*!■ ■mA/ BOUGHT WITH THE INCO FROM THE SAGE ENDOWMENT THE GIFT OF IS9I ME FUND i^\\\. ..la*, .a.H.3..^.1.5 Cornell University Library arV17868 A treatise on optics 3 1924 031 715 778 olin,anx Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031715778 A TREATISE OPTICS S. PAHKINSON, D.D.; F.E.S. FELLOW AND LATE TDIOB OF ST JOHN'S COLLEaE, CAMBEIDSE. FOURTH EDITION REVISED. HonUon : MAOMILLAN AND 00. 1884 {All Bights reserved.'] PKINTED BY C. J. CLAY, M.A. & SON, AT THE tJNIVEBSITY PRESS. PREFACE TO THE FIRST EDITION. The present work may be regarded as a new edition of the Treatise on Optics by the Rev. W. N. Griffin, which being some time ago out of print, was very kindly and liberally placed at my disposal by the Author. I have freely used the liberty accorded to me, and have rearranged the matter with considerable alterations and additions — especially in those parts which required more copious explanation and illustration to render the work suitable for the present course of reading in the University. The numerous diagrams which the subject requires have been inserted in the body of the work, instead of being collected in plates at the end, and are thus rendered more convenient for reference. I have appended a collection of examples and problems — distributed under the heads of the several chapters, as well as a miscellaneous set, — which are sufficiently numerous and varied in character to afford a useful exercise for the student: for the greater part of them I have had recourse to the IV PREFACE. Examination Papers set in the University and the several Colleges during the last twenty years. Subjoined to the copious Table of Contents I have ventured to indicate an elementary course of reading, not unsuitable for the requirements of the present examination of the First Three Days in the Senate-House. S. PARKINSON. Si John's CoLLsaE, October, 1859. The Fourth Edition has been carefully revised and many additions made to it : the collection of Problems in particular has received large accessions from recent Examination Papers. Si John's ColiiBob, May, 1884. CONTENTS. CHAPTER I. PAGE Laws of Propagation of Light: — Direct Reflexion and , Refraction ......... i Subject of tliis treatise, Geometrical or Formal Optics, Art. i ; Definition of self-lumirwus,medium,rays,penciloi\i^i,mcidLeriaedirectaxiAdblique, 2 — 7 ; laws of reJUxionandi refraction, 8 — lo; foundation of such laws, r I ; Def. of Geometrical and Principal Focus, focal length, aberration, 12, 13; Beflexion and. refraction a,t 3, plane surface, 14 — 19; Discussion of the Geometrical focus and conjugate foci of a pencil directly reflected at a spherical surface, 2 1 — 25 ; Geometrical and visible image, 26 ; Discussion of a pencil directly refracted at a spherical surface, 2 7—3 r ; Illustrations, — successive reflexions at parallel mirrors and inclined mirrors, 32—35 ; aplanatic surfaces, 36. CHAPTEE II. Illumination of Sv/rfaces . . .29 Measure and Zaio of illumination, 37 — 40; la'W of intensity of emission, 41; method of calculating illumination, 42, 43 ; objects appear equally bright at all distances, 44 ; Bjitobie's photometer, 45 ; Foucault's ^fto- Jomeier, 45*; example, 46 ; velocity of light, 47; Ji'oMcauZJ's Experi- ment, 47*. CHAPTER III. Aberration of small direct pencils . . .42 Measure of aberration, 48 ; Formula for aberration after refraction at a plane surface, 50, 51 ; after reflexion at a spherical surface, 52 ; after refraction at a spherical surface, 53 ; Least circle of aberration, its foiTnation, position and dimensions, 55, 56. VI CONTENTS. CHAPTEE IV. PAGE Focal lines of small ohliqti^ pencils, — Caustics . S3 Astigmatism, caustic cwrve, 57 ; mode of determining the form of a caustic, — catacaustics, diacaustics, 58 ; Def . of centres of thesur/ace and/ace,— incidence c«raiWcai and exccretnca J — oa;is of a surface, 59 ; obliqueyenoil reflected at a plane surface, 60 ; Formation of focal lines by an oblique pencil, — Def. of Primary and Secondary focal line, — circle of least con- fusion, 62 — 64; Illustration of (/comctricaJ and jjisiikimaje, 65 ; Posi- tion of the Foci of a small oblique pencil after reflexion at a spherical surface, 66 ; 'the same after refraction at a plane or spherical surface, 67 — 69 ; Position and dimensions of the focal lines and circle of least confusion, 70; Illustrations, — caustic after reflexion at a circle in certain cases, 71, 72*, caustic after refraction at a plane surface, 73. 74- CHAPTEB V. Successive refleonons aTid refractions ; — prisms — lenses . -73 Successive reflexions at plane surfaces, 77, 78 ; successive refractions at plane surfaces, — critical angle, 79 — 81 ; u. plate, 82 ; absolute and relative index of refraction, 83, 84 ; refraction through a plate, 85 — 87 ; Def. of edge, faces, reft acting angle, principal section of a prism, 88; deviation through a prism, jTiiTimumdeviatiou, 90 — 95 ; foci of a small pencil passing through a prism, 96 ; Def. of a lens, axis of a lens, forms of lenses, 97, 98 ; geometrical focus of a pencil passing through a lens, 99, 100; principal focus, focal length, of a lens, — concave and convexleus, — powcrof alens, loi — 104; a sphere regarded as a lens, — its focal length, 105, 106; relative motion of conjugate foci in the case of a lens, 107 ; focal length of a lens determined practically, ro8 ; cylindrical lenses, 108*; centre of a lens, 109; focal centres of a lens, no; Discussion of an oblique pencil passiug through a lens, 112, 113; combinations of lenses, 1 14 ; approximate determination of the focus of a small oblique pencil, 115 ; Lenses used in Lighthouses, 117. CHAPTEB VI. Eefraction through media of varying density ; — reflexion and refract/ion at a surface in any manner . . . .107 Absolute and relative retractiye index, 118; method of determining the path of a ray in a medium of varying refractive power under various CONTENTS. vii PASE circumstances, 120—125 ; Formulae for determining the course of a ray reflected or refracted at two or more surfaces in any manner, 126, 127. CHAPTEE VU. Spherical aberration of Imses ; — Eoccentrical pencils . 119 Aberration of a direct pencil, 1 29 ; form of a lens to produce minimum afteirafioM for rays parallel, — or not so, 130 — 132; eaampZes, 133; Ex- centrical pencil passing through one or more lenses — mode of defining it in simple cases, 134 — 137; equivalent lens, 138, 139; examples, Eamsden's and Huyghens' eye-pieees, 140. CHAPTEE Vin. Images and Caustics . . . -131 Image — geometrical or visible, — real or virtiial, — erect or inverted, 141 — 143 ; Example, image of a straight line formed by a lens, 144 — 146 ; Distortion, — linear, angular ajxd.tha,t oi curvature, 147; Deformation of an image relatively to the object, 148, 149 ; calculation of the distortion in the case of a spherical mirror, 150 ; Brightness of an image, igi, 152 ; length, &c. of a caustic, ^53, 154. CHAPTEE IX. On the Chromatic Dispersion 0/ Light . .148 Newton's experiment, 156 ; pure spectrum, how observed by Wollaston and Eraunhofer, 157, 158 ; fixed lines of the spectrum, 159 — i6t ; phos- phorescence andLflvxyrescence, 161"; specti"wm analysis, 162 (a), (/3), (7), (5) ; modeof determiningthere/racrtn^ angle of aprism, — iheminimum deviation of agiven colour — anitheindexof refraction, — testofthe law of refraction, 163 — 165; standard scale of colours, 166; Dei of devi- ation, dispersion, irrationality of dispersion, secondary spectra, 167; dispersionof twocoloursbyaprism, 168; dispersive power, 171; achro- matic combinations, — poss ible hut imperfect, 172, 173; conditions of achromatism for prisms, 174 — 176; for lenses, 177, 181 ; circle of chro- matic aberration, 182 — 184: ooloursof natural bodies, primary colours, 185. Table of refractive indices, 185*. viii CONTENTS. CHAPTER X. PAGE Of Vision atid Optical Instruments . .180 Tfte Mye, — retina, crystalline lens, uvea or iris, punctum ctscum, punctum luteum, 187 — 190; Binocular Yisiou, 191; Pseudoscope, 192; dimen- sions &e. of the eye, thaumatrope, anorthoseope, &c. 194 ; Defects of vision, long and short sight, 195 ; vision through a lens, 197 ; visual arapZe of anobject seen through a lens, 198; magnifying power ot alens, 200, ioi ; Telescopes, their object, 203 ; the Astronomical Telescope, its^eM of view, and magnifying power, — the ragged edge and stop, — collimator, 204 — 206; Galileo's Telescope, 207 — 210; Newton's Telescope, 211 — 213 ; Illustration of the use of Newton's Telescope, 213"; HerscAei's Telescope, 214 — 216; Gregory's and Cassegrain's Telescopes, 217 — 221 ; Defects of telescopes, 223 — 225; specula for reflecting telescopes, 226 — 228; the _/inder of a telescope, 229; object- glasses, 230; Eye-pieces, 231, 232; Astronomical telescope with Huyghens' or Eamsden's Eye-pieoe, 233 — 236 ; the erecting eye- piece, 237; cross-wires of an eye-piece, 239; Dynarmmieter, 241 — 243 ; Microscopes, 244 ; simple and compound, 245 — 248 ; Wollaston's microscopic doublet, 246; magnifying power of a microscope, 249 ; the Camera Obscu/ra, 250 ; the Camera Lucida, 252, 253 ; Hadley's . Sextant, 254^256; the Eeflecting Goniometer, 257; Brightness of the image in an Astronomical Telescope, 257*. CHAPTER XI. 0/ the Bainbow, <&c. . . . -237 Direct refraction through a sphere of water with one or two internal re- flexions, 2S9; discussion of the caustic formed byparallel rays incident upon a sphere of water and emergent after one internal reflexion, 260, 261 ; explanation of the Primary Rainbow, 262 ; the caustic after two internal reflexions vrithiu a sphere of water, 264 — 266; the Secondary Rainbow, 267; order of colours and dimensions of the rainbow, 269 272 ; halos, general remarks, 273; White Rainbows, p. 342. Examples cmd Problems . . . 2 ■; ■; For an Elementary Course the attention of the Student may be confined to the following articles : — 1—34. 37—41. 44, 45, 57, 58, 65, 74, 76—86, 88—92, 94, 95, 97— III, 114— ii7> 138 Def., 139—^43, 147—149, 186—222, ■233— 2401 244—252, 254. OPTICS. CHAPTER I. LAWS OF PEOPAGATION OF LIGHT: — DIRECT EEFLEXION AND REFRACTION. 1. When a material object is presented before us, we become by vision sensible of its existence and figure. In such a case light is said to be propagated from the object to our eyes, and the science of Optics has for its design the examination of the circumstances of such propagation. The science is divided into Geometrical and Physical Optics. In Geometrical Optics the circumstances of the transmission of light are computed on certain laws estab- lished by experiment ; in Physical Optics these laws are ac- counted for on hypotheses of the structure of bodies, and of the matter filling the space in which they are placed. In a similar manner, in geometrical or plane astronomy the phenomena of heavenly bodies are calculated on observed laws which their apparent motions are found to obey; in physical astronomy these apparent laws are shewn to result from the hypothesis of gravitation. The former branch of the science is the subject of the present treatise, wherein from certain laws established by experiment under simple circumstances the course of light under more complex circumstances is computed, and the results applied to the construction of optical instruments. These investigations will be conducted in independence of the physical branch of the subject — the experimental laws on which we commence being equally true, whatever be the nature of the hypothesis which professes to account for them. 2. Def. A body is said to be self-lvminous when it is capable in itself of making our eyes sensible of its existence. P. o. 1 2 LAWS OF PROPAGATION OF LIGHT. Thus the sun, the stars, a lamp, a red-hot iron, &c., are self-luminous, but by far the greater number of natural bodies possess no such property. Such bodies are luminous only by reflexion and require the presence of another lumin- ous body to render them visible. Thus when a lamp is brought into a dark room, the other bodies in the room become visible, and become more or less capable of illumi- nating others ; but this property ceases and they become in- visible when the lamp is extinguished or withdrawn. Ohs. When a luminous point or an origin of light is mentioned, we may understand it to be a minute portion of a luminous surface, in no direction perceptibly extended : and so by a. luminous line, we may understand such a surface perceptibly extended in one direction only. We shall regard them simply as &, mathematical point, and. a mathematical line respectively. 3. JDef. Any space or substance which light can traverse is called a medium,. Such as a vacuum, glass, water, &c. In this Treatise we shall consider only media which are not crystallized. 4. Def. When light emanates from a luminous point, we regai'd it as made up of rays, understanding by a ray 'the smallest portion of light which can be separately transmitted, stopped, or reflected; and we shall treat such rays as mathe- matical lines. In a uniform medium, it will be assunied that the course of a ray of light is a straight line; this law is shewn by experiment to be true in general. There are certain cases of apparent exception, such as are presented by the phenomena of diffraction, the explanation of which belongs to physical optics. 5. Bef. An assemblage of rays proceeding from a luminous point, is a pencil of light. The form of a pencil, unless an exception be expressly mentioned, will be regarded as a right cone having the origitt of light for the, vertex, and when this origin is infinitely distant, this cone becomes a circular cylinder as its limiting form. The geometrical axis of the cone or cylinder is called the axis of the; pencil. LAWS OF PROPAGATION OF LIGHT. 3 If the rays of a pencil of light produced in a direction apposite to that of propagation meet in a point, the pencil is divergent; if the origin be infinitely distant, its limiting form is a pencil of parallel rays;' if the rays produced in direction of propagation meet in a point, the pencil is convergent. The degree of divergence or convergence may be measured by the vertical angle of the cone which the pencil forms. 6. Def. When a pencil meets the surface of any sub- stance or medium, the incidence is called direct, if the axis of the pencil coincides with the normal to the surface at the point of incidence : in other cases the incidence is called oblique. 7. When light is incident on the surface of a medium different from that in which it is proceeding, a portion of it is scattered or dispersed over the surface and makes the surface visible ; another portion is in general reflected in the first medium (according to a law which will presently be stated) — and in certain cases (as when the medium upon which it is incident is translucent) a third portion enters this new medium according to another law, and is said to be refracted. TKhe course of the reflected and refracted rays where both exist, may be considered separately. 8. Lajw of Reflexion. When a ray is reflected at the surface of a medium, (i) The incident and reflected rays lie in onq and the same plane with the normal to the surface at the point of incidence, and on opposite sides of the normal. (ii) The angles which the incident and reflected rays make with the normal to the surface at the point of inci- dence are equal. 9. Law of Refraction. When a ray is refracted at the surface' of a medium, (i) The incident and refracted rays lie in one and the same plane with the normal to the surface at the point of incidence and on opposite sides of it. 1—2 4 LAWS OF PROPAGATION OF LIGHT. 'ii/V-'^ (ii) The sines of the angles which the incident and re- fracted rays make with the normal to the surface at the point of incidence, have a ratio depending only on the media between which the refraction takes place, and the nature of the light. 10. Thus if a ray QA be incident at A on the sur&ce of a medium (reflecting or refracting or both): WAN' the normal to the surface at A; AQ', AQ" the re- flected and refracted rays respec- tively; then will the lines. QJ., Q'A, Q"A, NAN' all lie in one plane, and the ^ QAN= z Q'AN. Also, if we write / QAN= , ^ Q'AN' = ^' then will the ratio -^ — v, be con- sm^ stant at whatever angle of incidence a ray of the same kind of light is refracted from one given medium into another. This constant quantity -^~, is usually denoted by the symbol jM, and is called the refractive index between the two media for the particular species of light considered. It is a parameter which varies, (i) if the nature of the light be altered, (ii) if the relation between the two media be altered. The relation between ^ and ^' is commonly written sin (fj = fi sin (f>'. i : It will in general be supposed in these pages that the refraction takes place into a denser medium, in which case /j, is greater than unity, and the angle of refraction less than the angle of incidence {i. e. (^' < 0}. 11. Remark. The process by which these and physical laws in general are established experimentally is this. Direct experiments render the law probable; such experiments however are seldom made with such minute accuracy as to prove the law exactly true;— next on supposition of the truth of the law in question, DIRECT REFLEXION AND REFRACTION. 5 the circumstances of more complex phenomena are computed, and when the results of these computations are found in re- peated instances of various kinds to agree minutely with observations, we have a very high degree of probability of the strict truth of the law. It is important to observe that of the truth of the laws of physical science we have only a moral certainty, — a certainty arising only from the improba- bility that an untrue principle should happen to explain suc- cessfully a great variety of phenomena. " The first law of motion in Dynamics, for example, we think probably true from experiments made on bodies on the Earth; it is proved by the agreement of the motion of the heavenly bodies cal- culated on supposition of its truth, with the motions which they are observed to have. On such foundations all the laws of natural philosophy rest. 12. Various illustrations of the law of reflexion have been given, more or less accurate ; perhaps one of the high- est confirmations of it is derived from the accordance of Transit observations of Stars made by reflexion at the surface of mercury with results obtained independently of reflexion. Of the law of refraction we shall speak hereafter when we come to explain the mode of determining the refractive in- dices of different substances. The law of reflexion seems to have been known in very early times as we find it laid down in the earliest writers on Optics ; the law of refraction was first accurately ascertained by Snell, about A.D. 1621, though it was first published by Descartes. 13. Direct Reflexion and Refraction. Def. Let Q be the origin of a pencil of rays whose axis QA is incident directly (Art. 6) at >4 on a plane or spherical reflecting or refracting surface. Then QA is the axis of the reflected or refracted pencil. Let QR be any ray incident at R whose direction after re- 6 REFLEXION AND EEPRACTION flexion or refraction (produced if necessary) cuts the axis in q, since the normal at li lies in the plane QAM. Then as It is taken nearer and nearer to A, the point q will a,pproach some point F in the axis, as its limiting position ; and by taking R sufficiently near to A the distance Fq may be made less than any assignable magnitude. This limiting position (F) of the point q is called the Geometrical Focus of the reflected or re- fracted pencil. The point F will in some cases be nearer the surface than q is ; in others more remote. Bef. The principal focus of a spherical reflecting or re- fracting surface is the geometrical'focus of a pencil of parallel rays incident directly upon the surface parallel to a fixed diameter, called the axis of the surface. Def. The focal length of a spherical reflecting or refract- ing surface is the distance between the ^surface and the prin- cipal focus,- — measured along the axis. The distance Fq is called the aberration of the ray Bq. If Ft be drawn perpendicular to the axis QA to meet Eq in t, it is sometimes conyenient to distinguish Fq as the longi- tudinal aberration and Ft as the lateral aberration. 14. A pencil of parallel rays will consist of parallStrays after reflexion at a plane surface. Let QR, Q'R' be any two rays of a pencil of parallel rays incident on a plane reflecting surface at the points R, R'. Let RS be the di- rection of the ray' QR after re- flexion. Draw RN, R'l^' perpen- dicular to the surface at the points R, R', and (i) if the planes QRN, Q'R'N' are coincident, draw R'S' in this plane parallel to R8; then since™ zN'R'S' = -NR8 = are parallel to \pq> / QRN=^ Q'R'N', AT A PLANE SUEFACE. and therefore R'S' is the direction of Q'E' after reflexion^ but (ii) if the planes QRN, Q'B'N' do not coincide, let R'S' be the intersection of the plane SER' with the plane Q'B'N'. Now ^^1 are parallel to {^; J; .-. the plane QRS is parallel to the plane Q'E'8', (Euc. xi. 15), St. liaeES st. line R'8', (Euc. XI. 16). Also because „ „| are parallel to ] ^, A^, ; .-. ^ QRN=^ QE'N', (Euc. xi. 10). Similarly, ^ SBF = ^ S'E'N'. But, ^ QBN= ^SEN, (Art. 8), .-. ^Q'R'N' = -8'R'N', and R'S' lies in the plane Q'R'N' ; therefore it is the direction of the ray Q'B' after reflexion ; and it has been proved to be parallel to ES. But QR, Q'E' are by supposition any two rays of the incident pencil; therefore the reflected pencil consists of parallel rays. ] 5. A pencil of parallel rays consists of parallel rays after refraction at a plane surface. Let QE, Q'R' be any two rays of a pencil of parallel rays incident on a plane refracting surface at the points R, R. Let SB be the direc- tion of the ray QR after refraction. Draw RN, R'N' perpendicular to the surface at the points R, R', and let S'B' be the intersection of the plane SRR' with the plane Q'R'N' (if these planes be not coincident). Now ^y[ are parallel to \n,j^, '> .-. the plane QES is parallel to the plane Q'R'S' ; .-. the straight line SR is parallel to the straight line S'R'. REFLEXION Also because p ^V are parallel to ■! p/ tit/ ; Similarly, .-. ^ QIiN= ^ Q!EN'. ^ 8BN=^ S'B'N'; = /isia(Si2iV = lj, sin S'B'N'. And 8' B' lies in the plane Q'B'N', therefore S'B' is the direction of the ray Q'R' after refraction, and it has been proved parallel to SB. If the planes SBB', Q'B'N' be coincident, and S'B' be drawn in this plane parallel to SB, it may readily be shewn that S'B' is the direction of Q'B' after refraction. Now QB, Q'B' are any two rays of the incident pencil,' therefore the refracted pencil consists of parallel rays. Bemarh. In the above figure the dotted lines ;Si^, S'B' are the directions of the refracted rays produced backward : it may not be amiss to suggest to the student that he will often avoid confusion in optical diagrams if he indicates by dotted lines any part of the directions of rays which are not actually traversed by the rays.' 1&. A pencil is incident directly upon a plane reflecting surface, to find its form after refleadon. Let Q be an origin of light from which a pencil whose axis is QA is incident directly at -4 on a plane re- flecting surface, QB any ray of the pencil incident at B, and reflected in direction BS in a plane passing through QR, and BN the normal to the surface at B. Since by the law of reflexion SB, QA lie in one plane, let them be produced to meet in q. Then since BN, qQ are parallel AT. A PLANE SUEFACK 9 ^ RQA = ^ QRN = ^ i^RN (Art. 8) = ^ RqA ; therefore the angles j^^ \ are equal to j^ „ each to each, and RA is common to the two triangles ; .■.Aq = AQ. Now QR is any ray of the incident pencil ; therefore the directions of all the rays of the reflected pencil (produced backward) pass through the point q. Hence the form of the reflected pencil is that of a cone whose axis is A Q, and vertex the point q — equidistant with Q from the reflecting surface, and on the opposite side of it. Cor. 1. Since the angles RqA, RQA are equal, the divergence of the incident and reflected pencils is the same. CoE. 2. If the incident pencil be convergent, a similar investigation will shew, that it will converge after reflexion to a point equidistant from the surface with the point of con- vergence of the incident pencil, and on the opposite side of it, — the degree of convergence being unaltered. CoK. 3. The direction of the ray QR after reflexion cuts the axis in q, and the same is true in the limit when R moves up to A ; therefore q is the geometrical focus of the reflected pencil. 17. The succeeding cases of direct reflexion and refrac- tion have greater difiSculty than the last investigation because in none of them will the pencil after reflexion or refraction pass accurately through a point. Our attention will at pre- sent be confined to plane and spherical surfaces. A pencil whose origin is Q and axis QA incident directly on a plane or spherical reflecting or refracting surface may be regarded as composed of a series of conical surfaces or shells of rays, as QRr, with a common vertex Q and com- mon axis QA. The rays of this conical surface will all be reflected or refracted similarly with respect 10 . CONJUGATE FOCI. to QA, and therefore their directions after reflexion or refrac- tion will form another conical surface with vertex q and axis Aq. Thus the reflected or refracted pencil will consist of a series of conical surfaces with a common axis, but dif- ferent vertices. The limiting position of the vertex q is the geometrical focus of the pencil. (Art. 13.) 18. The determination of the form of a direct pencil after reflexion or refraction will consist of two parts : (i) The determination of the geometrical focus. (ii) The determination of the vertex q of any one of the cones of rays above described. We will confine ourselves in the present chapter to the first of these parts — reserving the second for a subsequent chapter. Obs. As it is obvious that a ray of light would traverse the same path hacJcward if its course were reversed m any medium, it will be easily seen that Q (fig. Art. 13) would be the geormtrical focus of a pencil of rays emanating from F the geometrical focus of Q. In this point of view, the origin of a pencil and its geometrical focus after reflexion or refraction are convertible, and we shall^ for brevity, sometimes speak of them as conjugate foci. 19. _ To find the geometrical focus of a pencil after direct refraction at a plane surface. Let Q be the origin of a pencil whose axis QA is incident directly aA, A on & plane refracting surface, then QA is the axis of the refracted pencil. Let QE be any ray incident at R and refracted in a direction which, when produced backward, cuts the axis in q. Let F be the geometrical focus, or limiting posi- tion of q. Let AQ = u, AF= v, lines being considered positive when measured from A in a direction contrary to that of the incident light. EEFLEXION AT A SPHERICAL SURFACE. 11 Now RQA, RqA being equal to the angles of incidence and refraction of the ray QR, sin RQA = fx.sva. RqA ; AR_ AR - ■ RQ~'^- Rq' .•.Rq = H.RQ. In the limit when R moves up to ^, and q to F, AF = /i . AQ, or i; = /j,u, which formula gives the position of the geometrical focus of tlie refracted pencil. 20. Ohs. In this proposition the pencil is considered divergent, as will be done in other cases, but attention to the algebraic sign of u will make the result applicable to a con- vergent pencil. In such a case u is negative and ^) = yu.M = a negative quantity, indicates that the geometrical focus lies in a negative direction, i.e. behind the surface, at a distance from it determined by the numerical value of fiu. The stu- dent will have little difEculty in drawing a suitable diagram and investigating the case independently. 21. To find the geometrical focus of a pencil of rays after direct reflexion at a spherical surface. Let Q be the origin of a pencil of light whose axis QAis incident directly on a spherical reflecting surface of which is the centre : then AQ is the axis of the reflected pencil. (Art. 17.) Let QR be any ray incident ^i**^ at R, and reflected in direction Rq, cutting the axis in q; F the geometrical focus. Join OR. Let AQ = u, AO = r, AF= v, lines being considered posi- tive when measured from ^ in a direction contrary to that of the incident pencil. Now / QRO = ^qRO; .-. §- =^- .• (Euc vi. 3), 12 REFLEXION AT A or ultimately when B moves up to A, and q to F, AQ.OF^AF.OQ, or u(r — v)=v (u — r) ; 1_1^1_1 ' ' V r r u' .-.i+i-?, V u r a formula which determines the position of F. Cob. 1. If M be indefinitely great, or the incident pencil consist of parallel rays, the formula becomes 12 r - = - , or ?; = - , V r 2 which assigns the position of the principal focus of the re- flector, viz. at a point on its axis equidistant from the centre and the, sMr/ace. If F^ be the principal focus, AF^ = F^O = ~. 112 CoE. 2. Since - + - = - , we get by reduction V u r ° •' tiv-(u.+ v)'^ = 0; and.-. (« - Q (^ " 1) = (^ If jP, be the principal focus this result may be written QF^.FF^=^AF^\ or FF, : AF^ :: AF^ : QF^, a geometrical form of the relation which exists between the positions of the con^M^faie/ocz Q,.F. 22. The geometrical result of Art. 21, Cor. 2, can easily be obtained independently ; thus, suppose a ray Q'E incident parallel to the axis QA to be reflected in direction Bq (these lines Q'B, Bq' are not drawn in the figure), then in the two triangles Qq^B, Bq'q, we have "^ / q;Bq=^ QB(J=^BQq', SPHEEICAL SURFACE. 13 and ^ Rqq is common to the two triangles, which are there- fore similar, and we have q<^:q'R::qR: V = = . r. CoE. 2. To compare the degree of divergence of the pencil afier and before refraction we must take the limit of the ratio ■ RqA : ^ R QA when R approaches A ; ^, . ^. sinEff^ li- i 1 -SQ u- 4. 1 AQ u this ratio = -. — 57— r ultimately = -rj- ultimately = -r-^ = - . sm RQA •' Rq •' AF v If this result is negative it shews that the incident and re- fracted pencils are one divergent and the other convergent. 28. 01)s. The remarks in Art. 23 in the corresponding problem of reflexion will, mutatis mutandis, apply in the case of refraction of Art. 27 ; and the student is recommended as an exercise to draw the diagrams and go through the investi- gation for each case that can occur, — with the surface concave or convex and the incident pencil divergent or convergent. 29. The following mode of expressing the position of the geometrical focus with respect to the centre of the refracting surface is sometiines convenient. To find the distance of the geometrical focus of a pencil, of rays from the centre after direct refraction at a spherical surface. SF SPHERICAL SURFACE. 19 Let Q, be the origin of a pencil of light whose axis Q J is incident directly at ^ on a spherical re- fracting surface whose centre is 0: then QJ. is the axis of the refracted pencil. Let Qfl be any ray, incident at R and refracted in a direction which cuts AQvuq^; F\h& geometrical focus. Let OQ ='p, OA = r, 0F= q, lines being considered positive when measured from in a direction contrary to that of the incident pencil. _ sin QRO _ sin QRO sin ROq _ OQ.Rq '^ ~ sin qRO~ sin ROQ' sin qRO~RQ. Oq' In the limit /* = jq~~(JF' or, fi.AQ.OF=OQ.AF; i.e. iJi.q {p - r) =p {q - r), '^ \r pj r q " q p r ' a formula which determines the position of F with respect to 0. Obs. The result of this article is very convenient in the case of refraction through a sphere (see Art. 105). 30. To trace the relative change of position of the con- jugate foci of a refracted pencil. Suppose the refracting surface concave, then the formula conuecting the positions of the conjugate foci is fX, I _fi —1 V u r ' 2—2 20 MOTION OF CONJUGATE FOCI. where the lines v, u, r are supposed positive to the right. It is clear from the formula that if u increases or decreases, V must also increase or decrease, i.e. Q, F always move in the same direction. (i) Suppose M = 00 , i.e. the incident rays parallel, then v = /.-I r. and ^coincides with F^ the principal focus. (ii) As Q moves from an in- finite distance up to 0, F also moves up to 0, and Q, F will coincide at 0. Since when u = r, V also = r. (iii) As Q moves from up to A, F also moves from up to A, and Q, F coincide at A. (iv) When u becomes nega- tive, or the incident pencil conver- gent, let i^,, be a point such that AF = /.-I' then whilst Q moves from A to i'],, F will move from -il to — oo . (v) And lastly, when A Q be- comes > — — , V becomes posi- tive, and whilst Q moves from F^^ to —CO , F will move from + 00 toi^^. Similarly, the change of relative position may be traced when the refracting surface is convex. PARALLEL MIRRORS. 21 The remarks of Art. 26 apply also to this case. 31. The reflecting or refracting surfaces have been con- sidered spherical in the preceding articles, because such are the surfaces which generally occur in the construction of optical instruments. When the surface is any other figure of revolution, it is in general sufficient, when the pencil is not very large, to consider the surface the same as the sphe- rical surface generated by the circle of curvature at the vertex of the generating curve. The case of a reflecting paraboloid of revolution is how- ever of some interest, inasmuch as its properties are sometimes employed in large telescopes, like that of Lord Eosse. Illustrations and Examples. 32. A luminous 'point being placed between two parallel plane mirrors, to find the position of the images formed by suc- cessive reflexions at the mirrors. Through the luminous point Q let the line AQB be drawn perpendicular to the two mirrors A, B and produced indefinitely both ways. Then taking AQ^ = AQ, Q^ will be the geometrical focus or image of rays emanating from Q and reflected at the mirror A. The rays so reflected from A and diverging from Qj will be incident on the second mirror B, and if we take BQ^ = BQ^, Q^ will be the focus of the rays after reflexion 22 PARALLEL MIREOES. from B. The rays will then be incident upon A, diverging apparently from Q^ and be reflected at A, from a point Q^ such that AQ^ = A Q,^ and so on. Again, the rays diverging from Q and incident on the second mirror B will have a geometrical focus or image in Q', where BQ =BQ'; the rays diverging from this point Q' and incident on the first mirror A, will after reflexion at A diverge from a point Q" such that AQ" = AQ', and so on. Thus there are two sets of images, each infinite in number, all arranged on the line AB and becoming more and more distant after each reflexion. (The second set are not put in the figure to avoid confusion.) The distances Q Q^, QQ^. . . may be easily calculated. For putting QA = a, QB= b, AB = a + 5 = c, we have QQ, = 2AQ = 2a, QQ^ = BQ + BQ^ = QQ^ + 2BQ = 2a+2b = 2c, QQ, = AQ + AQ^ = QQ^+ 2AQ= 2c + 2a, QQ,= ... = 4c. And generally, QQ,„^, = 2«c+2a.' Similarly, for the second set of images, we should have QQ' = 2b, QQ"=2c, QQ'" = 2c + 2b, QQ- = 4c, (3Q'^"' = 2mc, QQ<'"'+" = 2nc + 26. If Q is midway between A and B,a = b, and the two sets will form a series which are successively at equal dis- tances (= c) from each other. Obs. The diagram will be sufficient to indicate the man- ner in which the images are seen by an eye in a given position, the small pencil received by the eye passing be- INCLINED MIRRORS, 23 tween the reflexions in the same manner as if it diverged from the successive images. A familiar illustration of the above may be noticed in a drawing-room where there are two mirrors on opposite walls, parallel to each other, — an interminable series of images of the objects in the room is observed. 3.3. A luminous point being placed between two plane mirrors inclined to one another at a given angle, to find the position and number of the images formed by successive re- flexions at the mirrors. Let the figure represent a section of the mirrors AjBhya, plane passing through the luminous point Q, and perpendicular to each mirror, and therefore perpendicular to their line of intersection — which is represented by the point 0. Describe a circle with centre and passing through Q. Draw QQj perpendicular to the first mirror A, and meeting the circle in i Qj. Then Q, Q^ are equidistant \ from the mirror A, and Q^ is the \. geometrical focus of rays diverging QaX:; from Q and reflected at the mirror A, i.e. Qi is the image of Q form- ed by reflexion at A. Draw Q^Q^ perpendicular to the second mirror B, then in like manner Q^ is the image formed by rays reflected from B, after reflexion at A ; the course of the rays between the two reflexions at A, B being the same as if they diverged from Q^. -Similarly drawing Q^ Q^ perpendicular to ^4 or ^ produced, we obtain a third image Q^ formed by rays reflected at J., ^, and again at A, and so on. Thus we get a set of images formed by ra,ys first reflected at A, then at B, again at A, and so on in succession. In like manner we should get a second set of images formed by va,js first reflected at £, then at A, again at B, and so on in succession. To avoid confusion we have not marked this second set of images in the figure. 24 INCLINED MIRROES. To determine the position of these images, let arc AQ = a = ^ QOA, ^QOB = ^, ^AOB = B = a+^. Then, QQ^ = 2QA = 2oi, QQ^ = AQ+AQ = QQ, + 2AQ= 28 + 2a, And in general in the first set of images QQ,„=2r»S, QQ,„,, = 2nS + 2a. Similarly in the second set of images Q', Q" ... QQ""' = 2«S, QQ""'^'^ = 2nS + 2^. The number of images is in this case limited ; for when we arrive at an image within the ^aOb, — i.e. the -^ AOB pro- duced backward, — the rays which proceed after any reflexion as if they emanated from this image — (Q^ in the figure), — can never fall upon either mirror again, since their directions have already intersected the plane of each mirror, — and conse- quently cannot be again reflected. (i) Suppose Q^^ the first image which falls within arc ah, then QQ^^ = 2nS must be not < arc QBa, but < arc QBh ; i.e, 2«S not < TT — a, but < tt + y3. (ii) Suppose Q^^^i the first image which falls within arc ah, then QQ^^^^= 2nh:+ 2a must be not < arc QAh, but < arc QAa; i.e. 2nS + 2a not < tt — /S, but < tt + a, or remembering that a + /3 = S, (2m -(-1)8 not Kir — a, but <7r+/3. These limits are the same, so that if the ?-"" is the first image which falls within ab, or the last of the first set of images, we have , TT — a T ^ -rr + S r not < ~^^ , but < — ^ ; ILLUSTRATIONS. 25 i.e. r is the integer which lies between ^^^^ and "^ ^ , or o o if — g — happens to be an integer, r is equal to this integer ; and we may remark that as the difference of these limits is equal to unity, there is only one value of r, and therefore no ambiguity. Cor. 1.' There will also be a number r' of images in the second set, and therefore r + r' images in all: r and / will either be equal, or differ by unity at most. buppose a > /3, then —g— , — ~ , — -^ , — k— are m order of magnitude, and if there is no integer between TT — yS IT + B — ^ — and — K — , then r ~r' = 1, but if there is an integer between — ~ and — ^ — , then r = r. o Cor. 2. If B be an exact submultiple of tt, then will 5" = -^ , and the last iniages of each set will coincide, — and the total number of images will be 2r — 1. A simple and well-known illustration of this last case (i.e. Cor. 2), is afforded by the Dehuscope, which consists of two small plane mirrors inclined at 90° to each other, so that S = -^ and therefore r = 2. Hence of any object placed within the quadrant included by the mirrors, the instrument presents three images symmetrically arranged in the three remaining quadrants. As an illustration of the general case, the student may discuss the following. Example. A luminous point moves about between two plane mirrors which are inclined at an ^ 27°. Prove that at any moment the mvmber of images of the point is 14 or 13 according as the angular distance of the point from the nearer mirror is greater or not greater than 9°. 26 APLANATIC 34. When an object as AB is viewed by reflexion at a plane mirror by an eye in any posi- tion, the image is seen by pencils which enter the eye as if they pro- ceeded directly from the image, and this image will appear in the same position whatever be the position of the eye; since all the rays pro- ceeding from any point of the object are reflected accurately from a point. But there is a curious perversion with respect to right and left of the relative position of the parts of the object and image. Thus when a person views himself in a looking-glass, the image of his right hand is apparently what would be the left hand of a man standing in the position of the image. This is not much remarked in consequence of the general symmetry of the right and left sides of a man. As a simple illustration of this perversion, observe the image "of a page of a book held before the mirror : or again, write a few words on a sheet of paper, take off the superfluous ink on a sheet of blotting paper, and hold the blotting paper in front of the mirror, when the original writing will be seen in its original order. 112 35. The relation - -f- - = - (Art. 21) which connects the V u r ^ ' conjugate foci of a reflected pencil expresses the law of re- flexion more directly than may at first sight be supposed. Since AB, ( = y) ultimately vanishes, it is very small com- pared with the other lines in the figure, M, V, r. Hence we may (neg- lecting cubes of small quantities and therefore putting sin 6 = 6) write / RqA = ^, ^ BOA=^ , ^MQA=^, and since ^ QB 0= ^ OBq, SURFACES. 27 .-. ^RqA-jLROq = ^ qUO= ^ ORQ = ^ EOA - ^ RQA ; ' ' V r r li' V r r u' 112 or - + - = - . V u r This equation which is approximately true when y is small, becomes strictly true in the limit when y =0. In a similar manner it may be shewn that the relation i^ _ i =^^ (Art. 27), when put in the form y-y=^{y-y\, expresses simply the law of refraction, sin0 = /isin ^', or , Si, If, — fi the refractive index from the first medium into the second. Draw Pn perpen- dicular to 8Q, and Pr perpendicular to HQ produced. Then we may regard Qn as the increment of SP, and Qr* as the decrement of HP, in passing from P to Q ; one of the two lines SP, PH being increased and the other diminished, since the normal at P must fall between 8 and H. cj), (j}' the angles of incidence and refraction at P, and PQ so small as to be regarded as coincident with its chord. Then S.SP= Qn= PQ.sm, 8 . HP==-Qr = -PQ. sin f ; .■.S.8P + fi.B.HP = PQ(sm(l>-fj.sia(l)') = 0, in the limit when Q moves up to P. 28 APLANATIG SURFACES. Hence, integrating /SiP + /t . -HP = (7 a constant (i). This equation is that of a curve which by revolution about 8R generates a surface such as' is required; and which is commonly called an aplanatic surface. Cor. 1. If the rays diverging from 8, are to diverge from a point if after refraction;"the curve required would be SP-i^.HP^C (ii). Cor. 2. If the surface is required which will reflect rays proceeding from 8, accurately to or from S, we should obtain for the forms of the generating curves SP+ HP = G, and SP — HP= G respectively, by proceeding in a manner similar to the above. These results can of course be obtained independently by direct investigation. They represent respectively an ellipse and a hyperbola ; and it may be easily shewn inversely that rays diverging from one focus of an ellipse converge to the other focus after reflexion at the curve, and that rays diverg- ing from one focus of a hyperbola will after reflexion at the curve diverge from the other focus. A particular case of either of these, is a paraboloid of re- volution in which rays incident on the concave side of the surface and parallel to the axis will be reflected to the focus, and vice versa. This property of a paraboloid has been em- ployed in forming specula for reflecting telescopes. Cor. 3. If the equation SP—fj, . HP = G were expressed in rectangular co-ordinates, it would give an algebraic curve of the fourth degree. In the particular case of (7 = 0, the surface will be a sphere. The discussion of aplanatic surfaces was originally pursued by Newton and Descartes, — in fact the class of curves included in the equation (i) are frequently called Cartesian ovals ; but with the exception of paraboloids referred to in the previous corollary, they are now become questions of curiosity. See Newton's Principia, Bk. I. § 14, Prop. 97. CHAPTER II. ILLUMINATION OF SURFACES. 37. When a pencil of light emanates from a luminous point and is propagated in a uniform medium if we suppose its intensity unaltered by the absorption of any portion of it by the medium, yet from other causes the illumination at any point of a surface exposed to the light is different in different positions and at different distances from the surface. 38. Def. The illumination at any point of a surface exposed to light is measured by a quantity /: ^Iic being the amount of illumination of an indefinitely small area k of the surface contiguous to the point in question, — some standard degree of illumination being referred to as a unit. Hence, if the same quantity of light fall on two very small areas, — Ik being the same for each, — the illuminations at any point of these areas are inversely as the areas. 89. When a small plane area is illuminated hy a pencil of rays emanating from, a point, the illumination at any point of the area cosine of angle of incidence (distance from origin)" Let Q J. 5 be a small conical ^„ pencil of light from an origin Q, g ^ AGJB, aCb a circular and oblique "''^^---——^ ____^ /Wc section of it through a point G "^^ in the axis of the pencil. liaCb be a small plane area illuminated by the pencil, (i) In all sections parallel to aCh, the quantity of light in the pencil being the same, the illumination at a point is inversely as the area of the section (Art. 38) ; i.e. inversely as (dist.)^ from Q, 30 METHOD OF MEASURING (ii) In all sections through at different inclinations to the axis, the quantity of light received being the same as that received \>y ACB, the illumination varies as the area inversely. But if the pencil be supposed so small that A OB may be regarded as the orthogonal projection of a (76, „, area A CB area a (Jo = ^^r- , cos ^ AUa .: illumination in a C6 = illumination in A CB. cos z ACa X cos z A Ga, and ^ AGa being the inclination of the planes AGB, aGb, is the angle between the perpendiculars to these planes at C, or is the angle of incidence of QG. Hence, when both the distance and angle of incidence vary togethei*, — the illumination at any point of the area cosine angle of incidence (distance)'' CoE. The illumination at any point of the area _ „ cosine / of incidence (distance)'' • ' where G depends only on the brightness of the illuminating point, and is the illumination at any point of a small area directly exposed to the pencil at a distance tmity from the origin. 40. Remark. The preceding result will apply to a curved surface illuminated by a pencil of light, since we may regard a very small portion of the curved surface contiguous to a given point on it, as coincident appreciably with the tangent plane to the surface at the point considered : — we may also regard the intrinsic brightness of any luminous surface as proportional to the amount of light emitted by a unit of area normal to the surface. 41. It is found by experiment that luminous surfaces appear to be of the same brightness at any point, whatever be the inclination of the surface at that point to the axis of the pencil by which it is seen. The Sun's disc, for example, is equally bright at all distances from the centre. The apparent ILLUMINATION. 81 intrinsic brightness of a bar of red-hot iron is not sensibly altered by inclining it obliquely to the eye. Thus if A'B be a small portion of a luminous surface viewed obliquely by an eye JE, the amount of light received from it will be the same as that received from a portion AB viewed di- rectly : AB being the section, perpendicular to the axis of the cone, of which E is the vertex and A'B the base ; i.e. the intensity of emission of light in direction A'E : intensity of emission perpendicular to the surface at A' :: area AB : area A'B = smd : 1, where is the angle which EA' makes with the luminous surface at A'. Hence it follows that the copiousness of emission of light from a luminous surface is proportional to the sine of the angle of emanation from the surface. 42. The following is a mode of calculating the illumina- tion at any point of a surface illuminated by a given surface of uniform brightness. Let ^ be a small plane area illuminated by a surface BG of uniform brightness. About A as centre describe a sphere, and let a line through A moving round the boundary of BC intersect the surface of this sphere in the curve be. Take also an element P of the surface BG, and let p be the corresponding element of the spherical surface formed as before. Let a, be the 32 ILLUMINATION. inclinations of AP to the illuminated plane and to the surface at P. If the element at P be regarded as an origin of light, illu- mination at A from it = C ^^ . area P . sin ^ (39, 41), and if the surface of the sphere be supposed of the same uniform brightness with B C, and the element at p be considered an origin of light, the illumination at A from it would be „sina = U — r-3 area p. Ap' ^ _ area P. "sin ^ areap ^""^ — aP — ^-Jf- i.e. the illumination at A from corresponding elements of BG and he would be equal, and therefore the illumination from BG would be the same as that from he. But the surface of the sphere at p being inclined to that of TT ^ at an ii = - — a, it follows that area ^ . sin a = area of projection of p on the plane of A ; .'. illumination at A from p G = -—.^ . area of projection oip on the plane of A ; .-. illumination at A from -BC= illumination at A from he (J = -j— 5 . area of projection of he on the plane of A. If therefore the area of the projection of he can be found the illumination at A may be known. Ex. If A be illuminated by a sky equally bright in all directions above the horizon, area of projection of 6c = tt . Ap' ; :. illumination at J. = irG. ILLUMINATION OF SURFACES. 33 43. Remark. If the illuminated surface be curved, then the small area A must be regarded as a small element of the tangent plane to the illuminated surface at A. The student who is acquainted with solid geometry will have little difficulty in applying the above process by means of the usual systems of co-ordinates to any case that may occur. If the illuminating surface BGhe not of uniform bright- ness, the quantity G which occurs in the above investigation will not be constant but will be a function of the position of P on the surface BG, — and the resulting illumination at A will in general have to be determined by integration. 44. Objects appear equally bright at all distances. The apparent intrinsic brightness of an object may be measured by the amount of light received from it by the eye, divided by the area of the picture on the retina of the eye. But this area is proportional to the apparent superficial magnitude of the object; i.e. to its real area A divided by the square of its distance D, or x -=^^, — moreover the apparent AG light or illumination is ' , where G is the real intrinsic brightness. Consequently the apparent intiinsic brightness A.C _ A j± . ij .a. „ which is not dependent on A or D. The apparent intrinsic brightness is therefore the same at all distances, and is pro- portional to the real intrinsic brightness of the object. This conclusion is usually expressed by saying that objects appear equally bright at all distances, which must be understood only of apparent intrinsic brightness, and the truth of which supposes no loss of light by absorption to take place in the media traversed. 44*. To compare the intrinsic brightness or illuminating power of two sources of light at moderate distances, we may cause the rays emitted by each to fall separately on two P. o. ^ 34 RITCHIE S PHOTOMETER. screens physically identical in character in a direction normal to the surfaces or pretty nearly so : the distances may then be varied until the illuminations are sensibly equal. Thus if A be the area of the projection of one luminous surface (a) on a plane perpendicular to the direction of the luminous rays, I) the distance of a from the screen, G the intrinsic brightness of a, the illumination on the screen may GA be measured by -^ ; if C", A', D' be similar quantities for another source of light yS and if the distances be varied until the illuminations from the two sources are equal we shall have GA G'A! -=^ = ■ ^,., , which will enable us to determine the ratio of Jr I) G:G'. 45. Various instruments have been constructed for com- paring the illuminating powers of two sources of light : it will suffice to describe two of them. Ritchie's Photometer consists of a rectangular box, about two inches square, open at both ends, of which ABGD is a A JS F C ^ /~ V Ji ^A^^^? -a, 1 /'■■■"■ J'\ \ /:■ \/ P // section. The inner surface is blackened so as to absorb ex- traneous light. Within the box inclined at angles of 45° to its axis are placed two rectangular pieces of looking-glass,. FG, FD, cut from one and the same rectangular strip to ensure the exact equality of their reflecting powers, and fas- tened so as to meet at F, in the middle of a narrow slit EFQ about an inch long and one- eighth of an inch broad, which is EXAMPLE. 35 covered with a slip of fine tissue or oiled paper. The rect- angular slit should have a slip of blackened card at F, to prevent the lights reflected from the looking-glasses mingling with each other. When we wish to compare the illuminating powers of two sources of light — (two flames for instance) — P and Q, they must be placed at such a distance from each other, and from the instrument between them, that the light from every part of each shall fall on the reflector next it, — and be reflect- ed to the corresponding portion of the paper EF or FG. The instrument is then to be moved nearer to one or the other, till the paper on either side of the division F appears equally illuminated. To judge of this it should be viewed through a tube blackened within, one end resting upon the upper part AB oi the photometer, the other applied quite close to the eye. When the lights are thus exactly equalized, it is clear that the total illuminating powers of the lumi- naries are directly as the squares of their distances from the middle of the instrument. Note. To render the comparison of the lights more exact, the equalization of the lights should be performed several times, turning the instrument end for end each time. The mean of the several determinations will then be very near the truth. By means of the above instrument we may obtain an easy experimental proof of the decrease of light as the in- verse square of the distance. For if we place four candles at F and one at Q (as nearly equal as possible and burning with equal flame) it is found that the portions of the paper EF, GF will be equally illuminated when the distances FF^ QF are as 2 : 1, and so for any number of candles at each side. 45*. In Foucault's Photometer, the two sources of light (a, yS) — ^gas lights for instance — which are to be compared, act separately on two different parts of a vertical plate of porcelain P Q sufficiently thin to be translucent. The opaque vertical screen RS, which separates the two illuminations one from the other, can be made to approach or retire from P ^ at pleasure. If such a position be given to it thait, the vertical planes, 3—2 36 VELOCITY OP LIGHT. drawn through olM, I3N which bound the portions illuminated severally by a and /3, inter- sect on the porcelain plate or nearly .so— the band MN il- luminated by both a and /8 can be made as narrow as we please — and if the distances of a and /3 be varied until the illuminations of the two parts of the plate PM, QN are sen- sibly equal, the intrinsic intensities of a, yS pared. Instead of the plate of porcelain — any uniform translu- cent membrane may be used : — or a plate of ground-glass, or of glass covered with a thin film of matter. Note. The student may consult Deschanel's Natural Philosophy — Professor Everett's edition 1882, Art. 1036, &c., on Measure of Brightness. can be com- 46. A plane surface touches a self-luminous sphere : to find the illumination of the surface at any point. Let A be any point in the plane surface which the sphere whose centre is G touches at P. Join CP. With centre A de- scribe a spherical surface, and let a straight line through A moving round the sphere G so as to de- fine the portion of it from which A receives illumination, intersect the spherical surface described about A in the small circle pq, the plane of which meets AG ov AG produced in c. Then projection on plane AP, of the curved surface pq, = projection of the plane circle pq = ■n- . cp' , — . (Hymers, Three Dim. Art. 81.) VELOCITY OF LIGHT. 37 Therefore illumination at A from the sphere (Art. 42) ^h-"^-- ■'■{%)-"■ Af Ap \ApJ ^ • \ACJ AC 47. Light is propagated with finite velocity. The eclipses of Jupiter's satellites are observed to happen sooner when, he is in geocentric opposition (and consequently nearest to the Earth), and later when he is in conjunction (or farthest from the Earth), than they ought to happen according to calculations made on supposition that he is at his mean distance from the Earth. The difference is accounted for by the hypothesis that light requires a finite time for its trans- mission. This supposition is confirmed by its explaining satisfactorily the apparent displacements of heavenly bodies, called aberration. The coefficient of aberration being the same for heavenly bodies at different distances, it appears that the velocity of light is uniform in the same medium. From observations on the aberration of stars, this velocity was supposed to be about 192000 miles per second in vacuum, or rather in that space which intervenes between us and the planets and fixed stars. Thus the time required for light to travel from the Sun to the Earth is about 8 minutes. There is reason to think however that this value for the velocity is too great: see next article. The discrepancies above referred to with respect to the eclipses of Jupiter's satellites were first noticed about A.!). 1675 ; but there is some uncertainty whether Roemer or J. D. Gassini first observed them. The finite velocity of light was employed by Bradley to explain the aberration of the stars, A.D. 1728. M. Fizeau was the first person who rendered the velocity of light sensible in experiments made on the surface of the Earth (see Comptes Rendus, t. 29, pp. 90, 132, July 23, 1849). A few months afterwards, the principle of a rotating mirror, which had been used by Prof. Wheatstone in measuring the velocity of electricity, was employed both by M. Mzeau and M. Foucault in experiments described by them in memoirs 38 VELOCITY OF LIGHT, presented on the same day to the Academy of Sciences {Gomptes Reridus, t. 30, May 6, 1850). A subsequent memoir by M. Fizeau is given in the Gomptes Rendus, t. 33. 47*. We proceed to give a short account of FoucauU's experiment. A beam of sunlight is transmitted by means of a reflector into a dark room through a small square aperture in the wiu': dow shutter. zm is a small plane mirror capable of revolving about an axis perpendicular to the plane of the paper in the direction of the arr' w ; z being the centre of a spherical mirror a" a' (of which only a small portion about a" is a reflector, subtending about 7° at z) ; zy the axis of a lens at y whose focal length /is such that a pencil of rays proceeding from a point a in the aperture after refraction through the lens y . would come to a focus at a' on the spherical mirror, but being reflected by the plane mirror zm actually comes to a focus at a" on the spherical mirror — the line a' a" being perpendicular to the plane zm: this pencil being reflected at a" pursues the same course reversed, so that after being again reflected by the plane mirror zm and refracted by the lens, comes to a focus again at a ; so that this arrangement of lens and reflectors produces an image of an object a coinciding with the object itself VELOCITY OF LIGHT. 