Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924083943872 In Compliance with current copyright law, Cornell University Library produced this replacement volume on paper that meets the ANSI Standard Z39.48-1992 to replace the irreparably deteriorated original. 1998 Cl^arnell UnioerBtty ffitbtara 3tt)ara, JStta fork FROM MATHEMATICS A TREATISE ON GEOMETRICAL OPTICS. EonHon: C. J. CLAY AND SONS, CAMBEIDGE UNIVEESITY PRESS WAREHOUSE, AVE MABIA LANE. ttatnliiiiBt: DEIGHTON, BELL, AND CO. Itijjig: F. A. BBOCKHAUS. A TEEATISE ON GEOMETRICAL OPTICS BY R. S. HEATH, M.A, D.Sc, FELLOW OF TRINITY COLLEGE, OAMBRIDOE, PKOFESSOR OF UATHEMATICS IN THE UASON COLLEGE, BIBMIKGHAM. CAMBRIDGE : AT THE UNIVERSITY PRESS. 1887 • lAll Sights rtterved.] CDambnDge : PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS. PKEFACE. In the following work an attempt has been made to give as complete an account of modern Optics, including the labours of Gauss, Listing, Maxwell, Helmholtz and Abb4 as could be com- pressed within the limits of a single volume ; at the same time much that is of interest and importance, both theoretical and practical, has been necessarily omitted. The subjects have been treated as far as possible in a natural order, beginning with the simplest and advancing to the most general and complex. Thus the reflexion and refraction of single rays of light are considered before the corresponding properties of pencils, and the complete approximate theory of lenses is given before the theory of caustics and aberrations, and these before the general theory of thin pencils. The detailed cbnsideration of heterogeneous media, which may be said to fall outside the province of optics, has been postponed to the last chapter. Gauss' theory of lenses has been worked out completely by elementary geometrical methods, so as to bring it within the reach of all students ; while, at the risk of some repetition, his own elegant analysis has been given in a separate chapter. Before treating the refraction of thin optical pencils after the manner of Maxwell, a short account of the general properties of all thin pencils which are not systems of normals, has been introduced. I append a list of the most important memoirs and treatises relating to this subject, most 62 VI PREFACE. of which I have been able to consult. The list has no pretence to completeness ; for numbers of memoirs have been so often incor- porated into text-books that it is not necessary to specify them in particular. I would mention Lloyd's "Treatise on Light and Vision," a most valuable though scarce work, as having been of special service ; the chapter on spherical aberration and the general description of the instruments have been derived mainly from this treatise. Cayley's Memoirs on Caustics, Maxwell's papers on Thin Pencils, Helmholtz' Physiologische Optik and Abba's papers on the microscope, have formed the basis of the sections dealing with these various subjects. Any notifications of inaccuracies or suggestions which may add to the usefulness of the work will be most gratefully received. R. S. HEATH. Mason College, Birminohaji. April, 1887. LIST OF MEMOIRS AND TREATISES. Refractimi of rays of Light. R. Rabau. Minimum aberration through a prism. Carl. Repositorium iv. 1868. M. Jenkins. Gfeometrical proof that the deviation of a refracted ray of light increases with the incidence. Messenger (2) li. 1873. P. M. Clark. Minimum aberration through a prism. Messenger iv. 1874. O. Airy. „ „ „ Messenger v. 1875. On systems of Lenses. Gauss. Dioptrische Untersuchungen. AbL Oottingen i. 1838 — 1843. Werke, Band v. A. Martin. Interpretation g^metrique et continuation de la theorie des lentilles de Gauss. Ann. de Chimie et de Physique (4) x. 1867. Listing. Ueber einige merkwiirdige Punkte in Linsen vmd Linsensystemen. Pogg. Ann. cxxix. 1868. V. V. Lang. Zur Dioptrik eines Systems centrirter Kugelflachen. Pogg. Ann. CL. 1873. J. LissAJons. Sur quelques constructions g^ometriques applicables aui miroirs et aux lentiUes. Comptes Rendus lxxix. 1874. A. Catlet. An elementary construction in optics. Messenger (2) vl 1877. H. ZracKEN-SoMMER. Ueber die Brechung eines Lichtstrahls durch ein Linsen-system. CreUe lxxxii. 1877. R. Most. Ueber ein dioptrisches Fundamental-gesetz. Pogg. Ann. SuppL vin. 1878. E, Pendlbbdrt. On equivalent lenses. Messenger vii. 1878. On the principal points of a system of lenses. Messenger IX. 1879—1880. C. Pendlebury. On lenses and systems of lenses, treated after the manner of Gauss. Cambridge, 1884. E. J. RouTH. Note on geometrical optics. Quart. Journ. Math. xxi. 1886. viii LIST OF MEMOIRS AND TREATISES. General Theorems and Caustics. \X. R. Hamilton. Systems of rays. Trans. Royal Irish Academy, vol. xv. 1828, XVI. 1830-31, xvir. 1837. H. HoLDlTCH. Caustic by reflexion at a circle. Qtiart. Journ. Math. I. 1857. The nth caustic hy reflexion at a circle. Ibid. II. 1858. Note on the incipient caustic. Ibid. iii. 1860. A. Catley. Memoir on caustics. Phil. Trans. 1857. Supplement to the memoir on caustics. Phil. Trans. 1867. Maxwell. On the cyclide. Quart. Journ. Math. vol. ix. 1868. O. RbTHiG. Der Malus'sche Satz und die Gleichungen der dadurch definirten Flachen. Crelle Lxxxiv. 1878. .J. Warren. Note on geometrical optics. Quart. Journ. Math. xn. 1873. G. F. Childe. Ray-surfaces of reflexion. Cape Town, 1857. Eay-surfaces of refraction. Quart. Journ. Math, xiii., xiv., 1875—76. Aberration. J. Herschel. Aberration of compound lenses and object-glasses. Phil. Trans. 1821. Airy. Spherical aberration of eye-pieces of telescopes. [1827]. Canib. Phil. Trans. III. 1830. Lord Ratleigh. Aberration of lenses and prisms. Phil. Mag. vol. ix., ser. 5, 1880. On the minimum aberration of a single lens for parallel rays. Proc. Camb. Phil. Soc. vol. IIL, part vill. 1880. Theory of thin pencils. KuMMER. Strahlensysteme. Crelle, vol. lvii. 1860. H. J. Sharpe. Systems of rays. Messenger, vol. iil 1874. Maxwell. On the focal lines of a refracted pencil. Lond. Math. Soc. vol. iv. 1871—73. On the application of Hamilton's characteristic function to the theory of any optical instrument symmetrical about an axis. Lond. Math. Soc. vol. VL 1874—75. On Hamilton's characteristic function for a narrow beam of light Lond. Math. Soc. vol. vl 1874—75. LIST OF MEMOIRS AND TREATISES. ix Dispersion and Achromatism. Robert Blair. Experiments and observations on the unequal refrangibility of light. Phil. Trans. Edin. 1791. Airy. On the principles and construction of achromatic eye-pieces of tele- scopes and on the achromatism of microscopes. [1824]. Camb. Phil. Trans. II. 1827. R. Leslie Ellis. On the achromatism of eye-pieces of telescopes and micro- scopes. Camb. Math. Joum. no. vi., vol. i., 1839. R. Pendlebury. On the condition of achromatism of a system of lenses. Messenger, vol. ix. 1879 — 1880. The Eye and optical instrmnents. Helmholtz. Physiologische Optik. Leipzig, 1867. Die theoretische Grenze fur die Leistungs-fahigkeit der Microscope. Pogg. Ann. Jubelband, 1874. Maxwell. On the general laws of optical instruments. Quart. Joum. Math. II. 1858. E. Hill. On a practical method of finding the magnifying power of a telescope. Messenger v. 1876. J. B. Listing. Ueber das Huyghens'sche Ocular. Pogg. Ann. clxii. 1871. J. J. Lister. On some properties in achromatic object-glasses applicable to the improvement of the microscope. PhU. Trans. 1830. Nageli u. Schwendener. Das Microscop. Second Edition, Leipzig, 1877. W. B. Carpenter. The microscope. Sixth Edition, 1881. The microscope. Enayd. Brit. vol. xvi. 1883. E. Abb^. On the microscope. Monthly Microscopical Journal, voL xiv. 1875. • Methods of improving the spherical correction. Journal of the Royal Micros. Soc. li., 1879. On the estimation of aperture. Ibid., vol. i. New Series, 1881. On the relation of aperture to power. Ibid., vol. ii., 1882. On improvements in the microscope with the aid of new kinds of optical glass. Ibid., part i., 1887. James T. Chance. On the optical apparatus used in Lighthouses. Proc. Inst, of Civil Engineers, vol. xxvi., 1867. Thomas Stevenson. Lighthouses. Encycl. Brit. vol. xiv. Ninth Edition, 1882. Young and Forbes. Experimental determination of the velocity of light. PhU. Trans., Part i., 1882. Bravais. On Halos. Joum. de FEcole Royale Poly technique, t. xviii. 1845. LIST OF MEMOIRS AND TREATISES. Treatises. Robert Smith. A compleat system of optics, to which are added remarks upon the whole. Cambridge, 1738. J. Herschel. Light. [1827]. Encycl. Metropolitana, vol. iv. 1845. Henry Coddixgton. Optics. Part i. Cambridge, 1829. Optics. Part II. Cambridge, 1830. HtriiPHREY Llotd. On Light and Vision. London, 1831. William N. Griffin. Optics. Second Edition. Cambridge, 1842. S. Parkinson. Optics. Fourth Edition. Cambridge, 1884. W. Steadman Aldis. Elementary Geometrical Optics. Second Edition. Cambridge, 1886. Lord Ratleigh. Optics. Evfiyd. Brit. vol. xvii. 1884. P. G. Tait. Light. ErMycl. Brit. vol. xiv. 1882. Light. 1884. Verdet. Cours de Physique, il 1869. Conferences de Physique, ii. 1872. CONTENTS. CHAPTER I. Nature and Properties of Light. ART. 1, 2. Nature and properties of light. 3. Law of Emission of Light. 4. Total quantity of light emitted. 5. 6. lUumiuation. 7. Objects appear equally bright at all distances. CHAPTER II. Reflexion and Refraction of Rats of Light. 8. Reflexion, refraction and scattering of light. 9. 10. Law of Reflexion.. 11. Analytical expression of the Law of Reflexion. 12. Incident and reflected rays make equal angles with any line in the plane. 13. Reflexion of projections of rays on a normal plane. 14. Successive reflexion of a ray at two mirrors. 15. Law of Refraction. 16. 17. Properties of the refractive index. 18. Critical angle. 19. Analytical expression of the Law of Refraction. 20. Projections of refracted rays on a normal plane. 21. 22. Deviation increases with the angle of incidence. 23, 24. A prism bends a ray of light from the edge of the prism. 25. Refraction by a prism in a principal plane. 26. Minimum deviation by a prism. 27. Prism of small angle. 28. 29. General refraction by a prism. xii CONTEXTS. CHAPTER III. Reflexion and Refraction of Direct Pencils. ART. 30. Reflexion at a plane .surface. 31. Successive reflexion at two parallel mirrors. 32. Successive reflexion at two inclined mirrors. 33. Reflexion of a finite bright body. 34. Direct refraction at a plane surface. 3.5. Refraction of rays from a finite body. 36. Reflexion and refraction at a spherical surface. 37. Reflexion of a symmetrical pencil at a spherical surface. 38. Principal focus of a mirror. 39. Reflexion of a small object. 40. Linear magnitude of the image. 41. Refraction of a direct pencil at a spherical surface. 42. Principal foci. 43. 44. Tracing the relative positions of conjugate foci. 45. Formula when any pair of conjugate foci are origins. 46. Image of a finite body. 47. 48. Geometrical constructions for conjugate foci and the emergent ray. 49. Linear magnitude of the image. 50. Helmholtz's formula. CHAPTER IV. Elementary Theory of Refraction through Lenses. 51. Classification of lenses. 52 — 56. Refraction by any thick lens. 57. Refraction by a thin lens. 58. Refraction by a sphere. 59. Positions of the cardinal points for different lenses. 60. Refraction by two lenses. 61 — 67. Refraction by any system of lenses. 68. Geometrical constructions for conjugate points. 69. Geometrical construction for the emergent ray. 70. Nodal points. 71. Another geometrical construction. 72. Case in which the initial and final media are the same. CONTENTS. XIU CHAPTER V. Refraction through Lenses. ART. 73. 74. Gauss's theory of refraotiou at spherical surfaces. 74. Principal points and planes. 75. Focal points and planes. 76. Listing's nodal iDoints. 77 — 79. Conjugate foci. 80, 81. Equivalent surfaces and lenses. 82. Case in which the cardinal points are at infinity. 83 — 85. Elementary theory of equivalent lenses. CHAPTER VL General Theorems. Caustics. 86. The reduced path of a ray of light is a minimum. 87. An orthotomic system of rays is always orthotomic. 88. Analjrtical proof of the preceding theorems. 89. Pi'operties of the characteristic function. 90. Surfaces which will reflect or refract to a point all rays issuing from a point. 91. Caustic curves and surfaces. 92. Character of a symmetrical pencil. 93. 94. Least circle of aberration. 95. Brightness of the least circle of aberration. 96 — 99. Caustic by reflexion at a circle. 100, 101. Geometrical investigations of caustics in two simple cases. 102, 103. Caustics after n reflexions at a circle. 104, 105. Secondary caustics. 106. Caustic by refraction at a plane. 107. Caustic by refraction at a circle. 108. Caustic by refraction at a circle when the incident rays are parallel. 109. Caustic by reflexion at an ellipse of rays issuing from the centre. 110. Length of the arc of a caustic. 111. Application of caustics to deduce the appearance of objects. 112 — 1 14. Curves of special illumination by reflexion at bright grooves. CHAPTER VIL Aberration of Direct Pencils. 115. Introduction. 116. Aberration by reflexion at a spherical surface. 117. Aberration by refraction at a plane. xiv CONTENTS. ART. 118 — 120. Aberration by refraction at a spherical surface. 121. Aberration in any lens. 122—128. Aberration in a thin lens. 129. Aberration in any system of thick lenses. 130. System of thin lenses. 1.31 — 133. System of two lenses. 134. Lateral aberration. CHAPTER VIII. On the General Form and Properties of a Thin Pencil. General Refraction of Thin Pencils. 135. Introduction to thin pencils. 136. Limiting points and principal planes. 137. 138. Focal points. 139. Focal lines. 140. Density of the rays. 141. Application of the theory to a system of normals. 142. Bounding surface of a thin pencil. 143. Circle of least confusion. 144. General section of a thin pencil. 145. Nature of an image. 146. 147. Elementary theory of thin pencils. 148. Circle of least confusion. 149. Elementary theory of oblique refraction of thin pencils at a plane. 150. Oblique refraction at a spherical surface. 151. Oblique reflexion at a spherical surface. 152. General form of the characteristic function for thin pencils. 153. 154. General refraction of thin pencils. 155. Application to a spherical surface. 156. Refraction of thin pencils by prisms. 157. Lens with cylindrical surfaces. 158. Oblique centrical refraction by a thin lens. 159. Approximate formulas. 160. Curvature of images. 161 — 163. Application of the characteristic function to the theory of any symmetrical optical instrument. 164 Characteristic function of a small pencil passing through a heterogeneous medium. CHAPTER IX. Dispersion and AcHROMArisM. 165, 166. Newton's experiment with prisms. 167. A pure spectrum. CONTENTS. XV ART. 168. Dark lines of the solar spectrum. 169. Different kinds of spectra. 170. Absorption spectra. 171. The measure of dispersive power. 172. Irrationality of dispersion. 173. Dispersion by a prism. 174. 175. Position of minimum dispersion. 176. Dispersion by two prisms with parallel edges. 177, 178. Dispersion by any system of prisms with parallel edges. 179. Achromatism. Secondary spectra. Experiments of Blair and Abbl 180. Condition for achromatism with two prisms. 181. Condition for achromatism in any system of prisms. 182. Chromatic difference of focal length in a thin lens. 183. The twofold nature of the correction for a system of lenses. 184. 185. Condition of achromatism of two lenses in contact. 186. ThQ same for any system of thin lenses in contact. 187 — 190. Achromatism of two lenses separated by an interval. 191. General theory of achromatism for any system of thick lenses. CHAPTER X. The Eye and Vision through Lenses. 192. The eye. The sclerotic and the cornea. 193. The choroid, iris, pupil, ciliary processes and the ciliary muscle. 194. The retina, yellow spot, fovea centralis and blind spot. 195. The crystalline lens. 196. The aqueous and vitreous humours. 197. 198. Measurements of the eye. 199, 200. Listing's numbers for the typical and reduced eyes. 201 — 203. Accommodation. 204. Periodic light. 205. Field of view. Theory of linear prospective. 206. Binocular vision. 207. 208. The horopter. 209. Impression of solidity. The Stereoscope. 210, 211. Theory of relief pictures. 212 — 215. Vision through a lens. 216—218. Spectacles. 219. Reading glasses. 220 — 224. Astigmatism. 225. Introduction to vision through any system of lenses. 226. Magnifying power of an instrument. 227. The eye-ring and eye-point. XVI CONTENTS. ART. 228. The field of view. 229. Case in which the eye-point falls within the instrument. 230. 231. Brightness of images. 232. Angle of divergence in a wide-angled aplanatic system. 233. Brightness when the pupil is not filled. CHAPTEE XI. Opticai, Instruments. 234. Simple microscope. 235. Coddington lens, Stanhope lens, Stanhoscope. 236. Doublets of WoUaston, Pritchard and Chevalier. 237. Sketch of theory of telescopes and microscopes. 238 — 240. The astronomical telescope. 241, 242. Galileo's telescope. 243, 244. On object-glasses. 245, 246. On eye-pieces. 247. Huyghens' eye-piece. 248. Ramsden's eye-piece. 249. The erecting eye-piece. 250. Magnifying power of a telescope with a compound eye-piece. 251. Field of view of a telescope with a compound eye-piece. 252. 253. Herschel's telescope. 254 — 266. Newton's telescope. 257 — 260. Gregory's telescope. 261. Cassegrain's telescope. 262. Aberration in Gregory's and Cassegrain's telescopes. 263 — 267. The compound microscope. 268. Magnifying power of the microscope. 269, 270. On the measure of the aperture of the microscope. 271. Recent improvements in the microscope. CHAPTER XII. Optical Instruments and Experiments. 272. The Camera obscura. 273. The Camera lucida. 274. Hadley's Sextant. 275. Fahrenheit's Heliostat. 276. Foucault's Heliostat. CONTENTS. XVU ART. 277. Silbermaun's Heliostat. 278—281. Lighthouses. 282 — 286. Determination of refractive indices. 287 — 290. Experimental determination of the cardinal points of an optical instrument. 291. Photometry. 292. Ritchie's Photometer. 293. Foucault's Photometer. 294. Rumfoi"d's and Bunsen's photometers. 295—297. Stellar photometry. 298. Methods of determining the velocity of light. 299. Fizeau's method. 300. Messrs. Young and Forbes' experiments. 301. 302. Foucault's experiments. CHAPTER XIII. Refraction through Media of VARrixQ Density. Meteorologicax Optics. 303. Introduction. 304 Equation of path of ray in a medium stratified in spherical surfaces. 305. Maxwell's Fish-eye problem. 306. Astronomical refraction. 307. Simpson's formula. 308. Bradley's formula. 309. Laplace's and Bessel's investigations. 310. Medium stratified in cylindrical surfaces. 311. 312. Mirage. 313. General heterogeneous continuous medium. 314—319. Theory of the rainbow. 320. Explanation of the primary rainbow. 321. Explanation of the secondary rainbow. 322. Rainbows of higher ordera. 323. Halos and similar phenomena. 324. Ice-crystals. 325. The halos of 22" and 46*'. 326. The parhelio circle. 327. The parhelia. The oblique arcs of LOwitz. 328. The parantheUa. 329. The anthelion. 330. The tangential arcs. 331. Figures of halos. CHAPTER I. The nature and general properties of Light. 1. Light may be defined as the external conditions which, acting through the instrumentality of the eyes, produce in the brain the sensation called sight. At various periods in the history of the science of Optics different and sometimes conflicting theories have been advanced to explain the nature of light. It is now usually supposed that light consists of vibrations of a highly elastic solid medium pervading all space, which has been pro- visionally called the aether. But, though opinions respecting the nature of light have been divided, there are a few of its leading properties which have been amply established by experience and are universally recognised as fundamental and dependent upon no hypothesis whatever ; and any theory on the nature of light has to furnish a satisfactory explanation of these properties before it can be accepted. The province of Geometrical Optics is to deduce by the methods of geometry the consequences of these general properties and thereby to explain the less obvious modifications which light undergoes, and to apply them to the construction of instruments for the improvement of our sight and the examination of objects too minute or too distant to be seen distinctly by the naked eye. 2. Any space through which light can pass, whether it be occupied by matter or not, is called a medium. In any homo- H. 1 2 PROPERTIES OF LIGHT. [CHAP. I. geneous medium light travels with uniform velocity in straight lines. Light consists of separable and independent parts. If part of the light proceeding from a luminous body be intercepted by an opaque obstacle, this will not in any way affect the remaining portion which is allowed to pass. Also, in general, light from two independent sources may travel along the same path without interference. These two experimental facts show that light is capable of a quantitative measurement. For the present we shall suppose that the light with which we are dealing is all of the same kind and homogeneous, and that its quantity or intensity is measured in terms of some fixed standard. When light travels through any medium of which we have cognisance, part of it is absorbed by the medium, and only part of it is transmitted. But in what follows, unless the contrary be stated, the media will be supposed to be perfectly transparent, that is, they will transmit the whole of the light incident upon them. It will often be convenient to consider the portion of light which travels along some particular line in the medium apart from the rest; such a portion of light is called a ray, and it will be supposed to have the form of an indefinitely slender cone, whose axis is the line in question. A collection of rays which during their course never deviate far from some fixed central ray is called a pencil of rays, and the fixed central ray is called the aosis of the pencil. If the rays of a pencil meet in a point, that point is called the focus of the pencil. As we shall have continually to mention the eye in the course of subsequent investigations, it may be well here to refer very briefly to the theory of the eye; the details of the theory cannot be given till later. The pencil of rays proceeding from a point and limited by the aperture of the pupil, is by the crystalline lens of the eye brought to a focus on the retina, and the point is seen by means of such an image on the retina. Each point of a surface gives a corresponding image, and thus we are enabled to form a mental picture of a surface. 3. Observation leads us to distinguish certain bodies, which may be called self-luminous, whose presence is necessary to excite our organs of sight. Bodies which in themselves are not luminous 2 — 3.] EMISSION OF LIGHT. 3 become luminous in the presence of a self-luminous body and are then visible to us. This distinction is immaterial for our present purpose however ; in treating of the emission of light from a body it -will not be necessary to consider whether the body is self- luminous or luminous through the presence of other bodies; the laws of emission are the same in both cases. Let dQ be the quantity of light emitted by a bright point or an indefinitely small element of a bright surface, within a small cone of solid angle dm, whose vertex is at the origin of light and whose axis is in a given direction, then the intensity of the emission of light in that direction may be measured by j^ . A bright body emits light in all directions, but the intensity of emission is different for different directions. The law of emission is given by a well-known experiment. Luminous bodies appear of the same brightness whatever be the inclination of the bright surface to the line of sight. Thus if a cylinder of silver be heated till it becomes luminous and taken into a dark room, it cannot be distinguished from a perfectly flat bar ; and similarly, a luminous sphere (like the sun as seen through a mist) appears like a flat disc. The same experiment is true of the intensity of the heat rays radiated from a hot body; in this form it is intimately associated with the Theory of Exchanges. This experiment shows that the intensity of emission of light from any element of a bright surface in any direction is proportional to the cosine of the inclination of the direction of emission to the normal to the element of the surface. For suppose that a bright body is viewed through a tube of small aperture ; when the tube is directed so that the element of the bright surface seen is normal to the line of sight, let the area of the element be w. Then when the tube is directed so that the normal to the element of the bright surface seen through the tube makes an angle 6 with the line of sight, the area of the element will be (0 sec 0. Letf{d) be the intensity of emission per unit area in a direction making an angle with the normal to the element ; then the whole amount of light transmitted to the eye when the element is inclined to the line of sight at an angle 6 is msec 6 . f{d). But this, by experiment is independent of 6, and therefore /((9) cccos e. 1—2 EMISSION OF LIGHT. [chap. I. 4. Let dS be the area of an element of the bright surface and let fj^dS denote the intensity of the light emitted in the direction of the normal to the element. Then fj, may be called the intrinsic brightness of the element. Let AB be the element, OZ the normal to it, and let OP be a direction making an angle 6 with the normal OZ, and such that the plane POZ makes an angle with a given fixed plane through OZ. Describe a small cone whose axis is OP, and let the solid angle of the cone be da. Then the quantity of light emitted within this small cone is fidS cos 6 dm, or /idS cos 6 sin 6 dO d<^. The whole quantity of light emitted by the element dS in all directions will therefore be fxdS 11 sin d cos dO d^, the limits of integration being ,^ = to <\> = 2ir\ 61 = to e = \-7r]' This gives on integration, fidS 27r . ^ or fiirdS. Hence if we denote the whole quantity of light emitted by the element per unit area by /u,', the intensity of emission per unit area in a direction making an angle with the normal will be f/JTr . cos 0, and therefore the intrinsic brightness is fj^'ltr. Similarly, if /i" denote the quantity of light emitted in all directions from a luminous point, the intensity of emission in any direction will be /i"/27r. 4 — 6.] ILLUMINATION. 5 5. If dQ be the quantity of light which falls on a small area dA of an illuminated surface surrounding a given point of the surface, then -j-j is called the intensity of illumination of the surface at that point. We shall now find the illumination of a small area dA due to an element of any bright surface dS. Let be the centre of the element of the luminous surface and G the centre of the illuminated area dA, and let 00 = r. Let 6 be the inclination of 00 to the normal at 0, and that of 00 to the normal at 0. Then if dA subtend a solid angle dm at 0, the quantity of light it receives will be fidS cos 6 da, where fi is the intrinsic brightness of the element. T, J , dA cos d> rsut dcD = = — -; r and therefore the quantity of light received by dA from the element dB is Td T A cos Q cos (f) /t dS dA j — -. This is symmetrical with regard to the two elements, and would therefore represent the quantity of light received by dS from the element dA, were it of intrinsic brightness /i. Let d(T be the solid angle subtended at by the bright element dB, so that J dB cos 6 then the illumination of dA due to the element dB is lxd(T cos ^. 6. The illumination of a small area dA due to any finite surface of uniform brightness may now be found. Take any small element of the bright surface about a centre 0, and as before let d] Ex. To find the illumination due to a spherical luminary. Let a be the semi-vertical angle of the cone whose vertex is at the centre of the area and which envelopes the bright sphere. The curve in which this cone cuts the sphere of imit radius is a circle whose radius is sin a. Hence if 5 be the zenith distance of the luminary, the illumination on a small horizontal area is I=lj.7ram^acos6. 7. Objects appear equally bright at all distances. The apparent brightness of an object may be measured by the whole quantity of light entering the eye from the object divided by the area of the picture of the object on the retina of the eye. Let P be any point of the object, p the corresponding point of the picture on the retina ; then it will be shown afterwards that 6 — 7.] ILLUMINATION. 7 the line Pp passes through the fixed point 0, the optical centre of the eye. Let *S be the area of a saiall object and s that of the picture and let OP = R,Op = r. Then S:R = s:r\ Now the quantity of light entering the eye is fiSro/E', where a is the area of the aperture of the eye. This may be written fj^ be the angle of incidence on B, , sin , ■H'bc sin = f • The preceding formula enables us to find the refractive index from glass to water. For fJ's«, = f-aa-fJ'aw that is, the refractive index from glass to water is f . Also, let /i, /ti' be the absolute refractive indices of the media A and £. Then if we denote the vacuum by the suffix v, fi^ = fi^. fi„,,. But fia^ is the reciprocal of fi^ or the reciprocal of fi, and therefore f^a» = f^ that is, the relative refractive index between any two media may he 17 — 18.] REFRACTION OF RAYS. 19 found by dividing the absolute refractive index of the second by that of the first. The law of refraction can now be more symmetrically ex- pressed in terms of the absolute refractive indices of the two media, fj, and /jl ; using the previous notation, the relation between the angles of incidence and refraction becomes fi sin

be given, ' the equation to determine <^' is, sin d) =—, sin d> . This value for sin ip' is always less than unity whatever the value of ' can always be found for any value of (f). Thus when a ray of light travelling in any medium is incident on a more highly refracting medium, the law of refraction always gives a direction for the refracted ray. But when the ray is passing from the medium B into the medium A which is less refractive, we may suppose ' given, and the equation to determine ^ is sin d) = - sin A' . If sin (J)' is greater than fi/fj,' the corresponding value for sin ^ becomes greater than unity ; so that the law of refraction fails to give a real direction for the refracted ray. The angle sin"' {fj^jfi), or, the greatest angle at which a ray of light proceeding in the more highly refractive medium can be incident on the other so as to be refracted into it, is called the critical angle between those media. When a ray of light is incident on a medium less refractive than the medium in which it is moving, at an angle greater than the critical angle, the whole of the light is found to be reflected ; the refracted part does not exist. This is known as total internal reflexion. 2—2 20 REFRACTION OF RAYS. [CHAP. II. 19. General formulae, giving the direction cosines of the refracted ray in terms of those of the incident ray and the normal to the refracting surface, may be constructed as in the case of reflexion. Let MQN be the normal to the refracting surface, PQR the path of the ray of light. Measure PQ, QR along the incident and refracted rays pro- portional to yu. and //.', the refractive indices of the media in which they are moving; and draw PM and RN perpendicular to the normaL Then since ^ sin ^ = fi sin ', the perpendicular PM is equal to the perpendicular RN. Now the projection of PQ on any line is equal to the projection of the bent line PMQ ; and the projection of QR is equal to that of the bent line QNR. But the projections of PM and NR are equal, since they are equal and parallel. Hence the difiference of the projections of PQ and QR is the same as the difference of the projections of MQ and QN. Let (p, q, r) be the direction cosines of the normal, (i, m, n) {I', m, n) the direction cosines of the incident and refracted rays respectively. Then since PQ, QR, MQ and QN are proportional to fi, y!, fj. cos ^ and fi! cos ^', respectively, if we take the difference of the projections on the three axes successively, we find the equations 19 — 20.] REFRACTION OF RAYS. 21 fd — III' = (/A COS (j) — fl COS ff>')p 1 fj,m — fi'm' — {jx cos <^ — /t' cos <^') q >. fin — fi'n' = (fi cos 4> — /i' cos (f>') r ) We may substitute in these equations the values of cos cj), cos (j)' in terms of the direction cosines, cos ^ = Ip -{■ mq + nr, cos = — <})', and the refraction becomes a reflexion. Substituting these values in the general equations for the direction of the refracted ray, the equations coincide with those already given for the corresponding problem relating to reflexion. All the subsequent theorems relating to refraction will give corresponding theorems for reflexion by making the same substitution /i = — fi. 20. There are two other useful theorems relating to the incident and refracted rays which may be proved from the preceding formulae, but the following geometrical proofs are simpler. The angles which the incident and refracted rays make with any plane through the normal to the refracting surface, obey the law of refraction. Also the projections of the incident and refracted rays on any plane through the normal are connected by a law of refraction, with a refractive index depending on the inclinations of the rays to the plane. For let AO, OB be any two refracted rays, and let the lengths oi AO, OB be taken equal to ^ and fi, the refracting indices of the two media, respectively. Then if AM, BN be drawn from A and B perpendicular to the normal to the refracting surface, AM, BN will be equal and parallel. 22 REFRACTION OF RAYS. [chap. II. Let PO, OQhe the projections of AO, OB on any plane through the normal, P and Q being the projections of the points A, B respectively. Then the triangles APM, BQN are equal in all respects. Let 7), J) be the acute angles which the incident and refracted rays make with the plane; ^, ^ the acute angles which the projections of these rays on the plane make with the normal. Then AP = fji,siarj, BQ = /j/ sin nj', and therefore, since AP is equal to 5(2, fi sin 1} = fi sin if. This proves the first theorem. Also OP = fi cos rj, OQ = fi cos 7]' ; and therefore, since PM is equal to QN, fi cos i; sin ^ = fi' cos r)' sin ^', which proves the second theorem. ^ 21. In any refraction, the greater the angle of incidence, the greater will be the angle of deviation. For if - '} = ^^^ tan ^ ((f> + (^'). But the deviation is equal to ^ - (f>'. If '. If we take logarithms and then differentiate, we see that d(f> d' tan ' ' and therefore defy is greater than d^' ; so that d (^ — ^') is positive. In other words the deviation increases with the angles of incidence and refraction. But further, the preceding equation may be written in the form d(l> _fji cos (f/ d(j} cos (f> ' and therefore I -7^ 1 = 5—, , \d \a(p I cos is represented by the arc J {Qq + Pp) ; and therefore, by subtraction, the increase in ' is represented by an arc ^ {Qq — Pp)- If we suppose ^, — rf)') . , . Ex. Snow that -: — j^ — ~-^ increases as increases. sin(<^ + 0') ^ 23. Any medium bounded by two plane faces meeting in an edge, is called a prism. The inclination of the faces to each other is called the refracting angle of the prism. At present we shall only consider the path of rays of light which pass through the prism in a plane perpendicular to both its faces, and therefore perpendicular to the edge of the prism ; we shall call such a plane a principal section of the prism. When a ray of light passes through a prism which is mare highly refractive than the surrounding m,edium, the deviation is, in all cases, from the refracting angle towards the thicker part of the prism. Let PQBS be the course of a ray of light through a prism in a principal section QOR. Draw the normals at Q and R meeting in L. There are three cases to be considered, according as the triangle OQR is acute angled, or contains a right angle or an obtuse angle. In the first case the rays PQ and RS lie on the sides of the 26 REFRACTION BY PRISMS. [chap. II. normals away from the vertex, and therefore the deviations both at ingress and egress will be away from the edge of the prism. In the second case let one of the angles of the triangle OQR be a right angle ; at that point of incidence there will be no deviation and at the other point of incidence the deviation is away from the vertex. In the last case, one of the angles, ORQ, is obtuse, the other o 23—24.] REFRACTION BY PRISMS. 