OA' ~ 
 
 , . 
 (u u) euu 
 
 OA'+ '- 
 
 ^ 
 
 OF' - ON' = OA' - - (oA f + ^-r-^1 = ON - OF 
 - 1 
 
 k 
 
 -fi ff 
 
 -u -euu 
 
 = /^ 
 
 equivalent focal length of the lens 
 
 (Conf. 72) 
 
THICK LENSES 
 
 63 
 
 90. From eq. 11 
 
 - g(X - OA)\ 
 
 l-k(X-OA) 
 
 = m" (x - t) + 
 
 l-k(X-OA) 
 Y 
 
 I - k(X - OA) 
 
 where , rj are evidently (since x = , y = rj satisfies the 
 equation) on the ray d), and evidently dependent only 
 on X and Y and not on m, 6; that is, every ray through 
 X, Y (the object) passes through , >/ (the image). 
 
 From OA = ON - -~, OA' = ON' - ^-^ 
 
 substituting these values in the expressions for , 17, we get 
 ON -X Y 
 
 whence 
 
 or 
 
 1 +k(ON - X) 
 1 1 
 
 X-ON 
 
 k- 
 
 '' ~Fi 
 
 P Pi F 
 
 " 1 +k(ON - X) 
 = -k 
 
 where p = distance from node N' to the image , and 
 pi = distance from the object X to the node N. This 
 shows that the nodal distances to object and image obey the 
 same laws as the thin lens distances. (Conf. 74.) 
 
CHAPTER IV 
 
 COMBINATIONS OF LENSES 
 
 91. Thin Lenses in Contact. 
 
 For the first lens = T 
 
 Vi U fi 
 
 for the second lens = 7 
 
 v* Vi / 2 
 
 whence, by addition = 7- + 7 
 
 v 2 u /i f z 
 
 1 
 
 ~ F 
 
 /i, /2 = focal lengths of the lenses 
 u = dist. of object from 1st lens 
 Vi = dist. of 1st image from 1st lens and of 2d 
 
 object from 2d lens 
 v 2 = dist. to image formed by 2d lens 
 F = focal length of combination 
 
 Example. A + lens, 2 in. focus, is cemented to 
 a lens, 9 in. focus. What is the equivalent focus? 
 
 Ans. Equivalent focus = 2f . 
 
 92. For a third lens, similarly 
 
 I I I 1 _L_ I 1 
 
 V, W"/1 + /2 /3~F 
 
 The power of the combination is the sum of the powers 
 of the components. (Conf. 42.) 
 
 For powers of lenses not in contact, see 96. 
 
 64 
 
COMBINATIONS OF LENSES 
 
 65 
 
 93. Thin Lenses not in Contact. 
 
 Taking A, a point on the refracted ray and in the front 
 focal plane of the second lens, 1 we can find its image, C, 
 through the second lens by known rays, as shown (or by 
 32) . But any other ray through A , as A B, must go through 
 the same image point, and thus we get the direction BC for 
 this A B ray after it has been refracted by the second 
 
 1*1 
 
 Horizontal 
 
 ens 
 
 Z* 
 
 A 
 
 ens 
 ackfocu$ of 
 Z'lens Back focus of 
 ^x^ combtnatLOn 
 
 ray 
 
 ^\^: __ 
 
 
 ofZ^lens focus tf^~^ 
 
 IUn ^ 
 
 ^^-Im'age ofJlby . 
 ^"^ known rays rwri zon- 
 tal and thru center 
 
 Diagram illustrating general case of two thin lenses, /* not equal 
 to m; hence front and back focal length of second lens will have 
 different values. Introduced for the purpose of getting a general 
 rule of construction for use in the next diagram, indicated by the 
 order of the letters. 
 
 lens. The point D, where it meets the axis, is the back 
 focus of the combination for incident parallel rays. 
 
 EXAMPLES 
 
 1. A candle is held 1 foot in front of a convex lens, 
 giving an image on a screen 4 inches behind it. A con- 
 cave lens is now placed in contact with it, and the screen 
 must be moved 8 inches further away to get the image. 
 What is the focal length of the negative lens? Check by 
 32. Ans. - 6. 
 
 2. A convex lens of 16 cm. focus, in contact with 
 a negative lens, gave a focal length of 48 cm. for the 
 
 1 In order to have its distance from the lens a definite and significant 
 value. 
 
66 THICK-LENS OPTICS 
 
 combination. What is the focal length of the negative 
 lens? Ans. - 24. 
 
 3. A concave lens of 8 cm. is combined with a con- 
 vex lens of + 6. What is the focus of the combination? 
 Ans, 24. 
 
 4. By the method of 32, 33, show that a compound 
 microscope (two positive lenses more than the sum of 
 their focal distances apart) must have the object without 
 the focal distance of the objective. (Conf. 33, Ex. 2.) 
 
 5. When the distance outside the focus in Ex. 4 becomes 
 infinity, the distance apart of the lenses becomes the sum 
 of the focal radii, and we have the celestial telescope. 
 
 6. Show by 32, 33 that a positive lens followed by 
 a negative lens, the distance apart being the difference of 
 the foci or less, will give a virtual image. (Ordinary opera 
 glass.) 
 
 7. If the distance apart in Ex. 6 is greater than the differ- 
 ence, there results a real image. (Telephoto, 113, 29.) 
 
 8. In the Huygens eyepiece (field lens focus = 3/, eye 
 lens focus = /, distance between lenses = 2/), show by 
 32 that rays incident upon the front lens pointed toward 
 a point between the lenses f / from the front lens will 
 emerge from the second lens parallel to the axis; that is, 
 will give a virtual image. 
 
 9. In the Ramsden eyepiece (field lens focus = eye 
 lens focus = /, distance apart of lenses = f /), show by 32 
 that parallel rays incident upon the front lens converge 
 to a focus //4 beyond the back lens. 
 
 10. By 32, 25, etc., show that the second lens of 
 Ex. 6 reinverts the image made by the + lens, thus giving 
 a final erect (virtual) image. 
 
 11. Similarly show that in Ex. 7 the real image is kept 
 inverted. 
 
COMBINATIONS OF LENSES 
 
 94. Back Focal Distance for Two Thin Lenses. 
 
 For construction of diagram, see 93. 
 
 - e d _d 
 
 e ~ d 
 
 67 
 
 By similar triangles 
 
 /i + / 2 - e 
 
 Back focus of 
 combination 
 
 g ~ f 2 
 
 
 Diagram showing the construction for back focal 
 distance when Hi = /*. The letters A, B, C designate 
 the same points as in the preceding diagram. 
 
 dis- 
 
 Therefore 33 = back focal distance of combination 
 tance from lens to focal point 
 
 / (/i ~ e) 
 
 /1+/2-6 
 
 95. Equivalent Focus for Two Thin Lenses. The two 
 
 l a lens 
 
 ^Mdal point of the 
 combination 
 
 Backjbcus of combination 
 
 Backfocusof2 A lens 
 Focus ofi*len& 
 
 - 
 
 thin lenses act like the two surfaces of a thick lens, and, 
 like them, have their corresponding nodal points deter- 
 mined in the same way (see 61), remembering that the 
 
68 THICK-LENS OPTICS 
 
 surfaces are now vertical plane surfaces. Add the con- 
 struction of 61 to the diagram of 94. 
 
 For construction of diagram, Conf. 93, 94, 61. 
 
 By similar triangles (lightly shaded in the diagram for 
 the second set) 
 
 d N' 
 
 /!+/- e ' f, 
 
 Therefore N' = - f ~. = nodal distance from corre- 
 
 Ji + J2 e spending lens 
 
 Therefore F = 33 + N' = F-vV-^- + 
 
 /!/ 
 
 -6 ' /!+/,- 
 
 = equivalent focal length 
 Ji -r/2 o f ^g combination 
 
 = distance from nodal point to focal point. 
 If we take into account the direction, we get 
 
 . . 
 
 and similarly 
 
 N - fl 
 
 1M - - j. . f 
 
 /i + /i 
 
 This section, with 61, enables us to find the nodal 
 points of a combination of lenses. 
 
 Ex. 1. In the Huygenian eyepiece (field lens focus 
 = 3/, eye lens focus = 2/, c = 2/), show that the front 
 focal distance (found by parallel rays through the eye 
 lens, from right) is - }/. (Conf. 93, Ex. 8.) 
 
 2. In the* Ramsden eyepiece (/i = /2, e = f/i), show 
 that the back focal length is J f\. That is, the combination 
 has the properties of an ordinary convex lens of //4 focus. 
 (It has the advantage, however, of being approximately 
 achromatic.) (Conf. 93, Ex. 9.) 
 
COMBINATIONS OF LENSES 69 
 
 3. By the method of 61, show that if two thin positive 
 lenses lie with crossed foci, and each lens within the focus 
 of the other, the resulting foci of the combination will lie 
 outside the lenses, and both nodal points between the 
 lenses. 
 
 4. In the preceding example, if each lens is without the 
 focus of the other, then both foci of the combination are 
 between the lenses, and both nodals are outside, and the 
 nodals are crossed. 
 
 5. By 32, 25, etc., show that a microscope composed 
 of a J in. objective and a 1 in. eyepiece, 6 inches apart, 
 for a person of 8 in. vision must have the object |J in 
 front of the objective. 
 
 EXAMPLES 
 
 1. Two positive lenses with a common focal length of 
 0.05 m. are 0.05 m. apart. What image results of a disc 
 0.01 m. in diameter placed 0.1 m. distant? 
 
 Ans. A real image 0.025 m. beyond the second lens. 
 Diameter of image = 0.005 m. 
 
 Note the crossing of the nodes. Draw a diagram and 
 compute the magnification by similar triangles. Conf. 
 diagram of 74, mutatis mutandis. 
 
 2. (Microscope ocular) Field lens, f\ = 2J, eye lens, 
 / 2 = If, c = 2i. Show graphically (see 95, 33, 77) that 
 the nodes cross, N' to just behind (right of) the field lens, 
 N to about an inch behind (right of) the eye lens (light 
 from left); that the posterior focus almost coincides with 
 the focus of the field lens ; the anterior focus not so closely 
 with the anterior focus of the eye lens. Calculate these 
 results by the formulae of 95. 
 
 96. Powers of Thin Lenses not in Contact. 
 From 95 
 
70 
 
 THICK-LENS OPTICS 
 
 Pi + P2 - 
 
 = power of the combination 
 
 [Pi, P2 = powers of the thin lenses 
 c = distance apart 
 
 Example 1. pi = 3 D, p 2 = 5 D, c = .025 m. 
 
 .'. Equiv. power = 3 + 5 - 15 X 0.025 = 
 
 7.625 diopters 
 Example 2. pi = p 2 = + 12 D, e = .02 m. 
 
 .'. Equiv. power = 12 + 12 - (12 X 12 X .02) = 
 
 21.12 diopters 
 97. Back Focal Distance for Light from the Right. 
 
 Diagram constructed exactly as in 94, mutatis mutandis. 
 
 f i + e ^ d & 
 
 By sim. triangles ^ , 
 
 Jl I /2T" 
 
 Whence i 
 
 e d + g / 2 
 / 2 (/i + e) 
 
 98. Equivalent Focus for Light from the Right. 
 
 /1/2 
 
 Similarly to 95 
 
 rr __ 
 
COMBINATIONS OF LENSES 
 
 71 
 
 Note the change of sign of e in the formulae when the 
 light comes from the right, due to the fact that e is nor- 
 mally positive. 
 