39 As the mirror zm is made to revolve, whenever its position during a revolution is such as to cause the pencil from a to fall upon a" and be there reflected, the image of a will ap- pear, — and for other positions of zm it will disappear. If the mirror zm do not make more than about 30 revolutions a second the intermittent af)pearances are distinct ; but when the number exceeds 30, the image appears steady and per- sistent — and coincident with a, so long as the plane mirror zm revolves with such a moderate velocity that its change of posi- tion is insensible in the short interval which is occupied by light in passing from z to a" and back again ; but if the zm is made to revolve rapidly (several hundred times a second) this is no longer the case, but the rays which were reflected from the plane mirror when in the position zm, after passing to a" and being there reflected, are again incident on the plane mirror when it occupies a position zm' : — the ^ mzm {=x) being the angle through which zm has turned in the time (2t) which has been occupied by light in passing from z to a" and back again to z. If we draw a" b' perpendicular to m'z, the pencil after reflexion at zm' and refraction through the lens will come to a focus at a point c in h'y produced, at a distance from y equal to ya ; — that is, c the image of a is deflected through a small but sensible space in a direction perpendicular to the axis of the lens and also perpendicular to the axis of rotation of the plane mirror. For convenience of observation, a piece of j^ate-glass vv which allows the direct rays from a to pass through it without obstruction is interposed at an ^ 45° to ay — and the returning rays are partially reflected at its front surface and viewed by an eye-piece E. The object a is a platinum wire drawn across the centre of the square aperture in the shutter, and the position of the eye-piece ^is adjusted so that the image of a formed by the apparatus when the mirror zm is quiescent coincides with the wire fixed in the principal focus of the eye-piece: — when the mirror zm is made to revolve rapidly, this image is seen displaced parallel to itself through a space equal to ac. 40. yELOCITY OF LIGHT. Calculation. Let e = ac = displacement of the wire, r = za" = radius of the spherical mirror, 71 = number of revolutions -of plane mirror in 1", 0, ^ the angles a'zV, dyV in circular measure, 'b = ay, 1 = yz, V = velocity of light, :.r + l = ya' ; then we shall have a? . 2j- vr = r, -=. — = 2,r = .. — , 2-n-n V , i-n-nr whence a: = . V Also d= ^ a'zh = 2 ^ a'a'h = 2 ^ mzm' = 2x= and arc a'b = rd = {r + r) 0, ox v = —, jr- which expresses the velocity of light in terms e {r + L) of quantities which can be measured. In the experiments made by M. Foucault r = 4 metres, h= 3 „ Z = 1-1818 „ /= 1-9 „ and when n = 800 it was found that e was = •0006"', which numbers would give v = 192540 miles nearly. From subsequent experiments however made with great care, M. Foucault (Comptes Rendus, t. 55, pp. 501, 792; unno 1862) concludes that i; = 298000 Momeires= 185172 miles per second — (a metre being = 39"37l inches:) — and h& estimates this to be the correct value within -^ part of the whole. VELOCITY OF LIGHT. 41 A careful repetition of Fizeau's experiment, made in 1874 by M. Alfred Comu, gave the velocity of light to be 186,700 miles per second. {Nature, xi. p. 274.) The Rumford medal was awarded in 1878 by the Royal Society of London to M. Comu for this and other Optical Researches. Careful experiments by Foucault's method made in 1879 by A. A. Michelson, an Officer of the United States navy, gave the velocity of light to be 186,380 miles ;per second. {Nature, xxi. p. 226.) Sir John Herschel says that we are authorized to con- clude that in estimating the velocity of light at 186,000 miles per second, we are within a thousand miles of the truth. {Familiar Lectures, p. 234.) By Foucault's apparatus the velocity of light in water can also be determined : for this purpose a tube AB filled with distilled water is interposed between the revolving mirror zm, and another concave mirror R, similar to a". The light re- flected along this, tube by the mirror sm.when it is in a suit- able position in the course of its revolution, will be reflected back again by R, and thus the light will traverse the column of water twice in its course. If the velocity of light in water be less than in air, the time of passage along the tube will be longer than for an equal length of air, and therefore displace- ment of the image observed at E would be greater than when no column of water is interposed, — and such was observed to be the case. For a complete account of these interesting experiments, the student may consult the volumes of the Gomptes Rendus above referred to; or Pouillet, Elements de Physique &c. Tome 2,7th edition, 1856; Jamin, Cours de Physique, Tome 3, 1869. CHAPTER III. ABERRATION OF SMALL DIRECT PENCILS. 48. It was stated at (18) that the determination of the form of a direct pencil after reflexion or refraction would con- sist of two parts, (i) The determination of the geometrical focus, (ii) The determination of the vertex q of any one of the cones of rays before described. We will here proceed with the latter of these parts in a few of the more simple cases. The equation , which gives the distance Aq will not in general admit of direct solution, but is solved by successive approximations, — the approximation being conducted accord- ing to powers of AR, the half-breadth of the conical shell in question, which in such pencils as occur in the computations of instruments is very small compared with the other lines involved in the equation. It will appear in the course of our calculations that AB? is the lowest power of AR involved in the equations. The square and higher powers of AR being at first neglected, a first approximate value of Aq is obtained; Next, by neglecting the cube and higher powers of AR, and substituting in the coefficient of AK', the approximate value of Aq before obtained, a second approximate value is obtained. By this means the value oi Aq may be determined to any degree of accuracy. It is found sufficient in the calculations which have reference to instruments to carry the approxi- mation as far as AR". ABERRATION OF A SMALL PENCIL. 43 49. We premise the following theorem for the purpose of shewing that an approximate value of a quantity to be determined, may be substituted in the small terms of the equation. Suppose V a quantity whos§^>yalue is to be found, and which is given implicitly by an eqiiatipn of the form v=r+f^py^ (A), where V involves known quantities only, and is independent of y, the small quantity by powers of which the approximation is conducted, — a,ndf(v) is a function of v and known quan- tities. Then by substitution f(v) =A^J^^!&rf\ =/(F) +/' ( V) .f(v) .f+... by Taylor's Theorem ; .■.v=-r+f(V).f+f'iV).f(v).y* + ... = F+/(F).y, if the approximation extend only to the square of y. It appears therefore that in the term of (A) which involves y, we may substitute the value of v obtained by neglecting y", and the result will be true to the order of y\ 50. When a pencil is incident directly on a plane refract- ing surface, to find the point where the direction of a given ray (jfter refraction cuts the axis. Let Q be the origin of a pen- cil whose axis QA is incident directly at J on a plane refracting surface, then QA is the axis of the refracted pencil. Let QR be any ray incident at R, and refracted in a direction which cuts the axis in q, the position of which is to be determined. l,etAQ = u,Aq = v'; lines being considered positive when measured from J in a direction contrary to that of the incident 44 ABERRATION OF pencil. Also let AR = y,a, quantity of which the cube and higher powers may be neglected. Now RQA, RqA being equal to the angles of incidence and refraction of the ray QB, sin RQA = /* . sin RqA ; .: Rq = fi . RQ, In the coefficient of y', for v' we may substitute fiu, the first approximate value of v, and the resulting equation will be true to the order oiy' (Art. 49); '^ \u ixuj 2 Cor. Hence it appears by comparing this result with that of Art. 19, that the geometrical focus is the approximate position of the point q, when powers of y above the first are neglected. 51. When a pencil is directly reflected or refracted at a surface, the aberration of any ray is the distance between the geometrical focus and the point where the direction of that ray after reflexion or refraction cuts the axis (13). The aberration of a pencil is the aberration of the extreme ray in any section of the pencil through its axis. A SMALL PENCIL. 45 Hence the aberration of the ray qR — v — fxu /J, ' 2u' This quantity is positive or negative, i.e. the aberration is from or towards the refracting surface, accoifding as ^4 > 1, or < 1, that is, according as the refraction takes place from a rarer medium into a denser one, or vice versa. 52i When a pencil is incident directly on a spherical reflecting surface, to find the point where the direction of a given ray after reflexion cuts the axis. Let Q be the origin of a pencil whose axis QA is incident directly at ^ on a spherical re- flecting surface whose centre is : then ^ ^ is the axis of the reflect- ed pencil. Let QR be any ray incident at R and, reflected in a direction which cuts the axis in q, the position of which is to be de- termined. Join OR. Let AQ = u, Aq = v', AO = r, lines being considered positive when measured from ^ in a direction opposite to that of the incident pencil. Also let AR = y, — a quantity of which the cube and higher powers may be neglected, — on which supposition y may be taken to represent either AR, or the 'perpendicular from R on AQ, at pleasure. Then since QRq is bisected hj RO; .■.^,=l§^o.QR.qO = qR.QO. Now QE' = RO''+ OQ^+iOR . OQ . cosROA = r' + (u-ry + 2r.(u-r)cos^- • , w -r „ = w' • y , 46 AEEEEATION OF after reduction, since cos BOA • = cos ^ = 1 - ^ , approximately; ..,g^=.{.-(l-l)|}. Similarly, S^i -'' {l - (;-?)|.}; If each side of this be divided hy Uv'r (^_l\{i-(^.'^\lX=.(l-l)U -(1-1)11 \v' rj \ \r u) 2mJ \r u) \ \r v'J 2v'\ ' 1,^1^1^1-1) (1,-1) (l^l)f V u r \r uj \v rJ \u vj 2 If we neglect powers of y higher than the first 112 — + - = -, or v' = v (Art. 21), V u r ^ ' which value we may substitute in the coefficient of y', and thei resulting equation will be true as fcir as y^ ; ...i+l.?+(i.nX v u r \r uJ r which gives the position of g-. V V .\r u) T 2 „,! \r uJ , r A SMALL PENCIL. 47 or putting ?;' = D in the term involving y^, we have aberration of ray Rq = v' —v= — ( ) . — ^ \r u] Cor. 2. If, as in Art. 24, the position of any ray is determined with reference to the centre of the surface, we should get -^ + - = 2cos^. Oq p Write Og = g', then r r — + - = 2cos^, 1. P and - + - = 2(Art. 24); IP 1 9. • — = - (1 ^ cos 6)=—, nearly, q r r = ^, nearly; -.•0 = y. aberration of ray qR = q .q=y-^ 53. When a pencil is incident directly on a spherical refracting surface, — to find the point where the direction of a given ray after refraction cuts the axis. Let Q be the origin of a pencil whose axis QA is incident directly at J. on a spherical refract- ing surface whose centre is 0; then QA is the axis of the refracted pen- cil. Let QR be any ray incident at J2,andrefracted in a direction which cuts the axis in q, the position of which is to be determined. Join ORi 48 ABEEBATION OF A SMALt PENCIL. Let AQ = u, Aq = v', AO = r, lines h6mg considered positive when- measured from ^i in a direction opposite to that of the incident pencil. Also let AR = y^^a quantity of which the cube and higher powers may be neglected. _ sin QRO ^ sin Q BO. sin R Oq ^ OQ.R q ^ ~ sin 2^0 " sin ROQ. sin qRO ~ MQ . Oq' or fi.RQ:Oq = OQ.Rq. Now . R/^ = RO^+ 0Q' + 2R0.0Q cos ROA = r' + (w — rf + 2r (m — r) cos - = w" — (m — r) — , nearly ; ...RQ^u\l-C-')l\. [ \r uj 2u) Similarly, S, -,/|l -(J- J) Ij; V u r \r V J \r uJ \v uJ 2 If the square of y be neglected, v the approximate value of v is given by the equation (i 1 /"■ — I V v, r ' which value of v' may be used in the coefficient of y^ and the result will be true as far a,s y^. LEAST CIRCLE OF ABERRATION. 49 v' l_/^-l + u i\Vi r fi' '\r uJ \r u J 2 ' which equation determines the position of q. Cor. /f_^ = /iIll.fl_iy.fl_/^ + l I) v /i v w V^* 2 ' therefore the aberration of the ray Bq' = v' —v /a' 'V^ M/ Vr u J 2 ■ ^ _ ytt-l \r ul n _ iJL + l \ f \ r M/ 54. O&s. In the preceding Articles, since powers of y above the second have been neglected, it is immaterial whe- ther we take for y the length of the arc AR, or the perpen- dicular distance of R from the axis of the surface. Least Circle of Aberration. 55. Let -42'' be the axis of a pencil of rays reflected or refracted directly at a spherical surface ; Hr, hr the extreme rays in any plane section of the pencil through the axis. P.O. 50 LEAST CIECLE which meet in the point r of the axis ; Ks, ks any other two rays in the same section meeting in the point s of the axis. Suppose hr (produced if necessary) to cut Ks in t and draw tm perpendicular to AF. Now (i) considering rays incident at different points along AH, when K coincides either with A or H,tmis=Q; there- fore for some position of .ff'in AH, tm is a maximum. (ii) If K have the position for which tm is a maximum, a circle with centre m and radius mt, in a plane perpendicular to AF, is the smallest space through which the whole pencil passes. For a circular section of the pencil to the left of m is larger in consequence of the converging cone Ksk, and a cir- cular section to the right of m is larger in consequence of the diverging cone trp. Suclfi a circle is called the Least Circle of Aberration of the directly reflected or refracted pencil. Ohs. In the preceding figure, the aberration is supposed to be towards the surface ; the student will have little difficulty in drawing a corresponding figure for the case when the aber- ration is, from the surface. In the following calculation of the position and dimensions of this circle, the cube and higher powers of the half-breadth of the pencil will be neglected, as has been done in former cases. 56. To calculate the position and dimensions of the Least Circle of Aberration after direct refleocion or refraction, at a plane or spherical surface. Let AK=y', AH=y, j-=rad. of spherical surface AK (fig. Art. 65). Draw KM perpendicular to AF. AM'=r vers. ^ = |- , nearly. OF ABEKEATION. 51 By similar triangles y' ms __ Ms _ 2r As y mi~ MK "y' ^ y~ ~ 2r ' w.s = mt \ y' 2rJ ' „ mr _ Mr _ Ar y mt~AH~Y~^' (Ar y \y 2r Now since mt, depending on the aberration, is at least of the order y'^, — therefore y' . mt and y . mt may be neglected. Also the difference between Ar and As being the difference of the aberrations of the two rays Hr, Ks., is of the order y'^ ; and therefore when multiplied by mt we may regard Ar and As as equal, since the error introduced by so doing is of a higher order than y'^. Hence we may take rs = rm + ms =mt . Ar ( - + - V«/ -3/ But (Art. 52, 53) the aberration of any ray in the same pencil xy'; .: Fs : Fr = y" : y\ .'. rs : Fr=y^ — y'^ :y^; ,^F,_l:ijl = rs = mt.ArJ-±f; y" yy Ar y Now mt is variable by the change of y' , and this value of mt will therefore be a maximum when y-2y'^0, i.e.y=|; 4—2 52 LEAST CIECLE OF ABERRATION. ■■'^^ = l-Try ^'^' rm = -7 . Fr (ii) ; which two equations define the position and magnitude of the least circle of aberration. In other words, the distance of the least circle of aberration from the geometrical focus is three-fourths of the longitudinal aberration of the pencil, and its radius is one-fourth of Fp, the lateral aberration of the extreme ray. CHAPTER IV. FOCAL LINES OF SMALL OBLIQUE PENCILS — CAUSTICS. 57. When a pencil of rays (Art. 13) is reflected or re- fracted at the surface of a medium, the reflected or refracted rays will not, in general, pass accurately through one point. This peculiarity is sometimes called astigmatism. The laws of reflexion and refraction would enable us to determine the direction of any particular ray after its reflexion or refraction, and, as in the case of any system of lines generated according to a determinate law, the consecutive rays after reflexion or refraction will each touch some surface as their envelope: this envelope being in fact the locus of their consecutive ultimate intersections. (Todhunter's Diff. Calc. Chap, xxv.) It will be sufficient for the purpose of illustration to con- fine our attention here to surfaces of revolution, the incident rays being either parallel to the axis of the surface or di- verging from some point in that axis. In such a case, the caustic surface will be one of revolution about the same axis as the surface of reflexion or refraction, — and a section of it by a plane passing through the axis will give the caustic curve, which curve is touched by each of the reflected or refracted rays which pass in that plane section. 58. The following would be the general process of deter- mining the caustic surface of a pencil for a given reflecting or refracting surface of revolution. In any section through the axis of the pencil, obtain the equation to any reflected or refracted ray referred to axes in that plane; this equation involves the co-ordinates of the point of incidence, which are connected by the equation to the reflecting or refracting curve. Thus the equation to the reflected or refracted ray involves only one independent parameter, and the locus of 54 DEFINITIONS. their ultimate intersections may be obtained in the same way as a common envelope. This envelope is the caustic curve, by the revolution of which about the axis the caustic surface is generated. Caustics formed by reflexion were formerly called catacaustics ; those formed by refraction, diacaustics. 59. Since the surfaces with which we have to deal are practically limited in extent, it will be convenient to give a few definitions before we proceed to examine the form of oblique pencils, and we shall suppose the surfaces to be either plane or spherical. Bef. Oblique incidence on a reflecting or refracting sphe- rical surface is either centrical or excentrical. A pencil is incident centrically when the axis of the pencil is incident at a definite point of the surface, called the centre of the face; in other cases it is incident excentrically. A distinction must be preserved between the centre of the surface and the centre of the face ; the former is the centre of the sphere of which the reflecting or refracting surface is a portion; the latter is a point on the surface itself, gene- rally the point with respect to which the surface is symme- trical. Def. The diameter of the spherical surface which passes through the centre of the face, is in general called the aods of the reflecting or refracting surface. The circumstance which renders the calculation of cen- trical pencils less complex than those of excentrical pencils is, that in the former case the point of incidence is a definite point of reference from which lines may be conveniently measured, but in the latter it is not so. 60. If a divergent pencil ie incident obliquely on a plane reflecting surf ace, it will diverge from a point after reflexion. Let Q be the origin of a pencil whose axis QA is incident directly on a plane reflecting surface AR'. Then after reflexion the pencil will diverge from a point q in QA FOCAL LINES OF SMALL OBLIQUE PENCILS. 55 produced, at a distance Aq = AQ (Art. 16). If now we suppose the whole pencil to be removed with the exception of the oblique pencil QER', the course of this portion of the pencil will remain unaltered, or the oblique pencil will after reflexion diverge from q. Similarly if an oblique pencil con- verging to q be reflected at the plane surface, it will after reflexion converge to Q. Cor. Since z R QR' = t RqR', the degree of divergence of the reflected pencil is the same as that of the incident pencil. 61. Other cases of oblique reflexion or refraction are not so simple as that of the preceding Article; we shall now shew that if the pencil be small, it will after reflexion or refraction converge to or diverge from two very small straight lines, perpendicular to one another and in different planes, so that. the direction of every ray passes through each of these straight lines. 62. To explain the formation of focal lines, when a small oblique pencil is reflected at a spherical surface, or refracted at a plane or spherical surface. Let QO be the axis of a pencil incident directly at C on a spherical reflecting surface, or on a plane or spherical re- fracting surface. If this pencil be supposed to consist of a series of conical surfaces of rays with QC for their common 56 FOCAL LINES OF SMALL OBLIQUE PENCILS. axis, since all the rays in any such surface will be reflected or refracted similarly about QC, the directions of the reflected or refracted rays will form a series of conical surfaces Hrh, Aqji, Ksk... having a common axis QG along which their vertices r, q^, s. . . are arranged. The consecutive intersections of these successive conical surfaces form the caustic surface (Art. 57). (i) If instead of the whole pencil we consider that por- tion of it only which is incident on the annulus of the reflect- ing or refracting surface, which would be generated by the revolution of .ff^ about QC,we have corresponding to this an annulus of the caustic surface through some point of which the direction of each ray of the conical shell of reflected or refracted rays now considered passes. If HK be small, this annulus of the caustic surface may be regarded as a circle in a plane perpendicular to QG, and whose diameter is repre- sented in the figure by q^t. (ii) Instead of the conical shell of light incident on the above annulus, let us now consider a small portion thereof incident about HK, by which we come to the case of a small oblique pencil whose axis after reflexion or refraction is The direction of every reflected or refracted ray now con- sidered will pass through some point of a small circular arc at q^, which may approximately be regarded as a straight line perpendicular to the plane QCA. This line is called the Primary Focal Line, and the point q^ the Primary Focus. Again, a section of the pencil by a plane through q^ parallel to the tangent plane of the surface at A, though actually a very elongated _/i^wre of eight — as indicated at Fin the figure, — may very approximately be regarded as a straight line, and is called the Secondary Focal Line, — and the point q^, where the axis of the reflected or refracted pencil cuts QG, is called the Secondary Focus. A section of the reflected or refracted pencil taken through the axis QC oi the reflecting, or refracting surface, would be strictly a straight line rs, — but it is convenient for our calculations to consider sections parallel to the reflecting or refracting element ^Z"— and to treat the section through q^ as the Secondary Focal Line, CIRCLE OF LEAST CONFUSION. 57 Bef. With respect to the small oblique pencil of which ■A-q^q^ is the axis, — the plane QGA which contains the axis of the incident pencil and also the axis of the reflected or refracted pencil, as well as QC the axis of the spherical sur- face, is called the Primary Plane: — and a plane passing through the axis Aq^q^ at right angles to this primary plane is called the Secondare/ Plane. Hence a small oblique pencil after reflexion or refraction converges to or diverges from two straight lines called focal lines ; — The Primary and Secondary focal lines being at right angles to the Primary and Secondary planes respectively. 63. Remark. When the aberration of a direct pencil is towards the surface, the primary focus of a small oblique pencil from the same origin is nearer to the surface than the secondary focus, and vice versa. The annexed figure will illustrate the reasoning of the preceding Article, when the aberration of a direct pencil is from the surface. The dotted lines in each figure are intended to represent the bounding rays of a section of the oblique pencil made by a plane through its axis Aq^q^ and perpendicular to the Primary Plane, — i.e. by the Secondary Plane. 64. Circle of Least Confusion. If a section be taken of the reflected or refracted pencil (Art. 62) by a plane parallel to the tangent plane to the re- flecting or refracting surface at A (Figs. Art. 62, 63), — when the plane is drawn through q^, the section, as we have seen, is 58 VISUAL PENCILS. approximately a straight line perpendicular to the primary plane. If this plane be supposed to move gradually from q^ to q^, — remaining parallel to itself, — the breadth of the section increases in the primary plane, and decreases in the perpen- dicular direction until at q^ the section becomes a straight line in the primary plane. At some point therefore in q^q^, the breadth of the section in and perpendicular to the primary plane is the same, and the section very nearly circular. This section of the pencil is called the Circle of Least Confusion. Obs. The caustic curve is the locus of the primary focus of small oblique pencils reflected or refracted at consecutive small elements of the surface. This suggests another method of finding the equation to the caustic, which is sometimes convenient. 65. We can now explain the nature of the pencils by which an eye sees the image of an object by reflexion or refraction. Let the figure represent a plane section through the axis of a pencil of rays diverging from a point Q, and reflected at the spherical surface A GB (or refracted at a plane or spherical refracting surface). The successive reflected rays will be successive tangents to the caustic curve Fq^, Fq, the cusp of which is at the geometrical focus F. An eye on the axis at E' will receive a small pencil diverging apparently from F, the geometrical focus, and the image of the point Q seen by E will coincide with the geome- trical image of Q (Art. 26). But the case is different with an eye not situated on the axis QC; for instance, an eye E VISUAL IMAGES. 59 receives a pencil of rays which is really a small oblique pencil reflected at some part A of the surface, the axis of this small visual pencil Hq^q^ being a tangent to the caustic curve in the plaae passing through the centre of the pupil of the eye and the axis QG oi the surface. This small oblique pencil diverges not from a point, but from two focal lines at q^, q^, these lines being at right angles to each other and not in the same plane. Hence the image of Q seen by E is more or less indistinct, and is seen in the direction of a tangent drawn from the eye to the caustic of Q in the plane EQG. Since the circle of least confusion is the nearest approach to a point which such a small oblique pencil admits of, we may perhaps regard the circle of least confusion as the image of Q, — and when an object of finite size is viewed by reflexion or refraction, we may consider the visible image to be the locus of the circles of least confusion of small oblique pencils emanating from consecutive points of the object; the position of any one of these being determined as above. These circles of confusion will overlap each other, and the image conse- quently will be more or less confused and indistinct. We may regard the comparative size of the circles of least confusion in different cases as a measure of the com- parative indistinctness of the visible image. It will sometimes happen that from an eye in a given position more than one tangent can be drawn to' the caustic curve in the plane QGE ; in such cases there will be more than one visible image. In the valuable optical diagrams published by Engel and Schellbach — (Sialle: first series, 1850, second series, 1856) — the locus of the primary focus is taken as the visible image ; this can only be received as an approximate position of the image ; but nevertheless the student will derive great advantage from consulting them. A simplified and less expensive edition of the first series has been published by the Rev. W. B. Hopkins, late Tutor of St Gatharine's Gollege. It will be a matter of convenience frequently for us to treat the primary focus as the point from which the visual pencil diverges, and regard the visual image as the locus of the primary foci of the small oblique pencils received by the eye from consecutive points of the object. 60 FOCAL LINES OF A Determination of the Position of the Foci of Small Oblique Pencils. 66. A small oblique pencil is Reflected at a spherical surface ; to find the distances of the foci from the point of incidence of the axis. Let Q be the origin of a small pencil whose axis QA is incident at A obliquely on a spherical reflecting surface whose centre is 0. Let Aq^ be the direction of the axis after reflexion, cutting QO produced in q^, the secondary focus (Art. 62): QjE another ray incident in the primary plane and reflected in the direction Hq^ which cuts Aq^ in q^, the primary focus. Join OA, OH, and draw Hn perpendicular to A Q. Let QA, HO intersect in K, and Hq^, AO in T. Let AQ = u, Aq^ = v^, Aq^ = v^^,AO = r, AOq^ = d, Aa-^M *^^ angles of incidence or reflexion of i"^ o- Then ^ HQA = -^ ultimately, AH.si.nHAQ .^. ^, AH. cosA .. , ^ = rrTi ultimately = ultimately ; .:Bcj)=^QHO-QAO = ^K-AQH-{^K-AOH) = AOH-AQH=^-^^i^^ r u Again, B<}>=q^HO-q^AO=^T-AOH- i^T- Aqji) = Aq^H-AOH = ^^^^^-^ v^ r ' Equating these values of S^, we get cos (i) 1 _ 1 cos (^ V, r r u ' REFRACTED PENCIL. 61 ■•■ - + '-- ^ ■ v^ u r cos ^ Further, r_ ^AO ^,\n [6 +J>) r _A0 ^ sm(0- Now Sd> = HQA = -j7^ approximately = • , If ji4 be the index of refraction from the first medium into the second, sin ^ = /i sin ^' (Art. 9), and by differentiating this we get the relation between the small corresponding variations S^, S^', viz. cos (j) . B(j) = fi cos ^' . S(f)'; therefore substituting the above values of B^, B^', Again fj, cos^ <^' cos' - refraction 9 +o' But Bcli=QHO-QAO = ^K-HQA-{^K-AOH) = AOH-HQA=^-^^^^, r u and Sf = q^HO -q^AO = ^T- Hq^A -(^ T-AOH) . ^^ ^ . AH AH cos 4>' = A OH - Hq.A = . And from the relation sin ^ = /w sin ^', we get by differen- tiation cos ' cos' (A w cos (j)' — cos (^ ,., or = '■ W- 64 CIRCLE OF '. r AO &m((i>'+e) ,,,-,/ X /J Again, — = -; — = V— ;i — - = cos + sm a cot u, r AO sin (0 + ^) , , • , ^ /j - = — -=- = i-^—p: — - = COS + sm cot a; u AQ sia ^ ^ T ' therefore remembering that sin ^ = ;ii sin ^', we have, by eliminating cot ff, M' 1 a cos rf)' — cos 1. The results are equally true for refraction from a denser into a rarer medium. We recom- mend it as an exercise to the student to go through the in- vestigations for the several cases that can occur, when the incident pencil is divergent or convergent, the surface concave or convex, and /* > 1 or /* < 1. 70. To calculate the position and dimensions of the circle of least confusion of a small oblique pencil reflected at a spherical surface or refracted at a plane or spherical surface. Let A be the point of incidence of the axis of a small oblique pencil on a spherical reflecting surface or a plane or spherical refracting surface. KMEN the section of the inci- dent pencil made by the surface which will approximately be an ellipse with its major and minor axes HK, MN, in and perpendicular to the primary plane. LEAST CONFUSION". 65 ■ Suppose mq^n, hqjc the primary and secondary focal lines, and ros, poq the breadths in and perpendicular to the primary plane of a section of the pencil through a point o of its axis by a plane parallel to the tangent plane to the surface at A. Let Aq^ = v^, Aq^ = v^, MN=\, ^ = angle of incidence of the axis of the pencil. Now if the incident pencil be small and its origin distant, it may be considered approximately a right circular cylinder, for the purpose of comparing the dimensions of the section KMHN : and this then being a section of the pencil by a plane inclined at an angle ^ — ^ to its axis, we have HK=,MN sec (f> = X sec . , By similar triangles pq _ oq., rs _ oq^ X ~ v^' X sec «2- P. 0. Xseccji 66 .CAUSTIC CURVES. CoE. 2. If the pencil be cylindrical at incidence, and be reflected at a spherical surface, m = oo , and (Art. 66) ; 1 _ 2 1 ^ 2 cos ^ Vj rcos^" v^ r . _r {1 + cos ■'■ ^°~r l+cos'.^ ' P1-^\ + cos'' sin^f^ a = \ sin" <}), a = \ . cos" ^ " CoE. 3. Since -f = -^ ~, and this latter quan- Ao — Vj^v^ ^ tity approaches unity as (p is diminished, ,*. ultimately AO = ^{v^ + v^, i.e. the centre of the circle of least confusion has the middle point between the two foci for its limiting position, and it may approximately be supposed to have this position when the obliquity (0) of the pencil is small. 71. Examples and Illustrations: Parallel rays are incident on a reflecting semicircular mirror, and in its plane ; to find the caustic curve. Let QB be any ray of the beam of parallel rays incident at R on the mirror AGB, whose centre is 0; CO the diameter of the mirror parallel to the incident rays. Describe a circle with centre and radius 0F = \. CO. Join no, and upon RT as dia- meter describe a circle, 0' its cen- tre; therefore 0'T=^.TO. Let RP be the direction of the ray QR after reflexion. Join O'P, PT. CAUSTIC CURVES. 67 Now ^PO'T=2PBT=2QEO = 2TOF, and 0T=20'T; .-. arc TF= arc TF. We may then suppose the circle MPT to roll upon the circle TF, the point P coinciding initially with F, the geo^ metrical focus of rays incident at C, so that P traces out the epicycloid FPA, FB,— the cusp of which is at F. And since EPT is a right angle and PTis the direction of the normal to the path of P, therefore PP is a tangent to the path of P, i.e. each reflected ray touches the epicycloid APFB, which is therefore the caustic curve. This caustic has a cusp at F, and touches the circular mirror at A and B. By the revolution of this curve about GO we should get the caustic surface of parallel rays reflected at a hemispherical mirror. Ohs. The caustics formed by reflexion may be easily shewn experimentally by taking a narrow strip of polished steel — (a piece of watch-spring for instance) — bent into any con- cave form. Place it upright on a sheet of paper, and let it be exposed to the rays of the sun, so that the plane of the paper passes nearly but not quite through the sun ; the caustic will be seen traced on the paper and marked by a bright, well- defined line; the part within being brighter than that without, and the light diminishing by rapid gradations from the caustic inwards. If the form of the spring be varied, varieties of catacaustics with their singular points, cusps, contrary flexures, &c. will be seen beautifully developed. The bright line seen on the surface of a drinking-glass nearly full o-f liquid, standing in the sunshine, is a familiar instance of the caustic of a circle. 72. To find the form of the caustic, when the reflecting curve is a circular arc, and the rays diverge from a point in its circumference. Let QR be any ray diverging from Q, a point in the circumference of the circular mirror ACB whose centre is 0. Join OR, and with centre and radius = OP = ^. OC describe a circle ; describe another circle on RT as diameter, so that its radius 0T= TO. 5—2 68 CAUSTICS BY REFLEXION RP the direction of QR after reflexionjoin O'P.Pr. Then z PaT= 2PRT = 2QR0 = TOF, and 0'T=OT; .-. arcTP = arc TF. Hence it will readily appear that the reflected rays all touch the epicycloid traced out by the . point P of the circle RPT, as this circle rolls upon FT fixed. This epicycloid is the caustic curve required ; it has a cusp at F, the geometrical focus of rays reflected at G, and touches the mirror at Q. We can easily find the polar equation to this caustic. Suppose P, Pjoined, and let PF=r, ^ PFG =9, OQ = a, since and 0F-- OT.= 0'T=0'P = ~, / FOT =■ WE Q = 2PR0' = - PO'T, PF is parallel to 00', and = ^ PFC = ^ FOT; •■3' ■00': ■■ PF+ 2F0 cos F0T= r + ~acoae; .:r = -a{l — cos 6), the equation to the caustic, which is a cardioid. 73. Rays diverging from a point are refracted at a plane surface, the caustic is the evolute of a conic section. (i) Suppose the refraction to take place from a denser into a rarer medium. Let QR be any ray diverging from Q and incident at R, on the plane surface. Draw QCB perpendicular to the surface, and make CB= QC. Describe AND REFRACTION. 69 a circle about QRB, and let LR be the direction of the refracted ray. Join QL, LB; the point of intersection of QG, LR. Since LR bisects z QLB, OB LB wehaYe^ = ^; OB _ BL + QL QL QB OQ QL' sin QLR "BL + QL QL sin QOL' And QLR = QBR = RQB = ^, the angle of incidence of QR QOL = ^' = z of refraction of QR ; QB sin d) . , ^ , ■'■mTQL=^i>'=^'^''^'''f'^^^- Hence BL + QL = ' = a constant quantity, and the locus of L is an ellipse of which Q, B are the foci, — and LR is a normal at i to the locus of L, since it bisects the angle be- tween the focal distances ; that is, the caustic curve is the evolute of -the ellipse whose foci are Q, B and axis major = — ^ , and eccentricity = /*. (ii) If the refraction takes place from a rarer into a denser medium, let LR be the direction of the refracted ray, produce LR to meet GQ pro- duced in 0, — then it may be shewn that L bisects the ex- terior angle QLB, and BL^BO; QL QO 70 ILLTJSTEATIONS. BL-QL BQ QB 00 sin OLQ sin / ■,\ "BL-QL QL sin LOQ sin ^' Hence the locus of i is a hyperbola, and OZ is the di- rection of the normal at L ; that is, the caustic curve is -the evolute of a hyperbola whose foci are Q, B, and axis major =^ BL-QL: QB , and eccentricity = fi. 74. Illustration. Thus, if Q be a radiant point, the rays from which are refracted at a plane surface ST into a rarer medium, the emergent rays will be tangents to a virtual caustic SF, FT, which is a portion of the evolute of an ellipse, one cusp of which is at F the geometrical focus of Q. The branches of the caustic touch the plane surface of the medium at 8, T, points determined by the condition that the rays QS, QT are incident at the critical angle. (Art. 80.) If we suppose the figure to revolve about QFA the axis of the surface, the curve SFT will trace out the caustic sur- face, to which every ray of the beam of refracted rays is a tangent. Rays incident beyond 8, T are internally reflected within the denser medium. ILLUSTRATIONS. 71 The caustic just obtained enables us to illustrate the de- formation of the visible image of an object 'situated in a denser medium than that in which the eye is situated, the media being bounded by a plane. The dotted curves in the figure are intended to represent the caustics of three different points of the object,— and to avoid confusion, only the axes of the pencils •which the eye receives are drawn. Again, suppose a surface of still water with a level hori- zontal bottom not very deep ; the bottom will not appear a 72 . PBOBLEM. plane, but will seem to rise on all sides, being shaped some- thing like a basin below the eye, and tending at distant points towards the surface of the water as an asymptotic plane. 75. If a small pencil of diverging rays he reflected at a concave spherical mirror, to find the limits of the distance of the origin from the point of incidence in order that the reflected pencil may converge to, or d/iverge from, both the focal lines. The distances of the primary and secondary foci of the reflected pencil from the point of incidence of its axis are given by the equations 1 2 1 1 _ 2 cos 1 v^ r cos ,-r :, ^ 1- J 2 cos and consequently > — ^— ^,... and it diverges from each line if v^ and v.^ are both negative, which will be the case if u , r cos «b J r Ti- 1 r cos — s— ^ 2 ^ •' 2 cos ^ 2 r and < ^ -, the reflected pencil converges to one of the fOcal lines and diverges from the other. If the rays reflected in the primary plane are parallel or V, be infinite, then u = — -— ^ and v^= — ^ . „ , ; the 2, ^2 sm (j) ■ latter is the polar equation to the locus of the secondary focus, when the origin assumes different positions in the same primary plane, so that the rays in the primary plane may be parallel. See a paper on the Cyclide by J. C. Maxwell, QuaHerly Journal of Mathematics, Vol. ix. p. 111. CHAPTEK V. SUCCESSIVE EEFLEXIONS AND REFRACTIONS; — PRISMS — LENSES. 76. In this section we proceed to examine the modifica- tion in direction and form which a pencil undergoes, after being reflected or refracted more than once at plane or sphe- rical surfaces. Def. When a ray has had its direction altered by reflexion or refraction, its deviation is the angle between its present direction and its original direction produced. 77. Successive Reflexions at Plane Surfaces. If a pencil he reflected once hy each of two plane surfaces to find the deviation of its axis : supposing its course to be in one plane perpendicular to the intersection of the surfaces. Let QRSTV be the course of the axis of a pencil reflected at Ji and 8 at two plane surfaces in a plane which cuts these surfaces per- pendicular in CA, CB. Then the angle VTv, measured from RTv, the direction of QR produced, to- wards the point F, — is the deviation of the axis of the pencil after the two reflexions. Draw Rm, 8n normals to the reflecting planes at R, S. 74 SUCCESSIVE REFLEXIONS. / Now, ^ vTr= / QRS - ^ EST = 2 A SBm -2 z BSn = 2 (ESB -8BC) = 2 ,ACB; i.e. the deviation of the axis of the pencil is double the incli- nation of the reflecting planes. Cor. The degree of divergence of the pencils is unaltered by the reflexions. (Art. 60. Cor.) ■ 78. When a ray is reflected at a plane surface, the in- cident and reflected ray are equally inclined to any the same line which is parallel to the reflecting plane. Let the plane which passes through QO, OR, the incident and >^ 1 /1\ ^ reflected rays, meet the reflecting plane to which it is perpendicular in the line iSOT. Through the point draw the line A OB in the reflecting plane, parallel to the given line. Take OR = OQ and through R, Q draw planes at right angles to the line SOT, and meeting AOB in B, A. Then remarking that each of the angles at S,Thsa. right angle, and that OR = OQ it will easily be seen that the triangles ROT, QOS are similar and equal, as are also BOT and AOS, and consequently RTB, QSA are so likewise; whence iROB = ^ QOA, i.e. QO, OR are equally inclined to AB, and the same is true with respect to any line parallel to -AB. Hence when a ray is reflected at two plane surfaces in succession, the inclination of the ray to the line of intersec- tion of the surfaces before the first and after the second re- flexion is the same; and a similar result may be inferred after SUCCESSIVE KEFKACTIONS. 75 reflexion at any number of plane surfaces in succession, if the .lines of intersection of the surfaces be all parallel. A familiar instance of raiys so reflected is afi"orded by the Kaleidoscope. Successive Refractions at Plane Surfaces. 79. In examining the eS"ect of successive refractions on a pencil, the following two considerations are employed : — (i) The geometrical focus of a direct pencil being the point of ultimate intersection of any refracted ray with the axis, the pencil after one refraction may ultimately be con- sidered to diverge from or to converge to this point as an origin, the pencil being supposed a very small one. (ii) If a ray be reflected or refracted in any manner in passing from one point to another, it is assumed that it might pass by the same course reversed from the latter point to the former. This is in accordance with a general law in Optics, that the visibility of two points from one another is mutual ; in other words, if a ray of light proceeding from A arrives by any course at B however often reflected or re- fracted, a ray can also arrive at A from B by retracing pre- cisely the same course in the opposite direction. Hence when a pencil is emerging from a refracting medium, we may when convenient reduce this case to the more familiar one of a pencil entering a refracting medium, by supposing the course of each ray reversed. Critical Angle. 80. If ^ be the angle of incidence of a ray of light, and <^' its angle of refraction into a denser medium, /i the refrac- tive index between the media, then sin (f>= fJ- sin (f)', or, sin ^' = - . sin ^. Now fi being > 1, (Art. 9), sin ^' is < ,1, and this equation gives a real angle of refraction for any given angle of inci- 76 CRITICAL ANGLE. dence ; and accordingly refraction into a denser medium is always possible whatever be the angle of incidence. If the refraction be from a denser into a rarer medium, and if ^' be the angle of incidence, 1, or tp' be not > sin"'- . Accordingly if the angle of incidence in the denser medium exceed this limit, it is found that refraction does not take place, but that the ray is reflected within the denser medium at the surface which separates the media. Def. The angle sin"'- . which the angle of incidence in the denser medium must not exceed, in order that refraction into a rarer medium may be possible, is called the critical angle of the media between which the refractive index is (i. ' The critical angle for water is about 48° 27' 40" crown glass 40° 30' for chromate of lead it does not exceed 19° 28' 20". 81. When light is incident at the surface of a medium at an angle greater than the critical angle the reflexion is total, and is much more brilliant than that obtained in any other way^ — as at the surface of quicksilver or of any polished metal. It may be exhibited in a simple manner by holding a glass of water above the level of the eye ; the under- surface of the water will appear very bright from the light internally reflected, and any object in the water-^a spoon for instance— r will be seen by reflexion at the under surface, more brilliantly than it would, by reflexion at any mirror. Or again, if a glass prism be held so that the eye may receive light pass- ing through it after internal reflexion at one of its faces, that face will appear as bright as polished silver. RELATIVE REFRACTIVE INDICES. 77 This property of internal reflexion is employed in the camera lucida, in diagonal eye-pieces for telescopes, &c., with great advantage. From the preceding we may explain some of the pecu- liarities which would present themselves to an eye under the surface of still water. All external objects would appear com- pressed within a conical space whose axis is vertical and ver- tical angle = 2 sin"'- = 97° nearly, the objects near the hori- zon being much distorted and contracted, especially in height. Beyond this conical space, objects within the water would be seen by reflexion at the surface. 82. JDef. A portion of a refracting medium contained between two parallel plane surfaces is called a plate. It is a result of experiment that when a ray of light passes through any number of media separated by parallel plane surfaces, if any two of these media be identical, the directions of the ray in them are parallel. This may be illustrated experijnentally by holding a plate of glass or any transparent substance before the object-glass of a telescope directed to a distant object or before the naked eye, — the apparent place of the object will be the same, whatever be the inclination to the visual ray at which the plate is held. 83. The experimental result just referred to may be em- ployed to obtain a relation between the indices of refraction of successive media, as follows : — Let A,B,C, She four media bounded by parallel planes, and let A and 8 be identical. Then if QR, Q'E be two rays parallel in A, their directions in 8 will be parallel after one of them QR has been refracted through B, C, and the other Q'R' refracted at once into G. But the rays might follow the same course in a reversed direction, in which case the angles of incidence from 8 to being the same, the angles of refraction are the same, and each of the rays when passing though G has undergone the same deviation from its direction in A. 78 REFRACTION Let ^j, ^2, ^3 be the angles of incidence on B, C, S, and let a/^^ denote the index of refrac- tion from any medium denoted by a into one denoted by /3. Then sin and a similar result might be ob- tained, connecting the indices of re- fraction of any number of media. 3 . 4 Ex. From air to glass /"■ = 5, from air to water fi =„. Hence from glass to water, index from air to water 8 g/^w" index from air to glass 9 " 84. It will readily follow from the previous Article that if a ray of light be refracted through any number of media bounded by parallel plane surfaces, its direction in any me- dium will have undergone the same deviation as if it had been refracted directly into that medium. "Kote. In all cases of refraction in which a single medium only is mentioned, the other is supposed to be vacuum. In such cases the corresponding value of /* is called the absolute index of refraction. It follows also from the preceding results that the index of refraction from a medium A into another B is the reciprocal of that from B into A — i.e. ^/j-b • bH'a = ^ '■ ^ conclusion which follows at once from the law of refraction combined with the remark in Art. (79). THROUGH A PLATE, 79 85. To determine the geometrical focus of a pencil after direct refraction through a plate. Let Q be the origin of a pencil whose axis QAB passes directly through a refracting plate. Let F^, F be the geometrical foci of the pencil after refraction at the first and second ' surfaces respec- tively. Let AQ = u, BF=v, AB = t From the first refraction Art. (19) AF^ = fji.u. . Now F^ being regarded as an origin, a pencil diverging from F^ after refraction at the second surface, has • F for its geometrical focus. Hence if the course of the pencil be sup- posed reversed, a pencil converging to F would after refraction into the plate have F^ for its geometrical focus j (Art. 79) .-.BF,- = ti.BF, or AF^ + t = fiv; .-. t = flV — flU, or « = ich determines the position ofi^. NoU. Since BF - -BQ = - /.-I . t, it folio ject appears nearer when viewed through a plate denser than the surrounding medium. 86. When a sraall pencil is refracted obliquely through a portion of medium bounded by two given surfaces, there is an apparent difficulty at the second refraction in consequence of neither the incident nor the refracted pencil having then a point of divergence or convergence. In the primary plane, however, the pencil on account of its smallness may after each refraction be regarded as con- verging to or diverging from a point, and we may thus apply — with reference to this plane— propositions which suppose 80 EEFEACTION the whole pencil to emanate from a point. Again, the rays incident in a plane perpendicular to th^ primary plane have, after each refraction, a point of convergence or divergence, and therefore we may use with respect to this plane the re- sults deduced in the previous sections. The form of the emergent pencil is thus determined by finding the foci of two sections of it, one by the primary plane, the other by a plane perpendicular to that plane. These foci are separated by an interval which is generally small in the cases which occur in the construction of optical instruments, where the pencils are of small breadth and obliquity. The results obtained on the above hypothesis are of course only approximate, and become less and less accurate the more numerous the successive refractions to which the pencil is subjected. 87r To determine the foci of a small pencil refracted obliquely through a plate. Let Q be the origin of a small pencil whpse axis QA8T passes obliquely through a plate, the surfaces of which it cuts at A and ;Si. Let Q,, Qjj be the primary and secondary foci of the pencil after one refraction, q^, q^ those of the emergent pencil. Let AQ = u, 8qi = v^, Sq^ = v^, AB = t, the thickness of the plate, 0, ^' the angles of incidence and refraction at A, and of emergence and incidence at 8. From the first refraction AQ= - — s-^. u. (Art. 67.) Now ^ cos ^oo^ ^ or v^ = u + t. ^, « (i). '■ fj, cos'0 ^ ^ Similarly AQ^ = /j,u, 8Q^ = fj,v^; .-. v^ = u-{- -, (ll). " ytt cos ^ ^ The results (i),. (ii) determine respectively the primary and secondary foci of the emergent pencil. Prismn. 88. Defs. A prism is a portion of a refracting medium bounded by two plane surfaces inclined at a finite angle to one another. The edge of the prism is the line in which these two surfaces meet, or would meet if produced. The two plane surfaces are called the faces of the prism and their inclination to one another is the refracting angle of the prism. A plane perpendicular to each of the faces, and there- fore to the edge of the prism, is called a principal section of the prism or of the two surfaces. 89. When a ray of light is refracted out of one medium into another, as the angle of incidence increase's, the deviation also increases. Let 6, <^' be the angles of incidence and refraction of the ray, then sin <^ = /* sin j>', — and supposing /* > 1, sin (^ — sin + Dce^ increased, since ^— ;r — is < tt ; 2 2 ' .•. tan "^ ^ is increased — — r being positive ; 0-£ •^-0' 2 ■^eing<^. ie. — the deviation is increased if the angle of incidence be increased : — the same result will follow if /i < 1. 90. The axis of a pencil which passes through a prism in a principal plane, is turned froia the edge of the prism; — the prism being denser than the surrounding medium. Let QRST represent the course of the axis of the pencil in a prin- cipal plane of the pris^m. Then the normals at R, 8, the points of inci- dence and emergence, must meet either within, without, or upon one face of the prism. (i) Let them meet within, as at ; then it is clear that QR being the incident and RS the refracted ray the deviation at R is from the edge of the prism ; similarly ST being the ray emergent at 8, the deviation at 8 is from the edge. Therefore the whole deviation is from the edge. (ii) Let the normals at R, 8 meet without the prism, Fig- 2. ■ Fig. 3. THROUGH A PRISM. 83 as at 0, — then it is clear that the angle of refraction at E (fig. 2)_is less than that at S; and therefore the deviation at i?, which is towards the edge, is less (compare Art. 89) than the deviation at S, which is from the edge, — and vice versa in fig. 3. Therefore on the whole the deviation is from the edge, (iii) If the normals at B, 8 meet on one face of the Kg- 4- Fig. 5. prism, then at that point the deviation of the axis of the pencil is from the edge, and there is no deviation at the other poiat, B ov 8 (fig. 4 or 5). Therefore on the whole the axis is turned from the edge, or towards the thicker part of the prism. Cor. If the prism be rarer than the surrounding me- dium these effects will be reversed, and the deviation of the axis will be towards the edge or from the thicker part of the prism. 91. When a pencil is refracted through a prism in a principal plane, to find the deviation of its axis. Let QR8T be the course of the axis of the pencil (figs. 1, 2, of previous Article) in a principal plane which meets the faces of the prism in 8 A, RA. Let QR produced to some point t cut 8T, or 8T produced backward, in r. Also let the normals to the faces at R and 8 meet in 0. Let = z trT the deviation of the axis, * = / 8AR the refracting angle of the prism, (i) If the normals at R, 8 intersect within the prism (fig- 1). and since the four angles of a quadrilateral are equal to four right angles ; .-. -I = TT - z SOR = z 08R + z. 0B8= <^' + ■^'. (ii) If the normals at R, 8 intersect without the prism (fig- 2), D = / 8RQ - z r8R = tt - ^ + ^' - (tt - i/r + i/r') = t|p _ T^' _ (^ _ 0')^ and the inclination of two surfaces being the same as that of their normals, * = z R08='7r - R80 -8R0=^!r'- (j>'. Therefore in both cases, » = t'±<^' (i), and D = i/r - i|r' + (^ - ^') or D = i|r + ^-i (ii). These results (i), (ii), combined with sin ^ = yu. sin <^', and sin^lf = fj, sin -«|r', are sufficient to determine analytically the circumstances of a ray passing through a prism in a prin- cipal plane. 92. 06s. 1. Since neither 4>' nor ■\Jr' can exceed a certain magnitude, viz. sin~^ -, it follows that the equation ^' +-\lr' = i will be an impossible one, if i exceed twice the critical angle of the substance of which the prism is com- posed. When i exceeds this limit the ray cannot pass ' through the prism in the manner supposed, but wUl be' internally reflected at the second surface. — It may of course emerge after one or more internal reflexions. Obs. 2. If (f)' and i/r' be considered positive or negative according as they fall on the side of the normals at R, 8 THROUGH A PRISM. 85 towards or from the edge of the prism A, and therefore <^ and i|r, positive or negative according as they fall on the side of the normals at R and 8, from or towards A, the two cases of the previous investigation may be conveniently included in one formula -which will always be algebraically correct, viz. D = y^ + ^ — % i = ^' + -yfr', and sin (}> = fJ' sin (j>', sin ■\]r = fj, sin yjr'. Cor. If (j) and yfr be each small,: — in which case « is small, — we have I> = -\lr +

• Also sin ; •'• cos 9 = /^ cos 9 -^ (m;, sm -v|r = /i sm -«|r' ; .-. cos A|r ^ = ^ cos •^ -^ (iv); d^' _ cos (j> d^ _ _ COS'vIr ■ ■ d^ ~ fj> COS 4>' ' dcf) fi cos ^jr' ' whence cos

-T77 = —f — 5-77 , — a positive quantity. dcji' 1^^ cos ^ cos ^ ^ ^ -^ Hence <^ = i^ makes D a minimum. That is, when the angle of incidence is equal to the angle of emergence, the deviation is less than in any other case. 94. We give another method of obtaining the condition of minimum deviation, which does not require the use of the Differential Calculus. "With our usual notation D = '\lr+^-t, i=(f,' + '\lr', sm^'. Eliminating <}>', yjr' we get sin <}) = fi sin + sin^ t = ^— 1^ + hzm^ = 1 — COS ((^ — -v|r) . COS {(j) + yjr), 2 sin sim|r = cos ((^ — ilr) — cos (^ + i|f). Whence (/u.^ - 1) sin^ i = {cos i + cos (^ - 1^)} {cos ^ - cos (^ + ilr)}. Now observing that i is a given quantity, we see that I) will be least when ^ + -«|r is least, i.e. when cos i— cos (^ + yjr) is least; this will be the case when cos« + cos (^ — -v^) is greatest, i.e. when cos (^ — i|r) is greatest, which it will be when