27 angle, OQR, being acute. Then the ray SR lies on the side of the normal towards the vertex, so that the corresponding deviation is towards the vertex, while at Q the deviation is away from the vertex. But the angle of refraction at Q is greater than that at R, the former being the exterior angle of the triangle QRL and the latter an interior angle. Hence the deviation at Q is greater than that at R, so that on the whole the deviation is away from the vertex. If the prism be less highly refractive than the surrounding medium, all these effects are reversed. 24. This theorem may also be proved by comparing the action of a prism with that of a plate. When a ray of light passes through a plate bounded by two parallel faces, it emerges parallel to its original direction. Let PQBS be the path of a ray through such a plate bounded by the faces AB, CD. Let RN be the normal at the second face. Now suppose the second face turned about R towards AB, in such a way as to make a prism whose edge is perpendicular to the plane of the ray. Let RN' be the new position of the normal to the second face and RS' the emergent ray. Then in the figure, the angle of incidence at the second face is increased; hence the deAna- tion at the second face is increased. The ray is therefore deviated towards the thicker part of the prism. 28 REFRACTION BY PRISMS. [CHAP. II. Similarly, if the second face CD be turned in the opposite direction, the deviation at the second face will be diminished and the same result will follow. 25. Let PQRS be the path of a ray through a prism whose edge is at 0, and whose refracting angle is t. Draw the normals at Q and R, LQM and LRW respectively, meeting in L. Let ^, ^' be the angles of incidence and refraction at Q, and let ■^, yjr' be the angles of emergence and incidence at R, respectively. We shall consider 4> and i/r as positive when they are measured from the normal towards the thicker part of the prism, so that <^' and yjr' will be positive when they are measured from the normals towards the vertex. In the figure ^, <}>', yjr, yjr' are all positive. By the law of refraction we have sin = /i sin <^') sin i^ = /Lt sin i|r'j ^ '' Also, the angles at the base of the triangle OQR are respectively ^ir - ', and Jtt - yjr, hence I + ^TT — ' and ylr' can never be greater than 7. If, therefore, the refracting angle of the prism be greater than 27, no ray can pass through the prism. If I be greater than 7, 0' and yfr' must always both be positive. Let D be the whole deviation of the ray as it passes through the prism. Then at the first refraction the ray is deviated through an angle <}> — <^', and at the second refraction it is further deviated through an angle -^ — •>|r'. Therefore D = t' ' i> > u^' . ^y V > -•■ / <- 4 "^ "Zsi^ V '^<^- The whole theory of the path of a ray of light in a principal section through a prism is contained in the equations (1), (2) and (3). 26. The deviation is a minimum when the ray of light passes symmetrically through the prism. Let (^„ be the value of for this symmetrical path, and let ^ gradually increase from 0„. Then ^' and i/r' increase and decrease, respectively, by equal increments; hence, since <^' becomes greater than ■^', the deviation at the first face increases faster than that at the second face diminishes, so that on the whole the total deviation increases. The same result is easily seen to be true even after ■y^' becomes negative (if it does become negative before <^ reaches ^tt). Hence as increases from ^^ the deviation continually increases. If <^ diminishes from c^,, then ■y\r increases from i^^, and we have only to consider the reversed ray to see that the same result follows. Hence, when the ray of light passes symmetrically through the prism, the deviation is a unique minimum. The theorem may also be proved by means of the formulae of the preceding article. The equations (1) are sin ^ = /A sin ^'| sin i|r = /i sin i/r' 30 REFRACTION BY PRISMS. [CHAP. II. and therefore adding, sin (f) + sin yjt = fi (sin - 1^) = 2/i . sin ^ (' + yjr') cos ^ ((^' - yjr'), xi. J. • • , /r. \ • 1 COS i( 6' — yjr') that IS sin i (x' + 1) = ii sin A t . ^-^^ f-r-' . ^ COS^C^-'f) Suppose that and -^ are unequal, say <^ is greater than yjr. Then the deviation ^ — ^' is greater than the deviation yjr — yjr ; therefore ^ — i/r is greater than <^'—y^', and therefore cos ^ (^'— ■>/^') is greater than cos ^ (^ — •<|^)- Similarly cos ^ (<^' — ilr') is greater than cos ^{ — yjr) it yJr be greater than (p. Hence in all cases in which + i) is greater than /i sin 1 1. But when = yfr, D is a unique minimum. Ex. 1. Show that sm^ ^ (2) + 1) = sin2 1 1 + .^l ' — T^^~K^,- , ■* ^ ' ''I - sm^ 5 (<^ - V') seo^ i t and thence show that the deviation is a minimum when = i/'. Ex. 2. Prove that (ju^ - 1 ) sin^ t = 4 cos s cos (« - 1) cos (s-(f>) cos (« - 0')> where 2»=<^ + <^' + 4. 27. When the refracting angle of the prism is small, then the deviation will be small. In this case Hence, sin (t + D — ^) = /x. sin (t — ^'), or, since i and D are small, (i + B) cos — sin ^ = /it cos ^' — fi sin ^' ; therefore D cos <}) = i{/i cos ^' — cos ^}, or ^^ J/tCOSf _^) ( COS ' will both be small, so that to the third order of small quan- 26 — 29.] REFRACTION BY PRISMS. 31 titles, the value of the deviation becomes which, to this approximation, is independent of ike angle of inci- dence. 28. We shall next suppose that the ray does not lie in a principal plane of the prism. Let the same notation as before be applied to the projections of the path of the light on a principal plane. Also let 77, r)' be the inclinations of the incident and refracted rays to the principal plane at the first refraction, ^, f ' the inclinations of the refracted and incident rays to the same plane, at the second refraction, respectively. Then by § 20 sin t] = fj, sin rj'] sin ^ = fi sin f J ' Also, ^ and 17' denote the inclination of the same ray to the same plane, and therefore f = tj' and ^ = i}. This proves that the incident and emergent rays are equally inclined to the principal plane, or to the refracting edge of the prism. Further, there are the equations of refraction sin cos 7] = jM sin ^' cos jj' sin i/r cos t) = /J. sin ■^' cos 17'J ' and ' + -yj/ = i. These equations contain the whole theory of the refraction of a ray through a prism. 29. We now proceed to find the deviation produced by the prism. If D„ be the deviation of the projection of the rays on a principal plane, we shall have D, = + y}r-i. Let OAB be the principal plane, OA, OB the projections of the incident and emergent rays on this plane, OP, OQ these rays themselves ; and let all these lines be terminated on a sphere whose centre is 0. Then the arc AB represents D„, and the arc 32 KEFRACTION BY PKISMS. [chap. II. PQ represents the complete deviation. Also the area AP and BQ are each equal to r], so that PQ will bisect AB in N. Then from the right-angled triangle PAN, we deduce the equation cos ^ Z) = cos \ Z>o cos 7), which determines the total deviation. From this equation we see that D is always greater than D^. Now the minimum value of X)„ is found by the same method as before; the deviation will be a minimum as the prism is turned about its refracting edge, when show that if D^, B^ D^ their minimum deviations through it are in Arithmetical Progression, then sin ^jOj _ sin ^D^ + sin \D^ 5. Two prisms of the same vertical angle but of different refractive indices are placed in contact with their edges parallel and their angles turned opposite ways ; prove that the deviation due to the system of a ray which is incident perpendicularly on the first surface of the system increases with the angle of the prisms. 6. A prism, refractive index /i' and refracting angle 60°, is enclosed between two others of refractive indices ji and angle 60", their edges being turned the opposite way to that of the first. Show that if a ray passes through without deviation, its course must be symmetrical, and that 3fl2 = ^'2 + ^' + l. 7. If n equal and uniform prisms be placed on their ends with their edges outwards, symmetrically about a point on the 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. Show 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^ 11*2-2,1 cos- +1 : j[i+l. 8. A battery of n similar prisms is so arranged that a beam of light after traversing them at minimum deviation comes out in position to traverse them again ; show that the angle A of the prisms is given by .A . fiv ^A\ Taking glass, for which ,i = |, determine the least number of prisms with which the result can be accomplished, and draw a sketch of the necessary arrangement. H. 3 34 EXAMPLES. [chap. II. 9. A direct- vision spectroscope is composed of three prisms, two of which are exactly alike and are placed each with a face in contact with the faces of the third and their vertices turned towards its blunt end. Find equations for the angles of the prisms and their refractive indices in order that a ray refracted through the three prisms may be able to emerge parallel to its direction of incidence. If the refractive indices of the two similar prisms and the third be ^6 and V3 + 1, respectively, and the angle of the third prism be 120", show that the angle of the two like prisms is tan""^ (6 + 3^3). 10. If 6, (f) be the angles of incidence and emergence of two parallel rays passing through a prism in a plane perpendicular to the edge; rfj, d^ the distances between these rays before incidence and after emergence, show that J = - ^ , where d6 is any small change in 6, and d<^ the corresponding Orn QiU change in 0. 11. Three plane mirrors are placed so that their intersections are parallel to each other, and the section made by a plane perpendicular to their inter- sections is an acute-angled triangle ; a ray proceeding from a certain point of this plane after one reflexion at each mirror proceeds on its original course ; prove that the point must Lie on the perimeter of a certain triangle. Prove that the ray after another reflexion at each mirror will proceed on its original path, and that the whole length of its path between the first and third reflexions at any mirror is constant and equal to twice the perimeter of the triangle formed by joining the feet of the perpendiculars. 12. Any number (n) of right-angled prisms are placed with their edges turned alternately in opposite directions, and a face of each in contact with a face of the next one, and a ray passes through with a minimum deviation. If and ijf be the angles of incidence on the first prism and emergence from the last, show that if w be even, and if ji be odd. 13. The section of a prism made by a principal plane is a triangle ABC. A ray faUs on AB making an angle

l/e\ and show that none of the emergent rays will be parallel to the axis if /i< 1/e^. 20. The interior of an elliptic ring is a perfect reflector, and an origin of light is placed in the focus S. Show that, if P^ be the point of the ?ith reflection of any given ray, and if A be the vertex nearer to S, then tan i ^2„-i=sin5 sin {(2re - 1) a - <^} , sin J 2)2,,= sin 5 sin2« is the angle which a plane through the intersection of the mirrors parallel to the incident ray makes with the plane bisecting the mirrors. CHAP. II.] EXAMPLES. 37 28. A ray of light is reflected a number of times between two plane mirrors, not in a principal plane ; prove that all the rejfiected segments of the ray are generating lines of a hyperboloid of revolution. 29. The siu'face 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 point be taken as origin of coordinates and the vertical line as axis of z, and if the equations of the slit are x = a, z= —c, show that the equation to the sheet of light in the water is a^{{x^+f) {x-af+x^{z+cf} =y.'^{x-af{a'^{x^+y^) + (?x^}. 30. A ray falls on a prism whose refracting angle is ^tt at a point P on one face and makes an angle 6 (in any plane) with the perpendicular from P on the refracting edge. If the ray can get through without internal reflexion, show that 51 Qq^'"^ = 2nt, Qq''""*^^ = 2m + 29, and in the second series, Q^tin) = 2««= Q?(2„+i) = 2nt + 2(9'. The number of images is limited; for when any one of the images falls on the arc ab, between the mirrors produced, it lies behind both mirrors, and therefore no further reflexion takes place. If the image g''"' be the first to fall on the arc ab, then, since this is one of the images which lie behind the second mirror, we must have the arc Qg'^"' > QBa ; that is, 2nt >Tr — 6, or 2n > . I If the first image which falls on the arc ah be one of those behind the first mirror, say Qg''^"'^'', we must have Qg''""*'' > QAh ; that is, 2ni-\-26>Tr- &, or 2m-\-6 + ff>'n--6, or finally, 2ra + 1 > . This is the same result as before, 2n being the number of images in the first case, 2n+l in the second. Therefore the whole number of images in the first series is the integer next greater than (tt — d)/t ; and, in like manner, the number of images in the second series may be shown to be the integer next greater than If t be a submultiple of two right angles, tt/c will be a whole number, and the number of images in each series will be tt/i, since 6/i and d'/i, are proper fractions ; so that the total number of images will be 27r/i. But in this case it happens that two of the images of the different series coincide. For if tt/i be an even integer, say 2n, then Qf^ + QiM = 2wt + 2wt = 27r, and therefore the images q''"'\ q^^„i coincide. And if ir/i be an odd integer, say 2w + 1, Q?'""^" + G?fe„^.) = 4w + 2 (5 + ff) = (4n + 2) t = 27r, and the images q^''"*'\ ^(jn+i) coincide. If therefore we include the radiant point in the number, the total nwmher of foci is 2'ir/i. This theory contains the principle of the kaleidoscope. 42 REFRACTION AT PLANE SURFACES. [CHAP. III. 33. From the case of a single pencil we may now proceed to consider the way in which any object is seen by reflexion in a plane mirror. When any object is presented to a plane mirror, every point of the object is emitting rays of light ; when the rays from any point are reflected at the mirror they will proceed as if from a focus on the other side of the mirror, such that the two corresponding foci are on the same perpendicular to the mirror and at equal distances from it. To every point of the object will correspond one such focus, and the aggregate of these foci is called the image of the object. The image will be similar to the object and equal to it in every respect, since corresponding points of the image and object are similarly situated with respect to the mirror. But since the faces of the image and object are turned towards opposite directions, the position of the object with respect to right and left will be inverted in the image. If the eye be placed so as to receive reflected rays, they will produce the same impres- sion as if they were radiating from a real object behind the mirror in the position occupied by the image. We may trace the rays by which the eye sees any point of the object, by drawing a pencil of lines bounded by the pupil of the eye, towards the corresponding point of the image as far as the mirror, and then joining the points of the section of the small pencil by the mirror, to the point of the object. 34. The pencils we shall now consider will be very slightly divergent, or in other wordg, the solid angles of the pencils will be very small. When the axis of the pencil coincides with the normal to the surface on which it is incident, the incidence is said to be direct ; in other cases the incidence is oblique. In general, the rays of the pencil after refraction or reflexion do not accurately pass through a point ; but there are many useful cases, where the incidence is direct, in which- the rays very approximately meet in a point. We shall now consider a few of these cases. A small pencil of light is incident directly on a plane refracting surface; to find the form of the pencil after refraction. 33 — 34.] REFRACTION AT PLANE STIRFACES. 43 Let the pencil diverge from a point Q, the axis of the pencil, QA, being normal to the plane refracting surface. Let QRS be the path of any ray of light, and let RS produced backwards meet the K »» R ^^'^^ --. A Q 1 axis in q. Then the angle AQR is equal to the angle of incidence of the ray, and the angle AqR, equal to the angle of refraction. But if /i, fjf be the refractive indices of the two media, the law of refraction, expressed in the usual notation, is /i sin ^ = fi sin a. 4. A pencil issuing from a point is incident upon a convex spherical refracting surface of index p. ; show that the distance of the point from its conjugate focus will be a minimum, when the distance of the point from the surface is to the radius of the surface as 1 : 1 + iJp. 5. A ray of light, traversing a homogeneous medium is incident upon a spherical cavity within it ; supposing the limit of the magnitude of the deviation of the ray, produced by its passage through the cavity to be 6, show that the index of refraction of the medium is equal to sec ^6. 6. Rays converging to a point Q fall on a spherical surface whose centre is C ; if, after one refraction, more than three rays in any plane through QG pass through the same point Q' on the axis QC, then will all the rays pass through the same point Q. 7. Parallel rays fall on a sphere, and emerge after one internal reflexion ; show that rays which are reflected at the same point of the surface are parallel after emergence ; show also that, when the refractive index is greater than 2, no two rays will be reflected at the same point. 8. Find the geometrical focus after direct refraction through a hoUow spherical shell bounded by two concentric spherical surfaces and filled with fluid of refractive index difierent from that of the shell. 9. Two spherical surfaces A, B have the same centre ; P is the geometrical focus of rays from a luminous point Q after reflexion first at the surface A and then at the surface B, and R is the geometrical focus after reflexion first at B and then at A ; show that OP, OQ, OR are in harmonic progression. 58 EXAMPLES. [chap. III. 10. A hemisphere of glass has its spherical surface silvered ; light is incident from a luminous point Q, in the axis of figure produced, on the plane surface ; show that if q is the geometrical focus of the emergent pencil, A the centre of the hemisphere, its vertex and /x the refractive index for glass, 1 12^ Aq AQ~ OA' 11. A ball of glass contains a concentric spherical cavity; show that, provided the radius of the cavity do not exceed the radius of the ball divided by the index of refraction jl of the glass, it wiU appear to an eye at any distance from the ball to be /i times greater than it really is. 12. A sphere of a refracting substance whose index is \/3 has a concentric spherical nucleus which is a reflector, whose radius is such that a ray which just enters the sphere grazes the surface of the nucleus. Prove that, if a ray, which is incident at an angle 60", return to the point of incidence after inter- nal reflexions, the path within the medium will be ^ of what it would have been if there had been no nucleus. 13. Explain why, in looking down the axis of a smooth gun barrel with an eye close to one end, a series of dark rings, images of the other end of the barrel, are seen on the surface, at distances from the eye equal to J, \,\... of the length of the barrel. 14. Two equal concave mirrors of radius r are placed exactly opposite one another at a distance a, supposed greater than 2r, apart. Rays emanating from a point on the line joining their centres are reflected alternately at the mirrors. Show that after an infinite number of reflexions the conjugate foci are distant \ s/(fl^ — Sar) from the middle point of the line joining the centres of the mirrors. 15. A transparent silver sphere is silvered at the back ; prove that the distance between the images of a speck within it formed (1) by one direct refraction, (2) by one direct reflexion and one direct refraction is 2/iaK!(a — c)-^(a-^-c — |io) (fic + a—3a), where a is the radius of the sphere, and c the distance of the speck from the centre towards the silvered side. 16. Six circles are placed with their centres at the angular points of a regular hexagon. How must a ray PQ fall on one so as to fall symmetrically on all the circles in order, the index of refraction from air to one of the circles being ^3 ? 17. A pencil diverges from a point P and passes directly through a transparent sphere whose centre is 0. If Q„ be the focus when it is not reflected inside the sphere, Q„ the focus when the pencil has been reflected 2n times inside the sphere, show that 0§o, OQ^i 0Q2...0Q„ form a series in harmonical progression, and that _1 L-i CHAP. III.] EXAMPLES. 59 18. 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 inversely as (y^ + Aa'^Y, where y is the distance of the point from the axis and 4a the latus rectum of the generating parabola. 19. 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 ; show that the total illumination of the inclined and vertical faces are in the ratio of 2 V3 : 1. 20. A luminous sphere rests within a hemisphere of twice its radius, the rim of which is horizontal ; find the 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, show that the illumination will be diminished in the ratio V5 - 1 : ij5 + \. CHAPTER IV. Elementahy Theory of Refkaction through Lenses. 51. 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. In general, the surfaces of revolution are spherical or plane. If these surfaces do not meet, the lens is supposed to be bounded by a cylinder having the same axis, in addition to the surfaces of revolution. The distance between the bounding surfaces, measured along the axis, is called the thickness of the lens. The thickness will generally be small in comparison with the radii of curvature of the bounding surfaces. Lenses are classified according to their forms. A lens bounded by two convex surfaces is called a double-convex lens. A lens bounded by two concave surfaces is called a double concave lens. A lens of which one face is convex and the other concave is called convexo-concave or concavo-convex, according as the light first falls on the convex or concave surface, respectively. The terms plano-convex, convexo-plane, plano-concave and concavo-plane need no further explanation. 52. We shall now consider the refraction of light through a single double-convex lens, the radii of whose faces are r, r'. The following abbreviations wUl be found to be convenient : 51 — 52.] REFRACTION THROUGH A LENS. 61 and let the thickness of the lens he /xc, fi heing the refractive index of the substance of which the lens is made, when that of air is taken to be unity. There exist two points on the axis of the lens, which are most useful in the determination of the positions of conjugate foci, and corresponding incident and emergent rays. They are a pair of conjugate foci, such that any incident ray passing through one of them, will emerge in a parallel direction through the other. These points are called the nodal points, and also from another property which will be pointed out later, the principal points of the lens. We proceed to find the position and properties of the nodal points. Draw any two parallel radii OQ, O'Q' of the spherical surface, and join QQ' meeting the axis in C. Then from the similar triangles OGQ, O'GQ', OG : 0'G = r : r, and therefore C is a fixed point. Any ray of light which in its path through the substance of the lens passes through G will emerge parallel to its original direction, because the tangent planes at Q, Q' are parallel to each other, and the lens will act on such a ray like a plate with parallel faces. If therefore we take N, N' the conjugate foci of G with respect to the two surfaces, a ray of light diverging from N will after the first refraction pass through G, and therefore after the second refraction will pass through N' and wiU emerge parallel to the original direction ; in other words N, N' are the nodal points. The point G is called the centre of the lens. The position of the nodal points can now be determined. The distance between the centres of the spherical surfaces is easily seen to be given by the equation 00' = r + r' -^vc, 62 REFEACTION THROUGH A LENS. [CHAP. IV. r and therefore, OC = -——, (r + r' - fic) r + r ficr r + r Thus .10=^, = ^, r + r f+f and similarly r + r' f+f Let h be the distance of N from A, h' the distance of N' from A', both distances being measured from the surface into the lens. Then, since N and C are conjugate foci, fi 1 _/i— 1 AG~h~^r' that is, T =*^— /^ - -2 ■ ^ cf f From this we deduce the value of h, namely, Similarly, h' = f+f-o- 53. There will be two images of a given object, formed by re- fraction at the two surfaces in succession, and we shall use a symmetrical notation for their positions along the axis. Let X, x denote the distances of the object and its first image, in front of, and behind the surface A, respectively ; and let y, y' de- note the distances of the final image and the first image behind and in front of the second surface, respectively. By the theory of a single refraction at a spherical surface, we get the equations 1 ^ /* _ /tj .(1). XX r If — / y y r and x' + y' = fj,c If planes be drawn perpendicular to the axis of the system at the nodal points, these planes are planes of unit magnification; 52 — 54] EEFRACTION THROUGH A LENS. 63 that is, any object lying in the first plane, will have an image in the second plane, equal in all respects to the object. This theorem may also be enunciated in a slightly different manner; the line joining the points where the incident and emergent rays meet the first and second planes respectively is parallel to the axis of the system. The two planes are called the principal planes, and the points where they meet the axis (in this case coinciding with the nodal points), the principal points. To prove this theorem, let /8, /S„ yS' denote the linear magnitudes of the object and its images, respectively. Then k X X y y so that ^ = -^ P yoD But at the nodal points x'jy' = rjr, and therefore by the equa- tions (1), each of these ratios is equal to x/y. Hence /3 = y8'. 54. If we eliminate x', y from the equations (1), we get 1 1 / a' / 2/ that IS = ^^—.+ -=^,. ^-/ y-f By reduction, this equation becomes ^y (f+f - 0) -fy (/ - c) -fx if- c) = cff. By means of this equation the positions of the /ocaZ points may be found ; these are points such that rays diverging from them are made parallel by refraction through the lens ; in other words they are the points conjugate to the points at infinity, in both directions. If we make y indefinitely large, we get the first focal point, x = g, where ^ f+f-o' 64 REFRACTION THROUGH A LENS. [chap. IV. Similarly, the other focal point will be given by the equation y = cf', where The distance between the first focal point and the first princi- pal point is equal to that between the second principal point and the second focal point, and this distance is called the focal length of the lens. If we denote this focal length by ^, we must have ^^g-h = g' -h', which gives or 11 ^__c Introducing these values g, g, (j> into the equation (2), it becomes, on dividing by/+/' — c, or («'-9)(y-g')=gg'+c ^\ f+f-c +'[• and therefore by reduction, {<»-9) {y-9')='- Let the distances of a pair of conjugate points measured respectively in front of and behind the focal points, be denoted by M, «; the values of w, v are then connected by the simple formula uv = (p\ p R< r' 1 l"^\ H Q' Q p:^ S / -^ s p' 55. The position of the point P' conjugate to a given point P may now be determined by a geometrical construction. Let F, F' be the focal points, H, E' the principal points. If we can trace two rays emerging from P after refraction by the lens. 54 — 57.] REFRACTION THROUGH A THIN LENS. 65 these will meet in the required point P'. For one of these rays choose the ray PR parallel to the axis, meeting the first principal plane in R; then the corresponding emergent ray will pass through R', where RR' is drawn parallel to the axis to meet the second principal plane in R'. But PR and QH are two parallel incident rays, and therefore after refraction they will meet in the focal point F' ; hence R'F' is the emergent ray. For the second ray choose the ray PF, meeting the principal plane in 8; then the emergent ray will be parallel to the axis, through the point S', the projection of S on the second principal plane. This determines the position of P- 56. Let yS, /3' represent the linear magnitudes of the object PQ and its image P'Q' as constructed by this process, reckoned positive if above the axis, negative if below. Then, by similar triangles, PQ : QF= 8H : HF. But PQ = ^,QF= u, 8H = P'Q' = - ^', and HF=; so that the relation becomes j8' V similarly ^"~^' 57. Two special cases may be noticed. First, suppose that the thickness of the lens is very small compared with the radii of its faces ; such a lens will be called a thin lens. In this case the points A, A' and C coincide, and the nodal points also coincide with these points. The equations then become I f X w r y 1/ s and x' + y' = 0. The quantities x, y will disappear on addition, and we get 11/ iNfl , 1) 1 - + - = U-l)\- + -\ = - a; As before, we have two focal points, each at a distance from H. 5 66 REFRACTION THROUGH A SPHERE. [chap. IV. the lens. If the distances of a pair of conjugate points measured from these focal points be u, v, so that u = x — ' then uv = '. 58. Next, suppose that the lens consists of a perfect sphere. In this case, we shall measure all distances from the centre of the sphere. Let X, X be the distances of the object and its first image, in front of, and behind, the centre, respectively, and y, y' the distances of final and first image behind, and in front of, the centre. Then we have n 1 M— 1 XX r Hence y y r x' + y' = Q X y fir 1) 1 = ^,say. P F O F' P' Let 0F= OF' = 4>, so that F, F' are the focal points. Then if P, P' be a pair of conjugate points, and PF= u, P'F' = v, the same relation holds between w, v as before, namely, uv = <^'. 59. We shall here trace the positions of the cardinal points for different kinds of lenses. I. Double convex lens. This is the typical case we have already considered ; the radii r, r are both considered positive. 57 — 59.] CARDINAL POINTS OF LENSES. 67 We shall suppose that in each case light passes from left to right, and shall distinguish the surfaces by' the figures 1, 2. The distances of the principal points measured from the surfaces towards the substance of the lens, are respectively h = h' = r + r'-(jM-l)c' cr' r + r' — {/ji—l)c' The distance between the principal points measured from left to right will he fj,c — h — h', that is TTZT' (fJ'-l)c(r + r'- /jLc) HH = ^^ '-j-^ Tv-^— = a, say. r+r' -(iJL-l)c ■' Also the focal length is = We shall suppose that the thickness of the lens is less than r + r', that is r +r' is > fic. Hence h, h', a and are all positive, and the arrangement of the points is as shown in the accompanying figure. In the limiting case in which one of the radii becomes infinite, the lens becomes a plano-convex lens. For example, suppose that r is infinite; then h' = 0, so that one of the principal points lies in the curved surface. II. Double concave lens. In this case, r and r' are both negative, so that h, h', a are all positive and <^ negative. The arrangement of the cardinal points is shown in the figure. If the radius of curvature of one of the faces becomes infinite, the lens is a plano-concave lens, and one of the principal points lies in the curved surface. 5—2 68 CARDINAL POINTS OF LENSES. [CHAP. IV. m. Convexo-concave lens. We shall consider r positive and r' negative. The case in which r is negative and r' positive, may be derived from this by supposing the light to travel in the opposite direction, and the positions of the cardinal points will be the same in each case. For convenience of reference we shall write down again the values of h, h', a and ^, with the sign of r' changed. These are cr h = h' = r — r' — {(JL — I)c' — cr' r — r' — (jfi — l)c' p,^ (fJ.-l)c(r-r'-fic) r—r' — (fi—l)c ' ,_ —rr' 'P-(^-l){r-r'-(fL-l)c}- We must consider several cases separately. (1) Suppose that r is less than r'. Then h is negative, a is positive, and positive. The positions of the centres of curvature of the surfaces and of the cardinal points are shewn in the figure. -b— t' This lens will be thickest in the middle and thinner towards the edges. (2) Suppose that r is greater than /, but that the centre of the surface 1 is behind the centre of the surface 2. This implies that r > r + fic, ov r — r > fi.c, and a fortiori r — r'>(ji — \)c. The value of h will be positive, a will be positive, but ^ negative. F' 59 — 60.] SYSTEM OF TWO LENSES. 69 The positions of the centre of curvature and the cardinal points for this case are shown in the figure. The lens will be thinnest in the middle and will get thicker towards the edges. (3) Suppose that r is greater than r', but that the centre of the surface 1 is in front of the centre of the surface 2. Then r — r is (/i — 1) c. Then ^ is negative, a is negative and h positive. The cardinal points are represented in the figure. The lens in case (3), is thickest in the middle. Summing up these results, we see that those lenses which have positive focal length are thickest in the middle ; also lenses thinnest in the middle have negative focal lengths. But the converses of these statements are not true, for there is one form of lens which is thickest in the middle but yet has a negative focal length. 60. The case of a system of two lenses may also be investigated in the manner of | 52 et seq. p H H^ R K^ i^ ^ Let H, H' be the principal points, and / the principal focal length of the first lens ; also let K', K be the principal points and /' the principal focal length of the second lens. Let P be any bright point, P, its conjugate focus after refraction by one lens, Q the conjugate focus of P,, after refraction by the second lens. Let X, x' be the distances of P, P, from the principal points H, H" respectively, the distances being measured in the usual way, and also let y, y be the distances of Q, P, from the principal points K, K' respectively, and let H'K = c, then x +y' = c. 70 SYSTEM OF TWO LENSES. [CHAP. IV. Ill Also - + - = 7 X X f \- y^y' /'] Eliminating x , y' we get the equation ' 1 1+1 r / «> / y .(1). which becomes, after reduction, ^ (f+f - c) -fy(f - c) -fx if- c) = 0. To find the position of the focal points, we make x, y succes- sively infinite ; making y infinite, we get the position of the first focal point, x = g, where ._ f{f'-c) f+f-o- 9 Similarly, iiy = g' be the position of the second focal point, fif-c) 9 f+f Let /8, )S,, /8' denote the linear magnitudes of the images at P, P,, Q respectively, then ^ = _^ 1 X X y y' The principal points are points of unit magnification, and therefore to find them we must make )S = /S' ; the corresponding abscissae are such as to satisfy the equation xjy = afjy' ; and by equations (1), each member of this equation is equal to ///'. Hence, reverting to the equation x' + y' = c, we find the values of af, y" to be X = ■/+/' _ cf 'f+f'} 60 — 61.J SYSTEMS OF LENSES. 71 If h, h' be the values of x, y, corresponding to these values of x, y', we get, by equations (1), 1 ./ + /' 1 ^ of /' h = - of f+f- and similarly h! = — -^ — %-, . /+/ -0 These points are the principal points of the system. If ^1) '"2 ''n-i ^® *^® ** radii of the surfaces, and, for brevity, suppose that iji w' A- 1 q^ n-I ' '1 ' B-1 Also, let the thicknesses of the media, measured along the axis be /i^t^, fif^ /^„-,d- 72 SYSTEMS OF LENSES. [CHAP. IV. Finally, let the distance of the object from the first surface be denoted by fiv, the distance of the first image also measured from the first surface by fi^v^, the distance of the second image measured from the second surface by /i^v^, and so on, and the distance of the last image measured from the last surface, by fij)^. We shall find the relations between these quantities, beginning at the end and reckoning backwards. The distances of the last two images reckoned from the last surface are easily seen to be, respectively, /*„«„, and /i„_j (i)„_, - «„.,); and since these are conjugate focal distances with respect to the last surface 1 1 I. This equation may be written in the form v„., = C+ j-- n In exactly the same manner it may be proved that '"^^ and therefore 1 1 1_1 Continuing this process backwards, we arrive at the equation — — — 1 1 Also the distances fiv, fi^v^, being conjugate focal distances with reference to the first surface, are connected by the relation or - = A;„ + — , 61 — 63.] SYSTEMS OF LENSES. 