 99. Back Focal Distance for Two Thick Lenses. 
 
 2 d lens 
 
 Tocus of combination 
 
 Focus of2 d lens 
 
 Focus of 
 J*len* 
 
 Diagram showing construction for back focus of thick lens, on 
 the same principle as the preceding diagrams; but note the horizontal 
 transference between the nodal planes in the second lens. The nodal 
 planes are designated by N,N f . For construction see 94. 
 
 By similar triangles 
 
 d d' 
 
 whence 33 = back focal distance 
 
 Ji 
 
 (with light from the left) 
 
 = distance from posterior nodal point of lens to 
 
 focal point 
 100. Equivalent Focal Length for Two Thick Lenses. 
 
 Similarly, as in 95 
 
 /1/2 
 
 F = 
 
 (with light from the left) 
 
 .+/-. 
 
 = distance from posterior nodal point of the 
 
 system to the focal point 
 
 remembering that F is measured from the nodal plane of 
 emergence for the combination, just as / 2 and /i were 
 
72 THICK-LENS OPTICS 
 
 measured from the nodal planes of emergence of the cor- 
 responding lenses. 
 
 Hence F = , / ! / 2 . - (with light from the right) 
 
 Jl T~ J2 ~T e 
 
 101. Nodal Distances for Thick Lenses. 
 
 As in 95, N'W = distance between N f of second lens 
 and nodal point of emission of 
 the combination, taking account 
 of direction 
 
 = distance from N of first lens to 
 nodal point of incidence of the 
 combination 
 
 /!+/.- 
 
 102. Graphic Check. Check for large errors, by sim- 
 ilar triangles as in 73, 68, except that there is no /* in 
 the formula. 
 
 103. Equivalent Thickness of Thick-Lens Combination. 
 yiW distance between the nodal points of the com- 
 bination 
 
 = e + NN' (1st lens) + N N' (2d lens) - -$ - $ 
 
 a a 
 
 [d=fi+f*- 
 
 = NN' (1st lens) + N N'. (2dlens) - ~ 
 
 a 
 
 = NN' (1st lens) + N N' (2d lens) - -^ 
 
 /1/2 
 
COMBINATIONS OF LENSES 73 
 
 F = focal length of Combination 
 /i,/2 = focal lengths of components 
 
 = distance between the N r of the one 
 component and the N of the 2d 
 component 
 
 This value should be used as a check in the computation 
 to compare with the value derived from the diagram by 
 introducing the various values, A N, etc. It must be used, 
 of course, in connection with the antecedent check of 68. 
 
 Example l.fi = 4, / 2 = 3, N N' (1st lens) = .15, 2dlens 
 = .2, c = 1.5. 
 
 Ans. F = 2.18, WW = - 0.059. 
 
 104. Power of Thick-Lens Combination. 
 
 From 100 P=-^ = T~^T'~rT = P^^~P^~ PW* 
 f Ji J2 /i/a 
 
 If both lenses are +, increase of c increases the equiva- 
 lent focal length and reduces the equivalent power. 
 
 If one lens is , so that the term p\p& becomes 
 positive, increase of e will shorten the equivalent focal 
 length and increase the power. 
 
 The value of WW (101) shows that the equivalent 
 thickness of a combination of two + lenses is reduced by 
 separating them, and may become zero or negative if e is 
 large enough; i.e. the two nodal planes will cross each 
 other, as is the case in many camera lenses (see Ex. 7, 
 106), microscope oculars, etc. 
 
 RESUME 
 
 105. Notice that the thick lens acts like a thin lens with 
 its surfaces (plane and perpendicular to the axis, because 
 we are considering only points near the axis) split apart 
 and separated the distance between the nodal (see 64) 
 
74 THICK-LENS OPTICS 
 
 points, so that if we should make the construction for a 
 thin lens and then split the diagram along the lens line 
 and pull it apart the distance between the nodes, we would 
 have the appropriate diagram for the thick lens. We call 
 this the equivalent thin-split. The nodal distance can, of 
 course, only be found by means of the equations of 67, 
 after e has been decided upon. 
 
 A combination of two lenses again acts like two thin 
 lenses (with separated faces, the nodal distance) removed 
 from each other the distance (c, see 95) between the 
 posterior nodal point of the first lens and the anterior 
 nodal point of the second lens (see diagram of 99), this 
 being the distance between the inner faces of the thin 
 lenses before the faces were split apart to act as a thin- 
 split. This distance, e, can be assumed as we please, and 
 then from the equations of 101 we can calculate the nodal 
 points which act as the front and rear face of the new 
 equivalent thin-split lens. And so on. 
 
 That is, we determine the equivalent thin-split lens of 
 the individual lenses, and then the equivalent thin-split 
 lens of the new combination, and so on, the final thin-split 
 lens being the equivalent of the combination. 
 
 Notice that in the equations of 94-100, /* has dis- 
 appeared because the medium between the two thin-split 
 lenses is air (for which /* = 1) or its equivalent, even if the 
 lenses be in contact. 
 
 106. Use of Formulae for Combined Lenses. 
 
 (a) Find /, /', F, A N, A' N' for each lens. (Conf. 76.) 
 
 f, 15 
 
 , 16, 72 
 A N, A' N', 67 
 
COMBINATIONS OF LENSES 75 
 
 /, /' = focal radii for surface refraction. 
 
 e = thickness of component lens. 
 
 F = focal length of the lens (or later of the combination, 
 or thin-split lens). ( 16, 72, 100.) 
 
 A, A' = anterior and posterior vertex of lens (or com- 
 bination of lenses). 
 
 'AN, A' N' = nodal distances from anterior and pos- 
 terior vertex of lens (or later of the combination). ( 67, 
 101.) 
 
 (ft) Select /i, /2 for the combination, then find (decide 
 upon) e, and then compute F, W, W for the combination. 
 ( 100, 101.) 
 
 9fl } <$l f = nodal distances from anterior (of the first 
 lens) and posterior (of the second lens) nodal points of 
 the components to the corresponding nodal points of the 
 combination. 
 
 /i, /2 = (used in the formula for the combination) the 
 F's of the components. 
 
 e = distance between the N' of the anterior component 
 and the N of the posterior component. must be taken 
 negative when the nodals cross. To calculate locate 
 the nodals roughly on a diagram, with the distances noted 
 thereon, and then derive the value of e. 
 
 (y) Then find A N -+- 9? = distance of anterior nodal 
 point of combination from the anterior vertex, A' N' + 9?' 
 = distance of posterior nodal point of combination from 
 the posterior vertex of the combination. A N + %l of 
 component = A N of the (next) combination, A' N' + W 
 of the component = A' N' of the combination. (See Exs, 
 7,8.) 
 
 Do not fail to use the checks of 14, 68, 73, 102, 101, 107. 
 
 Repeat this for the combination of this combination with 
 some other, and so on. 
 
76 THICK-LENS OPTICS 
 
 Great care must be taken in regard to + and signs. 
 Here most of the errors occur. 
 
 107. Graphic Construction (only available when there is 
 not too great disparity between the numerical values used) 
 after the foci and nodals of the component lenses have 
 been located (based upon the calculations of 16 for thin 
 lenses, and 67, 72 for thick lenses, because owing to 
 the disparity of the values used the intersections in the 
 graphic work are too acute to be of service). 
 
 Graphic construction is extremely valuable to check up 
 which side of the nodals (which limit e) the new nodals of 
 the combination lie. Does not need to be to scale so long 
 as the relation of greater and less is preserved. 
 
 IN THE CASE OF THIN-LENS COMPONENTS 
 
 Find the equivalent focus of the combination by starting 
 with a horizontal ray, its first refraction being found by 
 25-30. Where the refracted ray strikes the second lens, 
 find its new course by 32. The intersection of this new 
 course with the axis will be the focus of the combination. 
 
 Find the nodals (principal points) by the principle of 
 61 (illustrated in the diagram of 95). 
 
 IN THE CASE OF THICK-LENS COMPONENTS 
 
 Find the equivalent focus of the combination by starting 
 with a horizontal ray, its first refraction being found by 
 74. Where the refracted ray strikes the nodal line (inci- 
 dent, look out for crossed nodals) of the second lens, find its 
 new course by 77. The intersection of this new course 
 with the axis will be the focus of the combination. 
 
 Find the nodals of the combination by the principle of 
 61, prolonging the last ray (second refraction) back to 
 its intersection with the original horizontal ray. 
 
COMBINATIONS OF LENSES 77 
 
 EXAMPLES 
 
 1. Two thin positive lenses, 2 inches apart, of focal 
 lengths 6 and 9. 
 
 2 ' 6 12 
 
 A'N' + W = + 
 F = 
 
 6 + 9-2 13 
 
 -2-9 18 
 
 6 + 9 - 2 ~ 13 
 6-9 56 
 
 6 + 9-2 13 
 The nodals are between the lenses. 
 
 2. Thin lenses, 5 inches apart, a positive of 6 focal 
 length and a negative of 2. 
 
 A *r . c 5 6 + 30 
 
 AN + K -0 + 6 _ 2 _ 5= -y 
 
 A'#' + ft' = o + 7 5 i~ 2 ^ = - 10 
 
 F- 
 
 Notice that the nodals are outside the combination and 
 far in front, and that the back focal distance is only 2. 
 Distance between nodals = 25. 
 
 3. Same as in Ex. 2, but the combination reversed. 
 
 A N + ft = 10 A'N 1 + ft' = 30 
 Back focal distance = 42 F = 12 
 Distance between nodals = 25 
 
 4. Two thick lenses. 
 
 First lens, r = 6, s = 4, e = 3, /* = 1.5 
 / = 18, /' = 12, A N = J, A'N' = - | = - .889 
 
 F = -V 5 = 5.33, NN' = I 
 Second lens, r = 4, s = 2, e = 1, ft = f 
 
78 THICK-LENS OPTICS 
 
 /= - , /' = , AN = n = 1-05, A'N' = = .525 
 F = V T - = 5.614, NN' = T 9 s 
 
 Combination. Assume e = 2, thus making the lenses 0.062 
 apart. 
 /i = 5.33, / 2 = 5.614 
 
 _5.33_X J 5.614_ 29.939 
 5.33 + 5.614 - 2 " 8.947 
 
 A N + ft = 1.33 + 233 = 1.33 + 1.19 = 2.52 
 (located in the first lens) 
 
 A'N' + W = .525 + - " 8 47' = 525 ~ L25 = ~ - 
 (in the second lens) 
 9W = 0.804 
 
 5. Two thin lenses, 4 inches apart, a positive of 6 focal 
 length, and a negative of 6 focal length, the positive 
 lens in front. 
 
 AN = - 6, A'N' = - 6, F = 9 
 
 The nodals are outside the lenses, 6 and 2 respectively, 
 from the front face of the positive lens. yiW = 4. 
 