' the angles of incidence and refraction at the 1st surface, ^fTj^lr' emergence and incidence 2nd From the first refraction, ' cos' (f) Now the pencil emanating in the primary plane from Q^ diverges in that plane after refraction from q^ : hence if we suppose its course reversed, a pencil converging to §■;■ will converge in the primary plane to Q^ after refraction into the prism ; ■ „ a cos^ ylr' ^ cos i/r ^ cos" (b' . cos' -Jr ' C0S^^..C0S^l/r "■ ' Similarly AQ^ = fiu, AQ^= (juv^ ; ■'■k = '^ (ii)- These results (i), (ii), combined with the relations sin ^ = /i sin 0', sin ■\lr = fi sin yj/, and ^' + '\Jr' — i = the refracting angle of the prism, — are sufficient to determine the position of the foci. Cor. 1. From (i), (ii), we may obtain V, _ c^fcosy. _ ( ^^-l)tan''0+/. ' ■ V, cos''<^cos^./r" wnicn.ib-^^,_^^^^^,^_^^,. THROUGH A PRISM. 89 a result which shews that (ji being > 1) the primary or secondary focus is the nearer of the two to the prism, accord- ing as ^ is < or > t/t. Cor. 2. If the pencil at emergence diverge from a point, i.e. if v^ = Vj — we must have cos^<^' cosV cos(j) cosi/r Y r> which result is also the condition of minimum deviation. In this case the emergent pencil diverges from a point at the same distance as its origin from the edge of the prism, and the degree of divergence remains unaltered. This result is of importance in Newton's experiment hereafter described. Successive Refractions at Spherical Surfaces. 97. Bef. A lens is a portion of a refracting medium bounded by two surfaces of revolution which have a common axis, called the axis of the lens. Ohs. The bounding surfaces of a lens will, unless the contrary be expressed, be considered spherical — under which denomination plane surfaces are included as a particular case when the radius of the sphere is infinite. •If the surfaces of revolution do not meet, the additional surface which is required as a boundary to the lens round the edge, will be a cylindrical one, having its axis coincident with that of the lens. 98. Lenses are distinguished by different names, accord- ing to the nature of their surfaces. Thus in the figure : — 90 REFRACTION a is a double convex lens, b ... double concave, c ... convexo-plane, d ... concavo-plane, e ... plano-convex, f ... plano-concave, 9\ ,r... convexo-concave, h) Jc) ,(■■• concavo-convex, the forms g and Jc, in which the concave surface is less curved than the convex, are also known by the name of a meniscus. In these figures, light is as usual supposed to come from the right, and the order of the terms which are combined to form the designation of any lens is that in which it is incident on the two surfaces. Thus c and e are the same lens, but it is convexo-plane in the former case because light is incident first on the convex and then on the plane surface, — ^plano-convex in the latter case, because light is first incident on the plane and then on the convex surface. It may be convenient to speak of the surfaces in the order in which light passes through them, as the anterior and posterior surfaces. Obs. A pencil is said to be directly refracted through a lens, when the refraction at each surface is direct. 99. To. find the geometrical focus of a pencil after direct refraction through a lens, the thickness of which is neglected. Let Q be the origin of a pencil whose axis QAB passes directly through a lens, the thick- ness of which may be neglected, F^ and F the geometrical foci of the pencil after one refraction and at „, emergence. Let AQ = u, BF = v, r, s the radii of the first and second surfaces Q Fi THROUGH A LENS. 91 • of tlie lens respectively, lines being regarded as positive when measured in a direction opposite to that of the incident light. From the refraction at the first surface -j^-- = ^^^. (Art. 27) (i). Now a pencil converging to F would have F^ for its geo- metrical focus after refraction at B, (Art. 79) ; ■• BF^ V s ^'' therefore, if the thickness be neglected, or AF^ regarded as = BF^, we get V u ^ \r sj which result determines the position of F. Note. The points Q and F are called conjugate foci, with respect to the lens. 100. CoE. 1. If f be the thickness of the lens at the axis, since t = BF,-AF^, we might from (i) and (ii) obtain 1 1 t V s u r & a relation which is strictly accurate, and which may be used when the thickness is too considerable to allow of any of its powers being neglected. COK. 2. If the thickness be sensible, but small, and = t, since _^ = _i^- = _— ii =^fl--7^)iiearly, -^^-^ AF^{l.J^) 92 REFRACTION equation (ii) gives fi fit 1 _ fi — 1 \r- ^ \r sj fi\u r J or the effect of the thickness is to remove the point F far- ther from ^ by a distance equal to t [\ a-l fi \u r 101. Def. The geometrical focus of a pencil of parallel rays refracted directly through a lens is called the principal focus of the lens, — and the distance of this point from the posterior surface of the lens is the focal length of the lens. The focal length of a lens is commonly denoted by the symbol/! Hence we have This is the expression for the focal length on the sup- position that the thickness is neglected; if/' denote the focal length when the thickness is not neglected, we shall obtain from Art. 100, Cor. 1 — ^^(remembering that in this case u = CO ,v =f'), 1 1 t 1_^/U.-1 /i-1 fl' f s r and l=(^_l)g_l), THROUGH A LENS. 93 Whence ^i=)- fj, ' r r t T +- (iii), a relation between /and/'. If we neglect the square of - deduced from (iii), gives which accords with the result of Art. 100, Cor. 2, then the approximation r. Note. It will appear from the preceding, that when the thickness of a lens is not neglected the focal length does not remain the same when the order of the surfaces at which the refractions take place is reversed. 102. "We will next examine in what cases the focal length of a lens is positive or negative. Let T = CD the thickness of a lens measured parallel to the axis at a dis- tance Gn = y from the axis, t = AB the value of T when y = 0, i.e. at the axis; r, s the radii of the first and second surfaces. Then An = ~- , very nearly. Bm = 2s' r and Bm + T = t + An ; Hence /is positive or negative according as t is > or < t, i.e. according as the lens is thinnest or thickest, at the axis. 94 FOCAL LENGTH OF A LENS. Thus lenses may be divided into two classes distinguished by the sign of the focal length. A lens whose focal length is positive is called a concave lens ; — one whose focal length is negative is called a convex lens. Lenses may be constructed of an infinite variety of forms so as to have the same focal length — since there are two disposable quantities r, s, and only one relation connecting them, viz. ■ )-("-)(;-;)• 103. From a discussion of the equation 1_1_1 V u~f' due regard being had to the algebraic signs of the symbols u, v,f, the student will have little difficulty in deducing the following inferences : — ■ (i) Concave Lens. A divergent pencil incident directly on a concave lens diverges after refraction. A convergent pencil consists at emergence of diverging, parallel or converg- ing rays, according as its point of convergence is at a distance from the lens greater than, equal to, or less than the focal length of the lens. (ii) Convex Lens. A divergent pencil incident directly on a convex lens consists at emergence of diverging, parallel or converging rays, according as its origin is at a distance from the lens less than, equal to, or greater than the focal length of the lens. A convergent pencil converges after refraction. 104. Bef. The reciprocal (-7;) of the focal length of a lens is called the power of the lens. Since - , - measure the curvatures of the surfaces of the r s lens, we see that the power of a lens is proportional to the difference of the curvatures of the two surfaces. 105. A sphere may be regarded as a lens, and the formula for the transmission of a direct pencil through it EEFEACTION THROUGH A SPHERE. 95 deduced from Art. 100, Cor. 1 ; as however it is a case of some importance, we will obtain the formula independently. It will be convenient to measure lines from the centre of the sphere. To find the geometrical focus of a pencil of rays after direct refraction through a sphere. Let p be the distance from the centre of a refracting sphere of the origin of a pencil whose axis is refracted directly through the sphere, q^, q the distances of the geometrical foci of the pencil from the centre after refraction at the first and second surfaces respectively — lines being considered positive when measured from the centre in a direction contrary to that of the incident pencil ; r the radius of the sphere. Then from refraction at the first surface, i_/f = _/illl, (Art. 29) (i), q^ p r ^ and from refraction at the second surface, if the course of the pencil be supposed reversed, remembering that the radius is negative in this case, 1 jJL _IM — \ q~'q~ r ' q p r 9. P W^ which determines the position of the geometrical focus of the emergent pencil. 106. Def. The /ocaZ length of a refracting sphere is the distance from the centre of the geometrical focus of a pencil of parallel rays after direct refraction through the sphere. Hence if we write '/ for the focal length, 7 = - 2 ; ,111 and = ■>. 9 V f 96 MOTION OF CONJUGATE FOt!I. In a similar way the foml length of a hemisphere, mea- sured from the centre' of the spherical surface, would be = '—r , if the pencil is incident ^rsf on the plane surface ; but = -z y., if the pencil is incident _^rsi on the spheri- cal surface. 107. To trace- the relative change of position of the con- jugate foci of a pencil refracted directly through a thin lens. Suppose the lens convex, and let/ be the numerical value of the focal length, then the relation connecting the positions of the conjugate foci is ~ ~ ? ' ''^h.^^® the lines w, v are taken positive to the right ; — the direction from which light is supposed to proceed. It is obvious from the formula that u and v must increase and decrease together, algebraically: ^ i.e. Q,# always move in the same direction. Take two points F^, F^ on the anterior and posterior sides of the lens so that AF^ = AF^ =/. (i) Suppose u=cc, i.e. the incident rays parallel, then v=—f and F coincides with F^ the prin- cipal focus, on the posterior side of the lens. (ii) As Q moves from an in- finite distance up to F^, F also moves off to the left, and when Q coincides with F^, F has gone off to an infinite distance and the emergent rays are -parallel. (iii) When Q moves from F^ up to J., i^ appears on the posi- tive side of A , and moves up to- wards A, the emergent pencil being divergent and the focus F a virtual one, — and Q, F arrive at A simultaneously. FOCAI. LENGTH OF X LENS. 97 (iv) When u becomes nega- / tive or the incident pencil is con- f ._-:=s«^p--~^^ vergent, the emergent pencil is ^t^j} /^^-<4 ^ convergent, — and as Q moves off '^ \ from! A to an infinite distance, F moves from A up to F^. Similarly the change of relative position may be traced when the lens is concave. 108. Method of determining the focal length of a convex lens practically. Let a small bright object Q be placed further from the lens than its principal focus, q the a real image of Q on the opposite side ^ — -ft — -,,__ __ of the lens, '^'^^^-^^^JjS^^^^-^^'ir GQ = u, V n/= numerical focal length, Qg = «., then -1 + _L=^^ 1 11 or + - = - ; X — U U J .'. fx = ux- ■M' ...u- 2 • It appears from this result that the least positive value of X is 4f: hence if the image of Q formed at q he received on a screen, and the lens and screen be moved backward and for- ward till the distance of the image from Q is the least possi- ble, the focal length of the lens — withotit reference to alge- braic sign- — ^is one-fourth of this distance. If the lens be concave, let it be placed in contact with a convex lens whose focal length /' is such that the focal length F of the combination may be negative, the axes of the two lenses being coincident. Then if /' and F be each determined in the preceding manner/ is to be found from the formula {algebradc) P. o. 7 98 CENTRE OF A LBN'S. ^=i + l. (-A-rt. 114) F 1 or ^ = ^ / 1_1 the figure represent a 108*. Cylindrical Lenses. If section of such a lens perpendicular to its length, the image of a bright point Q will be extended into a line— slightly curved — and parallel to the length of the lens. Such lenses are occasionally employed, — for the purpose of extending the image of a bright point into a line of light, — as by Mr Huggins in his observation of stellar spectra. Centre of a Lens. 109. Def. The centre of a lens is a point in its axis where a line joining the extremities of two parallel radii of its two surfaces cuts the axis. Let OAB be the axis ; 0, 0' the centres of the first and second surfaces; ' H, 8 points in those surfaces where the normals RO, SO' are parallel. Join SB a.nd produce it, if necessary, to cut the axis in C, — C is the centre of the lens. Let r, 8 be the iradii of the surfaces, AB = t. Then by similar triangles y^, = - , r — AG r or s-t-AC rt whence AC= , which determines the position of C. The position of C evidently depends only upon the form of the lens. By observing the value of the expression for AC in dif- ferent cases, we shall arrive at the following results respecting the position of C FOCAL CENTRES. 99 (i) If the curvatures of the two surfaces of the lens are in opposite directions (as in a, b, fig. Art. 98), the centre lies within the lens. (ii) If one surface be plane (as in c, d, e,f) the centre lies on the curved surface. (iii) If the curvatures are in the same direction (as g, h, k, l) the centre lies without the lens — on the convex side in g, k — on the concave side in h, I. (iv) If the thickness of the lens be neglected, i.e. t=0, then AC=0, and the centre of the lens may be regarded as coinciding with the point A. Obs. G is the centre of similitude of the circles AB, B8. Focal Centres. 110. Def. The ultimate positions of the points on the' axis where the incident and emergent ray cuts the axis, when the direction of the ray between the two refractions passes through the centre of the lens, are called the focal centres of the lens. To find their position. Let mRST be the course of a ray, the direction of BS passing through G the centre ; m, n the focal centres ; r, s the radii of the first and second surfaces. (Fig. Art. 109.) Then in the limit when the ray coincides with the axis, we have from refraction at the first surface fi 1. /"• — 1 AC Am~ T ' and from refraction at the second surface yi 1 /"■— 1 BG~~B~n^ T"' and AC= , whence we get s — r A rt r, St s{s-r-t) + t' iM{s-r-t) + t' which determine the positions of m and n. 7- 100 ■ OBLIQUE PENCILS. 111. If a ray be refracted through a lens in such a manner that its direction between the two refractions passes through G the centre of the lens, its directions at incidence and at emergence from the lens will be parallel to one ainother. For if B and 8 be the points of incidence and emergence the surfaces at these points being parallel, the case is the same as that of a ray refracted through a plate whose surfaces touch the lens at R and S. When a pencil is refracted obliquely through a lens there will be an important difference produced, according as the pencil is refracted centrically or exoentrically, i.e. according as the direction of its axis between the two refractions does or does not pass through the centre of the lens. 112. . When a small pencil is obliquely and centrically refrOfCted through a thin lens, to find the distances of the foci of the emergent pencil from the centre of the lens. Let Q be the origin of a small pencil' whose axis QCST is refracted obliquely and centri- cally through a lens,-;— (7 being the point where the axis of the lens meets its first surface, which point is the centre of the lens if the thickness of the lens be neg- lected (Art. 109). L6t Q„ Q, be the primary and secondary foci of the pencil after one re- fraction, q^, q^ the primary and secondary foci of the emer- gent pencil. Let CQ = u, Sq^ = v^, Sq^ = %, r, s the radii of the first and second surfaces of the lens, 0, ^' the angles of incidence and refraction of the axis QG at G, and consequently the angles of emergence and incidence at 8. From refraction at the first surface /i cos" 4>' ops' 4> _ /^ cos 4>' - cos 4> "'CQ^ ~ u ~ r >(Art. b8> Now the pencil emanating in the primary plane from Q^ diverges in that plane after the second refraction from g^; OBLIQUE PENCILS. 101 hence if we suppose its course reversed, a pencil converging to q^ will in the primary plane converge to Q^ after refrac- tion into the lens ; fji, cos^ 0' cos^ 4> _t^ cos (f)' — cos (f> the points G, S being regarded as coincident ; .J. 1 _ /i cos 6' — cos (1 1> v^ u cos Similarly ^^ - ^ = ^ "°^ '^' " "°^ "^ , ' fj, 1 /4 cos 4>' — cos (^ (i)- . 1 1 " V. -- = (/icos^'-cos^)^---j (ii). Equations (i), (ii) determine the distances of the foci of the emergent pencil from S or G. 113. CoE. 1. The positions of the foci of the refracted pencil being thus known, the investigation of (Art. 70) gives the magnitude and position of the circle of least confusion. CoE. 2. If (j} be so small that its square may be neglected, 1=1 = 1 + (,_1)(1-1 v^ v^ u ^ \r s the same as for a direct pencil. Hence a pencil refracted centrically through a lens at small obliquity approximately converges to or diverges from a point at the same distance from the lens as the geometrical focus of a direct pencil with an origin at the same distance. CoE. 3. If we make a closer approximation neglecting powers of (f) above the third and putting

+ v^ ^\ 2/i ■ "^ Hence, practically the focal length of a lens is diminish- ed — or the power increased-^by inclining it slightly to the line of sight. CoR. 4. - If A be the distance of the circle of least con- fusion from the centre of the lens, the incident rays not being parallel, cos 4> 1 1 _ i;, cos ^ + ■Ug _ v^ V, A v^v^ (1 -I- cos 0) 1 + cos ^ = - f- - -^ / 1 \ ■y, Uj vj \1 + cos ^/ Combinations of Lenses. 114. To find the geometrical focus of a pencil of rays after direct refraction through a series of lenses in contact, whose axes are coincident. combination: of lenses. 103 Let Q be the origin of a pencil whose axis QG^C^... is refracted directly through a series of n lenses on a common axis, their centres being C^, C^... Let u be the distance of Q from C,; Wj, v^ the distances from C^, C^... respectively of the geometrical foci of the pencil after refraction through the first, second,... lens: — f^, f^... the focal lengths of the succes- sive lenses ; — lines being considered positive when measured in a direction contrary to that of the incident pencil. Then if the- thicljijess of each lens be neglected. 1 11 v. V. v.. 'A 7„ (A); + ■ - J ^ 2 (-^J = J suppose, which determines the position of the geojnetricfil focus of the emergent pencil ;— and shews that the power of a conibiha- tion of lenses in contact is equal to the sum of the powers of the several lenses,' Cor. If the leases be separated by finite interyals we have in place of equations (A), the following : \_ 1 ^1" »„ Vit«»-i /» 104 SMALL EXdENTEICAL PENCILS. By eliminating v^, v^'-'^n-i between these n equations v^ is determined. 115. If QC be the axis of an oblique pencil reflected at a spherical mirror or refracted centrically through a lens, we may generally — in the cases which occur in Optical Instru- ments where the obliquity is small — regard, (i) the reflected pencil as converging to or diverging from a point q, — Q, q being conjugate foci on the line QCq which passes through G the centre of the spherical surface j (ii) the refracted pencil as diverging from or converging to a point q which lies on the line QGq passing through C the centre of the lens,- — the point q being taken to be the position of the circle of least confusion — or else detel-mined from the equation--;^ — Tir\~i!> which will generally be vq L(^ J sufHciently approximate. When only a small part of such a pencil exists — as the small excentrical pencil Qrs — we may still regard this excen- trical portion as diverging from or converging to the same point Q after reflection or refraction. . LENSES USED IN LIGHTHOUSES. 103 The student will bear in mind that this supposition is only approximately true; but it is sufficiently so for the purpose of description and explanation of pencils passing through Optical Instruments, and we shall make frequent use of it hereafter. 116. The following construction for determining the posi- tion of q for an excentrical pencil is accurate to the same de- gree of approximation as is above supposed. Let i'^be the principal focus of rays, — proceeding from right to left" suppose, — either in the case of a lens or mirror ; Q the origin of the pencil, QR a ray parallel to the axis of the lens. or mirror and incident at R, QGq the ray which passes through. C the centre of the lens or mirror, — join RF, — then, if we neglect aberration, RF or FR is the direction of the ray QR after reflexion or refraction, — and on the same approximate supposition q will be the point in which QC and RF, — pro- duced if necessary, — intersect each other. 117. The forms of lenses now commonly used in Light- houses will perhaps be understood from the following brief description. The figure represents a section of the lens by a plane through its axis, with respect to which axis the lens is sym- metrical. 106 LENSES USED IN LIGHTHOUSES. The central part is the same as the section of a convex lens, the light being at its principal focus,^ — so that the rays at emergence from G are parallel to OG^ a, a... are sections of prisms, — the front surfaces of which are however somewhat convex, — so that the rays emerge from them parallel to DC, and b, b... another set of prisms from which the rays emerge parallel to OG after internal reflexion ; d, d is sl section of a polished spherical reflector whose centre is 0. (i) If we suppose the figure to revolve about 00, a hori- zontal axig, there will be fornied an annular lens, — and the beams transmitted through the prisms a, a, would be hollow cylindrical shells surrounding the central beaijj transmitted through the lens G. In this form of lens the e?:treme prisms b, b... are omitted, and when two or more such lenses are fixed in a polygon^,! frame which is made to revolve about a vertical axjs passing through 0,— rthe cylindrical beams transmitted through these annular lenses sweep the horizon and produce a revolving or periodic light. (ii) If wp suppose the figure to revolve through any given angle about a vertical a^is through 0, there will be formed a cylindrical lens, — and the light transmitted through the several parts of it will form a horizontal beam in the form of a sector of a circle embracing the part of the hori?;on within .which it is intended to be seeij, and constituting a fixed or permanent light. The rays diverging from in the direction opposite to that of the lens are returned by means of the reflector d, d — and from the fact that by means of the lens, prisms, and reflector, the rays which diverge from in every direction are rendered serviceable, this arrangement is called holo- photal. The light consists in general of a set of Argand lamps. For further information on this subject the student may consult the article Lighthouses,va. the Encyclopaedia Britannica. CHAPTER VI. EEFEACTION THROUGH MEDIA OF VARYING DENSITY: — ■ REFLEXION AND REFRACTION AT A SURFACE IN ANY MANNER. 118. When a ray traverses a medium the density of which varies continuously from point to point, we may regard the value of /i,- — the absolute refractive index, — rat any point as a function of the position of the point ; and the equation fi= G,a constant qiiantity, would belong to a surface at every point of which the absolute refractive index is the same, — and the form of this surface will indicate the law of stratification of the medium. By varying this constant continuously we should obtain the consecutive surfaces of equal refractive index in the medium ; and ii fi,fi + dfi be the absolute indices of refraction for two consecutive strata, the relative index from the former of the two into the latter will be . (Art. 83, 84.) When the change of density and therefore of refractive index is continuouSj the ordinary law of refraction will lead to a differential equation, the solution of which will enable us to determine the path of the ray. 119. Suppose a ray passing from a stratum A into another B, — fj,, fj, + dfi the absolute refractive indices of A and B, — ^, (p + 8 the angles of incidence and refraction at the common surface of A, B : — then we have (Art. 10), sm

= log G — kx; .: sin ^ = sin a . e""^, if = a when x ■■ And | = tan^; dy sin a . e""-^ "c^x V(l-sin'a.6-2'<'^)' The integral of which gives sin (a — Ky) = sin a ■ e""", ii y = when x = 0. The curve has an asymptote y = - • K 121. A ray passes through a medium, the value of fi at any point of which is a function of r the distance from a fixed point ; to find the equation to the path of the ray. The medium is stratified in spherical surfaces concentric with the fixed point G. Let ACP = e, GP = r be co- ordinates of any point P of the path of the ray, which will be in one plane passing through C. .+ (d^ + de)cos^ = (i), 110 EEFEACTIOIT THROUGH and remembering that •.d9 i . '^1' /■■\ this becomes ^ + cot + d^ ; .-. sin^ = ^^^ -sin{i^ + dj) + di^), whence = -^ sin ^ -f- (cZ^ -}- di/r) cos ^. 112 EEFBACTION IHEOUGH Now d^^^£.d. + ^y.dy, and J5=d.(tan-g), ■ '^'^ '^^'^ ' dx ^ {xy) suppose, dP F= cos i = cos i|r cos — sin ijr sin (f> cos w (iii), cos D = cos 2|r cos to . . .(iv). These four equations connecting the seven quantities <^, -f, 01, ^2, i 03, D, are sufficient to determine the circumstances of the ray, when any three of them are given. Cor. 1. We can easily shew that IP = IR, for cos IP = sin PA cos PAI, since AI = ? in the A PAI = sinP4.sinPJ5 = sin Q J.. sin QAB = sin Q,B . sin QBA = sin Q5 . cos Q,BI (= cos QI) = sin RB . cos QBI = cos RI. Hence, IP = IQ = IR, that is, the inclination of the ray to the line of intersection of the mirrors remains unchanged. OOE. 2. Further, since P/Q=7r-24/P, QIR=-7r-2BIQ; .: PIR = QIR - PIQ = 2 (AIP - BIQ) =2[(AIR+PIR)-('!r-BIR)] = 2{{AIR-{-BIR)-'7r+PIR] = 2 {(tt - i) - TT + PIR} ; :. PIR = 2i. AND KEFRACTION IN ANY MANNER. 117 That is, planes drawn through the line of intersection of the surfaces parallel to the direction of the ray before the first and after the second reflexion include an angle which is double the angle between the reflecting surfaces. 127. To find the direction of a ray after being refracted at two plane surfaces in any manner. Let radii of a sphere be drawn parallel to the direction of the ray at incidence on the first surface, after the first and after the second re- fraction, — their directions being con- trary to that of the pencil in each case, and let them meet the surface of the sphere in P, Q, li. Also let radii parallel to the normals to the first and second surfaces — drawn on that side of each surface on which the ray is incident — meet the sphere vn A, B: and a radius parallel to the intersection of the two surfaces, in /. Let DA — ^'1 ^^ angles of incidence and refraction at 1st surface, fzX) — - AB = i the angle between the two surfaces, PAB = 6^, QBA = 0^ the angles which the planes of first and second refraction make with the principal plane, i.e. the plane perpendicular to each surface, a) = AQB = the angle between the planes ot first and second refraction, /jifj/ the indices of refraction from the first to the second, and from the second to the third medium, 118 EEFEACTION IN ANY MANNEE AT ANY SITEFACE. B = PR the deviation of the ray; — then we have the relations sin a _ sin 6^ _ sin 0^ sin i sin ■y^' sin (^' ' cos i = cos 0' cos i/r' + sin ji sin y^r cos w, cos D = C0S(^ — 0')cOS('\|r — •\|r') + sin(^ — ^')sin (i|r — l|r') cos 0) and sin (^ = /4 sin ^', sin i|r' = yti' sin i/r. These equations are sufficient to determine the circum- stances of the transmission of the ray, when the quantities given render the problem determinate. CoE. If the first and third medium are the same, the case becomes that of refraction through a prism, and /i . /*' = 1. In this case we can shew that IP = IR. For cos IP = sin PA . cos PAI = sin PA . sin QAB = jx, . sin QA . sin QAB = /i . sin BQ . sin QBA = sin BE. sin QB A = sin BB. cos RBI = cos IR ; whence IP = IR, that is, when a ray is refracted in any manner through a prism, its directions before the first and after the second refraction are equally inclined to the edge of the prism. Note. Several analytical results defining the course of a ray reflected or refracted under different circumstances, will be found among the problems on this Chapter. CHAPTER VII. SPHERICAL ABERRATION OF LENSES; — EXCENTKICAL PENCILS. 128. In Chapter V. we have neglected the aberration arising from the spherical form of the surfaces of the lens through which a pencil passes. In the construction of lenses for telescopes, &c., it becomes a question of great importance to ascertain what particular forms can be given to a lens or a combination of lenses so as to destroy the aberration of a pencil passing through the instrument, — or, if this cannot be done, to reduce it within the narrowest possible limits. The formulae and calculations to which this enquiry leads are very tedious and intricate, and we only propose in this place to give one or two of the more simple investigations bearing on this subject, as an indication of the nature of the processes to be carried out, — referring the Student who has leisure and inclination to pursue the subject to works where it is more fully discussed. See a Paper by Sir John Herschel, On the Aberrations of Compound Lenses and Object Glasses. Phil. Trans., 1821. A Paper by the Astronomer Eoyal, On Spherical Aberration of the Eye-pieces of Telescopes. Camb. Phil. Trans. Vol. iii. Astron. Notices, Dec. 1862, Vol. xxiii. p. 69. Art. Light, in Encycl. Metrop. Coddington's Optics. 129. When a pencil is refracted directly through a lens, to find the point where the direction of any ray cuts the axis after refraction — and the aberration of the pencil. 120 SPHERICAL ABEEEATION Let Q he the origin of a pencil whose axis QAB is refracted direct- ly through a lens of inconsiderable thickness. Let QJtSThe the course of any ray whose directions after one refraction and at emergence cut the axis in g, and q respectively. Let AQ = u, Aq = v', r, s the radii of the first and second surfaces of the lens, — lines being accounted positive when measured in a direction contrary to that of the incident pencil, AB = y. Now from the first refraction, Aq^ 1 u r. fjL \r uj '^-^l)f,(Art.53)...(i). u If we suppose the course of the pencil reversed, a ray of a pencil converging to q after refiraction at £S cuts the axis in q^. Hence, neglecting the thickness of the lens so that B8 = AR = y, we obtain .f^ -1 + ya^ U v'J [s v' J 2 ^^^'' • • = (/^ ■ V u 1) + /i + 1 r uj \r u J \s v'J \s v' 2' In the coefificient of y^ we may use for v' its first approxi- mate value V given by the equation 1 u :(/.-!) (Art. 49); 1_1=(,_1)(1_1) V u ^ ^ \r sj + f^ 1 fji + W^ fl 1 \s V 1 f^ + i\]y' s V /J 2 ' OF LENSES. 121 which determines the position of the point q, and the aberration of the ray q8=v' — v f^-'i- ifi IV /I fi + i\ (1 iV/i f^+AUY f^ \s vj \s V J) Note. The position and magnitude of the least circle of aberration in this or any other case of combined direct re- fractions is given by Art. 56. 130. To investigate the form of a lens of given focal length in order that the aberration of a given direct pencil of parallel rays may be the least possible. The aberration of a pencil of parallel rays refracted directly through a lens the radii of whose surfaces are r, and s, — putting w = oo and y =/ in the expression for the aberration in the preceding article — 1 /I IV /I /i + l\ ... ^?-t-7Jt-7^j W> where {fi — 1)1 J = ;j,a given quantity (ii). The former expression is therefore to be made a minimum by the variation of r and s consistently with the latter con- dition. If we differentiate (i) with respect to - , and observe that by (ii) the differential coefficient of - is = 1, we get 3 2^ + 3\ -?-G--7){r-^': Q (1 1 V A 1\ "M^""^ (7^=^1X71 [s"/) Ij. (2 {n - 1) (At + 2) 2iJ?-^- 4 (/^-l)7l s / 122 MINIMUM ABERRATION S_ 2(fJ.-l)(fJi + 2) jS , s , 2 (At + 2) a(2fjt,+ l) ..... ""'^;-^+(7^r^ + 2/.--^-4 =' 2^.--;.-4 -("^)' also the above value of s makes the second differential co- efficient of (i) positive, so that the relation of the radii ob- tained in (iii) makes the aberration a minimum. 130*. If we call the minimum aberration Bf we shall obtain ^f_ m(V-i) f J 8(^-l)^(/. + 2)-/' The aberration tends to become less as n increases — but it remains considerable for all substances known in nature. If /i = 2, as in zircon, ^f—~T^'^-> If ii = 2'5, as in diamond, Bf— — ^rr,-T.. -^ 18 y See infra. Art. 246. See a short Paper by Lord Eayleigh " On the minimum " aberration of a single lens for parallel rays." Proceedings of Cambridge Philosophical Society, Vol. III., part viii., p. 373. 131. Ex. 1. Suppose fi = 1'.5, then - = — 6, 7 . _ 7 . "" and the value of the ininimum aberration will be found by substitution in the expression of Article 129 to be ^_15y' 14/- If the lens be a convex one, that is / negative, and the ratio of s : »' =f= 6 : 1, this lens is known by the name of a crossed lens. Ex.2. If 2/A'-;ti-4 = 0; s i.e. fi = 1'6861, then - = oo , FOE ANY PENCIL. 123 and the most advantageous form of the lens for collecting all the rays into a real focus is convexo-plane, having the anterior surface convex. This value of /i is nearly the refractive index of several precious stones and the more refractive glasses. 132. To find the form of a lens of given focal length in order that the aberration of a direct pencil diverging from a point at a given distance may he the least possible. The focal length of the lens being given, the value of v for a proposed value of u is known. Hence we must have \r uj \r u ) \s vj \s v s) f V xo' In consistence with these two latter conditions, let us assume 1 a+1 1 a-1 = a mmimum, whilst (m-1)(J- V 2/ ' « 2/ ' 1 x-^1 1 x-1 ^^^^ r-2(/.-l)/' s 2(^-1)/' so that X is the only variable quantity, — we get by sub- stitution, 1 iy/1 f. + i\ r u) \r u J -to Now we pass from '^ * I by changing the signs of T ; ^to- u \s vJ \S V I . g-,^^— — 3 {a; - (> - 1) a - /^}' {a; - (/^^ - 1) a - /^^} . . . (ii)^ 124 MINIMUM ABERRATION In adding (i) and (ii), terms of odd dimension in x and a disappear, and the sum becomes 4(;:^3{^^-4 (^+1) a.+ (;.-!) (3;.+2) .^^J^ , .which is to be a minimum by the variation of x ; ...0 = ^a;-2(/. + l)«; /i— J. . ,._ 2(/.'-l) ^ , which determines r and s, and consequently the form of the lens. The ratio of the radii = - = ^ - r X — I Cor. If the iacident pencil consist of parallel rays, J ^ s fj,(2/i + l) M = 00 and a = 1, - = ^r^ r > r 2/A^ — /i — 4 the same result as was obtained in the previous article. 133. Obs. The expression obtained above involves the quantity a depending upon the distance from which the inci- dent pencil diverges, so that it appears the best form of lens for diminishing the aberration will vary for different distances of the origin of the pencil. If the aberration for rays parallel at incidence on a com- pound lens of given focal length — consisting of several thin lenses in contact — be examined, it will consist of a series of terms similar to that in Art. (129), one term for each lens, and the condition that the aberration shall vanish will lead to an equation involving more than one unknown quantity — and consequently admitting of an unlimited number of solutions. As an instance of an aplanatic combination of two lenses of glass (fi= I'o), we may mention one calculated by Sir John Herschel. „. ,, (r = - 5-833 . First lens \ _ /= - 10, FOR ANY PENCIL 12-5 second lens in contact with the first, r = - 3-688, or -2-054, s = - 6-291, or -8128, /= - 17-829, or - 5-497. The first lens of the combination is a crossed lens, the second a meniscics — the focal length of the combination is — 6-407, or — 3-474 according as the first or second of the two sets of valuer for r, s,/in the second lens, are taken. 134. Ezcentrical Pencils. The mode of determining the form and course of direct pencils and of oblique centrical pencils when reflected or re- fracted has been sufficiently explained in the preceding pages. All pencils which do not belong to one or other of these classes are excentrical. For the investigations requisite to determine accurately the direction and form of such pencils the Student is referred to Coddington's Optics, Part I. We shall here give an approximate mode of defining the direction of the axis of an excentrical pencil which will be of service hereafter — the axis of the pencil being supposed in each case to lie in one plane with the axis of the reflecting or refract- ing surface — this being the ordinary case which occurs in Optical Instruments. 135. To find the course of the axis of a pencil after excen- trical reflexion at a spherical surface. Let QXA be the axis of a pencil incident at A on' a spherical reflect- ing surface of which is the centre of the surface and G the centre of the face, AT the axis of the re- flected pencil, so that X, Y are the points where the axis of the inci- dent and reflected pencil cuts GYX the axis of the reflecting surface. Let GO = r, OX = b, 07 = c', lines being considered positive when measured from G in the direction more nearly opposite to that of the incident pencil. A\soletAC=y, ^AYC = r], ^AXG=e. 126 EXCENTRICAL PENCILS. Then, as in (Art. 21) -y-^ = yyj , whence if powers of y above the first be neglected, and c re- present the first approximate value of. c', since in that case ' X, T may be regarded as conjugate foci, (Art. 22), , 112 CO r , tan rj h and = - ; tan e c equations giving c and tj, — quantities which define the direc- tion and position of the axis after reflexion. 136. To find the course of the axis of a pencil after ex- centrical refraction through a thin lens. Let h, c' be the distances from the centre of the lens of points where the axis of the pencil cuts the axis of the lens before and after refraction, r, s the radii of the first and second surfaces of the lens, lines being considered positive when measured in the direction more nearly opposite to that of the incident pencil. Also let y be the distance of the point of incidence of the axis from the centre of the lens ; e, rj the inclinations of the axis of the pencil to the axis of the lens before and after refraction, /the focal length of the lens. Then by an investigation similar to that of Art. (129), if powers of y above the first be neglected, and c represent the first approximate value of c', c' b \r s tan 77 _ & tan e c ' or, approximately, ---= (^-l) (^-- -j , tan 77 b tan e c ' which results give c and rj. EXCENTRICAL PENCILS. 127 137. To find the course of the axis of a pencil after excentrical refraction through a series of lenses which have a common axis. Let the axis of the pencil cut the axis of the lenses before and after refraction through the first lens at distances h^, c^ from its centre, and let f be the focal length of the lens. Let the same letters i, c,/with suffixes 2, 3 ... denote similar quantities relative to the second, third, ... lens, and if the thickness of the lenses be neglected, as a first approximation we have the system of equations 1 1 _1 1 1 _1 1 1_1 ,, , Ti,s orK~A' cr\-fro„-K-f}^''-^^^^' whence c„ may at last be found. Also if e be the inclination of the axis of the pencil to that of the lenses before incidence, Vi> Vij ••• Vn ^^^ inclinations after refraction through the several lenses, we have tanr]^ ^ h^ tan?;^ ^ h^ tan rj^ ^ \ tan e Cj ' tan ij^ c^'" tan 77„_j c„ ' , tan 17^ &,5, ...6„ whence - — - = ^-^ ; tan e c^c^ . . . c,. which results give c„ and r]„. Ohs. In each of the cases of the preceding three articles if the calculation were carried to a second approximation, we should have results of the form c'--c-^^y' tan^^6 tan e c where the quantities Ajf, B-if are terms introduced by the aberrations as calculated in previous articles. CoE. "We should find that J5 x j-^ : and if h be large and therefore c =/ nearly, we should have tan 7/ X -7. ^1 + B' ■j-J . 128 EQUIVALENT LENS. 138. Equivalent Lens. Bef. A lens is equivalent to a system of lenses on the same axis when an excentric pencil after refraction through it is inclined at the same angle to the axis as if it had been refracted through the system of lenses — the single lens having the position of that lens of the system on which the pencil is first incident. If F be the focal length of a lens equivalent to the lenses in the last proposition, and if the pencil after refraction through it cut the axis at an angle i; at a distance c, then using first approximations, we have tan,_6, l_l=^(^,t.l36), tan e c tan?; _ F tan 6 1^ F' tan7?„ _ 5,5, ...5„ tan 6 c^c^...c^ (Art. 137), If as is generally the case in the eye-glasses of telescopes 5j be very large, and j- may be neglected, we have i^=£i^» Note. See a paper On Equivalent Lenses by R. Pendlebury, M.A. — Messenger (^Mathematics, Vol. vil., p. 129, Jan. 1878. 139. We will give an independent investigation of the following example of an equivalent lens. To find the focal length (F) of a lens equivalent to a combination of two lenses — {focal lengths f,f) — on the same axis at a distance (a) from each other. Suppose the two lenses to be convex and their thickness neglected. Let a ray QB, parallel to the axis be refracted by the first and second lens in directions B8 and 8T. Then if we draw H V parallel to ST, A V will represent F the focal length of the equivalent bamsden's eye-piece. 129 lens, i.e. AVis the focal length of a lens which will produce the same deviation in a ray incident parallel to the axis, as the combination of the two lenses produces. Now AX =f, ; .-. BX=f^- a. Also 'BT'BX^fJ "■BT fjf,-a fJA-a)- Also A V= F, and by similar triangles A VR, BTS, F AR . ., , AX -g^ = -g^ = similarly -g^; . 1_ BX (f^-a)f^+f^-a ■■F AX.BT- f, -Mf^-aY f f :. F= „ ^y — ; the focal length required. /l +.72 ~ * Ohs. In the above each of the three lenses is treated as a convex lens, SknAf^,f^, F as their numerical focal lengths — if they were treated as concave lenses, we should obtain which is algehraically true in all cases of the combination supposed in the statement. 140. Ex. 1. Suppose /j =f^ and a = f^, then F= |_^. This combination represents Ramsden's Eye-piece — two convex lenses of equal focal length, placed on the same axis at a distance from each other equal to | the focal length of either. The lenses have generally one surface plane, the other convex, — and the convex surfaces turned towards each other. P.O. 9 130 HUYGHENS' EYE-PIECE. 2 Ex. 2. Suppose y; = 3/„ a =/, -/, ; .-/-^^ = a. This combination represents Huyghens Eye-piece — the focal length of the first, or field, lens being three times that of the second, or eye, lens, — each being convex and the dis- tance between them equal to the difference, or semi-sum, of their focal lengths. The field-glass is generally convexo- concave and the eye-glass convexo-plane. CHAPTER VIIL IMAGES AND CAUSTICS. 141. Def. If a luminous body be placed before a re- flecting or refracting surface, a pencil of rays will emanate from each point of the surface of the body, and will have after reflexion or refraction a geometrical focus or circle of least confusion, according as the incidence is direct or oblique. The locus of these foci or circles of least confusion for con- secutive points of the object will form a superficial figure, more or less resembling the object and called its geometrical image. If the rays be reflected or refracted more than once there will of course be an image formed after each reflexion or refraction. The consecutive intersections of the reflected or refracted rays emanating from any given point of the object will form a caustic surface, and an eye in any suitable position will re- ceive a small pencil of rays, the axis of which is a tangent to this caustic, and apparently diverging from the point of con- tact of this tangent with the caustic — or nearly so. The locus of these points of contact for consecutive points of the objects constitutes the visible image — the position of which, as well as its form and genera:l resemblance to the object, will vary with every change of position of the eye. See the illustra- tions given in Chapter ill. Art. 74. It will readily be seen that the difference between the visible and the geometrical image, both as regards position and form, will in general be greater, the greater the obliquity of the pencils by which the former is seen, — but when this obliquity is small and the extent of object viewed inconsiderable, the visible image will very nearly coincide with the geometrical, — and in such cases they may appreciably be regarded as coincident. This will be strictly the case at the centre of the field of view of a tele- 9—2 132 IMAGES. scope — and when the field of view is not large it may be regarded as true over its whole extent. When we speak of image simply, we shall in general mean the geometrical image. 142. Def. If an image consist of points through which the light actually passes it is called real; — in other cases vir- tual. Hence a screen placed in the position of an image will receive illumination only when the image is real. A familiar instance of a virtual image is that formed by a common looking-glass of an object in front of it: — the image of an object under water is virtual. The images formed by the object-glass of an astronomical telescope — by the large mirror of a Gregorian Telespope — by a camera obscura — are real. Further, a real visible object and its optical image diifer in this respect— from the former, light emanates ini every di- rection, and it can be seen in any direction, if nothing opaque is interposed between it and the eye,- — an image can only be seen when the eye is placed in the pencil of rays which go to form it, or diverge from it. If however the image be received on a screen, — such as paper or roughened glass — the light constituting the image illuminates the screen like a picture, and it can be viewed by the eye as if it were a real object. 143. Def. An image is erect when corresponding points of the object and image are on the same side of the axis of the reflecting or refracting surface — inverted, when they are on opposite sides. In the former case the corresponding parts of the object and image appear in the same relative directions in space — in the latter, the image is inverted as regards up and down, and also reversed as regards right and left. Considering the line of sight along the axis of the lens or mirror as the centre of the field of view, if the image were turned through 180° about this axis, the parts of the image would then be in the same relative position as the corresponding parts of tke object. The image of an object seen by reflexion at a plane mirror in a vertical position, suppose — is erect as regards up and down/ — but reversed as regards right and left. IMAGE OF A STRAIGHT LINE. 133 144. Geometrical image of a straight line placed before a lens, — cutting the axis. of the lens at right angles. Let PQ be the object, C the centre of the lens. We will sup- pose the image formed by centrical pencils; and the size of the object FQ to be small, so that we may take the formula J__J__1 ... Gq GQ f ^'^' to be that which connects the conjugate foci of any point, Art. (115). CP =a, Cq = r, f= focal length of the lens, then CO = r , and (1) becomes cos 9 1 cos rf) 1 f . --. ^=;?, orr = ^ , ' 1 + - cos = f, the image is elliptic, virtual, and diminished. (ii) Suppose the lens convex, /negative, a>f, the image is real, inverted, and elliptic, — G the further focus, — and magni- fied or diminished according as Cpx GP, i.e. as -^^> < a, or a <> 2/ f A ok pT ■""^ X,.---^ c p -^ V k 134 CURVATURE OF THE Obs. In a similar way the geometrical irbage of a straight line formed by reflexion at a spherical mirror may be ex- amined, and will be found to be a conic section. 145. Cor. 1. It may be useful to examine the more accurate value of the curvature at the vertex of the image, when the primary focus of an emergent pencil is taken to be the image of the corresponding point. Then Gq = v^,GQ- cos connected by the formula 1 1 _ M cos 0' — cos <^ v^ u cos'' (f> .{fj,— \)f (Art. 112) ; 1 _ cos <^ fJb COS 0' — COS ^ " v^ a cos^ ^ (/* — 1)/ Let Gp = v. then ^ff.= - + -,, V a f also Cn = Vj cos ^, qn ='v^ sin ^ ; ;. p = rad. curv. at » = ^ Lt. — — = -^z Lt. tj^ r-^—- '^ ^ 2 pn 2 V — Vj^ cos

\cos

'-cos [ ■^1 a +cosX;.-l)/l '^'^'"P sin^<^ ^ 1 L^. ^: 2 ' V cos (f) fj, cos ^' — cos OS (6 /A cos d) — cos = 0; IMAGE OF A STRAIGHT LINE. 135 P = IU.\ TT^i^ ■^ Z -(it COS — COSrf) | cos-./,(/.-l) -""^"^ = |Lt, ^ ^ Xl-fj-ll-f) ^^ ,. f = — ^ , after reduction. 146. CoE. 2. Suppose tlie image to be formed by the circles of least confusion. Let A= Cq = distance from C to circle of least confusion ^tyvOdt^ (Art. 70), v^ cos + 1;^ qn = A sin = 0, and is i_ ~ 1 1 .: 2, = Lt. M!= Lt. ^:^ = Lt. /^f\ '^ j3w Gp — Cn Gp-A cos = Lt. T^ . .. ^^^ "^ , , and tv- = 1 ultimately. CJp 1 cos^ up A Cp Alsoi = ^-^ + U(l + J)|(Art.ll3,Cor.4), .'. 2p = Lt. = T '^ 1 — COS0 •/-^ (-^)| :^-V--,.;2/ 136 DISTORTION. /■ I > ' 1 + 2/^ and .•. the curvature = -=(2+-)-2.. P \ 1^) f Note. If the image be formed by centrical pencils re- fracted through a system of thin lenses of same substance in contact, — the curvature of the image at the vertex would be (--Xt)^ which, as we see, depends only on the power of the combi- nation, — and not on the^orms of the lenses or the position of the object. The investigation of the curvature when the image is formed by excentrical pencils is more complicated — for which and other results for excentrical pencils the student is re- ferred to Coddington's Optics. Distortion. 147. The resemblance of an image to the object will seldom be perfect. If the image were exactly similar to the object, the ratio of the distance of two points of the image to the distance of the corresponding points of the object would be uniform for every combination of points that could be taken. When this is not the case, the image is distorted. Let us consider the image formed by a lens or a spherical mirror. If the ratio of the distances of two points of the image and of the two corresponding points of the object — measured in each case parallel to the axis of the lens or mirror — be not constant for different points, there is a dis- tortion in direction of the axis. This is commonly called distortion of curvature of the image. The image discussed in Art. (144) suffers this kind of distortion, and this particular example will account for the apparent convexity of the field of view of an astronomical DISTORTION. 137 telescope, siuce a plane surface in front of the object-glass would have for its image — at the principal focus of the object- glass — a surface concave towards the object-glass, and there- fore convex towards the eye-glass ; on which convex surface the objects contained in the original plane field observed, would appear to be distributed. Again, let the points of the image and object be referred to polar co-ordinates (r, 6) — the radius vector r being mea- sured perpendicular to the axis of the lens or mirror, and the ^ 9 being the angle which r makes with a fixed plane passing through the same axis. Then if the ratio of the radii vectores of any point of the image and of the corresponding point of the obj.ect is not constant when different points are taken — there is linear distortion of the image. And when the values of 6 are not the same for pairs of corresponding points of the image and object, — there is angu- lar distortion. When an image is formed by centrical pencils, neither linear nor angular distortion will have any existence. When the image is formed by eoocentrical pencils — as in the case of the eye-glass of a telescope — there will be linear distortion in the image : — and if the eye be not on the axis of the lens, or if the surfaces of the lens be not surfaces of revolution, there will be angular distortion also. Ohs. In all cases of oblique reflexion or refraction, the image does not in general consist of an assemblage of points, but of circles of confusion overlapping one another, and it is therefore called indistinct. This cause of imperfection of an image cannot be altogether obviated, and it will in general vary at different points of the image : — the magnitude of the circles of confusion at dif- ferent points might be taken as a measure of the comparative indistinctness. 148. Caustics. In Chapter III. we have indicated the general method which must be adopted to find the caustic formed by reflexion 138 DISTORTION BY or refraction in any case, and we have there given some ex- amples of their formation and of their use in enabling us to construct the visible image of an object. We will here give a few general theorems respecting caustics and a general ex- planation of the apparent deformation of an object seen by reflexion at a mirror, — or through a lens. 149. (i) Suppose an object is viewed through a convex lens. Let QR8E be the course of the axis of a pencil by which any point of it is seen. Then this pencil is deflected by the lens in the same way as it would be by passing through a prism whose faces coincide with the tangent planes to the surfaces of the lens at R and S. The edge of such a prism would be turned from the axis of the lens and the deviation of the pencil would be greater the greater the angle of the prism, — so that the angular dis- placement of points of the image from the axis of the lens would be greater than in proportion to the distance of the corresponding points of the object from the same axis. Hence the distortion of an object like fig. 1 would in character resem- ble that given in fig. 2. Kg. 1, Fig. 2. (ii) If an object be. viewed through a concave lens, a similar mode of reasoning will shew that the distorted image of fig. 1 would resemble the appearance of fig. 3. This accounts for the distortion of objects seen through spectacles. Kg. 3. A SPHERICAL MIREOE. 139 150. To estimate the distortion of a distant object viewed hy reflexion at a concave or convex mirror. Let E be the position of the eye, G the centre of the surface, Rj so that EG is the axis of visual reference. Let QRE be the axis of the small oblique pencil by which any point C of a distant object is seen ^ by E, q the position of the circle of least confusion of this pencil, AG^r, AE=x, AR = y; .-.AM^^. Then considering the rays from Q to be parallel, we have v=-^, ^. = 2^^> andi?2 = A,where cos 1 \=^^^^±±ll- = 'J^l (Art. 70), A v^v^ (1 + cos + se c0] 2/ -^"^'^'"^-^+¥ \ 2 ?-[ 1 + cos^: J r\ ^ j)^ j r l + l-g- if (^ &c. be neglected ; ^=R-e-3?|. r X by writing for -j- its Yalue from (i) and ,,. 12 1 putting -jy- r X in the small term ^f2x_ 1 + - '1 ly- fl 1\ \r x) \r xj^ 2 1 L '^ *■ J ?] 3 2 r X • ir) _ r ( _ (x — r) (3a; — 2r) if \ 1 e 2x —r\ 2x — r ' r^x] (ii). Again, Em = x-^- Mm = x -^ - Rq .cos r/ 2r -'"-2-r—2[^-t)='' 2r 2 2x-r f 2^-7^ f 2x-r ' 2rT? V 2xV (iii). BRIGHTNESS OF AN IMAGE. 141 If then D be the distance of the object, we have qm _Em tan7;_ r ( ^x^—r' y^ (a;—r)(3x—2r) y Dtane B 'tane 2Z)'l 'ix — r'lrsf 2x — t ' r'x 2B\ 6a;' — 8rx^ + ir'x — r' y" 2x — r ' 2r''x' = 2i) (1 -^ ^2/')' suppose, Now this ratio -p^ represents the ratio of the distances Jy Ijan 6 from the axis of the mirror — of a point seen in the image, and of the corresponding point of the object. The term Ay' inay be taken as measuring the distortion, and the algebraic sign of A will indicate the nature of the deformation of the image. Thus for example, suppose the object presents a series of horizontal and vertical parallel lines, fig. 1, Art. 149, — as a bookcase in a distant part of the room — the distortion becomes greater for greater values of y, i.e. for points more and more remote from the axis of the mirror, and is of the character pre- sented in fig. 2 or fig. 3, according as A is positive or negative. (i) For a concave mirror, if x be sufficiently great, — as for instance x > r, — A is negative, and the distortion is of the character of fig. 3 : the image being inverted. (ii) For a convex mirror, r is negative and A negative also for all values of x, and the distortion in this case also is as in fig. 3 — the image being erect. But in this case the numerical value of A will be greater than in the case of a concave mirror, and therefore the dis- tortion is more marked and disagreeable. 151. Brightness of an Image. Consider the image formed by a lens of a plane object of small but sensible magnitude : let d, d' be the distances from the lens of the object and image, A the diameter or aperture of the lens : / the intrinsic illuminating power of the object. 142 BRIGHTNESS OP AN IMAGE. Now the amount of illumination emanating from the oh- ject and intercepted by the lens may be measured by and this is diffused over the image ; .•. illumination at a point of the image _ /ttA^ area of object , area of object _ d^ _ ~ 4^" ' area of image ' area of image d!^ ' ,'. illumination at a point of the image = -^ ( 37) > which « apparent magnitude of lens (m area) as seen from the image. A similar result will follow for the brightness of an image formed by reflexion at a spherical mirror, and is, as we see, independent of the distance of the object. Thus the illumination of an image formed by a lens or mirror, — (supposing no light lost by reflexion or refraction or absorption by the screen which receives the image) — is the same as would be produced by the direct light of the lens or mirror if it were equally luminous with the surface of the object which emits the light. This supposes the object to be of sensible magnitude. When the object and its image are physical points — as a star and its image — the eye judges only of absolute light, and the brightness of the image is proportional to the density of rays concentrated in it, i.e. the brightness and apparent magni- tude of the lens as seen from the object. Thus in the case of a star whose distance is constant, the absolute brightness of the image is proportional to the area of the object-glass : hence the importance of large telescopes in observing very faint stars. 152. To measure the comparative density of the rays at different points of a catacaustic. CONDENSATION OP BAYS IN A CAUSTIC. 143 Let Q be the radiant point, Q the axis of the mirror, QR the axis of a small pencil QBS, of which q^, q^ are the pri- mary and secondary foci after re- flexion, ^ the angle of incidence, / the intrinsic intensity of rays diverging from Q. Then in the primary plane, condensation of rays at q^ RQ' which is obtained by comparing the breadths of sections of the reflected and incident pencils taken at small equal distances from q^ and Q — and assuming the condensation to vary in- versely as the breadth ; And in the secondary plane the condensation at q^ (and all along the primary focal line) oc therefore on the whole, condensation at q^ oc product of these two oc The above neglects the rays which may pass through q^ after reflexion at other points of the mirror, which however would vanish compared with the condensation of rays in the primary plane. The expression becomes indeflnitely large at the geo- metrical focus, since q^q^ is there = — that is, the condensa- tion of rays at the geometrical focus is very much greater than at other points of the caustic. A similar measure might be obtained in the case of a diacaustic, — but such measures cannot be expressed very simply in analytical ternas. 144 LENGTHS OF Note. Some cases of the more general problem of deter- mining the law of condensation of light at any point in a system of rays proceeding according to any law, and the law of illumination of a surface on which the system is received — are discussed in a paper On Systems of Rays in the Messenger of Mathematics, Vol. iii. p. 33, by Mr H. J. Sharpe. 153. Length of the curve of a caustic by reflexion or re- fraction. (i) By reflexion. Let S be the radiant point, SP, SQ, SR three consecutive rays, the first two intersecting in p after reflexion, the last two intersecting in q. p, q are ultimately points on the caustic, and pq is ultimately a small arc of the caustic. Let 8P= p, Pp = p', Ap = cr = length of caustic measured from some fixed point A on it. Draw Pr, Qs, perpendiculars from P on 8Q and from Q on Pp. Then if ^ be the angle of incidence at P, we have Qr = PQ sin ^ = Ps, nearly, and therefore if Ap be the small change of p in passing from PtoQ, Ap = Qr = Ps = Pp- Qp = Pp-{Qq-pq) = p' - (p' + Ap) + Ao- ; .-. Act = A/3 + Ap'. CAUSTIC CURVES. *145 This which is approximately true becomes strictly true in the limit ; therefore integrating o- = jO + p' + a constant, p, p' are supposed to be measured positive in the direction in which light is propagated. Ex. The whole length of the caustic in the exatople of Art. 71, is = radius of mirror : and in the example of Art. 72, the whole length of the caustic is = — - radius of mirror. (ii) By refraction. With notation similar to the preceding, fi being the index of refraction from first medium into the second, we have Ap = Qr= PQ . sin ^ = fiPQ sin tp,' = fMPs = fi{Pp-Qp) ^f,{Pp-{Qq-pq)} ^^{p'-{p' + Ap')+Aa}; .■.Aa=^ + Ap'; P / .•. o-=— +p+a constant. P.O. 10 146 ASYMPTOTE AND CUSP OF A CAUSTIC. Thus we see that the length of any catacaustic or dia- caustic corresponding to any pencil in one plane can be ex- pressed in algebraic terms. In each of the above cases (i), (ii), p, p are supposed to be measured positive in the direction in which light is pro- pagated. Analytical investigations of the above results are given in Herschel's Light and other works. Cor. When two consecutive reflected or refracted rays are parallel, it indicates an asymptote in the caustic. If ^ be any variable which defines the direction of an incident ray, ■^ the angle which the corresponding reflected or re- fracted ray makes with a fixed direction, — then for an asymptote in the caustic we must have -^ = 0. a