73 and therefore, finally V " t^ + k, + t^+k,+ + k„_^ + v^- 62. Let g/h, k/l, be the last two convergents of the continued fraction SO that, by the properties of such fractions, gl — hk = l; then the value of V will be given by the equation I ^ v„k + g V vj + h' It will be convenient to represent the distances of the object and its final image from the first and final surfaces, respectively, by f , f ' ; then ^= f^v, ^' = /i\, where ft! is written instead of /x„ for the refractive index of the final medium. The relation between ^ and ^' is ? ^'l + filh- or i;f r + l^'g^ - /^W - 1^1^'^ = 0- 63. The focal planes of the system are the planes conjugate to the planes at infinity. To find the first focal plane, we must make ^' infinite, then the rays will be parallel in the final medium. The corresponding value of f is J, fd f = ^ = 7i.say. Similarly, if we make f infinite, so that the rays are parallel in the first medium, the value of ^' becomes r = -¥ = 7.,say. The relation between f, f ' may now be written 74 SYSTEMS OF LENSES. [chap. IV. or that is (r-7.)(r-7.)= fifi'h fifilg k k' ^-I'lt-igl-hk], (r-7.)(r-%)=- Let u, u denote the distances of the conjugate planes from the focal planes, the same convention of sign being observed as before ; then M = 7j - f, m' = f - 7^. Also let /= - /a/A;, /' = - //A;. Then the relation between the abscissae of conjugate points takes the form 64. Let a, a,, a^ be the successive inclinations to the axis of a ray as it moves onwards through the diiferent media ; and let h, \, 6, be the distances from the axis at which it meets the successive spherical surfaces. Also let /ci tan a = ;8, yit, tan a , = j8, . . . p - — a 1 A.\ -^ Pi % Q A Ai In the figure, suppose that QAA^ represents the axis of the system, QP the incident ray, QiPP^ the course of the ray after one refr'action, produced backwards to meet the axis in Q^. Then AQ=h cot a = 6/i//3 . This relation may be expressed in the form fijAQ = ^/b. In exactly the same manner it maybe shewn that ^JAQ=PJh. But tijAQ — fjiJAQ^ = — (jj, — fi^jr, since Q, Q^ are conjugate foci at refraction at the spherical surface ; and therefore b b~ ""' or ^^ = ^ + kJ}. Also, referring back to the figure, it is easy to see that 6, = 6 + /u,j«jtanaj; that is 6, = 6 + ^^yS,. 63 — 65.] SYSTEMS OF LENSES. 75 In exactly the same manner it may be proved that K = K + tA) and so on. 65. By these equations all the quantities yS,, 6,, /S^. K ^^^J be expressed in terms of b and /S ; their values become /8. = {K ihK + i)+K}b + (*A + 1) iS, &c. The coefficients of b and /8 in these equations are easily seen to be, respectively, the numerators and denominators of the successive convergents to the continued fraction J 1^ J. 1_ J_ «"^«,+ k^+ t,+ k^+ +/;„_,- Denoting these convergents hj pjqiipjq^ the equations may be written in the forms, b,=pJ>+sA there being n spherical silrfaces. We shall denote the last two convergents by g/h, k/l, respec- tively, remarking that the quantities g, h, k, I are connected by the relation gl — hk = l, by the theory of continued fractions. Also, instead of the final values i„_„ ^„, yu,^ we shall write b', /S', /i'; then the last two equations of the series become b'=^gb + h^\ ^'=kb + l^ }■ If we solve these equations, and express 6, jS in terms of b', /8', we find by virtue of the relation gl-hk = l, 6= Ib'-h^ P = lb'-hff\ = -kb' + gl3'l 76 SYSTEMS OF LENSES. [CHAP. IV. 66. We shall next find the relation between the linear dimensions of a point and its final image. Let r), i;,, 7; J... denote the linear magnitudes of the object and its successive images ; then by Helmholtz' theorem jjLT] tan a = ya,?;, tan a, = fi^r]^ tan a^... ; that is, where 17' denotes the linear magnitude of the final image. The value of ^' has already been obtained in the form ^' = kb + l0. Now it is easily seen that 6 = — f tan a = — ^/S//*, and therefore ^=W-i But fil/k— f = 7, — f = M. with the previous notation, and /= —fi/k ; with these abbreviations, the preceding equation becomes The relation between the linear magnitudes of the object and image is therefore V _ w V f and from this we deduce — = — --. V f 67. If we take u = —/and therefore u = — /', these equations give 71 = 71' ; this shows that the planes u = —f, u = — /' are planes of unit magnification ; in other words, any ray passing through the system meets these planes in two points such that the line joining them is parallel to the axis. They are called the principal planes, and the points where they meet the axis, the principal points of the system. Let H, H' be the principal points, Q, Q' any pair of conjugate foci. Let QH = x, Q'H'=x', the distances being measured according 1 1 i r r Q F H H' F' Q' to the same convention of sign as before. Then the equation uu =ff' is equivalent to {x —f) {x' —f) =ff', from which we 66—69.] SYSTEMS OF LENSES. 77 deduce the equation X X The lengths // are called the principal focal lengths of the system. 68. We can now give simple geometrical constructions for the focus conjugate to a given point and for an emergent ray when the incident ray is given. Let F, F' be the principal foci, H, H' the principal points of the system. P R r' ""^^-^ H H '''"^~~~--^ S S' P Let P be a given point, it is required to find its conjugate focus. If we can trace the course of any two rays from P, we shall be able to find P". Take PF as one ray ; let PF meet the principal plane HS in S. Draw S8' parallel to the axis to meet the other principal plane in S' ; then the emergent ray will pass through S'. Also the rays FH and FS, since "they diverge from a point on a focal plane, will emerge parallel to each other ; if there- fore we draw 8'P' parallel to the axis, it will be the emergent ray corresponding to PF, and will pass through the required point. For the other ray, take PR, parallel to the axis, meeting the first principal plane in R. Draw RR! parallel to the axis to meet the other principal plane in R'. Then R'F is the corresponding emergent ray; produce R'F to meet 8'F in P', then P' is the point required. 69. The emergent ray may be constructed as follows : Let QPR be the incident ray, meeting the first focal plane in P, and the first principal plane in R. Draw RR parallel to the axis to meet the second principal plane in R! ; the emergent ray will 78 SYSTEMS OF LENSES. [chap. IV. pass through R'. Again, draw a parallel incident ray from F, meeting the first principal plane in S. Draw (SfS'T parallel to the axis meeting the second piincipal plane and the second focal plane in S', T respectively; S'T is the emergent ray corresponding to FS. But PR and FS are parallel, and therefore after refraction they will converge to a point on the focal plane at F'. Hence R'T is the emergent ray required. 70. The best construction is effected by means of two new points, called nodal points. These points have their abscissae such that u = — /', u = -f. Let them be denoted by N, N' ; then i\^, N' are conjugate to each other. They also have the property that an incident ray which passes through N will emerge from N' in a parallel direction. This may be shown by constructing the emergent ray cor- responding to an incident ray PN passing through the point N. 8 S T ^^^ F H ^-'N -^ Kr,.-^N' F' P R Let the points R!, T be constructed as in § 69, then the emergent ray will be the line joining R' and T. But if N' be the second nodal point F'N' = FH and therefore the triangles TN'F, SFH are equal in all respects. Again, H'N' = HN, and therefore the triangles R'N'N', RNH are equal in all respects. And therefore since F8, PR are parallel, the lines N'T, N'R' are in the same straight hne. This shows that the emergent ray corresponding to PN passes through N, and is parallel to the incident ray. If the initial and final media are the same we have f=f', and therefore the nodal points coincide with the principal points. 71. Let PQ be any incident ray through P. Let N, N' be the nodal points. Let PQ meet the first focal plane in Q. Draw N'Q' 69—72.] EXAMPLES. 79 parallel to PQ, meeting the second focal plane in Q' ; and draw Q'P' parallel to QN. Then P'Q' is the emergent ray. Join PN and draw N'P parallel to it to meet the ray Q'P' in F; then P' is the point conjugate to P. For draw RN parallel to PQ, N'R' parallel to QP'. Then the rays PQ and RN are parallel, and therefore will meet on the second focal plane, after refraction. But IfQ' corresponds to the ray RN, and therefore the emergent ray passes through Q'. Again PQ and QN are rays diverging from a point on the first focal plane, and therefore they will emerge parallel to each other. But QN will emerge parallel to itself ; hence the emergent ray Q'P' is parallel to QN. Finally, the ray PN will emerge from N' in a parallel direction, and therefore P' is conjugate to P. 72. In all cases of refraction through lenses used in air, the initial and final media are the same, and therefore ii = ij!, f=f'= — f-jle, and the relation between the abscissae of conjugate points becomes uu =f. The nodal points also coincide with the principal points, and all the constructions depend in a simple manner on the positions of four planes and the points where they meet the axis, namely, the two focal planes and the two focal points, and the two principal planes and the two principal points. EXAMPLES. 1. If an eye be supposed to consist of a sphere of fluid (radius r, refrac- tive index J^), in which is placed at a distance §r from the centre a convex lens whose axis coincides with the diameter and whose focal length and refractive index in air are, respectively, Jr and f ; show that the distance from the centre of the sphere for clear vision is f-Jf r. 80 EXAMPLES. [chap. IV. 2. A and B are fixed points, A being a luminous point and B the nearest point of a glass sphere with refractive index f». C, a point on BA produced, is the image of .4 as seen by an eye on AB produced beyond the sphere. Show n o that ACis least when the radius of the sphere is -jj- AB. 3. 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 /, is given by L — 1 _ ? 2/j - ft' - 1 m P /* 4. If a ray passes through a lena without deviation and if its directions before incidence and after emergence cut the axis of the lens in two points q, g', the limiting value of yg^ is a maximum or a minimum according as the thickness is equal to , _ , . where r, s are the radii of the surfaces of the V/i + l lens whose refractive index is /*, s being greater than r. 5. 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 / be the illumination of the screen where it cuts the axis and M F is what would be the illumination if the lens were removed, show that ■=; = , / ^ , , ' / (fu + uk + k {fu + uk + kff 6. If m be the linear magnifying power of a thin lens for an object at a distance u from the first surface, show that when the thickness t is taken into account, the magnifying power is increased by A* in which t^ is neglected ; r and s being the radii of the two surfaces of the lens, and u the distance of the object from the first surface. 7. One side of a plate of glass is accurately plane but the other (the front) is slightly curved, forming a sphere of large radius r. Show that if a pencil is refracted through it, its focus wiU be displaced through a distance \t-\ \ , where t is the thickness of the plate, and v, the distance of the focus of the incident pencil from the front surface. 8. Two thin lenses of equal munerical focal length / are placed on the same axis at a distance a apart, the one nearest the origin of light being concave and the other convex ; show that the least distance between an object and its final image is a+APfa. CHAP. IV.] EXAMPLES. 81 9. Two thin lenses of equal focal lengths, /, are placed on the same axis at a distance a apart. P, Q are conjugate foci of any pencil refracted directly through the lenses. Show that there exist two points (0) on the axis for either of which (a and being constants) ^td + tSi ^ constant, provided f -^ ia and > - Ja ; and that if / = Ja then -w-fi ~ 7^ 's copstant . 10. A thin lens has one face silvered so as to form a mirror. If Q be the image of a point P, formed by the mirror (by two refractions and one re- flexion), show that Q will be the same as if the lens were replaced by a spherical mirror whose radius R is given by the equation R s r ' r and s being the radii of the surfaces of the lens. 11. A spherical glass shell, whose outer radius is a and centre 0, has its spherical cavity filled with mercury, the cavity being just large enough to prevent light from being refracted through the sphere without reflexion at the surface of the mercury. A source of light is placed at a distance c from the centre ; prove that an eye, placed beyond on the line joining the source of light to 0, must, in order to receive any light, be at a distance from whose reciprocal is not greater than IX being the refractive index of the glass. 12. A system of 2re thin convex lenses of equal numerical focal length, /, are placed with their axes in the same straight line, and their centres at a distance 4/ apart, except the two middle ones, which are at a distance 8/ apart. Show that the focal length of a lens which must be placed midway between the two middle ones in order that the image of a bright point at a distance 4/ in front of the first lens may be formed at an equal distance behind the last lens is } , ^ /. 2m + 1 "^ 13. A luminous point P is placed in front of a vmiform sphere of glass (/i = 9), silvered at the back ; the distance of P from the sphere and the radius of the sphere are each one foot. Prove that, if the fraction X of the light which penetrates the sphere at or near perpendicular incidence, emerge again after reflexion at the back, the total illumination on a small screen, placed on the line from the centre of the sphere to P at a distance one foot from P, will be increased by this light beyond its value when the sphere is absent in the ratio l+Tj^X to 1. 14. Show that the image of an arc of a conic whose focus is at one principal point of a thick lens, is an arc of a conic whose focus is at the other. H. 6 82 EXAMPLES. [chap. IV. 15. A double convex lens is formed by two equal paraboloidal surfaces cut ofiF by planes through the focus perpendicular to the axis. Prove that for rays passing in the neighbourhood of the axis, the focal lengtl^picasured from the posterior surface of the lens is 2a/(fi^ - 1), and the distance between a bright point and its image is a minimum when it is 2a (/j. + l)/(/i - 1), 4a being the latus rectum of either of the generating parabolas, %nd /i the refractive index of the glass. 16. The two surfaces of a lens are formed by con^ntric «ind coaxal eUipse and hyperbola, of respective eccentricities e, e', which toath one another. Show that a small pencil of light incident directly, and diverging from, one focus of the ellipse, will converge to a focus of the hyperbola, if the refractive gg' i index of the lens be -; — . e -e 17. If m, m', m" be the magnifying powers of a combination of any nimiber of lenses on the same axis for objects at distances u, u', u" from the first lens, show that u'-u" u'' — u u — u' „ + ;-+ ^=0. m m m 18. If j; be the distance between two objects and a:' the distance between the corresponding images due to any system of lenses, and if m be the magnifi- cation of the first image and n that of the second, show that x' tl' — = — mn, X ft, where /x and /x' are the refracting indices of the initial and final media. 19. If (p, p'), (q, q^, (r, r*), be three pairs of conjugate points in any lens system, prove that I P^t P'' P _ pq • qr .pr .p'q' . q'r' .pV \p'^,py\~ ff where pq denotes the distance between the points ;;, q, with similar meanings for the other quantities, and/,/ denote the focal lengths of the system. CHAPTER V. Refraction through Lenses (continued). 73. The Theory of Refrax^tion through any number of media, bounded by spherical surfaces arranged symmetrically along an axis, was first successfully developed by Gauss ; we proceed to give an account of his method. We shall take the axis of x along the axis of the system. Let the abscissae of the vertices of the bounding spherical surfaces be a, a,, ctj , and their radii r, r^, r^ Let fi, fi^, fi^ be the refractive indices of the media, and for brevity, let [fi — fi^)/r = k„, (jJi't — f'^/r^ = k^ and let the thicknesses of the media reckoned along the axis, be fi^t^, fi,/, , so that a^-a = fijt^, a^ — a^ = /ij,,, &c. Let the equations of any incident ray making a small angle with the axis of the system be y ^^c.. M ■a)+b r z=- {w-a) + c I /* J and after refraction at the first surface let the equations to the ray be « = £• (x- a) + b' z = 'h(a;-a) + c' 6—2 84 gauss' theory of lknses. [chap. v. If X be the abscissas of the point of incidence P, and 6 the angle subtended at the centre of the spherical surface by the arc AP, measured from A, the vertex of the surface, then a; = a + r (1 — cos 6), and therefore, since both the incident and refracted rays pass through this point, 6+^r(l-cos0) = 6'+^r(l-cos5). Now y3, /8, are both small, and so is ^ ; we shall neglect the small quantities of the third order 6'^, and therefore we get b' = b. Similarly, c' = c. Thus b, c are co-ordinates of the point of incidence of the ray on the first surface. The direction cosines of the normal are (cos 0, — b/r, — c/r), and those of the incident ray (1, yS/y^t, y/fi.), and those of the refracted ray (1, /8,/^,, yjfi,), nearly. Expressing the relation between these direction cosines, by § 19, we get ^ - A = - - (m cos ^ - /4. cos .) = - ^^^^' J, to our approximation. The values of yS^, 7, may now be written 7,= 7 + V )' and the equations of the ray after its first refraction are completely determined. Let (6,, c,) be the co-ordinates of the point where the refracted ray meets the second surface, so that the equations to the refracted ray may be written in the form y = ^(x-a;)+b, ■2 = — (« - a,) -I- c, then if we compare these equations with the other forms of the same equations previously given, we find b-&a=b,-^a„ 73.] gauss' theory of lenses. 85 and therefore The same reasoning may be extended further ; the resulting equations will have the same forms as before, namely, 7» = 7i + ^iCi I ' and h = K+^A and so on. All the successive quantities Pfi^...hp^... may be expressed in terms of the first two quantities yS and h ; thus y3,=A„6+A ■'■f Pjiv PJi'i ^® t^® successive convergents to the continued fraction , J_ J 1 1_ J_ it is easy to see that the preceding equations are equivalent to the following: h=pp + qS, ^n=p^.^b+q,„-,^. We shall write the final pair of equations in the form ■ ^=kb + lfiy Then by the properties of convergents, gl—Kk = 1. By exactly similar reasoning it may be shown that c' =gc + hrf' y =kc + ly )' 86 gauss' theory of lenses. [chap. v. Solving these equations, we get, by virtue of the relation gl — hk = l, b= Ib'-h^') c= Ic — hy ] and , , , , f . The equations to the incident ray are now and those of the einergent ray ■a') + b' where '^,{z-a')+c' b' = gb + hm ^' = kb + ip]' c' = gc + hr/\ y' = kc + lyl' In the subsequent parts of the theory, it will be sufficient to confine our attention to one of the equations to the incident ray ; for the same reasoning will apply to the other equation in all cases. Similarly, we shall only use one equation to the einergent ray, 74. There are two planes perpendicular to the axis which possess the property that any incident ray meets the first, and the corresponding emergent ray meets the second, in points such that the line joining them is parallel to the axis. These planes are exceed- ingly useful in constructing the emergent ray corresponding to a given incident ray. We shall now prove the existence of su9h' planes and find their positions. The equation to the emergent ray may be written in the form kb + W, y- -Qc-a')+gb + h^, or y = ^.[l (x - a') + t^h] + b\^-^^^ +.^1 . /* M 73 — 75.] gauss' theory of lenses. 87 To find where the emergent ray meets the plane a; = a/, we have only to write x for x in this equation. Also the equation to the corresponding incident ray is y = - {x-a) + h. By properly choosing x and x, we may make these values of y coincide for all values of /S and h, that is, for all rays. Equating the coefiScients of /3 and h in the two equations, the necessary conditions are found to be I (x — a') + fi'h _ « — a 1 /i' ~ fj. \. k {x — a) + fig = fi j From the second equation we derive at once the value of x', r «' = «' + f (l-5') = P„say. Then, from the first and therefore x — a Z., . , l—\ * = «-^(l-0=i'.>say. These two planes are called the principal planes and the points where they cut the axis principal points. They have the property that the incident ray and the emergent ray cut the two planes, respectively, in points which are the projections of each other on these planes. 75. When the incident rays are parallel, the emergent rays will meet in a point ; and for all different systems of parallel rays, the corresponding foci lie in a plane perpendicular to the axis. Similarly, if the emergent rays are parallel, the incident rays must proceed from some point in a fixed plane perpendicular to the axis. These planes are called focal planes, and the points in which they cut the axis are called \he focal points. To find the position of the focal plane corresponding to incident parallel rays, we may suppose that /S is fixed while 6 is a variable 88 gauss' theory of lenses. [chap. v. parameter. The equation to the emergent ray being and b being a variable parameter, it is clear that the ray always passes through the fixed point determined by the equations y = ^\l(x-a') + fi'hl. The equation to the focal plane is therefore To find the equation to the other focal plane, we must make the emergent rays parallel, and therefore yS' constant and b' variable. By expressing the equation to the incident ray in terms of yS' and b', it may be shown in the same manner as before that the equation of the other focal plane is x = a + '^ = g„sa.y. The distance of the first focal plane in front of the first princi- pal plane is called the first focal length of the system ; and the distance of the second focal plane behind the second principal plane is called the second focal length. We shall denote these focal lengths by /and /'. Their values may be deduced at once from the abscissae of the focal planes and principal planes previously given. Thus Pi-9i = -^=f 76. There are two other points along the axes which have useful properties connected with the incident and refracted rays. They are such that if the incident ray pass through the first, the emergent ray will proceed in a parallel direction through the second. These points were discovered by Listing, who gave them the name of nodal points. 75 — 76.] gauss' theory of lenses. 89 To find the positions of these nodal points, let us enquire in what cases the emergent ray is parallel to the incident ray. The necessary condition is ^ = ^. This relation may be expressed in terms of h and /3, and becomes 1^ or 6 = ^f^_/in If this value be substituted in the equation of the incident ray, it takes the form From this equation it appears that whatever the direction of the incident ray, it passes through the point on the axis, for which it' III a; = a-^ + ^ = n„say. In the same manner it may be shown that the equation to the emergent ray takes the form ^=ff-«'-f+f and therefore the emergent ray passes through the point on the axis, for which The coordinates n^, n^ of the nodal points are therefore determined. From their values it is easily seen that The nodal points are therefore within the focal points, at distances from them equal to the focal lengths/' and/ respectively. In the very important case in which the initial and final media are the same, fi^fi and therefore f—f', and the nodal points coincide with the principal points. Other methods of finding the positions of these cardinal points will be given later. 90 gauss' theory of lenses. [chap. v. 77. When a system of incident rays all proceed from one point, the emergent rays also all ^jows through a point. These points are called conjugate foci ; also, one of them is sometimes said to be tlie image of the other. For we have seen how the incident and emergent rays depend entirely upon the values of ^, b, y, c. The conditions that the incident ray may pass through a point (^, rj, ^) are and a similar equation in (|, f ). Also the conditions that the emergent ray may pass through the point (^', -rj', ^) are and a similar equation in ^', f. It is possible so to choose (F. ■>?') f) ^ t*^ make the second conditions merely repetitions of the first. This will necessitate the equations If these conditions are satisfied, the points (^, j;, f), (f', -q', f) are so related that any incident ray passing through the former, will after refraction pass through the other point ; in other words, the two points are conjugate foci. From the equation we infer that a point and its image lie in the same plane through the axis. is 78. The relation between the abscissae of the conjugate foci fc (I _ «) (^' _ a') + /^r (f - a) - /.; (^ - a') - /t/A = 0. From this equation the positions of the focal points can be deduced, and by means of these points the relation can be much simplified. To find the focal points, we make first, the incident rays, and secondly, the emergent rays, parallel; this will be done by making successively ^ and f infinite. The 77 — 79.] gauss' theory of lenses. 91 abscissae of the corresponding images will be The relation between f, f ' may now be written (f - a) (r - a') - (fl'. - «^') (? - a) - (5^. - (») (r - «') = ^' , which may be expressed in the form U-o)-{9.-o)m'-o:)-{3,-a')]={g,-a){g,-a) + f^ (^-9d(^'-ff.) = -ff- If we denote the distances of the conjugate foci, respectively, in front of, and behind, the principal foci, by u, u', then u=g^ — f , u = ^ — g^, and therefore 79. Returning to the other coordinates of the conjugate foci, we find -1 = ^ = —5^2 f — !-2 ~ / '^ that is, ~ = "s = — iv • V K f Inverting, and remembering the relation between u and u, this equation may be written v r /■ From these equations it follows that if u = —f, and therefore u' = — /', then 11 = 1], ? = ?'• Thus, according to our previous definition, the planes m = — /, «' = — /' are the principal planes. If the distances of any pair of conjugate foci measured, respectively, in front of and behind the principal planes be denoted by X, x, then ■u, = x —f, u' = x —f, and the relation between x, x' is 92 gauss' theory of lenses. [chap. v. or, as it may be written, X X For the application of these various results to the geometrical construction of the corresponding incident and emergent rays, and of conjugate foci, we refer back to the previous elementary theory. 80. A single surface or a thin lens can always be found, which when placed with its vertex at the first principal point will refract the incident rays into exactly the same directions as the whole system of surfaces, so that if we imagine the interval between the two principal planes to be annihilated, this single refracting surface or thin lens, will give the complete emergent system. Such a surface or lens is said to be equivalent to the whole system of refracting surfaces. The system will be equivalent to a single refracting surface when the initial and final media are different; when the initial and final media are the same, it is equivalent to a thin lens. To prove these propositions, we refer the equations to the rays, to the principal points as origins. The equation to the incident ray may be written or finally, and the corresponding emergent ray has for its equation The value of y corresponding to the point where the rays meet the principal planes is y=— j^ = 6..say; then ^' = + kb, «• 79^-81.J gauss' theoky of lenses. 93 But this is the equation we should have obtained by a single refraction at a spherical surface, as ma}-^ be seen by com- paring it with the equations previously obtained. If fi, fj! be the refractive indices of the media, and r the radius of the surface, the value of k is k = r and therefore r = , - . k This value of r is measured from the vertex in the direction of the incident light. The system is therefore equivalent to a single refracting spherical surface whose radius is (jj, — /j^')/k and whose vertex is at the first principal point. 81. When the initial and final media are the same, it is necessary to suppose a thin lens to be placed with its vertex at the principal point The equations corresponding to the refraction through a thin lens are and therefore fi' = ^+{k, + kj b,. This will produce the necessary refraction, provided that k = k^ + k^. If fjb be the refractive index of the substance of the lens, r, r the radii of its two surfaces measured in the same direction as before. = {ix.-ii) r' 11 r r Let (^ be the focal length of the thin lens, defined in the usual manner, and expressed in terms of the refiractive index of the initial medium, then r(f')H)^ 94 hence or finally GAUSS THEORY OF LENSES. k=->^ [chap. V, 4> The focal length of the equivalent lens is therefore — fi/k ; it will be a collective lens if — fi/k be positive, and a dispersive lens if — fi/k be negative. 82. There is one case in which we cannot make use of the subsidiary points, and, as it occurs frequently in practice, it must be noticed. This is the case in which k vanishes ; for then the subsidiary points are all at infinity. When k = 0, the equations previously given reduce to gl=l, ^' = W, b'=gb + h^. . If the equations to the incident ray be y = - (x — a) + b z = - (x — a)+c those of the corresponding emergent ray will be y = ^{x-a')+gb + hJ3 z = ^ (x — a) + gc + hy If we put a = a' — ^-j~ , these last may be written z = ~{x-a) + gc We proceed as before to find the image of a point (f, 17, 5) ; the relations between (^, »;, f) and the coordinates of the image (f ', ij', J") are easily seen to be f^'v ^ ^T^ Uf -«) = /i o- fir} fi^ f-a fi^' 81 — 83.] EQUIVALENT LENSES. 95 these equations become on reduction and ftl (f -a) = fju'g {^ - a). Thus it follows that the linear (liinensions of the image are to those of the object in the constant ratio g : I or 1 : I, wherever the object be placed. The case now considered will present itself with a telescope arranged for seeing very distant objects by a long-sighted person. From the formulae, it is clear that a set of rays which are originally parallel will emerge parallel to each other. In the case of a telescope we shall have fj, = fi, and therefore by the previous equations, the tangent of the inclination of the initial ray to the axis is to that of the emergent ray in the ratio 1 : I. Hence, according to the usual definition, I or 1/g is the magnifying power of the telescope. If I be positive, the image is erect; if I be negative, it is inverted. Elementary Theory of Equivalent Lenses. 83. A lens is said to be equivalent to any number of lenses arranged at intervals along an axis when, if placed in a proper position, it will produce the same deviation in rays inclined at small angles to the axis of the system, as would be produced by the system of lenses. We shall first suppose the incident rays to be parallel to the axis of the system, so that the position of the equivalent lens is immaterial. The deviation produced by a thin lens may be found by supposing the lens to act like a thin prism formed by the tangent planes to the spherical surfaces at the points of incidence and emergence of the ray. The deviation will therefore be indepen- dent of the angle of incidence, for all small angles of incidence. To find the deviation, we suppose the incident ray to be parallel to the axis, and then the emergent ray will proceed to the principal focus of the lens. If y be the distance from the axis at which the ray strikes the lens, and f the focal length of the lens, the deviation is clearly d = — y/f, the lens being supposed collective. 96 EQUIVALENT LENSES. [CHAP. V. This expression will therefore represent the deviation caused by the lens iu any incident ray. Now suppose that there are n thin lenses whose focal lengths are, respectively ,/j,_/j. . ._/"„, arranged at intervals a,, a,,. . .a^, along an axis. For brevity, let k= — 1/f, for all suffixes. Let any ray originally parallel to the axis strike the lenses in succession at distances Vv Vi •••Vf, from the axis, and let 3,, 9j,...9„ be the total deviations of the ray, after passing through the several lenses. Then, using the value of the deviation just given, and expressing the distances y^, y,. . . in terms of the deviations, we obtain the equations From these equations it is easy to see that 3„ is the numerator of the last convergent of the continued fraction 2/x _L J_ J_ i i+ k^+ a,+ k^ + + k^' If F^ be the focal length of the equivalent lens, 9„ = — yJF^, = y,-S^„, say. Then K^ is equal to the numerator of the last convergent of the continued fraction 1 J. 1^ J_ J_ i+ k^+ a^+ k^ + +k/ The values of the first few numerators are 1, k^, ajc^ + 1, ajcjc^ +k^ + k^, a^aJcJc^ + a, {k^ + k^) + ajc^ + 1, afljcjcjc^ + aA (^1 + ^2) + fl^A iK + K) + h + K+ K from which we deduce the values a J\ Ji J\J % 8 J\ Jt J% J I >y2 J»' Ji vi Ji' JiJtJt These results might also have been obtained directly from the equations. 83—84] EQUIVALENT LENSES. 97 84. We may find a formula connecting two consecutive terms of the series K^, K^ which gives a ready method of calculating their values. For the last two equations are Eliminating y^ between these equations, we deduce 9« = (1 + «^«-l K) 9„-, + K Vn-l If now we substitute d = Ky, for all suffixes of d and K, we arrive at the equation which determines K^ as soon as K^_^ is known. For example, K^ = 0-+ a,k,) {k^ + k^ + k, + ajc, {\ + A; j + ajc^ (k^ + k,) + a^ajcjcjc,} + ^^ {1 + a, (k^ + k^) + ajc^ + a^ajcjc^}, which is equivalent to 1 1 1 1 1 ai/l,l,l\ f^ ,'^\ f^ .1] F. ^l V. ""/s ^l ~ I U ""/a ^fJ ~ "^ VZ ^fJ U ^fJ A \fi /j fJ AA V/i // Afi V2 /s/ /1/2 v/3 fj JiJiJaJi This is the value of the power of a lens equivalent to four given lenses, separated by intervals a^, a^, a^ from each other. H.v. Show that 1 -K -1 K~ 1 -a,.-i -1 1 -K-. 1 1 -a„ the determinant having (27i - 1) rows. 1 -1 98 EQUIVALENT LENSES. [CHAP. V. 85. If the incident pencil be of any form, the position of the equivalent lens is not immaterial, and must be found. Let the incident ray make an angle 9 with the axis ; then using the same notation as before, all the equations remain the same except the first, which is and therefore the final value of 3, will be the same as before, with ^1 + 9/2/1 written for ^. If the reciprocal of the focal length of the equivalent lens be denoted by K, since K involves AjjOnly in the first degree, the new value of K will be so that 9„ = Ky^ + d -rr- • Let the distance of the equivalent lens behind the first lens of the system be so ; then the incident ray will meet the lens at a distance from the axis equal to y^ + aid, and therefore the inclina- tion of the ray to the axis after refraction through it will be = Ky, + dil+Kw). Equating this value to the inclination 9^ we get 1+Ex = ^, dki sothat ^ = z(^-l)- This determines the position of the lens so that it may be equivalent to the given system of lenses. CHAPTER VI. Geneeal Theokems. Caustics. 