 6. Lens of Ex. 8, 72, 0.25 in front of lens of Ex. 9. 
 Whence 
 
 /i = 0.947, / 2 = - 1.875, e = 0.21 + 0.25 + = 0.46 
 
 -.1584- 
 
 .947 - 1.875 - .46 
 .158 - .314 = - 0.156 
 (in front of first lens) 
 
COMBINATIONS OF LENSES 79 
 
 A'N' + 31' = - 0.0625 + ~ M X . ( ~ fi L875) - 
 
 - I.ooo 
 
 - 0.0625 - 0.621 = - 0.6835 
 (in first lens) 
 
 7. Three lenses, in contact. Light from left. 
 First lens, r = - 4.29, s = - 1.20, /* = 1.5146, e = .230 
 whence/ = -- 12.627, /' = 3.532,^ = 3.157, A N = .2056, 
 
 A'N' = 0.0573. 
 Second lens, r = - 1.20, s = - 3.75, /* = 1.574, 
 
 e = 0.050 
 Whence/ = - 3.2906, /' = 10.2831, F = - 3.094, 
 
 AN = - 0.015, A'N' = - 0.047. 
 
 Third lens, r = - 3.75, s = - 1.8, /* = 1.517, e = 0.151 
 Whence / = -- 11.0033, /' = 5.2814, F = 6.528, 
 
 AN = 0.186, A'N' = 0.089 
 
 Combination of 1st and 2d lens, /i = 3.157, / 2 = 3.094, 
 
 e = - 0.0723 
 
 3.157 (- 3.094) 
 
 Whence F = Q = - = -71.76 
 
 0.2056 +- 
 
 0.2056 - 1.68 = - 1.4744 
 A'#' + ft' = - 0.047 - 1.65 = - 1.697 
 
 Notice that e is negative because the N' of the first lens 
 and the N of the second lens are crossed. 
 First combination, combined with 3d lens. 
 
 /i = - 71.76, / 2 = 6.528, e = 1.697 + 0.186 = 1.883 
 
 - 71.76 X 6.528 
 
 Whence F = 
 
 - 71.76 + 6.528 - 1.883 
 
80 
 
 THICK-LENS OPTICS 
 
 
 - 468.4492 _ 
 - 67.115 
 
 1.883 (- 71.76) 
 
 - 67.115 
 - 1.474 + 2.013 = 0.539 
 
 A'N' + W = 0.089 + 
 
 0.089 + 0.186 = 0.275 
 8. Three doublets combined, see diagram. 
 
 Line of vertices 
 
 ...i-\ .......;.-..._ ...-^.j ^4 Line of lens nodals 
 
 jvn \' Line of doublet nodals 
 
 k- ...... 2.00--- -* 
 
 . . 
 
 ....... -2.77---'- ------- * 
 
 3.96-- 
 
 -/.04-M 
 
 Nodals of first combi- 
 nation 
 
 Nodals of second com- 
 bination 
 
 1 8 
 
 First lens of 1st doublet. r = GO, s = 1, e = -,//, = - 
 
 o 
 
 Whence 
 
 i. oo 
 
 - 1 
 
 00 
 
 L _8_l 8 1 + - 16 
 
 3 2 
 
 A'N' = 
 
 Q , ' '_f 1 8 \ 
 8 ( oo + / - e) 
 
 = 
 
 -8 
 
 8'oo-|-i 8*1-0-0 
 
COMBINATIONS OF LENSES 81 
 
 3 
 
 Second lens of 1st doublet, r = I, s = 1, e = 1, /A = - 
 
 2 
 
 Whence / = 2 - 1 =3 
 
 2-3-3 
 r = u- 
 
 A'N' = 
 
 3 (3 + 3 - 1) 5 
 
 2-1-3 2 
 
 3 (3 + 3 - 1) ~~ 5 
 
 -1-3-2 _ 2 
 3-5 5 
 
 562 
 VsJ doublet. f\ = -, / 2 = r-, c = - 
 o 5 o 
 
 = 2 - 3077 
 
 ^' + 9J '=- + r-= - + = 0.15385 
 
 2 8 
 First lens of 2d doublet, r = oos = 4, e = ->, = - 
 
 3 5 
 
 Q 
 
 Whence / = - oo = oo 
 5 
 
 , 84 32 
 
 ; ~5*3' ~T 
 
 5 . <x> (- 3f) - 5 32 = - 20 
 
 8 (oo -3^2 -|) 3. 8 3 
 
 A N = - - QQ = 2 5 _5_ 
 
 5 ' 8 ' "oo - V - ~ 3 ' 8 ~ 12 
 
82 THICK-LENS OPTICS 
 
 15 - 32 
 
 A ' N' 
 
 oo 
 
 4 3 
 
 Second lens of 2d doublet, r = 4, s = 4, e = ~> /u. = - 
 
 o .Z 
 
 Whence / = | - 1 2 = 12 
 z i 
 
 2 12 -12 72 
 " 3 ' 12 + 12 - I " 17 
 
 42 12-3 J*_ 
 
 3 ' 3 ' 68 = 17 
 
 A'N' - ~ 4 2 12 3 8 
 
 ~3~ ' 3 ' ' 68 " " T7 
 
 20-72 8 
 
 Second doublet, fi = - -^ > / 2 = jy > == 
 
 - 90 79 1 
 
 Whence F = -f - g ^ ^ -g = 9.73 
 
 3 "17 17 
 
 A ~ 20 
 
 A AT . c 5 . 17' 3 5 40 
 
 A AT + 91 == 12+ - 20 . 72 "8" = 12 + 37 = L498 
 3 h 17 17 
 
 A' N' + ft' = ^ + ^y^l = - .4705 + . 6868 = 0.2163 
 
 3 8 
 
 First lens of 3d doublet, r = <x>,s = 10, e = > P = 
 
 - 8 10 - 80 
 
 Whence / = oo,/ r - - - 5 = - 
 
COMBINATIONS OF LENSES 83 
 
 F 5 
 
 
 - 80 
 
 1 -5-10 
 
 8 
 
 
 3 
 
 c 
 
 80 3 3 
 
 
 
 50 
 
 3 4 
 
 
 
 3 
 
 
 AN 3 
 
 5 
 
 8' 
 
 00 
 
 15 
 
 j i\ . 
 4 
 
 00 . 
 
 32 
 
 4 8 3 oo 
 
 3 3 
 
 Second lens of 3d doublet, r = 10, s = 10, e = ~ , /* =Q 
 
 O If) 
 
 Whence / = . ~ . 2 = 30 
 z i 
 
 -3 -10 
 
 f "2 r 
 
 2 30-30 400 
 
 ~ 3 '30 + 30 -f == 39 
 
 3 2 30 _ 20 
 = 2 ' 3 ' 30 + 30 - f ~ 39 
 
 - 3 2 30 2 - 20 
 
 A IV 
 
 Tfa'rd doublet. 
 Whence F 
 
 A \T 4- 9f> 
 
 2 
 
 / - - 
 
 3 117 
 - 50 
 
 39 
 400 
 
 20 
 
 .400 
 81 
 
 4- 
 
 Ji 3 ' /2 
 - 50 400 
 
 39' 
 
 1 
 
 39 
 5 
 
 3 
 
 24 
 
 39 - 
 
 .6923 
 - 20 
 
 50 
 3 
 
 400 
 
 20 
 39 
 
 = 4R87S - 
 
 39 
 50 
 
 39 
 
 3 
 
 _ 
 
 "3 39 39 
 1.23456 = 1.70331 
 
84 THICK-LENS OPTICS 
 
 - 30 
 A'N' + 9fe' - -^ + 0.75975 = - 0.51282 + 
 
 oU 
 
 0.75975 = 0.24693 
 
 Combination of second and third doublet, so that the posterior 
 nodal point of the second doublet coincides with the 
 anterior face of the third doublet. 
 This makes * = 1.70331, /i = 9.73, / 2 = 24.6923 
 
 9.73 X 24.6923 240.253 
 
 9.73 + 24.6923 - 1.70331 32.7190 
 7.343 
 
 Whence F = 
 
 A N + ft = 1.498 + 32 7 ' = L498 + 
 
 1.277 = 2.775 
 '* + V = 0.24693 + " 
 
 0.247 - 1.285 = -- 1.038 
 
 Combination of first doublet with preceding combination, so 
 taken that the posterior nodal plane of the doublet coin- 
 cides with the anterior face of the second doublet. 
 This makes c = 2.775, /i = 2.3077, / 2 = 7.343 
 
 2.3077 X 7.343 16.948 
 
 2.3077 + 7.343 - 2.775 
 
 AN + N = 1.08173 + 2.3077 X + 2 ' 3077 = L082 
 = 1.082 + .918 = 2.000 
 
 A'N' + W = - 1.038 + ' = - 1.038 - 
 
 = - 1.038 - 2.921 = - 3.959 
 
COMBINATIONS OF LENSES 85 
 
 9. Camera lens, composed of three lenses, light from left, 
 First lens. ^ = 1.6103, r = 1.264, s = 1.48, e = 0.105, 
 air space = 0.232. 
 
 Second lens. /* = 1.61(|3, r = - 2.09, s = - 0.553, 
 e = 0.358, air space = 0.0053. 
 
 Third lens, n = 1.524, r = - 0.5325, s = - 2.8, 
 e = 0.110. 
 
 , . 1.6103 X 1.264 OOC1 
 
 /ens. /i = -- o~6103 -- = 3 ' 3351 
 
 - 1.6103 X 1.48 
 0.6103 
 
 _ - 3.3351 X 3.9050 
 " 1.6103 X (- 0.6749) " 
 
 3.3351 X 0.105 
 "1.6103 (-0.6749) ' 
 
 _ - (- 3.9050) X 1.05 _ 
 1.6103 (- 0.6749) 
 
 Second lens. 
 
 1.6103 X (- 2.09) 
 /1= 0.6103 
 
 
 -1.6103 (-0.553) 
 0.6103 
 
 - 5.5145 X 1.4591 
 1.6103 (-4.4134) 
 
 T - 5.5145 X 0.358 
 
 = ' 2778 
 
 1.6103 (- 4.4134) 
 
 _ - 1.4591 X 0.358 _ 
 ~ 1.6103 (-4.4134) ~ 
 Third lens. 
 
 = 1.524 (- 0.5325) _ _ 
 " 0.524 1 ' 
 
86 THICK-LENS OPTICS 
 
 - 1.524 (- 2.8) 
 /2= 0.524 
 
 - 1.5487 X 8.1435 
 1.524 X 6.4848 
 
 . ,. - 1.5487 X 0.110 nm7Q 
 
 AJV= 1.524 X 6.4848 ' 173 
 
 - 8.1435 X 0.110 _ _ 009064 
 1.524 X 6.4848 
 
 First combination, of 2d and 3d lens. 
 
 /! = 1.1322, / 2 = - 1.2762, c = - 0.0854 
 AN = 0.2778, A'N' = - 0.0906 
 
 Therefore A N + % - 0.2778 + 
 
 0.2778 + 1.6499 = 1.9277 
 
 A'N' + V = - 0.0906 + ~ <~ **W<- a 854 > = 
 
 u.uooo 
 
 - 0.0906 + 1.8598 = 1.7692 
 1.1322 (- 1.2762) 
 
 - 0.0586 
 Second combination, of 1st lens and 1st combination. 
 
 /i = 11.9837, / 2 = 24.6561, = 2.5369 
 AN = - 0.3222, A'N' = 1.7692 
 
 Therefore A N + * - - 0.3222 + 1L98 ! 
 
 - 0.3222 + 0.8915 = 0.5693 
 A'N' + ' = 1.7692 + - 
 
 1.7692 - 1.8345 = - 0.0653 
 11.9837 X 24.6561 
 
 34.1029 
 
 = 8.664 
 
COMBINATIONS OF LENSES 
 
 87 
 
 Look out for negative e, 
 To calculate e, locate the 
 
 Excellent for care in signs, 
 caused by the crossed jiodals. 
 nodals on a rough diagram. 
 