-F t>-0 V-u Water at 15" iJ I'OOO 1-33095 1-33171 1-33357 1-33585 1-33780 I-34I27 I-34417 Crown Glass, No. 9 2-535 1-52583 1-52685 1-52959 I-533OI 1-53605 1-54166 1-54657 FUntGlass,No.i3 i-m 1-62775 1-62968 1-63504 1-64202 1-64826 1-66029 1-67106 Oil of Turpentine 0-885 1-47050 1-47153 1-474+3 i-47835 1-48174 1-48820 1-49387 SPECTRUM ANALYSIS. 155 Fraunhofer's observations as given in his Essay on the De- termination of Refractive Powers, &c. — the refractive index of a fixed line being understood to mean that of a ray which would suffer the same deviation if it existed. Grown glass is made of fine white sand and kelp or pearl ashes — it is colourless and is the best kind of glass employed for glazing windows, plate glass, &c. Flint glass or crystal is of a pale green hue and is made of silica, lead, and potash in proportions of about \, J, ^ — but varying in different specimens. The proportion of lead is increased to increase the refractive power which increases with the density of the medium. See table of refractive indices, p. 179. 162 [a). The analysis of light by means of the spectrum produced by a prism — what is commonly called spectrum analysis — has become a subject of great interest and careful observation since the important discoveries of Kirchhoff, who (a. d. 1859) propounded as a natural law that if a vapour when sufficiently heated possesses the property of emitting light of certain refrangibilities, that vapour at a lower tem- perature has a tendency to absorb, or refuse a passage to, light of the same refrangibilities which may be incident upon it : — and he demonstrated the law experimentally in the case of certain metals — Sodium, Lithium, Potassium,, <&c. This law with respect to light is analogous to the law of exchanges with respect to heat, as laid down by Dr Balfour Stewart, which asserts that th6 relation between the amount of heat emitted and that which is absorbed at any given temperature remains constant for all bodies ; — and that the greater the amount of heat emitted the greater must be the amount of heat absorbed. In order however that this law of exchanges as thus formulated with respect to heat-giAdng rays may hold good for luminous rays, Professor Roscoe insists that the emissive and absorptive powers of substances must be compared at the same temperature {Spectrum, Ana- lysis, p. 214. 2nd ed. 1870). 162 (/3). The spectra of light emanating from different substances, when decomposed or analyzed by a prism, may differ in many important respects from each other. Fol- 156 SPECTRUM ANALYSIS. lowing Mr Huggins, one of the most careful and accurate of experimenters, we may classify all the spectra which can present themselves in three general groups or orders. (i) A spectrum of the first order is continuous, unbroken by darh or bright liaes. Such a, spectrum is afforded by light emitted from an opaque body in a state of incandes- cence — and almost certainly from matter in the solid or liquid state. A spectrum of this order gives us no information of the chemical nature of the body from which the light emanates. (ii) A spectrum of the second order is discontinuous, consisting of colour lines of light, separated from each other. Such a spectrum is afforded by light emitted from luminous matter in the state of gas. Each element and every com- pound body that can become luminous in the gaseous state without suffering decomposition, is distinguished by a group of lines peculiar to itself. Thus the vapour of Sodium, gives for its spectrum the bright double-line D, in the orange-yellow. Lithium gives a bright red line between B and G, and a weak yellow line between G and D. Potassium, gives a red line coinciding with A, and a pale violet line not far from H. .Thallium gives a bright green line not far from E. Hydrogen gives three lines coincident with G,_ F, 0, in the red, hluish-green and violet. Nitrogen and oxygen give more numerous lines than hydrogen. Iron gives a very complex system of lines — and other substances give lines more or less numerous, and in most cases besides the few predominant lines there are faint traces of others. (iii) In the third order of spectra the continuity of the coloured light is broken by darA lines. These dark spaces do not arise from the source of light, but from the absorption of definite colours — (or rates of vibration, if we are regarding the physical theory of light) — by vapours through which the light has passed. Such spectra are formed by the light of the sun and stars. SPECTRUM ANALYSIS. 157 162 (7). It was shewn by Kirchhoff that if vapours of terrestrial substances come between the eye and an incan- descent body, they cause groups of dark lines — and that the group of dark lines produced by each vapour is identical in the number of lines, and in their position in the spectrum with the group of hright lines of which the light of the vapour consists when it is luminous. By this discovery Kirchhoff interprets the dark lines in the solar spectrum : — he infers that of the light emitted from the incandescent body of the sun, rays of various refrangibilities are absorbed in passing through the glowing ■ vapours of various substances present in the Sun's atmosphere : and he concludes with certainty that iron, sodium, magne- sium, hydrogen, calcium, barium,, copper, zinc, &c. are present in the Sun's atmosphere in a state of luminous gas : — but he finds no trace of gold, silver, antimony, mercury, alumi- nium, tin, lead, lithium,, &c. See Roscoe, Spectrum, Analysis. Also, Report upon the present state of our knowledge of Spectrum Analysis, presented to theBritishAssociation,1880: — Also, Professor W. G. Adams, Nature, Vol. xxii., p. 411. 162 (S). It is obvious that if a comparison be made of the spectrum of the light emanating from an unknown source (as the Sun, a star, or a nebula) with the spectra of different terrestrial elements, as standard spectra, im- portant information may be obtained as to their chemical constitution. The spectra of several fixed stars have been observed with great care by Mr Huggins and Dr W. A. Miller, and consist in most cases of very numerous bands of dark lines, a comparison of which with the solar spectrum shews that many terrestrial elements which are present in the Sun are also present in the several stars : that some which are present in the Sun are not present in particular stars, and vice versd — for example, hydrogen is present in the Sun's atmosphere and in that of Aldebaran, but is wanting in that of Betelgeux. Mr Huggins has also examined the spectra of several nebulae, and the results are very remarkable. The spectra of some of them are of the second order, indicating that the matter of which they are composed is in a gaseous state. Eor fall information, however, on the various branches of 158 DETEEMINATION OF THE spectrum analysis the student may consult Roscoe's and Huggins' Spectrum Analysis. 162 (e). If a pencil of sun-light be analyzed by a prism and examined with respect to the thermal or heat-giving power, and the chemical or actinic power, as well as the hrniinous power, it is found that what may be called ' the thermal spectrum, the colour spectrum, and the chemical spectrum, overlap each other: — the thermal spectrum ex- tending considerably beyond the red end of the colour spectrum, and the chemical spectrum considerably beyond the violet. In the following diagram the relative extension of the three several spectra are represented by the horizontal line O^B...OT: — the letters A, B, G...H corresponding to the several lines in the' colour spectrum (Art. 160): — and the ordinates bounded by the curves a, /3, 7, representing the intensities at the corresponding points of the thermal spec- trum, the colour spectrum, and the chemical spectrum severally. 163. The following two propositions will indicate the nature of the observations requisite for determining the index of refraction of a ray of any colour — the ray being defined by its position relative to the fixed lines of the spectrum. 164. To measure the refracting angle of a prism. Let the prism be firmly fixed to a graduated circle pro- vided with verniers, — the edge of the prism being at the INDEX OF EEEEACTION. 159 centre of the circle, perpendicular to its plane, and turned towards the object-glass of a telescope fixed to the circle. Let the circle be turned until the image of a well-defined distant object A, seen by reflexion at one of the faces of the prism, and viewed through the telescope, coincides with the intersection of the cross- wires at the principal focus of the object- glass of the telescope — and read off the verniers attached to the circle. Turn the circle about its centre till the image of the same object seen by reflexion at the other face of the prism coincides with the intersection of the cross- wires — the effect will be the same as if we supposed the circle to be stationary, and the object A to revolve about the centre of the circle into a position A' such that it is seen by reflexion at the second face of the prism, and read off the verniers again. The difference of the readings — or the angle through which the circle has been turned — is equal to twice the angle of the prism. 165. To measure the minimmn deviation of a ray corre- sponding to one of the fixed lines, out of air into any, mediwm formed into a prism; and thence to determine its index of refraction. Let be the centre of a graduated circle moveable round an axis perpendicular to its own plane on a fixed circle carrying verniers, Cc a telescope the axis of the object- glass of which passes through the axis of the circle and is parallel to the plane of the circle, — A the intersection of the plane described by the axis of the telescope with the slit perpendicular to the plane of the circle, by which light is admitted. The telescope is provided with cross- wires at such a distance from the object-glass that when it is pointed to A, the image of A may be at the inter- section of the wires. Place the prism with its edge perpen- 160 DETERMINATION OF THE dicular to the plane of the circle, so that its faces may be equidistant from the axis of the circle. Turn the prism until the incident ray and the given emergent ray make equal angles with the faces of the prism — i.e. until the deviation of this particular ray is a minimum, or the image of the fixed line stationary. The prism retaining this position, turn the telescope till the image of the line coincides with the inter- section of the wires and read the verniers. Now remove the prism and turn the telescope until the intersection of the wires coincides with the image of A and read the verniers again. The difiference of the two readings — or the anglp through which the circle has been turned between the two observations — ^is the minimum deviation of the given ray. If D be this deviation, t the refracting angle of the prism, jjL the index of refraction for the given colour, or fixed line, then with the notation of Art. (95), . D + i . sm — ^— ^ ^ s]n . t smg from which fi can be calculated numerically. 166. Obs. The refractive index of a fluid or a gas for a given fixed line may be found in the same manner by en- closing the fluid or gas in a hollow prism of glass, the sides of which are plates with their surfaces accurately parallel. The deviation of the axis of the pencil then arises entirely from refi-action through the fluid. To obtain the index of refraction from air into vacuum, the hollow prism must be exhausted : the deviation in this case will be negative, and the value of /t < 1. Of course the refractive index from vacuum into air is the reciprocal of the value of fi thus found : the values of the absolute refiractiye indices can then be calculated (Art. 84). From vacuum, into comm,on air, fj, = 1-000294 nearly ; — the density of the air at the earth's surface being to that of water as •0013 : 1, nearly. IRRATIONALITY OF DISPERSION. 161 Note. When the prism is not sufficiently perfect to ex- hibit the fixed lines, we must select by estimation the par- ticular part of the spectrum for which the index of refraction is required, — and measure the deviation as above. As the spectrum produced by a given species of light — sun-light for instance — refracted through a given substance always presents the same phenomena, we may regard it as a standard scale of colours which can be reproduced at any time, and in which any particular tint may be defined and identified by reference to the fixed lines. Thus the value of fj, for any particular medium or sub- stance is not invariable, but is susceptible of every possible value between certain limits : but rays corresponding to each particular value of fi will conform to, the laws of reflexion and refraction which have been discussed in the previous chapters on the supposition of light being homogeneous. 167. Def. A ray of white light being decomposed by refraction at any surface into a beam of coloured rays, the angle between any coloured ray, — or ray corresponding to a fixed line, — and the original white ray produced, is the devia- tion of that colour or fixed line. The difference of the deviations of two colours, or fixed lines, is the dispersion of those colours or fixed lines. A discussion of the values of [i obtained in Fraunhofer's manner leads to the conclusion,- — that the ratio of the dis- persion of any two colours to the dispersion of the extreme colours of the spectrum, is not constant when different media are used. This fact is called the Irrationality of disper- sion. Hence if two prisms be formed of different substances and of such refracting angles as to give spectra of the same total length, i.e. equal total dispersion, — and they be placed so as to refract a pencil of white light in opposite directions, — in the emergent beam the extreme red and violet will be united but the intermediate colours will not be completely so, but will give rise to a spectrum of faint colours and small P.O. 11 162 DISPERSION OF TWO breadth. Such spectra which exist in consequence of the irrationality of dispersion are called secondary spectra. 168. When a ray of white light is refracted through a prism in a principal plane, tojmd the dispersion of two colours of given refractive index. Let t be the refracting angle of the prism, 4> ^^^ angle of incidence of the white ray, cji' the angle of refraction of the coloured ray for which fj, is the refractive index, — and -f-', ■yjr angles of incidence and emergence of the same at the second surface, — D the deviation of this colour. Then sbi = /JL sin ', sin ■\lr = fi sin i^' ; j) = cl> + ylr-i, t = <^' + -.|r' (Ajt. 92, 06s. 2). If fi + Bfi be the index of refraction for another of the colours into which the incident ray is separated by refraction, Sfi being small — D+SB the deviation of this second colour — then will SB be the dispersion required. Treating fi as an independent variable, since it is suscep- tible of all values between the limits corresponding to the extreme colours, being constant, dfi '^ dfL dfj. , d'yJr . , , , , , d-Jr' cos y -ji- = sm Y> + fi cos Y-^ , = sin (f)' + fjL cos /r' COS ff>' COS ^' COS yfr therefore the dispersion -SB=^.Sfi = -, . Sfi. dfi cosy- cos ^ '^ COLOURS BY A PEISM. 163 169. COE. The above expression for the dispersion cannot vanish for any position of the prism. It admits how- ever of a minimum value, which will happen when cos ifr cos ^' = maximum. When such is the case, fi being constant in this problem, tan y^ . d-\}r + tan ^' . d(f)' = ; but d(p! = — dyjr', and from sin i/r = yti sin i/r', we have cot yfr . d-^jr = cot yjr' . d-^\ whence tan'' T|r = tan 0' tan ■^'. a relation which must obtain when the dispersion of two colours is a minimum .; this combined with sin ^ = /It sin ^', ^' + i^r' = i, sin.y^ = jj, sin f^', is sufficient to determine the direction of the incident ray when the dispersion is a minimum. 170. To determine the position of any part of the spectrum seen through a prism. Let Q be the origin of a small pencil whose axis is obliquely refracted through a prism in direction QA8 in a principal plane of the prism. Let g'j be the primary focus of the emergent pencil; then to an eye in AS the given rays, will appear to diverge from a line passing through q^ parallel to the original slit. And with notation of (Art. 92, Ohs. 2), sin = < that of Q according as ^ is > = < i/r, 171. JDef. If fi^, fi^ /jt, be the indices of refraction for the extreme red and violet rays capable of producing a sensible impression on the eye of the observer, and for rays of mean refrangibility out of air into any medium, the quantity is called the dispersive power of the medium. This quantity is frequently denoted by the letter ■sr. If the medium be formed into a prism with a small refracting angle i, and if D^ D, D be the deviations for the extreme red and violet rays and for rays of mean refrangibility of the axis of a pencil which passes through the prism in a principal plane and at a small angle of incidence and emer- gence, then D, = {lM,-l)i (Art. 92, Cor.) Z>=(;.-l)z; or the dispersive power of a medium formed into a thin prism is equal to the ratio of the total dispersion of the axes of the extreme red and violet pencils to the deviation of the axis of the pencil of light of mean refrangibility. For an account of dispersive powers of second and higher orders to which the irrationality of dispersion gives rise, the student is referred to Herschel's Treatise on Light, (Art. 438). Achromatic Combinations. 172. A combination of prisms or lenses is said to be achromatic when the dispersion of the pencils of ; light re- CONDITIONS OF ACHROMATISM. 165 fracted througli them is reduced within the narrowest possible limits. We proceed to consider the conditions which must be satisfied in such combinations. A pencil of sunlight after refraction does not in general converge to or diverge from a point, for two reasons ; (i) from the unequal refrangibility of the different species of light of which it is composed, and (ii) in consequence of the finite breadth of the pencil and the curvature of the refracting surface. These causes of aberration being independent of one another may be separately considered. It will therefore be supposed for simplicity in the following articles, in investigat- ing the conditions of achromatism, that no spherical aberra- tion — (i.e. aberration arising from the curvature of the re- fracting surfaces,) — exists in the pencils which we consider. Moreover we shall consider it sufficient to obtain the conditions of achromatism for the axis of a pencil, since if these conditions are satisfied for the axis, they will be so very approximately for the other rays of the pencil,— supposing it small. "When we speak of a colour it must be understood to be defined by its position among the fixed lines of the spectrum — or to correspond itself to a fixed line. 173. The possibility of an achromatic combination arises from the fact that the dispersion of a ray and the deviation of any particular colour — or the mean deviation, if for conveni- ence we take this as a definite measure of the deviation of the ray — produced by a refracting' medium are not propor- tional. A comparison of the results given in the table (page 154) will prove this. If dispersion were proportional to deviation for different media, then any combination which would destroy dispersion, would also destroy deviation ; and in consequence would be useless for the purposes for which such combinations are designed. For instance, a combination of lenses in a telescope- would have no magnifying power, if it did not produce devia- tion in the axis of a pencil transmitted through the telescope; but since in different media dispersion is not proportional to deviation, media can be found which produce the same disper- sion in opposite directions of a given colour relatively to an- 166 ACHROMATIC PRISMS. other, — ^but at the same time, diffei^ent deviations in opposite directions in the axis of a pencil. If a pencil then be refract- ed through these media, the two colours in question can be united, while the axis of the pencil suffers a deviation equal to the difference — or algebraically, the swm — of the deviations which the media would separately produce. If irrationality of dispersion (Art. 167) had no existence, then in providing a combination such that two given colours should not be separated, we should simultaneously unite lights of all species. But since the colours are disproportionately dispersed in different media the other colours will in such a case be very nearly but not exactly united.. A pencil there- fore refracted through, an achromatic combination will illu- minate a screen with light still slightly coloured and give rise — as we have before stated — to a secondary spectrum (Art. 167). The fixed lines in this spectrum do not generally preserve the same order of succession which they have in the primary spectra. A combination of different media achro- matic for all kinds of light being thus in general unattainable, it is customary to unite rays which are powerfully illuminat- ing and also differ much in colour — ^tke rest remaining but partially united. 174. Achromatic Prisms. A pencil of light passes through two prisms of srtvall re- fracting angles — passing in a principal plane of each — to find the condition of achromatism. Let t, i be the refracting angles of the prisms ; n, // the indices of refraction for a given colour (A)'. Then the devia- tions which the prisms would separately produce for this colour are (fi — l) t and {/j/ — 1) i. (Art. 92. Obs. 2.) Hence the total deviation for this colour = (/.-l)t + (/*'-l).'. li fi + Sfi, fi' + Bfi' be the indices of refraction for another colour {B), then the total deviation for this colour will be = (fi + Sfi-l)c + {iJ,' + Bfi'-l)i,'. TWO OR THREE COLOURS UNITED. 167 If then the deviations be the same for the two colours A and B — i.e. if the combination be such as to unite these two colours, — ^we must have = S/i.t + S/i'. t' (i), the condition required. The ratio of i to i being given by this equation (i) for any proposed pair of values of S^ti, S/i' — that is for any two given colours, we can determine the values of i and t' if an- other relation between these be given, — suppose, for example, the combination is required to produce a given amount of deviation in a given colour, then we must have (/* — 1) t + (jt4'— 1) t' = given deviation (ii). And then (i), (ii) are sufficient to determine i and i. Note. The two colours A and B only will in general be united, since in consequence of irrationality, the ratio of d(i to djj! is in general different for each pair of colours. From equation (i) it is seen that the refracting angles i, t' have contrary signs — -which indicates that the edges of the prisms must be turned towards opposite, parts. CoR. Similarly if there were three prisms the condition that two colours A and B should be united after refraction through them, will be = dfj,.i + dfj.' .I' + d/j," .i" (iii), where t, l, t" are the refracting angles of the prisms, and dfi, dfi, dfi" the difference of the indices of refraction for A and B in the substances of which the prisms are severally composed. If dfi^, d/jL^', d/j," be similar quantities for A and a third colour C, then A and G will be united if = dfj,^.i, + dfj,^ .I' + dfj," .i" (iv). If the conditions (iii), (iv) co-exist, then the three colours A, B, G would be united, — and a third condition, like that 168 ACHROMATIC PEISMS. of (ii) in the present Article, will render the values of i, i, i" perfectly determinate. Similarly, if there were n prisms disposable, we might so determine their refracting angles as to unite n colours. 175. The following is a more general case of achromatism with two prisms. A pencil of light passes through two prisms — the axis of the pencil passing in a principal plane of each : to find the condition of achromatism. Let I be the refracting angle of the first prism, /* its re- fractive index for a given colour, ^, <^' the angles of incidence and refraction at the first surface of this first prism, and ■^, ■^' the angles of emergence and incidence on the second surface of the axis of a pencil of light corresponding to this colour ; .;. sin0 = /isin^', sin i/r = ^ti sin i/r', ~ = tan 6, —^ . 176. Note. Since only a limited number of colours can be united by an achromatic combination, — -it will be best to select colours for this purpose which have a brilliant illu- miaating power and also are as nearly as may be comple- mentary (see Art. 185) — since by this course a far greater concentration of light and a more approximate union of the remaining colours will be produced, than if we attempted to unite the extreme red and violet rays which are too little luminous to render their union a matter of importance. For example, if two colours only are to be united, the best to be selected for this purpose are those defined by the fixed lines G and F, or D and i^ are similar to ■! , „ ; ^ cv _ Cv cr _ Cr vr Cr + Gv V , .•.ab =i. w = -^, « / which determines the diameter of the circle. COE. If the incident rays be nearly parallel v is nearly equal to /, and the radius of the circle of chromatic aberration will be independent of the focal length of the lens and vary as its aperture. 184. In a telescope with a given eye-glass the magnify- ing power varies as the focal length of the object-glass — (see next chapter), — while the size of the circle of chromatic aber- ration remains the same if the aperture remains the same. In viewing a white field through the telescope, the circles of chromatic aberration for different points would overlap and form a white field bounded by a coloured border of constant breadth so long as the aperture of the object-glass remains COLOURS OP NATURAL BODIES. 177 the same. Hence the greater the magnifying power, the less would be the area of the coloured border in proportion to the whole apparent field of view given by the object-glass. To secure this advantage, astronomers before the discovery of the achromatic object-glass were in the habit of having re- fracting telescopes of gre9,t length,- — even as great as 150 feet. Huyghens in particular was celebrated for his skill in making large glasses. {Herschel, Art. 458.) Since the discovery, however, of the means of constructing a compound achromatic object-glass, the necessity for such large instruments has been obviated, and they are now con- structed of much more moderate dimensions, and the size of the instrument reduced within a more manageable compass. As we have seen (Art. 178) the conditions of achromatism depend only on the focal lengths of the component lenses — not at all on their /orms or the order in which they are placed. By a suitable arrangement of these latter quantities — i.e. forms and ordei — the conditions requisite for the destruction of spherical aberration (before referred to. Art. 130) can be secured, and a compound object-glass constructed which shall be at the same time both aplamatic and achromatic. 185. Colours of natwal bodies — Primary colours. The colours of the spectrum given by a prism — as in Newton's experiment — could again be combined into white light by an equal prism set in the opposite di- rection — or any one or more colours might be insulated (as in fig. Art. 157) and separately examined or combined. From experiments of this kind it appears that all the colours of the spectrum must be combined in their natural propor- tions in order to produce white light — but that any shade of colour in nature may be imitated by a combination of the tints of the spectrum with a brilliancy unequalled by any artificial colouring. P. o. 12 :^78 PRIMARY COLOURS. If any substance be exposed in the prismatic spectrum, it appears to be of the same colour as that part of the spectrum in which it happens to be placed, but its hue is much more brilliant and vivid when placed iu a part of the spec- trum analogous to its own natural colour. Hence we are led to concluds'that the colours of natural bodies are not qualities inherent in themselves,-but arise from their aptitude to absorb and reflect different kinds of light. Thus a substance ap-. pears green — suppose — in consequence of its disposition to reflect green rays alone, the other prismatic colours of which the white solar light falling upon it is composed, being stifled or absorbed by it. lu' forming his theory of the composition of colours, Mayer regarded all colours as resulting from the combination of three primary colours, — red, yellow and blue. And he gave as results of his experiments the different proportions in which these colours entered into combination to produce others. Dr Young assumed red, green And violet as primary colours, and states that white light is composed of these complementary colours in the proportions of two parts of red, four of green, and one .of violet. It has been shewn by Helmholtz and Maxwell that the following mixtures of two complementary colours, when brought together into the eye, produce the effect of whiteness; — violet and greenish yellow; indigo and yellow; blue and orange; greenish blue and red. Sir D. Brewster regards red, yellow and blue as primary colours, a certain portion of each existing at every point of the spec- trum — the colour being determined by the predomiaating colour at that point, mixed with white light: — but this theory of Brewster's has been proved to be fallacious, for Helmholtz has shewn that the green ray, for example, is not made up of blue and yellow light superposed, and we cannot separate anything else but green out of it. Hence we conclude that each particular ray has its own peculiar colour, and that light of each degree of refrangibility is monochromatic. See Dr Young's Lectures on Natural Philosophy; Trans. Boy. Soc. Edin. Vols. xii. xv.; Chevreul, Gercles Chromatiques, Helmholtz, Handbuch der Physiologischen Optik, in Karsten's Allgemeine Encyclop. der Physik. TABLE OF REFRACTIVE INBICES. 179 See also Sir John Herschel's remarks on Newton's Theory of the Colours of Natural Bodies in Art. Light, Encyc. Metrop. Arts. 1134 &c. 185*. We give a short table of refractive indices — or values of fi, from vacuum, into different substances — for rays of Tnean refrangibility. The fixed line E of the spectrum, which is in the green space and nearly the mean ray may conveniently be taken as the line or colour of reference. Vacuum Atmospheric air at freezing teTn^erature and pressure ^^g^^'Qii =o"''7 according to BioT Ice from i "3070 to Water, fresh: see p. 154 „ salt (? sea) Alcohol (s.G. 0-866) Nitric Acid (s.o. 1-48) •. Fluorspar i "433 to Oil of Turpentine, s.G. o'BSs, ray E Alum Oil of Olives Canada balsam Brazil pebble, s.o. 2'62 Glass, crown I'S^S to plate I '500 to „ flint '. r576 t° Glass, tinged red with gold „ lead I, flint 2 „ lead 3, flint 4 Gum Arabic Iceland spar i'4887 to Eock crystal i'547 *" Topaz I •6102 to Euby I "601 to Zircon i'g6i to Garnet 'Diaxaoni, various specimens 2-439 to 1-000294 i'3ioo 1-336 1:343 i'37i 1-410 1-436 1-47835 •■457 1-470 i'532 i'532 1-563 ^ I-540 ^ 1-642 ^ 1715 , 1-724 1-732 1-512 1-6636 1-562 " 1-652 1-779 2-015 1-81S 2-775, 12—2 CHAPTER X. OF VISION AND OPTICAL INSTRUMENTS. 186. We proceed to give a description of the optical structure of the eye, and the manner in which vision takes place — and we shall then go on to describe some of the more important optical instruments. 187. Description of the Eye. The human eye consists of transparent substances en- closed in two coats nearly spherical in form. The figure represents a section of it by a plane through a line AGD, called the axis of the eye, with respect to which the surfaces of the coats are symmetrical. The exterior coat EDF, called the sclerotica, is homy and opaque, — except the front part A, which is transparent and slightly protuberant beyond the nearly spherical surface of the rest, and called the cornea. The second coat, interior to this, is called the choroides; it is opaque but has a circular aperture GH behind the cornea, — called the pupil — whose centre is in the axis of the eye. Over the back of VISION. 181 the eye there extends a black velvety substance, called the pigmemtum nigrum, perfectly incapable of reflecting light — and in this is embedded a delicate tissue of nerves called the retina, which communicate with the brain by means of the optic nerve K. B is a, soft transparent jelly-like substance — called the crystalline lens — in the form of a double convex lens having its axis coincident with A OD the axis of the eye, and held in its place by tendons springing from the choroides. The spaces between the cornea and the crystalline, and between the crystalline and the retina, are filled with transparent fluids, called respectively the aqueous and the vitreous hu- mours. The refractive index of each of these humours out of air is nearly that of water: — the refractive index of the crystalline is a little greater. Note. The values given by Sir J. Herschel are for the aqueous humour . . . . fi = 1"337 vitreous huTaour . . . . /jl = 1'3S9 crystalline lens, mean . . . /ii = l'384. The values given by Prof. M'Kendrick are for the aqueous humour \ „ _ jns _ i'3379 vitreous humour J ' ' ^ crystalline lens (? mean) . . A* = ir = 1"4545 188. To explain the manner in which vision takes place. Let the axis of the eye be directed to a point of an object PQ. A pencil from any point P falls upon and is refracted • by the cornea. Of this pencil a portion limited by the aper- ture GH is again refracted by the aqueous humour, the crystalline lens and the vitreous humour, and is made _ to converge very nearly to a point p on the retina. The im- pression thus made on the retina is communicated to the brain and produces the sensation of vision of the point P. 189. The spot K where the nerves of the retina pass through the coats of the eye, is insensible to vision — and from this cause is called the blind spot, or punctvm ccecum. The following simple experiment is given by Sir J. Herschel for 182 VISION. shewing the existence of this point. "On a sheet of black paper or other dark ground place two white wafers having their centres three iaches distant. Vertically above, that to the left hold the right eye at 12 inches from it, and so that when looking down on it the line joining the two eyes shall be parallel to that joining the centres of the wafers. In this situation, closing the left eye and looking full with the right at the wafer perpendicularly below it, this mily is seen, the other being completely invisible. But if removed ever so little from its place, either right or left, above or below, it be- comes immediately visible, and starts as it were into exis- tence. The distances here set down may perhaps vary slightly in different eyes." 189*. The central part of the retina is more sensitive to light than the more anterior parts of it. About the point where the axis of the eye meets the retina is aii oblong yellow spot, which is the most sensitive to light, and is chiefly employed in distinct vision. This yellow spot, — or pwrvdwrn luteum — has a horizontal diameter of about 'OS inch, and a vertical diameter of about "003 inch — and corresponds to a visual angle of from 2° to 4°. The central part of the spot, or fossa — is believed to possess the most acute sensibility — the diameter of it being only about "008 irich. The great mobility of the eye enables it to bring the images of successive points of an object on to the more sen- sitive parts of the retina very quickly; — and as the im- pression produced by light on the retina continues about ^ of a second after the light has ceased to act, by this means a distinct visual picture of an object is acquired. 190. The front part of the choroides surrounding the pupil is called the uvea or iris, which is differently coloured in different individuals, and by an involuntary action is capable of expanding or contracting the pupil within the limits of about "25 and '09 inches : — so as to admit a larger or smaller pencil of light as the object viewed is less or more brilliant. This involuntary action may be observed by any one watch- ing his own eye as he walks towards a mirror, near which is placed a bright object — as a lamp for instance. VISION. l83 The aperture of the pupil in the human eye is always circular — but in animals of the cat kind the vertical diameter appears to remain invariable, the contraction taking place in a horizontal direction, — in the eye of the horse, on the con- trary, the horizontal diameter of the pupil remains nearly constant, whilst it admits of contraction in a vertical direction. The eye is able to adapt itself automatically to objects at different distances, so as to make pencils of different degrees of divergency converge nearly to a point on the retina. That this focal adjustment of the eye for different dis- tances is effected — in part at least — by a change of form of the crystalline lens, — has been shewn by Prof Helmholtz ; — ■ who by carefully observing the images of a bright object reflected at the anterior and posterior surfaces of this lens, found that as the eye adjusted itself for different distances these images underwent a change in form and position which could only be accounted for by a change of curvature of the surfaces of the lens : and this is probably attended by a change of form in the cornea likewise. From the accurate measures which have been made it further appears that the surface of the cornea is a prolate spheroid, and the anterior and posterior surfaces of the crys- talline lens are oblate spheroids: — the density also of this lens is found to increase from the outside towards the centre, the tendency of which is to correct the aberration by shorten- ing the focal length for rays which pass near thp centre. The forms of the several surfaces, as might be expected, vary in the eyes of different animals — in fishes the crystalline lens is nearly spherical. This power of adaptation however does not enable the eye to see objects within a certain distance — in general about eight inches — but of an object sufficiently brilliant at any greater distance distinct vision can be obtained unless the obliquity or divergence be such that the point p does , not fall on the retina. This power of adaptation depends largely on habit. " A North American savage has a most perfect vision of very distant objects, but can hardly distinguish one held within arm's length." Coddington. 184 BINOCULAR VISION. Eays which are convergent at incidence on the eye cannot be brought to convergence on the retina. The eye when employed in a natural manner is achro- matic in a remarkable degree ; there is no appearance of coloured fringes about the edges of an object, and in looking at any variegated object the colours do not run into one another, or produce chromatic confusion; and this is true not only for those parts of the object which, appear near the centre of the field of view, but also for those which are seen by pencils of considerable obliquity. Experiments however by Wollaston, Fraunhofer, and others have shewn that this achromatism of the eye is not absolutely perfect. 191. Of Binocular Vision, or vision with two eyes. When an object is viewed by both eyes, an inverted image is formed upon the retina of each eye, but only a very small part of the object is seen distinctly at one and the same time ; — a portion of it, for instance, which would sub- tend an angle of a few degrees at the centre of the eye. The point at which the axes of the two eyes intersect is the point which is seen most distinctly, — and the extreme mo- bility of the eyes within certain limits enables them to change this point of regard from one part of an object to another with great ease and rapidity. And although we see distinctly only a small space at once, yet the visual pictures on the retina give us cognizance of the general relations of surrounding objects to a much larger extent. Thus if we are regarding a particular picture in a room we are simul- taneously conscious of the relative position of other pictures, • the furniture, &c. of the room, and thus a general impression is formed which is rendered more accurate and complete by making the surrounding objects successively points of regard. Further, as the view of external objects presented to either eye is not -strictly one and the same, the pictures formed on the retinae are dissimilar; and this dissimilarity combined with the relative degrees of light and shade in the objects observed, and the variation of inclination of the axes of the eyes, materially assists the mind in forming its impression of projection or solidity, and of relative distance PSEUDOSCOPE. 185 of the objects contemplated ; the mind being prepared by a long course of experience, tactile, muscular and otherwise, to form definite conclusions as to figure, &c. from definite visual sensations. Thus of the two pictures of a scene prepared for a stereo- scope, neither by itself gives us any impression of solidity j but when they are both viewed through the instrument, the two eyes receive the dissimilar pictures in the same way as if they were actually contemplating the scene portrayed; the illusion is complete, and the figures start into relief as if we were looking at an actual scene in nature. This result does not follow unless the pictures presented to the two eyes agree .(approximately at least) with the pic- tures which would actually be presented by a real scene; and the difficulty of making the two pictures coalesce is in- creased when the mind is not familiar with the objects or the kind of objects intended to be represented. 192. For the purpose of shewing the important share which the mind has in interpreting visual sensations, Prof. Wheatstone, — to whom we are indebted for the invention of the stereoscope, — has contrived a Pseudoscope, an instrument consisting of a pair of prisms with certain adjustments so that the image thrown upon either retina may be such as, without the intervention of the instrument, would naturally be received by the other. If we put the action of the mind out of consideration, we should expect that everything viewed by this instrument would be turned inside out, elevation would become depression, and vice versa ; — in fact, it ought to give what Prof. Wheatstone calls the converse of relief. This effect, indeed, follows easily for objects whose original and converted forms are familiar to the mind, — as in a seal and its impress, — but in other cases the mind admits the converted form with greater difficulty, according as the converted form is less familiar and probable. 193, It has sometimes been felt as a difficulty, that ob- jects appear erect, although the image of them oh the retina is inverted. Erect and inverted are. simply terms implying relative position, and as our optical knowledge of all external 186 DIMENSIONS OF THE EYE. objects is derived from the picture of them on the retijoa, there is the same relative displacement of all objects as regards their disposition in space. Thus whilst the picture of men on the retina exhibits them with their heads downwards, at the same time heavy bodies appear to fall upwards. The mind forms an estimate of all the relative parts of the picture simultaneously, and its relation to external objects is judged of by experience and habit. As however a discussion of the physiology of vision is beyond our purpose, the student who wishes to pursue the subject may consult an interesting article on Binocular Vision in the Edinburgh Review for October, 1858, and the authorities there referred to. Also Edinburgh Review for October, 1881, article on " Lectures on the Recent Progress of the Theory of Vision by Prof. Helmholtz," translated by Dr Pye Smith, 1873. Also Eye-sight, good and bad, by Robert Brudenell Carter, F.R.C.S., London, 1880. 194. It may be interesting to know the dimensions of the human eye, — they will of course vary m different indivi- duals, but the following are given as belonging to an eye of average size for a person in middle life. The axis of the eye measured from the outer surface of the cornea to the retina = "95 inches, and the portion of this length occupied by the cornea = "04 of an inch, the aqueous humour = '11, crystalline lens = "17, vitreous humour = '63. Interior transverse diameter of the eye . . "90 inches. Vertical chord of the cornea "46 Horizontal chord of the cornea "49 Chord of the crystalline lens "37 Radius of external surface of cornea . . . 'SS Radius of anterior surface of crystalline . . "33 Radius of posterior surface of crystalline . '24 The centre of the eye for optical purposes is a point nearly in the centre of the pupU, in the plane of the iris. The angle between the axis of the eye and the line joining the centre of the punctum caecum with the centre of the eye is about 14°, the breadth of the punctum csecum is about ^ inch, subtending an angle of 5° at the centre of the eye. DIMENSIONS OF THE EYE. 187 By the revolution of the eye in its socket, its axis has a range of about 55° in every direction about its mean posi- tion. The impression produced by light on the retina continues about ^ of a second after the light has ceased to act, so that if a bright object be whirled round in a circle, the period of its gjTation being less than this, it will appear as a continuous bright circle. With respect to the magnitude of the minimum visibile, or apparent magnitude of the least visible object, it is usually stated that an object is invisible to most eyes un- less it subtends an angle of at least one minute. This how- ever will vary for different eyes, and the brightness of the object must materially influence the . question, since we find that the fixed stars produce a distinct and vivid impression on the retina, although the angle which thev subtend to the eye is so small as to be incapable of being measured by the best instruments. If a series of equidistant holes be pierced along the circumference of a disc of cardboard, which is made to turn about its centre, and we look through these holes, we shall have successive views of exterior objects, very brief and at short intervals : we shall see them in the situations they occupy at the instant they are perceived. If they are fixed, each impression will be identical with the preceding one ; we shall see them without displacement and without interrup- tion, only their brightness will be diminished. If they are in motion, we shall perceive them in successive positions, as if they had jumped from one to another. Suppose, for instance, there is fixed behind the revolving cardboard a figured sheet divided into as many sectors as there are holes — say six — through hole no. 1 we view sector no. 1, — when hole no. 2 arrives, the sector no. 2 will take the place of the preceding one — and so in succession. In each of these sectors let there be represented, for example, a blacksmith whose figure is the same in each, only his arm which holds a hammer is depressed in no. 1, a little raised in no. 2, still more raised in no. 3, — at the highest eleva- tion in no. 6. We pass on rapidly to no. 1 again — and the effect is that the eye which receives these impressions in 188 DEFECTS OF VISION. succession seems to see the hammer raised little by little, and suddenly to drop. In a way similar to this may be explained various amusing toys, the tliawmatrope, phena- kistiscope, the wheel of life, the anorthoscope, &c. If a groiip of objects be in motion in a dark room, — for example, a picture on a cardboard whirling about an axis — and the darkness be illuminated by an electric spark, the objects do not seem to be in motion, but stationary in the position they occupied at the instant: — shewing that the duration of the illumination is so brief that no sensible change of position of the group takes place during it. The numerical results of this Article are taken from Lloyd's Treatise on Light and Vision. Perhaps the most complete Treatise on the Physiology of the Eye is Helmholtz's Optique Physioldgique, 1867. The student may also consult generally the article Eye by Professor M^Kendrick, in the Encyo. Britt. 9th ed. 1878. Also Professor J. D. Everett's edition of Deschanel's Natural Philosophy, 1882, part iv. Optics : — which contains numerous excellent diagrams. 195. Defects of Vision, An eye which produces too great refraction of a pencil incident upon it brings pencils from distant points to conver- gence at points so far before the retina as to produce no dis- tinct impression upon it. This defect is called short sight. On the other hand, for an eye which cannot sufficiently refract a pencil, the least distance of distinct vision is greater than eight inches, pencils from points within this least dis- tance being brought to convergence behind the retina. This is long sight. 196. Vision through optical contrivances depends on the fact that if a pencil diverging from a given point fall on the eye, it is immaterial whether that point be an actual source of light, or whether the rays have been made to converge to it and afterwards to diverge. An image therefore is visible in the same manner as a luminous object in the same posi- tion would be, with this limitation, — that from any point of a luminous object rays diverge in all directions, but from VISION THROUGH A LENS. 189 any point of an image rays diverge only in directions cor- responding to the directions of those rays which form that point in the image. In general explanations of vision through optical instru- ments, spherical aberration may from its smallness be disre- garded. Pencils may be considered after reflexion or refrac- tion to converge to or diverge from a point, and an excentrical pencil may be supposed to have the same point of divergence or convergence as the centrical pencil from the same origin. Further, when an object is at such a distance as to be conveniently seen, the pencil received by the eye from any , point of it will from its smallness have a very small degree of divergence, — so small that for the purpose of general descrip- tion of an instrument we shall regard the pencils which emerge from the eye-glass to consist of parallel rays. The slight adjustment requisite to give distinct vision of the image must be performed by each observer for himself. 197. Vision through a lens. Let PQ be a small luminous object, G the centre of a lens whose axis is GQ, ^ the centre of the pupil of an eye whose axis coincides with the axis of the lens. A pencil of light diverging from a point P of PQ falls upon the lens, and after refraction may be considered as diverging from sbme point p in GP, or in GP produced, or as converging to some point p in PG produced. Thus pq an image of PQ is formed. 190 VISION THROUGH A LENS. If Eq be not less than the least distance of distinct vision of the eye, then of the pencil diverging from p, the pupil selects . a portion prs which has been refracted excentrically through the lens, and by this pencil the point p is visible. Thus the image pq will be seen by the eye. COK. If PQ be very near to the principal focus of the lens so that the image pq may be very distant, the excentrical pencil pE by which the point p is seen may be considered to consist of rays parallel to pC. [Gom/pare Art. 115.) 198. To find an expression for the visual angle wnder which a small object is seen through a lens. Let EG be the axis of the lens, E the eye, PQ the object, pq its image, EC = x, CQ = u, PQ = y, EQ=^d, /= focal length of the lens, z pEq = <^, / PEQ = a = angle which PQ subtends at the eye. Then VISUAL ANGLE THROUGH A LENS. 191 ±=1+1 M_Oq_ f Cq f^u' PQ GQ u+f - y Also tan a = -^— ; u + oc tan u + X (i)- U + X (l + -7.J ,(ii). tana 1/ -L /». I I J a pq be at the distance of distinct vision A = Eq suppose, then 1 1 ,..., A — X f u The results (i), (ii), (iii) enable us to discuss the apparent magnitude for different relative positions of the object. If the rays emerge in a state of parallelism, then u = — /,, the lens must be a convex one — i.e./ negative — and we shall have tan (/> = — ^ from (i). Hence the less/ is, the greater will <]> be, — and this suggests a reason why ^ is called the power of a lens (Art. 104). 199. In the results of the previous article, if > a the visual angle of the image is greater than that of the object, and the latter may be said to be magnified, and conversely. But it will be useful to compare the magnitudes of the iniag& and object on the supposition that they are both observed at any the same distance from the eye — that of distinct vision,, for instance — and we give the following definition. 192 MAGNIFYING POWER OP A LENS. Definition. When an object is seen through a lens the magnifying power of the lens is the ratio of the angle which the image subtends at the eye — to the angle which the object would subtend at the eye if it were in the position of the image and viewed directly. This mode of estimating magnifying power is equivalent to comparing the linear magnitudes of the image and object. 200. To find the magnifying power of a lens for given positions of the object with respect to the lens. The object being supposed small the angle which pq subtends at the eye = ^ , and PQ viewed at distance Eq would sub- tend an angle Eq' If then m ^ denote the magnifying power of the lens, h PQ GQ- ^""^Gq'CQ-f' 201. To examine the values ofm. vmder different circum- In the figure of last Article (i) If/ be positive — or the lens concave — Cq is positive and the image erect. Also Gq is 1, or the image is magnified. In the latter case, the image is magnified (i.e. m > 1 nume- MAGiNIFYING POWER OF A LENS. ]93 rically) if ^> 2 — or the distance of the object from the lens less than twice the focal length, — otherwise the image is diminished. 202. A convex lens having the effect of producing an erect and more distant image of a near object, assists the eye of a long-sighted person, — and a concave lens by producing an erect and nearer image of a distant object, assists a short- sighted person. This explains the use of spectacles and eye-glasses. It was remarked by Dr Wollaston that the best form of lenses for this purpose is concavo-convex — since the indistinctness for oblique pencils arising from aberration is much less in a lens of such a form than in an equiconvex or equiconcave lens of the same power : — in other words, the field of distinct vision is larger in the former than in the latter. If an object be viewed through a convex lens, — the object not being farther from the lens than its principal focus — the divergence of the pencil by which any point of it is seen, is greater as the object is nearer to the lens. Other defects of vision arising from the want of symmetry in the refracting surfaces of the eye, may be corrected or diminished by using eye-lenses of various forms — adapted to each particular case : for example, lenses one surface of which is spherical and the other cylindrical. Telescopes. 203. When an object is at a great distance the small pencil from any point of it which is selected by the pupil of the eye may not have sufficient illuminating power to make a sensible impression on the retina : and further, the parts of many distant but visible objects cannot be distinguished be- cause the distance between the parts subtends at the eye no appreciable angle. Now if an image of such an object be formed near the observer by a lens or reflector, and pencils converging to, or diverging from, points in this image be refracted through a lens to the eye, the condensation of rays in the pencil from any point of the object may be sufficient to P. 0. 13 194 ASTRONOMICAL TKLESCOPE. render that point visible — and the directions of the axes of the pencils from different points may inchide at the eye appre- ciable angles. In other words, the large lens — or reflector — serves to condense into each point of the image formed by it a larger number of rays than conld be received by the unas- sisted eye, and so makes the image more brilliant than the object ; — the eye-glass magnifies this image and also renders a larger extent of it visible at once than could be so without such assistance. Such is the principle of the Telescope. There are two classes of telescopes — ^refracting and re- flecting telescopes — so named from the manner in which the image of the object viewed is formed ; — in the former class the image being formed by a lens, in the latter by a reflector. We will first describe these instruments in their simplest construction — deferring the explanation of the additions and modifications by which the vision of objects through them is improved. See Art. 151 on brightness of images, also Art. 257*. 204. The Astronomical Telescope, AGB is a convex lens, — called the object-glass, — fixed in a tube, and acb a convex lens, — called the eye-glass, — fixed in another tube which slides in the former — the common axis of the tubes being the common axis of the lenses. The focal length of the object-glass is numerically greater than that of the eye-glass, and when the instrument is in adjustment for viewing very distant objects, the distance between C, e the centres of the lenses is the sum of their focal lengths. If the axis of the lenses be directed to a point Q of an object PQ, which is so distant that a pencil incident on the object-glass from any point of PQ may be regarded- as con- ASTRONOMICAL TELESCOPE. 195 sisting of parallel rays — the pencil from a point' P after refrac- tion through the object-glass converges very nearly to a point p in PC produced (Art. 197), — Cp being equal to the focal length of the object-glass, and thus pq an inverted image of PQ is formed. This image — from the adjustment of the instrument — is at the principal focus of the eye-glass ; and therefore a pencil diverging from a point p of the image con- sists after excentrical refraction through the eye-glass of rays parallel to pc, (Art. 197), and suitable for givmg distinct vision of the image of p formed by the eye-glass, to an eye applied to the eye-glass. Thus an inverted image of PQ is seen through the Tele- scope ; or rather, the image presents a picture of the object turned half-round about the axis of the Telescope {see Art. 143). 205. Field of View — i.e. the space that can 'be viewed with the telescope at one and the same time. If Pj be a point of the object such that the ray PA of the pencil from it is refracted in the line Ab, which joms opposite parts of the object-glass and eye-glass, then every ray of this pencil — «,nd also every ray of a pencil from any point nearer to Q than Pj is, — falls upon the eye-glass and is refracted to the eye. Again if P^ be a point in the object such that the ray P^B of the pencil from it is refracted in the line Bb, which joins corresponding parts of the object- glass and eye-glass, then of this pencil this ray alone falls upon the eye-glass, and is refracted to the eye. Of a pencil from a point between P^ and P^ a portion reaches the eye ; — which portion is less and less as the point is more and more distant from Q, and no ray of a pencil from a point more distant than P^ from Q is refracted by the eye-glass. Hence on looking through the telescope, points whose an- gular distance from Q exceeds PfiQ are more and more faint as this distance is greater, — and points whose angular distance from Q exceeds PJyQ are invisible. This gradual fading away of objects at a distance from the centre of the field is known as the ragged edge of the field of view. It is remedied by putting a stop — or diaphragm with a circular aperture 13—2 196 ASTRONOMICAL TELESCOPE. - — at the position of the image pq, so as to destroy that part of the image which would be seen by partial pencils. The angular extent of the uniformly bright field of view is then the angle subtended at G by the diameter of the aperture of the stop. If a moving object— as a star moving, say, from left to right — be observed through the telescope held in a given position — the image of the star traverses the field of view in the opposite direction, from right to left. Note. See Art. 257* for some useful remarks on the brightness of images formed by an Astronomical Telescope. 205.* When a small object is placed at a distance from a convex lens equ^l to the focal length, the cones of rays emanating from the several points of the object are trans- formed by refraction through the lens into cylinders of parallel rays— =and the object viewed through the lens appears to be at a very great distance : such a lens — or combination of lenses — suitably mounted and used for the purpose, of pro- ducing the same effect as a distant mark is 6alled a colli- mator, and is much employed for adjusting instruments in an observatory. 206. An expression for the radius of the stop {p) and the angular radius of the field of view (^) can easily be obtained. Let "^"h be the focal lengths and half-breadths of the object-glass and eye-glass respectively. Join Ab cutting pq in p — through p draw a line parallel to Cc ; then these two lines with the intercepted portions of the lenses — (treated as straight lines), — will give us two similar triangles, whence we have A ~ /o ' therefore p =- ^''^; ~ -^f" , atid rf, = £=AyirA GALILEO S TELESCOPE. 207. Galileo's Telescope. 197 ACB is a convex lens called the object-glass, fixed in a tube, and acb a concave lens_ called the eye-glass, fixed in another tube, which slides in the former, — the common axis of the tubes being the common axis of the lenses. The focal length of the object-glass is numerically greater than that of the eye-glass, and when the instrument is adjusted for view- ing very distant objects, the distance between C, c the centres of the lenses, is the difference of the focal lengths. If the axis of the Telescope be directed to a point Q of an ©bject FQ, which is so distant that a pencil incident on the object-glass from any point of it may be considered to consist of parallel rays, the pencil from a point P after refraction through the object-glass converges very nearly, to a pointy in PC produced, — Cp being equal to the focal length of the ob- ject-glass — and thus pq an inverted image of PQ would be formed. Of the pencil converging to any point p of this image, the eye-glass (which is about the size of the pupil of the eye) selects the portion prs .which has been refracted excentrically at the object-glass,— and since by the adjust- ment of the instrument pq is at the principal focus of thp eye-glass, this portion of the pencil after centrical refraction through the eye-glass consists of rays parallel to cp, and suit- able for giving distinct vision of the image of p formed by the eye-glass to an eye applied to the eye-glass. Thus an erect image of PQ is seen through the Telescope — since the rays from the extremities of the object have not crossed each other before entering the eye. , 198 GALILEO'S TELESCOPE. 