86. If a ray of light pass from a point A to another point 5, through any number of media, undergoing any number of reflexions and refractions, then the actual laws of reflexion and refraction are such as to make S (^/s) a minimum, where p represents the length of the path of the ray situated in the medium whose refractive index is /x. Conversely if we assume the path of light to be such as to make 2yu.p a minimum, we are led to the actual laws of reflexion and refraction. The expression S/i/a is frequently called the reduced path. We shall first prove this general theorem for a single reflexion and a single refraction, and afterwards extend it to any number of reflexions and refractions. Let AFB be the path of a ray of light which travels in a homogeneous medium from a point -4 to a point B, undergoing one reflexion at a surface GB ; then the total path between A and B is a minimum, that is, AP + BB is less along the actual path than along any consecutive path as AQB. For a variation of P perpendicular to the plane APB, this proposition is clearly true. Let AQB be a consecutive path in the plane APB. Then the difference AQ - AP is equal to the projection of PQ on AP; and similarly the difference BP - BQ is 7—2 100 GENERAL THEOREMS. [CHAP. VI. equal to the projection of PQ on PB. But these projections are equal, because AP and PB are equally inclined to PQ. Thus, AQ + QB = {AP + PB), -which proves that the increment of the total path vanishes, and therefore the total path is a minimum. A similar theorem holds if we take the path from A to B, supposing the ray to suffer a refraction at a surface OD. Let /i, fi be the refractive indices of the two media, then /lAP +ijf PB is a minimum for the actual path. B Draw the normal PN, and let the angles of incidence and refraction be and /jfBP-/jfBQ = ,i'PQsm'. Hence the whole variation, fiAQ + ^'BQ - {fiAP+ /jl'BP) = PQ (fj, siatf) — fi sin ') = 0. This shows that for the actual path, fiAP + fi'PB is a minimum. The previous theorem is a particular case of this; we have only to put /Jb'=—fj. to deduce it from the more general theorem. Next, suppose that the ray of light in its passage from A to B undergoes any number of refractions or reflexions. Let p be the 86—87.] GENERAL THEOEEMS. 101 length of the path in any medium whose refractive index is /u.. Then it has been shown that S/a/j is a minimum for separate variations of the points of incidence between consecutive media ; and therefore by the principle of superposition of small variations, it will be a minimum when simultaneous variations are admitted. The actual path, therefore, makes S/xp a minimum between any two points. If the variations of refractive index be gradual, the same principle holds good, and the path of the ray of light is such that ffids is a minimum. 87. Another important proposition, enunciated by Malus, easily follows from the preceding. Any system of rays originally normal to a surface, will always retain the property of being normal to a surface after any number of reflexions or refractions. The general theory of systems of lines will be given later ; but we may remark here that a doubly infinite system of lines is not in general a system of normals. Let ABODE, A'B'G'D'E ... be a series of rays normal to a sur- face at A, which undergo any number of refractions and reflexions. Measure off along these rays distances to E, E , such that 2/i/3 is the same along each ray ; then we shall show that the rays are finally normal to a surface EE. Join AB and ED. Then 2/i/5 along ABODE is the same as along A'B'O'D'E. But by what has been shown .above for any ray and its consecutive it follows that 2/i/3 along A' BODE is the same as along A'B'O'D'E, and therefore the same as along ABODE. Take away the common 102 GENERAL THEOREMS. [CHAP. VI. parts ; then if fi, fi belong to the final media, there remains the equation, fiA'B + fjlED = fiAB + fi'DE. But, since AB is normal to the surface AA', A'B = AB ultimately, and therefore, DE" = DE; that is, EE is perpendicular to DE. The same may be proved for every point E near E, and thus the surface EE' near E is perpen- dicular to the ray DE, and by similar reasoning to every other ray of the system. 88. These theorems may also be proved analytically. Let a ray of light pass through several media of different refractive indices. Let (a, y8, 7) be the direction cosines of the inci- dent ray, and (x, y, z) a point on it; then (a, y8, 7) may be regarded as known functions of x, y, z. Let this ray meet the first surface in the point (f„ 7;,, fj), and let (a^, /S^, 7,) be the direction cosines of the refracted ray in the second medium and (a;.^, y^, z^ be a point on the refracted ray. Then at the first refraction the direction cosines of the rays are connected by the equations /^ - /^2"2 = 0* cos ^ - /i, cos + J^ ; then this equation may be written u sin { — ^ff) = k sin 2(f> ; P Q that is T 4- . , = 1, cos 9 sm (p where p^ (u+p)co.^e 2k ^~ 2k ■ The arbitrary parameter a enters into this equation only through , which gives ^ - Q -^ 33i>~smr^~^''^y' then if we substitute the values of P and Q in the equation of the ray, we find that X = 1. Eliminating ^, the equation to the envelope is P^ + Q^ = l. 96 — 97.] CAUSTIC BY REFLEXION AT A CHICLE. Ill Expressed in polar coordinates it is {(u + p) cos ^6}^ + {{u - p) sin {6]^ = {2kf. To rationalise the equation P* + Q^ = 1, we cube both sides; it becomes P + Q^ + 3P^Q^(PUq^) = 1, or I - P'' - Q^ = 3P^ qI Cubing the equation again, we get finally, (l-P'-Qy = 27P'Q'. The equation may easily be transformed into Cartesian co- ordinates. For and therefore P' + Q' = ^ j-s, + -^ + - cos e\ Also, PQ=g^(c'-r-Osin^ 8rV Hence the equation to the caustic becomes {(4,0" - a^) («" + f) - 2a^cx - aV}' = 27a*cy {a? + y^- cj. 97. If in the equation to the caustic we put 6 = 0, and there- fore Q = 0, P = (u +p)/2k, it becomes (i-py=o, and therefore u+p= ±2k, and each of these points is a triple intersection. Expressed in terms of a and c, the distances of these points from the centre of the circle, are ac ac r = ^ , r = — ■ '2c -a' 2c + a' 112 CAUSTIC BY REFLEXION AT A CIRCLE. [CHAP. VL These points are cusps on the caustic. Again, if we wish to find the points of intersection of the caustic with the circle u=p, we get Q = and P =p/k cos ^6, and the equation to the caustic gives (1_P«)» = 0, which reduces to p cos ^0= ±k, or a cos ^0= ±c. Each of these points is a triple point of intersection ; the reflected ray, therefore, touches the circle r = c, and the correspond- ing incident ray is perpendicular to 00. These triple points of intersection are cusps, and the tangent at the cusp is perpendicular to the radius vector ; they are imaginary if c is greater than a, that is, if the luminous point is outside the circular reflector. 98. To find the directions of the asymptotes, we must make M = 0, in the equation to the curve. The values of P and Q are then i-_Z a and from the equation to the caustic, we derive the equation 1-- ic' = 27 6-|?-^^> that is, 27aV sin' = (4c' - a')'. This equation gives the directions of the asymptotes, and shows that they are imaginary if c be less than J a. We shall now find the length of the perpendicular on them from the origin. Dififerentiating the equation and afterwards putting u = 0, we get after dividing by common factors, i \-ja cos i0 -^ sin i0\ 1 ■ -Ta sin h0 -■§ cos i0[=O. 98—99.] CAUSTIC BY REFLEXION AT A CIRCLE. 113 This gives dd I (sin 10 cos ^0)^ [cos^ ^6 - sin^ ^6} + ^ (cos'' ^d - sin* ^6} = 0, or ~ (sin ^0 cos ^(9)* = - - (cos^ ^0 + sin^ ^0) 1 /2c\l 2c U/ ■ Hence '^ (sin 0r = - I f'-f , d0^ ' c \aj du /4c;'' — a' d0V-T-=-'- If we denote the length of the perpendicular from the centre on the asymptote by •cr, we get V 3 The asymptotes are imaginary, if c be less than ^a ; and when c = ^, they coincide with the axis of x. 99. We shall next find the points of intersection of the caustic with the reflector; for this purpose we shall use the Cartesian equations. If we make x^ +y' = a^ in the equation to the caustic it becomes (3aV - a* - 2a^cxy = 27aV (a' - c7 {a' - »'), or, by expansion and division by a*, 8a»cV - cV (15a* - IBaV - 27c*) + 6cx (3c= - a^f + a' + 18aV-27aV = 0, which may be written in the form {ex - ay {Sa'cx + a* + 18aV - 27c*} = 0. Hence the caustic touches the reflector at the points given by the equation ex = a^ which are the points of contact of the tangents drawn from the luminous point to the reflector if the luminous point be outside the reflector, and are imaginary if this point be inside. The other point of intersection is determined by the equation _ 27c*-18aV- a* ^~ 8a=c H. S 114 CAUSTIC BY REFLEXION AT A CIRCLE. [CHAP. VI. This value of x is numerically less or greater than a, according as c is greater or less than a; that is, according as the luminous point is outside or inside the reflecting circle. The shapes of the caustic curves for different positions of the luminous points are shown in the following figures. In the first figure the incident rays are parallel; the other figures represent the caustic curve as the luminous point approaches nearer and nearer to the centre of the reflecting circle. Fig. 1. c = oo. Fig. 2. oa. Fig. 3. c=a. Fig. 4. c\a. Fig. 6. c=\a. Fig. 6. c = \a. 99—100.] CAUSTIC BY EEFLEXION AT A CIRCLE. 11.5 100. The caustic by reflexion at a circle may be found by elementary geometry in two cases, first, when the incident rays are parallel, and secondly, when they diverge from a point on the circumference of the circle. When the incident rays are •parallel, the caustic is an epicycloid formed by the rolling of one circle upon another of twice its radius. For from the centre C of the reflecting circle, draw the radius CA parallel to the incident rays ; then the caustic is symmetrical with regard to the line CA. Let SP be any one of the incident rays, reflected by the circle at the point F in the direction PQ. Join GP ; then by the law of reflexion, GP will bisect the angle SPQ. With centre G and a radius equal to half the radius of the given circle, describe the circle BR bisecting the radii GA, GP in B, It, respectively. On PR as diameter describe another circle meeting the reflected ray in Q, and join QR. Since SP is parallel to GB, the angle SPG is equal to the angle PGB ; and therefore the angle QPR is equal to the angle RGB. The angle QPR is subtended at the circumference of the circle by an arc QR ; and the angle RGB is subtended at the centre of the other circle by the arc RB, and the radius of the second circle is double the radius of the first, and therefore the arc QR is equal to the arc RB ; and if the circle PQR were to roll along the circle RB, the 8—2 IIG CAUSTIC BY REFLEXION AT A CIRCLE. [CHAP. VI. point Q would finally coincide with B. Now as Q begins to move, the point of contact R is for an instant fixed, so that the motion of Q is perpendicular to QR ; and therefore the reflected ray PQ touches the curve described by Q. This is true whatever the position of the point P. The locus of Q is an epicycloid, and this is the caustic curve required. 101. If the incident rays diverge from a point in the circum- ference of the reflecting circle, the caustic curve is a cardioid, or, in other words, the caustic may be described as an epicycloid in which the rolling circle is equal to the fixed circle. Let be the origin of the incident rays, 00.4 the diameter of the reflecting circle ; then the caustic curve will be symmetrical about the line OCA. Let OP be any incident ray which is reflected at P by the circle in the direction PQ. Join CP ; then by the law of reflexion, CP will bisect the angle OPQ. With centre and radius equal to one-third of the radius of the given circle, describe a circle meeting CA and CP in B and R, respectively, and on PR as diameter describe another circle cutting the reflected ray in Q ; join QR. The radii of the two smaller circles will be equal to each other. Now, since the triangle GPO is isosceles, the external angle PGB is double of the angle GPO, and therefore double of the angle QPR. Hence the arcs RB, QR subtend equal angles at the centres of their respective circles, and therefore these arcs are equal. If the circle PQR were to roll along the circle RB, the point Q would finally come to B. As the circle PQR begins 100—102.] CAUSTICS BY REFLEXION. 117 to roll, the point of contact R is for a moment stationary, and therefore Q begins to move perpendicular to QR along PQ. From this it follows that the reflected ray touches the curve described by the point Q. This is true whatever the position of the point P^ The locus of Q is a cardioid, and this is the caustic required. 102. There are two cases in which we can find the caustic after the rays have been reflected at a circle any number of times ; first, when the incident rays are parallel, and secondly, when they diverge from a point in the circumference. Let a ray be reflected any number of times at a circle ; and let (?„(?j be the first path of the ray across the circle, making an angle yfr^ with the positive direction of the axis of x, and let the angle GfiA be denoted by 0^. Let 0, i|r be corresponding angles for the nth reflected ray. Then the equation to this ray will be y — csin6 = tan ^^{x — c cos 6), or y cos sfr — x sin i^ + c sin {'<{r — &) = 0, where c is the radius of the circle. But if the angle OGfi^ be denoted by (f>, we have e = 0^ + n (it - 2(j>} = 0, + n7r - 2n and i/r = ilr, + n (27r - 2<^) = V^o + 2»7r - 2n0. Hence the equation to the nth reflected ray becomes ■X sin {f„ - 2n^) - y cos (f, - 2n^) = (- 1)" c sin (v|r„ - (9„). First, let the rays be incident parallel to the axis of x ; then 118 CAUSTICS BY REFLEXION. [CHAP. VI. we may write d„ = (/>, •<|^o = tt, and the equation to the reflected ray is X sin 2h^ + y cos 2n; we thus get tl^^ equation iB cos 2n + y cos {2n + 1) ^ = (- 1)» c sin . t'he envelope of this line may be found as before. Differentiating with respect to the variable parameter, we get the equation a; cos (2?i + 1) ^ - y sin (2ft + 1) (^ = (- 1)" .^-l-j c cos ^ ; and these two equations give ( — l)"c X = ^ ' {{n + 1) cos 2n^ — n cos (2n + 2) ^} , ( — VTc y =-^2;^^ { - (^i + 1) sin 20 + »i sin (2ft + 2) be the angle of incidence and ^' the angle of refraction at Q ; then the angle P08 = (f>', and (f> = Z H8Q = Z HPq = Z SPO. 122 CAUSTICS BY REFRACTION. [CHAP. VI. Hence SO : SP = sin : sin ', and therefore yu, SO = fi' SP But since the angle P is bisected, HO : HP = SO : SP and therefore fj, HP = fi SP. By addition, ^i.SH = /j,' (SP + HP). Thus the locus of P is an ellipse whose foci are S and H; and PQ is normal to the ellipse, and therefore the ellipse is an ortho- tomic curve. The evolute of this ellipse is the caustic required. If the second medium is more highly refractive than the first, it may be shown in the same way that the caustic is the evolute of a hyperbola whose foci are S and H. 107. To find the caustic by refraction at a circle for rays issuing from a point. Let be the centre of the refracting circle, jS the origin of light, and SQ an incident ray, QR the corresponding refracted ray. Describe a circle through S and touching the radius OQ in Q ; let this circle meet OS in H, and the refracted ray in P . Then OH . OS = OQ", and therefore If is a fixed point. Also, from the similar triangles OQS, OHQ, QS : HQ = OS : OQ, which is a constant ratio. Let be the angle of incidence, (j)' that of refraction. Then if OQ be produced to T, : sin ', which is a constant ratio, and therefore also QH : QP is a constant ratio. Now by Euclid vi. D, QH.SP + QS.PH = SH. QP; and therefore if 8P = p,PH=^ p, QH QS QPP + QP'^^ or mp + nip = c, say. The locus of P is therefore a Cartesian oval, of which ;Si and H are foci. Also since PQ divides the angle between the radii vectores into two parts whose sines are in the ratio of the chords QS, QH, that is in the ratio ni' : m, PQ is the normal to the curve. Hence the caustic is the evolute of a Cartesian oval of which S and H are foci. 108. This construction fails when the rays are parallel. The equation to the caustic in this case may however be found by a different analysis. Let ' be the angles of incidence and refraction of any ray parallel to the axis of x, so that sin ^' = h sin 0, where k = /i/fj,'. Then if we take the centre of the circle as origin, the equation to the refracted ray is y — a sin ^ = tan {') [x — a cos ^), or y cos (i^ — ^')—x sin (^ — (j>') =a sin (f>'. Now let + b^ cos^ ' , _ 2ba^ sin <^ a^ sin^ (j) + b' cos^ ' where Q^ 2p - x — u _ Zph-^ ;i---acos0 a^fi^ ' 2)-3 since p^ (a^ + 62 . . r^) = a262^ from the ell ipse. Clearing of fractions, we get X (a^ + b^ -Sr^) = u {a^ + b'^ -r^) — 2r^ a COS (f>. But u {a? + 62 _ r2) = M (a2 sin^ <^ + 6^ cos^ <^) = 2a62cos(^, and therefore .r {a^ + b-- 3r^) = 2fl! cos <^ (6^ - j-2) = - 2a cos' (a^ - 6-'). 126 LENGTH OF THE ARC OF A CAUSTIC. [CHAP. VI. Similarly y {a'- + b^- Sr^) = 2a sin (a^ - r^) = 26 sin' <^ (a^ - 6=). ©^ Hence, by division, tan <^= - — 7 • 0" Again, eliminating (p between these two equations, we get which becomes |(^^ + f-\ \ (a^ + 6^ - 3j-2) = 2 (a^ - h'-). Butr2 = a2cos2<^ + i2sin2^ = ^^4 ~. eA0) Hence, finally, {©'- (l)T[<«'^'-'{©'Kf)'} -{"■©'-(!)'}]-<•-'• "»' '•■ {©' + (I)')' [©' «" - "•> + (!)' «•■ - »■']-"' - '■■ which is the caustic required. 110. To find the length of the arc of a caustic. The length of the arc of a caustic of any orthotomic system of rays in one plane can always be found. For the caustic is the evolute of the orthogonal curves. Suppose a system of rays issuing from a point, or normal to a given surface, to be reflected and refracted any number of times. For each ray, form the function 2/t/9, and let F= S/xp. Let the final medium be of refractive index /t, and let F= V^ be the equation to an orthogonal curve in this medium, say the curve PQ. 109—112.] CURVES OF ILLUMINATION. 127 Let AB be any arc of the caustic, and let PA, QB be the rays touching at A, B. Then the arc AB = QB—PA, by the properties of evolutes. Also V^ = V, + ^PA, Vs = V,+ fjiQB; and therefore by subtraction, Vb- Vj^ = /j, (arc AB). 111. We can now, by means of caustics, indicate more accurately the manner and position in which an object under water is seen by an eye outside. Suppose for instance that the water had a horizontal level bottom not very deep. Let P be a point on the bottom, let us trace the pencil of rays by which an eye sees the point P. Draw the normal PM and consider rays in the plane EPM. Construct the caustic in this plane which is touched by refracted rays originally diverging from P. We must draw the two extreme tangents to this caustic which will meet the eye, and then these will bound the part of the pencil which traverses the air ; if we join the points where these tangents meet the surface to P, the joining lines will bound the pencil as it passes through the water. The two tangents to the caustic meet at the point of contact of either of them, very nearly. Thus to an eye outside the point P appears to be at p. Curves of illumination. 112. If light be incident on a series of bright curves or grooves dra^vn very close together, so that the reflected light may be received by the eye of a spectator, he will see one or more curves 128 CURVES OF ILLUMIXATION. [CHAP. VI. of special illumination drawn across the grooves. This is very commonly seen when bright rods, such as the spokes of a wheel of a bicycle are revolving in the light of the sun. We shall now consider how these bright curves are produced, and how their forms may be investigated when the reflecting curves are given. Every point of a reflecting curve will scatter light, and will in this way make itself visible to the eye ; but there will be one or more points on the curve which will reflect light to the eye according to the regular laws of reflexion. More light will reach the eye from such a point of the curve than from any other, and the point will therefore appear brighter than the rest of the curve. The locus of these bright points will be a bright curve, whose form is required. Let the system of reflecting curves be represented by the equation {x, y) = a, where a is an arbitrary parameter ; and let the incident light proceed from- a given luminous point Q. Let E be the position of the eye of the spectator, and P any point on one of the reflecting curves. Draw the tangent to the curve in its own plane. Then, if the reflecting curve be a small groove or a thin rod, an infinite number of tangent planes can be drawn to the groove or rod, all passing through this tangent line. If one of these can be drawn so as to reflect a ray of light proceeding from Q in the direction of PE, then P will be a bright point. In order that this may be possible, we must be able to draw a normal to the groove at P, which shall lie in the plane QPE and bisect the angle QPE. If this can be done the rays QP, PE will make equal angles with the tangent line at P; and, conversely, if this condition be satisfied it is easy to see, as in § 12, that the other two con- ditions are also satisfied. Let {x, y, 0) be the coordinates of the point P, (/, g, h) those of the point Q, and (a, h, c) those of the point E. Then if I, m be the direction cosines of the tangent at P, and expressing the condition that the lines QP, EP make equal angles with the tangent at P, on opposite sides of it, we get the equation (/- x)l + (g-y)m (a-x)l+{b-y)m _ J{f- xf + {g- yf + A= + J {a - xY + (6 - yj + c» ~ " 112 — 114.] CURVES OF ILLUMINATION. 129 The ratio I : m may be eliminated by means of the previous equation, and we get the equation -Jif-^y + iff-yf + h:' J{a-xf + {h-yr + c' ' and this is the equation to the bright curve required. 113. The same equations may be arrived at in a shorter manner by means of the theorem that the whole length of the path of light from one point to another point is a minimum. For if P be the bright point, it follows that the length of the path QP + PE must be a minimum, subject to the condition that P shall always lie on the curve ^ {x, y) = a. If {x, y, 0), (/, g, h), (a, b, c) be respectively the coordinates of P, Q and E, QP + PE= J{x-fy+(y-gY + h' + J{x-aY + {y-bf + c\ To make this a minimum, subject to the condition just expressed, we equate to zero the first differential of each equation; we there- fore get the equations ^^ dx + ^dy = 0, (x-f)dx + (y-g)dy (x - a)dx + (y - b)dy ^^ Ji'^-fr+iy-gr+t'-' j{x-ar+{y-bT+c' Eliminating the ratio dx : dy between these equations, we obtain the equation to the bright curve, in the form {x-f)4>y-{y-g)4>, (x-a)c}>„-{y-b), ^^ Ji«>-fT + Q/-9T + l^' Jix-ay + iy-by + c' 114. As an example of the foregoing process, let us find the form of the bright curves seen on the spokes of a bicycle wheel revolving in the sun-light. Take the axis of the wheel as the axis of z, and suppose the direction of the sun's rays to be defined by the direction-cosines {I, m, n), and let the coordinates of the eye be {a, b, c). Then supposing the wheel to be plane, the equation to the reflecting curves will be of the form y = x tan 6 ; and therefore, if we express the fact that the incident and reflected rays make equal angles H. 9 130 EXAMPLES. [chap. VI. with the line whose direction-cosines are (cos 6, sin 6, 0), on opposite sides of it, we find „ - /, (a — x) cos6 + (h — y) sin 6 . Jia-xy + ib-yy + c" Eliminating 6, the equation to the bright curves becomes (Ix + myr [{a - xf + {b- yf + c'} = {{a-x)x + {h- y) y]\ EXAMPLES. 1. A luminous point is placed at a distance h in front of a plane refracting surface. Show that the orthotomic surfaces of the rays within the medium are those formed by the revolution of the curves about the axis of ^, a being a variable parameter, the origin being the foot of the perpendicular from the luminous point on the plane boundary, and the axis of X normal to that boundary. 2. Rays emanating from the focus of a parabola are reflected from the cvolute of the parabola, show that the caustic is the evolute of a parabola. 3. If rays emanating from the vertex are reflected from a parabola, the caustic is the evolute of a cissoid. 4. When the luminous point is the centre, the caustic by reflexion of the involute of a circle is the evolute of the spiral of Archimedes. 5. Rays emanate from the pole of a plane curve whose equation is given in the form f{r,p)=(i (I), show that the equation to the katacaustic will be the result of eliminating r and p between (1) and the equations dr 'Jr'^-p'^+s/r^-p'^ = «^ (2), P^^'^^^ (3). Conversely, if the equation to the katacaustic be <^('->')=0 (4), the equation of the reflecting curve will be obtained by eliminating r', p' between the equations (2), (3), (4). CHAP. VI.] EXAMPLES. 131 If the reflecting curve be the involute of a circle, represented by the equation r^=p^ + a% the equation to the caustic will be r^ (8a2 -^2)2= 4a2 {16a* +p^ {4a'' -f)] . If the reflecting curve be the hyperbolic spiral, r5=aor(r2 + a2)j)'=aV2, the equation to the caustic will be r{a-p) = ap. If the caustic be a circle r=a, show that the reflecting curve is determined by the equation aV^ = 4p2 ( J.2 _^2). 6. A luminous point is placed in front of a thick plate of glass with parallel faces ; show that the caustics produced by the successive reflexions and refractions at the surfaces of the plate are the evolutes of two series of equal and similarly placed prolate quadrics of revolution, each of which has at least one focus coincident with one of the successive reflexion-jmages of the point due to the faces of the plate considered as plane mirrors situated in air. 7. If rays from a luminous point be reflected at a parabola, show that the katacaustic has three asymptotes, reflected from joints at finite distance, except when the luminous point is on the axis, when there are only two, and that then the abscissa of the point where the asymptote is reflected is one-third the abscissa of the luminous point. 8. Prove that, if rays of light proceed from a point and be reflected at a conic whose plane contains the radiant point, the reflected rays are all normal to a bicircular quartic which has the radiant point as double point. If the radiant point be the centre of the conic, show that the equation of the quartic may be written £ r^=Acos26+B. 9. A lumir«3&s 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 r = acos \6. 10. Eays proceeding from a luminous point in the pole of the spiral r=ae^ ax& reflected at the curve ; show that the caustic is a similar spiral. Also show that the spiral will be its own caustic if T (2»ir-a)cota Jseca = e^ where n is any positive integer. 11. Eays parallel to the axis of y are incident on the reflecting curve y=^, show that the equation to the caustic is the catenary . y=^{e*+i.+ e-'»^"}. 9—2 132 EXAilPLES. [chap. VI. 12. A ray proceeding from a point in the circumference of a circle is reflected n times at the circle ; prove that the point of intersection with the consecutive ray similarly reflected is at a distance from the centre equal to o/(2n.+ l)i\/l+4n{n + l)sin^6 where a is the radius of the circle, and 6 the angle of incidence of the ray. Prove also that the caustic surface generated by such rays is the surface of revolution generated by an epicycloid in which the hxed circle has the radius a/{2n + 1), and the moving circle the radius nal(2n+ 1). 13. Prove that the dfcstic for a pencil of parallel rays refracted at a circle of radius unity and refra^on-coefficient /i, is given by the equations /i^(l -/i2)^=|i2cos'<^ + (/i^-sin2^)* ; ;i^=sin'^. 14. Light emanating from a point is reflected at a curve so that the caustic is a circle whose centre is and radius equal to a. Prove that the curve must belong to one of the families r {'Jr^ — a^ . ,a\ fl+const.=- + \^ + sm-i-L a \ a r) 15. Rays issuing from the centre of a given circle are refracted at a curve so that the refracted rays are all tangents to the circle. Find the equation to the refracting curve. 16. Rectify the caustic in the case of rays parallel to the axis of x falling on the reflecting curve * sin {ay) =er: I 17. At a point on the inside of a polished hollow right 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 that they are all epicycloids, the radius of the rolling circle for the «th curve being 7ia/(2«, + 1), and that of the fixed circle a/(2n+ 1). 18. The surface of a hollow right cone is grooved with an infinite number of circular grooves. A bright point is placed on the surface ; prove that an , eye situated on the opposite generating line will see a bright curve which lies on a sphere of radius abc/{a''-b^), passing through the vertex of the cone, where a, b are the distances from the vertex of the bright point and the eye, and c is the distance between them. 19. If a series of fine smooth grooves be cut in a plane surface in the shape of concentric circles, the bright curve formed by reflection of the light from a luminous point, and seen by an eye situate in the plane through the luminous point and the axis of the circles, will be a circle. 20. Fine polished wire with circular transverse section is disposed alon^ the meridians of a sphere whose axis is directed to the sun. Prove that those reflected rays which have a common direction normal to the axis proceed from ciirves in which the sphere is met by an elliptic cone, the planes of whose circular sections are inclined at half a right angle. CHAP. VI.] EXAMPLES. 1-33 21. In a hollow ellipsoidal shell small polished grooves are made coin- ciding with one series of circular sections, and a bright point is placed at one of the umbilics in which the series terminates ; prove that the locus of the bright points seen by an eye in the opposite umbilic is a central section of the ellipsoid, and that the whole length of the path of any ray by which a bright point is seen is constant. 22. A bicycle wheel in which the spokes are perpendicular to the axis is placed in the sun and spun rapidly. Show that the equation of the bright curve seen on the spokes by an eye in the axis of the wheel produced is of the form r^ (sec^tf sec^a — 1) = oi^, a denoting the angle between the direction of the sun's rays and the plane of the wheel, and a the distance of the eye from the wheel. 23. A man standing on the sea-shore sees the light of a star reflected on the surface of the sea when it is covered with gentle ripples travelling in all directions, find the equation to the boundary of the bright patch on the water, considering the undisturbed surface of the sea to be a horizontal plane. Find the condition that this patch should reach to infinity. If z be the zenith distance of the star, and the tangents to the boundary of the bright patch from the man's feet contain an angle 45, show that if the patch do not extend to infinity, the angle which it subtends at the man's eye in a vertical plane passing through the star is 4 tan"i (sin 6 tan z). CHAPTER VII. Aberration of direct pencils. 115. When rays of light diverging from a point are incident on a plane reflecting surface, we have seen that after reflexion all the rays pass accurately through another point, which was called the conjugate focus of the given point. But when a pencil of rays diverging from a focus is incident directly on a plane refracting surface or a spherical reflecting or refracting surface, it is only the rays in the immediate neighbourhood of the axis of the pencil which after reflexion or refraction can be considered as passing through a point; the other reflected or refracted rays touch a caustic surface. We shall suppose that the incident pencil meets the reflecting or refracting surface within a circle of small radius y, which we shall call the aperture of the surface. Let q be the focus of the rays which are in the immediate neighbourhood of the axis, and let Pq' be the extreme ray after reflexion or refraction, cutting the axis in c[, and a plane through q perpendicular to the axis in t. Then qc[ is called the longitudinal aberration of the ray P^, and qt its lateral aberration. These 115 — 116.] ABERRATION IN A SPHERICAL MIRROR. 135 aberrations may be expressed approximately in terms of the aperture when the aperture is small. We need only find the aberration for pencils which diverge from points on the axis. For if-a pencil diverge from a point not on the axis, the image will lie on the line joining the origin of light to the centre of the reflecting or refracting surface ; and this line will be the axis of the pencil and the longitudinal aberration along this line will be known. This must be projected on the original axis by multiplying by the cosine of the inclination of the line to that axis. This inclination will be very small, so that when multiplied by the small aberration its cosine may be taken to be unity ; and therefore to our present approximation, the longitudinal aberration is the same for all points lying in a plane perpendicular to the axis. 116. To find the aberration of a pencil directly reflected at a spherical surface. Let QPR be the path of the extreme ray, and let PR produced backwards meet the axis in q. Let be the centre of the sphere, QAO the axis of the incident pencil, and let OQ =p, Oq' =p', OA = r, and let the angle POA be denoted by 0, and the angle of incidence of the ray QP, by ^. Then r sin {(f> — 6) p sin (f) ' r _sin{^ + 0) _ p sin <}) ' 136 ABERRATION OF DIRECT PENCILS. [CHAP. VII. ..hence ':+':, = ^in ( = - (1 — cos y). p, p r"- In this equation, powers of above the second may be neglected, and p' is very nearly equal to Pq'; the equation may therefore be written p' r' and therefore, since y = r sin ^ = r0, approximately, we get This is the value of the longitudinal aberration of the extreme ray. We notice that qq has the same sign as r, and therefore if we stand at the mirror and look towards the centre, the caustic points in this direction in all cases. When the rays are incident parallel to the axis of the mirror, q will be at the principal focus of the mirror, so that Oq will be /, where/ is the focal length of the mirror ; in this case the longitu- dinal aberration will be , y^ m =- ¥' 117. To find the aberration of a pencil directly refracted at a plane surface. Let QFR be the course of the extreme ray, so that AP is the radius of the aperture. Let PR produced backwards meet the 116 — 118.] ABERRATION OF DIRECT PENCILS. 137 axis in ([, and let AQ, = u, Aq' = u', AP = y. Then if <^, ^ + M?^.}=''(«-^){> + 2-^.)- 118^121.] ABERRATION IN LENSES. 139 By means of equation (2) this reduces to [u-p fJLlt V (3), ■P v-p) where on the right we may suppose v to have its first approximate value «„. This formula contains the whole theory of aberration at a spherical surface. 120. It is generally more convenient to measure the distances from the surface, and not from the centre. The formula (3) may easily be transformed in terms of the new variables. Let a, /3 denote the reciprocals of AQ, Aq, measured from the surface to the right ; then 1_1_1 a p u' and -= . /:i p V The equation (2) now becomes a-p = p.{^-p) (4), and the equation (3), But by differentiation, ~^ = j > and therefore dv= -^d^ = — . _ y d^. Hence dl3 = yW (^ - pf (0 - fi^) , or d0 = ^{0-py(0-fi^)y- (5). If yS be eliminated by means of the equation (4), this result becomes '^'^ = ^^(''-")°^^"^'' + ^)"^2/' (6). 121. To find the aherration in the refraction of direct pencils by lenses. Let a, denote the reciprocals of the distances from the first surface, of the points where the axis is met by the incident and refracted rays, respectively, and let ^, a' denote similar quantities 140 ABERRATION IN LENSES. [CHAP. Vll. with reference to the second surface, and let 6 be the reciprocal of the thickness of the lens, and p, p the curvatures of the bounding surfaces of the lens ; then, when the aperture is made very small, the relations among these quantities are a. - p = p. [^ - p), a - p = p [0' - p'), ^ - ^' = ^ • Let y, y be the radii of the apertures of the first and second surfaces respectively; then since a' may be regarded as a function of the two variables yS' and y , we must have <'«'-(S)''^+(|)*'. /' where the differential coefficients are partial. But if we differentiate the second and third equations, we get Let the variation of /8 due to the change of aperture at the first surface be denoted by yey'; and supposing fi' fixed, let the variation of a' due to the change of aperture at the second surface be denoted by Ky'^; then d/8 = wy", and therefore i^-^i^y and also ^^ j dy' = Ky"" = «' ^, y\ since y' '• y= 1//8' : 1//3, by similar triangles. Substituting these values in the expression for da', we get J , , f /8" ,^\ «a = 2r V« ST + « o^l • We have now to substitute the values of k, k as found in the previous investigation. It was there shown that d^ = Kf = \{fi- py (fi - ^a) f; and if we eliminate p by means of the equation a — p = fi {fi — p), we find the value of k to be 121 — 122.] ABERRATION IN LENSES. 