 108. Magnifying Power of a Microscope (Compound). 
 
 Magnifying power = - - = 1st magnif. X 2d magnif. 
 o % 
 
 (55) 
 
 / = focal length of the eyepiece 
 D = least distance of distinct vision 
 v = dist. from objective to image 
 u = dist. from objective to object 
 
 = size of object 
 
 1 = size of real image 
 
 7 = size of virtual image 
 
 109. N. B. In the microscope the magnifying power is 
 the ratio between the virtual image and the object, because 
 both are seen at the same distance, the distance of distinct 
 vision. In the telescope, however (see next section), the 
 virtual image and object are not seen at the same distance 
 and the comparison must be made on a different basis; 
 viz., comparison of the angles under which the virtual 
 image and the object are seen. The distance at which 
 the virtual image is to be considered depends upon the 
 " set " of the eye of the observer. A person with a very 
 flexible eye can vary the distance from far to near, which 
 
88 THICK-LENS OPTICS 
 
 produces a .slight variation in the angle. A far-sighted 
 or presbyopic eye has the eye " set " for the far distance, 
 and therefore for the slightly smaller angle subtended by 
 the virtual image, which, however, is practically the same 
 angle. Ratio of angles could have been used in the micro- 
 scope also. 
 
 110. Magnifying Power of a Telescope. 
 
 Angle under which object would be seen by the naked 
 eye = a, practically. 
 
 Angle under which object would be seen by the tele- 
 scope = /3. 
 
 O Jj 
 
 Therefore, magnification = - = approximately, since 
 the angles are small. 
 
 BU.836. - = o- 1-B-' 
 
 F F D +f F 
 
 and - = y D = mag. = y nearly 
 
 For the eye looking at a landscape, - is approximately 
 
 F 
 
 1, and the magnification = j 
 
 = distance of distinct landscape vision 
 
 = focal length of the eyepiece 
 
 = focal length of the object of glass 
 
 Example. Object glass of telescope is 20ft. focal length. 
 With a \ inch eyepiece, what is the magnification? 
 Ans. Mag. = 480. 
 
COMBINATIONS OF LENSES 
 
 89 
 
 111. Magnifying Power of Opera Glass (Galilean tele- 
 scope). 
 
 a = angle of object at eye of observer, practically 
 ft = angle of image 
 D = least distance of distinct 
 
 vision 
 
 F = focal length of object glass 
 / = focal length of eyepiece 
 
 Magnification = - (practically exact for distant objects) 
 
 D f F 
 
 -- nearly 
 
 - 
 
 ] 
 
 Example. If the object glass is 4 in. focus, and the eye- 
 piece 1| in., what will be the magnifying power and the 
 distance between the lenses? 
 
 Ans. 8/3; 2.5. 
 
 112. Magnifying Power of Camera. (See 47, 53.) 
 
 For telephoto camera, see 120. 
 
CHAPTER V 
 
 TELEPHOTO LENS 
 
 113. In 29 we found that a negative lens interposed 
 in the path of converging rays so that the aerial object 
 fell within the focal distance of the negative lens gave a 
 real image beyond the aerial object. 
 
 Jl 
 
 t 
 
 v%^ JIUVK fwul distance 
 
 Equivalent focal 
 
 This is the principle of the telephoto lens, the aerial 
 object being the real image made by the camera lens. 
 As shown in the diagram, the result of interposing the 
 negative lens is to give an image as if made by a long 
 focus lens in the position A. But a long focus lens gives 
 a large image, and usually requires a long camera box; 
 i.e. the long back focal distance. By reference to the 
 diagram it will be seen, however, that in the case of the 
 telephoto lens, the back focal distance is very much less 
 than the focal distance. Hence instead of a very long 
 focus lens with its correspondingly long box, we have the 
 combination of two lenses, a -f and a , and the same 
 effect with a very much shorter box. See 107, Ex. 2, 
 
 93, Ex. 7. 
 
 90 
 
TELEPHOTO LENS 91 
 
 114. Focal Length of a Telephoto Combination. 
 
 By 94, 95, focal length = F = , ~ ^ ' f 2 
 
 Jl 12 
 
 back focal distance = 2 ^ = FB 
 
 Ji - 12 - 
 
 /i = focal length of + lens 
 / 2 = of 2d lens 
 
 e = distance between the nodes, emergence of 1st lens, 
 incidence of 2d lens 
 
 115. Telephoto Magnification. 
 
 90? = increase of magnification due to combination as 
 compared with the + lens alone 
 size of image made by combination 
 size of image made by converging lens 
 focal length of combination 
 focal length of converging lens 
 
 I /'/ /if 2 , -fa 
 
 " /I (/I + /* - )/! (/I - *2 - )/! / - fj ~ 
 
 i i = ) i 
 
 , 
 
 2 (/I f 2 ) f 
 
 back focal distance 
 
 _ 
 
 num. val. of focal length of neg. lens 
 Therefore F B = back focal distance = f 2 (2ft 1) 
 F = equivalent focus of combination 
 
 2 
 
 Notice that FB for a given 2ft is affected only by the 
 negative lens used. 
 
 EXAMPLES 
 
 1. For m = 3,/ 2 = - 3, we get F B = 3 (3 - 1) = 6. 
 
 2. Rays forming a real image are intercepted by a con- 
 
92 THICK-LENS OPTICS 
 
 cave lens of 12 in. focal length at a distance 8 in. from the 
 screen. How far must the screen be moved to be in focus 
 again? 
 
 Ans - Tt = I ~ ~ /. = 24, /. 24 - 8 = 16 = distance 
 LA o V 
 
 to be moved. 
 
 3. For /i = 6,/ 2 = -3, 
 
 1. If F B = 12 then 2ft = 5 F = 30. 
 
 2. If <m = 3} then F B = 7J F = 21. 
 
 3. If F B = 7} then F = 21. 
 
 4. Dallmeyer's Telephotographic lens, /i = 6, / 2 = 3. 
 
 Since F must be +, e > 3, say 3|. 
 
 Hence AW = - 30, A'W = - 15, F = 24, F B = 9. 
 
 5. In Ex. 5, suppose FB = 12, what should e be, and 
 what will be the value of F? 
 
 Ans. e = 3f , F = 30. 
 
 6. /i = 9J, / 2 = -- 13. 
 
 Ans. AW = - 5,V, AW = - 6H, F = 16&, F B = 9A- 
 
 7. /i = -- 13, / 2 = 9|. 
 Ans. 
 
 Notice that this is the lens of Ex. 6 turned around, and 
 observe the large increase of FB when the negative lens is 
 in front. 
 
 116. Focal Distances. 
 
 fA fA l/i 2 fs 2 
 
 r = -; ^ = r- = 7 = m T 1 
 
 /i /2 
 
TELEPHOTO LENS 
 
 FNi~ 
 
 4/*_^ 
 d d 
 
 I Ffi means the distance 
 |_ between F and /i, etc. 
 
 _= 
 
 d ~ d 
 
 = m p = distance of front focal point of the combina- 
 tion from the front focal point of the 
 positive lens 
 
 ( A Ni, because ANi is essentially negative, and FNi essen- 
 tially positive, and the sign is needed in order to give 
 them the same combinative sense) 
 
 Similarly ^j- = = distance of back focal point of the 
 combination from the back focal 
 point of the negative lens 
 Hence Distance between front focal point of the system 
 
 and the front lens (nodal point) = mF + /i 
 Distance between back focal point of the system 
 
 F F 
 
 and the negative lens = h /2 = $2 
 
 m m 
 
 117. For a single lens, 47 " 
 
 v ==+/= dist. to image 
 
THICK-LENS OPTICS 
 
 u = - (Nf + /) = dist. to object 
 
 
 
 -^r = magnification = - 1 
 
 N = reduction factor 
 Similarly for the telephoto combination 
 
 / = dist. of image from neg. lens (nodal point) 
 F , F 
 
 = N + m- f * 116 
 
 = dist. of object from pos. lens (nodal point) 
 
 -(NF + mF+fi) 116 
 
 N = reduction factor for the system 
 F = focal length of the system 
 
 = focal length of + lens 
 / 2 = focal length of lens 
 
 -f,--/, 
 
 E = bellows extension 
 
 Example, fi = 6, / 2 = - 3, e = 3J. The results are 
 given in the diagram. 
 
 118. Focal Radius in Terms of the Magnification, 
 
 N 
 
 [47 
 
 Therefore 
 
 -GH 
 
TELEPHOTO LENS 95 
 
 But v - F = E - F B ( See diagram) 
 
 therefore F B = E - - -^ 
 
 But 
 
 F = m F B + /i 
 
 [115 
 
 whence F = 
 
 mE+fi 
 
 J = bellows extension 
 
 /i = focal length of + lens 
 
 N = reduction factor 
 
 If N = oo (i.e. object very distant), this becomes 
 F = mFs + /i, as before 
 
 EXAMPLES 
 
 1. /i - 10, / 2 = - 4, with the object 80 distant, to 
 reduce to J size. 
 
 Mag. produced by 1st lens = Q _ 1Q = ^ [See 47 
 
 .'. M =mag. of 1st image = | [M | = J 
 
 ... # = 4 ( - 1) = 10 [ 47 
 
 V> -10 + 10 = 
 
 2. If /i = 8, / 2 = - 3, and object is 60 in front, N = 2. 
 
 1st mag. = y 2 ^, M = J 3p-, FB = 5 iS ^ = 
 
96 
 
 THICK-LENS OPTICS 
 
 3. /i = 10, / 2 = - 4, with the object 167J in front of 
 the + lens, to reduce to J size. 
 
 M = <% 3 E = 27J F = 35 
 Other facts are shown in the diagram below. 
 
 flO 
 
 JV, 
 
 A'W = - 25 F = 35 F B = 10 
 u = - 105 v = 52J 
 
 4. Positive lens of 8 in. focus and negative lens of 3 in. 
 focus with the object 60 in. away, to reduce to | size. 
 Mag. of + lens = T V, M = \ 3 -, E = V, F = V- 
 
 119. Distance to Object for a given Magnification, == 
 
 M = magnif. of 1st image due to 2d lens 
 final image E + f 2 
 
 1st image f 2 
 
 rr 
 
 # = extension of bellows = FB + -TT 
 
 (Conf . 47) 
 [ 118 
 
TELEPHOTO LENS 97 
 
 FB = F -^ [ 115 
 
 - M = -~ = 1st reduction X mag. due to 2d lens = final 
 reduction 
 
 u = dist. of object = fi (n + 1) fi A 
 
 L ' r=7i' 5 47 
 
 Find FB, then E, then M, then n, then M. 
 Example. F = 24, /! = 6, / 2 = - 3, N = 5 
 .-. FB = 9, J = 13|, M = 5J, n = 28 
 .-. w = 6 (28 + 1) = 174 in. 
 120. Reduction Factor. 
 
 From 118 N = _T -- =j. For TV, see 119, 117. 
 ~r Ji " 
 
CHAPTER VI 
 REFLECTION AT SPHERICAL SURFACES 
 
 NOTE. This chapter is introduced on account of some experi- 
 mental observations. 
 
 121. The angles of incidence and reflection are equal. (See 
 
 any text-book on physics.) 
 