208. Field of View of a Oalilean Telescope. If Pj be a point in the object such that the ray P^A of a pencil from it is refracted in the hne Aa, which joins corre- sponding parts of the objefct-glass and eye-glass, then of the pencil from this point, or from any point nearer to Q, a por- tion falls upon the eye-glass sufficient to fill it. Again if P^ be a point in the object such that the ray P^A of the pencil from it is refracted in the line Ah, which joins opposite parts of the object-glass and eye-glass, then of this pencil this ray alone falls upon the eye-glass and is refracted to the eye. Of a pencil from a point between P^ and P^ the portion which reaches the eye partially fills the eye-glass, and is less and less as the point is more distant from Q ; also, no ray of a pencil from a point more distant from Q than P^ is refracted by the eye-glass. Hence there will be a ragged edge to the field of view, which in this Telescope is incurable — because the image formed by the object-glass is virtual, and therefore cannot be limited by a stop. Ohs. The vision through the telescope will be most dis- tinct when the refraction through the eye-glass is centrical, and hence the size of the eye-glass ought to be nearly the same as that of the pupil of the eye : — if it be much larger and the eye be not applied at its centre, the refraction through it will be excentrical, and the distortion and chro- matic dispersion increased. Note. When an object is viewed through a Galilean telescope, the parts of the image seen will appear in the same relative positions as the corresponding parts of the object : — there is no inversion as regards up and down, nor reversion as regards right and left, — and the image of a moving object will move in the same direction as the object itself moves. Hence this telescope is very convenient for observing land objects. 209. Def. When an object is viewed by a telescope the magnifying power of the telescope is estimated by the ratio of the angle which the image seen subtends at the eye to the angle which the object would subtend at the eye if viewed directly. MAGNIFYING POWERS. 199 210. To find the inagnifying power of ike Astronomical Telescope — or of Galileo's Telescope. Since the point p of the image pq (figs. Arts. 204, 207) is seen by a pencil whose axis after refraction through the eye- glass is parallel to pc, and the point q by a pencil whose axis is qc, the image of PQ subtends at the eye an angle pcq. And PQ would subtend at the eye an angle PcQ, — or PCQ since the object is very distant. Hence ■ magnifying power = ^^^^f tan pcq ■ , ^ =tiK7^^PP^°^™^*"^y' cq _ focal length of the object-glass focal length of the eye-glass Ohs. This result gives approximately the linear magni- fying power at any point of the field of view : — though it is strictly true near the centre of the field of view only. Further, the instruments are supposed to be in adjust- ment, so that any pencil from a very distant point emerges from the eye-glass as a pencil of parallel rays : practically the eye-glass must be pushed in a little, in the case of either telescope, to suit a short-sighted eye, — so that from this cause the magnifying power will be slightly different for dif- ferent observers. 211. Newton's Telescope. ACB is a concave spherical reflector — or speculum — 200 newton's telescope. whose centre is 0, and whose axis GO coincides with the axis of the tube at the extremity of which it is placed, DEF a small plane 'mirror inclined at 45° to the axis of th,e tube ; ach a convex eye-glass placed in a tube which slides in an aperture of the former tube : the axis of the two tubes are at right angles, — the plane of DEF is perpendicular to the plane of the axes of the tubes — and the axis of the lens ach coincides with that of the tube in which it is placed. If the axis GO be directed to a point Q of an object PQ which is so distant that a pencil incident on A GB from any point of it may be considered to consist of parallel rays, the pencil from a point P after reflexion at this mirror converges very nearly to a point ^' in PO produced, Op being = | CO (Art. 115), and there f'q an inverted image of PQ would be formed — or rather, the image p'q' would be a picture of the object turned half-round about the axis of the spherical reflector. This pencil being reflected again by DF will con- verge very nearly to a point P, the length of path of any ray to p being equal to that to p' (Art. 60). Thus pq an in- verted image of PQ is formed, and the position of the eye- piece is such that this image may be at its principal focus. Hence the pencil diverging from any point p of the image consists after excentrical refraction through the eye-glass of. rays parallel to pc, and suitable for giving distinct vision of the image ofp formed by the eye-glass to an eye applied to the eye-glass. Thus an inverted image of PQ is seen throiigh the tele- scope. 212. Field of View of Newton's Telescope. If Pj be a point in the object sixch that the ray of a pencil from it which, after reflexion at A CB, is incident on the small mirror at D, is reflected by DF in Db the line joining opposite parts of the small mirror and eye-glass, then of a pencil from P^ — or from any point of the object nearer to Q than Pj — every ray which falls on the small mirror is refracted by the eye-glass to the eye. Again if P^ be a point in the object such that the ray of a pencil from it which is incident on the small mirror at F is reflected in Newton's telescope. 201 Fb the line joining corresponding parts of the mirror and eye-glass, then of the pencil from P^ this ray alone reaches the eye. Points between P^ and P^ appear more and more faint the farther they are from Q, — and points at a greater distance from Q than P^ is are invisible. 213. Magnifying power of Newton's Telescope. Let p'q' (fig. Art. 211) be the virtual image of PQ formed by the large mirror : pq is similar and equal to p'q'. Now pq subtends through the eye-glass the angle -^ , and PQ sub- cq p'q' tends to the naked eye the angle POQ or j~-, . Hence magnifying power = -cj . -^, s, J i, tr cq pq = %-' cq _ focal length of large reflector focal length of eye-glass Rote. The form of the small mirror must be an ellipse in order that it may intercept as little of the incident light as possible, and just reflect all the light incident upon the curved mirror. A rectangular prism of glass is sometimes used instead of a plane mirror; the passage of a pencil through it is indicated in the figure. 213*. To illustrate the use of a Newton's Telescope by an observer — -fig. Art. 211. — Suppose the axis of the telescope CO to be -directed to an object in the South— e.g. a group of stars — the eye-tube being on the West side of the large tube, so that the observer is looking Eastward and his left hand is towards the speculum end of the large tube of the telescope — and his right hand towards the open end of it. The virtual image at p'q would be a picture of the object turned half- 202 herschel's telescope. roimd about the axis of the large tube, — which axis is also the axis of the speculum. This is transferred by the small plane mirror to pq — forming there a real image : and when viewed by the observer the East and West parts of the object would be seen reversed, — i.e. the East and West parts of the object would appear towards his right and left hands respectively — and the picture compared with the. object would be inverted as to up and down. If a point of the object move across the field of view — say from East to West, the corresponding point in the picture will move from right to left of the observer — -i.e. from the open end towards the speculum end of the large tube, and parallel to its axis. If the Eye-tube be placed on the East side of the large tube — the small plane mirror being suitably adjusted — the obsefver will now look Wes^-ward — that is, he will be turned half-round with respect to his former position : a star moving from East to West will still appear to the observer to inove from his right to his left — i.e. from the speculum end towards the open end of the large tube. See a Letter of George Hunt in " the Observatory," No. 57, Jan. 2, 1882. 214. Herschel's Telescope. . A CB is a concave spherical reflector, or speculum, whose centre is 0, and whose axis CO is inclined at a small angle to the axis of the tube at the extremity of which it is placed, acb a convex eye-glass in a sliding tube attached to the inner surface of the larger tube, the axes of the eye-glass Cc and that of the large tube being in the same plane with and equally inclined to CO. If the axis of the large tube be directed to a point Q of herschel's telescope. 203 an object PQ, which is so distant that a pencil incident on ACB from any point of it may be considered to consist of parallel rays, the pencil from a point P after reflexion at the mirror A CB converges very nearly to a point p in PO pro- duced, Cp being = J CO (Art. 115), and thus pq an inverted image of PQ is formed. The position of the eye-glass is such that this image is at its principal ' focus, and therefore the pencil diverging from any point p of the image consists, after excentrical refraction through the eye-glass, of rays parallel to pc and suitable for giving distinct visionof the image formed by the eye-glass. This arrangement of the Eye-glass and speculum makes the Telescope a front-view Telescope, and an inverted image oi PQ is seen through the telescope — but not reversed as to right and left with respect to the observer — who is turned half-round, so to speak, with respect to the object PQ. Note. This construction of the reflecting telescope was originally proposed by Le Maire in the early part . of last century, — but Sir W. Herschel was the first who made any extensive use of it. 215. Magnifying power of Herschel's Telescope. Since / pGO = ^ PCO, and ^ qCO = ^ QGO (Art. 8) ; .-. .pCq = , PGQ. Now pq viewed by the eye-glass subtends the angle pcq, and PQ viewed by the naked eye subtends the angle PCQ, or pCq ; ■■■ magnifying power = ^^^ = - focal length of mirror ~ focal length of eye-glass ' 216. The principle of Herschel's Telescope is the same as that of Newton's — the only object of the plane reflector in the latter being to throw the image unaltered in form into another position where it may be more conveniently viewed. The speculum in Herschel's Telescope is designedly of large 204 GREGORYS TELESCOPE. aperture, so that when a faint star is observed a pencil suffi- ciently large to make it visible may be received by the spe- culum, and reflected to the eye. The advantage of Herschel's construction over Newton's arises from there being no part of the incident pencils stopped by the back of the small mirror, and no loss of light from a second reflexion. The ragged edge of the field of view in Herschel's Tele- scope may be remedied by a stop placed at the principal focus of the eye-glass — and the angular diameter of the field of view will be the angle which the aperture of the stop subtends at the centre of the face of the speculum — or approxiTnately, the angle which the breadth of the eye- glass subtends at the same point. A similar method of suppressing the ragged edge '&n.i estimating the field of view, will apply to Newton's Telescope. 217. Gregory's Telescope. ACB, DEF are two concave spherical reflectors, or spe- cula, with a common axis GE, which is the axis of a tube at the extremity of which A GB is placed, — this nairror bei,ng much larger than BEF and of larger radius. The concavities of the mirrors are turned towards one another, and the "prin- cipal focus of AGB is between the centre 0' and principal focus / of DEF. In a tube which is "fixed in an aperture at the centre of AGB is a convex eye-glass ach, the axis of the eye-glass coinciding with that of the reflectors. If the axis of the reflectors be directed to a point Q of a cassegeain's telescope. 205 very distant object PQ, a pencil from any point of it P after reflexion at AGB converges very nearly to a point j) in the straight line produced which joins P with the centre of A GB, and thus pq a real inverted image of PQ is formed at the principal focus of AGB. Since this image is between the centre and principal focus of DEF, a pencil diverging from any point p after excentrical reflexion at DEF con- verges to a point p' in the straight line produced which joins p with 0', the centre of DEF (Art. 115) ; and thus p'q^ an erect image of PQ is formed. This image being at the principal focus of the eye-glass is in a suitable position fpr being distinctly seen through the eye-glass, — and thus an erect image is seen through the telescope. The parts of the image seen will appear in the same relative positions as the corresponding parts of the object : — there is no inversion as regards up and down, nor reversion, as regards right and left :■ — and the image of a moving object will m,ove in the same direction as the object itself moves. Hence this telescope is a convenient one for observing land objects. Compare Note, p. 198. Note. The relative position of the principal foci of AGB and DEF is determined by ^ consideration of the relation 112 - + - = -, V u r 2 1 for since it is requisite that Eq (i.e. v) should be large, must be small and positive, i.e. Eq must be a little greater than Ef. Ohs. In Gregory's and Cassegrain's telescopes the ad- justment for different eyes is performed by shifting the small mirror by means of a screw, as is shewn in the figure, — in each of the other telescopes, it is performed by moving the eye- glass backward or forward. 218, Cassegrain's Telescope. A CB is a concave and DEF a convex spherical reflector with a common axis GE, which is the axis of a tube at one 206 cassegeain's telescope. extremity of which A CB is placed. The mirror A CB, which IS much larger and of greater radius than BUF, has its con- cavity turned towards the convexity of DBF, and the prin- cipal focus of ACB IS between ^ and / the principal Lus 7 i- i^if-*''^® ^^^''^ "^ ^^^'^ ill an aperture at the TllV- S^.'' f.^'^^'^'n ^^"-S^-^'^ ««^' ^^^ axis of the eye- glass being that of the reflectors. •' If the axis of the reflectors be directed to a point O of a very distant object PQ, a pencil from any point P, after re- flexion at AGB, converges very nearly to a point p in the line which joins P with the centre of AGB, and then va an inverted image of P^ would be formed at the principal focus ot ACB Since this image is between DFF and its principal focus,_the pencil converging to any point p after excentrical ^^TZ i^-/- '=°'^^^^^P« to a point p' in the straight line produced which joms 0' the centre of H^Pwith p and »V an inverted image of PQ is formed. This image bSng at ?h'e principal focus of the eye-glass is in a suitable position for being distinctly seen through the eye-gla^s, and thus an in^ verted image is seen through the telescope. ..^^n% .T^e relative position of the principal foci of AGB Tlnrilte ^ ^ consideration similar to that of T«1?^ remarks on the image seen through an Astronomical Telescope.-(p. 195, line 13... p. 196, line 6,)-apply to the image seen through a Cassegrain's Telescope ^^^ MAGNIFYING POWER OF GREGORY'S TELESCOPE. 207 219. Magnifying power of Gregory's Telescope. Take the figure of Art. 217. 'LQi Cq=Oq = F\ . , , ^, ,. (large . Ef= 0'f=f ] = ^'"'^^ ^^-^g^^^ of l^^s^j mirror. cq =/e = focal length of the eye-glass, numerically. qf=x = distance between the principal foci of the two mirrors. Let Q be the point of the object to which the axis of the Telescope is directed, — the image of any other point P by reflexion at A GB is at p — and the image of p formed by re- flexion at DEF is at p — the straight line pp' passing through 0' the centre of i)^i^: The portion FQ of the object would subtend at the eye the angle FOQ, and the image of PQ subtends at the eye the angle "pcq. Hence .« . ^ p'cq' tan p'cq' m = magmfymg power = j^^ = I^jTqq approximately, _ PS Oq _ Oq p'q' cq' ' pq cq' ' pq ' But Oq=Cq = F,cq'=f,^^=^, 0'q==f-x, " pq a>' and m = -^—^ . Obs. This expression is strictly accurate at the centre of the field only, but may be taken as approximately true over the whole. Ifote. The expressions for m in this article and the fol- lowing one are equally true for Cassegrain's Telescope. 208 FIELD OF VIEW As the adjustment of the Telescope for different . eyes is effected by moving the small mirror by means of a tangent screw, the above expression is a very convenient one, since it involves 00^ — the variable quantity in the instrument, — in a very simple form. 220. An approximate expression for the linear magni- fying power may be obtained independent of x. Since C^g' is in general small compared with Oq, we will regard g' as coincident with C, approximately. We then have -^, + -^^ = ^, or •+. whence J'+/-l- x=^-^^^^ =^- +f; 00 so :. Fx + x' =f. If x^ be neglected as being small compared with the other terms of this equation, we have x _r F' :. m = F.f F^ a result which in general is sufficiently approximate. 221. Field of view of Gregory's Telescope. The field of view may be limited either by the eye-glass or the small mirror. QF GREGORY'S TELESCOPE. 209 The following approximate expressions will be practically sufficient. (i) Suppose the field limited by the eye-glass. Through 0' the centre of DEF and a the edge of the glass draw the line aO'p cutting the image of PQ . formed by the large mirror in p — and let PC be the ray which would be re- flected at G the centre of ACB to p. Then of the pencil from P which fills the large mirror more than half falls on the eye- glass and is refracted to the eye. Let ^ =pGE = PGQ = angular radius of field of view thus defined, ac = y^ = half breadth of eye-glass, f^ its focal length, JSF = y^= small mirror, Cq = F, Ef=f, /(? = «= 4t . approximately (Art. 220). TVT ^ sin i> O'p Now Jy- = -. y^fZy- = 77^ ^ aOc sm VUp Gp but O'q JALz^ = I (/_ x) (Art. 219) ; * / since x is small compared with F and / (ii) If the field be limited by the small mirror — let the ray PG after reflexion at G fall upon the edge of the small mirror, then of the pencil which falls upon the object-mirror from any point nearer to Q than P is, more than one-half reaches the eye, P. 0. I't 210 DEFECTS OF TELESCOPES. If ^j = POQ be tHe angular radius of the field of view thus defined, we have Cor. If the breadths of the eye-glass and small mirror are such that the field of view limited by them severally is the same, then ^ = ^^ and y^ _ f-V' . F+f FiF+f.)' since _/^ and /are small compared with F. 222. The telescopes have here been described in their simplest forms, for the purpose of explaining the principle of their construction. We proceed to notice briefly the defects of such instruments which render their modification by com- pound object-glasses and eye-glasses necessary, whereby these defects are diminished while the principle of the telescope is unchanged. 223. I. The Astronomical Telescope. (i) Let light be considered homogeneous. A pencil from any point after oblique centrical refraction through a single object-glass converges to two focal lines, and the image — or assemblage of circles of confusion — is indistinct and curved with its concavity towards the object-glass. A direct pencil also is refracted with aberration. The compound object-glass commonly used consists of lenses in contact, — and therefore by no arrangement of their forms can the indistinctness and curvature of the image be diminished (Art. 146). All that can be done therefore is to construct the lenses so as to produce the least possible aberration in a direct pencil of parallel rays. If however a distinct and flat image were formed by the object-glass, yet this image viewed through a single lens by DEFECTS OF TELESCOPES. 211 excentrical pencils, would be indistinct, curved, and also dis- torted (Art. 147). These defects are lessened by properly- adjusting the forms of two or more lenses which form a compound eye-piece, or ocular. The three defects cannot be entirely removed together ; — each therefore is diminished as far as possible according to the artist's judgment, and with reference to the use for which the telescope is intended, (ii) Let light be considered as composed of different species, and let spherical aberration be disregarded (Art. 172). A pencil of such light refracted centrically through a simple object-glass is divided into pencils converging to a series of points in their common axis, and thus a series of coloured images differing slightly in position is formed. The most vivid of these images are united by a compound object-glass of two or more lenses, the focal lengths of which are properly taken. Again, an achromatic image viewed by a single eye-lens will from unequal refrangibility be confused, — and the con- fusion will be of a worse kind than that produced by the objeet-glasSj because in the latter case the coloured pencils from the same point have a common axis, — but in the present case the refraction being excentrical, they have not a common axis, and the coloured points corresponding to any point of the image formed by the object-glass are spread over the field. To remedy this confusion the focal lengths of the lenses forming a compound eye-piece are so adjusted that the axes of pencils of the most vivid colours belonging to the same point of the object emerge to the eye parallel to one another, — in which case such pencils, if they be small, affect the eye in the same way as if they were coincident. Obs. It is worthy of notice that the conditions of achro- matism affect the focal lengths of the lenses combined in an object-glass or eye-piece : — the conditions of diminished in- distinctness, curvature, and distortion of the image have re- ference to the forms of the lenses. See Astron. Notices, Vol. xxviil. p. 202; Vol, xxiv, p. 195 ; Vol. XXV. p. 22. See an article in the Times Newspaper of March 21, 1881, 14—2 212 PEFECTS OP TELESCOPES. for a popular account of the large Refracting Telescope lately made by Mr Grubb of Dublin for the Observatory of Vienna. 224. II. Oalileo's Telescope. In the Astronomical Telescope the refraction through the object-glass is centrical, — through the eye-glass excentrical ; but in Galileo's the reverse is the case. Hence what has been said of the defects of a simple object-glass and eye-glass in an Astronomical apply respectively to the eye-glass and object-glass of a Galilean telescope. In this telescope, fur- ther, the chromatic dispersion through the object-glass is more unpleasant than that of the eye-glass, — and the eye- glass produces distortion in the image. 225. III. The Reflecting Telescope. In the Reflectiug Telescope there is spherical aberration from the curved reflectors which produces indistinctness and curvature of the image,— and in the telescopes of Gregory and Cassegrain distortion, — in consequence of the excentrical re- flexion at the second mirror. These defects are found to be lessened if the large reflectors be made not exactly spherical, but figures generated by a conic section about its axis — para- bolic or slightly hyperbolic. The defects of the single eye- lens which are lessened by a compound eye-piece are the same as have been mentioned in the Astronomical Telescope. 226. In the Astronomical Telescope since the pencils pass centrically through the object-glass and excentrically through the eye-glass, the flsld of view depends only on the aperture of the eye-glass ; — ^the aperture of the object-glass affecting only the brightness of the field. In Galileo's Telescope, on the contrary, where the refrac- tion through the object-glass is excentrical, the fl^ld of view depends upon the aperture of the object-glass — and this is the reason why this telescope is not so generally used for astro- nomical purposes as for a perspective or opera-glass where small magnifying power is required. For with a high mag- nifying power, and a field of any considerable extent, the ' extreme pencils would be refracted by the object-glass at such SPECULA FOE TELESCOPES. 213 a distance from its axis as to make their chromatic dispersion unpleasant and with diiEculty diminished. Moreover, the brightness of the field "in Galileo's telescope is nearly the same as that of the object, for (fig. Art. 207), the breadth of the visual pencil at the object-glass rs : its breadth at the eye-glass :: focal length of object-glass : focal length of eye-glass, i.e. in the ratio of the magnifying power to unity ; in other words, the quantity of light received by the eye from any small area of the object is greater than what would be re- ceived from the same area without the intervention of the instrument, in the same ratio that the area is magnified — and therefore its apparent brightness is unaffected. This telescope exhibits objects erect, — which is of great advantage for the purposes for which it is generally em- ployed. 227. Specula for Reflecting Telescopes. The alloy used as a speculum metal is composed of 126'4 parts of copper to 58'9 of tin — from which proportions how- ever different experimenters deviate more or less. The compound is very brittle and the casting, grinding and polishing specula of large size are matters requiring great care and judgment. The action of the atmosphere upon them for two or three years reduces their brilliancy so much as to render it necessary to re-polish them at such short intervals. In the celebrated Telescope of Lord Eosse, the specula are made of the alloy above mentioned and are of 6 feet aperture and 53 feet focal length — being mounted so as to admit of being used, by a slight adjustment, either on the Newtonian or Herschelian principle. Dr Steinheil states that he has recently discovered a method of coating a glass mirror with silver and polishing the metal side, so as to obtain a metallic speculum of greater reflective power than any hitherto known : according to his estimate the following is the comparative brightness for light 214- SPECULA FOR REFLECTING TELESCOPES. reflected at an angle of 45° — direct light being considered = 100. Brilliancy. Loss of Light per cent. Direct light 100 O'O Silver mirror 91 9" Metallic mirror, Lord Rosse's] alloy ^- 67-18 :.• 32-82 Object-glass by Frauahofer,] _„ „ . . transmitted light y" •■•••'- See Astron. Notices, Vol. xix. p. 56, Vol. xxvL p. 77. The student may also consult Lord Rosse's memoirs in the Phil. Trans, for 1840 and 1850. Also a Description by Mr Lassell of a maxihinefor polishing specula, &c. with an account of a 20 feet Newtonian Telescope, &c. in the Memoirs of the Roy. Astr. Sac. Vol. XVIIL See also the Article Telescope in Rees' Gyclopcedia and in the Encyc. Britann. The reflecting telescope constructed by Newton- was of ^ery small dimensions, — the speculum being only of about 1 inch aperture, and the focal length 6 inches, with a mag- nifying power of 38. It is still preserved in the apartments of the Royal Society. 228. In the Gregorian Telescope it is essential that the small mirror should work truly on the axis of the large one, an adjustment which it costs some trouble to secure, and which is very easily deranged ; it is also readily affected by a very slight tremor of the stand of the telescope : — but notwith- standing these disadvantages it is in great favour with many observers from its compactness and the convenience resulting from its shewing objects erect. The Cassegrain construction is not much used: — it is however the construction employed in the large reflector of four feet aperture recently erected at Melbourne Observatory, Australia — made by Mr Grubb of Dublin. OBJKCT-GLASSES. 215 229. A large telescope has generally a small field of vie-w-, and there is often difficulty in direct- ing it to a proposed object: to diminish this, a small telescope, called a finder, of small power and large field of view is often attached externally to the tube of the large telescope near the eye-end of it ; the axes of the two being parallel. A considerable extent of space being thus within the field of the finder the instrument can be moved till the object proposed coincides with the centre of the field of the finder, — i.e. at the intersection of its cross-wires — and it is then at the centre of the field of the large one. 230. Object-glasses. The compound object-glass commonly used in a refracting telescope consists of a lens of crown glass in contact with a lens of flint glass. The conditions which the combination has to fulfil are that it shall be achromatic for given kinds of light — (see Art. 178) — and also aplanatic or free from spherical aberration. (See Art. 132.) (Astron. Notices, Vol. xxiv.) Achromatic object-glasses have also been constructed of three lenses in contact, consisting of a concave lens of flint glass between two convex lenses of crown glass. The conditions of achromatism can thus be satisfied for more species of light than in the former case: the forms of the lenses will be determiiied on principles similar to those just referred to. Such object-glasses are now not frequently employed in con- sequence of the difiiculty of centering the lenses so that their axes may exactly coincide. Object-glasses of three lenses have occasionally been made in which the middle one con- sisted of some liquid, but they have been found not to Ipe very durable in consequence of a chemical change in the liquid. Besides the references given in this and the preceding article the student may consult Lehrhuch der Analytischen Optik von I. C. E. Schmidt, for a niethod proposed by Gauss 216 EYE-PIECES for taking into account the thickness of the lenses in a large object-glass, — and Grunert, Optische Untersuchungen, also Gamb. Phil. Trans. Yol. vi. 231. Eye-pieces, or Oculars. The defects of a single eye-lens — referred to in Art. 223, viz. indistinctness, linear distortion, as well as that of curva- ture, and chromatic dispersion — are diminished by employ- ing a compound eye-piece. Those in most general use are (i) Uuyghens' Eye-piece, (ii) Ramsden's Eye-piece : they are also known as the negative and the positive eye-pieces respec- tively. From the expression for tan t; (Art. 137) we see that the distortion produced by a single lens ac -^^ . With a view of diminishing the distortion, Huyghens proposed to construct an eye-piece of two separated convex lenses — dividing equally between the two the deviation produced in an excentrical pencil. 232. To find the distance between two lenses in order that an excentrical pencil incident parallel to the axis may suffer an equal amount of deviation at each lens. Let QR8T represent the course of the axis of the pencil, f , f^ the numerical focal lengths of the lenses A, B, which we will sup- ., pose convex ones, AB = a. Then using first approxima- tions only, we have deviation at i? = z RXA, S= c XST= A STB-^RXA, if these be equal, we get ^ STB = 2 ^ SXB, and the angles being small, tan STB = 2 tan SXB; .•.BX=2.BT. J 1__1 BT BX fj Also, BX=f. POSITIVE AND NEGATIVE. 217 But BX=AX-AB=f^-a; .'■ a =/j — /j the distance required. The construction adopted by Huyghens in consistence •with this condition was an eye-piece of two convex lenses whose focal lengths are in the ratio of 3 : 1, the less powerful lens being placed as a_/?eW-glass— i.e. nearest the object-glass of the telescope — and the distance between the lenses being the difference of their focal lengths. • It is a remarkable coincidence — undesigned by the in- ventor — that if the lenses be of the same material, this con- struction fulfils simultaneously the condition of achromatism of an excentrical pencil, viz. « = H/x+/.) (Art. 181). The Huyghenian — or negative — eye-piece is therefore achromatic. 233. Vision through an Astronomical Telescope with a Huyghens' Eye-piece. Let A GB be the object-glass of an Astronomical Tele- scope directed to a very distant object PQ:- DEF the first ]= 1 7^ A. ^ — " !i.W' '1 P *^W ==ii: J C — — \ y B lens, or field-glass, and acb the second lens, or eye-glass, of a Huyghens' eye-piece, the focal length of DEF being three times that of acb, and the distance Ec being the difference, or semi-sum, of the focal lengths without reference to sign. A pencil from a point P of the object after refraction through the object-glass would converge very nearly to a poiat p, Cp being equal to the focal length of the object-glass, — but being excentrically refracted by the field-glass converges to a point 218 ASTRONOMICAL TELESCOPE 'p' in F/p, and thus f'q^ an inverted image of PQ is forined. The position of the eye-piece, is such, that §■' is the bisection of Eg, and therefore this image is at the principal focus of the eye-glass. Hence a pencil from any point f of the image after excentrical refraction at the eye-glass consists of rays parallel to p'c, and suitable for giving distinct vision of the image formed by the eye-glass to an eye applied to the eye- glass. Thus an inverted image of PQ is seen through the telescope, 234. The compensation between the two lenses which renders Huyghens' eye-piece achromatic admits of a simple general explanation. The deviation of the axis of a pencil of light produced by a convex lens is greater the greater the distance from the axis of the lens at which the axis of the pencil is refracted : for this axis is refracted in the same degree as it would be by a prism whose surfaces touch the lens at the points where the axis of the pencil is incident and emergent, and therefore the deviation is greater as the refracting angle of such prism is greater (Art. 92). Now when a pencil of light refracted by the object-glass falls on the field-glass, it is separated by it into a series of coloured pencils whose axes follow different courses, — the deviation of the axis of the red pencil being least, and that of the violet greatest. The axes of the pen- cils do not cut the axis of the lenses between the lenses, and thus the axis of the red pencil falls on the eye-glass at the greatest distance from the axis of the eye-glass, and conse- quently is most refracted by it ; the axis of the violet falling nearest to the axis of the eye-glass is least refracted by it. Thus the pencils from the same point in the object which are least and most refracted by the field-glass are respectively most and least refracted by the eye-glass, and consequently, by a proper selection of the lenses, may be parallel when they enter the eye. 235. The position of the eye-piece is determined by the condition that a direct pencil after refraction through the field-glass may have the principal focus of the eye-glass for its geometrical focus — i.e. a direct pencil must at incidence on WITH HUYQHENS' OR EAMSDEN'S EYE-PIECE. 219 DEF be converging to a point q such that after refraction through DEF it converge to q, the principal focus of ach, hence q' must be the middle point of Ec, and Eq must there- fore (Art. 99) be equal to half the focal length of DEF, or three-fourths the distance Ec. The focal lengths and position of the lenses of a Huy- ghens' eye-piece being determined, the forms of the lenses are to be chosen with reference to the use of the telescope. The conditions for diminishing distortion, indistinctness, and curvature of the field being different, it is a matter of judg- ment which of these defects are most to be avoided. Accord- ing to Mr Coddington {Optics, q. v.) these defects will be obviated as far as possible by making the field-glass a menis- cus, having its jadii in the ratio of 11 : 4 ; and the eye-glass a crossed lens, the radii being as 1 : 6, — the forms indicated in the figure. 2.36. Ramsden's Eye-piece. This eye-piece, sometimes called the positive eye-piece, is a combination of two separated convex lenses, for the purpose of diminishing the effects of spherical aberration — which can be effected better by two lenses than by one of equivalent power, because more disposable quantities are thus introduced into the calculations of the forms of the lenses. The lenses are taken of equal focal length, and the dis- tance between them is two-thirds of the focal length of either. This eye-piece is not achromatic. The passage of a pencil through an Astronomical Tele- scope provided with a Ramsden's eye-piece is indicated in the figure. 220 THE ERECTING EYE-PIECE. The position of the eye-piece, when in adjustment, is de- termined by the condition that a direct pencil at inciden6e on the field-glass must be diverging from a point q^ such that after refraction through DEF it may diverge from a point q coincident with the principal focus of ach : hence since Ec = \ of focal length of aoh, therefore E(i = ^ of the same focal length, and therefore (Art. 99) Eq = \ of the same. The considerations which lead to the forms of the lenses are similar to those mentioned in the case of Huyghens' eye- piece. The lenses are generally of the forms in the figure — the field-glass being plano-convex, the eye-glass convexo- plane. 237. The Erecting Eye-piece. The inversion of the image by an Astronomical Telescope, when furnished with either of the eye-pieces already described, renders it unsuitable for viewing terrestrial objects. To remedy this, an Erecting Eye-piece of four lenses is commonly used. The manner in which an object is seen through a tele- scope with this eye-piece will be sufficiently understood from the course of a pencil traced in the figure. The distances, forms, and focal lengths of the lenses are adjusted to diminish as far as possible chromatic and spherical aberration. The re-inversion of the image is sometimes effected by an eye-piece of three lenses; but in all the erecting eye-pieces that I have. met with, the correction for chromatic dispersion has been very imperfect. Note. When an object is viewed through an Astronomical Telescope furnished with an Erecting Eye-piece the parts of the image seen will appear in the same relative positions as the corresponding parts of the object : — there is no inversion as regards wp and down, nor reversion as regards right and left — and the image of a moving object will move in the same CROSS WIRES. ,221 direction as the object itself moves. Such an arrangement makes the telescope very convenient for observing land objects. Compare tiofe, p. 198. 238. In the description of the eye-pieces they have been supposed to be employed in an Astronomical Telescope. The same eye-pieces are however employed with the Reflecting Telescopes. In Galileo's telescope a single eye-lens is gene- rally used because the refraction through it is centrical. The investigations of the field of view will still be true in tele- scopes with compound eye-pieces, if the field-glass of the eye- piece be used in them instead of the eye-lens. The determi- nation of the magnifying powers will also hold good, if the simple eye-lens be supposed such as will refract an eccentrical pencil in parallel rays at the same inclination to the axis of the lenses as it has at emergence from the eye-glass (Art. 138) — i.e. if we substitute the equivalent lens for the eye-piece. On the theory of eye-pieces, see Grunert, Optische Unter- suchungen, Theil III., and two memoirs by Mr Airy in the Cambridge Phil. Trans. Vols. II. and iii. Also on the eye- piece for correction of atmospheric dispersion, see Monthly Notices, vol. 29. p. 333, and vol. 30. p. 57. 239. Since the field of view of a telescope is of finite extent, it is necessary to have certain points in the field to which an object observed for the purpose of measurement may be referred. This is in general attained by fixing in the tube of the telescope a set of fine parallel threads in a plane perpendicular to the axis of the lenses, and one or more threads at right angles to this set, which if placed at one of the images formed by the telescope are, like that image, dis- tinctly visible through the eye-glass. Such a set of threads are commonly called cross-wires or spider lines: — a line joining the central point of the set with the centre of the object-glass is the line of sight of the telescope, and with this line the optical axis of the object-glass (Art. 97) ought to coincide. In Huyghens' Eye-piece the wires would be placed at the principal focus of the eye-glass, and therefore would be dis- 222 PEACTICAL METHODS OF torted by excentrical refraction through that lens alone ; while the image seen would be distorted by excentrical refraction through the field-glass and eye-glass, and consequently in a different degree from the wires. In, this case the position of any point of the field would be estimated incorrectly by referring it to the wires, and thus Huyghens' Eye-piece can never be used in a telescope intended for measuring. In Ramsden's Eye-piece the image given by the object- glass is formed in front of the field-glass, and at this image the wires are placed. The image and the wires are thus each seen by two excentrical refractions, and are therefore distorted in the same degree, so that the position of a point in the former is correctly estimated by referring it to the latter. This therefore is the eye-piece in telescopes used for obtaining measurements. Galileo's Telescope can never be employed in measuring, because the image is a virtual one behind the eye-lens. 240. In an Astronomical or Galileo's Telescope, by moving the eye-glass inwards so as to bring it nearer to the object-glass, the pencil from any point of the image emerges from the eye-glass in a state of divergence (Art, 103), and is therefore adapted for a short-sighted eye. In viewing a near object such that a pencil from any point of it cannot at incidence on the object-glass be supposed to consist of parallel rays, but has a sensible divergency, the eye-glass must be moved outwards. The " corresponding adjustments are effected in the tele- scopes of Gregory and Oassegrain by moving the small mirror by a fine screw. 241. To determine practically the magnifying power of a telescope. If the Hght of the sky fall upon the object-glass or large mirror of a refracting or reflecting telescope, a real image of that lens or mirror is formed by the eye-piece in the same manner as of a self-luminous object in the same position. The magnifying power of the telescope is approximately equal to the quotient of the diameter of the object-glass or FINDING THE MAGNIFYING POWEE. 223 mirror divided by tlie diameter of its image thus formed. The diameter of the former can be directly determined, and the latter can be measured by a contrivance of Ramsden's, called a Dynamometet — whence the numerical magnifying power of the instrument is obtained. 242. The fact that the linear magnifying power of a tele- scope is equal to the ratio of the diameter of its object-glass — or large mirror — to the diameter of the bright image of the same, may be separately proved for each telescope with any given eye-piece. It may suffice for us to shew its truth in one case — and we will take Gregory's Telescope with a simple eye-lens. Let^gj* be the image of the large mirror AGB formed by the small mirror whose centre is E, very nearly at its prin- cipal focus, p'qr the bright image of pqr, which the eye-glass whose centre is c forms, and which from the largeness of GE may be considered as the principal focus of the eye-glass. Then if F,f^,fl be the focal lengths without regard to sign of the large niirror, small mirror and eye-glass severally-— by triangles which are very nearly rectilinear and similar, we get diameter of mirror _ OE pr Eq ' and ^ = — , ; pr cq 224 . MICROSCOPES. diameter of mirror _ GE cq p'r Eq ' c<( = — tH^ . y , nearly, = ^^— , nearlj--, = magnifying power of the telescope (Art. 220). This method is not applicable to Galileo's Telescope, be- cause the image of the object-glass formed by the eye-lens is virtual. Note. A simple and elegant proof for any telescope of the statement at the beginning of this Article is given by E. Hill, M.A., Oxford, Cambridge and Dublin Messenger of Mathematics, Vol.'v. p. 84, 1871. 243.' The Dynamometer above referred to (Art. 241), as originally used, consisted mainly of a slip of mother-of-pearl ■with a scale of tenths of an inch engraved on it, on which the bright spot was received so that its diameter could be read oif at once. Much greater accuracy is now attained and an eye- piece with a divided field-glass is employed, one half of this lens remaining fixed and giving a bright circular image of the object-glass in a permanent position — the other half lens can be moved by a screw transversely to the axis of the instru- ment, and the bright image formed by it is brought into contact with the other fixed image — first on one side of it and then on the other : the difference of the readings of the graduated head of the screw in these two positions affords a means of determining the diameter of the bright image within a. five-hundredth part of an inch. 244. Microscopes. Some objects are so minute that when they are viewed by the naked eye at the least distance of distinct vision, the dis- tances of their parts subtend no appreciable angles, and there- fore cannot be discerned. In these cases it is advantageous to view an image of the object, instead of the object itself, — and an instrument for this purpose is called a microscope. MICROSCOPES. 225 Microscopes are called simple or compound, according as a real image of an object viewed by them is not or is formed. 245. A single lens, or a sphere, forms a simple microscope. If an object be placed nearer to a convex lens than its principal focus, an erect and magnified image of it may be seen by an eye on the axis of the lens (Art. 201). Also if a small object PQ be placed nearer to the centre of a refractiag sphere than its principal focus, a pencil diverging from a point P will after direct re- fraction through the sphere diverge very nearly from some point p in CP produced, and pq an erect image of PQ is thus formed. This image, if its distance from an eye close to the sphere be not less than the least distance of distinct vision, may be seen by the eye by direct pencils. The distinctness will be much improved, if the visual pencils be restricted to pass nearly through the centre of the sphere — which can be effected by filling up a groove: a, b, leaving only a small circular opening at the centre of the sphere — the effect of spherical aberration will be thus almost entirely obviated. 246. Since the minimum aberration (see Arts. 130, 130*) for parallel rays is less for substances of high refractive power, it is of great advantage to construct lenses for this purpose of substances for which fi is large; — for example, there is much less aberration in a lens of zircon (fj, — 2) than in a lens of crown-glass. The diamond has a still higher refract- ing power than this (see p. 179), which combined with its low dispersive power makes it the most desirable substance to be used for this purpose. A simple microscope preferable to a single lens is com- posed of two convex lenses separated by, a small distance on a common axis. If an object be placed nearer to the first P. o. 15 226 COMPOUND MICBOSCOPE. ■lens than its principal focus, so that a virtual image of it may be formed by each lens, the image formed by the second lens will be distinctly seen by an eye whose axis is the axis of the lenses, and whose distance from this image is not less than the least distance of distinct vision. This is the- principle of Wollaston's Microscopic Doublet. 247. The compound refracting microscope is an Astrono- mical Telescope adapted for viewing near objects. ACB is a convex lens called the object-glass, and acb a convex lens called the eye-glass, fixed in a tube whose axis is the axis of the lenses. The distance of the centres of the lenses admits of being altered for the purpose of adjustment. If the axis of the lenses be directed to a point Q in an object PQ which is farther from the object-glass than its principal focus, the pencil from a point P after refraction through the object-glass converges very nearly to a point j> in PC produced, and thus pq an inverted image of PQ is formed. The position of the eye-glass is such that this image is at its principal focus, and therefore the pencil from any point p ©f the image consists after excentrical refraction through the eye-glass of rays parallel to pc, and suited to give distinct vision to an eye applied to the eye-glass, and thus an inverted image of the object is seen through the microscope. 248. Compound object-glasses and eye-pieces are com- monly employed for reasons similar to those which render them necessary in telescopes. CAMERA OBSCUEA. 227 For a full account of the microscopes of different makers, the mode of mounting and illuminating the object under ob- servation, &c., and the recent development of microscopic science, see The Microscope, its History, Construction, and Applications, by Jabez Hogg, tenth edition, 1882 ; also The Microscope and its Revelations, by Dr Carpenter; How to work with the Microscope, by Dr Bea'le ; also the important articles Microscope in the Penny Cyclopcedia and in the Encyc. Britann. 249. Bef. The magnifying power of a compound Mi- croscope may "be estimated by the ratio of the angle which the image seen subtends at the eye to the angle which the object would subtend at the eye, if placed at the distance of distinct vision and viewed directly. Thus li K = distance of distinct vision, OQ = u,f., i'" focal lengths of AOB,.aab without reference to sign, we have angle subtended .by the image == ^-^ =^, PO angle which, object would subtend at distance K=~^ :, _K p±_K Cq ••^~F'Pq~F^ u' TT. , • n 1 1 1 i-( fu But numerically 7=r+-=T'; ^'.uq—— — -„:, •' Cq u J ^ u — / ••"^-Fiu-fr 250. The Camera Obscy,ra. If in an aperture in the wall of a darkened room there be inserted a single convex lens,. or a combination of lenses of considerable negative focal length, a real image of external objects is formed at a distance from the lens, If this image 15—2 228 CAMERA LtlCIDA. be received on a screen, eithet directly or after the direction of the pencils has teen altered by reflexion at a plane mirror, an inverted picture of external objects is visible. The annexed diagraia represents a box constructed on the same principle, the iriiage formed by the lens Tjeing received after reflexion at a plane mirror, on a piebe of oiled paper, or ground glass, from which extraneous light is shaded by a lid which can move up and down. 251. If an object be placed before a convex lens, Or combination of lenses, at a distance a little greater than that of the principal focus, and be illuminated by the sun or a powerful artificial light a real inverted anS magnified image of the object is formed, and if received on a screen in a darkened room will be seen as a picture on the screen. This is the principle of the Solar Microscope and the Magic Lantern. 252. The Camera Imcida of Dr WoUaston. AB CD is a section of a quadrilateral prism of glass made by a plane perpendicular to the -four planes which bound it; the z A = 90°, thez(? = 135°,andz5=^Z> = 67"30'. The surface AD except a small por- tion near D is blackened so as not to allow the passage of light. Let PQ be a luminous object placed before the side AB. The axis of a pencil from a point P of this object after passing nearly perpen- dicularly through AB is incident on BC at, an angle exceeding 1 !» K A ^ ^P-i •B H I- u CAMERA LtrCIDA. 229 the critical angle — which between air and glass is about 41° 49', and therefore is totally reflected ; in a similar manner it is totally reflected at CD, and then emerges through AD. li pq be a screen — of paper for example— and if a pencil from a point p of it after refraction through the prism near to D emerge in the same direction with the pencil from P, then if the screen be sufficiently distant, the image of P and the point p of the screen are seen together by an eye at D, and a representation of the object PQ is Yisible ou the screen. This instrument is sometimes used for tracing the eleva- tion of a building, copying designs on an altered scale, &c. There exist several different constructions with the same object, which it is unnecessary to describe here. 253. The following investigation will shew that the pic- ture of PQ seen on the screen is the same as the projection of PQ upon a plane parallel to AB. Let radii of a sphere parallel to the edges of the prism, and to the normals to its four surfaces p in the order according to which light falls upon them, meet the surface of the sphere in /, A, B, G, D'. Then / is / \ C the pole of the great circle ABCD, and / I J A AC = l = BD, BC = '^. ^ '' ^-^^^ 8 4 ' Also let radii parallel to the axis of a pencil before and after refraction into the prism, before and after refraction out of the prism, meet the sphere in P, Q, 8, T, these radii being drawn in the direction opposite to that in which the light proceeds. Let great circles through I and these points meet the circle ABGD in p, q, s, t respectively. Produce DI to D' and join lA, ID'. Draw the great circles AQP, D'ST, and also the great circle Ir through the direction of the axis of the pencil after one reflexion. Then Bq + Br^TT, ox Aq + Ar = -i , Cr+ Cs = TT, or Us ■{■ Ar = -r ; 4 230 hadley's sextant. .•.Aq = D's. Also IQ = IS, (Art.. 78) ;, .-. z QAI = z SD'I, , sin TD' sin PA and ^ cnv — fJ' = -^ — rT"^ ; sin SD "^ sin QA therefore the triangles IP A, ITD' are equal in all respects ; .-. IT=IR (i), and z TID' = a PI A ; TT /, TIP='i (ii). From (i) ft appears that the axis of a pencil refracted and reflected by the prism of a Camera Luci'da has at inci- dence and emergence the same inclination to any edge of the prism, and from (ii) it appears that planes ■ parallel to the edges of the prism drawn through the axis of the pencil at incidence and emergence are perpendicular to each other. The effect of the- prism therefore is merely to turn through 90° — - about an axis parallel to the edge of the prism — the plane of the axis of any pencil, whilst in this plane the axis preserves the same direction as before relatively to the same edge. Hence the picture seen on the screen pq is the same as the projection of the object PQ (fig. Art. 252} upon a plane paral- lel to AB. 254. Hddlej/s Sextant, or Quadrant. AC, AB are bars of metal united at A, the centre of a ITS CORRECTIONS. 231 circular arc of from 50° to 70°, to which they are attached at G and B. AD another bar turning about a hinge at A, and carrying a pointer D and vernier along the arc BG. At F and A are two plane reflectors whose surfaces are perpendi- cular to the plane ABG : the former is fixed to AG, the latter is moveable with AB, and is parallel to F when the pointer D coincides with the point E of the arc GB. Hence in any other position the angle DAE is the inclination of the mirrors to one another. Of the mirror F the lower part only is sil- vered, so as to allow the passage of direct rays close to the edge of this reflecting part. ^ is a sihall telescope attached to AB, the axis of its lenses being parallel to the plane ABC and passing through the division between the silvered and unsilvered parts of F. The instrument is used to measure the angular distance between two distant points. Let P, Q be two points whose angular distance is re- quired. The plane ABC being brought into the same plane with' them, and the telescope pointed to Q, let ^Z> be moved until P seen through the telescope by a pencil reflected in. succession at A and F appears to coincide with Q, which is seen directly. In this case the deviation of the axis of the pencil is the angular distance of P and Q. But the deviation of the axis is double the inclination of the mirrors (Art. 77), or double the angle DAE. Hence if EG be graduated from E as the zero point, every half degree being marked as a whole one, the reading coiTesponding to the position of the pointer D will be the angular distance of P and Q. Obs. The mirrors F and A, are called the horizon-glass and the index-glass respectively. The angle measured by the instrument is the inclination of PA to QFG — if the points P, Q be distant this will coin- cide sensibly with the angle which they subtend at G, i. e. at the eye of the observer. 255. Note. If when the pointer D^s&tE the zero point, the planes of the mirrors, supposed perpendicular to the plane ABG, are not accurately parallel, the angular distance of two objects determined by the instrument will be affected- 232 REFLECTING GONIOMETEE. with a constant error called the indeoo-error. This correction, which must be made to observations made by the quadrant, is equal to the reading of the limb when the mirrors are exactly parallel — which is the case when they are so adjusted that a very distant bright point, as a star, seen distinctly through the telescope, coincides with its image formed by reflexion at the two mirrors. , 256. To find ttt,e correction to the angular distance of two objects observed by a sextant, wherein the aocis of the telescope is not exactly perpendicular to the intersection of the plane mirrors. In the figure constructed as in Art. (126) let IP, IR he nearly quadrants and equal to the angle between the axis of the tele- scope and the intersection oi the plane mirrors. Let 6 be the read- ing of the instrument, or doiiible the R . inclination of the mirrors. PR = + 8 the angular distance of the objects, IR = '^,-a = IR 2 Now PIB = d (Alt. 126, Cor. 2) ; .'. cos {0+S) = sin^ a + cos^ a cos 0, which gives the correct angular distance of the objects. Also a and S being' small, we have approximately 8 = — a' tan ^ , the required correction to the reading of the limb. If Jibe the number of seconds in the angle a, the cor- rection in seconds = — n'' . sin 1" . tan ^ . 257. The Reacting Goniotneter. The goniometer is an instrument for measuring the angle between two plane faces of a crystal, and consists of a circle BRIGHTNESS OF IMAGES. 233 of metal turning about an axis perpendicular to its plane. The rim of the circle is graduated, and is read by a pair of verniers in opposite positions. Let the crystal be attached to the circle, so that the plane of the latter is perpendicular to the intersection of the faces of the former whose inclination is required. Bring the circle into such a position that the image of a well-defined straight line perpendicular to the plane of the circle formed by reflexion at one of the faces of the crystal coincides with another well-defined straight line which is seen directly, and read the verniers : — window-bars will answer the purpose of these straight lines very well. Turn the circle until a similar coincidence is made between the same straight line seen directly, and the image of the other formed by reflexion at the other face of the crystal, and read the verniers again. The semi-sum of the differences of the two readings of each vernier is the angle through which the circle has been turned, and is equal to the angle between the normals to the two faces of the crystal and supplemental to the inclination of the two faces. For a full description of the construction and use of this valuable instrument, which was invented by Dr WoUaston, see the article Crystallography in the Encyclo'p. Metrop. Also for a description of many optical instruments used in surveying and other practical operations see the Treatise on Mathematical Instruments in Weale's Series. 257*. On the brightness of images produced by the As- tronomical Telescope, and the Intensity of their light. Since every point of an object viewed through the Tele- scope must appear as a point whatever may be the magnifying power, — the intensity of the illumination of the several points 234 BRIGHTNESS OF IMAGE IN THE of the image will depend upon the quantity of light which pro- ceeds from each point of the object and reaches the eye. The brightness however depends upon the impression of the whole image upon the eye. Let a beam of rays (regarded as pa- rallel) from a point of a distant object fill the object-glass (breadth D), and emerge as a pencil of parallel rays from the eye-glass (breadth d), the breadth of the emergent pencil being S — the whole being supposed to fall on the eye-glass, — •57 the breadth of the pupil of the eye, m the magnifying power of the telescope — which. = -k whether a single eye-lens or an eye-piece be used ; and suppose for the present that in- is not < S and S not > d : also let 1 : a be the ratio in which light is diminished by absorption in its passage through all the lenses of the telescope. Now a beam of rays at incidence on the object-glass and at emergence from the eye-glass occupies circular areas of dia- meter D, d respectively — hence the brightness of the image formed by the eye-glass —, the brightness rapidly diminishes as the square of m. On the other hand, / or the intensity of the light, is constant as soon as m = or > — , provided that the field of ■ST view always includes the whole of the magnified object. / can therefore Become very great when D is great, and this is the reason why exceedingly faint stars can be seen through a telescope with a large object-glass. The diameter to- of the pupil (which may be assumed to be about 0-2 of an inch) is not only different in different observers, but also varies with the absolute intensity of the light of the object viewed — e.g. it is less when we view the Moon, greater when we view Saturn ; less when we view the Moon through a telescope of 5 inches aperture than through one of 2 inches aperture. The sky or ground of the heavens has a certain degree of brightness not only in daytime, in twilight and moonlight, but even at night in the absence of the Moon. This bright- ness of the sky also diminishes in the telescope as a ^ .