141 The value of k may be found by substituting /8' and a for a and y8, respectively, and l//i for /a ; and therefore Substituting these values of k, k in the expression for di, it becomes rf«' = 2-(^^{f (/8-a)M^-/.a)-|;(^'-aT(^'-^a': which is the general expression for da! for any lens, of whatever thickness it may be. The quantities /3, y8' may be expressed in terms of a, p, a', p', and then we get a value and therefore f 124. Several cases may be considered, in order to compare the advantages of lenses of particular forms. In a piano-spherical lens, having its plane side towards the incident light, p = 0; and therefore, omitting the dash from the curvature of the second surface, d<}> = -ip,'(jj^-i)pY. And in this case (fs = — (jj, — 1) p ; therefore ''i'=i{i^J^y- In a piano-spherical lens, having its curved side turned towards the incident light, p' = 0, and d )=(/^-i)(j-y=l(j-j-'). knd therefore the thickness is equal to y'lf. This gives a more definite meaning to the preceding results. 125. We shall next find the form of a lens which will refract a pencil of light issuing from a given point, and bring it to a focus at another given point, with the minimum of aberration.. In this problem a and a' are given, and /3 is the variable whose value is to make doL a minimum. We must therefore choose /8 so that (/i -I- 2) yS' - (2/tt + 1) (a -l-a') /8 -F/t (a= + aa' -f a") may be a minimum. This expression may be written in the form 4-^^^ [{(/* + 2)2^-(2/.-f-l) (a-}-a')r-F V0. + 2) (a' + aa'+a") ■(2/ct-H)'(a-Ha7J, 144 ABERRATION IN LENSES. [CHAP. VII. and therefore, for a minimum, (^ + 2)2/3 = (2/x + l)(a+a'), and the expression reduces to -i^ ((/.= + 2p) (a' + aa' + a") - (^» + ^ + J) (a + a')'} = ^ ^^''^ " i^ <''' " "^' ~ ^^ - ^^' ""'^- The minimum value of doL is therefore /^(g -g) , f M-i , , _ ^, _ , To find the form of the lens, we have only to substitute the value of yS obtained above, in the equations and we get where (p -l)p= p^ {p -l)p' = p.^ p =px' + qa p =pz +q«.' , V + M „ i-a i-a']' J 2m' - M - 4 P- 2{p -l)(M + 2)' "^ 2(^-l)(/* + 2) The curvatures of the bounding surfaces are therefore deter- mined. 126. The form of the lens will depend upon the positions of the point from which the light is proceeding, and that at which the rays unite. When the incident rays are parallel, a = and a' = = \^ , p' = q(f>. Accordingly, the ratio of the curvatures 125 — 127.] ABERRATION IN LENSES. 145 of the surfaces is independent of the power of the lens, and is p' _ £ _ 2/i'' — ^ — 4 p p~ 2fj? + fi ' When /t = f , this ratio becomes P 6- The curvatures of the two surfaces of the lens lie in opposite directions, so that the lens is either double convex or double concave ; and the curvature of the posterior surface is ^th of that of the anterior. Such a lens is called a crossed lens. If the index of refraction be such as to satisfy the equation 2/j,^-fj,-4< = 0,or fi = l(l+ V33) = 1-686 nearly, which is about the value of fj, for the more highly refracting kinds of glass, then p' = 0, and the lens will have its posterior surface plane. For a crossed lens, the aberration for parallel rays is — {^y^/f; while for a piano-spherical lens whose curved surface is turned towards the incident light, the aberration is — ^y'/f The piano- spherical lens is therefore nearly as good as the crossed lens; it is much easier to make, and is therefore much more commonly used. When piano-spherical lenses are used as objectives for micro- scopes, the rays diverge from a point very near to the surface of the lens, and emerge nearly parallel to each other, so that it is the plane surface which must be presented to the object. 127. The aberration of any thin lens can be expressed in a simple form in the notation of the last article. For a lens of minimum aberration, it was shown that Assume therefore for any lens, (2/t-H) ,,.,H' -1 6, 2(ji + 2)^ ' iJL + 2 and for brevity, let /i + 2 (/*-l) 10 146 ABERRATION IN LENSES. [CHAP. VII. then if we make these suhstitutions in the expression for the aberration, it becomes da' = \m (a' - a) f [n {a! - a)" - aa' + 6=}. The curvatures of the surfaces may be expressed in terms of a, a',e by means of the equations (fi—l)p = fi^ — a, (/Ji — l)p'=/j.l3 — a', and using p, q, m with the same meanings as before, the values of the curvatures are found to be p=p% p' = pa ' + qa + me] i + qa +me)' 128. If we wish to make the lens aplanatic, that is, such that the aberration vanishes, the equation of condition is ^ = aa' — n (a — a)'. A primary condition is therefore that a and a' have the same sign ; and further, they must be such as to make aaf >n(a' — a)'. These conditions can never be fulfilled for parallel rays; for then (i = 0,a' = \ which gives an imaginary value of e. 129. Aberration of a pencil directly refracted through any number of spherical surfaces arranged symmetrically along an axis. Let /A, fjL, ft!' ... be the successive refractive indices of the media; let a, fi be the reciprocals of the distances of the point and its first image, respectively, from the vertex of the first surface ; and let p be the curvature of this surface, and let dashed letters denote the corresponding quantities for the other surfaces in succession, all these distances being measured from left to right. Then the following relations hold between these quantities : /(«'-p') = m"(/S'-AI m , /." (a" - p") = f."'{^'-p"),l ^ ''• If we denote the reciprocals of the thicknesses of the media 127 — 129.] ABERRATION IN LENSES. 147 between the surfaces, measured along the axis, by 6,6'...,-we must have also 1_1_1 \ /8 a:~e' J. _ J. 1 /3' |2e ^- (a + ^)| + m'f ^2e' g - (a' + ^^=0. Now if we differentiate the values of p, p", in terms of a, /8, a', ^', e, e, we get de m'|' + p' + g' = These values of m-j-,m' -r- must be substituted in the pre- ceding equation. For brevity, let 131 — 134.] ABERRATION IN LENSES. 151 and then the equation becomes {l€ + m(a + ^)]+4>' {I'e' + m' (a' + /3')} = 0. The quantities e, e' are therefore completely determined, and from them the values of the curvatures of the different surfaces of the lenses may be found. 133. When the incident rays are parallel, a = 0, /8=«' = <^, /3' = <^ + <^' and therefore the equations of aplanatism are m^ {e" + n<^^] + m'^ {e'^ - ^ ((/> + ^') + H""] = <>, ^ {U + m^} + ^' {ZV + m' (2^ + ^')} = 0. From these equations e, e' may be found, and the resulting values substituted in the equations p=p + me, p" = g'(j) + p' ((ji + + q'{<^ + ') + m'e, and then the compound lens is completely determined, and will be aplanatic, not only for parallel incident rays, but also for rays diverging from a point whose distance is finite and con- siderable. We notice that these equations determining the radii of the surfaces do not involve any relation between the focal lengths of the two component lenses, and may be satisfied whatever the values of M+fi^=0, and we then get z (a' + 13') = - (ax + ^y) = -{ea^+(f + f)al3 + gl3% If S denote the angle made with the plane of xz, by a plane through the principal ray and parallel to the ray (a, y8, 7), we must have tan S =^/a, and therefore z = — [e cos" S + (/+/') sin 8 cos S + 5^ sin" S}. This can be made to assume a simpler form by getting rid of the middle term. Thus, let S = a + ] — sin ^ cos ^ {(/ + /') cos 2(o - {e — g) sin 2ft)}. 156 THEORY OF THIN PENCILS. [CHAP. VIII. Choose then disappears ; the coefficient of cos'^ becomes — ^ {e (1 + cos 2to) + (/+/') sin 2(o + g(l- cos 2ft))}, ov —^{e+g +k), and the coefficient of sin'0 may be reduced in the same way. Thus the equation takes the form z = r^ cos'^ + r^ sin'^ where 'ri = -hie+9 + k), 'r^ = -h{e + g-k). Thus the point of shortest distance lies between the fixed limits, determined by the equations z = r^, z= r^. These points are there- fore called the limiting points of the ray, and the corresponding planes parallel to two consecutive rays, namely, the planes ^ = 0, <^ = Jtt, are called the principal planes of the ray. 137. There are two rays of the system whose shortest distances from the principal ray vanish, to a first approximation. The length of the shortest distance between the ray (a, ^, y) and the principal ray, is \x + fiy. This will therefore vanish, to a first approximation, if \{ea+fP)+^L{f'a + g^) = 0. Eliminating the ratio \ : /a, and expressing a, /8 in terms of 8 as before, this becomes /' cos" Z + {g — e) sin S cos h — /sin" 8 = 0, or (5r-e)sin28 + (/ + /')cos2S=/-/. If we substitute ft) + for 8, this equation takes the form cos 2^ {(/ + /) cos 2m + {g- e) sin 2m] + sin 2^ {{g - e) cos 2ft) - (/ + /) sin 2 = 0, which gives (^ = or \ir. The focal points therefore coincide with the limiting points, and the focal planes are at right angles to each other. Every ray, therefore, of a system of normals, passes through two straight lines, each perpendicular to the principal ray, and lying in planes through the principal ray, which are perpendicular to each other. The two focal points coincide if / = 0, /' = ; and in this case all the lines of the small pencil meet in a point. 142. In order to form an idea of the character of a small pencil, it will be useful to find the equation of the bounding surface of the pencil in some particular cases. We shall suppose that the pencil meets the orthogonal surface in a small ellipse, whose axes are in the principal planes. For axis of z, we choose as before the principal ray, and the point where this ray meets the orthogonal surface shall be the origin, and the primary and secondary focal planes shall be the planes of xz and yz, respectively. Then if the focal distances be denoted 141 — 143.] THEORY OF THIN PENCILS. 161 ^y ^1) ^2) the equations of the bounding curves of the pencil are a'^b^-'l (1), z = 0] ::;■} <^). ;:sl ®- Every generator of the bounding surface must meet these bounding curves. Let the equations of any generator of the bounding surface be y=Cz + D] Then, since this line intersects the curves (1), (2) and (3), A, B, C, B must satisfy the relations B' B^ , Av, + B = Cv, + B = Oj The last two equations serve to determine A and G in terms of B and D, and the equations of the generator may be written : = 5 1 If now we eliminate the remaining constants B, B, the equation to the bounding surface is obtained in the form 143. By giving z different values, we may determine the form of the bounding curves of normal sections of the pencil. These ■will in general be ellipses, but they may be circular. They will be circular if z be chosen so as to satisfy the equation ..(i-i)-.t.(i.£)- H. 11 162 THEORY OF THIN PENCILS. [CHAP. VIII. Thus there will be two circular sections of the pencil, one of which always lies between the focal lines. For suppose that v^ is greater than «,, then if we take «e-o='C-3=- we get a circular section, and its position and radius are given by the equations a + b _a b r a b _ It is easy to see from the first of these equations, that z lies between «, and v^^. This circle is sometimes called the circle of least confusion, as will be explained later. 144. But every small pencil has not a circular section of this form ; the preceding argument only applies to symmetrical pencils. For instance, if the curve in which the pencil cuts the orthogonal curve be an ellipse whose principal axes do not lie in the focal plane, we may take as the equation of the bounding curve aa? + 2hxy + by" -■ z- in which case the equation of the bounding surface of the pencil may be shown to be 2hxy by" _^ V vj \ vj\ vj \ vj Whatever constant value we may give to z, the equation to the section can never take the form of a circle. In the case of an optical instrument, however, the orthogonal surfaces are always surfaces of revolution about the axis of the instrument, and the first focal plane of a small part of a symmetrical pencil will be a meridian plane, so that the small pencil has the necessary symmetry,, and therefore has a circular section. 143 — 146.] ELEMENTARY THEORY OF THIN PENCILS. 163 145. It has been seen that when a small pencil emerges from an optical system, it will not in general have a single focus through which all the raj's of the pencil pass, but all the rays of the pencil pass through two focal lines; we must next enquire into the nature and position of the image of an object afforded by such pencils. We may suppose one of the small pencils to be received on a screen. If the screen pass through the first focal line, the section of the pencil will be a small line, say a horizontal line. If a number of such pencils, originally diverging from the several points of the object be received on the screen, each pencil will give rise to a short horizontal line on the screen ; so that the breadth of the image will be greatly exaggerated as compared with its length. In the same way, if the screen be placed at the other focal line of the pencil, each point on the object will be represented in the image by a small vertical line, and the length of the image will be exaggerated as compared with its breadth. Both these images are defective, in that they distort the shape of the object. The best image is given when the screen occupies the position of the circle of least confusion; for then, corresponding to each point of the object, there appears a small circular patch of light on the screen, and if the circle of least confusion be very small, this will not seriously impair the value of the image. The image is therefore taken to be the aggregation of the overlapping circles of least confusion. The size of the circle of least confusion may be taken to represent a measure of the indistinctness. 146. The nature of a small pencil may be investigated in a less satisfactory manner, by Elementary Geometry. When the system of rays is symmetrical about an axis, we have seen that the orthogonal surface is a surface of revolution. We are now going to consider a small pencil of the system whose rays meet the orthogonal surface within a small area. We shall suppose this bounding line of the pencil at the orthogonal surface to be a small ellipse of semi-axes a, b, whose centre is C Let MGN, PBS. . . be lines of curvature, of the system which consists of circles whose centres lie on the axis. Then rays from all points of the line MGN will meet the axis in the same 11—2 164. ELEMENTARY THEORY OF THIN PENCILS. [CHAP. VIII. point q^; and since the area is small, MCN may be taken to be a straight line ; and therefore all rays passing through the line MCN, lie in the plane q^MN and meet in q^. Similarly all the rays which meet the orthogonal surface in the line PES lie in a plane qPS and meet the axis in a point q. Let these planes meet in the line mq^ri. When the line PRS moves up to coincide with MCN this line mq{n, takes up a definite limiting position ; and since the area on the orthogonal surface is small, all the planes such as qPS very nearly pass through this limiting position of the line mq^n. This line is called the primary focal line, and the point q^, where the principal ray meets it, is called the primary focus. Thus all the rays of the pencil pass through the primary focal line very nearly, and will be treated as if actually passing through it. 147. Also all the rays of the pencil meet the axis, and therefore the axis might be taken as a secondary focal line ; but this is not convenient, because this line is not perpendicular to the principal ray of the pencil. A section of the pencil by a plane through q^ perpendicular to the principal ray does not differ much from a straight line, the actual shape of the section is a curve with two loops, like a slender figure of eight. For since all the rays from MCN pass through q^, the. breadth of the section at q^ vanishes. But the rays from the line PRS meet in q, and therefore they will have diverged again before meeting 146 — 148.] ELEMENTARY THEORY OF THIN PENCILS. 165 the plane of section, giving a section of small breadth prs ; and similarly, if we consider a line of curvature below MCN, the rays from it meet on the axis beyond q.,, and therefore will not have met when they reach the plane of section. It appears there- fore that the figure bulges out both above and below q^; but it will be proved that the breadth of the section is a small quantity of the second order, and the breadth will therefore be neglected, and the section will be considered to be a straight line. The breadth of the section may be found from the figure by the consideration of similar triangles. Let 6 be the inclination to the axis of the principal ray ; then pr _ qr _ q^r cot 6 'PR~qR~ ' qR ' since the angle qrq^ is very nearly a right angle. Al«o 2£ _ Mj _ %-^i OR Cq, V, ' where v^, v^ denote the distances Cq^, Gq^, respectively ; and there- fore, substituting the value of q^r, we find PR.GR v.,-Vi , . pr = 5:5 — . cot 0. ^ qR v^ The greatest values of PR and OR are respectively b, a, and qR is equal to v^ nearly, so that pr is of the order ab (w^ — v^/v^v^. But a, b, are very small compared with v^, v^, and therefore the breadth pr is always a small quantity of the second order. The line is called the secondary focal line, and the point q^ the secondary focus. The plane through G and the axis is called the primary plane, and the plane through the principal ray, perpen- dicular to the primary plane, is called the secondary plane. Thus the primary plane contains the secondary focal line, and the primary focal line lies in the secondary plane. 148. If we assume that when we choose a section between the two focal lines such that the breadth of the section in the primary plane is equal to that in the secondary plane, this section is circu- lar, the position and magnitude of the circle of least confusion may be found by elementary geometry. 166 CIRCLE OF LEAST CONFUSION. [chap. VIII. Let mhnk be a section perpendicular to the principal ray and let hok, mon be the breadths in the primary and secondary planes. respectively, and suppose each of these is equal to 2r, and let z = Co. Then by similar triangles, hk:HK=oq,:Cq,\ and also iim : MN=oq^ : Cq, that is. r a r V V, v,-z V, These equations determine the position and radius of the circle of least confusion, and give the same results as before, namely, a + h __ a h r ah When the section of the pencil by the orthogonal surface is circular, then a = b, and the formula to find the circular section is 2 z 1 1 ■■- +- and therefore 2^ is a harmonic mean between v^ and v^. 149. We shall now investigate by elementary geometry the positions of. the primary and secondary foci of a small pencil diverging from a point, after reflexion or refraction at a plane or spherical surface. When a small pencil is reflected obliquely at a plane surface, we have seen that the reflected rays all pass accurately through a point, so that the two foci coincide. In this case there is no circle of least confusion ; the image of a point is a point, and the definition is perfect. 148 — 149.] OBLIQUE REFRACTION AT A PLANE. 167 Next, let the pencil be refracted at a plane surface. Let QP be the principal ray of the pencil, qJP the direction of the ray after refraction. Let the angles of incidence and refraction of this ray be, respectively, <^ and '. Let QP' be a consecutive ray in the primary plane and let this ray after refraction meet qJP in y,. Then, ultimately, when P' coincides with P, q^ will be the primary focus. If QA be drawn normal to the plane, and if the refracted ray produced backwards meet QA in q^, then q^ will be the secondary focus. Let QP = u, q^P = t)^, g-^P = v^. Then fi sin = fi sin ^', and therefore, fj. cos - and similarly d(j>' = a; cos ^' .(1). If we substitute these values of d, d<^' in the previous equation and divide both sides by x, it becomes fjL cos" _IJ-' cos" ' u V, AP u ' AP V. • Also sin = ■ sin x'> ^ with the axis, respectively. Then, if ^, (j)' be the angles of incidence and refraction, ^ = ^ — %, 4' — ^ ~ X- By the law of refraction, fi sin <^ = /a' sin ' ; and therefore fi cos ^ d dx — /J^' cos ^' d^ = (ji cos /u, d^ = X cos ^7*'i' ^'^^ ^^ — ""I''' j ^'^*i therefore fi cos" (^ fj! cos'' ' _fi' cos — jjf cos ^' w j^j r ^ '' This equation determines the value of y,. 149 — 152.] REFRACTION OF THIN PENCILS. 169 Also, by equating areas of triangles, we get the identity A q^PO - A QPO = A q^PQ ■ and if we express these areas in terms of the lengths u, v^, r, this identity becomes v^r sin (f>' — ur sin ^ = uv^ sin (^' — ^). By virtue of the law of refraction, /u. sin 4> = f^ sin + z sin <}>) '] '2B C ' ^J ^•^^• and this is the general form of the characteristic function for a small pencil. 152—153.] REFRACTION OF THIN PENCILS. 171 153. To find the relations between the constants A, B, 0,6, , A', B', C, 6', ' for the incident and refracted pencils. Let the normal to the refracting surface be taken as the axis of z, and the plane of incidence as the plane of xz, so that if cf) be the angle of incidence of the principal ray, the characteristic function for the incident pencil will be of the form (3). The characteristic function for the refracted pencil will be of the same form with" dashed letters for the constants. Let the equation to the surface in the immediate neighbourhood of the point of incidence be The characteristic function is continuous at this surface, and therefore V=V along it. In the equation F-F = 0, substitute the value of z from the equation to the surface ; then neglecting powers of x, y above the second, it becomes K- -fi^' + (^ + lb + ^) (/^cos ^-/x'cos^') - a; (/A sin (^ - ix' sin ^') fa? cos" (A y' xy cos ^\ , (0? cos' ^ y' .o^y c os \ _ 25+~a^J+'' I 2A' +2l'+ ~C~~)-^- This equation must be true for all small values of x, y. Equat- ing to zero the several coefficients, we find K = K', fi sui = fi sin <})' (4); fi cos' <}) fi cos' <})' _ fj. cos 4>~ H' <^QS — fJ-' cos ' _fJ' cos _ f^ If the focal lines of the incident pencil lie, respectivel}', in and perpendicular to the meridian plane, 1/0 = 0, and therefore 1/0' = 0, so that the focal lines of the emergent pencil also lie, respectively, in and perpendicular to the meridian plane. The same is true if the incident rays diverge from a point; for in this case a=b, so that 1/0 = 0, and A = B. If, as before, we denote the distance of the radiant point from the surface by u, and the distance of the foci by v, v', the equations 154 — 1.56.] REFRACTION THROUGH A PRISM. 173 take the form II cos" (^ fji! cos' ' _fi cos — fjf cos ^' u V r ' /J, /a' _ /"■ cos (j) — fjf cos (f)' u v' r ' which agrees with the result previously obtained. 156. To find the focal lines of a small pencil after refraction through a prism. We shall suppose the axis of the pencil to lie in a plane perpendicular to both faces of the prism, and that the pencil passes through the prism so close to the edge that the length of its path inside the prism may be neglected. Let <^, ^' be the angles of incidence and refraction of the principal ray at the first surface, ■y and i|r. the angles of incidence and emergence at the second surface. The plane of incidence is the same for both refractions. Let the characteristic function for the incident pencil, when the axis of z is in the direction of the principal ray, be ^ ^^\ 2A 2B Cj' and let F, be the corresponding function for the pencil after one refraction, so that and after the second refraction let the function be the positive direction of the axis of z being in each case opposite to that of the transmitted light. Then, at the first refraction the equations connecting A, B, G, A^, B^, (7,, are cos' ^ _ /A cos" <^' "^2 Ar~ B B, cos 4> _fj' cos 4>' 174 REFRACTION THROUGH A PRISM. [CHAP. VIII. and at the second refraction the equations connecting A^, J5,, (7,, and A', F, C, are /x cos' y{r' COs'^lr ~ A' • A' 1 ~ B" fl cos l|r' cos y^ C, ~ G ■ Eliminating A^, B^, G^ between these two sets of equations, we arrive at the equations 1 cos" ^ A cos''' 1 cos" ^{r A' cos" 1^' 1 B^ 1 ^B" 1 cos ^ 1 cos ■yjr G cos ^' G' cos 1^' ' These equations completely determine all the circumstances of the emergent pencil. If 1/(7 = we must have l/C = also ; that is, if the focal lines of the incident pencil be, respectively, parallel and perpen- dicular to the edge of the prism, the focal lines of the emergent will also be, respectively, parallel and perpendicular to the edge of the prism. If the incident pencil diverge from a point, so that 1/C7 = 0, and A = B, the emergent pencil will not in general diverge from a point, but from a pair of focal lines, respectively parallel and perpendicular to the refracting edge ; but if the principal ray pass through the prism with minimum deviation, we shall have ' = yjr', so that A = A', and therefore A' = B'. In this case, the emergent pencil will diverge from a point. The distances of the two foci from the edge of the prism are, respectively, A and A', measured both in the direction of the passage of light ; thus the foci are equally distant from the refracting angle of the prism. 156 — 157.] REFRACTION THROUGH A CYLINDRICAL LENS. 175 157. Next we shall consider the refraction of a small pencil through a thin lens both of whose surfaces are cylindrical, but such that the generating lines of the two cylinders are inclined to each other at an angle 22. Let the radii of the surfaces be a, h, and in the case considered, let these radii and all distances be measured in a direction opposite to that of the incident pencil. Let the axis of the lens be taken as the axis of z, and let the plane of yz bisect the acute angle between the generating lines of the two cylinders. Let the circumstances of the pencil be originally expressed in the manner of § 152 by the constants A, B, G, and after one refraction by J.,, £,, Oj, and after the second refraction by A', B', C. Then the values of P, Q, R for the first surface are P = Q = R = cos a a sin'' a a sin a cos a a and for the second surface the corresponding values are cos' a P' = R' = sm a sin a cos « Moreover the angles of incidence and refraction are very small, so that cos ^ = 1, cos ^' = 1 approximately, and therefore the equations connecting the constants are, ^ < 1 (/x- A - 1)003" a a 5, 1 i^- B~ a C, 1 (/.- - 1) sin a. cos a a 17C REFRACTION THROUGH A CYLINDRICAL LENS. [CHAP. VIII. and also 1^ A' 1 B' 1 C" A,' (1 — /i) cos' a ' h (1 — /Lt) sin' a ' h (1 — /u.) sin 1 cos a If we eliminate A^, £,, Cj, by adding the corresponding equations of these sets, we get finally i-^ = (/._l)cos'ag--J) i,--i=(^-l)sin'a(l-^) 11,,.. /I 1\ ^,-^ = (^-l)smacosa(^- + ^j 1 B C Now if M, V be the distances of the two focal lines from the lens, and if 6 be the angle between the plane of yz and the focal line corresponding to u, measured from the axis of y towards that of x, then 1 cos" e sin' e A' u ^ V 1 sin= e cos" e B' u V ■; rii = { 1 sin ^ cos 6 C \u v) and if the incident pencil be diverging from a point at a distance x from the lens, then 1 1^ A X 1 1 B X h" Hence, subtracting the first two equations, g-i) cos 2^ = 0.-l)cos 2.(1-1); 157 — 158.] OBLIQUE REFRACTION THROUGH A LENS. 177 by the third equation. A lens of this kind does not bring a pencil radiating from a point to a single focus, and therefore it is said to be astigmatic, and the measure of the astigmatism may be taken to be the value of 1/m ~ 1/v. To find this value we must square and add the last two equations ; and so we obtain 1 1 / IN /i i 2 7 = (m— 1) \/ — + 15 r COS 42. M t) ^^ ' \ a ¥ ab If the second surface be plane, 1/v = 0, and therefore 1 1 ^-1 Such a lens has been called by Professor Stokes an astigmatic lens, and the line through the centre of the leus in the direction of the generators of the cylindrical surface the astigmatic axis of the lens. 158. To find the focal lines of a small pencil after oblique centrical refraction through a thin lens. The central ray of the pencil passes through the centre of the lens, and will therefore emerge parallel to its original direction. Let ^ be the angle of incidence and final emergence of the principal ray, ^' the angle of refraction within the lens. The plane of incidence is the same for both refractions. Let the characteristic function for the incident pencil, when the axis of z is in the direction of the incident ray and the plane of xz the plane of incidence, be ^ and let the corresponding functions after one and two refractions be, respectively, H. 12 178 OBLIQUE REFRACTION' THROUGH A LENS. [CHAP. VIIL /i being the refractive index of the substance of the lens, and the direction of the axis of z being in each case opposite to the direction of the transmitted light. Then at the first refraction, cos' ^ /i cos' (^' _ cos — fi cos ' 1 /i _ cos <}> — fJ- cos <^' cos fJ, COS 4*' _ n and at the second refraction, /t cos' ' cos' /li cos /JL 1 fi COS ^' — COS <^ B. ~B'^ »•' • fl cos ' COS ^ _ r. c. c where r, r' are the radii of the spherical surfaces. Eliminating A^, B^, C^, the relations between the constants of the incident and emergent pencil are cos' ^, - -g = (/. cos \ = = (u cos A' - cos A) ( > I . In the case in which ^ and ^' are small, we may substitute 158 — IGO.] CURVATURE OF IMAGES. 179 COS ^ = 1 — J^", COS ^' = 1 — ^', and therefore cos<^'=i-|;. Substituting these values for cos0 and cos(/)', we get /i cos (/)' — cos ^ = (/i — 1) j 1 + ^ L If, therefore, we call the focal length of the lens /, so that v' n f\ ^^,m\ 160. One of the defects of an image of an object as seen through a lens, is that it appears curved, when the object is plane. We proceed to find the curvature of an image close to the axis ; in other words, we shall compare the curvatures of sections of the object and its image made by any plane through the axis of the lens. This curvature of the image will depend partly upon the obliquity of the pencils proceeding from the points of the object more remote from the axis. Let u be the distance of the point of the object under consideration from the centre of the lens, and <^ the obliquity of the principal ray of the pencil. Then, in general, there are two focal lines at distances v, v, from the centre, where 1 1 1 (, and subtract the latter, and then we get qn QiV Cp.Gq UP.CQ f = -j\l-^-l-Tc4>^ :-}(|h-A.)<^'. But Cp.Cq = Gp', ultimately, and therefore Gp.Cqi^'={Gp.,\>Y = pn\ and similarly, GP .GQf= [GP . ^f = PN\ so that we get qn pn' QN _ (2i+l) PN' 2/ • And if p, p be the radii of curvature of the object and image, respectively, by Newton's theory of curvature l/2p' = qn/pn^, and 1/2/) = QN/PN", ultimately, so that 1 1^ (2A; + 1) P P~ f ' This gives the relation between the curvatures of an object and its image. The curvature of the image is independent of the position of the 160 — 161.] THEORY OF ANY SYMMETEICAL INSTRUMENT. 181 object ; for the relation between the curvatures of the object and its image is independent of the distances u, v. If the object be plane, so that the curvature of the section of the object is zero, 1/p = 0, and therefore 1 (2/j+l) P f ■ The radius of curvature has therefore a sign opposite to that of/ The value of h depends, as has been shown, on the focus chosen to represent the image ; for a primary focus, & = 1 + 1/2/i ; for a secondary focus, k = l/2fi, and for the circle of least confusion k is the mean of these values. For the geometrical focus, k = 0, and therefore JLl _1 P P f 161. The characteristic function maybe applied to investigate the theory of any optical instrument symmetrical about an axis. For, take two fixed points 0, 0' on the axis, in the first and final media, respectively, as origins, the axis of the telescope being the axis of z, the positive direction of the axis at being opposite to the direction of the incident light, and the positive direction of that at 0' being in the same as that of the emergent light. Consider the part of the characteristic function between two points {x, y, z), {x', y, z) on any ray, situated respectively in the first and final media. Let the ray through these points meet the planes of x\) at 0, 0' in the points (^, r)), (^', r\). Then if V be the value of the characteristic function from the point (^, t)) to the point (f', r{), and F„ the value from to 0', F may be expanded by Taylor's Theorem in the form, V ^0^^ ^^+v ^^+^ ^^.^v ^^, + terms involving higher powers of ^, 17, f, 77'. 182 THEORY OF OFl'ICAL INSTRUMENTS. [CHAP. VIII. In all the coefficients, it is supposed that ^ and 7; are made equal to zero after differentiation, so that the coefficients are constants. We may therefore write the value of V in the form + I af + c^V + h W + I a'r + cW + i ^ V + Let U be the characteristic function from the point {x, y, z) to the point {x', ?/', z) ; then from the definition of this function U=V + (,{{x-^f + (y- rif + ^f + /.' [{X' - ^y + iy' - r,J +/f . + iaf +0^^7 + 1 &»?' + pW + q^v + r^'v + svv' + Ja'r + cW + |6V+ But since the function is symmetrical with regard to the axis of z, we must have /=0, 5-^0, / = 0, g' = 0. c = 0, c' = 0, q = o, r = 0, a = b, a' = b', p = s. The characteristic function, therefore, reduces to U= V, + ,,z + /.V + 1^ |(^ - ^y + {y- rif ■ +p(?r+w)+ the square roots being expanded by the Binomial Theorem. The characteristic function is stationary for small variations of fi V, f. v'> and therefore, rejecting higher powers of small quantities, 161.J THEORY OF OPTICAL INSTRUMENTS. 183 Solving the first and third equations to find f, f , we get ^f=f («'-5)-'v ^ j where i) = (a+fj(a' + '^)-p', and similar formul?e hold for 17, tj' in terms of y, y'. Hence i)(.-?)={.(a'+Q-/}. + ^ ' But the characteristic function U may be written in the form, +^(«+^)f+i'r}+ir(Ff+(«'+f^)f} + similar terms in y, ij = F.+/.^+/.v+g(«.-i)+|^'(<.'-r) + similar terms in y, 17. Substituting the values of x- ^, m — f' in terms of x, x this becomes 184 THEORY OF OPTICAL INSTRUMENTS. [CHAP. VIII. + a similar terra in. y, rj. I^et g= -5 — — -" 9=, - p — aa p —aa /•_ MjJ j^,_ tLlL^ •'~ p'-aa" ^ p'-aa" now that these letters have ceased to enter into the investigation with other meanings. Then, if in the expression for U, we multiply the numerator and denominator of the fraction by zz', and divide them by aa —p", it easily reduces to r7_ TT , „- , ,/ ' , , ^ (^' - g') ^^ + /^' (^ - g) ^" + (// +/m) xx '-' — ' "1" A** T" /* •2 T -J 7 r-r-; r, ttt, i^-ff){^ -9)-J/ . . fJ^jz- g) f + H''{z- g) y" + iff,' +/» yy ^ {^-g){^'-g')-ff 162. If (I, 1)1, n) be the direction cosines of the incident ray, through the point {x, y, z), by the properties of the characteristic function we learn that dV dV dV I : m : n = -r- ■ -t- ■ -;—■ ax dy dz If we put, for brevity z-g=ul z -g =u] and remember that ffj.' =/V. these ratios become I _ m _ n u'x +/V u'y +f'y' uu — Jf ' Similarly if (Z', ml, n') be the direction cosines of the emergent ray, we may show that ;' / / ' TO _ M ual +fx uy' +fy uu' — ff ' Now suppose that one of the two points, say (x, y, £), remains fixed while the direction of the ray through it changes ; then the 161 — 162.] THEORY OF OPTICAL INSTEUJIENTS. 185 first set of these equations determine where any ray meets the plane z = z . But if X , y', /, be chosen so that the denominators of the ratios vaaish, that is, if u'x = -fz'' '^i-'y = -f'y'\, uu= ff, then this set of equations will be satisfied for all values of the ratios I : m : n. In other words, whatever be the direction of the incident ray through (x, y, z), it will pass through the point (*'. y, z), as determined by these equations. It is easy to see from the equations giving the ratios V : in : n that this property is reciprocal ; hence {x, y, z), {x , y, z) are conjugate points. Their coordinates are connected by the equations uu'=ff, X _2/_ u Id y' "7 x' -t-. u' X y ~ f. These equations at once give the positions and characteristic properties of the cardinal points of the system of which we have given an account in connection with Gauss' theory of any system of lenses arranged symmetrically along an axis. Thus if we make u infinite, we get u = 0, and vice versa; so that rays which are parallel in the first medium all pass through a point on the plane u = 0, in the second' medium ; in other words w' = is the equation of the second focal plane. Similarly M = is the equation of the first focal plane. Also if we take u = —f, and therefore u = — /', we find X = af y = y' This proves that any ray meets the planes u = — /, u = — /', in points, such that the line joining them is parallel to the axis of the system. These are therefore the planes of unit magnification, or principal planes. From these cardinal points we may deduce the positions of the nodal points, and apply any of the constructions previously 186 THEORY OF OPTICAL INSTRUMENTS. [CHAP. VIII. given for determining the position of a focus conjugate to a given focus, or the direction of an emergent ray corresponding to any incident ray. 163. These constructions fail in the case in which if = aa! , for then the values of the coordinates of the cardinal points and the principal focal lengths become infinite. The characteristic ' function becomes a.i-' + I'pxx' + aal' U= V. + imz + im'z' + 1- uz a z , - + ^- + 1 + a similar term in y, y. If we use this form of the characteristic function, the direction cosines of the incident and emergent rays are given by the equations I m 11 ax+px' ay+pq' r .