 By geometry, a line bisect- 
 ing the angle of a triangle 
 divides the opposite side into 
 
 u ._ _. ^ segments proportional to the 
 
 adjacent sides; hence, since 
 
 the angle of reflection is equal to the angle of incidence 
 
 a _ c 
 b~d 
 
 But for points near the axis, d = u, c = v, whence 
 
 a v r v v 
 r = - or - - = - 
 o u u r u 
 
 1 1 2 
 
 or - + - = - 
 
 v u r 
 
 For u = oo, i.e. parallel rays from a distant object, 
 Vf = 2 = ft called the focal distance, whence 
 
 1 1 _ 2 _ 1 
 v u r f 
 
 This is a general equation, applicable to convex or con- 
 cave surfaces, attention being paid to the values of the 
 letters, measurements to the right being positive. 
 
 98 
 
REFLECTION AT SPHERICAL SURFACES 
 
 99 
 
 122. In reflection, it simplifies the questions of signs 
 if we suppose the light to come from the right, thus making 
 u positive for real objects and negative for aerial objects, 
 with the following tabulation of results. 
 
 Divergent pencil 
 Real object 
 
 Convergent pencil 
 Aerial object 
 
 concave mrror 
 
 convex mirror 
 concave mirror 
 
 convex mrror 
 
 >2 
 
 Negative v means the image is virtual. 
 
 123. Graphic Construction for the object in various 
 positions; determined by known lines, focal and central. 
 
 .Aerial Urnaae^ Object 
 
 object Focus * 
 
 The order of the letters denotes the order of construction. 
 
 Caution. This is accurate only near the vertex, and 
 is introduced here in order to illustrate the method. In 
 actual construction, symbolize the surface by a straight 
 line perpendicular to the axis, as was done in the case of 
 
100 THICK-LENS OPTICS 
 
 a thin lens. For tracing any ray, we have similarly to 
 32, rs > <j> \ \ to c, where s = surface, c = line through 
 center of curvature. 
 124. Magnification. 
 
 TV/T -c 4.- Image v r v 
 
 Magnification = ^, . - = - - = - 
 
 Object r u u 
 
 EXAMPLES 
 
 1. A candle flame 1 cm. long, 36 cm. in front of a concave 
 mirror, whose focal length is 30 cm., gives what image and 
 where? 
 
 Ans - + ^ .'.' = 180. Mag. = = 5 
 
 2. A flame 2 in. in front of a positive lens of 1 in. focus, 
 and plane mirror } in. behind the lens, reflects the rays back 
 through the lens. Show that the real image will be J in. 
 from the lens. First image is 2 behind the lens or 1J be- 
 hind the mirror. Second image (aerial) 1 J before the mirror 
 or 1 before the lens. Third image is J before the lens. 
 
 3. How far from a concave mirror must an object be 
 placed to be magnified n times? 
 
 Ans. v = nu (real image); v = nu (virtual image). 
 
 11 1 (n 
 
 For real image, -? = - H --- .'. u = - - For vir- 
 
 ' / v nu n 
 
 11 1 (n-l)f 
 
 tual image, -. = .'. v = - 
 
 f u nu n 
 
 4. Gas flame 10 in. from wall. Required real image on 
 the wall three times as large. What mirror and where? 
 Ans. v = 3 u' u = dist. from mirror to object; 10 -f u = 
 
 3u. .'. u = 5. -7 = -= + ^ - .'. f = 3f . Result, a concave 
 j 5 15 
 
 mirror, 3f focal length, 5 in. from object. 
 
REFLECTION AT SPHERICAL SURFACES 
 
 5. u = 10, / = - 30. 
 
 Ans. v = - *-. Mag. = - f . 
 
 6. Object = 1 in., u = 18, / = + 15. 
 Ans. v = 90. Mag. = 5. 
 
 7. u = if. 
 
 Ans. v = /. Image virtual = twice object 
 
 8. Mag. = 12. Object 11 from screen. 
 Ans. 12 u u = 11. /. u = 1, .'. / = if. 
 
 101 
 
CHAPTER VII 
 EXPERIMENTAL OBSERVATIONS 
 
 To get thoroughly satisfactory results requires care, 
 experience, and a trained eye. The average untrained eye 
 cannot see things as they actually exist. Make several 
 observations and take the mean of them. 
 
 125. To Find Radius of Curvature of a Surface. 
 
 Telescope \swface 
 
 "<=L^z ------- \- ------ ..... ----- -fk 
 
 j* ....... _ .................... ^ ......... _ ........ _______ J\ 
 
 Scal 
 
 The lights (or other suitable objects) produce two images, 
 the distance apart of which is observed through the tele- 
 scope on the scale. 
 
 2 Al 
 
 r = radius of curv. = 7 -- x^ (for convex surface) 
 
 L -2 
 2AI 
 
 (for concave surface) 
 
 L + 21 
 
 I = distance of the images apart 
 L = actual dist. of the lights 
 A = dist. from surface to line of the lights 
 
 The less the curvature the greater A must be to get 
 accurate results. In the case of biconcave or biconvex 
 lenses it is easily seen which images are from the front or 
 back surface, by their inverted or erect position. 
 
 102 
 
EXPERIMENTAL OBSERVATIONS 103 
 
 Proof. The distance a, at which the virtual image of 
 the line L is formed behind the surface, is given by (see 
 
 121). 
 
 11,2 Ar 
 
 - = -T- + - > whence a = =-T ; 
 a A r 2 A -f- r 
 
 X r a 1 A 
 
 L A + r A A -{- a 
 
 1 r 
 
 whence =- = 
 
 [A = length of image 
 
 L 2 (A + r) 
 
 2A1 
 
 whence r = = ^-r 
 
 Similarly for a concave surface. 
 
 (Second method.) Make object (small illuminated disc 
 in screen) and image coincide by reflection from concave 
 mirror, or by interposing a positive lens in front of the con- 
 vex mirror. 
 
 For concave mirror, r = 2/ equals the distance of the 
 mirror surface from the image (object). 
 
 For convex mirror, find image of disc when mirror is re- 
 moved. The distance of this point from the front of the 
 mirror before removal is r. 
 
 If object and image do not coincide, / can be calculated 
 from the u and v distances. Check graphically by diagram 
 1 of 40. 
 
 Or for a concave mirror / can be measured directly by 
 first rendering the rays parallel by a positive lens; the dis- 
 tance of the image from the mirror is /. 
 
 126. By Matching with Surface of Known Curvature, 
 using the scale on both as in 125. 
 
 This matching can be roughly done by holding the sur- 
 faces in the hand, and observing the image of some bright 
 object, window, lamp globe, etc. 
 
104 THICK-LENS OPTICS 
 
 127. Radius of Curvature of Surface of Small Curvature. 
 
 Focus a telescope on a scale at a distance A from the 
 object glass. With the telescope thus focussed, let the 
 image of an object reflected from the surface to be tested 
 be clearly seen when the distance between the object and 
 the surface is a, and that between the surface and the 
 object glass of the telescope is e. Then 
 
 A - e 
 
 rad. of curv. = r = 2 a 
 
 T-> f ^ v e -\- A ^ T . , . 
 
 Proof, j- = = , L' being practi- 
 
 2 
 
 cally at the surface [Sim. triangles 
 
 Whence r = 2 a -: 
 
 A e a ) 
 
 The best value for e is about -~ 
 
 Positive r denotes a concave surface ; negative r a convex 
 surface. 
 
 The absence of parallax between the cross wires of the 
 telescope and the image is the test of distinct vision. 
 
 FOCAL LENGTH OF A THIN POSITIVE LENS 
 
 128. With the Sun. Distance of the image of the sun 
 from the lens = /. 
 
EXPERIMENTAL OBSERVATIONS 105 
 
 129. Lens Distances. /=-^-. (See 16.) Distances 
 
 u v 
 
 to the left are negative, u = distance from lens to object, 
 v = distance to image. For accurate focussing, see 156. 
 Solve graphically by diagram 1 of 40, for check. 
 
 130. With a Telescope. Focus the telescope on some 
 distant object. Place the lens in front of the object glass. 
 Look through the telescope, without altering its length, 
 at some plane object (a newspaper), adjusting the distance 
 for distinct vision. The distance of the object from the 
 lens = /, because the lens sends parallel rays into the 
 telescope already set for parallel rays. 
 
 131. By Different Positions of the Lens. 
 
 / = - u pi . I = distance between image and object, 
 
 a = distance between the two positions of the lens when 
 giving a distinct image, the object and screen remaining 
 fixed. 
 
 Proof. The distances of object and image from the lens 
 are \ (I + a) and \ (I a), whence ( 17), 
 
 1 22 41 
 
 / 1 - a ' 1 + a I 2 - a 2 
 
 132. From Equality of Object and Image. Distance 
 between object and image = 4 jf. (See 37, Ex. 2.) 
 
 133. Comparison of Images. A candle (or illuminated 
 aperture) is placed a distance a from a screen and the 
 image focussed on the screen. On moving the lens towards 
 the candle another image is formed which is m times as 
 large as the former. 
 
 The focal length = 
 
 (1 + 
 

 106 THICK-LENS OPTICS 
 
 Proof: By 131, / = -^~ 
 
 \b = distance between 
 2 positions of the lens 
 
 - 1 
 
 2/ 
 From the 1st eq. 6 2 = a 2 4 a/ 
 
 From the 2d eq. 6 = (a ~ 
 
 m + 1 
 
 (a 2 -4a/+4/ 2 ) (m - I) 2 
 
 (m + I) 2 
 
 (g2-4a/+4/ 2 )(m-l) 2 
 (m + 1) 2 
 
 ma 2 
 
 (m - I) 2 
 
 2 am 
 
 (m - I) 2 
 
 / ma 2 4 a 2 m 2 
 
 VI^ny^Tm"-- I) 4 
 
 M 
 
 2am a / , 4m 2 _ m (m 2 + 2m + 1) 
 
 (m - l^ + m^T V^ ^(m-l) 2 " 
 
 2 am + a (m + 
 
 (m - I) 2 
 
 a Vm (m + 1 - 2 Vm) = /- /Vm - IV 
 
 (m - I) 2 \m-iy 
 
 Vm 1 \ 2 a Vm 
 
 (Vm - 1 
 
 Q.E.D. 
 
EXPERIMENTAL OBSERVATIONS 107 
 
 FOCAL LENGTH OF THICK POSITIVE LENS 
 134. From Highly Magnified Image. 
 
 I = length of a division of the scale, 
 L = length of the image. 
 
 Proof: 1 + 1 = 1. (See 16.) ^ = - Whence/ = 
 
 V U J l U 
 
 v j -; Since v is very large, small errors in its measure- 
 L + I 
 
 ment or mistakes in locating the nodal point (to which it 
 should be measured) do not materially affect the result. 
 
 .Lens 
 
 White screen - 
 
 wttfiyreatly magnified dlumlnatefL scale 
 
 image *" (onj?lass)~ 
 
 Screen and scale may be interchanged, with diminished 
 image, using a lens to read the image, in which case 
 
 L 
 
 135. Swing of Camera or Lens Carrier. Swing the 
 camera or lens carrier horizontally on a 
 
 
 table (guide by a flat stick with a small 
 
 nail through one end) until a distant ver- c 
 
 tical object is focussed on two vertical \'/ 
 
 lines (short) on the extreme edges of the :'/ 
 
 ground glass screen. Mark the angle of // 
 
 swing on the table. In the angle plot c/ 
 
 the distance a, between the lines on the 4 = distance be - 
 
 ' . tween lines on the 
 
 screen, perpendicular to the bisector of gcreen> &=f OC aldis- 
 
 the angle. The bisector, 6, will equal tance 
 the focal length. 
 