^ j and therefore the ratio of the brightness of an observed object to the brightness of the sky remains constant for all magni- fying powers. This is the reason why for considerable magni- fying powers we do not observe a correspondingly great decrease of brightness. But if we call this brightness of the sky b, although the ratio B : h remains constant, our eye can nevertheless no longer distinguish the difference {B — h) of the brightness of the object and the sky when this difference is very small. Hence faint nebulae, tails of comets, &c. become invisible under high magnifying powers. The intensity of the light of the portion of the sky which we see in the telescope varies inversely as m^ nearly. This intensity of the light of the field may be so great as wholly to prevent our seeing objects of feeble intensity. This is the reason why with the comet-seeker (a telescope of large aperture and small magni- fying power) we cannot see stars, even of the first magnitude, 236 . BRIGHTNESS OF IMAGE. in the daytime, when we can see them without difficulty with telescopes of much, smaller aperture and greater magnifying power. This also explains why with high magnifying powers we often discover very faint stars which are wholly invisible in the same telescope with lower powers. The more perfect, a telescope is, the more nearly will the image of a star resemble a bright point ; and according tp the above, we may without hesitation always employ for the observation of fixed stars the highest magnifying powers. The substance of this article is the explanation of the working of a telescope given by Olben. See Chauvenet!s Astronomy, Vol. n. p. 16. Consult also Deschanel's Natural Philosophy, Prof J. D. Everett's Edition, 1882, pp. 1050— 1056,— on Measure of Brightness and Effective Brightness and Intrinsic Brightness. CHAPTER XI. OF THE RAINBOW, ETC. 258. If a pencil of light be refracted into a sphere, when it is incident on the interior surface of the sphere, a portion of it emerges and another portion is internally reflected ; this latter portion being again incident on the interior surface is partially reflected and partially refracted; and so on con- tinually. The intensity of light in the pencils which thus succes- sively emerge rapidly decreases. 259. Let G be the centre of a sphere of water, the re- fractive index of which out of air for rays of mean refrangi- bility is 1-335, or ^ nearly. Let a pencil of parallel rays of o homogeneous light whose axis is in direction AG he incident directly on the sphere. Take Gq, = ^AG, Cq.^^A C, Gq,=^A C, (Arts. 29, 24), then ?i?2?3 are the geometrical foci of the pencil when re- fracted' into the sphere, reflected at £he internal surface and emergent from the sphere respectively. 238 SUCCESSIVE REFRACTIONS AND REFLEXIONS Again, if we take ■-AC, Cq,= ^^.^^, ^^, 5 Gq, = 3AC, Cq, = lAC, Cq,= ^ AC, Cq, = ^AC, the points q^q^^qi being respectively on the side of -0 indicated in the figure. Then q^q^q^qt are the geometrical foci of the pencil, when first refracted, once reflected, twice reflected, and emergent after two internal reflexions jespectivefly. 260. Suppose a system of parallel rays incident on a P jr.- V refracting sphere — as a rain-drop — -and emergent after one internal reflexion. Let pqrse be the course of one of these rays incident parallel to the diameter PACT, and passing in a plane which contains C the centre -of the sphere, and let us for the .present confine our attention to rays which pass in this plane. Let ^ ' be the angles of incidence d,nd refraction at q, 6 = arc Ir, i. e. .the angle which Ir subtends at G, ■^ = As; produce es backward to meet fq produced in t, then ^ etv = i) = deviation of the ray ,pq. Now 'the deviation at each of the refractions at q aad s is-^ — 0', and at the -reflexion at r is tt — 2^', and these deviations are all in the same direction ; A 2) = 2(.^-^') + 'r-2f = 7r + 2(<^-20') (i). IN A SPHEEE. 239 In order to examine whether D admits of a maximum or minimum for different values of (p, we have, remembering that sin i = u sin ', and .'. -tt = ^ > * T- r- -r ' dcj) /M cos 2 cos ^ a9 \ d^/ \ fj- cos 9 d^ fi cos 9 1 \/i cos 9 / J Let ^j be the value of (p which makes -j-r = 0, then 2 cos COS 9, and ;•. > ; and . . -5-7^ IS positive ; i.e. D is a minimum when ^ = ^^, and there is only this one value of (j) for rays incident on the same side of PA which makes D a minimum. Hence, considering rays incident parallel to PA, as <^ increases from up to ^ , the corresponding deviation dimi- nishes from tt, when ^ = to 77 - 2 (20/ - (^1) when ^^ = sin"' */(— ^) - and afterwards increases from this value up to 27r — 4 sin"' - , TT as ^ increases from <^^ up to -^ . 240 REFaACTION Further, = ^ Icr = 2(j>' - tj> = '!^^ . Hence increases as D diminishes, and vice versd, and is a mawimwn when D is a minimum. Again, if v^ be the distance from q at which two con- secutive rays after refraction into the sphere intersect, we have from the formula. /tt cos" ^' cos" _ /M cos (j)' — cos 2r cos 2 (/M cos ^' — cos < 2 cos < d(j)' d(j}^ if increases from up to ^^. 261. From the preceding discussion we draw the follow- ing inferences. Itpqrse be the ray which passes with minimum deviation, then a small pencil incident at q has r for its primary focus after refraction, and emerges at s as a pencil of parallel rays in some direction se. As ^ increases continuously from up to 0^, i/r increases and D diminishes continuously, i. e. the rays incident on the arc Aq emerge from the arc As in a state of divergence. P. o. 16 242 EXPLANATION OF THE If fg be the extreme incident ray (for which ^ = -^ j , and we take the arc AsK equal to the maximum value of i/r, then the rays incident along qg after refraction into the sphere are incident on its surface within the arc Ir and emerge through the arc sK : and since the deviation of the ray which emerges at s is a irdnvmum, the rays which emerge froni sK will by their consecutive intersection after emergence form a caustic curve, of which se is the asjrmptote, — the ray emergent from the extreme point K being that which was incident at ^ ^,^. The line se will also be an asymptote to the virtual caustic formed by the rays emergent along As, produced backward. If we now suppose the whole figure to revolve about PAG as an axis we arrive at the case of a beam of parallel rays incident on the sphere, and the surface traced out by se will be a conical asymptote to the caustic surface formed by the emergent rays. If the figure represent a plane section through the axis of the beam PGI and an eye E', — if the , eye be situated within the space included by AP and se, it will receive a small pencil of rays, transmitted through the sphere in the manner supposed, — the divergence of the visual pencil being less and less as the angular distance from PA is greater and greater, till when the eye is so situated that se meets it, the visual pencil consists of parallel rays, and the impression received is more vivid than for any other position of the eye : if the eye be beyond the space included by PA and se it receives no light transmitted in the manner we are considering. 262. We proceed to explain the formation of a Primary Rainbow. Let G be the centre of a small spherical drop of rain fall- ing in the air, and let a beam of sun-light fall upon it, which from the distance of the sun regarded as a point may be considered to consist of parallel rays. Suppose light homogene- ous, — then of this beam a small pencil having pq for PRIMARY RAINBOW. 243 its axis after being refracted into the sphere and once inter- nally reflected, may emerge in a divergent state in the direc- tion tE (Art. 261) and fill the pupil of an eye E, creating the sensation of a bright point in the sky in the direction Et of the species of light which is considered. Through E draw EB parallel to pq, then since the de- viation of pq — and consequently the angle tEB — depends merely on the angle of incidence of pq, and since all rays from the sun may be considered parallel, therefore all drops of rain whose centres lie in a conical surface of which EB is the axis and tEB the semi-vertical angle, will transmit to the eye a similar pencil of divergent rays. A drop of rain at less angular distance from EB than G will transmit to the eye a small pencil with greater deviation and greater divergence than tE, and therefore producing the sensation of a less bright poiat in the sky, — and the divergence of pencils which reach the eye at a greater angular distance from EB than A is less than that of the pencil tE, until when the deviation- is a mioimum, the pencil is nearly one of parallel rays, and the impression produced by it on the eye is greater than that of any other pencil. The impression therefore to the eye produced by pencils transmitted in the way supposed from the drops of rain in a distant shower, — the observer being between the sun and the shower, — would be an illuminated sky of the colour of light, which has been considered, the brightness being greater at greater distances from the line EB, until it is bounded by a circle whose centre is in EB, — beyond which there is no colour, and comparative darkness. For each species of sun-light this would be the appearance, the bounding circles for different species differing slightly in position, since the amount of minimum deviation depends upon the refractive index of the light (Art. 260). Th.^ result of superposing these illuminations of different colours will be white light within a certain distance of the line EB termi- nated by a narrow circular hand of vivid prismatic colours arranged in concentric circles about the line EB. Beyond this band there is no illumination of the sky by pencils trans- mitted to the eye in the manner now considered, and it will appear comparatively dark. 16—2 244 PRIMARY RAINBOW. Further, considering the finite extent of the sun, the bands of colour such as have been described resulting from pencils proceeding from the several poiats of the sun, will overlap each other, — the breadth of the band will be increased by the sun's apparent angular diameter, and the colours will be more or less miKed ; but the extreme colours of the result- ing band will be unchanged. This phenomenon produced by pencils which have been once internally reflected in the rain-drops is called the Pri- inary Rainbow. 262*. Note. The question may suggest itself, whether two observers or two eyes — even if very near each other — see the same rainbow. The pencils of rays transmitted to two eyes with minimum deviation cannot be identically one and the same pencil : — so that the rainbows seen by two eyes are not in identically the same position in space ; — even if the two eyes are those of the same observer. Again, a rainbow seen apparently reflected in water is not an optical image of the rainbow seen in the sky directly : — but is a reflected image of the rainbow which would be seen by an eye below the water vertically below the eye of the observer, — and at a distance below the surface of the water equal to the height of his eye above it. 263. Some writers suppose that the pencils received by the eye in a state of divergence, — those namely which emerge through the arc As in Art. 261 — are too faint to produce any sensible impression on the eye, and that only those pencils which pass with minimum deviation, and so enter the eye in a state of parallelism, give a sufficiently vivid sensation. This supposition afibrds the same explanation, as above, of the rings of prismatic colours which constitute the rainbow, but it leaves out of consideration as insensible the faint illumination of which in the previous article we have supposed the bow to be the boundary — which being of a neutral tint is not very striking. Since, however, no rays can reach the eye from drops outside the bow, except such as have been refiected from their SECONDARY RAINBOW. 245 surfaces, whilst from drops within the bow the eye receives rays which have been internally reflected as well as rays re- flected from the external surfaces of the drops, we have an explanation why the space within the bow must appear brighter than the space without it. The fact that the rain-drops are in falling motion does not interfere with the phenomena, — the place of a falling drop 4Deing immediately supplied by another yielding the same kind of pencil of transmitted rays. 264. Again, suppose a system of parallel rays incident on a refracting sphere, as a rain-drop, and emergent after two internal reflexions. Let pqrste be the course of one of these rays incident parallel to the diameter PACT, and passing in a plane which contains C the centre of the sphere, considering only rays which pass in this plane. Let (f), = 2 (^ - ^') + 2 (tt - 2f ) = 27r + 2(') ; d(j) \ /J' cos

j = sin~'^[ — ^ j , and afterwards increases from this value up to Stt — 6 sin"' — , as 6 increases from 01 up to -„ . Further ■y^ = arc IqAt = 6^' le of which mal fjL cos j^ If 0„ be the value of (j)^. THE SECONDARY RAINBOW. 247 265. If pqrste be the ray which passes mth minimum deviation, it follows that the change of deviation for the consecutive ray being inappreciable, — a small pencil of which pq is the axis will emerge as a pencil of parallel rays of which te is the axis; — and by employing the formula of Art. 68, it may be shewn without much difficulty that the primary focus after the first refi:action is at a distance from q g along 2^ = T qr, — ^that between the two reflexions the rays are parallel, and that after the second reflexion the primary focus is at a distance firom t along ts = -j ts. 266. From the preceding discussion we draw the follow- ing inferences. If pqrste be the ray which passes with minimum devia- tion, then a small pencil of which pq is the axis, emerges as a pencil of parallel rays, of which te is the axis. As — ^' is the deviation produced in the axis of the pencil by each of the two refractions, and tt — 2^' that pro- duced by each of the p reflexions, and these deviations are all in the same direction ; .-. D = 2(-^') +p(7r-2(j>')=p7r+2 {4>-(p + 1) f }, and sin ^ = /* sin "' .■.0.1-(, + l)^', and P = cos ^ — fi cos ^' d^ ' .'. fi cos ; OEDER OF COLOUES. 251 therefore -7-^ is positive, or D, increases as fi increases — that is, the minimum deviation of the axis of a pencil of any colour is greater as fi is greater. The minimum deviation therefore in any rainbow is least for red, and greatest for violet. Considering then the figure of Art. 262 — in the primary rainbow, the red circle is highest and the violet is lowest. In the secondary bow (figure, Art. 267) the red circle is lowest and the violet is highest. These conclusions agree with the results of calculations given in the succeeding articles. 271. If we take the values of /x for extreme red and violet rays to be 1'331 and 1"344 respectively, we shall ob- tain, — in the primary bow, tovred (j>, = 59''S2', (^,' = 40°21', D, = 137"40', angular radius of red = tt — Z>j = 42° 20' ; foT violet (^, = 58H4', (/>/ = SO'SO', D, = 139»28', angular radius of violet = Tr — D^ = 40''32'. Hence angular breadth of the bow which would be formed by rays proceeding from one poiat of the sun's disc = 42»20' - 40°32' = 1°48', nearly. Taking account of the pencils proceeding from different points of the sun's disc, the breadth of the bow will be in- creased by the sun's apparent diameter, i. e. by about 32'. Hence angular breadth of the primary rainbow = 1"48' + 32' = 2° 20', nearly. It appears that the violet circle will be the lowest and the red the highest in the primary rainbow. 252 DIMENSIONS OF THE RAINBOW 272. In the secondary how, tor red <^, = 71°54', <^; = 45°34', Z>, = 230''24', angular radius of the red = D, — tt = 50° 24' ; for violet <^, = 7r29', / = 44''52', Z>, = 23:3°46', angular radius of the violet = -D, — tt = 53°22'. Hence the total angular breadth of the bow will be = 53°22' - 50° 24' + sun's diameter = 2°58' + 32' = 3»30'. It appears that the red circle is lowest and the violet highest; 273. The investigations of this Chapter must be received as general explanations rather than as exact calculations of the phenomena of the Rainbow. A pencil of parallel rays inci- dent on a rain-drop, and emergent after one or more internal reflexions, forms a caustic surface, — and the small oblique part of the pencil which enters the eye is determined by draw- ing from the eye a tangent to the caustic. Now we have assumed in the preceding explanations that the illumination is a maximum when the eye is so situated as to receive a pencil whose axis has passed with minimum deviation, but if we consider the diagram of Art. 261, it is obvious that if the eye be situated within this limiting direction se, as at E', — one tangent can be drawn from the eye to the caustic formed by the rays emerging from sk, and another to the virtual caustic formed by the rays emerging from sA. And thus there may be several directions of maximum illumination, none of them coinciding strictly with the direction es. The intensity of illumination of the sky in different direc- tions has been calculated on the principles of Physical Optics, (Cambridge Philosophical Transactions, Vol. vi.). It is thus found that in the case of the primary bow the principal maxi- mum of illumination lies a little within the position which the geometrical theory gives : .also that withiu this there is a series of inferior maxima becoming in succession smaller. Hence GENERAL EEMARKS. 253 when compound light is considered there will be in addition to the principal bow a series of interior bows decreasing in brilliancy, — to these the name of spurious or supernumerary bows is given. Two or three of the spurious bows of the primary rainbow may sometimes be seen in nature. The results of theory on this subject have been tested by measurement, by allowing a very thin column of water to run from a vessel, and receiving by a telescope artificial light which has been internally re- flected in this column. (Cambridge Philosophical Transac- tions, Vol. VII.) There is frequently observed in northern countries, and sometimes in our own climate, a regular and complex series of luminous curves surrounding the sun or moon. There is commonly discerned, 1°. a first circle or halo AA', red within,_ violet without, concentric with the sun and making with him an angle of 22° or 23°; 2°. a secrnid circle or halo BB', similar to the preceding, placed at 46° from the sun ; 3°. a diametral horizontal portion BB' of a very large 254 REFERENCES. circle, the parhelic circle, on which is seen opposite to the sun a bright point, the anthelion; 4°. at the points of junction of this circle with the two halos in B, E, A, A' increased intensity of luminosity, which have been taken for images of the sun; 5°. at G, C, D, D' horizontal arcs, tangents to the circular halos:— at G, G' they have little brightness, — at D, D' they are very vivid and constitute the most brilliant part of the phenomenon ; 6°. last a vertical white line DZK making a cross with BB'. In most instances there is only seen a portion, more or less extended, of the whole phenomenon. M. Bravais, beyond all others, has given the most com- plete explanation of these appearances — on the hypothesis of pencils of light transmitted to the eye after one or more intemar reflexions through small hexahedral crystals of ice, which are suspended in the air by ascending currents, — especially in the cold mornings of spring and autumn. The above picture and description is taken from M. Jamin, Cov/rs de Physique, Tome ill., 1869, to which the student is referred for further information. The student may also con- sult Dr Young's Lectures, or Moigno, Optique Moderne, Vol. i. — and the memoirs of M. Bravais on the Rainbow, Parhelia, Halos and the optical phenomena which accompany them, in the Journal de I'Ecole Royale Polytechnique, Tom. xviii. EXAMPLES AND PEOBLEMS. CHAPTER I. Laws of Propagation of Light — direct reflexion and refraction. 1. A circular disc, six inches in diameter, is placed with its plane parallel to a vertical wall, and a person places his eye anywhere in the plane of the disc ; shew that a plane circular mirror, 3 inches in diameter, will if placed in a proper position on the wall just enable the person to see the whole image of the disc. 2. A fish is floating in a cubical glass tank filled with water, with its head in one comer and its tail diagonally op- posite; describe the appearance which will be presented to an eye looking towards the corner in the direction of the length of the fish, and in the same horizontal plane with it. 3. The image of a stick immersed in water is iuclined to the horizon at an angle of 45°; find the inclination of the stick, f being the index of refraction from air into water. 4. The faces of two walls of a room, meeting at right angles, are covered with plane mirrors ; shew that a person will be able to see but one complete image of himself in either wall. 5. A sportsman is shooting at a fish in the water; is it necessary for him to aim above or helow the fish ? 6. From the equation connecting the distances of the conjugate foci from the point of incidence, deduce the equation connecting the distances of the conjugate foci from the centre of the reflecting surface. 256 EXAMPLES AND PROBLEMS. 7. A pencil of parallel rays is incident directly upon a spherical refracting surface, and after refraction converges to a point at a distance from the surface equal to three times the radius^ — find the index of refraction (i) when the surface is concave, (ii) when it is convex. 8. If a luminous point be seen after reflexion at a plane mirror by an eye in a given position, there is a certain space within which the image of the point can never be situated, however the position of the plane mirror be changed, — find this space. 9. A speck within a solid glass cube is viewed directly through each face in succession; prove that the six images will form the angular points of an octahedron, not generally regular but having all its diagonals equal. Also compare the volume of the octahedron with that of the cube. 10. A bright point on a glass plate is viewed by an eye close to the plate, and at a given distance from the point; find the direction in which the w* image is seen after successive reflexions within the plate. 11. If the direction of a ray proceeding from a point P on the circumference of a circle, after refraction at the curve pass through a point on the circumference at an angular distance IT ^ firom P, find the point of incidence; — \/S being the refract- o ing index. 12. A small object is placed in one focus of a prolate spheroid of glass, which is silvered at one of its poles; an eye being placed near the other pole, views the object directly, and by reflexion : if v^, v^ be the distances of the images seen from the eye, shew that 11 4. v^ v^ latus rectum ' Shew further that only one image will be seen it fie > 1, — and supposing fie <1 compare the magnitudes of the images. 13. A luminous point is placed at the focus of a prolate spheroid, the index of refraction from which into air is equal CHAPTER I. 257 to its eccentricity. Find the direction' after refraction of a ray falling on the further side of the spheroid. How must the direction after refraction of a ray falling on the nearer side be determined ? Explain the discontinuity. 14. A luminous point is placed at one comer of a cube of some refracting substance; shew that there will be three images formed by refraction, situated at the angular points of an equilateral triangle, and that these images will be simulta- neously visible, if the eye- — supposed to move in a plane per- pendicular to the diagonal passing through the luminous point — lie within a certain hexagon. 15. A coin is placed at the bottom of an empty hemi- spherical basin of given radius, and is just not visible to an eye looking over the edge — when the basin is filled with water, the whole of the coin is just visible to the eye in the same position — find the diameter of the coin. 16. A luminous point is placed in front of a refracting medium bounded by a plane transparent surface; prove that, if the bounding surface turn in any manner about a given point in its own plane, — the geometrical focus of the rays after refraction into the medium, will always be on the sur- face of a sphere. 17. Two rays emanate from a point in the circumference of a reflecting circle, in the plane of the circle: supposing that their w* points of incidence are coincident, prove that the angle between their original directions is any one of a series of w — 1 angles in arithmetical progression. 18. A speck is situated just within a glass sphere; shew how much of the surface of the sphere must be covered, in order that the speck may be invisible at all points outside the sphere on a line drawn from the speck through the centre. 19. If the angle of a hollow cone, polished internally, be any sub-multiple of 180°, a cylindrical pencil of rays incident parallel to the axis will, after a certain number of reflexions, be a cylindrical pencil parallel to the axis, and of the same diameter as the incident pencil, P. 0. 17 258 EXAMPLES AND PKOBLEMS. 20. A luminous point is placed within a reflecting sphere, and the light which falls directly upon the most distant point of the surface is repeatedly reflected. Shew that the distances of the geometrical foci from the centre decrease in harmonic progression. 21. An eye is placed close to the surface of a sphere of glass {ft. = f ) which is silvered at the back ; shew that the image which the eye sees of itself is three-fifths of the natural size. 22. A room has its walls covered by mirrors, shew that a man may see himself by reflexion at four of the w'alls^ if he look in a direction parallel to either diagonal. In what direction must he look in order to see himself after refleixion at three of the walls ? S3. A circular disc exactly fits a hole in a wall and revolves in its own plane about its centre; 2w equidistant radii are drawn on it, and the space between every alternate pair removed : supposing the image of every object to remain the t** part of a Second upon the retina of the eye, find the small- est angular velocity which may be given to the disc consist- ently with the eye having an unobstructed view through the hole. If the disc itself be looked at under these circumstances, what will be the appearance presented ? 24. A candle is placed at a given distance in front of a vertical plane circular mirror on a line perpendicular to the horizontal diameter at its extremity; shew that the boundary of the reflected light, which falls on a wall of which the plane is perpendicular to that of the mirror, is a parabola ;— and de- termine its latus rectum. ■ 25. A luminous globe falls from a point above the Earth's surface in a dark night : shew that it will look like a bright falling column, elongating as it descends. 1 :i CHAPTER I. 259 If Cj, c^, Cg be the lengths of the apparent column at the ends of times t^, \, t^, from the commencement of the fall,^ .prove that, gravity being considered constant, and the resist- ance of the air being neglected, ^1 (p2 - C3) + K (C3 - cj + 1, (c, - c,) = G. 26'. A thin plate of glass is placed parallel to a screen, and there is a luminous point in any given position on the other side of it ; the glass is cracked in one spot in the shape of two straight lines forming a cross, the planes of the cracks being perpendicular to that of the glass : shew that there wiH be two corresponding figures of a cross thrown upon the screen, the one bright and the other dark : shew also that the lumi- nous point may move in a certain fixed line without pro- ducing motion in the figures. 27. A stick of given length is suspended vertically in a room. A candle is carried past ithe stick in such a way that the shadow of the stick falls partly on one wall, and partly on the ceiling, — the extremities of the shadow on each lying in two straight lines : find the locus of the candle. 28. An opaque sphere is placed upon a plane, and in the diameter passing through the point of contact a luminous point is placed, — its distance from the sphere being equal to the radius, — ■ Prove that the area of the shadow cast on the plane is three times that of a great circle of the sphere. If a transparent liquid, whose refractive index is ^3 be placed above the plane, so as just to cover the sphere, shew that the area of the shadow will then be reduced to twice that of the great circle. 29. If a concave mirror revolve round an axis through its centre, slightly inclined to the axis of the mirror, find the appearance on a screen placed at the farther focus, when a series of electric sparks passes in quick succession through the nearer focus in the axis of revolution. 17—2 260 EXAMPLES AND PEOBLEMS. 30. Three candles are placed in a room, and the two shorter being lighted throw shadows of the third upon the ceiling ; if the directions of these shadows be produced, where will they meet ? • - 31. If a globe be placed upon a table, the breadth of the elliptic shadow cast by a candle (considered as a luminous point) will be independent of the position of the globe. • 32. The centre of a spherical ball is moveable in the Tertical plane which is equidistant from two candles on a table : find its locus when the two shadows on the ceiling of the room are always in contact. 33. In Art. 24 — if Q, q be conjugate foci, — and on any chord AB oi the reflector a point G be taken, and QG, qC cut OB produced if necessary in R, S; then R, 8 will be conjugate foci. 34. The sides of a triangle are reflective, and a lumi- nous point is placed within it : prove that the area of the triangle formed by the images varies as the rectangle con- tained by the. segments of any chord of the circumscribing circle passing through the luminous point. 35. If a luminous point move between the centre and surface of a refracting sphere, prove that the distance be- tween the point and its geometrical focus will be greatest when the distance of one from the centre is equal to the distance of the other from the surface. 36. A person whose height is h, observes vertically be- neath his eye, an object at the bottom of a clear pool: he then removes to a distance d, keeping his eye on the object, when his line of vision makes 45" with the surface ; shew that if /i^ = 2'5, the depth of the pool =2{d — h). 37. A pencil of rays is directly refracted through a hemisphere of glass (radius = 7*) jo=the distance of origin of light from the incident surface, shew that the position of the geometrical focus will be unaltered when the hemisphere, is reversed if r' + (/x -i- l)pr - /i Ott - 1) p= = 0. CHAPTER I. 261 ■ 38. AB is the diameter of a polished semicircular arc APB. A ray of light proceeds from a point Q in the tan- gent at A, and after reflexion at P and B returns to Q. If the length of the ray's path be 2 feet, the mirror's diameter is very nearly 7 '35 inches. 39. P is a point in a diameter AB of a sphere. If P be the origin of a pencil of rays and u, v the distances of the geometrical foci after direqt reflexion at A and B, shew that iuv + 4r' — 2ur =3{u + v) r. 40. In Art. 33, if the luminous point, together with its images, form a regular pentagon, shew that the mirrors must be inclined to one another at an angle 72°. 41. A cube of glass stands in the sunshine. Prove that in general light emerges from it in twenty-eight directions. 42. In Article 33, if the angle between the mirrors be 80°, determine the positions of the point for which there wilt be 5 images, and those for which there will be 4. 43. A transparent sphere is silvered at the back, prove that the distance bet weesi the images of a speck within it, formed (i) by one direct refraction, (ii) by one direct reflex- ; , J- . e ^- ■ 2iJ.ac{a-c) ion and one direct retraction, is = -. — ^ ^~, — ^ , {a+c— (xc) {jiG+a — 2c) a being the radius of the sphere, and c the distance of the speck from the centre, measured towards the silvered side. 44. In one side of a triangle, the interior of which is capable of reflecting light, there are two holes ; determine by a construction the position of a point from which rays may enter so as each to pass out after one reflexion. 45. A bright point is moved about an ellipse whose focus coincides with the centre of a reflecting sphere, of such a mag- nitude that the directrix corresponding to the focus meets the sphere. Prove that the positions for which the geometrical focus of the point is a point in the same ellipse, are those in 262. EXAMPLES AND PEOBLEMS. which the ellipse is cut by the diameters of the sphere which are drawn to the points where the directrix meets it. 46. A luminous point is situated in a plane which cuts at right angles the line of intersection of two plane reflectors, in the point 0; shew that if the straight line joining the two images of the point, touch a circle the centre of which is 0, the locus of the luminous point is also a circle. 47. A man 6 feet high stands in front of a looking-glass which rests on the ground and leans at an z 30° against a wall, from which he is 10 feet distant. What must be the length of the glass that he may just see his whole person ? 48. Two rays are incident upon a spherical reflector in the same principal plane, and the angle between their direc- tions before incidence is ^, and after reflexion is —/, and shew that none of the emergent rays will be parallel to the axis if ^a < -^ . 52. A ray is refracted through a sphere, its shortest dis- T til tance from the centre of the sphere being — the radius — CHAPTER I. 263 shew that if n be large the total deviation of the ray n 53. If I, m, n be the direction cosines of the normal to a mirror, (X, /j,, v), (A,', /j,', v')' those of the incident and re- flected ray, then — ; — - = - — — = = 2 ((\ + mu, + nv). 54. The inner surface of a hollow square is polished, and a luminous point is placed at the intersection of the diagonals; shew that the number of distit-ct images formed after n re-, flexions is the sum of the series 4 (1 + 2 + 3 + .. .+n). 55. Q is a luminous point situated on AB the diameter (2r) of a hollow sphere (centre 0) polished at A and B:f and /' are the foci of the pencils reflected at A and B respectively — shew that 4 //'=-r 4* OQ' 56. Explain the principle of the Kaleidoscope: — an4 shew that when the mirrors are inclined at 60°, the area of the field of view is to the area of the transverse section of the tube as 27r + 3 \/3 : TT. 57. The locus of the image of a luminous point reflected in a plane mirror is a circle, prove that the mirror always touches a conic section. 58. The radii of the centre and inner surfaces of a spherical refracting shell are B, r respectively. Prove that the distance from the centre of the geometrical focus of a pencil of parallel rays after refraction through the whole shell is 1 fi Br 2fi-l'B-r' 264- EXAMPLES AND PROBLEMS. 59. A ray emanates from a luminous point P and after two reflexions at a reflecting circle returns to the point again, shew that p + 1 where 6 is the requisite angle of inbidence, p the distance of P from the centre, and q the distance of the centre from the point .where the' ray in its course crosses the diameter on which P lies. 60. In the case of two plane mirrors (Art. 83) if S lie between - and ^ , prove that there will be 2n + 1 images when the angular distance of the luminous point from the nearer mirror is lesS thaff'the lesser of the two quantities TT — nS, (w + 1) S — TT ; and that when it lies between these two quantities there will be 2n or 2 (w + 1) images according as the former or the latter is the lesser of the two. 61. If a ray of light after reflexion at each of the sides of a triangle in succession retrace its path, shew that it must proceed along the lines joining the feet of the perpendiculars drawn from the angular points to. the opposite sides. 62. A luminous point is in the centre of an equilateral triangle; shew by considering the course of a ray parallel to one side, that the distance of the image from the luminous point for 2w reflexions is na, and for 2w + 1 reflexions is a being a side of the triangle. 63. A ray is incident on a refracting sphere; shew that after one . internal reflexion it will emerge parallel and re- versed, if cos 0' = ^ ; and after two internal reflexions parallel and proceeding, if cos i\/2; shew that when it is looked into by an eye situated anywhere on its axis produced, the whole of the inner curved surface will glisten brightly as compared with the inside of the opposite end. 69. Two plane vertical mirrors intersect at right angles and a person looks into the angle formed by them._ Prove that, supposing no light can be reflected at the line of junction of the mirrors, he will see' only one eye in the mirrors, and that if he shut either eye the image seen will be that of a closed eye. 266. EXAMPLES AND PROBLEMS. 70. A hollow globe of glass has a speck on its interior, surface; if this be observed from a point outside the sphere on the opposite side of the centre, prove that the speck will- appear nearer than it really is by a distance ^ =- 1, pro- vided that t the thickness of the glass is equal to the radius of the internal cavity. 71. Explain why it is that writing paper soaked in oil becomes semi-transparent. 72. If a, /8 be the distances of any two geometrical foci Q, g from the surface of a spherical refracting medium, and Q', c[ be any other two geometrical foci, shew that 73. Two circular plane reflectors, the diameters of which are a, /3, are placed so that the line joining their centres is perpendicular to the plane of each, and a bright point is placed midway between the centres; the greatest number of images of the point visible to an eye looking over the edge of the larger reflector, is expressed by the greatest integer in /3 + a Chapter II. Illumination of Surfaces. 1. Two candles of unequal brightness and height are placed upon a horizontal- table, — shew that the locus of the point on the table which is equally illuminated by the candles is a circle. 2. Why does the apparent brightness of a light seen by night — (neglecting such disturbing causes as absorption by the air) — remain the same at all distances ? CHAPTER II. 267 3. A plane drawn through a given point is illuminated by two selt-lummous spheres; find the position of the plane when the illumination at the given point is a maximum. 4. If a candle be placed upon a table at a given dis- tance from a point in that table, what must be the height of the candle that the brightness at that point may be the greatest possible ? 5. The shadow cast by an oblate spheroid resting on its- vertex on a horizontal plane, in the sun,— is an ellipse ; and the spheroid stands in its focus, 6. If the direct rays of the sun pass through a small hole and fall perpendicularly on a screen, the spot of light on the screen will be circular ; but if the hole be of con- siderable size, the shape of the spot will be generally similar to that of the hole : explain this. What phenomenon explicable on the same principles has been observed in the shadows of the foliage of trees during a partial eclipse ? 7. A plane mirror revolves about a vertical axis in its own plane, and an eye in the same horizontal plane with a small jet of light observes its image : shew that the appear- ance will be the same as if the jet had crossed a field- defined by a hole in a screen in the place of the mirror, with a velocity double of that which it would have had if it had moved as if attached to the mirror. If the jet rapidly contract and expand, what will be the appearance of its image when the velocity of the mirror is very great ? 8. A plane mirror in the form of a circle whose rad. = a turns very quickly about a diameter. The eye and a luminous point are at equal distances d from the centre of the face of the mirror, and lie in a plane through the centre perpendi- 2(58, EXAMPLES AND PROBLEMS. cular to the diameter about which the mirror turns. Prove that the eye will see a line of light subtending an angle (a cos a> 2 sin d % where 2a is the angle subtended at the centre of the mirror by the line joining the eye and the luminous point. 9. Two semicircular self-luminous plates are placed with their diameters upon a plane, and with their planes per- pendicular to the line which joins their centres A and 5; find the position of that point on the line AB where the illumi- nation of the plane upon which the semicircles are placed is least. 10. A small plane touches a self-luminous paraboloid of revolution at its vertex and is then moved parallel to itself along the axis produced ; prove that the illumination of the plane varies inversely as its distance from the focus. 11. A uniformly bright sphere is placed in a hollow paraboloid, the centre of the sphere coinciding with the focus ; shew that the total illumination -=tan\/- n. V a a-Vx Ja for a portion of the surface corresponding to a length x of the axis. 12. A circular window of radius c is made in a wall running north and south. Shew that the area of the illumi- nated portion of the inner floor caused by the sun's rays is TTC^ cot a sin /3 — -where a, yS are the altitude and azimuth of the sun and the niagnitude of the sun's disc is neglected. 13. A lumiQOus point is situated at the centre of the base of a hollow perfectly reflecting cylinder of very small radius, and a horizontal screen is held over the cylinder at a height above its upper end which is half as great again as the height of the cylinder. Prove that a series of alternately darker and brighter rings is formed on the screen, the breadths of which are equal to the radius and diameter of the cylinder respectively. chaptUe II. 269 14. A'briglit point is placed at the pole of a curve de- fined by the equation i = -+ycos^"«0; r a shew that the illumination of the whole curve x - , a 15. Supposing the sky on a cloudy day is as bright at every point as the moon's disc is at night, compare the light of such a day with that of a night when the moon is at full and shining : the moon's angular radius being = 15'. 16. A very narrow band of uniform breadth le is bent into the shape of an elliptical hoop. If a luminous sphere of rad. a and brightness / be placed with its centre at one of the foci, shew that the total quantity of light received on the hoop is — -J — /, where I = lat. rect. of the ellipse. 17. A circular disc in which the brightness at distance r from the centre = /Sr, illuminates a small plane placed per- pendicularly to the line joining it with the centre of the disc, and at a distance equal to the radius of the disc (a) ; shew that the illumination "^"Kf-O- 18. A curve is illuminated by a bright point on it : if the illumination at each point of the curve vary as r-""^ find the equation of the curve. Ex. n = ^. 19. If (7 be the centre of a sphere of radius r, and of in- trinsic brightness B, shew that the illumination produced by the sphere at the point P of another surface is T^J^^pjCOS^, where or < Vf • 21. A pencil of light consists of two kinds, which are differently absorbed in passing through the same medium ; in passing through a plate of a medium A of a unit of thickness the intensities of the emergent light for each unit of intensity of the incident light are a, /3 respectively, — and the cor- responding quantities for a medium B are a', ^'. Find the relative thicknesses of two plates of the two media, in order that the character of the light may not be changed by trans- mission through them. *(& *ic * ^P * 22. Light emanating from a luminous circular disc, placed horizontally on the ceiling of a room, passes through a rectangular aperture in the floor : ascertain the form and area of the luminous patch on the floor of the room below. Shew that neither the shape nor the area of the patch will be affected by any movement of the disc along the ceiling. 23. Suppose the interior surface of a hollow sphere to be self-luminotis and non-reflecting, biit to scatter a certain por- tion, say — , of the light which falls upon it, the intensity of scattered light following the law of emanation ; — find an ex- pression for the apparent brightness. 24. A narrow self-luminous rectangular lamina is placed with one end at the edge of a circulai- plate ; the lamina is at right angles to the plate and its plane passes through the centre of the plate : find the whole illumination on the plate. If the length of the lamina be equal to the diameter of the plate, its intrinsic brightness and breadth being given, prove that the illumination varies as the diameter of the plate. 25. A ray of light is incident upon one of two reflectors, inclined to each other at an angle — , in a direction parallel to a line which is at right angles to their intersection and bisects CHAPTER II. 371 the angle between them ; supposing the intensity of a ray- reflected at an ^ (/) to be to that of the incident ray as e cos ^ : 1, shew that the intensity of the ray after it has suffjered n reflexions will be' to that of the incident ray as 26. Find the quantity of light received by an inflnite plane placed at a given distance from a self-luminous sphere. 27. A pencil of parallel rays is incident parallel to the axis of a refracting prolate spheroid, so as after refraction to converge to the focus — shew that the illuminating power of any small pencil will vary as SP'\ (SP — HP), where Pis the point of incidence on the spheroid and 8, H the foci. 28. If a surface be illuminated by a uniform bright sphere, and the illumination at any point be a function of the distance from the centre of the sphere, shew that the surface must be one of revolution. 29. A right cone, the radius of whose base is to its height as 1 : V2, stands on a table and its surface is uniformly self- luminous: shew that the illumination of a point on the table at a distance from the axis of the cone equal to its height is 30. A luminous point is placed at one of the foci of a semi-elliptic arc bounded by the axis major : prove that the whole illumination of the arc varies inversely as the latus rectum. 31. A window, the height and breadth of which are respectively h and 2a, reaches to the ground : shew that the illumination of a point of the floor at a distance x in front of the centre of the window varies as cot ' - - „■, .r. _.v cot ' ^ [---, -1^ a? _^j._i l(V-\-cc 32. If Cj, %, C3 be the lengths of the meridian shadows of three equal vertical gnomons, on the same day, at three dif- 272 EXAMPLES AND PKOBLEMS. ferent places on the same meridian, — prove that the latitudes \j, Xj, \ of the places are connected together by the equation ••tan(\-\3)^ ^•tan(X3-\)^''=-tan(X,-X,) "' 33. ■ The illumination on a small plane area produced by a luminous straight line of small but sensible thickness k placed parallel to the area at a distance d from it, is i"^ {a + /3 + sin (a + /3) cos (a - jS)} ; where a, /S are the angles of incidence of the extreme rays which fall on opposite sides of the normal to the plane area. 34. The interior surface of a sphere of radius b has a uniform intrinsic brightness B, and is non-reflecting, but Jth scatters — of the light which falls upon it, the intensity of scattered light following the law of emanation. Concentric with this sphere, and within it, is a black sphere of radius a. Prove that the illumination at any point of the bright sphere is 1 + B. — (^-S(-f5 35. Shew that the directions of the shadows of parallel straight rods thrown on a plane by a luminous point all meet in a point. Determine its position : and shew that it will trace out a curve of area A -. — if the luminous point sin a ^ describe a curve of area J. on a second plane, where a, ^ are the angles made by the rods with the planes respectively, 36. Shew that the least shadow which can be formed on a plane by a parallelepiped of equal edges intercepting the rays of a luminous point at an infinite distance = a" (4 cos' X - 1) tan' X ; CHAPTER II. 273 and the greatest, on a plane perpendicular to the incident light = 2 73 a' sin' X. The parallelepiped has one of its solid angles formed by three plane obtuse angles each = 2X, and a = length of an edge. 37. ^ and i? are two luminous points whose intensities are as n : 1 . P is a point in an ellipse of which they are the foci. Shew that the illumination of the curve at P is a max- imum or a minimum, when Determine which it is, and shew that for such points AP 15.. must be > ^ and < ^ major axis. Shew also that by increas- ing the value of n the above value of AP increases. 38. . A luminous point is placed within a triangle ; prove that the total illumination of the sides is a minimum when the illumination of each side is proportional to the area of the triangle which has that side for base and the luminous point for vertex. 39. A triangular prism whose nine edges are all equal, is placed with one of its rectangular faces on a horizontal table and illuminated by a sky of uniform brightness, shew that the total illumination of the inclined and vertical faces are in the ratio 2^3 : 1. The foci of an ellipse are luminous points. Shew that the illumination at any point P AG^-BG'' + CP' QC CD' 40. A luminous point is placed on the axis of a truncaited conical shell ; prove that the whole illumination of the surface of the shell varies as where a^, a^ are the radii of the circular ends of the shell, and Cj, Cj the distances of the luminous point from their planes. 18 P. o. 274 EXAMPLES AND PROBLEMS. 41. A circular cone stands on a horizontal table. Vertically above the summit is a luminous point from -which a small pencil falls on the parts of the cone close to the vertex. Shew that there will be a circular bright ring on the table, of radius tt, — ^, , where r is the radius of the base and h the height of the cone. 42. A regular tetrahedron stands on its base, and is exposed to a sky of uniform brightness; shew that the integral illumination of each of the upper faces is •„.--- , where B is the brightness of the sky and b = edge of tetrahedron. 43. A vessel in the form of a right cylinder, polished on the inside, stands upright in the sunshine. It has an opaque cover in which there is a hole. Prove that, provided this hole be not intersected by a certain vertical plane, there will be a circular space in the base of the cylinder on which no light falls, whatever be the length of the cylinder ; and that this is equally true after any amount of fluid has been poured in. 44. A polished hemisphere is placed with its base on a plane, and receives light in a direction perpendicular to the plane ; prove that the illumination at any point of the plane by reflected rays x r^ , cj> being the angle which the ray reflected to the point makes with the plane. 45. A bright point is placed at the focus of a reflector which is in the form of a paraboloid of revolution ; prove that the illumination, from the reflected light, of any point of a plane perpendicular to the axis of the reflector varies in- versely as (t/ + 4(1')'^ — where y is the distance of the point from the axis, and 4a is the latus rectum of the generating parabola. 4f). In M. Foucault's experiment for determining the velocity of light, if the radius of the spherical mirror be 3^ yds., and the revolving plane mirror be at the principal CHAPTER III. 275 focus of the lens, the focal length of which is jr— yds., and the image of the platinum wire be moved through a distance of •009 of an inch when the mirror revolyes 880 times in 1", determine the velocity of light in miles per 1". Chapter III. Aberrations of small direct p&ncils. 1. A pencil of parallel rays is incident on a convex re- flecting surface — j/, r being the semi-aperture and radius of the mirror, — the diameter of the least circle of aberration 2. A small pencil is directly refracted at a plane surface, — the radius of the least circle of aberration = , ., . ^ . 8/a' m' 3. A pencil of parallel rays is refracted directly at a spherical surface, — the radius of the least circle of aberration 4. A mirror of given aperture and focal length, and of small curvature, has the form of a prolate spheroid; — shew that the aberration for parallel rays varies inversely as the major axis. 5. A segment of a paraboloid of revolution is cut off by a plane perpendicular to the axis so as to form a plano-convex lens: when a pencil of rays parallel to the axis is refracted through the lens, find the aberration in terms of the thick- ness, — which is so small that its square may be neglected. 6. If the law of refraction were assumed to be ^ = /ii^', shew that the approximate error in finding the point where a ray incident near and parallel to the axis of a spherical refracting surface cuts the axis after refraction, would be -— . ^ . •— , — using the ordinary notation. 6/x /A — 1 r 18—2 276 EXAMPLES AND PROBLEMS. 7. If X be the distance from the principal focus of a spherical reflector of the centre of the least circle of aber- ration, for a pencil of parallel rays incident on the reflector, and y be the radius of the circle, prove that 27r'/ = 64a>' approximately— where r is the radius of the reflector. 8. Prove that the aberration of a pencil of parallel rays incident directly on a spherical refracting surface, is less than it would be for a reflecting surface of the same shape, if the index of refraction be > 2. 9. The aberration of a ray which passes through a plate of glass of thickness t is (1 . ^ A i> 4> heing the \ V/A — sm (/)/ angle of incidence : hence shew that if /i" = 2 there is .no aberration to the third order of small quantities. 10. A ray proceeding from a point on the circumference of a circle is reflected n times at the circle, prove that its point of intersection with the consecutive ray similarly reflected is at a distance from the centre = ;r -, Jl + 4in(n+\)s,vc!^6'd 2w + 1 ^ ' being the angle which the ray before reflexion makes with the diameter. 11. A pencil of rays is refracted directly through a hemisphere, the distance of the origin from the plane surface which is that of first incidence being — , shew that the aberra- tion of a ray incident at a distance y from the axis r being the radius. CHAPTER IV. 277 Chapter IV. Focal lines of small oblique peiKils. 1. The distance between the focal lines of a small oblique pencil after refraction at a plane surface is = (/i — ) tan''^ . ii. 2. A small pencil of parallel rays enters a reflecting sphere, and after n internal reflexions emerges through the same aperture; find the angle of incidence and the position of the primary focal line at emergence. 3. An eye is placed under and, close to the surface of a clear and stagnant fluid, and a vertical straight rod of given length is placed at a given distance from the eye; determine the form of the image, and the altitude of its highest point above the surface of the fluid. 4. If an eye be placed in air close to the surface of a clear stagnant fluid, shew that the apparent form of a circular arc in the fluid, whose centre coincides with the place of the eye, and whose plane is perpendicular to the surface, is de- fined by the equation p sin^ , r=a /i'-cos'^' where, a = radius of circle, fi = refractive index, and the radius vector r drawn from the position of the eye makes the angle with the surface : — -the image of any point being supposed to coincide with the primary focus. 5. A small pencil of homogeneous light is incident obliquely on a plane refracting surface; shew that if the obliquity be small, the distance of the point of incidence from the centre of the circle of least confusion will be a har- monic mean between its, distances from the foci. 278 EXAMPLES AND PROBLEMS. 6. From the formula - + - = ? for finding the v^ u r secondary focus of a pencil of light reflected obliquely at a .1 r 1 1 1 2 /I 1V«« concave mirror, deduce the lormula - + - = -+ — V u r \r uj r for the aberration of a direct pencil. 7. A small pencil of rays suffers a series of reflexions within a spherical surface, prove that = --{n-r), where c is the length of each successive chord of the sphere described by the axis of the pencil, and m„ is the distance of the primary focus from the middle point of the chord after the n'" reflexion. 8. A straight rod lying at the bottom of a river is viewed by an eye at a given height above the surface of the water, — determine the form of the curve in which it is seen projected at the bottom of the water. 9. A small pencil of rays proceeding from the centre of an ellipse is reflected at the curve, determine the point of incidence that they may be in a state of parallelism after re- flexion; and shew that the problem is impossible unless the eccentricity is > -r^ . 10. If Q be a luminous point placed in the interior of a reflecting circle, shew that the primary focus of a small pencil of rays incident at a point R will be at an infinite distance if QR = 5-—; . chord bisected in Q : — shew that this is only possible if the distance of Q from the centre is greater than half the radius. 11. On the polished inner surface of a sphere parallels of latitude are drawn; shew that, to an eye at the pole, these lines after n reflexions will appear on the surface of a sphere whose radius is to that of the reflecting sphere as n-t-1 : 2n + l. CHAPTEE IV. 279 If meridians were drawn on the sphere, shew that after n reflexions, they would appear to an eye at the pole to be situated on the surface r sin (2n+l) e = p sin (2n +2)0: where r is the distance from the eye, p the radius of the sphere, and & the angle between r and axis. 12. Prove the following geometrical constructions for determining the position of the primary focal line — (i) after oblique reflexion at a spherical surface, (fig. Art. 66), Q the origin, 0, A the centres of the surface and of the face of the reflector, Aq^ the direction of the ray after reflection. Draw OF perpendicular to QA, FG perpendicular to AO, join QG and produce it to meet Aq^ in q^ which will be the primary focus. (ii) after refraction at a spherical surface, (fig. Art. 68), Q the origin, Aq^^ the refracted ray. Draw QF, FG perpendicular to AQ, AO respectively :— join GO meeting Aq^ in K, — draw KV, Vq^ perpendicular to AO, Aq^ respectively, — then q^ is the primary focus. (iii) Adapt the case (ii) to the case of oblique refraction at a plane surface. 13. A small pencil of rays suffers a series of reflexions within a polished spherical surface ; if the' rays are initially parallel, the primary focus after the w"" reflexion divides the chord of the surface which is the axis of the pencil in the ratio 2n—l: 2n + l. 14. A pencil, refracted obliquely at a plane surface, passes through a small square stop of area c' parallel to the plane at a distance a from the origin of light, shew that (i) the circle of 280 EXAMPLES AND PROBLEMS. least confusion is approximately rectangular; (ii) its distance from the plane is a harmonic mean between the distances of the foci ; (iii) its area is cV /cosV — cos^c^Y a" l^cos"^' + cos^c/)/ ■ 15. A rectangular hyperbola («" — y' = a^) is placed in a vertical position in water, the conjugate axis being in the surface : shew that the appearance of the hyperbola to an eye at the centre is the curve {/.^ {a? + f) - f] {^' {X' + f) - 22/f = a/.V, the image being formed by primary foci ; fx, = refractive index from air to water. Will the asymptotes be seen ? ■ Caustics. 16. If Q be the focus of incidence, QA the axis of the pencil, i^the principal focus of the mirror, and if the perpen- dicular from Q on AF and the line from Q perpendicular to QA cross AF sX equal distances from F, shew that the primary and secondary foci will be at equal distances from and on opposite sides of the mirror. 17. In a spherical mirror, if the origin of light be in the principal focus, and the pencil of small obliquity, ptove that (^1 ~ ■"a)" ~ '^^i''2 iiearly. 18. Rays parallel to the axis of y are incident on the reflecting curve y = e^, the catacaustic is the catenary 19. Rays are incident on the parabola y' = 4iax perpendi- cular to its axis,^^the equation to the catacaustic is- /9a — xV ax CHAPTER IV. 281 Q or, if the focus of the parabola is the origin, a = r cos'' ^ . , o Discuss the form, &c., of this curve. 20. Rays are incident on a circle from a point S within the circumference, and CO is drawn perpendicularly. from the centre G on any ray SQ reflected at Q to the point F of the caustic ; prove that the locus of P may be traced by means of the equation 1 J_ A ^/{OF'-G0')~SO'^Q0- 21. Rays emanating from a point S are reflected at a plane curve — if SY be the perpendicular from 8 on the tan- gent to the curve at any point P, and 8Y be produced to a point Q such that QY=SY, then will the caustic curve be the evolute of the locus of Q. The same problem may be stated thus : if with each point successively of the reflecting curve as centre, and its distance from the radiant point as radius, we describe a series of circles, — the envelope of all these circles will be a curve, the evolute of which will be the caustic. 