( ^%^+l) px + a'x' py-\-a']/ , laz aV N' By the same reasoning as before, it may be shown that the relations between a pair of conjugate points are determined by the equations ax + px' = ' ay+py' = The first two of these equations may be expressed in the forms of y' a - = !. = - P. X y a' ■ It appears, therefore, that the linear magnification is constant for all positions of the object, namely, X a ay. 162 — 164.] THIN PENCILS IN HETEROGENEOUS MEDIA. 187 Also, for any rays whatever, we have shown that fjd III' ax + j)^ px + ax' ' and by virtue of the relation p^ = aa', this may be written fd _ fi'l' a p) We may also prove a similar equation between m and m', and therefore the direction cosines of the incident and emergent rays are connected by the relations V _'ni' _ fi p I in ft a _ fjJc ~' 7 • A* The interpretation of this equation is that /i^■/yu,' is the angular magnification ; and therefore the angular magnification is constant. Rays which are parallel in the first medium are parallel also in the final medium. This is the case of an astronomical telescope directed towards a star when focussed to suit a normally sighted person, 164. To find the characteristic function for a small pencil after passing through any heterogeneous medium. We shall suppose the initial and final media to be homo- geneous, of refractive indices fi and /i', respectively. Let be a fixed point on the principal ray in the first medium and 0' a fixed point on the same ray in the final medium ; we shall take 0, 0' for origins and the directions of the principal ray for axes of z in the two media, the positive direction of the axis of z in the first medium being opposite to the direction in which light is travel- ling, and in the second medium being in the same direction as that of the light. Let any ray meet the plane of xy at in a point (x, y), and that at 0' in a point («', y'). Then if Z7be the value of the characteristic function from the point {x, y) to the point («', y'), and [/"o the value from to 0', U may be expanded by Taylor's, theorem in the form 188 THIN PENCILS IN HETEROGENEOUS MEDIA. [CHAP. VHI. dU dU ,dU ,dU da^+^d^+^'d^' + ydj' ,(PU d'U , , ,d:'U "^d^' + '^'^dMiz + ^y df "^ dxdy , d'U , d^U , d^U , d'U +^-" d^'^'^y d^^'^yndy-^yy s^' , ^d'JJ , , d'U , „d'U +^' d^'+^^d^'+^y w + terms involving higher powers of x, y, x', y'. This may be written U=U, +fx + gy +/V + g'y' + ^ ax^ + cxy + if hy^ +pxx' -r qxy + rx'y + syy' ■\-^a'x' + c'x'y' + \h'y'^+ ..., the constant coefficients being functions depending on the nature of the medium and supposed known. Let the characteristic function of the incident pencil be and that of the emergent pencil then, noting the directions of the axis of z in the two cases, we must have U=V+V'; that is, K + K'=U, +fx + gy +f'x'+g'y' + i aa;' + jxy + ^ jif + pxaf + qxy + rx'y + syy' + la'x" + y'xy' + i^'y"+ where a = a + and a.', fi', y' denote similar expressions for the second medium. 164.] THIN PENCILS IN HETEROGENEOUS MEDIA. 189 Now by the properties of the characteristic function this equation must still be true if we let the points {x, y), {x', y') undergo small displacements ; by differentiation we arrive at the following equations : ox + 7y +px' + qij +/= ' 7,r + ^y + rx' + sy' + r/ = px + ry + a V + ^'y +/' = qx + syJr ix + Py' +g' = _ But since x, y, x , y, all vanish together, /=0, g = Q, /' = 0, g'=0. Solving the first two equations for x, y iu terms of x', y, we find a:S = (r7 - p^) x' + {sy - q^) y, yB = (py- ra) x' + (qy - sx) y', where S = ayS — 7*. Hence, substituting these values into the other two equations, we must have x' (a'S + pry — p^^ + rpy — r'a) + y (y'B + psy — pq^ + qry — rsa.) = 0, a/ (y'S + qry —pq^ + spy — sra) + y' (yS'S + qsy - g'/S + qsy - sV) = 0. These equations must be true for every ray, that is, for all values of x', y, and therefore a'S =p^P — 2pry + r'a) y'B =pq0 — {ps + qr) 7 + rsx \ . ^'B = (f^-'2,qsy + s\] By means of these equations the coefficients a', /S', y are determined ; these serve to determine the constants A', B', C and all the circumstances of the emergent pencil. The coordinates (x, y), (x', y') are connected together by a linear transformation. If therefore the small pencil be cut by the plane of xy at in an ellipse, then the section of the pencil by the plane at 0' will also be an ellipse. This proves that if at any point the normal section of the pencil be elliptical, all its normal sections will be elliptical. 190 EXAMPLES. [chap. VIII. EXAMPLES. 1. A ray passes through a right-angled prism with minimum deviation, meeting the face of the prism at a distance y from the edge. Show that the foci of the emergent pencil will be separated by a distance y^/2 (/i^ - l)//i. 2. The image of a straight line perpendicular to the axis of a convex lens at a very great distance from it approximates to a parabolic curve, whose equation is 2(.r+/) + (2-|-J)^=0, the centre of the lens being the origin, and the circle of least confusion being taken to be the image of any point. 3. A small pencil falls obliquely on a looking-glass, and emerges after reflexion at the silvered back. Show that it proceeds from two focal lines, whose distance from one another is 2< cosset)' - cos^ and <^' are the angles of incidence and refraction at the front of the glass, and t the thickness of the glass. 4. A double convex lens has faces whose radii are each 8 inches, and its refractive index is (^3+1) /^2 ; find its power when its axis is inclined at an angle of 30° to the line of sight. 5. A pencil of rays diverging from a point P, whose position is variable, is incident on a refracting sphere at a given point in a given direction ; if § be the corresponding primary focus after refraction through the sphere, G the position of § when the incident pencil consists of parallel rays, F that of P when Q is at an infinite distance, prove that PF.qG^^"^"^-^^^^ 16 sm2(^-<^) where a is the radiiLs of the sphere, <^ the angle of incidence on the sphere and <^' the angle of refraction. Show how to find the corresponding theorem for a pencil refracted through any number of spheres, the axis of the pencil lying always in one plane. 6. A small pencil of parallel rays is refracted centrically through a double convex lens, the radii of whose surfaces are each equal to r, and whose thick- ness is t ; show that, if the square of t be neglected, the distance of the primary focus from the point of emergence of the pencil will be r sin ' cos^ t sin ' cos^ (f> 2 sin (<^ - <^') 4 sin cos' <^' ' '\ r being the radius of the sphere. 8. A ray of light QAST passes through a plate of glass bounded by parallel planes, A being the point of incidence and S the point of emergence ; prove that if q^, q^ be the primary and secoiidarj' foci, and 8q^ = v^ '%2=^2) AQ = %i, then t co.s^ (A t:, = u+- fj. cos and <^' the angles of incidence and refraction of the axis of the pencil which passes through the prism with minimum deviation, then if the length of the path of the pencil inside the prism be so small as to be negligible compared with the other lengths involved, show that = — =-7 (u. cos rf)' - cos ' l sin •^ = iJ. sin i|r' ^ . ' + d(j)', yp-' + di]r', ip- + d^jr denote the corresponding angles for any other ray, whose refractive index is /i + d/j,. Then by differentiating the relations between the quantities ', yfr, i|r', /A, we have cos ' + fi cos ' d(p' cos -^ d-^jr = dfj, sin yjr' + /j. cos yfr' dyjr' d' + df = 0. From these equations we may eliminate 9^' and d-^' by multiplying the first equation by cos •^' and the second by cos <^' and adding ; in this way we get cos <^ cos i/r' 9(^ + cos l/r COS 0' 9l/r = 9/i, {sin ^' cos l/r' + Sin T|r' COS ^'} = 9/isin(^' + -«/r'). 172 — 174.] DISPERSION BY A PRISM. 203 and therefore cos (f> cos l|r' d(f) + cos T^ COS ' must be a maximum. Equating to zero the first differential of this expression, we easily deduce the equation tan y^df + tan ^' dj>' = 0. If in this equation we write d(f>' = — dyfr', it becomes tan 1^ d-yfr = ta,n ^' 9\|r'. But the angles ^jr, yjr' are always connected by the relation sin '>{'' = /J. sin yfr', and if we take logarithms of both sides and then differentiate, we get cot ylrd\}r — cot ■^' d\}r'. 204 POSITION OF MINIMUM DISPERSION. [CHAP, IX. We cau now eliminate the ratio dyjr : 9i|r', and we thus arrive at the equation tan' i/r = tan ^' tan yjr'. By combining this equation with the equations siii ifr = /x sin ■^■' and ^' + -i|r' = t, the position of minimum dispersion is completely determined. The elimination of yjr and ^' may be conducted in the following manner. The relation between i/r and -^^ when expressed in terms of tangents, becomes tan'i/r _ jjHa^yjr' l+tan'i/r ~ l+tan'-f' ' and by substituting for tan' yjr its value tan (f>' tan yjr', we get ,^ ,, tan<^'(l+tan'-f') u' t&nyjr = - , , , , , ,, , '^ ^ 1+ tan ^ tan yjr / » i\i ,' tan ' = 0, when cos = 1; after this 0' becomes negative, and cos ' diminishes from unity to cos (7 — t). Also, cos yjr increases at first up to unity, when -\^ = 0, and then yjr becomes positive and increases up to Jtt, so that cos yjr diminishes from unity down to 174—177.] DISPERSION ])Y PRISMS. 205 zero. Thus at first the dispersion gets smaller, then it attains a minimum, and afterwards increases without limit. Since the dispersion may be indefinitely increased by adjust- ing the position of the prism with respect to the incident ray, it follows that the dispersion jvoduced by a imsm, whose refracting angle is ever so great, may be counteracted by the dispersion of another prism of the same material, whose refracting angle is ever so small. 17G. To find the dispersion produced by two prisms whose refracting edges are imrallel. Let I be the refracting angle of the second prism, and fj.' the refractive index for the standard ray of the substance of which it is composed ; and let <^", ^"', yjr", ■^"' be the angles corresponding to ^, ^', ■^, -y of the first prism. Then we have the following relations between the angles sin ^ = /i sin <^' , sin -v|r = /x sin ■^' , sin <^" = /i' sin ^"' , sin i|f" = y! sin ■^"' , ^ + ^' = i, <^"' + ^"' = i'. Also, if the inclinations of the adjacent faces be 6, then i|r + ^ ' = ^. Taking variations of all these quantities corresponding to a ray whose refractive indices are /4 + 9/.t, /i' H-^/^', we get cos ^ cos 1^' 3(^ + cos -v^ cos <^' 9T|r = 9/i sin i \ cos <^" cos i|r"' 3^" + cos •\|r" COS <^"' 9i/r" = 9/A'sin I) ' ^^jr + ^^" = 0. To eliminate 9-^ and 9^", we have only to substitute their values from the first equations into the last, and the equation thence resulting will contain only 90 and 9if-", and it will therefore be a relation between the dispersions of the initial and final rays. 177. We may proceed in the same manner to find the disper- sions produced by the combination of any number of prisms whose axes are parallel. For brevity, let sin I cos 4> cos y}/ Q = p == ~ . ^ COS l/r cos ' ' COS (j)' COS ifr ' 206 DISPERSION BY PRISMS. [CHAP. IX. where the angles , ', y}r, yfr', i belong to any one of the prisms, and suppose that there are n prisms, distinguished by symbols with suffixes 1, 2, B...n. Tlien if we apply the equations alreadj-^ obtained to each prism in succession, we obtain the following equations Hi +^4>, =0 If we multiply the first set of equations by 1, p„, J>„P„-i, PnFn-iPn-5. >P„Pn-i Pii ^^^ ^'^'^ them, all the angles but dyjr^^ and 9^, will disappear by virtue of the second set of equations; so that H« +PnPn-X---P2 P. 9^1 = 9/^„ g„ + 9/«„-l ff,,-! P^ + 9/^,.-. qn-l P„Pn-l+--+^/^iqxPn Pn-l ■ ■ ■ Pi- This is the relation between the angles of dispersion of the pencil at incidence and emergence. 178. When the refracting angles of the prisms are small, and the ray passes nearly perpendicularly through them, the value of the dispersion assumes a very simple form. For if t, i', i" ... be the refracting angles of the prisms and fi, /j,', fi" ... the refractive indices of the standard ray for the different substances of which they are composed, the deviation produced by the first prism will be and there will be similar expressions for all the subsequent deviations. If we denote the whole deviation by A, we get by addition, A = (/.- 1) t + (/-I) t' + (/'- 1) c" + ... 177 — 179.] ACHROMATISM. 207 and therefore for any other ray the dispersion will be given by the equation ^^ = ^^li + ^^Ji' i,' + d,x,"i"+... Let IS, ct', -bt" be the dispersive powers of the different substances for this ray, then the value of the total dispersion becomes SA = •sr (/li - 1) t + ro-' (/ - 1) t' + ... 179. If a ray of light be made to pass through two prisms in succession, it is always possible to adjust their refracting angles, so that the dispersion produced by the first may be counteracted approximately by the second, and consequently that the emergent ray may be without colour. This Newton conceived to be impossible, without at the same time making the deviations of the two prisms counteract one another, so that the whole deviation of the pencil would disappear. He appears to have been misled by an accidental circumstance in an experiment in which he counteracted the refraction of a glass prism by enclosing it in a water prism ; he had mixed sugar of lead with the water to increase its refractive power and this gave it a higher dispersive power also, and it so happened that the emergent ray was colourless, when by properly adjusting the angle of the water prism the emergent ray was made parallel to the incident ray. From this he concluded that the dispersion of all substances was proportional to the deviation of the mean ray, and that therefore the dispersion could never be destroyed so long as any refraction took place. This made him despair of improving refracting telescopes, and led him to turn his attention to the application of mirrors to these instruments. Newton's mistake was first discovered by a gentleman of Worcestershire named Hall, who made the first achromatic telescope. This discovery, however, was allowed to fall into oblivion, until the experiment was again tried by Dollond, an optician in London, who found that the dispersion could be corrected without destroying the refraction, and therefore that Newton's conclusion was not correct. We have seen however that different coloured rays are not dispersed in the same proportion by different substances ; or in other words, that the spectra formed by prisms of different 208 SECONDARY SPECTRA. [CHAP. IX. substances are not geometrically similar. Hence, if the prisms be arranged so as to unite two rays (for example, the extreme red and the extreme violet rays) in the emergent beam, there will be still a small dispersion of the otlier rays. Thus the beam instead of emerging quite colourless, will form a second but much smaller spectrum ; this is called the secondary spectrum. Also, it will be found that by using three prisms of three different materials, three rays of the emergent beam (for example, the red, gi-een and violet) may be united; but still, owing to the irrationality of dispersion, the other rays will not be quite united, and there will be another still smaller spectrum called a tertiary spectrum ; and so on indefinitely. In theory, therefore, it is impossible to attain perfect achromatism, without the use of a very large number of different media; j'et in practice these successive spectra rapidly grow fainter and become insensible ; so much so, that it is seldom deemed necessary to combine more than two rays. The two rays selected will not be the extreme red and violet rays, because these are comparatively faint ; it is better to combine the two rays whose brightness and difference of colour are greatest, such as a ray from the yellow-orange and one from the green-blue. The first successful attempt to get rid of the secondary spectra was made by Blair ; an account of his work was published in the Phil. Trans. Edin., 1791. He found that in the spectrum of hydrochloric acid, the more refrangible part of the spectrum, green to violet, was much more contracted, and the less refrangible part of the spectrum more dilated, than in most metallic solutions; and by mixing the chlorides of antimony and of mercury in suitable proportions with hydrochloric acid, or with salammoniac, he obtained a fluid which, while having a different absolute dispersion from crown-glass, gave a spectrum geometrically similar to that of crown- glass. When a combination of two lenses or two prisms was constructed out of this fluid medium and crown-glass, in such a way that in the emergent beam of light two differently coloured rays should be united, the emergent beam was absolutely without colour. Blair's object-glasses were considered as of singular merit at the time, but through certain inconveniences attending lenses made of fluid media they never came into use. What Blair effected with fluid lenses. Professor Abb^ of Jena 179 — 180.] ACHROMATISM OF PRISMS. 209 claims to have now achieved hy his discoveries of new kinds of glass. In 1881, Professor Abb^ assisted by Dr Schott, commenced the work of examining the optical properties of all glasses, that is, of all known substances which undergo vitreous fusion and solidify in non-crystalline transparent masses. The work was continued till the end of 1883, and directed towards the solution of two practical problems. The first of these was the production of pairs of kinds of flint and crown-glass, such that the dispersion in the various regions of the spectrum should be, for each pair, as nearly as possible proportional. The second problem was the production of a greater multiplicity in the gradations of optical glass, in respect of the two chief optical constants, the index of refraction and the mean dispersion. The first problem has been satisfactorily solved, with the result that achromatic lenses of a much more perfect kind than have ever before been attainable are now being manufactured ; and the second has also been successfully carried out, and a whole series of new glasses of graduated properties are at the service of the optician. 180. We shall now find the condition that when a ray of light from the sun falls upon a combination of two prisms, the emergent ray may be colourless. To do this we shall investigate the condition that two of the brightest rays may be united in the emergent beam, and shall suppose that the secondary spectrum is so small that it may be neglected. Let one of the rays be chosen as the standard ray, and let the refractive index of the other ray be expressed by means of a small variation from that of the standard ray, as before. Then, since the incident and emergent rays are united, we shall have d<)) = 0, d-\{r" = 0, in the equations of § 176 ; these equations will therefore be cos 1^ cos " from these equations, we find dfi sin I cos " cos yjr'" -\- djj! sin i cos i^ cos (/>' = 0. H. 14 210 ACHROStATISM OF PKISMS. [CHAP. IX. The angles yjr, <})', (f>", yjr'" which occur ia this equation are conrrected by the relations sin ijr = /J, sin (t — '), sin " = fi sin (t — i/r"') ; so that if fi, p! be given, there are four independent angles entering into the equation of condition, namely, i, l, ifr and (f)". Hence if the first prism be given in position so that the light falls upon it at a given angle of incidence, the angles t and ■yfr will be given, and i and (f)" will remain arbitrary. The equation of condition may therefore be satisfied in two ways, either by fixing the position of the prism and varying its refracting angle, or by varying its position when the refracting angle is given. If the prisms are of the same material, so that dfi. = dfi', the emergent beam may still be achromatised in either of the two ways. If the prisms are both placed in their position of minimum deviation for the standard ray, the equation of condition for achromatism assumes a simpler form. For in this case, cf>' = '>jr' = ^i and (j)'" = ~ilr"' = ^i ; so that the equation becomes dfj. sin I cos <})" cos ^«' + 9/i' sin i cos (p cos Jt = 0. If now we divide each side by 2 cos ^i cos Jt', and notice that sin (^ = /4 sin Jt, sin (f)" = /a' sin ^i, this equation reduces to fj/dfj- tan " = 0, where <^ and (j)" are determined by the two preceding equations. 181. When there are n prisms, the condition of achromatism may be found in the same way. To unite two rays, one of which is chosen as the standard ray, and the other a ray whose refractive indices for the different media differ from those of the standard ray by small increments, we must make 9^^ = and 9i|r^ = in the equation of § 177, and then the condition reduces to 9/^„ q„ + dij,„_, g-,,., 2\ + diJ,„_, q,^_^p„ p„^, +. . .+ 9/i. q,p„ p„_^. . .p^ = 0. When the ray of light passes nearly perpendicularly through a series of prisms of small refracting angles, this equation assumes the simple form 180 — 182.] ACHROMATISM OF LENSES. 211 With n given prisms, it is possible to form a combination which will unite n rays of the spectrum. For suppose that the substances and the refracting angles of all the prisms are prescribed, and further that the first prism is fixed in position, so that the light falls upon it at a given angle of incidence, then the inclinations of the adjacent faces of consecutive prisms will still be at oar disposal. These {n — 1) arbitrary angles will enable us to satisfy the (n — 1) equations which express the conditions that {n — 1) rays of the spectrum may emerge in the same direction as the standard ray. If the prisms be made of the same material, dfi„ = ^/J,^_,^ =...=d/J,^, and therefore if the combination be achromatic for one pair of colours, all the coloured rays will be united. In this case, perfect achromatism may be secured by one relation among the angles at our disposal. Achromatism of lenses. 182. By the proper combination of lenses the dispersion of differently coloured lights may approximately be destroyed ; just as in the case of light passing through two prisms, the dispersion produced by one lens may be approximately counteracted by that produced by a second lens, so that the emergent rays may be with- out colour. We shall first confine our attention to the approximate theory of lenses, in which the thickness of the lens is neglected and the principal points considered as coinciding in one point called the centre of the lens. For the accurate theory of lenses becomes in this case much complicated by the fact that the principal points of the lenses, from which all distances are usually measured, them- selves vary in position according to the refractive index of the particular ray we are considering. In all cases we shall let /i, be the refractive index of the standard ray, and /a + 9/i the refractive index of any other ray. The focal lengths of the lenses will be supposed to be expressed in terms of the refractive index of the standard ray. It will be useful to find the change in the focal length of a lens, as the ray changes from the standard ray, to any other. The 14—2 212 ACHROMATISM OF LENSES. [CHAP. IX. value of the focal length of a double convex lens, the radii of whose bounding surfaces are r, s, respectively, is given by the equation, where /i is the refractive index of the substance for the standard ray. Giving a small variation to fi, so that it becomes fi + d/j,, this equation gives _ a/i_ 1 and therefore, if we denote the dispersive power of the medium by XT, the variation of the focal length is determined by the equation KM- 183. When an image is formed by a lens or system of lenses which is not achromatic, the light being not homogeneous, it will be affected by dispersion in the lenses in two particulars ; first, the different coloured images will be distributed in different positions along the axis of the system, and secondly, the coloured images will have different magnitudes. In certain cases both these defects can be removed, in other cases only one of them can be removed, and to choose which correction shall be made, it will be necessary to consider the use to which the system is to be applied, so as to remove the defect which is of most consequence. For the object-glass of a telescope two lenses are used, and are placed close together so as to act as one lens. Then a point and its image always lie on the same line through the centre of the lens, so that if the lenses be corrected so that the differently coloured images all lie in the same plane perpendicular to the axis, they will all have the same magnitude. It will therefore be necessary only to make the first correction, and then the other will be satisfied. These object-glasses are usually made of a double convex lens of crown-glass outside, combined with a double concave lens of flint-glass, which has a higher dispersive power than crown-glass. It is easy to see in a general way how the correction may be 182 — 184.] ACHROMATISM OF LENSES. 213 effected. By the convex lens the coloured images will be formed at different distances along the axis, the violet image being the nearest to the lens, and the red image the most remote from it. The effect of the concave lens on these images will be to throw them farther away from the lens, and the effect on the violet image will be stronger than that on the red image. By a proper adjustment of the lenses, the final violet image may be made to coincide with the final red image, or any two other colours may be united in the final image. If the lenses were of the same kind of glass, in order that the dispersion produced by the one should be neutralized by that produced by the other, the lenses would have to be such that the deviation produced by the two lenses would also destroy each other, and therefore the combination would not produce an image at all. But it has been seen that for different kinds of glass tlie dispersion is not proportional to the deviation, but that flint-glass has a higher dispersive power than crown-glass, so that it is possible to destroy the dispersion without destroying the deviation. IS-i. We shall now investigate the condition that a combina- tion of two lenses made of different kinds of glass, placed close together, may be achromatic for two given colours. We shall suppose that one of the colours is the standard colour, and that the focal lengths of the two lenses are /,/', respectively. There will be two images ; the first being the image of the object formed by the first lens, and the second being the image of this first image formed by the second lens. Let x, x be the distances of the object and the first image in front of, and behind, the centre of the first lens, y, y the distances of the first and second images in front of, and behind, the centre of the second lens, respectively. Then X X f 111 - + - = w . y y f If we neglect the thicknesses and the distance between the lenses, y =- x, and therefore 1111 - + - =7+7;. ^ y f t 21-i ACHROMATISM OF LENSES. [CHAP. IX. The condition that the system should be achromatic is that y should be the same for the two colours ; and therefore, since x is independent of the colour, ,/) -'(/■)-• that is, ->- + ^ = 0. This is the condition of achromatism for the combination. . This condition is independent of x and y, so that the combina- tion will be achromatic for objects at all distances. It is immaterial in what order the lenses are placed. In the construction of microscopic object-glasses, achromatic couples of this kind are very generally used, each consisting of a plano-concave lens of flint cemented to a double convex of crown, the plane face being exposed to the incident light. 185. If the combination of two lenses in contact has been over-corrected for dispersion, that is, if the violet image formed by the two lenses be at a greater distance from them than the red image, the defect may be removed by slightly separating the two lenses. The distance between the lenses must be very small, or else when the coloured images are corrected for distance they will not be corrected for magnitude. The same equations hold good as before, namely. 111 X X f 1 1 1 y y f )- and besides these there is the equation x' +y' = a, a being small. Then supposing the coloured images to be formed at the same distance, dx = 0, and dy = 0, and therefore dx tr f'~f 18i— 186.] ACHROMATISM OF LENSES. 215 and since dx' + dy' = 0, Substituting for y' its value a - x', and neglecting squares of the ratio a : x , this equation reduces to „ ■E7 fl 1) TO- ■CT Thus the distance a is not independent of the position of the object ; but when the combination is used as the object-glass of a telescope, the distance of the object x is very large compared with the focal lengths. Neglecting, therefore, the reciprocal of x, the equation of condition is ^7=7-^7 = or, since -j- is very nearly equal to — j, 2at3- TO- ■ut' This shows that if the correction is to be possible, ■nr j f -\- ■ur' I f must be negative. But if in the first form of the combination, dy be the change of y due to the change 9/a in the refractive index, dy •37 m' and therefore "by must be positive. The violet rays will therefore form an image at a greater distance from the lens than the red rays ; that is to say, the original lens was over-corrected. 186. If three thin lenses, formed of media of different dis- persive powers, be combined into a single lens, the system may be made achromatic to a higher degree of approximation; the coloured images formed by three different kinds of light may be united. More generally, if »i lenses form a combination, whose thickness may 216 ACHROMATISM OF LENSES. [chap. IX. be neglected, the system will unite the images formed by rays whose refractive indices are /x and fx + dfj., provided that x(=) = o. This may be proved in the same way as before. The equation of condition can be satisfied for n — \ systems of values of 8/u, and therefore the images corresponding to n lines of the spectrum may be united. 187. When the two lenses forming a combination are sepa- rated by an interval, it is impossible simultaneously to effect the two corrections for dispersion. For let X, x be the distances of the object and its first image in front of, and behind, the first lens, y', y the distances of the firet and final image in front of, and behind, the second lens, respectively, and let /3, ^^, be the linear magnitudes of the object and its images. Then the following ratios must hold : -^=_£ ^' y i and therefore ^^^ /8' a/y- If the coloured images corresponding to refractive indices ytt, /i + d/j,he formed at the same distance and also have the same magnitude. and and therefore , = 01 Sy = or 186 — 188.] ACHROMATISM OF LENSES. 217 But x +y' = a, wLere a denotes the distance between the lenses ; so that it is necessary that dy', dx' should both vanish. In other words each lens must be achromatic of itself. This can- not be effected unless each lens of the combination be itself an achromatised couple of lenses in contact. 188. It is often necessary, however, to correct a system of two lenses separated by an interval, for errors due to dispersion, as far as possible ; so that we must choose which of the two corrections shall be effected, and which left. It is then usual to make the coloured images have the same magnitude ; for the eye is a better judge of the magnitude of an object than of its distance. Using the same notation as before, the condition is that ^/8> But we have seen that (f)=»- ^_xy^ 'i\H' 111 by virtue of the equation -+- = -=■,. ^ y y f Also x' + y' = a; and therefore ^ _x ta — x' N _x la N X \f A/' V / ^ X x + a ax or finally, ^, = l-j jr- + j2, ■ Equating to zero the variation of this expression, we get XTJ! [x + a) sr' _ ax {is + •ar') This is therefore the condition for the partial achromatism of the two lenses. In general, it is not independent of the position of the object. 218 ACUROMATISM OF LENSES. [CHAP. IX. 189. If we consider the inclinations of rays to the axis of the instrument, instead of the magnifying power, it will be seen that we have ensured that two coloured ra3's diverging from the object will emerge parallel to each other. For if a, a! be the inclinations to the axis of the original and final rays, cutting the axis at the points determined by x, y, we may see directly from a figure, or by Helmholtz' theorem relating to the magnifying power, that /S tan a' xy' tan a x'y ' so that if the condition previously found be satisfied, then 9/ = ; and the final raj's emerge parallel to each other. 190. The most useful application of this condition is to the achromatism of eye-pieces. The rays strike the eye-piece excen- trically diverging from the image formed by the object-glass. The images formed by the lenses of the eye-pieces are formed exactly as if the rays diverged from a real object, except that the rays from any point of the image do not fill the whole of the lens. The centre of the object-glass is usually very distant as compared to the focal lengths of the lenses of the eye-piece. If we make x very large in the previous equation of condition, it becomes •ET ■bt' _ a (■sr -|- in-') or / / // TO-/"' -t- ct'/ ■07 -)- •07 There is a special advantage in making the lenses of the same kind of glass, because then if we make two coloured images coincide, all the coloured images will be united. The condition for achro- matism then becomes f+f or in words, the distance between the lenses must be half the sum of their focal lengths. 191. The conditions for achromatism for any system of lenses, thick or thin, arranged along an axis, may easily be deduced from Gauss' theory. 189 — 191.] ACHROMATISM OF LENSES. 219 For the relations between the coordinates of a point and its image may be written in the forms /, (f _ a) (r - a) + ix'cj (^ - a) -fil{^'- a') - /./.'A = 0, V' ^ / ' using the notation of § 77. If we suppose the point (f, 77, f) to be fixed, for perfect achromatism the coordinates of the conjugate point ought to be independent of the particular ray chosen ; and this for all values of f, T], ^. These conditions may be fulfilled for two rays by making a@-o. 3,-0. These conditions are equivalent to and d(^,l\= 0. The quantities g, h, Ic, I are connected by the equation gh — hk=l which may be written in the form from which we deduce the condition 9 ( — j = 0. This can only be realised perfectly by mafing the first and last media the same ; this is actually the case in most optical instruments. The conditions then reduce to any three of the following % = o, d(fih) = o, a(-) = o, dl^O. If we refer back to the values of the coordinates of the cardinal points of the system, it is easy to see that the preceding 220 EXAMPLES. [chap. IX. coaditions are equivalent to making the positions of the principal points, and the focal length of the system, the same for the two colours. EXAMPLES. 1. Show that at a single refraction at a, plane surface the dispersion is proportional to the tangent of the angle of refraction. 2. If two media be such that the increments of refracting indices for each species of homogeneous light be proportional to those indices themselves, a ray of light may be refracted at their common surface without dispersion. 3. If a ray be incident on one face of a triangular prism, and after enter- ing the prism be reflected five times internally at the sides of the prism taken in order, show that it will emerge from the first face in a direction which makes the same angle with the normal as the incident ray, and that a pencil which so passes through the prism is not coloured. 4. If n, V be the indices of refraction for the red and violet rays, respectively, for crown-glass, and /i', v be the indices for the same rays for flint-glass; and if two thin lenses be constructed, one double convex of crown- glass with each surface of radius r, and one double concave of flint-glass with its surfaces of radii r and «, and they be placed in contact so that the light ia incident on the surface of radius s ; then the combination will be achromatic ii r + 1 -.is = fi - V : iL - v . 