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 D. VAN NOSTRAND COMPANY 25 PARK PLACE NEW YORK SHORT-TITLE CATALOG OP JJuMkathms n* OP SCIENTIFIC AND ENGINEERING BOOKS This list includes the technical publications of the following English publishers : SCOTT, GREENWOOD & CO. CROSBY LOCKWOOD & SON CONSTABLE & COMPANY, Ltd. TECHNICAL PUBLISHING CO. ELECTRICIAN PRINTING & PUBLISHING CO., for whom D. Van Nostrand Company are American agents. Descriptive Circulars sent on request. JULY, 1912 SHORT-TITLE CATALOG OF THE Publications and Importations OF D. VAN NOSTRAND COMPANY 25 PARK PLACE Prices marked with an asterisk (*) are NET All bindings are in cloth unless otherwise noted ABC Code. (See Clausen-Thue.) A i Code. (See Clausen-Thue.) Abbott, A. V. The Electrical Transmission of Energy 8vo, *$5 oo A Treatise on Fuel. (Science Series No. 9.) i6mo, o 50 Testing Machines. (Science Series No. 74.) i6mo, o 50 Adams, H. Theory and Practice in Designing .8vo, *2 50 Adams, H. C. Sewage of Seacoast Towns 8vo, *2 oo Adams, J. W. Sewers and Drams for Populous Districts 8vo, 2 50 Adam, P. Practical Bookbinding. Trans, by T. E. Maw. i2mo, *2 50 Addyman, F. T. Practical X-Ray Work . 8vo, *4 oo Adler, A. A. Principles of Projecting Line Drawing 8vo, Vol. I. Theory of Engineering Drawing Aikman, C. M. Manures and the Principles of Manuring. . . 8vo, 2 50 Aitken, W. Manual of the Telephone 8vo, *8 oo d'Albe, E. E. F. Contemporary Chemistry i2mo, *i 25 Alexander, J. H. Elementary Electrical Engineering i2mo, 2 oo Universal Dictionary of Weights and Measures 8vo, 3 50 " Alfrec." Wireless Telegraph Designs Allan, W. Strength of Beams under Transverse Loads. (Science Series No. 19 ) i6mo, o 50 ! Theory of Arches. (Science Series No. u.) i6mo, D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 3 Allen, H. Modern Power Gas Producer Practice and Applica- tions 1 2mo, *2 50 Gas and Oil Engines 8vo, *4 50 Anderson, F. A. Boiler Feed Water 8vo, *2 50 Anderson, Cap. G. L. Handbook for the Use of Electricians .8vo, 3 oo Anderson, J. W. Prospector's Handbook i2mo, i 50 Andes, L. Vegetable Fats and Oils 8vo, *4 oo Animal Fats and Oils. Trans, by C. Salter 8vo, *4 oo Drying Oils, Boiled Oil, and Solid and Liquid Driers . . . 8vo, *5 oo Iron Corrosion, Anti-fouling and Anti-corrosive Paints. Trans, by C. Salter 8vo, *4 oo Oil Colors and Printers' Ink. Trans, by A. Morris and H. Robson 8vo, *2 50 Treatment of Paper for Special Purposes. Trans, by C. Salter i2mo, *2 50 Annual Reports on the Progress of Chemistry. Vol. I. (1904) 8vo, *2 oo Vol. II. (1905) 8vo, *2 oo Vol. III. (1906) 8vo, *2 oo Vol. IV. (1907) . 8vo, *2 oo Vol. V. (1908) 8vo, *2 oo Vol. VI. (1909) 8vo, *2 oo Vol. VII. (1910) 8vo, *2 oo Argand, M. Imaginary Quantities. Translated from the French by A. S. Hardy. (Science Series No. 52.) i6mo, o 50 Armstrong, R., and Idell, F. E. Chimneys for Furnaces and Steam Boilers. (Science Series No. i.) i6mo, o 50 Arnold, E. Armature Windings of Direct Current Dynamos. Trans, by F. B. DeGress 8vo, *2 oo Ashe, S. 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