22. Assuming the formula for the position of the pri- mary focal line in reflexion at a spherical surface, — prove that if Q be the radiant point within the circle, QP the ray cor- responding to an asymptote of the caustic, and P' the point in which PQ produced meets the circle, then QP' = 3 . QP : and hence deduce a geometrical construction for the asymp- totes of the caustic. 23. A radiant point being placed in front of a looking- glass, find the caustic curve formed by the rays emergent from the glass after reflexion at the quicksilver. 24. Parallel rays are incident upon a cylinder in a direc- tion perpendicular to its axis ; shew that the equation to a section of the caustic surface is {^' -l)y = [{iM^aY - x^]^ + {a' - {^L'xY}^ 282 EXAMPLES AND PROBLEMS. the centre of the corresponding section of the cylinderj whose radius is a, being the origin, and its diameter perpendicular to the incident rays, the axis of x, — n the refractive index. 25. Kays proceeding from a luminous point in the pole of the spiral r = ce"'^°', are reflected at the curve, the cata- caustic is a similar spiral. Further, the spiral will be its own catacaustic if ^ sec a = eC^"""")*"""^ where n = any positive integer. 26. Eays fall on the cycloid jf = a vers"' - + v'(2a« — «') parallel to the axis of x, the catacaustic is two cycloids of half the dimensions of the original given by the equation + y + Y = ^/((2'^ -cc){x- a)] + a cos^'y/ (^^) . 27. Rays are incident on a concave spherical mirror (rad. a) parallel to the axis : the caustic will be defined by the equation {4(^' + 2/^)-a'}= = 27ay, the centre of the sphere being the origin, and the axis of x parallel to the incident rays. 28. A pencil of rays emanates from the origin of the curve /J r = a cos" - , prove that the length of the caustic formed by n + 1 reflexion at a loop of the curve is 4 ^ a, — and the radius ^ n + 2 ' of curvature of caustic „n(K + i) . e „.,e = 2 -. jrrr,- a sm - cos - {n + 2)' n n at the point corresponding to 9. Q 29. The catacaustic of the curve r=^a sec' ^ , the pole o being the radiant point, is a semi-cubical parabola. CHAPTER V. 283 80. If a^ be the co-ordinates of a radiant point, the caustic curve formed by reflexion at the circle x' + y' = c' is [4 (a^ + ^) {x' + f) -c'[{x + ay + (2/ + ^Y]Y = 27c* {^x - ayf {o^ J^f-^- ^y, Cambridge and Dublin Math. Journal, li. pp. 128, 236. The student may refer to several papers on the Caustics by reflexion at a circle, by Mr Holditeh, in the Quarterly Journal of Mathematics. Chapter V. Successive reflexions and refractions — prisms — lenses. 1. Draw a line between two plane mirrors which may be the direction of a ray proceeding from a given luminous point after reflexion at either of the mirrors. 2. When a ray of light is reflected at any surface, the incident and reflected rays are equally inclined to any plane which passes through the normal at the point of incidence. 3. What is the greatest apparent zenith distance which a star can have as seen by an eye under water ? 4. A pencil passes directly from air into water, through a plate of glass, find its geometrical focus, assuming _3 _4 5. What must be the inclination of two mirrors in order that a ray incident parallel to one of them may, after two reflexions, be parallel to the other ? 6. A ray of light passes from one medium through three others bounded by parallel planes, the refractive indices taken in order being >J^, ^JQ, V2, V6 ;— if the angle of inci- dence at the first refraction be ^ , find the direction of the 4 ray in each of the plates. 284 EXAMPLES AN"D PROBLEMS. 7. If the air near the surface of the ground be less dense than at small altitudes above it, there will be observed inverted images of distant horizontal objects. Hence explain the Mirage. 8. If rays are incident upon a common looking-glass in an oblique direction from a candle, — one faint image is ob- served before the principal image, — and a row behind it diminishing rapidly in brightness. Explain this. 9. If a transparent plate lie between two media of un- equal densities each less than that of the plate, and if a ray, passing from the denser of the external media into the plate, fall at the critical angle on the third medium, — the angle of incidence at the first surface of the plate will be the critical angle from the first on the third medium. 10. A ray passing through a point Q is incident upon a refracting plate; q is the intersection of the emergent ray, produced backwards, with the normal to the plate through Q ; if the angle of incidence be = tan"' jj, and t the thickness of the plate, prove that 11. A rod, inclined at any angle to a plate of glass, is seen by an eye on the opposite side of the plate ; shew that the length of the image of the rod formed by geometrical foci is equal to the length of the rod. Is the image formed by the refraction at the first surface of the same magnitude as either ?. 12. A glass plate of sensible thickness is in contact with the surface of still water, find how much the image of a point in the water will be elevated. Find also the area of the upper surface of the glass which is effective in transmitting rays. CHAPTER V. 285 13. At the centre of one end of a glass cylinder a black spot is marked, shew that to an eye at the centre of the other end a number (n) of rings will be visible, — where n is the greatest integer in ^^^ .^ b being the length of the cylinder and a its radius. 14. _ If n equal and uniform prisms be placed on their ends with their edges outwards, symmetrically round a point on a table, find the angle of each prism in order that a ray refracted through each of them in a principal plane may describe a regular polygon. Shew that the distance of the point of incidence of such a ray on each prism from the edge of the prism bears to the distance of each edge from the common centre, the ratio of y /a'" — 2/A cos - + 1 : fi + 1. 15. If a be the minimum deviation for a prism of ^ i, and /3 that for a prism of ^ 2i, prove that . a sm- 2 sm 4 «+/? 16. If D be the minimum deviation for a prism whose refractive index is /ll and angle t, prove that eot ^ + cot -^ = 11 cosec -^ . 17. A luminous point is seen through a triangular glass prism, by means of pencils once internally reflected, trace the course of the axis of a pencil from the luminous point to the eye, — | being the index of refraction from air into glass. 18. A prism is held before the eye with the edge ver- tical, and one face perpendicular to the axis of the eye ; how much in azimuth of the horizon will the observer be pre- vented from seeing by the interposition of the prism ? 286 EXAMPLES AND PROBLEMS. 19. A small pencil diverging from a given poini is re- fracted through a prism in a principal plane and very, near the edge — if the prism be turned about its edge, form the equations necessary for determining the curves on which the foci lie, and shew that these curves intersect. 20. Rays are incident at one point of a prism in all directions in a principal plane, — shew that if the refracting angle be > sin~* - , rays incident from that side of the normal which is towards the edge of the prism will not pass through, — and examine what rays will pass through. 21. If there be a small speck upon the middle of one of the sides of an equilateral prism, and a person place his eye close to the opposite edge in the plane drawn through the speck perpendicular to the axis of the prism : shew that he will see two specks, and find the apparent angular distance between them. 22. A small pencil of rays is transmitted through a prism in a principal plane — find the focus of emergent rays in the primary plane, taking account of the thickness of the prism ; — and deduce from the result the expression for the place of the focus when a pencil is transmitted through a plate which is bounded by parallel plane surfaces. 23. Two equal prisms of the same refracting substance are placed in contact : shew that a ray of given species will be just incapable of transmission .through the compound prism thus formed when the mutual inclination of their edges is 2 sin"^ ( — -. — j , a. being the refracting angle of each prism. 24 Can objects be seen — (without internal reflexion) — through a prism whose refractive index is | and refracting angle a right angle ? 25. Two prisms are in contact with their edges perpen- dicular to one another; a ray perpendicular to the edge of CHAPTER V. 287 one prism being refracted through the combination emerges from the second perpendicular to its edge : determine the deviation, and shew that the problem is impossible if the angle of either prism exceed the critical angle. 26. The deviation of a ray refracted through a prism of small angle in a principal plane ^, <^' being the angles of incidence and refraction at the first surface. In what case will the deviation be = (/i — 1) i, nearly ? 27. If a ray of light QACS be refracted through a prism IKL in a principal plane, and if the vertical angle KIL — ix, QAK=d, ACL = (j), and the whole deviation of the ray = S, then will tan l - - j tan -^ = tan (d-\ — ^ — j tan —^ . 28. A small pencil of rays traverses a prism in a prin- cipal plane vnth mininaum deviation. The length of the path of the axis within the prism being c, and i the angle of the prism, — shew that the distance between the primary and secondary foci after emergence = (tan ^1 c. 29. A ray passes through a prism in a principal plane, the deviation being equal to the angle of incidence, and each of them equal to twice the angle of the prism, — shew that the latter angle = cos"^ ^ I — ^ — j . 30. When a ray is refracted through a prism in a prin- cipal plane, prove with the notation of Art. (91) that sin {D + i- cot a — 1 where a = critical angle of the substance of the prism. Further, the angular space within which rays may be inci- dent in order to pass through the prism is cos"' /^^^y ~"A 35. The ends of a glass cylinder are worked into convex spherical surfaces whose radii are each equal to the length of CHAPTER V. 289 the cylinder ; prove tliat the geometrical focus of a pencil after direct refraction through the ends of the cylinder is determined 2 1 2 1 by the equation = , where u and v are measured ■' ^ V u T from the face nearest the origin of light and r = length of the cylinder. 36. If a convex lens be moved between an object and a screen, find the condition that a real image may be formed on the screen. Shew that in this case there will be two positions for which a real image will be formed, and if m^ m^ be the magnification of the images in these positions m^ n\=l. 37. A glass lensf/A = ^j of given focal length is placed under water (/^ = o) ! ^^^ *^® geometrical focus of a pencil of parallel rays directly incident upon the lens. 38. The least distance between an object and its image formed by a plano-convex lens of glass is 1'2 inches, find the radius of the spherical surface ( /* = sj ' 39. A person views an object through a convex lens with both eyes ; trace the pencil by which each eye sees the object. 40. A convex lens, hjeld 12 inches from a wall, forms on the wall a distinct image of a candle ; when the lens is held 6 inches from the wall, it is found that, to produce a distinct image of the candle on the wall, the distance of the candle from the lens must be doubled ; — find the focal length of the lens. 41. From a prolate spheroid, formed of glass, portions are removed by planes passing through the two foci and per- pendicular to the axis ; if a luminous point be situated at one of the foci, and the eye placed close' to the other, describe the p. o. ^^ 290 EXAMPLES AND PROBLEMS. appearance which will be presented, so far as it is due to pencils which have suffered one internal reflexion. 42. Rays issuing from a luminous point in its axis are incident upon a thin lens. A portion of those that enter the lens is allowed to proceed at once through the second sur- face ; a second portion however does not escape until it has been twice internally reflected; a third portion four times reflected, and so on. Shew that a row of images will be formed whose distances from the lens are in harmonic pro- 43. In the annular lens of a Lighthouse, — ^investigate the relation between its focal length and the radius of curvature of a section of one of its concentric rings by a plane through its axis. 44. If a ray of light be incident upon a refracting sphere and the directions of the incident and emergent rays produced meet the surface in the same point ; find the angle of incidence , 3 + V5 when /i = — 2 — . 45. A sphere composed of two hemispheres of different refractive powers is placed in the path of a pencil of light in such a manner that the axis of the pencil is perpendicular to the plane of junction and passes through the centre ; deter- mine the geometrical focus of the refracted pencil. 46. Find the geometrical focus of a pencil of rays re- fracted through a hollow glass sphere, whose, external and internal radii are r, r' respectively. 47. Shew what kind of lens will be required to enable a person to see distinctly under' water. If an eye be under water, shew how to determine the course of the ray by which any object above the water is seen. 48. A. plano-convex and a plano-concave lens of equal CHAPTER V. 291 power are fitted together so as to form a plate ; — if the lenses be slightly separated, remaining co-axial, examine the change they will produce in the divergency of a small pencil of diver- gent rays incident directly (i) on the negative lens, (ii) on the positive lens. 49. Two plano-convex lenses of the same size and form are placed so as to make a double-convex lens : find the focal length of this lens, the refractive indices of the two substances being /* and jj!. 50. A convex lens is moved between a candle and a ver- tical screen, the distance of which is greater than four times the focal length of the lens : prove that two real images of the candle will be formed on the screen : — and if the ratio of their magnitudes is m, find the focal length of the lens. 51. The focal length of a convex lens in air is 15 inches. When its lower surface is immersed in a certain fluid, a ver- tical pencil of parallel rays is brought to a focus at a depth of 24 inches : when the other surface is immersed, the depth of the focus is 40 inches ; and when the lens is wholly im- mersed the focal length is 60 inches. Find the refractive indices of the lens and fluid, and the radii of the surfaces of the lens. 52. A pencil of rays is directly incident on a double concave lens the second surface of which is silvered ; find the geometrical focus of the pencil after emergence, and the form of the spherical surface which, placed at the same distance as the lens, would reflect rays to the same focus. 53. If the radius of the anterior surface of a concave glass speculum of inconsiderable thickness {c) = a, aind if 13c the radius of the second surface = a -|- -x- , the image of a dis- tant object formed by reflexion at the first surface will coin- cide with the image formed by refraction at the first surface, then by reflexion at the second surface, and by refraction again at the first. 19—2 292 EXAMPLES AND PEOBLEMS. , 54. A pencil of rays is directly refracted through a series of lenses separated by jEimte intervals a^, a^, -..a,,.!, the axes being coincident. Shew that if ^ , j- j- be the focal lengths of the lenses, the geometrical focus will be given by the equation 111 11 v..= K + a,,^^ + K-i+"'\ + u 55. If r, s be the radii of the surfaces of a lens of thick- ness t, and if f = il + -\{s — r), shew that a pencil will be refracted through the lens without aberratioDy if the origin be at a distance (1 + /u.) r from the lirst surface, and that the di- vergency of the pencil will be unaltered by the refraction. 56. If jj, be the refractive index of a sphere, r = its radius, fir = distance of the origin of light from the centre, prove that the extreme incident rays on emergence intersect a screen touching the sphere at the point opposite to the origin of light in a circle, whose radius is '57. If /be the focal length of a lens when the thickness is neglected, shew that when the thickness is sensible, but small, the principal focus will be very approximately at a dis- tance / from the first surface of the lens if the ratio of the radii of the first aiid second surfaces be J [jl — 1 : V/*- 58. A concave mirror of radius r has its centre in the centre of a convex lens, and the axes of the two coincide. If / be the focal length of the lens and if rays proceeding from a point at a distance u from the lens emerge after a second refraction as a pencil of parallel rays, prove that 1 2_2 u r~f CHAPTER V. 293 59. A person finds that he sees distinctly in a vertical plane at a distance of 5 in. and in a horizontal plane at a dis- tance of 6 in. ; he therefore uses an eye-glass the front surface of which is spherical and the posterior surface cylindrical, the axis of the cylinder being vertical ; determine the curvatures of these surfaces in order that he may see distinctly an object placed at a distance of 8 in. from his eye, the eye-glass being supposed to be close to the eye. fi = l-o. 60. Two convex lenses, of focal lengths a and b, are placed at a distance c ; if P and Q be conjugate foci, F, G the respective positions of P and Q when Q and P are respectively at an infinite distance, prove that PF.GQ=( ""i y. 61. A double convex lens having radii r, s and a small finite thickness t is placed with its posterior surface in contact ■ssith a fixed ideal plane ; shew that when the lens is reversed the position of the focus for parallel rays will be altered by the quantity r — s t r + s' fjb 62. A lens is placed in the centre of a concave mirror, the axes being coincident, a pencil is incident directly on the lens, and after refraction is reflected at the mirror, and again refracted through the lens, prove that the last geometrical focus is the same as if the pencil had been once reflected at a mirror coincident with the image of the concave mirror formed by the lens. 63. If P and Q be two small equal objects in the axis of a lens and on opposite sides of it, the angle subtended at P by the image of Q is equal to the angle subtended at Q by the image of P. 64. In Art. 91. If , i/r be the angles of incidence and emergence, D the deviation, prove that cos ((^ - -i/r) (cos i — cos (D + i)} = cos i. cos (D + %)-! -^/i^sin'^■. 294 EXAMPLES AND PROBLEMS. 65. Two convex lenses have the same axis, shew that the image of a point on the. axis at a distance d from the nearest lens will be unaltered in position by reversing the combina- tion, if the distance between the lenses be {A-d){A-d)- 66. Two parallel rays are incident at an angle ^ on one face of an isosceles prism and emerge at right angles, one of them having been reflected at the base ; if i be the angle of the prism and /i the refractive index, prove that sin 2<^ = (1 — fi? sin^ i) sec i. 67. In different prisms formed of the same substance, prove that as the refracting angle increases the minimum de- viation also increases. 68. If the minimum deviation for rays incident on a prism be a, the refractive index cannot be less than sec - , and the ^ of the prism cannot be > tt — a. 69. A pencil of light passes through two prisms whose edges are parallel in a principal plane of each ; if the pencil passes through each with minimum deviation, the angles of incidence on the first and emergence from the second prism being <^, <^^ respectively, the two rays for which the refractive indices are fi, (1 + Sfi, and /u,^, fji,^ + Bfj,^ will emerge parallel if — tan d) = — - tan A, . ix ^ Ml 70. The minimum deviation of a ray refracted through a prism of 60° is 90° ; shew that the refractive index of the prism for that ray is 1'93 nearly. 71. A rod situated in a plane perpendicular to the edge of a prism {i) is viewed through the prism ; find the incli- CHAPTEE V. 295 nation of its image to either face of the prism : and shew that the image will be parallel to the rod, if cot ^a - 1 j cot (« + !)= A*, where a is the z which the rod makes with the plane bisecting the ^ between the faces of the prism, /* the refracting index of the prism. 72. BOA is a diameter of a sphere and a spherical portion described from centre A, with radius AO, is cut out; prove that if u be the distance from in the line BO produced of an origin of light, and v the distance from of the geometrical focus after refraction through the solid BO, /.^ i_(fi-iy 73. The minimum deviations at the three angles of a triangular prism of a ray of index fj, are S^, B^, B^ ; prove that fJ' - iJ? (cos -^ + COS -^ + cos ^ j + A' (cos ' ' + cos ' „ + cos ° g ' I - COS - — ^ = . 74. Verify the following construction for finding the focus conjugate to a point P on the axis of a lens 0, every- thing being supposed positive: produce OF to Q, making PQ equal to the focal length of the lens ; on as diameter describe a circle, and let it be cut in E by the circle described with Q as centre and QP as radius ; draw BN perpendicular to OQ: measure OS in direction of and equal to Pif : 8 is the required focus. 75. Two prisms with refracting angles each 45° are placed with one face of each in contact, and their edges at right angles. If the refractive index =^2, the minimum deviation of a ray which passes through the pair of prisms will be 30°. 76. Two prisms whose refracting angles are right angles and refracting indices /*, /m' are placed so that one fe,ce of 296 EXAMPLES AND PROBLEMS. each is in contact : their edges are parallel and their refract- ing angles opposed. Prove that the minimum deviation of the compound prism is sin~^(/i'^ — fi'). 77. If a ray pass through a lens without deviation and if its directions before incidence and after emergence cut the axis of the lens in two points q, q', the limiting value of qq' is maximum or minimum according as the thickness is equal to . (s — r) V/A V/I+ 1 where r, s are the radii of the surfaces of the lens, /i the refractive index and s> r. 78. If i = angle of a prism, /j, = sec a, D = minimum deviation, then . i . D . D + 2i tan a sm ^ = sm — sm — ^ — . Shew that if i = 2 sin"'' - , where u. = refractive index for mean rays, then an eye placed against the prism will see a faiat arch of red. 79. The section of a prism made by a principal plane is a triangle ABC. A ray falls on AB making an angle c^with the normal (measured towards the edge) and, after internal reflexion at BG, emerges from AC. Shew that with the usual notation D = '-ylr' = B-G. 80. If m be the linear magnifying power of a thin lens, for an object at a distance u from the first surface, shew that when the thickness t is taken into account, the magnifying power is increased by m'' . — n +'- vA ~ , — in which f is neglected : r, s being the radii of the two surfaces of the lens, and u the distance of the object from the first surface. See Arts. 99, 200. 81 . Parallel rays fall on a thick lens and converge to B, CHAPTER V. 297 while if A be the origin of light, the emergent rays are parallel. Shew that if U, V be any pair of conjugate foci, then AU.BV-- firs ,/*(/. -l)(r + s)+(/.-l)^i ■where r, s are the radii of the bounding surfac es, and t the thickness of the lens. ■82. A ray passes through a right-angled prism with minimum deviation, meeting the face of the prism at a distance y from the edge. Shew that the foci of the emergent wv'2 pencil will be separated by a distance - — - (^^ — 1). A' 83. Prove that a concave lens can be constructed such that the direction of every ray of a pencil proceeding from a certain point shall after refraction at the first surface pass through the centre of the lens, — that in this case there will be no aberration at the second refraction, — and that the only effect of the lens will be to throw back the origin of light a distance (/j. — l)t,t being the thickness of the lens. 84. ABGD is a double convex lens (axis BD) whose thickness t = ij. (sum of radii of surfaces). P is a dark spot within the lens at the point of intersection oi AC and BB ; Q, B the images of P seen from two points in the axis on different sides of the lens. Then, if/ be the focal length of a thin lens which has the same radii and refractive index as the lens ^5 CD, ^^-;I^2 + (/.-l)(;.-2)^- 85. The distances of the conjugate foci from the first and second focal centres of a lens are connected by the equa- tion 1 1 /I 1 .fi-l t\ 298 EXAMPLES AND PROBLEMS. Chapter VI. Refraction through media of varying density. 1. The index of refraction at any point of a medium x_ bounded by a plane is e", where x is the distance of the point from the plane ; find the equation to the path of a ray inci- dent at an angle a. on the bounding plane, and shew that it will have an asymptote perpendicular to the plane at a distance ex from the point of incidence. 2. The index of refraction (/i) in a medium varies from point to point, being a function of the distances x and y from two planes at right angles to each other ; a ray traverses the medium in a plane perpendicular to these two planes ; if log {fi) =f (xy), prove that the curvature of the path of the ray varies as 3. A ray of light which passes through two media bounded by parallel plane surfaces, will emerge parallel to its first direction, if the deviation in passing out of one me- dium into another under a given angle of incidence be sup- posed proportional to the difference of the densities of the media. 4. The index of refraction at any point of a medium cc (distance)"* from a given plane, prove that the path of any ray in the medium is a cycloid. 5. The refractive index at any point of a medium = ( - where r is the distance of the point from a given point. If when r = a the direction of a ray of light be inclined at 3 • . an -^ J TT to the radius vector, shew that the path of the ray is a cardioid. CHAPTER VI. 29S 6. If the refractive index of a medium at any point be proportional to its distance from a fixed point, the path of the ray will be a rectangular hyperbola. If it be proportional to its distance from a given plane the path of the ray will be the curve 2a; c I a -I — = -e" + -e ' , a a c a and c being constants. 7. The refractive index at any point of a plate at a distance x from its first surface is given hy fj, = f (x) ; ii v be the distance from the first surface corresponding to a distance X in the plate, prove that the position of the focus is given by V and a; being estimated positive in the same direction. Ex. li fi = 1 + ax, find the change in the position of the focus in passing through a plate of thickness c. 8. A prism is of such material that the density of the plane section through the edge at an ^ ^ to one of the faces is /lie" *"" " ; — ^if a ray be incident perpendicularly on this face at a distance a from the edge, shew that the distance r_ of the ray when incident at an -^ ^ upon a plane section is given by the equation a __ *_!H^rcos_(^_+a)\™^""' r ~ I cos a J 9. A ray of light passes through a medium of which the refractive index at any point is inversely proportional to the distance of that point from a certain plane. Prove that the path of the ray is a circular arc of which the centre is in the above-mentioned plane. 10. In Art. 122, shew that the projection of the radius of curvature of the path of the ray on the normal to the surface of equal density at any point of the path is equal to .dx) \dy. 300 EXAMPLES AND PROBLEMS. 11. When a ray traverses a medium of variable density, shew that when it is normal to a stratum of equal density, there will in general be a poiiit of inflexion in its path ; point out the exceptions, and discuss the nature of the path in these exceptional cases. 12. In a medium bounded by parallel planes the index of refraction is the same at equal distances from both planes, a and is = fie ", where d is the distance from the plane bisect- ing the whole thickness (2a) of the medium. If a pencil of light pass directly through it, find the distance between the foci of the incident and emergent light. 13. ABC, ABD are the faces of a triangular prism of small refracting angle (a), and the refractive index out of air at any point P in the prism = /* (1 + m&), where m is constant and 6 is the angle between the planes ABO and ABP. Shew that if JD be the deviation of a ray which is incident per- pendicularly on ABC, and is refracted through the prism, i) = (^,l)« + ^\ 14. A medium is bounded by the planes of x and y, the refractive index at any point being given by /* = e"" ; two rays are incident on it parallel to the axes respectively, and at distances c from the origin ; shew that if they intersect, it will be at an angle whose circular measure is -^ 5 . 1.5. A ray of light passes through a medium bounded by parallel planes, the density of which varies in such a manner that the index of refraction at any point = 1 4- kx, where so is the. distance of the point from the plane on which the ray is first incident. The angle of incidence being a and the point of incidence the origin, shew that the path of the ray is de- fined by the equation cos= I . e''" " -1- sin" I . e^"' " = H- ifca;. CHAPTER VI. 301 16. A vessel of depth a, the top and bottom of which are horizontal planes, is filled with a transparent fluid, the refrac- tive index of which at a depth z below the surface is 1 + - . a Two small holes being made in the top, a ray of light enters at one hole, is reflected at the bottom and emerges at the second hole ; shew that the distance between the holes must not be > 2a logs (2 + V^)- 17. Shew that ii fj? — 1 = kp for any state of density of the Earth's atmosphere, the following differential equation of the trajectory of a ray of light coming from a heavenly body results independently of any Theory of Light, viz. : Tn a sin z . dr da = r\/{r^ — a' sin'^ z + hr^ p) ' z being the apparent zenith distance of the body at the Earth's surface, a the Earth's radius, and r, 6 polar co-ordi- nates referred to the Earth's centre and to the radius drawn to the place of observation. 18. ABCD are four points, abed are their respective images formed in any manner by refraction through any number of surfaces ; AB, CD, ah, cd are bisected at right angles by the same straight line, shew that AB.CD _ ah . cd '^'AC + AD~^^ac + ad' where ^j, fi^ are the indices of refraction of the media in which ABCD and abed lie respectively. 19. A ray is propagated through a medium of variable density in a plane with respect to which the medium is symmetrical, — if the refractive index at any point whose polar co-ordinates are r, 6 be Xre"*, and •^ be the angle which a ray passing through the point makes with the radius vector at the point, prove that r = ae " , a being a constant. 20. Supposing the atmosphere to consist of spherical 302 EXAMPLES AND PROBLEMS. strata of uniform density, prove that if the observed zenith 'distance of a star at a given station be z, and the tangent to the path of the ray which reaches the eye in passing through the stratum whose refractive index is /^ and radius r, makes an angle % with the vertical at the given station, . where /i„, r„ are the values of fi, r at the given station. 21. A refracting medium consists of spherical surfaces which have a common tangent plane : the internal and ex- ternal radii being a, h respectively. The refractive index of any surface varies inversely as the diameter of that surface. Shew that if a small pencil emanate from the centre of the interior surface and be refracted directly through the shell, the distance of the geometrical focus of the emergent pencil from the exterior surface is 6 [ 1 + tan lege - ] . 22. If the path of a ray cut at a constant angle a the surfaces of equal density in a variable medium, prove that /A = /x.(| e* **'"', where ^ is the inclination of the path to a fixed line. 23. A ray traverses in one plane a medium in which the density is a function of the distance from a fixed point. Its path is an equiangular spiral about this point a^pole. Prove that the path of any other ray which traverses the medium also in a plane is an equiangular spiral. In general if the path of the first ray be known,, prove that that of the second belongs to a certain family. 24. If the surfaces of equal density be symmetrical with respect to the plane of x, y — and if in going from one to the next in the plane oi{x,y) — (in which plane the ray is propa- gated) — the index /* may be put in the form /=(a; + a)/(?/'e'), CHAPTER VI. 303 shew that the parabola y'" = 4aa; is a possible path for the ray to trayel in. 25. Prove that in a transparent paedium of variable index of refraction fi, the differential equations of a ray whose length is s are d I dx\ diM d f dy\ du, d / dz\ da ds V dsj dx' ds \ dsj dy' ds \ dsj dz If jtt = /i ' 1 — 1-2 j , shew that the equation of a ray in the plane of xy is y. = J}? — a' sin ( - + a j , where a and a are arbitrary constants. If X and x' are the values of x at any pair of intersections of consecutive rays, shew that a^ jtan (- + ol\- tan T- + a j I + (¥ - a') (x - x') = 0, and that if x' corresponds to the wth intersection from x, a; — «' is > (n—1) Tra but < nira. 26. Prove that in a transparent medium whose refrac- tive index /t is a function of x the distance from a plane, the curvature of a ray at a point of its course where it is parallel to that plane is = — t- . H ax X 27. If the path of a ray of light he y = ae° shew that the index of refraction at any point is determined from the equation 28. A point of light is placed at the origin of co-ordinates in a medium where the refractive index is given by /i = e-"*. Shew that an eye placed at the point {x,y) will see the origin of light by means of a small pencil of light — one of the focal lines of which lies on the axis of x and the other at a distance 304 EXAMPLES AND PKOBLEMS. V from the eye, where wVe"* cos'' Ky = 2 (cos koc — cos Ky) — "whilst the axis of the pencil makes an / i|r with the axis of x, where cot i/r sin Ky = cos Ky — e-'"'. 29. A ray is propagated through a medium of variable density in a plane with respect to which the medium is sym- metrical, prove that the differential equation of the path is dp _ fdf£\ _ s/r^ -p^ fdiji\ Find the general form of /x if this path be the equiangular spiral p = r sm. a. ; — and prove that along the spiral ij, the angle of incidence of the first incident ray, and 9 the angle between the plane of inci- dence of the first incident ray and a plane perpendicular to the intersection of the reflecting surfaces. 35. There are three plane reflectors, two of which are at right angles to each other, and a ray of light is incident upon the third and reflected successively by each of them ; — it is required to shew that the angle between the first incident and last reflected rays is equal to twice the angle of incidence upon the first surface. 36. An oblique cone of rays falls upon a narrow reflect- ing annulus cut from the surface of a cone by planes perpen- dicular to the axis ; find the general form of the reflected image thrown on screens at different distances perpendicular to the axis of the cone. p. o 20 306 EXAMPLES AND PROBLEMS. 37. A ray of light is incident parallel to the axis of a reflecting elliptic paraboloid whose equation is l^ + - = 2- c at a point (a, /3, 7) ; shew that the equations of the reflected ray are x — a y — ^ z — 7 ^v+f^y.! ^ 27 (!) Each reflected ray will pass through each of two parabolas lying in the principal planes of the paraboloid. 38. A ray is incident at a point x, y, s of a surface m = 0, and there is both a reflected (i) and refracted (ii) ray, — if I, m, n be the direction-cosines of the normal to the surface u = at X, y, z, — a;S7 those of the incident ray, % the angle of incidence and reflexion, i' that of refraction — the equations to the reflected and refracted rays are severally X-x _ Y-y Z-z ... ~ (1). 21 cos, i — a. 2mcos^— /3 2w cos 4 — 7 X-x _ Y-y _ Z-z ?sin('i— 'i')+asini' msin(i— i')+/Ssini' nsin(i— i')+7sini' ^ ^' 39. A ray of light is incident from the centre of an ellip- soid, the inner surface of which is polished, and the equatior of which is prove that the equations to the reflected ray will be X — x F— y Z — z CHAPTER VI. 307 where x, y, z are the co-ordinates of the point of incidence, and Further,— Vrove that all rays which after reflexion pass through the line oo = y=z, were before reflexion in the surface of the cone defined by the equation 40. If a ray be reflected at two plane surfaces, its direc- tion before incidence being parallel to the plane bisecting the angle between the two planes, and making an angle 6 with their line of intersection, prove that sin -^ = smd sin a, a being the angle between the planes and D the deviation. 41. A pencil of rays is incident parallel to the axis a; of a refracting ellipsoid whose greatest and least axes are a and c respectively ; find the nature and limits of the two curvilinear boundaries of the portion of the plane x, y through which all the rays will pass ; — and shew that if e, e, e" be the eccen- tricities of the principal sections in the planes xy, xz, yz re- spectively, and IJ' = - , the boundaries will be two ellipses whose semi-axes are ae"^ he"^, and -7- , he" ; but if /i = - , all Q the rays will pass through a portion of the arc of an ellipse, whose semi-axes are ae, he" included between the vertex and CCL& a double ordinate at a distance -1- from its centre. 42. The surface of a piece of water is covered, except one narrow slit in the form of a straight line ; a luminous point is placed in a given position above the surface ; — if this be taken as origin of co-ordinates and the vertical line as axis of z ; and x= a, z= —g are the equations to the slit, — shew that the equation to the sheet of light in the water is o^[{x' + f){x-ay + x\z + cy}= fi\x-af{a\x' + y') + c'x'']. 20—2 308 EXAMPLES AND PROBLEMS. 43. Rays diverging from a point a, h, c are reflected at the curve z = 0,f{x, y) = 0, the normals to the polished part of which all lie in the plane s = ; shew that the equations to the ray reflected at the point [x, y, 0) are X-x Y-y Z 2p(h-y)-{f-l){a-x) 2p{a~x) + {f-mi-y) if+l)c where p = -^, — and the sheet or surface formed by the re- flected rays will be found by eliminating x, y between these equations and f (xy) = 0. Ex. If / (so, y) = y' — 4tax and & = 0, the ray surface of reflexion is r^ {c-Z) = 4a {aZ+ cX). ^ * * yp -(& The following esamples of ray -surfaces ofrejlexionfoiiaedi as in problem (43) are taken with modifications from an interesting work on Say Surfaces of Re- flexion, by Prof. Childe, Cape Town, 1857. I am indebted to the same work for the problems 5 1 — 54 of this Chapter. Lateral and vertical ray-surfaces of reflexion are the names given to surfaces formed as in problems 43 and 5 1 respectively. 44. In problem (43) (i) The equation to the plane passing through the inci- dent ray and the normal to the reflecting curve is X- a; -f (Z- 2/) j3 - {a - a; -f (6 - 2/) p} - = 0. c (ii) The equation to the plane passing through the re- flected ray and the tangent to the reflecting curve is (X-x)p-{Y-^y)^\{a-x)'p-Q>-y)\?- = ^. c 45. In (43) if the incident rays are all parallel, their direction being defined by the direction-cosines I, m, n, the equations of the reflected ray become X-x ^ Y-y _ Z 2mp-l{f-l)~2lp + m (/ - 1) ~ (p"" -|- 1) ?i " CHAPTER VI. 309 In this example, if the ratio I : m remain constant whilst n varies the re- sulting ray-surfaces of reflexion wiU have for their envelope the cylinder whose axis IS paraUel to the axis of z, and whose trace on the plane of xy is the ordi- nary caustic formed by the reflexion of rays all incident in the direction defined by (J, m, o). 46. If/ {xy) = 3^ - a; tan a = 0, the ray-surface of reflexion (either lateral or vertical) is ^ Z sin « — y cos a c a sin a — b cos a 47. If/ {xy) =:x''+f-p'' = a. circle,— the ray-surface of reflexion is {c^ {w' + f) - {c^ -f W) ^Y =^p'{z + cf {{ex - azf + {cy - hzf}. 48. In the preceding example if the incident rays are all parallel, and their directions defined by {I, m, n), the ray- surface of reflexion is (i) y {x^ + f) -{- (Z^ -f m') z'Y = p" {{nx - hf + {ny - mz)'], or writing m = — which does not sacrifice any generality of form — this becomes {n' {a? + f) - ivy = p' { (ncc - Izf + nY} (ii) If the direction of this system of parallel rays be varied, subject to the conditions m = 0, V + 1-? = 1, only — the envelope of the ray-surfaces given by (i) is ip' - a?-f) {8 {x' -f f) + pY = 27 pV. This surface is an epicycloidal cylinder whose axis is parallel to the axis of z, and whose trace on xy is the epicycloid generated by a circle of radius - rolling on a circle of radius - ; the cusps of the epicycloid being on the axis .4 2 of X. It is in fact the epicycloid of Art. 71. 49. Find the envelope of the lateral ray-surfaces of reflexion, the reflecting curve being (as in 47) — the circle al' + f-p'^Q, and the locus of the radiant point {a, h, c) a circular ring defined by c = constant, a^ + b^ = k^. 310 EXAMPLES AND PROBLEMS. Result. The double cone given by tlie equation c /v/(*^ + y^) +KZ=±p (a + c). 50. If the reflecting curve (as in 43) be a cycloid z = 0, y='^ vers"^ -»J(2a.x — x^) ; and the incident rays are all parallel to the plane of xz, their directions being defined by (I, O, n) — the lateral ray-surface of reflexion is = no. vers "' (!yJS +^[(^" + '^^^ {^^ - '^ ^"^ - 2='^5]- This surface is discussed at length in the work above referred to. -'I'- -jf -jl'. jj- jfc, 51. Rays diverging from a point {a, h, c) are reflected at a polished curve whose equations are z= 0,f (x, y) = 0, the normals to which are all parallel to the axis of z : ;shew that the equation to the surface formed by the reflected rays — the ray-surface of reflexion — is , f cX+ a Z cY+ hZ \ _ •^V c + Z' c + Z J- 0. 52. In the preceding question, if the incident rays are all parallel — their direction being defined by the direction- cosines I, m, n— the ray-surface of reflexion will be . (nX + IZ nY+mZ \ _ 53. If the polished curve in problem (51) be a circle, viz. f{xy)=^Q = x^ + f-p\ the ray-surface of reflexion is {ex + azY + {cy + hzf = p^{2 + c)\ 54. Find the envelope of the ray-surfaces of reflexion CHAPTER VII. 311 formed by the curve z = 0, x" + f - p' = 0, as in problems 51 and 53, when the locus of the radiant point a, b, c is (i) A circular ring defined by the equations c = constant, a' + b^ = k^ = a constant. , Result. The double cone defined by the equation c ^/(oe' + f)+xs = ±p (z + c). (ii) A circular ring defined by c = const., and (a-a.y + {h- /3)' = K^ a, ^, k constants. Result. Two oblique cones defined by {ex + azY + {cy + ^zf = {(p + «) ^ + pc]\ (iii) A luminous spherical surface concentric with the reflecting circle and defined by the equation a' + 6" + c" = k". Result. Two right cones defined by the equation y{x^ + f)±pY = ^^.z\ (iv) A luminous paraboloid of revolution, viz. a^ + V = imc. Result. The envelope is a cone, p'ia>' + f) = {mz-py. Chapter VII. Spherical aberration of lenses ; — Excentrical pencils. 1. A pencil of parallel rays is directly refracted through an equiconvex lens of 20 inches focal length and 4 inches aperture, {fj. = 1'5) — the aberration of the extreme rays = ^ inch. 2. A man stands opposite to a convex mirror fixed to one of the walls of a room ; what appearance as to size and position will his image, seen in the mirror, present to him ? 312 EXAMPLES AND PROBLEMS. 3. If the thickness of a concavo-convex lens be equal to (yti + 1) times the distance between the centres of its spherical surfaces; shew that a point may be found in its axis, from which if rays diverge and fall upon the concave surface, they will diverge accurately from a point after emergence. 4. The form of a lens lfi = -^] being such that the aber- ration for parallel incident rays being a minimum, — express the aberration in terms of the focal length of the lens and the breadth of the incident pencil. 5. A pencil is refracted directly through an equiconvex lens, of focal length /^ the origin and geometrical focus being equidistant from the centre, — the magnitude of the aberration -^' (a -A - 2f-[l^-^j- 6. When a small pencil of parallel rays is centrically and obliquely refracted through a thin lens, the magnitude of the circle of least confusion is unaffected by the form and sub- stance of the lens, and depends only on the breadth of the incident pencil, and the inclination of its axis to that of the lens. 7. A direct pencil of rays converges to a point at the extreme distance from a lens at which an absence of aber- ration in a converging pencil is possible : determine the relation between the radii of the lens that there may be aberration in the refracted pencil . (/* = f ) • Result. Witli the notation of Art. 132, we must have /A— 1 /A — I which gives _ 2 {iJ?-i ) ai= II sj (fi,- if a? - iJ, (ii.+ 2) _ X — J I'. + i the limiting value of a is a = ± yJtS!^ — V. _ j. J:^ _ j. ^.^g . no CHAPTER VII. 313 the pencil being convergent at incidence, u is negative, and therefore the nega- tive value of a must be taken, and we obtain «^- I cc=2 a= - a-27, /i + 2 * " and - = = ^—i- = — nearly. s x+i 227 17 •' 8. A pencil of parallel rays falls directly upon an equi- eonvex lens: shew that if the lens be split in two by a plane perpendicular to its axis and the two parts be separated by an interval equal to | the focal length of either, the convergency of the emergent pencil is double that in the former case when the lens is entire. 9. Three convex lenses are placed on the same axis ; shew that the focal length [F) of the equivalent single lens is given by the equation /./.+/./s+/aX-«(/.+/3)- &(/.+/.) + «& =-^=, where f^f^f^ are the focal lengths of the three lenses, and a, b the distances between the first and second, and the second and third respectively. 10. Two lenses of the same material (/j, = Vo) the radii of whose faces taken in order are r^s^r^s^ are placed on the same axis at a distance a. The conditions that the system may be achromatic for a direct pencil of parallel rays and at the same time that the spherical aberration at each lens may be a minimum are given by the equations s^ =. _ 6r^, (70a - 24rJ r, = (70a + 144rj) 5, = (7a + 12rJ^ 11. A pencil of parallel rays whose axis before refraction cuts the common axis of two lenses (focal lengths f^,/^ at a distance a) at a distance d from the centre of the first lens — the focal length F of the equivalent simple lens is given by the equation F-f^A fSA dj- 314 EXAMPLES AND PROBLEMS. 12. A plano-convex lens is in contact with a concavo- plane lens on the same axis, the refractive indices being ytt, /a', and r the radius of the common spherical surface. A ray "which cuts the axis at a small ^ e, and at a distance d from the compound lens, is refracted through it. Prove that the deviation of the ray is (fi — fi) - e. 13. Two lenses, the thicknesses of which are t, t', radii r, s, r', s', and indices fi, fi, having a common axis, and touching each other in the axis, are traversed by a ray so that its directions before incidence and after emergence are parallel. Shew that if co be the distance of the ultimate intersection of the axis and the path of the ray between the two lenses, from the point of contact of the lenses, IM-1 t fl'-l t'\l fji, r fi s ) (o \r s /M rsl ^ '\rs /* rsj Chapter VIII. Images. 1. A lighted candle is placed before a concave spherical mirror, the candle being perpendicular to the axis of the mirror and in the same plane with it; find how the image of the flame moves as the candle bums. 2. A short object is placed perpendicularly on the axis of a concave spherical refractor, and at a distance from it f equal to - , / being the focal length ; prove that the linear magnitude of the virtual image is half that of the object. 3. A person regards the image of himself in a large concave mirror, trace the changes of the image in magnitude, CHAPTER VIII. 315 position, and distinctness, as he walks from a considerable distance along the axis up to the surface. 4. The distance of a luminous point from a concave spherical reflector is less than half its radius; determine (i) the points of incidence at which consecutive reflected rays will be parallel, (ii) the position of the luminous point along the axis which renders the distance of these points from the axis a maximmn or minimum. 5. A circular luminous ring, radius a, is placed on the ceiling of a room ; — in the floor is a hole in the form of an equilateral triangle of side h ; shew that the area of illumi- nated patch on the floor of the room beneath = ira" + V36' + Gab, the rooms being of equal height. 6. P8p is a focal chord of a parabola, S a luminous point; shew that the illumination at P : illumination at p :: (Spf' : (SPf. 7. Two candles (P, Q) of equal intensity situated at the same distance from a wall, throw upon it shadows (m, n) of a small object ; prove that the illumination in these two sha- dows are as Prf : Qm^. 8. Two candles are placed on a horizontal table, the one lighted, the other not,— the latter being the taller and fixed : the other is moved about on the table so that the shadow of the first on the ceiling is of constant length. Shew that the locus of the foot of the candle is a conchoid. 9. The circumference of a circle is luminous except a very small arc which serves as a reflector ; find the image of the luminous part when the image is defined to be the locus (i) of the primary focal lines, (ii) of the secondary focal Imes, (iii) of the circles of least confusion. 10. The curvature at the vertex of the image of a straight 316 EXAMPLES AND PROBLEMS. line formed by refraction through a sphere = 2 - — - . curva- turiB of the sphere. 11. A parabola whose latus rectum = 21 is placed before a concave spherical refracting surface, the focus and axis of the parabola being coincident with the centre and axis of the surface. The parabola is convex to the surface, and an image of it is formed by direct pencils. Shew that if./ be the distance of the principal focus from the surface (/ being > l) the image will be a confocal hyperbola whose eccentri- . . f city is = fzri' 12. In the flat roof, of a building there is a square hole exposed to the light of a uniform sky. The floor of the up- permost room has also a square hole in it under the former but its angles opposite to the sides of the former : — determine the form of the whole illuminated surface of the floor of the room below, and the illumination at any point. 13. A straight line passes through the point of intersec- tion of three rectangular reflecting planes, and is equally inclined to each of them ; find the series of images produced, and distinguish between those resulting from one, two, and three reflexions respectively. 14. The image of a straight line perpendicular to the axis of a convex lens at a very great distance from it, ap- proximates to a parabolic curve, whose equation is 2(^-1-/) -(-(24-^)^=0, the centre of the lens being the origin, and the circle of least confusion being taken to be the image of any point. 15. Shew that the image of the surface f {x, y,z) = made by reflexion at the plane mirror loa + my ■\-nz = ^ is f{«' + lp,y + mp, z + np) = 0, p being such a quantity that P Ix + iny -f- nz +^ = 0. CHAPTER VIII. 317 16. An image of a very distant object is formed by a plano-convex lens, the pencils before incidence on the lens passmg through a small diaphragm {B), whose centre is in the axis {ABO) of the lens : being the centre of the con- vex surface, and A the point where the axis meets the plane surface of the lens. If AO == fj. . AB, the image formed by the lens will be distinct. N.B. Distinctness will be secured if the refraction at the second surface— the convex one — be direct. 17. A stop is placed on the axis of a concave spherical reflector at a distance from it equal to four times its focal length: to measure the distortion at any point of the image of a very distant object to which the axis is directed. With a figure and notation similar to that of Art. 150 (the position of E and Y must be reversed)— if Y be the stop or diaphragm, AR = y, a being the pomt of the image seen corresponding to the point Q of the object, we shall obtain 2m=rtaae(^i + |!), now the part of the object whereof qm is the image has a size proportional to tan e; hence the distortion at any point of the image is measured by the term involving y^ in the expression r i-m 18. -A small light is placed at the focus of a perfect reflector in the form of a paraboloid of revolution : prove that the brightness, due to reflexion, at any point within the volume of the paraboloid, varies inversely as the square of the focal distance of the end of the diameter through the point. 19. The caustic by reflexion at the parabola 2y' = 3cx, the vertex being the luminous point is represented by the equation 8c^x — 6c V — ^ = 0. 20. If S be the luminous point, SZ the polar subtan- gent of the curve (Art. 153), ZJ^ the perpendicular on the corresponding tangent to the caustic, then the locus of if is an involute of the caustic. 21. A glass scale divided into equal parts is placed at a distance a from a concave mirror at right angles to the axis ; 318 EXAMPLES AND PROBLEMS. to an eye placed at a distance h beyond the scale, n divisions of the scale appear to cover n + 1 divisions of its inverted image ; shew that if r is the radius of the mirror 11 h - = - + r a 2na (a + 6) ' 22. In Art. 153. li Pp = -.SP for all points of the curve of reflexion, shew that the curve of reflexion is fr\ 2 . n — 1 , = sin —^r— 6. c/ 2 23. If the origin of light he on the surface of the sphere, the caustic surface formed by rays which have undergone n reflexions is the surface of revolution whose generating curve is an epicycloid formed by a circle of radius ^ :t rolling on a circle of radius 2n+l' 24. If the origin be the source of light, and the distance between the point of incidence and the corresponding point 6 (r) in the caustic by reflexion be j, , ., shew that the form of r (r) the reflecting curve is given by \dd) ^"^ <\>{ry 25. The focal length of a double equiconcave lens (/i = f ) is 5 inches : prove that the distance from the lens of images of a distant object formed (i) by reflexion at the 1st surface, (ii) by one reflexion at the 2nd surface, (iii) by two reflexions at the 2nd surface are 2^ inches, 1\ inches and \ inch re- spectively : — and compare the sizes of the three images. 26. At a point on the inside of a polished hollow circular cylinder of radius a is placed a luminous point: explain the formation of a series of bright curves on any plane at right angles to the axis of the cylinder ; and prove CHAPTER VIII. 319 that they are all epicycloids, the radius of the rolling circle for the nth curve being -^ — n . and that of the fixed circle 7; ^. ^ 2n+l' 2n + l 27. A luminous point moves along a diameter of a reflecting circle of radius a ; prove that the two cusps of the caustic, which are not on that diameter, move on the curve d r = a cos ^ . 28. Parallel rays fall on a curve whose equation is given in "the form s =/('ijr), where 1^ is the angle the normal makes with a fixed line parallel to the direction of the rays. Shew that the radius of curvature of the caustic is given by the equation 4 j lies between tan ' -and tan"' - . Supposing there to be no internal reflexion, shew that to an eye placed close to the centre of a circular end there will appear a dark ring fringed with colours ; — indicate the colours which will compose the fringes, — a the altitude of the cylinder being greater than r the radius of the base. 22. When a spectrum is measured in Fraunhofer's man- ner, to find the angle subtended through the telescope be- 326 EXAMPLES AND PROBLEMS. tween the fixed line A in its axis, and the fi^ed line B, having given fi^ = V3, f^B — I^a= '001, the refracting angle of the prism being = 60° and the distance of its edge from the slit and the object-glass being 3 and 10 times the focal length of the object-glass respectively, — the telescope mag- nifying 20 times. With the notation of Art. i68, the angle which the lines A and B subtend at the prism = -, 5u= '002 in circular measure, after reduction. cos (p cos \j/ If /= focal length of the object-glaaS, the linear distance o| A and B in the image formed by the prism =3/. (•oo2) = 'oo6 ./. This distance subtends at the object-glass, the Z , and therefore thecof responding part of its image formed '^' -006 by the telescope, and viewed by the eye subtends at the , eye the Z . 20 = •00923 in circular measure = 3 1'44" nearly. 23. A prism-shaped piece of glass whose transverse sec- tion throughout is a quadrant of a circle has the curved sur- face silvered. A narrow cylindrical beam of compound light is incident perpendicularly; on one of its faces. Describe the successive appearances presented to an eye looking through the other face as the light is moved from the edge of the prism, in a plane perpendicular to it and passing, through the eye, especially marking the order of the colours. Shew that the limits of the distance of emergence from the edge of the prism for a ray of light whose refractive index into the prism is /t are '^\/2;IT2^^^'^V2;^- 24. In Art. 68 prove that for rays of different refran- gibility, the locus of the primary focus will be a curve of the third degree having a cusp at the point of incidence on the refracting surface. 25. In Newton's experiment (Art. 156), if the screen be placed in a plane perpendicular to the direction of light before incidence on the prism, prove that the length of the CHAPTER IX. 327 spectrum, for a given position of the edge of the prism, will be proportional to (f^v - H'r) sin i cos" D cos (Z> + t — ^) coa'^' ' i being the refracting angle, n„ fi^ the refractive indices for extreme rays, .sg). 81. Prove that for minimum dispersion (of a pencil passing through a prism in a principal plane) Ij? sin (3^' - 2i) = Oi' - 2) sin ^' ; ^', i, /jL having their usual meaning. 32. "Fraunhofer shewed that in a telescope with two lenses a very fine wire placed inside the instrument iu the focus of the object-glass is seen distinctly through the eye- piece, when the telescope is illuminated with red light ; but is invisible by violet light even when the eye-piece is. in the same position." Oanot's Physics. Give the probable reason of this phenomenon. It is explained by the achromatism of the eye being imperfect. Art. 190. 33. A pencil of compound light passes excentrically through n thin lenses separated by finite intervals. If the system be achromatic then '■■{^y^.i^y^^ where 6, is the distance of the point of intersection of the axis of the pencil before incidence on the r"^ lens (whose focal length is/^ and dispersive power to-,) from the centre of that lens. Obtain from this result the ordinary expression for the ratio of the dispersive powers of two lenses separated by an interval a for a pencil of light whose axis cuts the axis of the lenses at a distance b from the centre of the first lens, viz. OT J _ _ (g -H h)f H- a b ■B7, bf^ + ah CHAPTER X. 329 34. Having given ^,, = 1-545, /t^= 1-525 and the focal length of a lens for rays of mean refrangibility = 4 in. and its breadth = 2 zw., shew that the diameter of the circle of chromatic aberration = -0128 in., nearly, for a pencil of rays parallel at incidence. (Art. 183.) 35. If a small pencil of light pass directly through a plate of thickness h, the index of refraction being / [-),x being measured from the plane of incidence, and c varying slightly with the colour of the light, — shew that the chromatic aber- ration on emergence is /f-j being supposed equal to unity. Chapter X. Vision through Lenses, i&c. 1. A person can see distinctly at the distance of 6 inches, find the focal length and nature of a lens which will enable him to see distinctly at a distance of 18 inches. 2. A concave lens is placed directly between an eye and a screen, determine how much of the screen will be visible through the lens. 3. A wafer is viewed through a convex lens, of 8 inches focal length, placed half-way between it and the eye; if the diameter of the lens be ^ inch, that of the wafer \ inch, and the distance of the wafer from the eye 8 inches; the whole of the wafer will just be seen. 4. An eye is placed close to a sphere of glass, a portion of the surface of which, most remote from the eye, is silvered, — ^prove that assuming eight inches to be the least distance 330 EXAMPLES AND PROBLEMS. of distinct vision, the eye cannot see a distinct image of itself unless the diameter of the sphere be at least ten inches in length. 5. A short-sighted person moves his eye-glass gradually from his eye towards a small object; shew that the linear magnitude of the image will keep increasing during the motion, and that the angle subtended by the image at the eye will be least wheti the eye-glass has advanced half-way towards the object. 6. At what distance from the eye must a concave lens be placed that the apparent linear magnitude of a small distant object may be diminished one-half? 7. Explain the following facts; — an object seen with both eyes appears single; — we form erroneous ideas of the size and distance of an object in an unusual situation; — it is very difficult to judge of the exact place of an isolated object seen with one eye only. 8. A small object is viewed through a sphere of water (/A = f ; radius of sphere = ^ inch), — being placed at a dis- tance of ^ of an inch from the sphere; find the magnifying power. 