5. A small pencil of parallel rays of white light, after transmission in a principal plane through a prism, is received on a screen whose plane is perpendicular to the direction of the pencil ; prove that the length of the spectrum will be proportional to (ji,- fir) sint-T-cos^ 2) cos (/)-f-t-^)cos0' ; where i is the refracting angle, (p, ^ ?)i = 1 + 7. . In convex lenses f is positive, and therefore the object will appear magnified ; in concave lenses / is negative, and therefore the object appears to be diminished b}' the lens. H. IG 242 VISION THROUGH A LEXS. [CHAP. X. 214. If in the formula for tan 6 we substitute for x' its value ia terms of x, it becomes /3 tan 6- - + ?-/ By giving different values of x, f, it is easy to see from this formula, in what manner the visual angle changes when the positions of the object and eye vary. We notice that this formula is symmetrical with regard to x and f, so that the positions of the eye and object may be inter- changed in every case without altering the visual angle. When the eye is at a principal focus the apparent magnitude is independent of the position of the object ; and similarly when the object is at a principal focus its apparent magnitude is independent of the position of the eye ; the apparent magnitude being, in both cases, equal to that under which the object would be seen by the naked eye, when at a distance equal to the focal length of the lens. For if we make either x =f, or ^ =/, we get tan 6 = ^jf. Again, when either the eye or the object be close to the lens, the apparent magnitude is that under which the object would be seen by the naked eye. For in these cases, we must make ^ or a; very small ; and therefore tan B = fi/x or yS/f . 215. But in all cases the visual angle will necessarily be limited by the aperture of the lens ; so that the greatest value of tan 6 will be equal to y/f nearly, where y denotes the semi-aperture of the lens. The greatest linear extent of object, visible through a lens in any position, may be called the field of view. Its magnitude is at once ascertained by equating the value of tan as previously found to y/^ ; the greatest value of /8 is accordingly ^ = ^^^1-7)- The linear extent of the field of view, therefore, varies as the aperture of the lens, other things remaining the same. When the object is in the principal focus of the lens x =/, and therefore /3 = yf/^. When the eye is in the principal focus of the lens, the linear extent is equal to the aperture of the lens, what- ever the position of the object. For if f =/, j8 = y. 214—217.] SPECTACLES. 243 When the object is close to the lens, that is, when x is very- small, the value of yS becomes very nearly equal to y ; so that the extent of the field is in this case independent of the position of the eye. On the other hand, when the eye is close to the lens, that is, when ^ is small, the field becomes very great. Spectacles and Reading Glasses. 216. The distinctness of objects as seen by the naked eye depends on the accurate convergence of the rays of different pencils to points on the retina. We have seen that the eye is furnished with a mechanism for adapting itself for seeing distinctly objects at different distances. A normal eye when not actively accom- modated is adapted for rays coming from a distant object, or for parallel rays ; and it must be accommodated for seeing objects which are near, the range of distinct vision extending from five or six inches to infinity. Eyes for which the greatest distance of distinct vision is finite are called short-sighted, or myopic; these eyes can only bring divergent pencils to a focus on the retina. On the other hand, eyes which can bring to a focus on the retina not only parallel rays but convergent pencils are called long-sighted or hypermetropic. The defects in these eyes depend on the length of the axes of the eyes ; in a short-sighted eye, the axis is too long, and in a long-sighted eye it is too short. In both short-sighted and long-sighted eyes the accommodating mechanism may be quite perfect. When this is the case, their defects may be entirely remedied and the eyes made normal by the use of spectacles. 217. Let the range of distinct vision by the naked eye extend from points distant a, h from the eye. In a normal eye, h will be infinite ; in a short-sighted eye h will be finite and positive, and in a long-sighted eye h will be finite and negative. Suppose the eye to view an object through a lens of focal length /, placed close to the eye, f being positive for a collective lens, and negative for a dispersive lens. Then if x, x be the distances from the eye (or from the lens) of an object and its image, respectively, measured outwards, 11^1 X x'~ f 16—2 244 SPECTACLES. [CHAP. X. The rays striking the eye will appear to diverge from the image ; aud therefore the rays may be brought to a focus provided X lies between the limits a, b. If we substitute for x' the values a, b in succession, the corresponding values of x will be the limits of the range of distinct vision through the spectacles. When the accommodating mechanism is perfect, we have only to choose /, so that the farther limit of distinct vision is at an infinite distance. We must therefore make x infinite when x' = 6, and thus we find the focal length of the spectacle glass, namely, /= — b. The nearer limit of the range of distinct vision becomes ab x= ; b — a and therefore the range of distinct vision through the spectacles will extend from ab/{b — a) to infinity. In a short-sighted person b is finite and positive, and therefore / is negative ; he must therefore use dispersive lenses, generally double concave lenses, whose focal length is equal to the greatest distance of distinct vision by the naked eye. Thus if the range of distinct vision extends from 3 to 6 inches from the eye, the use of a concave lens whose focal length is 6 inches, will cause the range of distinct vision to extend from 6 inches to infinity. On the other hand, in a long-sighted eye b is negative and therefore y is positive. For example, if the range of distinct vision extend from 12 inches outwards through infinity to — 12 inches, the spectacles chosen must be collective lenses of 12 inches focal length ; substituting these values in the general formula, we find that the range is then from 6 inches to infinity. Practically, these glasses may be chosen by making the person look at a distant object ; then the weakest concave glasses which will enable a short-sighted person to see this object distinctly, and the strongest convex glasses which will enable a long-sighted person to see it distinctly, are the glasses suitable to the eyes. The limiting points of the range of distinct vision may be measured by making the person look through suitably chosen convex lenses, so that the points in question are brought within 12 inches from the eye, and then their distances can be measured on a divided scale. They are generally not the same for both eyes, so that the two eyes require different, glasses. 217 — 219.] READING GLASSES. 245 Short-sighted persons who have to do delicate work, have sometimes to bring things close to the eyes ; in this case they should use rather weaker concave glasses, than those prescribed above. For the same purpose, achromatised prismatic glasses, which are thicker towards the sides next the nose, and thinner towards the sides next the temples are used, because the objects can then be seen with less convergence of the axes of the eyes. 218. As the age of a person advances, the eye gradually loses its power of accommodation ; it is supposed that the outer layers of the crystalline lens lose their elasticity, so that the lens becomes less capable of changing its form and curvature. This defect is known as presbyopia. It is entirely different from the defect described above, called long-sightedness ; though aged persons are sometimes said to be long-sighted. The structure of aa eye does not alter with age, so that a person with normal eyes, will still see distant objects when he becomes old ; but the range of accommo- dation of the eye is then less than before, so that it cannot bring to a focus on the retina pencils of rays issuing from points very near to it ; in other words, the nearer limit of distinct vision has receded from the eye. Presbyopic eyes therefore need convex glasses to enable them to see near objects, as in reading or writing ; but they must be laid aside to look across a room or at a distant view. Usually the glasses are chosen so as to bring the nearer limit of distinct vision to 10 or 12 inches from the eye. For very aged persons, whose sight has lost its keenness, it is sometimes advisable to use spectacles which will bring this nearer limit to within 8 or even 7 inches from the eye, so that objects may be seen under a greater angle. From what has been said, it is evident that presbyopia may exist along with the other defects previously mentioned. Both long-sighted eyes and short-sighted eyes can be made normal by the use of spectacles, as we have seen. When presbyopia sets in, these eyes will need two pairs of spectacles, one for walking, and another for reading and writing. 219. A convex lens of considerable aperture and magnifying power is often used as a reading glass, or for viewing the details of small objects. Such a glass may be used bj^ both short and lone-sighted people. For suppose that the glass is placed, so that the object is in the principal focus of the glass, then the rays 246 ASTIGMATISM. [CHAP. X. emerging from the lens are parallel. If the glass be now moved a little nearer to the object, the emergent rays will diverge, and can be brought to a focus on the retina by a short-sighted eye ; if on the other hand the glass be moved a little farther away from the object, the emergent rays will converge and will be adapted for distinct vision by a long-sighted ej'e. In each case the magnifying power will be 1 + X// where X is the least distance of distinct vision, and /the focal length of the lens. Astigmatism. 220. In addition to the defects already mentioned, the eye is often not symmetrical about its axis, and also, like other optical instruments it is subject to spherical aberration. The defects of the image arising from these causes are all included in the term astigmatism. If a person look at a distant small bright point, while his eyes are accommodated for a nearer object, we should expect that there would be a small circular patch of light on the retina ; but often, instead of a circular spot of light, he sees a star-shaped figure, with from four to eight rays. The shape of this figure is different in different individuals, and generally different for the two eyes. If the light be white light, the parts near the edges of the bright parts of the figure are tinged with blue, while those in the centre are a reddish yellow. If the light be weak, only the bright parts of the figure are visible and several images of the bright point are seen. The appearance of a star or a distant light, as a figure with streaming rays, is connected with these appearances. Similarly if the eye be accommodated for an object at a greater distance than the bright point, another star-shaped figure is seen ; and even if the eye can accommodate itself to the light, provided the light be sufficiently strong, the same appearances present themselves. If the object be a fine line of light, the appearances can easily be deduced from the preceding phenomena; several images of the line are seen, most eyes seeing at least two. Some of these phenomena are due to moisture on the cornea, and can be changed by blinking with the eye-lids. But most of them are caused by irregularities in the crystalline lens. Bonders 219 — 221.] ASTIGMATISM. 247 investigated them by allowing light to pass into the eye through a small slit, which he moved into different meridian planes. He found that usually every sector of the lens brought the incident rays to a focus, but that the foci for different sectors did not coincide. Also each sector did not bring the rays very accurately to a focus, but those rays which were incident near the axis of the eye, had their foci farther off than those which were incident near the edges of the pupil. These phenomena are called irregular astigmatism, and they cannot be removed. 221. The defects of the image, arising from want of symmetry in the curvature of the refracting surfaces, and especially of the cornea, are called regular astigmatism. Regular astigmatism occurs in almost all eyes. The eye is not in general accurately accommo- dated for horizontal and vertical lines at the same time. If a person look at a figure consisting of a series of lines all radiating from a point, when any one of the lines looks clear and sharply defined, the others do not, and as the distance of the figure changes, first one line and then another is well defined. Or if he look at a figure consisting of a series of concentric circles drawn at small intervals, a peculiar star-shaped appearance is present in the figure. In the white rays, the edges between the black lines and the white intervals are clear and well defined, and they are less and less clear till we reach the blacker rays. If the accommodation of the eye be allowed to change or the figure be moved to a greater or less distance from the eye, other parts of the figure become clear, and it seems as if the clear rays moved to and fro. This may be explained by the theory of thin pencils already developed. If we trace back a thin pencil from a point on the retina through the eye into the air, the emergent pencil will be orthogonal, and will have two focal lines at right angles to the axis of the pencil, lying in planes through the axis of the pencil which are perpendicular to each other. For the state of accommodation of the eye, the positions of these focal lines are the best positions for seeing lines parallel to their directions respectively. If the distances of these focal lines from the eye be p, q, then 1/p — Ijq may be taken as a measure of the astigmatism of the eye in that state. If a lens be placed in front of the eye, it will not change the value of l/p — 1/q ; so that 248 ASTIGMATISM. [CHAP. X. we may assume that 1//) — Ijq retains very nearly the same value ■when the state of accommodation in the eye changes. 222. This astigmatism may be corrected by a properly chosen astif^matic lens. If we place before tlie eye a lens such that when a pencil of light issuing from a bright point pass through it, the emergent pencil has a pair of focal lines coinciding with those already found for the eye, then when the pencil enters the eye it will be refracted so that the rays meet in a point. The lenses used for this purpose have one surface cylindrical. It was shown in § 157 that the measure of the astigmatism of a cylindrical lens, the radii of whose surfaces are a, and h, and the angle between the generators of the two cylinders 2i, is = (u - 1) A/ -, + Ta 7 COS 47, u V ^ \ a b ab or, in the case when the posterior surface is plane, II V a ' A lens of this latter form may always be chosen to cure the astigmatism of the eye ; the radius of the cylinder must be such that p q~ a - 223. If two astigmatic lenses, one concave and the other convex, be united so that their plane surfaces coincide, we reproduce the conditions of the first case. When the radii of the cylindrical surfaces be equal, a = b, and therefore 1 _ 1 _ 2{fi-l) sin 2% u V a ' The lenses may be arranged so that they can be turned about their common axis, so that a may be made to vary ; then the astig- matism of the combination may be made to vary from zero, when 22 = 0, up to 2 (/i - l)/a, when 2a = Jtt. This furnishes a very simple method of measuring the astigmatism of an eye. The person is made to view through a convex lens a figure formed by a series of lines diverging from a point, and the figure is moved farther and farther away till the farthest point is reached for which the spectator can see any of the 221 — 225.] VISION THROUGH LENSES. 249 lines distinctly. A pair of crossed astigmatic lenses is then interposed, and turned about their common axis until all the lines are visible ; the astigmatism of the eye is then corrected by that of the combination ; and the measure of astigmatism is determined by observing the angle between the generating lines of the cylindrical surfaces. 224. The direction of the focal lines of the eye are determined by observation ; if we choose the axes of co-ordinates to coincide with the focal planes, .0=0, using again the notation of § 157. Hence sin 2^ = 0, which gives a = or Jtt. If we choose the generating line of the cylindrical surface to coincide in direction with the more distant focal line, a = Jtt, and i_i^_0^-i) p q a ' This shows that a is negative, and therefore that the cylindrical surface is convex. If this surface is to be concave, we must make a = 0, or the direction of the generators of the cylindrical surface must coincide with that of the nearer focal line. The second surface of the spectacle glass, instead of being plane may be made spherical. This is equivalent to uniting the astigmatic lens with an ordinary lens one surface of which is plane. This will not alter- the measure of the astigmatism, and by choosing the curvature of the spherical surface properly, the lens may be made to correct astigmatism and also myopia or presbyopia at th6 same time. The use of cylindrical lenses to cure the defects of astigmatism was discovered by Airy. On vision through any number of lenses. 225. To determine in what manner an object will be seen through any system of lenses, it will be necessary to trace the course of a pencil of rays from any one point of the object; the several foci of this pencil determine the positions of the several images of the object. The field of view, the effective apertures of the lenses, the visual angle under which the object is seen, and the best position for placing the eye, are all found by considering the course of the axes of these several pencils. These axes all 250 MAGNIFYING POWER. [CHAP. X. pass through the nodal points of the object-glass, or the first lens of the system, and may therefore be considered as a pencil of rays issuing from the second nodal point of this lens ; the extreme rays of this pencil determine the field of view and the effective apertures of the several lenses' after the first. We have already investigated the position of the focus of a small pencil issuing from a point and traversing any number of lenses bounded by spherical surfaces. In such an optical system the initial and final media are the same, and therefore the principal points coincide with the nodal points, and the two focal lengths are equal. 226. The magnifying power of any telescope may be defined to be the ratio of the angle under which an object is seen through the telescope, to the angle under which it would be seen by the naked eye. This definitioii supposes that the linear dimensions of the object and image are to be compared, and not their areas. This is sometimes expressed by saying that an instrument magnifies a certain number of diameters. The axes of the extreme pencils which enter the object- glass, determine the angle under which the object would be seen, were the eye placed at the centre of the object-glass ; and in the case of a telescope, this does not differ sensibly from the angle under which it would be seen by the eye in its position when looking through the instrument ; for the distance of an object is usually very great compared with the length of the telescope. With the notation which we used in stating Gauss' general theory, let the axis of one of the extreme pencils be determined by the quantities /3, b; and after refraction at the several surfaces, by ^,, 6j, ^j, 6jj .. . y8', b'. Then since the axes of all the incident pencils pass through the first nodal point of the object-glass, 6=0, very nearly ; for the first nodal point will be very close to the surface of the object-glass. Substituting this value of b in the equation we find that ^ = '> and therefore the magnifying power of the instrument is repre- sented by I. 225 — 227.] MAGNIFYING POWER. 251 227. The image of the surface of the object-glass as seen through the instrument is called the eye-ring. Every ray which passes through the instrument will emerge within the eye-ring, at the image, namely, of the point at which the ray strikes the object-glass. If the instrument be directed to an illuminated surface, or to the sky, each point of the eye-ring receives light from all points of space whose rays can traverse the instrument, that is, from all points of space which can be seen by help of the instrument. If the eye be placed so that its centre is at, or close to, the centre of the eye-ring, it will therefore embrace the entire field of the instrument. The centre of the eye-ring, is therefore the best position for the eye, and is called the eye-point. We have seen that the axes of the several pencils striking the object-glass form a pencil issuing from the second nodal point of the object-glass ; the final focus of this pencil as it passes through the instrument, will be very near. to the centre of the eye-ring; and therefore the eye will receive as many of the axes of pencils issuing from the object as possible. To find the magnitude and position of the eye-ring, we have only to make ^ = a ; then ^' = a-!^. Also for this point, , kh . , it-u — i '„' ^^9-^^ I v' 1 so that — = -j, since gl — kh = 1. V ' Remembering that I is the magnifying power of the telescope, this equation shows that the magnifying power of the instrument is equal to the ratio of the radius of the object-glass, to that of its image as seen through the telescope. This gives a practical way of measuring the magnifying power of a telescope. The telescope is pointed to a bright surface, and the diameter of the eye-ring is measured by a graduated scale and lens, forming a micrometer. The diameter of the object-glass can also be measured, and the ratio of the latter to the former gives the magnifying power. 252 FIELD OF VIEW. [CHAP. X. 228. The Field of view is determined by the axes of the extreme pencils which meet the object-glass. The effective apertures of the various refracting surfaces may be obtained by finding the corresponding values of b. If /3 be the inclination of the extreme axes of the incident pencils, then, since b vanishes, the effective apertures will be determined by the equations, /3, = y3, + A;A, b, = b^ + ^,t^, /93 = /S, + A-A, 6, = 6, + ^3^3, By these equations b^, i^-.-V are determined. If we add the first set of equations, we arrive at the result, which may be written, {i-\)^=Kb, + Kh+...+K_,b,,_, This shows that the field of view is continually increased by adding more convex lenses ; for corresponding to both surfaces of a convex lens k is positive. If any lens have its aperture diminished, the values of all the other apertures and of the field of view are diminished in the same proportion. It is useless to make the aperture of any of the lenses greater than its effective aperture. 229. Sometimes it happens that the position of the eye-point falls within the telescope, that is, in front of the outer surface of the eye-lens ; this will be the case when hjl is positive, for then ^ — a! will be negative. The eye cannot then be placed at the eye-point, but is put as close to the eye-point as possible ; it will therefore be placed close to the eye-lens. In estimating the field of view, the radius of the pupil must then be used instead of the radius of the outer surface of the eye-lens. Brightness of Images. 230. It has already been proved that if /8, /8' be the linear dimensions of an object and its image by refraction at a spherical 228 — 230.] BRIGHTNESS OF IMAGES. 253 surface separating two media of refractive indices fi, fi!, and if a, a! be the angles of divergence from the axis of two corresponding rajs in the two media, then /i/S tan a = ya'/S' tan a'. In other words, the value of /n^S tan a. is unchanged by the refraction ; and therefore after any number of such refractions the same law must hold ; that is, /ijS tan a — yu.' /3' tan a', where y!, yS', a', refer to the final medium. To the first order of small quantities, we may write a for tan or, a! for tan a', and therefore the formula becomes, /X2,S = yu,'a';S'. Now let dSi, dS', be small elements of the object and image, standing perpendicular to the axis, so that dS : dS' == ^-^ : ^" ; and let da be the small solid angle bounding a pencil diverging from dS, and let da/ be the corresponding solid angle for the emergent pencil. Then if / be the brightness of the object, the quantity of light issuing from the object is, L = IdS dco ; and if /' be the brightness of the image, the quantity of light emerging from the system will be L' = rd8'Seo'. On the supposition of no absorption of light by the media, L = L', and therefore Ids dco = I' dS' dco'. But d8 : dS' = ^' : ^'\ and also dco : dco = a" : a!'. Hence I a' ^ == I' oT ^'^ Keferring back to the preceding equation, connecting a, /3 with a, y8' we arrive at the equation J 2 r' . '2 : jjL = 1 : fjL . 254 BRIGHTNESS OF IMAGES. [CHAP. X. If the initial and final media are the same, as is nearly always the case with optical instruments, that is, the brightness of an image formed by rays which make small angles with the axes, after refraction through any optical system, is equal to the brightness of the object. 231. But this law must also hold, for any wide-angled aplanatic system, without the restriction that the rays shall make very small angles with the axis. For if it did not hold, it would be possible to arrange an optical instrument in such a way that a person could make an object look brighter than it does to the naked eye, which contradicts all experiments made with different kinds of refracting media. And if this were possible for light, it would also hold for heat rays, whose laws of emission and refraction are the same as those of light; and this would contradict the law of the equality of radiation between bodies of equal temperatures. Hence we infer that the law of brightness I : //,= = /' : fi!' holds for all aplanatic systems. 232. We are now able to extend the law connecting the angles of divergence of the initial and final pencils to wide-angled systems. For consider, as before, the quantity of light sent out by an element of bright surface dS, placed perpendicular to the axis. The intensity of the emission of light in any direction varies as the cosine of the deviation of that direction from the axis. Hence the whole quantity of light sent out within a cone of semi-vertical angle a will be L = Ids I "cos e . 27r sin 6 d0, •'0 or L = TT Ids sin' a. The corresponding formula for the emergent beam will be L' = IT I'dS' sin' a: 230 — 233.] BRIGHTNESS OF IMAGES. 255 Equating these values, we find that IdSsin'a^rdS'sin'a'. But I : fi^ = r : fj.'\ and also dS : fi'' = dS' : ;8'=; hence the equation becomes fi^ sin a = /i'/8' sin a'. This expresses for wide-angled aplanatic systems, the relation between the angles of divergence of the initial and final pencils, which was found before for pencils of very small divergence. When the angles of divergence are very small, it is immaterial whether we write a, or tan a, for sin a. This law was enunciated independently by Helmholtz and Professor Abb^ of Jena in 1874. 233. It often happens in the case of instruments of high magnifying power, that the emergent pencil does not completely fill the pupil of the eye, and then the brightness of the image on the retina will be less than when the pupil is quite filled with the rays. For let 7„ be the brightness when the eye is filled with the rays, and let \ be the distance of the image from the eye ; then the section of the pencil by a plane coinciding with the pupil of the eye, will be ttX'' sin'' a' ; and therefore if p is the radius of the pupil I : /, = 'n-Vsin'a' : irp'; which gives J =^o {-) T/f^'W'^^^ "■• The last medium before the eye will be air, so that /t' = l, and /37/3 is the magnifying power, which is denoted by m, and therefore - _ - \' /i.' sin' a p m an equation which we shall find useful in dealing with optical instruments. 250 EXAMPLEP. [chap. X. EXAMPLES. 1. A stereoscope is constructed of two glass prisms (fi = ^) with their edges coincident, and placed so that the faces of each are equally inclined to the plane on which the two pictures are placed, and at a distance of 6 in. The eyes of an observer are 2^ in. apart ; find their distance from the prism when the axes of the pencils from the middle points of the two pictures have minimum deviation and cross at the poiut half-way between them, the points being 4 in. apart. Show that the angles of the prisms must be nearly tan ~' |. 2. Three convex lenses of focal lengths /,, f^ /, are separated by intervals a, b ; find the magnifying power of the combination, and prove that it is independent of the position of the object if (/l-«)(/3-6)+/2(/l+/,-«-6) = 0. 3. The light after passing through an optical instrument symmetrical about an axis is reflected by a plane mirror perpendicular to its axis so as to pass through it again in the reverse direction ; show that the compound instrument so formed is equivalent in every respect, if spherical aberration be neglected, to a simple spherical mirror, with its vertex in the position conju- gate to the plane mirror and its centre of curvature at the corresponding principal focus. 4. ■ If in any optical instrument formed of lenses and mirrors on the same axis, m is the magnifying power when the instrument is adjusted for an eye which sees clearly with the incident light parallel, and if the eye-glass (focal length /) is moved till the instrument is in adjustment for an eye whose distance of distinct vision is 5, show that the magnification is increased by mf/S. 5. A cylindrical beam of light is incident on the object-glass of a telescope; find the distribution' of the illumination in the focal plane and show from your result that the resolving power of a telescope increases with its aperture. CHAPTER XI. Optical Instruments. 234. We have already treated the theory of vision through a single lens and its application to spectacles and reading-glasses. The next optical instrument in the order of simplicity is the simple microscope. We have seen that when an object is placed at the focus of a convex lens, the rays of the several pencils will emerge parallel to each other, and therefore each pencil will be brought to a focus on the retina without effort; and in this position the angle under which it will appear to the eye is the angle it would subtend at a distance equal to the focal length of the lens. Consequently the image will be distinct and magnified. A lens of high power thus used is called a simple microscope. If yS denote the linear dimensions of the object, the tangent of the visual angle will be ^Jf, while the tangent of the angle under which it would be seen by the naked eye at the least distance of distinct vision is /8/X,; the measure of the magnifying power is therefore Xjf. Single lenses answer very well so long as the focal length is not smaller than one inch ; but when higher powers are required, combinations of more than one lens are preferable. "^ 235. A form of simple magnifier, which possesses certain advantages over a double convex lens, is that commonly known as a."Coddington lens." The lens is spherical, but the rays are made to pass nearly through the centre of the lens. The first idea of it is due to Wollaston, who proposed to unite two hemi- spherical lenses by their plane sides, with a stop interposed, the central aperture of which should be equal to one-fifth of the focal H. 17 258 SIMPLE MICROSCOPES. [CHAP. XI. length. The same end was shown by Brewster to be attained more satisfactorily by grinding a deep groove round the equatorial part of a spherical lens, and filling it with something opaque. The great advantage of this lens is that the oblique pencils as well as the central pencils, pass normally into the lens, so that they are but little subject to defects of aberration. The Stanhope lens consists of a cylinder of glass with its ends ground into spherical convex surfaces of unequal curvature ; the length of the cylinder is so' arranged that when the more convex end is turned towards the eye, objects placed on the other end shall be in the focus of the lens. This furnishes an easy way of mounting light objects for examination. A modified form of the Stanhope lens, in which the further surface is plane, has been used extensively in France for the enlargement of minute pictures photographed on the plane surface ; it is called a " Stanhoscope." 236. Wollaston was the first to use a combination of two lenses instead of a single lens ; this combination is still known as Wollaston's Doublet. It was suggested by an inverted Huyghens' eye-piece, to be described presently. It consists of two plano- convex lenses whose focal lengths are in the proportion of 1 : 3, the plane surfaces being turned towards the object, and the lens of shorter focal length being placed next the object. The distance between the lenses can be adjusted to suit different eyes, but is usually | of the shorter focal length. Pritchard, who made doublets which magnified 200 to 300 diameters, performing excellently, made the distance between the lenses equal to the difference of their focal lengths, while the latter could vary in ratio from 1 : 3 to 1 : 6. A better doublet was invented by Chevalier, who placed two plano-convex lenses of equal focal lengths but of different diameters, very close together, the larger being the nearer to the object, and between them he fixed a diaphragm. In this way he obtained more light and admitted a greater distance between the lens and the object. Triplets have been constructed on the same principles. The combination with sufficient care of three plano-convex lenses gives even better results than doublets. They can be made com-: 235 — 238.] THE ASTRONOMICAL TELESCOPE. 259 paratively free from aberration both spherical and chromatic. But they are so much inferior to the modern compound micro- scope that they are only used for rough observations or for dissecting. 237. The refracting telescopes and the compound microscope, in their simplest forms, consist of two lenses. The lens placed nearer to the object receives rays directly from the object and forms a real inverted image of the object; this lens is called the object-glass, or the objective. The inverted image is viewed by the eye through the other lens, which is called the eye- glass or eye-piece; this eye-glass alters the divergence of the small pencils which form the first image, so that they can be brought to a focus on the retina without efifort, and increases the visual angle under which the image is seen. In general, an eye is accommodated for rays emerging parallel to each other ; the eye- glass is therefore placed so that the first image is in the principal focus of this lens. In microscopes, however, where the magnifying power is very important, the instrument is arranged so that the final image is at a distance of about 10 inches from the eye ; this distance is conventional, but is chosen once for all, so that the magnifying powers of different instruments may be compared under like circumstances. The Astronomical Telescope. 