108 THICK-LENS OPTICS 
 
 136. Movement of Screen. Focus on very distant 
 object. Then focus on a near object, making the image 
 
 and object equal (same size discs with parallel lines, one 
 pattern covering the other). / = the distance the screen 
 is moved. (Conf. 37, Ex: 2; 47, Ex. 15.) 
 
 137. Movement of Screen. Focus for very distant 
 
 object. Focus on a near object for the image = -= of object. 
 
 (s = number of units on scale, d = number of units on 
 screen covered by the s units of the scale.) 
 
 Th f - I = distance moved by screen 
 
 d [between the two focussings 
 
 Proof: f = V \~ v = dist - to ima S e - See 47 
 
 m + 1 [m = magnification 
 
 B /. 
 
 m + 1 
 
 TTT , . a sa , ., d 
 
 Whence / = = -y [m = magmf . = - 
 
 m d s 
 
 138. Angle of Vision. / =- (See 2.) h can be 
 
 tan ct 
 
 Hstant object 
 
 Distant object 
 
 measured by a scale or a sliding lens, a must be measured 
 by an instrument, or tan a can be found by 157. 
 139. Unit Screen Movement (Lionel Laurence) 
 
 / dmn 
 \/ m _ n = 
 
 if m = 2n, n = 1 
 
EXPERIMENTAL OBSERVATIONS 
 
 109 
 
 D f ! ! ! i K f ( f + n) . 
 
 Proo/: f - + fc + a = r whence b = - >l - d 
 
 [74 
 
 h "TT -f B 
 
 zi4_ LLIJ 
 
 Object with Object with 
 image at 4 image at B 
 
 11 
 
 f (f + m) 
 
 - 
 
 whence 
 
 / dmn 
 Vm n 
 
 at 
 
 140. Measurement of Image. At a distance a 
 least 100 /, so as to 
 rank as a distant 
 object, set off at 
 right angles two 
 marks \ a distant 
 from the center line. 
 The distance between the two images on the screen will be 
 \ f. This distance is most accurately measured by a sliding 
 lens (microscope) focussed on the aerial image. 
 
 Any other submultiple of a can be similarly used. 
 
 If a is not large enough to be ranked as distant, then the 
 
 distance apart of the images is ^ instead of ^ 
 we can find / as follows : 
 
 av v 2 
 
 a + v 
 
 v 
 
 a v 
 
 From this 
 
 This result is accurate to within about 5 per cent. 
 
 141. Comparison with Standard Lens, with distant 
 object. 
 
110 THICK-LENS OPTICS 
 
 _ focal length of standard X length of image by lens 
 J length of image by standard lens 
 
 (See Ex. 16, 47.) 
 
 142. Double Focus, with Equal Distances to Object and 
 Image. 
 
 Find F and F f for parallel rays. Place object and screen 
 at distances somewhat less than J the distance between 
 F and F', and move by small equal increments until the 
 object and image are equally distant from F and F' and 
 
 Mdal planes 
 Object I \ ^_ Image 
 
 Focus Lens Foeiis 
 
 the image distinct. These distances will be the focal length 
 required. (See 47, Ex. 15, in connection with 74.) 
 
 143. Double Focus, with Unequal Distances to Object 
 and Image. 
 
 |<- X- *~, 
 
 Serein with, F 
 
 illuminated scale. 
 Ob/eet 
 
 Determine F, F' for parallel rays, and then determine 
 x and y for distinct image. Then 
 
 / = focal length = \/xy [See 75 
 
 If the principal focus is within the combination, interpose 
 between object and combination any good short focus posi- 
 tive lens and adjust till the image is reflected back by a mir- 
 ror to coincidence with the object (or slightly to one side by 
 tilting the mirror). Then the image will be at the prin- 
 cipal focus of the combination, since the emerging rays are 
 parallel and reflected back parallel by the mirror. Remove 
 the combination and find the image made by the positive 
 lens. Determine this point (on the mounting) relative to 
 
EXPERIMENTAL OBSERVATIONS 
 
 111 
 
 the combination before it was removed : it will be the prin- 
 cipal focus (F) of the combination. Find F' in the same 
 way. Then find x, y, as before, and/= Vx y. (Microscope 
 ocular.) 
 
 144. By Movement of Lens. Same method as in 131. 
 
 Diagram showing the extreme rays from a point; in the two positions. 
 e = distance between nodals; a = distance between the two posi- 
 tions of the lens; I = distance between the screen and object 
 
 a 
 
 = v u 
 
 
 u 
 
 + v + e = I 
 
 
 
 I a e 
 
 1 + a - e 
 
 
 2 
 
 2 
 
 i 
 
 2 
 
 2 
 
 / 
 
 l+a - e ' 
 
 I a e 
 
 - / 
 
 
 70 1 9 
 
 I + d 
 
 
 A 72 i vcr y sr 
 
 [74 
 
 - a 2 I 2 + a 2 
 
 [67 
 
 i i/ i c/ 
 
 Notice the correction induced by the thickness of the 
 lens. *(Conf. 131.) 
 
 FOCAL RADIUS OF A NEGATIVE LENS 
 
 145. With Sun. If lens is deep and not too small, 
 focal length = b. (Conf. 47, Ex. 14.) 
 
112 
 
 THICK-LENS OPTICS 
 
 This is an uncertain method, on account of the indis- 
 tinctness of the bright patches. 
 
 r mcwes of 
 holes, Hct in. 
 apart 
 
 Card with 
 Wholes a in. apart 
 
 146. With Sun. / = 
 
 Ad 
 
 d = diameter 
 
 of the lens aperture, D = diameter of the circle of light 
 cast by the lens when in the path of the sun's rays, A = 
 distance of the screen from the lens, 0.0094 = 2 tan of the 
 apparent diameter of the sun. 
 
 If the lens is deep and not too small, we can write, 
 neglecting the 0.0094 A, 
 
 Ad 
 J ~~~ D - d 
 
 which becomes / = A, if D = 2 d, as in 145. 
 Proof. 
 
 Extreme ray from P, thru one edge 
 Hand 
 
 D 
 
 Extreme ray from Q thrw 
 one edge o/ihelens 
 
 C Virtual im< 
 qfsuris di 
 
 By similar triangles 
 
EXPERIMENTAL OBSERVATIONS 113 
 
 147. With Stronger Positive Lens. Combine the two 
 lenses and find focus F. Find focal length of positive 
 lens, F'. Then ( 91) 
 
 F F' 
 
 f = focus of negative lens = -=7 ^ 
 
 r r 
 
 The positive lens should be so chosen as to make the 
 difference F' F as large as possible. 
 
 148. With Positive Lens and Comparison of Images. 
 Focus with a positive lens and measured image. Interpose 
 negative lens and measure new image. Call the magnifica- 
 tion over the positive image, M . 
 
 Move the negative lens a small distance, D, nearer the 
 screen, and measure the image, calling its magnification 
 over the positive image, M' '. 
 
 Then focal length of neg. lens = f ^, _ 
 Proof: j + 1 = M, j + 1 = M r 
 Therefore, since identically -^ 
 
 then f = M^lM 
 
 d, d f = distances from negative lens to screen. 
 
 If we move the combination instead of the negative lens, 
 and call M', M the actual magnifications of the images for 
 two positions of the object, this method will apply to a 
 microscope objective, or to the ocular (positive or negative) 
 by inverting it. 
 
 149. With Positive Lens and Comparison of Images 
 (Lindsay Johnson method). 
 
 Focus with positive lens and measure image. 
 
 Interpose negative lens and measure new image, adjusting 
 
114 THICK-LENS OPTICS 
 
 until the magnification over the positive image is 2; call 
 the distance between screen and negative lens, a. 
 
 Move the negative lens until the magnification over the 
 positive image is 3. Call the distance between the nega- 
 tive lens and the screen, b. 
 
 Then F = focal radius of the negative lens = b a. 
 Proof: By preceding case 
 
 b a b a ' 
 F = M^M = JTTT2 = 6 ~ 
 
 TO LOCATE THE NODAL POINTS 
 
 150. (a) Determine the focal radius ( 134-144) and 
 lay off this distance from the focal point, marking the 
 result on the mounting. (Conf. 57.) This is the node of 
 emergence. 
 
 (6) Locate the point (by twisting the lens around a 
 vertical ayis: on the optical bench) around which the lens 
 can be turned on a vertical axis without displacing the 
 image. This is the node of emergence. 
 
 If the nodal of emergence is beyond the center of rotation, 
 the image will move in a contrary direction to that of the 
 back of the lens, and vice versa. 
 
 This point can also be determined by reflecting the image 
 back through the lens to coincidence with the object by 
 means of a mirror (or slightly to one side by tilting the 
 mirror). When a slight movement of rotation produces no 
 movement of the reflected image the axis of rotation is at 
 the nodal, and moreover the focus of the lens is the distance 
 of the axis of rotation from the object (image). 
 
 Reverse the lens and repeat the operation, to find the 
 other nodal: both cases. 
 
EXPERIMENTAL OBSERVATIONS 
 
 115 
 
 MAGNIFYING POWER: TELESCOPE 
 
 151. Visual Comparison of Images. Distant object. 
 
 Mag. power = N = number of clapboards (divi- 
 n sions on a scale) seen with 
 
 one eye (naked) which are 
 covered by n clapboards seen 
 with the other eye through 
 the telescope 
 
 152. Visual Comparison of Images. Near object. 
 Focus the telescope on a very distant object, and then 
 
 fix in front of the object glass a thin convex lens of low 
 power (a spectacle glass of about 2 m. focal length). 
 
 The telescope is then pointed to a scale at such a dis- 
 tance that the divisions appear well defined, focussing by 
 moving the scale, not the telescope tube length. 
 
 N a 
 Mag. power = j- 
 
 \N, n = same as in 151 
 
 a = dist. from scale to object glass of telescope 
 b = dist. from scale to eye of observer 
 
 Proof. With the lens in front of the telescope, we have 
 practically a large microscope, in which the magnification is 
 
 &+/ F 
 
 f 
 
 - ( 108) 
 
 b = dist. at which the scale is seen 
 / = focal length of eyepiece 
 F = focal length of object glass 
 a = focal length of intruded lens 
 
 But the magnification of the telescope is ( 110) 
 
116 THICK-LENS OPTICS 
 
 6 + / F \b = distance of dis- 
 
 F 
 
 hi 
 
 / b tinct landscape vision 
 
 a 
 
 To convert the first into the second we must multiply by 7- 
 
 u 
 Q.E.D. 
 
 MAGNIFYING POWER OF A MICROSCOPE 
 
 153. Visual Comparison of Images. With one eye 
 (naked) count the divisions on a scale 10 in. (25 cm.) from 
 the eye which are covered by one or more divisions on a 
 scale seen through the microscope. 
 
 
 
 Mag. power = 
 
 n 
 
 N = number of divisions seen by 
 the naked eye at 10 in. 
 
 n = number of divisions seen in 
 the same space through the 
 microscope 
 
 A convenient way for observing N is to use a camera 
 lucida on the eyepiece, with the naked-eye scale 10 in. 
 from the eye, through the camera lucida. 
 