9. A concave lens is moved from contact with a small object up to the eye, shew that the apparent magnitude of the image seen by the eye will first diminish and then in- crease, — and that it will be a minimum when it is midway between the object and the eye. 10. An object viewed through a convex lens in two different positions appears in each case equally magnified, but one image is erect and the other inverted, — shew that their mean distance from the lens is equal to its focal length. 11. An object is viewed through a convex lens in two different positions, so that in each case the image appears equally magnified, but in one case erect and in the other inverted; if the nearer position be determined by u, and/ be the focal length, shew that the distance between the two positions is = 2 (/— u). CHAPTER X. 331 12. Shew that a defect in the eye of such a nature that the rays of a pencil incident in a horizontal plane are differ- ently refracted from those in a vertical plane, may be remedied by the use of a lens of which one surface is cylindrical and the other spherical. Find the foci of a pencil of parallel rays in the two principal planes, when the radii of the sphere and cylinder are 3^ and 4J inches respectively. 13. A closed hollow cylinder, about two inches long, has in the middle of one end a very small hole, and in that of the other a circular aperture of about the same diameter as the pupil of the eye. A pin is so placed in the plane of the aperture that its head is near the centre. When the flame of a candle is viewed through the tube, the aperture being held close to the eye, the flame appears upright and the pin inverted. Explain the phenomenon. Telescopes, &c. 14. The angular radius of the uniformly bright field of view in Galileo's telescope — mth the usual notation — is 15. The aperture of the object-glass of an astronomical telescope being 5 inches, and that of the pupil of the eye -^ of an inch, find the least magnifying power for which the whole of a pencil incident on the object-glass can enter the eye. 16. The apertures of the eye-glass and object-glass of an astronomical telescope of which the magnifying power is 50, are respectively 3 and 2 inches; find the diameter of a stop that will completely intercept the ragged edge bordering the field of view. 17. What are the several effects of covering the central part of the object-glass, and of the eye-glass of an astrono- mical telescope ? 332 EXAMPLES AND PEOBLEMS. 18. Why is the apparent brightness of a star increased by the use of a telescope, whilst that of a planet is not ? 19. In the simple astronomical telescope, when the apertures of the two lenses are proportional to their focal lengths, the field of view (as seen by whole pencils) is a single point. If a convex lens of the same aperture as the eye-glass but of any given focal length, be placed in contact with the eye-glass, determine whether the field of view is thereby in- creased or diminished; (the telescope being always supposed adjusted for distant objects). 20. The focal length of the object-glass of a simple astronomical telescope is 15 inches, and its breadth is 3 inches; find the focal length of the eye-lens in order that on looking at a distant point the pencil of rays may just fill the pupil of the eye, — the breadth of the pupil being one-fifth of an inch. 21. In an astronomical telescope — with the usual nota- tion — shew that for a person who can see distinctly at a dis- tance a, the diameter of the stop should be /o/. + «^(/o+/e) ■ 22. In an astronomical telescope, object-glass of 40 inches focal length, with an erecting eye-piece whose lenses begin- ning with the field-lens are 1'2.5, 21, 1, '5 inches respec- tively; — the distances between the first and second lenses being 2 inches, and between the second and third 1^ inches, and the distance between the field-lens and object-glass 41 inches, — find the magnifying power, and the position of the remaining lens when the instrument is in a. perfect state of adjustment. 23. In Art. (200); the distance between the two positions of the object for which the images are magnified m times, one 2/ erect and the other inverted, is — . m CHAPTEB X. 333 24. In Galileo's telescope, if F, F^, F,„ be the magnitudes of the field of view visible by half pencils, by whole pencils and of the entire field visible respectively, shew that <2.F = F^^F,.. 25. "What effect is produced by viewing an object very close to the eye, but through a fine pin-hole ? ' 26. Three convex lenses of focal lengths /j/^/, are sepa- rated by intervals a, b, find the magnifying power of the combination, and prove that it will be independent of the position of the object if (/.-'^)(/3-i)+/.(A+/,-a-&) = 0. 27. Explain why in looking through a moderately thick hedge, we obtain a better notion of the objects on the other side of the hedge by running alongside it, than by standing still. 28. A Galileo's telescope is adjusted so that a pencil from an object 289 feet from the object-glass emerges as a parallel pencil ; the focal length of the object-glass is 1 foot, and that of the eye-glass 1 inch ; shew if the axis is directed to the sun and a piece of paper held 23 inches from the eye- glass an image of the sun is formed on the paper. The sun's apparent diameter being cot"' 120, what is the size of this image, and is it erect or inverted ? 29. A looking into B's, eye sees four images of a candle. The first is very bright, small and erect — the second is fainter, larger and erect, — the third is still fainter smaller and in- verted, — and the fourth is inverted, of a dull reddish brown colour, and indistinct, but rendered more distinct if A uses a concave glass. When B adjusts his eye to an object very close to his eye, the first image remains unaltered, the second and third are diminished, and the fourth requires a stronger concave glass to render it distinct. How are these images formed, and what do they indicate about the adjustment of the focal distance of the eye ? 30. If the bright globe of a lamp be looked at for some time, and the eyes be then turned towards a dark part of the 334 EXAMPLES AND PROBLEMS. room, the image remains, but changes colour from yellow through orange, red and violet, and disappears with a greenish blue tint. Explain this. 31. Shew how any telescope may be used for discovering approximately the distance (supposed not very great) of any visible objects. Graduate the micrometer screw by which the length of the axis of an astronomical telescope when used for such a purpose might be ascertained. Suggest means of obviating errors due to changes in form of the observer's eye, on the supposition that these changes are not instan- taneous. 32. In an astronomical telescope with an eye-glass of focal length/, if the instrument be in adjustment for a person who sees distinctly at a distance a, shew that the distance through which it has to be moved for a person who sees dis^ tinctly at distance h is 33. The focal lengths of the large and small mirrors of a Gregorian telescope being 36 and 1^ inches, and the dis- tance between them 38 inches, find the position of the image formed by the small mirror. Also, if the focal length of the eye-glass be 1 inch, deter- mine approximately how far the small mirror must be moved in order to adapt the telescope to an eye which sees most dis- tinctly at a distance of 11 inches. 34. -Calculate the magnifying power and field of view of a Gregorian telescope from the following data — focal lengths of large mirror, small mirror, field-glass, eye-glass = 24, 2, 3, 1 inches severally, distance of field-glass and eye- glass 2 inches, field bar aperture = | inch. 35. The focal lengths of the large and small mirrors of a Cassegrain telescope are 24 and 1 inches, and the distance of their principal foci ^ inch, — find the position of the eye-lens, the focal length of which- is 1 inch, and the magnifying power. CHAPTER X. 335 36. The object-glass of an astronomical telescope has a focal length of 50 inches, and the focal length of each lens of the Ramsden's eye-piece is 2 inches ; find the position of the eye-piece Avhen adjusted for ordinary eyes, and the mag- nifying power of the telescope. Result. — The distance tetween the ohject-glass and field-glass of the eye- pieoe=50-5 inches and the magnifying power=— ^ . 37. The focal lengths of the larger and smaller mirrors of a Gregorian telescope are 32 and 3 inches, and the dis- tance between their principal foci = ^ inch ; the focal lengths of the lenses of the Huyghens' eye-piece are 3 and 1 inch ; find the relative position of the eye-piece and mirrors when the instrument is adjusted for ordinary eyes, and the magni- fying power. 38. To an eye placed at the aperture of the large mirror in Gregory's telescope there will appear an inverted image of both mirrors near the smaller; and if the axis of the smaller be slightly disturbed the images will be shifted to- wards that part of it which is most inclined from the larger. Prove this property, and explain its use in the practical ad- justment of the telescope. 39. In a Newtonian telescope the longer diameter of the small plane mirror is 2 inches, and the diameter of the objecti-mirror 8 inches, find approximately what fraction of each incident pencil is stopped. 40. The object-glass of an astronomical telescope has an aperture of 1 foot and the magnifying power of the instru- ment is 240. Shew that the brightness of the image is to that of the object as 1 ; 25, if aperture of pupil be J inch. 41. The diameters of the eye-glass and object-glass of an astronomical telescope are 1 inch and 6 inches, and their focal lengths 1 inch and 20 inches respectively. If the axis be pointed to a rod of indefinite length at a distance of 150 feet, how much of it will be seen through the telescope ? 42. A Gregorian telescope being adjusted so that the pencils of rays erderge from the eye-glass in a state of paral- 336 EXAMPLES AND PROBLEMS. lelism, shew that to suit an eye which sees distinctly at a distance a, the small mirror must be moved towards the larger one through a space = .^,^i,.^.,^ , where / /' are the numerical focal lengths of the small mirror and eye- glass, and X the distance between the principal foci of the large and small mirrors. 43. In an astronomical telescope with Eamsden's eye- piece, F,f being the focal lengths of the object-glass and of each lens pf the eye-piece, — the magnifying power of the 4 or < m : and that the angular breadth of this portion will be (m' ~ v) ffy where _/i^' are the focal lengths of object-glass and eye-glass, and y the breadth of the eye-glass. 57. A person who reads small print at a distance of two feet finds that with a pair of plano-convex spectacles he can read it at a distance of one foot — what is the radius of the curved surface of either lens, /* — f ? 58. The lenses of a common astronomical telescope whose magnifying power is 16, and length from object-glass to eye- glass 8^ inches, are arranged as a microscope to view an object placed | of an inch from the object-glass; find the CHAPTER X. 333 magnifying power, the least distance of distinct vision being taken to be 8 inches. 59. Kays parallel to the axis fall on a reflecting curve y=/(^) ; shew that the point of the caustic corresponding to a:, y is dod' dad' If ^' are the angles of incidence and refraction of the axis QP of the pencil which is incident at P, QP=u and GP = a. 5. If bb' be the breadths of the p*'^ and q*^ rainbows respectively and B the sun's apparent diameter, shew that 6. The angular breadths of the primary and secondary rainbows at the eye of the observer are respectively, IC^T-^'^' and|(^y.S/., nearly, where S/i is the difference of the refractive index for the extreme colours, and /* the index for mean rays. 7. When the rays emerge parallel after two refractions and one reflexion within the drop, shew that if (f) denote the 342 EXAMPLES AND PROBLEMS, angle of incidence, /t the refractive index and D the devia,tion, then 2cos(/) = /^cos(^+2-4,). 8. Having the following approximate data, obliquity of ecliptic = 23° 30', latitude of London = 51° 30', for rain-water, cos"' // — — - j = 76° 40', and cos-' - ^ {^^^ = 46° 40', shew that in the latitude of London, no portion of a tertiary rainbow can be seen by an observer, whose back is turned towards the sun, if the sun be distant from the summer solstice by an angle greater than that determined from the equation sec = 2 cos 11° 30', 9. Find between what hours of the day on the 21st March a rainbow will be visible to an observer at the equator -^having given M = |, log.0 2 = -3010300, log,„ 3 = -4771213, ■ isin21° = 9-55462l7. 10. How are white rainbows accounted for ? In consequence of tlie finite size of the sun's disc, tlie colours of the raiu- bowa given by pencils of light from different points of it, overlap each other — and the traces of colour may under exceptional circumstances become very slight. For instance when the sun-light passing through a cloud of ice- crystals in the upper strata of the atmosphere, -and being reflected at the surfaces of such crystals, reaches the eye after being refracted through rain-drops in a lower . stratum of the atmosphere,— the source of light is spread over a large spherical angle — there is no sharp edge to the bow, — the bands are broader and fainter — ^fhere is little trace of colour — and the rainbow is white or nearly so. Such phenomena are not of frequent occurrence. See an account of one seen by M. Cobhu on November 28, 1883, and reported by him in the Comptes Bendus, Tome 97, p. 1530. No. 27 (31 December, 1883). MISCELLANEOUS PROBLEMS. S48 Miscellaneous Problems. 1. AO is a radius of a sphere (reflecting internally) whose centre is 0, at the bisection of ^ a luminous point is placed. Supposing light to fall first on the side of the sphere towards A, and calling v^^, v^^^^ the distances from A of the {^iif", and (2re + I)*'' images, shew that 4W-1 4,1 + 3 2. If a circular disc whose circumference is studded with bright points be made to roll with great rapidity within a circle of double the diameter, shew that the appearance will be presented of a number of rectilinear rays of light diverging from a common centre. Find the number of the rays. 8. A pencil of parallel rays is incident on the curved surface of a cylinder, and the reflected rays fall upon a screen which is parallel to the ends of the cylinder, shew that the area of the screen which is illuminated by reflexion is TT (a^ — If) tan^ a+%c{a — h) tan a, — where c is the radius of the cylinder, a, h the distances of its ends from the screen, and a the inclination of the incident pencil to the axis of the cylinder. 4. A man standing on the banks of the Cam, beside Trinity bridge, observes that, the inverted image of the con- cavity of the arch receives his shadow exactly as a real inverted arch would do, if it were in the place where the image by reflexion appears to be. Explain this. 5. A ray is incident parallel to the axis of a polished prolate spheroid, and after two reflexions becomes again parallel to the axis. Shew that if {x^y^ {x^^ be the two points of reflexion, ^^=^:— and find the points when the path of the ray in the spheroid is a rectangle. 6. A plane luminous ellipse throws light on a small plane area parallel to it and situated in the line drawn through 344 MISCELLANEOUS PROBLEMS. the centre of the ellipse perpendicular to it. If a, b are the semi-axes of the ellipse, and c the distance of the small area, ab the illumination x v{K+cO(&^+cor 7. A person haviag a very small fragment of a concave reflector, the principal radii of curvature of it being p, p', wishes to place it so that it may be considered part of a paraboloid of revolution, the positions of the focus and axis of which are given. Shew that it must be placed with its plane of least curvature passing through the focus, at the distance \^{pp') therefrom, and with its tangent plane in- clined to the axis of the paraboloid, and to its focal distance at an angle whose sine is . / ( ) ■ 8. It has been remarked by writers on optics that " the concavity of the heavens appears to the eye to be a less por- tion of a spherical surface than a hemisphere," — and that " the appa,rent distance of its parts at the horizon is generally between three and four times greater than the apparent dis- tance of its parts overhead." How do you account for this appearance of the sky ? By what method are the proportions here mentioned determined ? Shew how these circumstances may be used in accounting for the apparent change in the size of the sun or moon, or in the distance of two neighbour- ing stars, as they ascend from the horizon to the meridian. 9. When a cylindrical china jar, standing upon the ground, receives the sun's rays obliquely, a bright curve is observed to form itself at the bottom of the jar, and it is found that the shape and dimensions of this curve are not affected by the varying elevation of the sun : account for this latter circumstance, and determine the nature of the bright curve. 10. A plane mirror, moveable about an axis in its own plane parallel to the axis of the earth, revolves from east to west with half the sun's apparent diurnal motion. Shew that the direction of the reflected rays of sunlight wiU not be sensibly altered during the day. MISCELLANEOUS PROBLEMS. 345 11. The sun's light is refracted through a prism, the edge of which is vertical,— find the position of the refracting surfaces m order that for a given altitude of the sun the deviation of the rays of a given colour may be a minimum. — If z be the sun's zenith distance, i the refracting angle, X the angle of first incidence reduced to the horizon, /y; the index for the given colour,— shew that the minimum devia- tion D is given by the equations = /^sin|.y/|l+(l-l,)cot^.}. sm -^ = sm ^ sm sma; 12. One half of a circle is a bright reflecting surface, the other half dull ; a luminous point is placed at the bisection of the dull part. Shew that the bright surface is divided into two portions each of which is equally illuminated. 13. A narrow flat polished steel-wire is bent into the form of a circle, so as to form a cylinder of indefinitely small height ; a luminous point is placed so that the perpendicular from it to the plane of the ring falls within the ring. Shew that all the rays after reflexion pass through a straight line of finite length. Calculate the length and position of this line, and point out the modification of the problem when the perpendicular does not fall within the ring. 14. If a plane surface be placed parallel to the plane of the ring — (Prob. 13) — and below it, — the bright curve on the plane surface has for its equation r = a-\-h cos Q, the point where the line passing through the luminous point and the centre of the ring meets the plane being taken as the pole. Trace this curve and find the condition that it may have a cus.p. 1-5. The directrix of a parabola is reflected at the curve, the exterior rim of which is polished ; shew that the image is a curve lying between the parabola and its evolute and bisecting all the lines which are common normals to the one and tangents to the other. 346 MISCELLANEOUS PROBLEMS. 16. A small plane area is illuminated by a bright circu- lar ring of given dimensions, — the line joining the centres of the area and of the ring being perpendicular to the plane of the ring ; find the position of the area, so that the illumina- tion may be the greatest possible. — If the area be replaced by a bright point, find its position so that the greatest quantity of light may be thrown on the ring. 17. A sphere of glass encloses a concentric sphere of an opaque substance and is placed on a horizontal table, a bright point is placed very near, but not close to its highest point; find the least magnitude of the opaque sphere which will pre- vent any ray from reaching the table ; — if the magnitude be slightly less than this, find the diameter of the circular bright ring on the table. 18. A ray of light is reflected a number of times between two plane mirrors — not in the principal plane ; prove that all the reflected segments of the ray are generating lines of a hyperboloid of revolution. 19. A plane mirror revolving about a vertical axis in its own plane, receives and reflects a small sunbeam, which after reflexion forms a spot of light on the horizontal floor. De- termine the motion of the spot on the floor, and shew that its image as seen in the revolving mirror is stationary. 20. A small plane mirror is placed at the principal focus of a telescope, nearly perpendicular to its axis, and the tele- scope is directed approximately to a distant luminous object ; shew that the rays reflected at the mirror will, after repassing the object-glass, return in the exact direction in which they came, in spite of the small errors of adjustment of the mirror and telescope. 21. A surface exposed to a luminous sphere is such that the illumination at any point oc (distance)"" of the point from the centre of the sphere. Examine the cases when ?i=0, 1, 2, 3. 22. A small pencil of rays emanates from a certain point on a parabola the axis of the pencil being an ordinate to the MISCELLANEOUS PROBLEMS. 347 curve. If the primary focus after reflexion at the curve be on the parabola, prove that the angle between the axis of the incident and reflected pencil is 2 sin"' —= ^2 23. A ray proceeds from any point of the curve X' y in any direction, and is reflected by the surface a? , y' z' ^ 1- — 4-— = 1 • shew that after reflexion its direction will again intersect the curve. 24. Find the direction in which light must be incident on one face of a triangular glass prism in a principal plane, in order that a ray after two internal reflexions and emergence at the same face, may proceed in the same direction as a ray directly reflected without entering the prism. If the edge of the prism be placed parallel to the earth's axis, and the prism fixed so that the above direction is parallel to the plane of the meridian ; — shew that when the sun's disc crosses the meridian, two images of the sun may be seen moving in opposite directions and crossing each other. What is the principle and use of the Dipleidoscope ? 25. An atmosphere consists of concentric spherical strata, the density in any stratum varying inversely as the square of its radius ; prove that the path of a ray incident on a certain stratum at an angle whose secant is the refractive index of ■ that stratum, will be a reciprocal spiral, — and that at every stratum the secant of the angle of incidence will be equal to the refractive index. Also determine the path for a greater angle of incidence than the above. , : N.B. If /M be the refractive index for a gas for a ray i entering from a vacuum, it is found by experiment that /m'-I :' oc density. 348 MISCELLANEOUS PROBLEMS. 26. A small plane perpendicular to one of the diagonals of a cube at a point in that diagonal produced is illuminated from the uniformly bright surface of the cube ; shew that the illumination is measured by the expression or ^ • • o • "" 3i . -. — n sm a sm « sm - , sm t* 3 where 6 is the angle between two adjacent edges of the pyra- mid (vertex 0) circumscribing the cube, and a, /S are their inclinations to the diagonal. 27. Two parallel and equal lines AB, CD are traced on paper, and viewed by the two eyes placed so that the line joining the centres of the eyes is parallel to AC; shew that the eyes may be so adjusted that the images of AB, CD may appear superposed, and that the resulting image will appear parallel to the plane of the paper, nearer to the eyes and smaller than AB or CD. Under the same circumstances if AB, CD be inclined at a small angle, their images may appear superposed, and the resulting image will appear to project from the plane of the paper. Explain this, and shew why the parts of two pictures which, when viewed with the stereoscope, appear to project towards the eyes; frequently appear, when superposed by the naked eyes, to project in the opposite direction. 28. The shadow of a ball is cast by a candle upon an inclined plane in contact with the ball; prove that as the candle burns down, the locus of the centre of the shadow will be a straight line. 29. If the index of refraction of a medium at a point distant r from a fixed point be a/ 1 + ^, prove that a ray of light traversing the medium will describe a central conic section whose eccentricity will be > or < 1 according as the upper or lower sign be taken. 30. On what law of density will the total refraction of a ray of light through the earth's atmosphere be the same as through a homogeneous atmosphere ? MISCELLANEOUS PROBLEMS. 349 . 31. A vertical window-bar of an opposite house, seen through a pane of glass, appears a zig-zag line ; describe the inequalities in the thickness of the pane of glass indicated by the deviations from the vertical, and give reasons for your conclusion. Rain is streaming down a pane of glass in parallel vertical lines ; what will be the appearance of a circular arch seen through it ? 32. The shadow of a given ellipsoid thrown by a lumin- ous point on the plane which passes through two of the prin- cipal axes, has its centre on the curve in which the same plane intersects the ellipsoid ; shew that the equation to the locus of the luminous point is a'^¥ \c' J' where a, h the semiaxes of the ellipse through which the plane passes, and c the remaining axis, are taken for the axes of X, y, z respectively. 33. A polished hemisphere is placed with its base in con- tact with a plane, and a cylindrical pencil of rays falls upon its convex surface in a direction perpendicular to the plane : the illumination at any point of the plane is proportional to sin^0 2 4- sin ^ ' where <^ is the angle which the ray reflected to that point makes with the plane. 34 A uniformly bright cylindrical column, of radius a, and of indefinitely great height, stands upon a plane ; shew that the illumination of a disc of the plane, of radius r, con- centric with the column, 35 Denoting two mirrors by a = 0, ^S = 0, what must be the angle between them, that the ray parallel to a-/3 = may, after reflexion at each mirror, be parallel to a +/3 = ? 350 MISCELLANEOUS PROBLEMS. 36. The illumination at any point of an equilateral hy- perbola, the centre being the origin of light, oc (distance)"*. 37. Shew that the space-penetrating power in an astro- nomical telescope is measured by f J. .•^j , where i'^and / are the focal lengths of the object and eye-glass respectively. 38. The magnifying power of a telescope is _ breadth of visual pencil at the object-glass breadth of visual pencil at the eye-glass ' supposing no light lost at any lens. 39. A blinking cat looks at a very little fish, which is in the axis of a thick closed glass cylinder full of water ; first, when the axis of the cylinder is placed in a vertical .position, and secondly when it is horizontal : in both cases the fish is at the same distance from the cat's eye, the line joining them being horizontal and perpendicular to the axis of the cylinder. Prove that the cat will believe the little fish to be less remote in the former than in the latter position of the cylinder :— and ascertain the difference between the apparent distances in terms of the thickness of the glass and the radius of the internal surface of the cylinder. 40. An erecting eye-piece of four lenses has the focal lengths of the lenses /j, .^./g,^ and their intervals a^, a^, a, ; if/ is the focal length of the equivalent simple lens, 1_1 1 1 , 1 ^/'l , 1 , i\ , /"l.iUJ:.! JsJi \Jl Jv JlJiJiJi 41. A luminous sphere rests within a hemisphere of twice its radius, the rim of which is horizontal; find the MISCELLANEOUS PROBLEMS. 351 whole illumination of the interior surface of the hemisphere ; — and if the sphere be raised so that its lowest point just coincides with the centre of the hemisphere, shew that the illumination will be diminished in the ratio V5 - 1 : VS + 1. 42. A pencil of rays is refracted directly through a series of concentric spherical strata ; if fj.^, fj,^ be the indices of re- fraction from vacuum into air and into the r"' stratum (count- ing from the outermost), c,. the outer radius of the r"" stratum, and % v^ the distances of the initial focus and of the focus after r refractions from the common centre, shew that 43. A ray of light is incident perpendicularly on one of the faces of a prism of which the density varies in such a manner that the coefficient of refraction at any point is fie^, — fi being constant, and the angle which a plane through the point and the edge of the prism makes with that face upon which the ray is incident. If a be the refracting angle of the prism, <^ the angle of incidence on the second face, shew that (f> is determined by the equation cos ^ — sin ^ = e*"^". 44. The focal lengths of the large and small mirrors of a Gregorian telescope are 18 and |-f inches respectively, and their distance from each other is 19 inches ; find the position of the image formed by the small mirror, and the magnifying power, if the focal length of the eye-glass = ^ inch. 45. The density at any point of a prism (vertical ^ = 2/S) varies as its distance from the nearest face of the prism. If a ray pass through it in a principal plane, its distance from the edge at the points of incidence and emergence being a, and its nearest approach to the edge being c, the deviation D is given by the equation sin [/3 +=-) =sin ;S. e2(«-«»'P' - 352 llIISeELLANEOUS PEOBLEMS. 46. A spherical mirror is to be reduced by grinding to a, parabolic one of the same focal length ; find approximately what thickness parallel to the axis must be ground away at any point. 47. In Art. 122, if /t =/(ar) shew that (i) the curvature at any point of the trajectory (ii) Find the form oi f(x) in order that the trajectory may be a hyperbola. (iii) Jifix) = a + /3 sin - , determine the trajectory, c 48. A small pencil is directly refracted through a medium for which the index of refraction at any point is /i = -T (as in Art. 122). Shew that the distance of the geometrical focus. from the origin of light will be 2ka J' where t is the whole thickness of the medium traversed. 49. The distance between a luminous point and the centre of the pupil of the eye is (D), and the radius of the pupil (r), which is supposed very small compared with B; shew that the illumination- of the pupil is -^ cos a (^1 - ^ -^^ + -g- p sm'' aj nearly, where a is the angle which I> makes with the axis of the eye, and higher powers of -^ are neglected. 50. The axis of the object-glass of an astronomical tele- scope is inclined to the axis of the telescope at a small angle a ; shew that in order to adjust the telescope for viewing distant objects, the eye-glass must be pushed in through a space = .or .f. 2/j, ■'' MISCELLANEOUS PROBLEMS. 353 where/ is the focal length of the object-glass, and /* its re- fractive index. 51. The extremity of the shadow of a vertical post stand- ing on one side of a pond falls on a point A at the bottom of the pond. This point cannot b§ seen by a person on the opposite side, being just hidden by the sun's image in the water. Given the tan {sun's altitude) = f , the height of the post above the water =12 feet, that of the eye = 18 feet, the breadth of the pond = 70 feet, and /a = | ; prove that the depth of the pond is 20 feet. 52. P, Q are the foci of incident and emergent rays of a small pencil passing directly through two thin coaxial lenses ; shew that two fixed points M, N can be found in the axis of the lenses such that _1 1__1 QN PM~G' and find C in terms of the focal lengths and distance of the lenses. 53. The ends of a glass cylinder are worked into portions of a convex and concave spherical surface, — radii r, s re- spectively, — having their centres in the axis of the cylinder ; shew that the distance of these surfaces, in order that an eye placed at the concave surface may see the image of a distant object distinctly, must = — — =-^ ; — and the magnifying power will be = - • 54. If N, n, F be the focal centres and principal focus of a lens, the distances of the conjugate foci measured from two points N', n', — so situated in the axis that nn'=p. NN' = (1 —p) . nF, — are connected by the equation £_-! = J^- V pu nF ' p being any constant quantity. p. 0. 23 354 .MISCELLANEOUS PROBLEMS. 55. A hollow paraboloidal vessel, whose upper rim is circular, is placed with its axis vertical upon a horizontal plane, and exposed to the sun's rays. The boundary of the illuminated part of the interior of the vessel is a plane curve, whose projection on the plane of the rim is a circle of the same radius as the rim, and whose centre is distant from the centre of the rim a space equal to the latus rectum of the vessel multiplied by the tangent of the sun's altitude. 50. The distance between two lenses of a common astro- nomical telescope in perfect adjustment is a, the magnifying power being - , It is found that the object-glass can be achro- matised by means of a certain lens placed in contact with it, and that if at the same time a concave lens of focal length h be placed in contact with the eye-glass the instrument will still be in adjustment ; shew that if A^, A^ be the dispersive powers of the media forming the compound object-glass, How is the character of the telescope altered by the above arrangement ? 57. If a string be wrapped round a glass prism whose section is an equilateral triangle, so as to be always inclined at the same angle to the axis of the prism, the portions of the string seen by internal reflexion will appear to be parallel to the portions seen directly. 58. If the earth, supposed spherical, were covered to a depth h with water, h being small compared with r the earth's radius — shew that the height to which a person must be raised above the surface of the water in order to see as far below the horizon as when he was on the surface of the earth is ^ — j-^^ — — - nearly, /i being the index of refraction for water. 59. A ray of light emanating from an umbilicus of an ellipsoid is reflected at the surface to the opposite umbilicus : prove that the length of the path is constant, and equal to MISCELLANEOUS PROBLEMS. 355 twice the distance between the extremities of the greatest and least axes. 60. A trapezium ABGD, of which the side CD is pa- rallel to BA, and the angles at A and B are each 60°, is the base of a right prism of glass. Prove that the prism may be used as a stereoscope, if the observer look in at the face AB ; one picture being in contact with the face BG and the other opposite to (7D at a distance which is to the distance of G from J.i? as 1 : fjb. Prove also that the magnitude of the picture will be the greatest possible when AB is four times G.D. 61. A penny being placed with a flat side on an ordinary looking-glass and a candle lighted and placed on the same side of the looking-glass as the penny, but so that the per- pendicular from it on the looking-glass does not meet the penny, ,part of the penny's principal image is observed .to be much brighter than the rest by an eye so .placed that the image of each point may be regarded as its geometrical focus. Explain this and determine the position of the shaded part of the image. 62. Q is a luminous point situated on the circumference of a perfectly reflecting circle, QP any incident ray, P Q' the chord in direction of the reflected ray, p the point of inter- section of PQ' with the consecutive reflected ray ; prove that Q'p=2Pp. 63. In Art. 70 — if the reflecting iportion of the surface, the radius of which is r, be approximately circular of radius a, and if ^ be the angle of incidence, u the distance of the point of incidence from the origin of light, shew that the radius of the circle of least confusion is equal to au sin'"" <^ w (1 + cos"" 0) — r cos ^ ■ :. 64. An opaque sphere attached by a string to the bottom of a vessel of water floats just immersed, the surface being exposed to the rays of the sun, whose zenith distance is a. 23—2 356 MISCELLANEOUS PEOBLEMS. Describe the form of the shadow at the bottom and the colours with which it is fringed. If the water be just deep enough for the bottom to have' no absolute shade, its depth below the sphere : radius of sphere :: cos J {0^-\-6^ : sin \ {6^ — 6^,6^d^\iemg the ap- parent zenith distance of the sun to an eye under water for violet and red rays respectively. 65. If light of intrinsic brightness I pass through an absorbing medium bounded by planes perpendicular to its direction, the' materia,l of which is-such that the absorption of a. unit of brightness; at a distance x from^ the plane on which the light is incident is/(/») for a unit of thickness of the medium at the point — shew that the intrinsic brightness on emergence through, the plate whose thickness is a- is Jg/(,''logU-/«)(te_ 66. A bright line is placed parallel to the axis of a polished cylinder. Shew that the curve which is seen on the cylinder by an eye placed in the plane xy at the point a^, lies on the surface ¥ [{a + xf + (/3 - yf + z"] = {fix - ayf {(6 - x)' + f] + 6y^^ h being the distance- between the bright line and the axis of the cylinder which is the axis of z. 67. There are two confocal reflecting ellipses ; a ray proceeds from a point P of either of them in a direction pass- ing through one of the foci, and is continually reflected be- tween' the curves. If after 2n — 1 reflexions it returns to the point P, the length of the path = n times the difference of the major axes. 68. If GA the diameter and GP any chord of a lem- niscate be reflecting mirrors, shew that a ray incident on GP at P in the direction of the tangent at P, will retrace its course after three reflexions. 69. A varnished sign-board swings under a vertical sun. Shew that the envelope of any particular reflected ray is a circle. AJEISCELLANEOUS PKOBLEMS. 357, Also find the size of the bright patch on the ground for any position of the sign-board. 70. If the Moon were a transparent refracting globe, of foca,l length equal to the distance from the Earth, shew that during a total eclipse of the, Sun, the light and heat at any place at the instant of totality would be increased about fourfold. 71. The north bank of a canal -which runs east and west is vertical. The Sun is shining in the south and waves are travelling along the canal. Shew that on the bank will be seen a series of bright waves running along, whose shapes will be epicycloids, if the hollows of the canal waves be circular arcs. What difference will be made bythe altitude of the Sun ? 72. Assuming that light of dll colours has the same ve- locity ia a vacuum, Shew that by reason of the earth's atmo- sphere, refraction and aberration ought each to produce an infinitesimal spectrum in the image of a star. Taking Cassini's hjrpothesis of a homogeneous atmosphere, find the position of a star for which these spectra destroy each other ; shew that abetration = tan (zenith distance), and that if two colours unite so will all. 73. Four luminous points are placed in a medium in a plane perpendicular to its plane surface. In what positions will an eye placed close to the surface see only two images ? 74. A fay of white light is incident in a principal plane on one side of a prism and emerges at the opposite side after reflexion at the base. Shew that if the ray of medium re- frangibility suffer no deviation at the opposite side, the violet or red rays will emerge nearest to the edge, according as the side of the prism on which the ray falls is greater or less than the other. 75. In Art. 122 — if the surfaces of equal density be such that in going from any one to the next in the plane of {xy) 358. MISCELLANEOUS PEOBLEMS. the expression for the refractive index /* may be put into X the form fi' = (x + a)/(t/V), shew that the parabola y^ = iax is a possible path for a ray to travel in. 76. In front of a paraboloid of revolution which has its external surface polished, a circular ring of radius R is placed, with its plane perpendicular to the axis and its centra at the intersection of the axis and directrix of the generating para- bola ; shew that r being the radius of the image of the ring, 3 and that if i? = latus rectum = 4a, then r=a. 77. If /tt be the index of refraction from vacuum into air of the same density as that of the atmosphere at the distance r from the earth's centre, and. if p be the value of dfi' dr fi ' r ' at the earth's surface, D the diiference between the refraction at the horizon and at a very small apparent altitude A, prove that D^ _p_ A~i--p' 78. The angles at the base of a triangular prism are (0 — <^) and 2^, where sin ^ = /a sin ^ ; a ray of light falls on the shorter side of the triangle ; the angle of incidence is 6 on the side of the normal next the vertex ; shew that the ray after reflexion from the base and from the other side will emerge from the base in a direction parallel to its original direction, and that unless sin^ 6 > sin ^, the second reflexion will not be total. 79. A convex lens of focal length a is placed at a distance a in front of a concave mirror of focal length 6. Shew that an object and its image formed by rays which have passed twice through the lens and have undergone an intermediate reflexion, are always at equal distances on opposite sides of a point dis- MISCELLANEOUS PROBLEMS.. 359 a= tant a — gT ill front of the lens, and that the image is equal to the object but inverted. 80. The solid angle subtended at an eye under water by a given portion of a very distant object is to the angle sub- tended by the same portion when the eye is out of water as cos sin 6' to cos 6' sin 6, the angles of incidence and refrac- tion being 6 and 6'. Prove this, and also shew that the num- ber of rays from each point of the object which enter the pupil of the eye under water, is to the number which enter it out of the water, supposing the aperture to be the same, as cos 0' to cos 0. Hence infer that the whole sky if uniformly bright will also appear uniformly bright to an eye under water : and that, while the apparent magnitude is diminished, the brightness, omitting loss of light by transmission, is increased, but in a less proportion than that of the diminution of magnitude. 81. In one of the faces of an isosceles prism of flint-glass a cavity is made bounded by a spherical surface, and in con- tact with it is placed a double convex lens of crown glass one of the surfaces of which exactly fits the cavity; a plano-con- vex lens of crown glass is placed in contact with the second face of the prism, the centres of the two lenses being in a plane perpendicular to the prism. A pencil is incident directly on one of the lenses and after internal reflexion from the back of the prism emerges directly from the other, find the condition of achromatism. 82. The sides of a quadrilateral which can be inscribed in a circle are reflective; if a luminous point be placed on a diameter of the circle bisecting at right angles one of the diagonals of the quadrilateral, prove that the distance between two of the images is equal to the distance between the other two. Also prove that if the luminous point be placed at the intersection of the diagonals, a circle can be inscribed in the quadrilateral formed by the images. If d be the distance 360 MISCELLANEOUS PEOBLEMS. between the centres of the two circles, R, r their radii, and the angle between the diagonals of the quadrilaterals, shew that d'' = E'- Br cosec0. 83. A small lens and a luminous object on its axis are moved in such a manner that a point of the image remains fixed in position, and it is found that the defect of the bright- ness at this point from the maximum varies as the square of the sine of the angle turned through by the line joining the lens and point. Shew that the lens describes a conic section. (See Art. 151.) 84. A brass plate grooved with an infinite number of concentric circles is placed horizontally in the sun; shew that an observer whose eye is in the vertical plane throixgh the sun and the centres of the grooves will generally see a bright straight line and an arc of a circle on the plate. If the plate be not horizontal, what will be the appearance ? 8.5. Two horizontal straight edges are held between a window and the eye, at a short distance from the eye. The upper one is fixed and is slightly further from the eye than the lower. If the lower be moved up parallel to itself when they approach each other, the upper appears to meet the lower; account for this. 86. In a Galilean telescope, if m be the magnifying power, / the focal length of the object-glass, 2a, 26 the breadths of the object-glass and eye-glass, find the field of view as limited by half-pencils at least on the, object-glass. If this be 2a, and remain constant while the magnifying power receives a small increment Sm, shew that the focal length of the object-glass must be diminished by the amount cot a . Sm. {m-iy 87. A plane luminous curve in a medium (fj.) is viewed by an eye placed at the plane bounding surface. The eye MISCELLANEOUS PROBLEMS. 361 being the origin and the initial line the normal to the surface, shew that, if p =/(sin 6) be the polar equation of the curve, o = -L /• f^J^] V - (1 + /^' )sin'g is the equation of the image. 88. The axis of a telescope bisects at right angles a straight horizontal scale, one yard long divided into inches, and passes through a vertical axis 35 inches from the scale, in the plane of a mirror vfhich is capable of turning about it. Determine the angle between two positions of the mirror in one of which the division marked 4, while in the other that of the division marked 33, appears on the cross wires of the telescope. 89. Four convex lenses whose focal lengths are a, h, b, a are placed at intervals a + b, 2b j- ,a + b on the same axis, shew that the emergent ray is in the same straight line with the incident ray, 90. In making with an Astronomical Telescope an obser- vation for which it is essential that the brightness of the image on the retina should be at least a hundredth part of that of the object, shew that the highest magnifying power which can be obtained is 1000, the diameter of the object- glass being 25 inches, and that of the pupil of the eye I inch. What is the highest magnifying power that can be used without any diminution of brightness ? ^ 91. A transparent hollow cylinder stands on a table, and on the inside of the cylinder is wrapped a narrow band of bright reflecting foil in the form of a helix which just makes one revolution. In the centre of the upper rim of the cylinder is placed a luminous point: find the equation of the curve of light on the table, and trace the curve; — prove that 362 MISCELLANEOUS PEOBLEMS," the illumination by reflected light at a distance r from the axis of the cylinder varies as 2ft +r 1 »- {^W + (2a + rff where 2A is the height and a the radius of the base of the cylinder. 92. Given a curve and' the origin of a pencil of rays, to find (i) whether the caustic curve has asymptotes, and (ii) the position of such asymptotes if they exist. If a parabola have contact of the second order with a curve at a given point, its focus being 5, shew that the caustic curve formed by reflexion of a pencil of rays diverging from B will have an asymptote corresponding to this point. 93. If 0, ^ be the angles of incidence and emergence of two parallel rays passing through a prism in a principal plane; d^, d^ the distances between those rays before incidence and after emergence, shew that where B9 is any small change of 6 and S(j) the corresponding change of (p. Shew from this that the position of minimum deviation is that of most distinct vision through a thin prism. 94. Find the position and radius of the circle of least confusion when a small pencil is obliquely and centrically refracted through a thin lens. A double convex lens has faces whose radii are eacht Jb + 1 8 inches and its refractive index is ■ ; find its power . v^ . when its axis is inclined at 30° to the line of sight. (See Art. 112, Cor. 4.) MISCELLANEOtrS PROBLEMS. 363 95. A luminous point is placed in contact with the base of a hemisphere of glass- — refractive index fi and radius a ; a sheet of paper is held parallel to the base on the other side of the hemisphere. Shew that if the distance of the luminous point from the centre of the hemisphere be > - , there will be a dark band on the paper bounded by two hyperbolas. 96. The image of a right line is formed by centrical pencils through a thin lens. If p^ p.^ p^ be the curvatures of the images formed by the primary focus, the circle of least confusion, and the secondary focus respectively, shew that 2P2 = P1 + P3' 97. In a sphere of glass there is a cavity the boundary of which is a sphere described on a radius of the former. A small object is placed at the point where the glass is in- definitely thin, and is viewed by an eye at the other extremity of the diameter passing through this point — find where the image is formed and determine its magnitude. If /^ = | shew that the image is one-fifth as large again as the object. 98i A and B are fixed points, A being a luminous point and B the nearest point of a glass sphere with refractive index ijl. G a, point on BA produced is the image of A as seen by an eye on AB produced beyond the sphere. Shew that AG is least when the radius of the sphere is ^±^AB 2 — fi 99. Light is incident upon a refracting medium, the index of refraction at any point of which is a function of the . distance from a fixed plane; find the differential equation of the path. (See Art. 120—2.) If the axis of x be perpendicular to the plane and the index of refraction be /Xj tan - , shew that the equation of the 0/ 364 MISCELLANEOUS PROBLEMS. path of the ray which is incident at the point f-j- , OJ at an angle -r- is 2sin- = ce'' +75-6 " , where c = --.= + -7= . 100. If a pair of lenses on the same axis be achromatic for rays incident on the first parallel to the axis, then /*!' f-i^fv/i heing the indices and. focal lengths of the lenses, and a the distance between them. 101. If T= 0, N= be the equations to the tangent and normal at any point P of a reflecting curve, i^, tj the co-ordi- nates of a radiant point, shew that the equation to the ray reflected from the curve at P is where T', N' are what T and N become when f, 97 are substituted for the current co-ordinates. 102. Prove that the magnifying power of a thin double convex lens, the radius of each surface being p, when the space between the lens and an object at distance a is filled with fluid of index fju is given by 1 _ a 2fi — fj! —\ m p' fjf 103. Light parallel to the axis falls upon a convexo-plane lens; shew that the aberration for the extreme ray for refraction through the lens is equal to (j,-i y{p, + i) + i I 2/.>-l) >' MISCELLANEOUS PROBLEMS. 365 where y is the radius of the circular plane face, and r that of the spherical surface. 104. A luminous point is placed on the axis of a concave lens at a distance u from it. The light falls on a screen at a distance k behind the lens and perpendicular to the axis of the lens. If I is the illumination of the screen where it cuts the axis, and if T is what the illumination would be if the lens were removed, shew that r {fu + UIC + Kff ' 105. A small pencil of parallel rays is refracted cen- trically through a double_convex lens the radii of whose surfaces are each =r, and whose thickness is t; shew that, if the square of t be neglected, the distance of the primary focus from the point of emergence of the pencil wUl be r sin (/)' cos"^ t sin (f>' cos"^ 2 sin {(j)- (p') 4sin cf) cos"0' ' ^, ' being the angles of incidence and refraction. 106. Two conical shells of light given by the equation 2^= ±001/ are incident on the reflecting surface xyz = const. ; shew that the reflected rays pass through two straight lines at right angles to each other. 107. Three lenses are placed so as to have a common axis, the second being equidistant from the first and third ; if the combination be such that the image of a luminous point is always at the same distance from that point, prove that either the focal lengths of the first and third are equal, or the focal length of the second is equal to a quarter of the distance between the first and third. In each case find the constant distance, and in the second case shew that it is equal to the distance between the first and third lenses. 108. Rays are incident from the centre upon a reflecting ellipsoid at points situated on a central circular section: prove that the reflected rays pass through a diameter of the % ellipsoid perpendicular to this section. 366 MISCELLANEOUS PROBLEMS. 109. A portion of a medium, the refracting index of which at any point whose distance from a fixed point is r* is -j , is bounded by three planes mutually at right angles passing through 0: a ray of light is incident on one of the plane faces at a given point, find the length of its path in the medium. 110. A pencil.travelling in amedium is refracted through n other media (whose boundaries are concentric spheres); if p, p' be the distances from, the centre of the incident and emergent pencils, then ^(^ M - "^ I ^' ~^^ I ^' ~ ^'' 1 ... I ^"-' ~ ^" ^ i) where jjl^, fi^,...fi^ are the refractive indices fi-om the original medium into the others commencing from the outside, r^, r^,...r^ the radii of the bounding surfaces. 111. Shew that of a pencil from a point in the ragged edge of the field of view of an astronomical telescope there is lost a portion whose breadth is equal nearly to a . i/^, where a is the distance between the lenses, and T|r is the angle sub- tended at the centre of the objecJ;-glass by the distance of the point from the boundary of the field of view. 112. A plane mirror and a concave mirror are placed opposite one another on the same axis at a distance apart greater than the radius of the concave mirror: a person stand- ing with his back to the plane mirror, but close to at, observes the three brightest images of a candle he holds in his hand: he moves the candle forward, till it coincides with the nearest image, prove that the other .two images will coincide also at the same time. He then moves the candle still further forward a distance x, till it coincides with another image; prove that at this instant the first image will disappear, and if a be the dis- tance between the mirrors the radius of the concave mirror is x + a ± J{x -I- af — Sax MISCELLANEOUS PROBLEMS. 367 113. The focal lengtli of the object-glass of an astronomi- cal telescope is 40 inches, and the focal length of four convex lenses forming an erecting eye-piece are respectively f, ^, |, | inches, the intervals between the first and second, and be- tween the second and third being 1 inch and ^ inch respec- tively ; find the position of the eye-lens, and the magnifying power when the instrument is in adjustment. 114. A man standing on the seashore sees the light of a star reflected on the surface of the sea, rwhen it is covered with gentle ripples travelling in all directions ; find the equa- tion to the boundary of the bright patch on the water, con- sidering the undisturbed surface- of the sea to be a horizontal plane. Find the condition that this j)atch should reach to in- finity. If 2 be the zenith distance of the star , and tangents to the patch from the man's feet contain an angle 4^, shew that if the patch does not extend to infinity, the angle which it sub- tends at the man's eye in the vertical plane passing .through the star is 4 tan"' (sin 6 tan z). 115. A luminous .point Q moves round a closed curve surrounding the focus^of an ellipse, and is seen by reflexion at the ellipse by an eye at the other focus S. Prove that two images will always be seen, and that if q be an image seen by reflexion at the ellipse at P, SP.Pq _ HP.PQ 8q ~ HQ • Also, if H be the centre of the curve' traced out by Q, and a line through 8 meet the loci of the .two images in q^, q^, and SQ' be taken on this line a harmonic mean between Sq , Sq , Q' will trace out a fixed curve whose area is to that of the ellipse as 2 + 3e^ : 8, where e is the eccentricity of the ellipse. 116. A pencil of rays is incident directly on the plane surface of a medium, the other boundary of which is such 368 MISCELLANEOUS PROBLEMS. that the length of the path of every ray within the medium is c : prove that the distance of the geometrical focus of the pencil after emergence from the origin of the pencil is ac(/A-iy_ ii'a + c ■where /* is the refractive index, and a the distance of the origin from the first surface of the medium. 117. A ray of light passes through a medium whose index of refraction varies continuously ; prove that d f dx\ _ djji. ds \ ds) dx ' s being the length of the path of the ray to a point whose co- ordinates are xyz. If in air /[i — 1 varies as the density and if /u, at a certain 3400 place is , and if the height of the homogeneous atmo- sphere be five miles, prove that when the temperature is con- stant the effect of refraction on distant horizontal objects is to increase the earth's apparent radius as found from the dip from 4000 to 5230 miles: and that if the temperature over a frozen sea increase about 6°F. for every hundred feet of ascent, objects may be seen reflected in the sky. 118. The equation of the boundary of a comet or nebulous body being given, together with the density at any point of its interior, it is required to find the law of brightness of its apparent disc, supposing that each particle sends the same quantity of light to the eye of a distant spectator. Ex. A spherical body of radius a, the density at any distance r from the centre being A >J{a? — r^). Conversely. From the law of brightness of the apparent disc, and the density at any point and known distance of the body, find the equation of the boundary, — supposing it to be symmetrical with reference to a plane perpendicular to the direction of vision. MISCELLANEOUS PROBLEMS. 36,9 Ex. The density of the body being the same as in the foregoing example, the brightness at any apparent distance R from the centre of the disc varies as a' — m^R^. See Quart. Jour. Math. Vol. iii. p. 364. 119. In the phenomenon of Solar Halos how are the posi- tions and colours of the inner and outer circles surrounding the Sun, and the positions of the Parhelia, theoretically ex- ' plained ? Why are the Parhelia not exactly coincident with the inner circle ? 120. Within a reflecting circle on the same side of the centre are two parallel rays, one dividing the circumference into arcs which are as 3 : 1, — the other dividing it into arcs which are as 8 : 1 ; find the least value of n such that after each ray has suffered n reflexions, they may be again parallel. 121. Given the directions of three plane mirrors in space, construct a straight line such that if light from it be reflected by the three mirrors in succession, the third image shall be parallel to the straight line. Walton and Mackenzie, Solutions of the Cambridge Problems, 1854, p. 73. 122. Rays are incident parallel to the axis of a; on a re- flecting curve, and the equation to the catacaustic is y =f[x), the equation to the reflectiag curve will result from elimi- nating t from the two equations "" fit) i+v[i+{/'(or] Boole's Differential Equations, p. 257. 1 23. Supposing a very large number of hexagonal crystals of ice to descend continually in the atmosphere with their axes vertical and faces turned in all possible directions, — prove that the reflexions of the light of the sun or moon from the vertical faces will cause the spectator on the earth's surface to see a horizontal circle of white light of the same breadth and altitude as the luminary. What phenomenon may be ex- plained by this result ? p. 0. 24 370- MISCELLAJSTEOUS PROBLEMS. 124. When a large number of crystals of ice of the form of hexagonal prisms with plane ends perpendicular to their axes are suspended in the air, it is found that many of the simple crystals are. joined together at their ends or sides so that the axes are parallel. Hence shew that (i) those of the compound crystals which have the axes horizontal, and are situated opposite to the sun at the same altitude, send light to a spectator by two reflexions at vertical planes inclined at an angle of 90° to each other, whatever be the orientation of the planes ; (ii) those which have the axes vertical, and are situ- ated in an azimuth of 120° from the sun at the same altitude, send light to the spectator by reflexions at planes inclined to each other at an angle of 120°. What phenomena observed sometimes to accompany Solar Halos are explained by these results ? OAMBBIDGE : PRINTED BY 0. J. OLAT, M.A. & SON AT THE UNIVERSITY PKESS.