 154. To find the Work done by the Ocular. Focus on 
 the object with the ocular in place. Remove the ocular 
 and with the aid of a small lens and piece of ground glass 
 (see 156) find the position of the real image made by the 
 objective. Measure its distance from the top of the tube, 
 and note the corresponding place on the ocular. Having 
 previously determined the focal length of the eye and field 
 lenses of the ocular, calculate (graphically or algebraically) 
 the reduction caused by the field lens. (The position of 
 this reduced image should be just inside the focus of the 
 eye lens. See diagram of 37.) Multiply this by the 
 magnification caused by the eye lens. This product is 
 the final action of the ocular. 
 
EXPERIMENTAL OBSERVATIONS 117 
 
 If the first focussing is done on a scale, on or in the posi- 
 tion of the ground glass, it will give the magnification caused 
 by the objective. 
 
 The product of the two magnifications should equal that 
 found by 153. 
 
 (Second method.) Place a small rectangular opening of 
 known width at one end of a tube 10 in. long, in the other 
 end of which is placed the ocular to be measured. 
 
 At the image of the rectangular opening (slightly above 
 the ocular) place a scale and with a lens read the width of 
 the image on the scale. The reduction from rectangular 
 opening to image, inverted, will be the magnification of 
 the ocular. 
 
 , , width of rectangular opening 
 
 Mag. power = r-rrr j~r- r^ y- 5 
 
 width of image on the scale 
 
 Proof. The details are left to the reader, with the fol- 
 lowing guide. Make a diagram (skeleton, 25, 77) of the 
 ocular and by 61, illustrated by 95, construct the nodal 
 lines. Starting with the virtual image (call it A) 10 
 inches to the left of the right-hand lens of the ocular, derive 
 from it the aerial object (call it B), just outside the lens, 
 which produced the virtual image. The ratio of magnifica- 
 tion between these will be the magnifying (reduction by 
 first lens, and final magnification by second lens) power of 
 the ocular. For, a real object, the rectangle, at A, since 
 these are conjugate points, will give a real image at B, 
 which can be measured by a scale, and the ratio between 
 this A and B will be the same as before. 
 
 Notice, due to the crossing of the nodals, the aerial 
 object B gives a virtual image at A, since B is to the left 
 of the H nodal line and within the F focus, though outside 
 the lens. (See 95, Ex. 4.) 
 
118 THICK-LENS OPTICS 
 
 When the objective is in operation the aerial object is 
 generally within the ocular combination, instead of outside, 
 and the virtual image at infinity. But this does not 
 materially alter the angle under which the virtual image is 
 seen, and therefore not the magnifying power. 
 
 Check this by locating A and B through the lenses, 
 instead of through the nodal planes. Pass A toward the 
 left to infinity, and note how B passes through the lens 
 to a point between the lenses; the point to which the 
 image made by the objective is deflected by the first lens 
 of the ocular. The aerial image at this point gives the 
 virtual image at infinity seen by the eye of the observer. 
 
 See also method of 143. 
 
 155. To Find the Index of Refraction. Sight the ob- 
 jective of a microscope on a well-marked, hard (celluloid) 
 white surface, or piece of scratched glass, and read the 
 scale on the limb of the microscope (vernier attachment 
 necessary), the aperture of the objective having been 
 diminished by slipping over the objective a cap (paper) 
 with a small central opening (1 mm. = ^V in- diameter). 
 
 Having marked the center of the lens whose index of 
 refraction is desired, by a small circular ink spot with a 
 clear center on the upper surface (in order to locate the 
 center through the microscope), slip the lens under the 
 objective and focus on the center of the upper surface. 
 The difference of the readings is the thickness (t) of the lens. 
 
 Then carefully focus, through the lens, on the marked 
 white surface on which the lens rests. The difference of 
 the last two readings will be a, t a being the amount 
 the objective must be raised from focussing on the white 
 surface, due to the interposition of the lens. Then 
 
 - 1 r - a 
 a r t 
 
EXPERIMENTAL OBSERVATIONS 119 
 
 or if the lens is merely plane glass 
 
 _ t_ 
 
 = index of refraction 
 
 t = thickness of the lens 
 
 a = amount the objective must be lowered from focus- 
 sing on the top of the lens, to bring the mark on 
 the white surface into focus 
 
 r = radius of curvature of the upper surface of the lens 
 
 N.B. To get satisfactory results, very great care and 
 a fine vernier are necessary. 
 Proof. 
 
 Diagram showing the lens in place under the ob- 
 jective. = center of curvature of the upper surface. 
 S = marked spot on the white surface on which the 
 lens rests, a = distance the stage must be raised or 
 the objective lowered in focussing from the top of the 
 lens to the marked spot S (through the lens), a = 
 angle of incidence, /8, of refraction, d = semi-opening 
 of lens, c = semi-diameter of visible part of lens. 
 
 Note. a, /?, and y are supposed to be so small (due to 
 the narrowed opening of the objective) that their cosines 
 = 1 and their sines = a, ft, y, respectively, as also their 
 tangents = a, /?, y. (See any trigonometry, Functions of 
 Small Angles.) 
 
120 
 
 THICK-LENS OPTICS 
 
 sin (y + a) _ t_ 
 sin (y + ft) ~ a 
 
 sin y cos a + cos y sin a _ ^ _ 
 sm y cos ft + cos y sin ft a c 
 
 [6 
 
 - cos a 4- sin a 
 
 - cos ft + sin 
 r 
 
 [sin y = 
 
 sin a 
 
 sm a 
 sin ft 
 
 c 
 
 - -\-srn ft 
 r 
 
 , I + sin ^ 
 
 i T 
 
 a sin ft 
 
 [ 
 
 COS a 
 
 cos 
 
 tc 
 
 a r sin /: 
 
 + -- 
 a r sin 
 
 ra 
 
 L raa ^ 
 
 a raa tc + ta ) 
 
 t r a 
 
 a r t 
 
 ra sn 
 
 raa 
 c 
 
 fna = a, 
 ex 
 /D 
 AJ ^"^ 
 tj^ 
 
 a, etc. 
 
 tan (y + a) 
 
 _d 
 
 ~ y i ,,. 
 
 c 
 
 tan y 
 
 r 
 
 
 d 
 
 d c dr cu 
 
 a 
 
 u 
 
 * ~ u r ur 
 
 C 
 
 _d 
 
 
 a 
 
 u 
 
 
EXPERIMENTAL OBSERVATIONS 121 
 
 If r = oo, /a = - 
 
 Check your determination of /* by using it to calculate 
 F of the lens, and test by observation. 
 
 Similarly for a concave surface, we would have [since 
 the angle (y + a) becomes replaced by (a y) and 
 
 PRACTICAL SUGGESTIONS 
 
 156. To Focus Accurately. Set a fine pin (needle) in 
 line with the lens, and with the eye in line fix the vision 
 on the pin. Adjust the lens backward or forward until 
 a motion of the head slightly sidewise does not alter the 
 position of the pin on the image seen through the lens. 
 If on moving the head the pin moves across the image in 
 the same direction, the image (and therefore the lens) is 
 too close, and vice versa. 
 
 Better still, use a short focus lens, focussing it on the 
 edge of a translucent (transparent) screen (piece of cellu- 
 loid). Then move the lens until the image appears dis- 
 tinct, testing similarly by the motion jy 
 of the head. 
 
 157. To Measure an Angle with- 
 out Angular Instruments. Deter- 
 mine three points, A, at the vertex T~ 
 
 of the angle, B, in line with one side of the angle, and C, 
 in line with the other side and at right angles to AB. 
 
 This may be done on the ground, or on a table or top 
 of a level box. Then 
 
 90" 
 
122 THICK-LENS OPTICS 
 
 BC BC 
 
 tan A =35 or sin A -= -^ 
 
 whence the angle A can be found by a table of tangents or 
 sines. 
 
 158. Make the experimental observations with mono- 
 chromatic light (as well as white) by using red, blue, green 
 screens (colored glass). 
 
 Use no alcohol or other solvent on mounted lenses, 
 except in an emergency and with the greatest care. Cleanse 
 greasy lenses with a weak solution of washing soda, rinsing 
 with clean water. 
 
 Keep all lenses covered from dust; keep a cover over 
 the eyepiece of the microscope when not in use. 
 
 Keep lenses out of the sun as much as possible they 
 will gradually discolor. The cement may overheat. 
 
 Clean lenses with the greatest care, lightest pressure, 
 softest cloth free from dust and grit, with a circular motion, 
 never across the lens. Use soft camel's hair brush when 
 feasible. 
 
 159. Home-Made Optical Bench. A little skill can 
 make a home-made optical bench, as shown, with which 
 fairly good work can be done. In the absence of anything 
 better, lenses fastened on the tops of corks with pins will 
 do roughly good work. 
 
 a, board on edge (J or in. wide), or graduated yard 
 stick. 
 
 6, pedestal to keep board upright, two or more. 
 
 c, sliding piece on top of board, with center line for read- 
 ing distances moved. 
 
 d, center line of the various pieces. 
 
 e, rotating piece, pivoted to c at center, by small 
 screw or pivot, exactly in center, and in line with 
 center line d. 
 
EXPERIMENTAL OBSERVATIONS 
 
 123 
 
 /, lens carrier sliding on e, so as to allow bringing the 
 nodals over the center of revolution. 
 
 g, g, sloping sides to accommodate different sized lenses. 
 
 h, h 
 
 grooves in which to slip the lens so that it will be 
 held upright: h is used for a short focus lens, when 
 
 i would be too far from the end of /, the nearest 
 approach of the screen. 
 j, center line of h h, a known distance from d. 
 Suggestions as to how to put the pieces together are 
 indicated by dotted lines. 
 
 Modified carriers (without rotation) should be provided 
 for holding screens, reading lenses ( 156), etc., to be used 
 in connection with the one above. 
 
124 
 
 THICK-LENS OPTICS 
 
 -5 
 
 Kg] 3 
 
 Fig 14 C 
 
 Fig. 15 
 
 -r 
 
APPENDIX 
 
 THIS contains a series of progressive propositions giving 
 succinct methods of construction and interpretation, with 
 important theorems for surfaces and lenses, culminating in 
 propositions XIII-XVIII, which give general discrimina- 
 tions for locating the nodals and foci for different forms of 
 lenses. 
 
 These propositions afford a valuable general check upon 
 the calculations and graphical constructions, guided as they 
 are by the numerical relations between the surfaces. 
 
 The calculator cannot have too many checks, as he will 
 quickly discover when he essays an independent and uncor- 
 roborated investigation. 
 
 These propositions give a complementary point of view 
 to that in the body of the text, valuable and almost in- 
 dispensable, especially for those making original numerical 
 investigations. 
 
 SURFACE REFRACTION 
 
 Notation. Prolong is the prolongation on the right of 
 the surface (light is always supposed to come from the left) 
 of the ray impinging on the left side. 
 
 Emerge is the position of the ray after refraction through 
 the second surface from the denser to the rarer medium. 
 
 r = the radius of the surface of incidence, the surface to 
 the left of the denser medium. 
 
 s = the radius of the surface of emergence, the surface to 
 the right of the denser medium. 
 
 e = thickness of the lens. 
 
 H f , F f = nodal of emergence and corresponding focus. 
 
 H, F = nodal of incidence and corresponding focus. 
 
 [For convenience of illustration, /* is assumed = f . If /* 
 
126 LENS OPTICS 
 
 is not equal to f , in the results which follow substitute 
 ^ for 3, and - r for 2.] 
 
 I. To trace an incident ray, surface refraction. Figs. 
 1 and 2. (Conf. 33.) Note carefully the order of the 
 letters, which indicate the order of construction; and the 
 formula for construction, rs -> <fo 1 1 
 
 II. To trace an emergent ray, surface refraction. Figs. 
 3 and 4. 
 
 III. (Cor. to II.) Fig. 5. Emergent rays from a surface 
 with a + radius, with the prolongs convergent to a point on 
 the axis 
 
 beyond the 3s point, are bent upward, above the hori- 
 zontal ; 
 
 at the 3 s point, emerge horizontal; 
 within the 3 s point 
 
 and beyond the s point, are bent upward above the pro- 
 long, emerging convergent: 
 
 and within the s point, are bent downward, below the 
 prolong, emerging convergent. 
 
 IV. (Cor. to III.) Emerges originating from points on 
 the axis to the left of a + radius surface are divergent, bent 
 upward from the prolong. 
 
 V. (Cor. to III.) Fig. 6. H3s>3r-e (positive me- 
 niscus, convex to the rays), then incident horizontal rays 
 emerge convergent, but 
 
 rising from the prolong, if s < 3r e', 
 
 falling from the prolong, if s > 3 r e. 
 
 If 3s < 3r e (negative meniscus, convex to the rays), 
 then the incident horizontal rays emerge divergent, rising 
 from the prolong. 
 
 VI. (Cor. to II.) Fig. 7. Emerges from a radius sur- 
 face, originating from a point on the axis to the left of the 3 s 
 point, are bent downward to convergence: 
 
 from the 3s point, are bent downward to horizontally; 
 
 from a point within the 3 s point, are divergent, falling be- 
 low the prolong, if from without the s point; rising above 
 the prolong, if from within the s point. 
 
 VII. (Cor. to VI.) Rays convergent to a point on the 
 
APPENDIX 127 
 
 right of a radius surface have their emerges bent down 
 below the prolongs. 
 
 VIII. (Cor. to VI and I.) Fig. 8. If8s-e>8r (neg- 
 ative meniscus, concave to rays), then incident horizontal 
 rays are divergent; 
 
 falling from the prolong, if s < 3 r + e; 
 
 rising from the prolong, if s > 3 r + e. 
 
 li 3s e < 3r (positive meniscus, concave to the rays), 
 then incident horizontal rays are convergent, falling below 
 the prolong. 
 
 IX. (Cor. to VIII.) Fig. 9. In a positive meniscus, con- 
 cave to the rays, (s < r + e/8), the nodal of emergence is 
 outside the lens to the right, and F' to the right of that. 
 
 X. (Cor. to VIII.) Figs. 10, 11. In a negative menis- 
 cus, concave to the rays (s > r + e/3), the nodal of emer- 
 gence is on the outside of the lens to the left if s <3 r + e\ or 
 in the lens if s > 3 r + e. In either case F' is to the left 
 of H'. 
 
 XI. (Cor. to V.) In a positive meniscus, convex to the 
 rays, since 3 s > 3 r e, the nodal of emergence is outside the 
 lens to the left if s < 3 r e; inside the lens if s > 3 r e. 
 
 In either case the focus F' is to the right. 
 
 XII. (Cor. to V.) In a negative meniscus (3 s < 3 r e), 
 convex to the rays, since the emerge rises from the prolong, 
 the nodal of emergence is to the right and outside, and F' 
 to the left. 
 
 XIII. (Cor. to I and IV.) Double concave lens. 
 Emerges resulting from incident horizontal rays are bent 
 upward from the prolong and H' is within the lens, F f to the 
 left. Similarly as to H, F, mutatis mutandis. 
 
 XIV. (Cor. to I and VII.) Double convex lens. Inci- 
 dent horizontal rays are bent down by the first surface and 
 the emerge falls below the prolong, therefore H' is within the 
 lens, and F' to the right. Similarly as to H, F. 
 
 XV. (Cor. to IX and XL) Positive meniscus, concave 
 to the rays. H f is outside the convex with F' further out- 
 side, measured with the rays. 
 
 H is outside the convex if r < 3 s e. 
 H is inside if r > 3 s e. 
 
128 LENS OPTICS 
 
 F is measured from H toward the concave, against the 
 rays. 
 
 XVI. (Cor. to X and XII.) Negative meniscus, con- 
 cave to rays. 
 
 H' is outside the concave if s < 3 r + e. 
 
 H r is inside if s > 3 r -\- e. 
 
 F' is measured from H' against the ray. 
 
 H is outside the concave with F measured with the rays. 
 
 XVII. (Cor. to X and IX.) Positive meniscus, concave 
 toward the rays. 
 
 H' is outside the convex if s < 3 r e. 
 
 H f is inside if s > 3 r e. 
 
 F' is measured with the rays. 
 
 H is outside the convex with F measured against the rays. 
 
 XVIII. (Cor. to XII and X.) Negative meniscus, con- 
 vex to rays. 
 
 H' is outside the concave, with F' measured against the 
 rays. 
 
 H is outside the concave if r < 3 s + e. 
 H is inside if r > 3 s + e. 
 F is measured with the rays. 
 
 SURFACE REFLECTION 
 
 (The order of the letters in the diagrams indicates the order of oper- 
 ations. Conf. : 7.) 
 
 Reflection from convex surface. Figs. 12, 13, 14. Image 
 virtual and smaller. (Conf. notation of prop. I.) 
 
 Reflection from concave surface. Figs. 15, 16. Image 
 real. 
 
 Reflection from concave surface. Fig. 16. Object out- 
 side the center. Image real, inverted, smaller, inside the 
 object. 
 
 Reflection from concave surface. Fig. 17. Object in- 
 side the focus (r/2). Image virtual, erect, larger, behind the 
 mirror. (For object between the center and focus, image 
 is real, inverted, larger, outside the object.) 
 
INDEX 
 
 A, A', 75 
 
 AN, A'N', 75 
 
 AN + W, A'N' + 91', 75 
 
 Analytical investigation, 55 
 
 Axis of Surface, 3 
 
 B 
 
 1,67 
 
 Back focal distance, two thick 
 
 lenses, 71 
 
 two thin lenses, 67 
 light from right, 70 
 
 Camera magnification, 89 
 Center, optical, 9, 43 
 Circle of confusion, 35 
 Combinations of lenses, 64 
 Condenser, best position, 32 
 Confusion, circle of, 35 
 Copying, 31 
 
 D 
 D, 25 
 
 Diagrammatic investigations, 9 
 Diagrammatic procedure, 10 
 Diopters, 25 
 
 e, 41, 42, 125 
 e, 67, 75 
 Emerge, 125 
 
 Emergence, principal point of, 39 
 Enlarging, 31 
 Equation, of line, 56 
 refracted ray, 57 
 
 Equivalent focus, two thin lenses, 
 
 67 
 Equivalent focal length, two thick 
 
 lenses, 71 
 Equivalent thickness, thick-lens 
 
 combination, 72 
 Equivalent thin-split, 42, 74 
 Exposure, 33 
 Eye, 48 
 
 F, F r , 7, 75, 125 
 /, 5, 34, 75 
 /', 7, 75 
 / number, 34 
 
 /i /z focal length of lenses of com- 
 bination, 67, 75 
 Focal length, radius, two thick 
 
 lenses, 71 
 
 telephoto combination, 91 
 thin positive lens, by sun, 104 
 by lens distances, 105 
 with a telescope, 105 
 by different positions of lenses, 
 
 105 
 by equality of object and image, 
 
 105 
 
 by comparison of images, 105 
 thick positive lens, by highly 
 
 magnified image, 107 
 by swing of camera, 107 
 by movement of screen, 108 
 by angle of vision, 108 
 by unit screen movement, 108 
 by measurement of image, 109 
 by comparison with standard 
 
 lens, 109 
 by double focus, 110 
 
130 
 
 INDEX 
 
 by movement of lens, 111 
 of negative lens, by sun, 111 
 by stronger positive lens, 113 
 by positive lens and compari- 
 son of images, 113 
 Focal points, 62 
 Focus, back focal, two thin lenses, 
 
 67 
 
 equivalent, two thin lenses, 67 
 to accurately, 121 
 Formulae, use of, 54 
 
 use of for combined lenses, 74 
 
 Graphic check on calculation, 22 
 Graphic construction, 76 
 
 tracing any ray, 54 
 
 oblique ray, 15 
 
 H 
 
 H, H', 41, 42, 48, 60, 125 
 Human eye, 48 
 Hyperfocal distance, 34 
 
 Index of refraction, 1, 118* 
 
 L 
 Line, Equation of, 56 
 
 M 
 
 M, 29 
 
 H, index of refraction, 2 
 Magnification, convex lens, 28 
 camera, 28, 89 
 compound microscope, 87 
 
 by visual comparison of images, 
 
 116 
 
 simple microscope, 37 
 ocular, 111, 116 
 opera glass, 89 
 surface reflection, 100 
 telescope, 88 
 
 by visual comparison of images, 
 115 
 
 Microscope, simple, magnification, 
 
 37 
 compound, magnification, 87 
 
 N 
 
 N, 31 
 W, W, 75 
 Nodal, 49 
 
 Nodal distances, thick lenses, 72 
 Nodal plane, images in, 47 
 Nodal points, 42, 61 
 
 construction for, 42, 44 
 
 calculation for, 45 
 
 to locate, 114 
 Number of the lens, 25 
 
 Oblique rays, diagram for, 15, 54 
 Ocular, magnification, 111, 116 
 Opera glass, magnification, 89 
 Optical bench, home made, 122 
 Optical center, 9, 43 
 
 P, P', 25 
 
 Point of emergence, 39 
 
 Powers of distances, 25 
 
 lenses, 25 
 
 not in contact, 69 
 thick-lens combination, 73 
 Principal points, 39, 60 
 
 point of emergence, 39 
 
 plane, 41, 60 
 Prolong, 125 
 
 Q 
 
 Q, Q', 25 
 
 R 
 
 r, 5, 7, 125 
 Radius of curvature of surface, 102 
 
 by matching, 103 
 
 small curvature, 104 
 Reduction factor, negative lens, 36 
 Reduction factor, telephoto, 97 
 Reflection at spherical surfaces, 98 
 
 diagrammatic construction, 128 
 Refracted ray, equation of, 57 
 Refraction, index of, 1, 118 
 
 surface, 1, 2, 126 
 
INDEX 
 
 131 
 
 s, 6, 7, 125 
 Sine, 1 
 
 Sines, Law of, 2 
 Slowness factor, 34 
 Spectacles, 26, 27 
 Standard formula, 18 
 Surface refraction, 1, 2, 125 
 Equation for, 4 
 
 Tangent, 1 
 
 Telephoto combination, focal length, 
 
 91 
 
 focal distance, 92 
 focal radius, 94 
 distance for given magnification, 
 
 96 
 
 lens, 90 
 magnification, 91 
 
 reduction factor, 97 
 Telescope, magnification, 88 
 Thick lens, 39 
 Thin lenses, in contact, 64 
 
 not in contact, 65 
 Thin-split, 42, 74 
 
 U 
 
 U.S. number, 33, 34 
 
 Use of formula --- = - 18 
 v u f 
 
 Vertex of Surface, 3 
 
 W 
 
 w, 5 
 
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