238. The common Astronomical telescope, the construction of which was first explained by Kepler, consists primarily of two convex lenses fixed in a tube. In the figure, BAG is the lens which is turned towards the object, and it is therefore called the object-glass. This lens forms an inverted image pq, of the object, corresponding points of image and object lying on the same line through A, the centre of the object-glass. Bq, Aq, Cq are three rays diverging from any one point of the object which, after refraction by the object- 17—2 260 THE ASTRONOMICAL TELESCOPE. [CHAP. XI. glass, are made to meet in q, the corresponding point of the image. These rays after crossing at q, fall upon the convex lens hac, called the eye-glass, and after refraction they are in general made to emerge parallel to each other. This will be effected by adjusting the position of the eye-glass, so that the image ])q shall lie in its principal focus. Let f, f be the focal lengths of the object-glass and eye-glass, respectively. Then the angle qA-p is the angle which the object subtends at the centre of the object-glass, and this will not differ sensibly from that subtended at the eye. By the naked eye, there- fore, the object is seen under an angle whose tangent is — yS//", where ;S is the linear dimensions of the image. Also, the image •pq will be seen through the lens at an angle whose tangent is /S//"', wherever the eye be placed, supposing pq to be in the principal focus of the eye-glass. The magnifying power is therefore 239. The field of view is defined by the axes of the extreme pencils which are transmitted by the eye-glass. It will therefore be the angle which the eye-glass subtends at the centre of the object-glass. "Wherefore, if h' denote the semi-aperture of the eye-glass, and @ half the field of view, © = In order to take in the whole extent of this field the eye must be placed at the point in which the axes of the extreme pencils, diverging from the centre of the object-glass, meet the axis of the telescope on their final emergence. The place of the eye is there- fore the focus conjugate to the centre of the object-glass as seen through the eye-glass. If x be the distance of this point outside the eye-glass. In the construction of the instrument, the tube is prolonged to f so that !r — -L. 238 — 240.] THE ASTRONOMICAL TELESCOPE. 261 the required distance and is there furnished with an eye-stop, and in looking through the instrument the eye is placed close to the end of the tube. 240. The field of view as defined by the axes of the extreme pencils is not the entire extent of the visible field, as determined by any rays whatever transmitted through both the lenses. For if we join the extremities and the centre of the object-glass to one extreme point of the eye-glass, and let the joining lines meet the common focal line in rqs, all the rays from the object-glass which fall within ps strike the eye-glass; but only half the rays which meet at q are transmitted by the eye- glass, while only one ray of those meeting in r will meet the eye-glass. Thus all the field within ^s is seen by full pencils, while that between the lines As and Aq is seen by parts of pencils, the part exceeding half the pencil in each case ; and the part of the field between Aq and Ar is seen by parts of pencils, the parts being less than half the pencil in each case. Let ©', ©" be the values of half the bright field, and half the total visible field, respectively. Let the line mnh be drawn through the extremity of the eye-glass parallel to the axis of the telescope ; then by similar triangles. Cm : mb = sn : nb. If we denote ps by y, and the semi-apertures of the lenses by b, V , respectively, this relation becomes, b + V _ b'-y f+f'~ f ' X,- V, ■ fb'-f'b which gives, y = -yTf^ ■ But y =f ®' ; wherefore 262 GALILEO'S TELESCOPE. [CHAP. XL To find the value of ©", we have only to change the sign of b ; and therefore "/(/■+/)■ If h'/b =/'//, that is, if the apertures of the lenses are propor- tional to their focal lengths, ©' vanishes ; in this case the bright- ness of the field decreases from the centre to the circumference. If b'/b be less than /'//, the value of 0' becomes negative, and no part will be illuminated by full pencils. The field as determined by the axes of extreme pencils, is limited by the line Aq, and therefore by elementary geometry, or by the values previously obtained, ©' + ©" = 2©. The field is limited practically to the bright field ©', by means of a circular stop, which is placed at the principal focus of the object-glass, whose radius is fb'-fb This will exclude the images of all points formed by partial pencils. In an Astronomical telescope there is usually fixed a network of fine wires, vertical and horizontal, the plane of the wires being the focal plane of the object-glass. The image of the object given by the object-glass will then lie in the plane of the wires, and the image and the wires are viewed together through the eye-lens. By the aid of these wires the position of the image of any point can be accurately measured. Galileo's Telescope. 241. This telescope, called after its inventor, Galileo, was the first whose construction was explained on theoretical principles. 240 — 241.] GALILEO'S TELESCOPE. 263 It differs from the astronomical telescope chiejQy in the form of its eye-glass, which is a double concave lens, and is placed between the object-glass and its principal focus. A pencil of light diverging from the object is brought to a focus by the object-glass ; but before the rays reach this focus, some part of the pencil is caught by the eye-glass. In the annexed figure, BAG is the object-glass, hac the eye-glass, and pq is an inverted image of the object formed by the object-glass, corresponding points of the image and object lying on the same line through A, the centre of this lens. Bq, Aq, Dq, are three rays diverging from any point of the object, and after refraction they are made to converge to the point q, the corresponding point of the image. These rays fall upon the eye-glass and after refraction they are, in general, made to emerge parallel to each other. This will be effected when the eye-glass is so adjusted that the image pq is in its principal focus. When directed towards distant objects, pq is also in the principal focus of the object-glass, so that the distance between the lenses is then equal to the difference between the focal lengths of the two glasses. Let y8 be the linear magnitude of the image pq, and _/ /' the focal lengths of the object-glass and the eye-glass, respectively. Then the angle under which the object is seen by an eye placed at A is equal to the angle qAp, and this will not differ sensibly from the angle under which it will be seen by the eye in its proper position. The tangent of this angle is — ^/f. Also the image pq will be seen through the lens under an angle whose tangent is — /S//'. The magnifying power is therefore Thus the magnifying power is the same as in an astronomical telescope, the focal lengths of whose lenses are the same as in this instrument. The latter has the advantage of being shorter ; for the distance between the lenses in this adjustment is equal to the difference between their focal lengths, whereas in the former it is equal to their sum. A more important advantage which this instrument possesses is that through it objects are seen erect and not inverted, as in the Astronomical telescope. This is readily seen by following 2Gi Galileo's telescope. [chap, xl the course of the axes of extreme pencils as they diverge from the centre of the object-glass. When they meet the eye-glass they are made to diverge still more by it ; and therefore the pencil flowing from the uppermost part of the object will proceed to the lower part of the retina, and vice versa; and therefore the object is seen in the same position as by the naked eye. On this account the instrument is convenient for viewing terrestrial objects. The ordinary opera-glass consists of a pair of Galileo's telescopes placed with their axes parallel, and arranged so that the distance between the lenses can be altered so as to adapt the telescopes for seeing objects at different distances. 242. The field of view in this instrument is very limited. For the axes of the pencils flowing from the several parts of the object, diverging from the centre of the object-glass, will diverge still more after refraction by the concave eye-glass, and therefore, for the most part, they will fall without the pupil of the eye and be lost. In order that the eye may receive as many as possible of these axes, it must be placed as near as possible to the point from which the axes diverge. This point, which is the eye-point, lies within the instrument, and therefore the eye cannot be placed at it, but will be placed close to the eye-glass. The effective aperture of the eye-glass is therefore reduced to that of the pupil, and it is useless to make the eye-glass of much greater aperture than the pupil. The value of the field of view as determined by the axes of the extreme pencils which strike the eye-lens, is therefore equal to the angle subtended by the pupil of the eye, at the centre of the object-glass. If b' be the semi-aperture of the pupil of the eye, and @ denote half the field of view, then /-/'• But this is not the total visible field as seen by any rays whatever; this may be found as before. The aperture of the pupil being small compared with that of the object-glass, few, if- any, whole pencils after refraction by the object-glass will fall within the pupil ; it is therefore usual to regard those pencils which completely fill the pupil, as whole pencils, and those which OBJECT-GLASSES. 265 241—243.] do not fill it as partial pencils; so that the field of view is limited by the object-glass. We shall suppose that the semi-apertures of the lenses are b, b' ; the latter being equal to the semi-aperture of the pupil. Let the extremity of the object-glass be joined in succession to the extremities and centre of the eye-glass, and let the joining lines meet the line pq in r, q, s respectively. Then pencils converging to any point within pr, will fill the eye-lens ; the pencils converging to points in rq, will more than half-fill the lens, while pencils converging to points in qs will not half-fill the eye-glass. Hence if B' be half the bright field, and ©" half the whole visible field, @' = pr 7' 0"=-^. But, by similar triangles, it may be shown just as in the case of the Astronomical telescope that so that pr ©' = ^fb-fb'. f-f ' fb-fV fif-fy The value of ©" is obtained by changing the sign of b', and therefore In this telescope the principal focus of the object-glass is virtual, and therefore no stop and no network of fine wires for measuring, can be used. Object-glasses. 243. We shall next apply the preceding theoretical considera- tions to the construction of good object-glasses. 266 OBJECT-GLASSES. [CHAP. XI. One advantage of a telescope over the naked eye, in viewing a distant object, is the quantity of light which the instrument admits. The eye admits a small cone of rays issuing from each point of the object, just sufficient to fill the pupil; whereas a telescope admits a cone large enough to fill the whole object-glass. Thus a telescope enables us to see stars which are too faint to be perceived by the naked eye. The larger the aperture of the object- glass, the more light will be admitted. The first requisite of an object-glass is therefore a wide aperture. We have seen that the brightness of an image is equal to that of the object ; so that when the light from the image completely fills the pupil, just as light from the object does, they will appear of equal brightness. But when the magnifying power of the instru- ment is large, the emergent pencil never fills the pupil. When the telescope is directed towards a bright surface the emergent pencil fills the eye-ring. Let r be the radius of the eye-ring, and p the radius of the pupil; then, as has been remarked, r is usually smaller than p, and the apparent brightness will be less than the brightness of the object in the proportion of the areas of the eye-ring to that of the pupil. The brightness is therefore given by the equation -•©■• But if m be the magnifying power, m = hjr, where h is the semi- aperture of the object-glass. Hence Thus the brightness depends on the magnifying power and on the aperture of the object-glass ; and if the magnifying power be large, the aperture of the object-glass must be large too, otherwise the brightness of the image will be impaired. In Galileo's telescope the eye is placed close to the eye-lens, and the pupil is filled when points are seen by full pencils, and therefore the brightness of the image is very nearly equal to that of the object, and it does not depend on the aperture of the object- glass. But in this instrument the field of view depends on the aperture of the object-glass. This aperture, however, cannot be made very large, because the refraction through the lens is 243 — 244.] OBJECT-GLASSES. 267 excentrical, and if the aperture be large, the extreme pencils will be refracted at such a distance from the axis as to make the chromatic aberration considerable. 244. Object-glasses are usually made of two lenses, a convex lens of crown glass being combined with a concave lens of flint glass. The pencils of light are incident centrically on the first lens, and if there were an interval between the lenses, the incidence on tlie second lens would be excentrical ; this would be disadvantageous, and the two lenses are placed close together. We have therefore four quantities at our disposal, namely, the radii of curvature of the four surfaces of the two lenses. The focal lengths of the two lenses are immediately determined by two essential conditions. These are, that the combination must have a given focal length, and must be achromatic. Let /, /' be the focal lengths of the lenses, and F the focal length of the combination. Then 1^_ 1 1^ y-y + f Also the condition for achromatism is 7 + 7 = »■ These two equations determine / and /', so that no other condition can be satisfied which involves relations between the focal length's. We shall next consider errors due to aberration. The defects of an image formed by a single lens are (i.) Distortion due to curvature of the image. (iL) Indistinctness due to obKquity in the more remote parts of the field. (iii.) Indistinctness due to spherical aberration. In the case we are considering there is no linear or angular distortion, because the incidence is centrical, and therefore the object and image are parallel sections of the same double cone. It has been shown that if p, p be the radii of curvature of the object and its image in the central parts of the field, 1 1 m + \ Ik' + l] 268 OBJECT-GLASSES. [CHAP. XI. We cannot therefore remove the defect of curvature, because it would involve a relation between the focal lengths of the consti- tuent lenses. Next, consider the effect of obliquity. After oblique centrical refraction at a lens, the pencils do not converge to a focus, but to two focal lines. To remove this defect it will be necessary to make the two focal lines after refraction through the two constituent lenses coincide. It has been shown that if v, v' be the distances of the focal lines from the centre of a single lens, for a pencil issuing from a point at a distance u, ----_-^-^. + f^l-f-2^ IS U f { 2yx where / is the focal length of the- lens, and ^ the obliquity, which is so small that powers of j> higher than the third may be neg- lected. To make the focal lines coincide after refraction through both the lenses, it will be necessary that This again involves a relation between the focal lengths which cannot be satisfied ; so that except when <^ = 0, we cannot ensure distinctness. The defect due to spherical aberration is treated by making the aberration vanish for parallel rays. It has been shown how the aberration of the combination depends on two quantities e, e which are independent of the focal lengths. To make the aberration for parallel rays vanish.it is necessary to impose a single relation between the quantities e, e, and we can still satisfy one more condition. Two courses are open to us. We may make the lenses fit together, so that they may be cemented together into one lens ; or, as was shown in the chapter on aberration, we may make the aberration vanish not only for parallel rays, but for rays flowing from a point at a distance finite but considerable. 244 — 245.] EYE-PIECES. 269 Eye-ineces. 245. la tlie Astronomical telescope instead of a single eye- glass it is usual to use a combination of two lenses separated by an interval. The introduction of a third lens between the object-glass and the eye-glass will increase the field of view of the instrument. For this reason it is usually called the field-glass. The incidence of the pencils on the field-glass is not centrical, so that no advantage is gained by placing it close to the eye-glass. The two lenses of an eye-piece are therefore separated by an. interval. We have therefore five quantities at our disposal, namely, the four radii of curvature of the four surfaces of the lenses, and the distance between the lenses. If f, f be the focal lengths of the two lenses, a the distance between them, the focal length of the equivalent lens will be given by the equation The focal length F of the combination will be a given quantity, so this is to be considered as one relation between the constants. By far the most important defect of the image given by a single lens is that due to chromatic aberration. For a combination of two lenses separated by an interval, it is not possible to remove entirely the defects of this chromatic aberration. The defects of the image are two-fold, the coloured images are not in the same plane perpendicular to the axis of the telescope, and they are not of the same magnitude. Either of these defects can be removed but not both ; and the first defect is of the less consequence and is therefore neglected. It is best to make the lenses of the same kind of glass, for then if the combination be achromatic as regards two colours, it will be perfectly achromatic, because there will be no irrationality of dispersion. It has been shown in § 190 that the condition for this imperfect achromatism for two lenses of the same kind of glass is This is a second relation between the constant?, 270 EYE-PIECES. [chap. XI. The errors of spherical aberration are more complicated than those which occur in the object-glass, because the pencils are incident on the eye-piece excentrically and with considerable obliquity. The first defect is indistinctness, due to the spherical aberration of the more oblique pencils. When the central part of the image is at the proper distance from the eye-piece, the marginal parts will be too distant ; and therefore if the central portion of the field be distinct, the marginal portions will be indistinct, and if the eye- piece be pushed inwards in order to see the marginal portions, the central portion will become indistinct. The second defect is curvature of the image. The third defect is linear and angular distortion. The axes of the extreme pencils proceeding from the centre of the object-glass will, by the spherical aberration of the eye-piece, meet the axis of the telescope at a nearer point than those of the central pencils. The ratio of the visual angles will therefore be greater in the extreme parts of the field than at its central parts ; these extreme parts of the field wiU therefore appear unduly enlarged, and the object will appear distorted. .A fourth defect is due to the astigmatism of oblique pencils ; the pencils after passing through the instrument will have two focal lines, and the image of a point formed on the retina by such a pencil will be in general an ellipse, becoming however sometimes a circle. 246. Without entering into details connected with these defects, it will be understood that the errors will, in general, be reduced by diminishing the aberrations of extreme pencils, and that if the forms of the lenses be given, this effect will be produced by increasing their number and dividing the refraction. The resulting aberration, other things being equal, will be least when the whole bending of the ray is equally divided among the lenses. The condition for equal refraction is easily obtained. W^e shall confine our attention to two lenses. Let a ray, originally parallel to the axis meet the two lenses at distances y, y' from the axis. Then the deviations produced by the lenses are yjf, and y'jf, so that we must have ylf=y'lf. But if 6 be the inclination to the axis of the ray between the lenses 245—247.] EYE-PIECES. 271 y' = y-ae, and 5 = ^; therefore this gives /=/(i-|). or finally, «=/-/■ This condition, expressed in words, is that the interval between the lenses must be equal to the difference of their focal lengths. This is the principle on which Huyghens' eye-piece was constructed. The preceding conditions only relate to the focal lengths and positions of the lenses, and are independent of their particular forms. The aberrations will depend largely on their forms; but the different defects previously mentioned in general require different and sometimes opposite forms for their correction. It is therefore necessary to sacrifice the perfection of the instrument in one respect to improve it in another which may be of more importance for the particular object for which it is intended. The theory of this part of the subject is however very troublesome, and it is but little attended to in practice. The lenses employed are almost invariably plano-convex or equi-convex lenses. 247. If we combine the condition of achromatism with the condition for equal refraction at the two lenses, we get the two equations « = H/+/')1 «=/-/ r From these equations we deduce /=3/, a = 2/. The eye-piece will therefore consist of two lenses, the field-glass having a focal length equal to three times that of the eye-glass, and the distance between them equal to twice the focal length of the eye-glass. This is the construction of Huyghens' eye-piece, invented bj' him to diminish the effects of aberration, by making the deviations of the rays at the two lenses equal. It was afterwards 272 HUYGHENS EYE-PIECE. [chap. XI. pointed out by Boscovich, that it also possessed the advantage of being achromatic. ; ________ p 1 ^■^ A The eye-piece is usually made with plano-couvex lenses, the plane faces being next the eye. Rays proceeding from the object- glass would meet in q, qp being in the principal focal plane of the object-glass ; the rays are caught by the field-glass before reaching q, and are brought to a focus at q\ which is in the focus of the eye-glass, so that the rays will emerge parallel to each other. Let A, B be the centres of the lenses, AF the focal length of the lens A ; then since AF = 3/', AB = 2/, the point F is also the principal focus of the lens B. Since c[p' is in the focus of the lens B, Bp' = ^AB. Also, since p, p' are conjugate foci with respect to the lens A, _1 !__ ]_ Ap' Ap-3f' and Ap'=f; therefore Ap = ^f' = lAB. Thus p is the middle ■ point of AF. Thus the field-glass must be placed between the object-glass and its principal focus, at a distance equal to half its own focal length from the latter. This eye-piece cannot be used in telescopes where measurements by means of spider-lines or fine wires are to be made. For the principal focus of the object-glass is virtual. The wires could not be placed at the image qp, because there will be distortions in the image of the wires due to the eye-glass, while the image of the object will be distorted by excentrical refraction through both the field-glass and the eye-glass ; so that the wires and the image will appear distorted in different degrees, and therefore the position of a point in the field would be estimated incorrectly by referring it to the wires. In all telescopes graduated by wires, for measurement, the field -glass must be beyond the principal focus of the object-glass; then the image and the wires if distorted at all are distorted equally, and therefore no error will result in the measurement. 247—248.] RAMSDENS EYE-PIECE. 273 248. la the common astronomical eye-piece, known as Ramsden'.s eye-piece, the two lenses are of equal focal length, and therefore the condition of achromatism requires that the distance between them should be equal to the focal length of either. But in this arrangement, the field-glass being exactly in the focus of the eye-glass, any dust which might happen to lie on it or any flaw in the glass would be magnified by the eye-glass and confuse the vision. The distance between the lenses is therefore made a little less than the focal length of either ; and thus, though the eye-piece is not achromatic, the departure from perfect achromatism will not be great. The lenses are usually plano-convex lenses with their curved surfaces turned towards each other, and the interval between them two-thirds of the focal length of either. Rays proceeding from the object-glass converge to a focus at q in the principal focal plane of the object-glass, and after crossing at q meet the field-glass. Their direction is then altered, so that they diverge from the point q', and this point is made to lie in the focal plane of the eye-glass, so that after refraction at the latter, the rays emerge parallel to each other. Let A, B be the centres of the two lenses, and let AF = / the focal length of either, then AB = |/. Also since ^p' is in the principal focus of the lens B, Bp' =/, so that Ap' = ^f. Also p and p' are conjugate foci with respect to the lens A, and therefore \ ]_^1. Ap Ap' /' and Ap' = ^f, therefore Ap = J/. Thus the field-glass is placed beyond the focus of the object- glass at a distance from it equal to one-fourth of its own focal length. The radii of the lenses are arranged so as to remedy as many of the defects of aberration as possible, and the indistinctness H. 18 274 THE ERECTING EYE-PIECE. [chap. XI. arising from this cause in this eye-piece is much less than in any of the other ordinary constructions. 249. There is another eye-piece in common use, known as the erecting eye-piece ; it is used for terrestrial objects. A terrestrial telescope differs from an astronomical telescope only in having an erecting eye-piece, instead of an ordinary eye-piece. One form of erecting eye-piece is shown in the figure. A and ^^ B are two convex lenses of equal focal length, placed at any distance from each other, ^gj is the image as formed by the object-glass. The lens A is adjusted so that pq lies in its principal focus ; then it is easy to see that the lens B will form an image p'q', of pq equal to it in magnitude but turned upside down, and the distance Bp' will also be equal to the focal length. Besides these two lenses there is an ordinary Huyghens' eye-piece which must be adjusted to the image p'q', just as in the astronomical telescope. The distances between the four lenses are fixed ; they are usually fitted into one tube, and adjustments for different distances are effected by pushing in or drawing out this eye-tube. This eye-piece will not be achromatic, because the focal lengths of the two first lenses will not be the same for all colours. 250. The position of a compound eye-piece when arranged for distinct vision, and the magnifying power of the instrument, may be found by considering the images formed as the rays pass through the instrument. We shall suppose the object to be very distant, and that the instrument is arranged so that the rays of the emergent pencils are parallel to each other, and therefore the first image will be in the principal focus of the object-glass, and the last image will be in the principal focus of the eye-lens. Let x, a; be the distances of these images in front of, and behind, the field-lens, respectively, and let /3, /S' be the linear magnitudes 248 — 250.] TELESCOPE WITH COMPOUND EYE-PIECE. 275 of these images, and a, a' the initial and final inclinations of the axis of the extreme pencil. Then, if / /', /" be the focal lengths of the three lenses, and a, a' the intervals between them, a=x+f ] a' = x'+f"]' 1 111 and _+ = X X J-' The relation between the intervals a, a is therefore 111 If we clear of fractions and add f' to each sid© of this equation, it takes the form To find the inclinations of the initial and final pencils, the equations are a ■■ /' a = and also X /8' and therefore the Kiagmifying power is a' m = - = a 0' f - ^ f" or «i = y f But x'_ X = 7-' and therefore m = -/{^ ^ f" f'f"\ This formula might have been found direetly by substituting for the two lenses of the eye-piece the ecpuvalent lens, and then using the result already obtained for the magnifying power of the astronomical telescope consisting of twO' lenses. IS— 2 276 TELESCOPE WITH COMPOUND EYE-PIECE. [CHAP. XI. In exactly the same way it may be shown that If we multiply these equations together we get the relation between the intervals already mentioned. The object will be inverted, as in the ordinary astronomical telescope, unless a be greater than f-\-f'; that is, unless the distance between the first two lenses be greater than the sum of their focal lengths. If the focal lengths of the lenses be given, and also the magnifying power, the intervals between the lenses are deter- mined ; for, from the preceding values of m, we get «=/+/' +^4. «'=/'+/"+ ^^- 251. The field of view is determined in the same way as in the ordinary astronomical telescope, supposing it to be governed by the first two lenses. The aperture of the third lens will then be chosen so as to allow of all the rays to pass through. Thus if the field be determined by the axes of the extreme pencils, corresponding to the field of view ©, let the semi-apertures of the field-lens and eye-lens be, respectively, h', b". Then a Also if a be the inclination of the axis of the pencil after refraction at the field-lens, a' — © will be the deviation produced by that lens, and therefore a' = ©-|. Also b" = h' + a'a'; therefore b" ==b' + a'@-^ = ©|a + a'-^]. which determines the aperture of the eye-lens. The aperture of the eye-lens corresponding to the greatest visible field and the bright field may be found in the same way. 250 — 252.] REFLECTING TELESCOPES. 277 In Galileo's telescope, the incidence on the eye-lens is centrical ; the e\'e-lens used is therefore always a single concave lens, or an achromatised pair of lenses in contact. Reflecting Telescopes. 252. If instead of a convex object-glass, a concave mirror be used to receive the rays proceeding from an object, an image of the object will be formed by the mirror, which, if the aperture be sufficiently large, may be viewed directly by means of an eye-piece placed in a suitable position, as in the case of the telescopes pre- viously described. Such is the principle of Sir W. Herschel's telescope, which is the simplest of the reflecting telescopes. In order that the head of the observer may intercept as little light as possible, the axis of the mirror is slightly inclined to the axis of the tube in which it is fixed, and thus the image is thrown near the edge of the tube, where it is viewed through an eye-lens, or eye-piece, the observer having his back to the object and look- ing down into the tube. The obliquity of the incident pencil to the axis of the mirror will produce a slight distortion of the image, but the errors due to this cause are scarcely appreciable in the very large instruments to which this construction is alone applicable. We shall suppose that the object is very distant, so that the image formed by the mirror will be in the principal focus of the mirror ; and also that the instrument is to be adapted to the use of eyes with normal sight, so that the emergent rays must be parallel, and therefore the eye-lens must be placed in such a position that the first image may lie in its focal plane. Now the angle which the object will subtend at the centre of the mirror, and therefore at the eye, will be equal to — ^jF, where ^ is the linear magnitude of the image, and F the focal length of the mirror. And the angle under which the image will be seen by the eye- will be ^jf, f being the focal length of the eye-lens. The magnifying power is represented by the ratio of the latter to the former, and therefore F 278 herschel's telescope. [chap. xr. This iostrument therefore gives an inverted image. Oh- The arrangement of the mirror and eye-lens are shown in the figure. BAG is the large spherical reflector, AO being its axis, and its centre ; AP is the axis of the tube and Aa the axis of the eye-lens, and these two lines are equally inclined to AO, the axis of the mirror. Bq, Aq, Cq, are three rays which are brought to a focus at q by the large reflector ; the rays afterwards meet the eye-lens and finally emerge parallel to each other. The focus q and the corresponding point of the object lie in the same line through 0, the centre of the reflector. 253. The field of view in this telescope as determined by the axes of the extreme pencils is obtained by joining the extreme edges of the eye-lens to the centre of the large speculum. The distance between the lens and the centre is F —f, nearly ; for .40 = 2^, and Ap= F and the inclination of Ap to AO is very small. If therefore a denote the semi-aperture of the eye-lens, and half the field of view, = The focal length of the eye-piece in these instruments is very small in comparison with that of the mirror so that the field of view is very nearly eqvnl to the angle subtended by the eye-lens at the vertex of the large reflector. The entire visible field is formed by joining the corresponding edges of the eye-lens and the reflector ; the intercepted portion of the perpendicular erected at their common focus will be the linear magnitude which is illuminated by any ray whatever proceeding from the object, and the angle subtended by this perpendicular at the centre of the reflector will be the field of view. The determination is exactly the same as in the astronomical 252 — 254.] jjewton's telescope. 279 telescope, and we may adopt the result previously obtained, namely, 0' = Af+aF F{F+fy where A denotes the semi-aperture of the object-mirror, and @' half the extreme field. If we neglect fin comparison with F, and substitute m for F/f, this becomes which, in instruments of large magnifying power, does not differ widely from the results previously obtained for the mean field. Herschel's great telescope was constructed in 1789 ; it was 40 feet in length, and the great reflector was 50 inches in diameter. The quantity of light obtained by this instrument was so great as to enable its inventor to use eye-pieces of far shorter focal length than any previously used. Lord Rosse's telescope has a speculum of 53 feet focal length and 6 feet diameter. Newton's Telescope. 254. The principle of the front view, as previously described, can only be used in instruments in which the aperture is very considerable, and to instruments of moderate aperture it is wholly inapplicable. In the telescope invented and constructed by Newton, the rays reflected by the object-reflector are received on a small plane mirror placed between the object-mirror and its principal focus. The plane of the mirror is inclined to the axis of the telescope at an angle of 45°, and the rays which tend to form an image in the principal focus of the object-reflector are reflected laterally and form an image near the side of the tube, equal and similar to the former, and similarly placed with regard to the plane mirror. This image, whose plane is parallel to the axis of the tube, is viewed through an eye-piece placed at the side of the instrument. Instead of a plane mirror, Newton used a rectangular isosceles prism of glass, through the sides of which the rays enter and emerge perpendicularly, being reflected totally at the hypotenuse. The reflexion at the hypotenuse being total, there is a much 280 Newton's telescope. [chap. XI. smaller loss of light in the reflexion than in the reflexion at a metal speculum. The arrangement of the mirrors and the eye-lens is shown in the figure. BAG is the object-mirror, B'A'C the plane mirror, and bac the eye-lens. Rays BQ, AQ, CQ are reflected by the large mirror to a focus Q, where PQ is the principal focal plane of the reflector. But before they reach Q they are reflected by the small plane mirror and meet in q ; after crossing at q they strike the eye-lens and emerge parallel to each other. The point Q and the corresponding point of the object lie on a line through the centre of the large reflector ; also the image PQ and the second image qp are symmetrically placed with regard to the mirror B'A'C, and qp is equal in magnitude to QP- If F,f denote the focal lengths of the object-mirror and the eye-lens, and e, e denote their distances from the centre of the plane mirror, then in the figure, A'P = F-e\ A'p=e'-f\' since the first image is in the principal focal plane of the large mirror, and the last in that of the eye-lens. Hence, since A'P = Ap, we get e + e' = F+f. This is the condition of distinct vision with parallel rays. The magnifying power may be found just as in the case of Herschel's telescope ; the value of it is, as before, F 254—255.] Newton's telescope. 281 255. The small mirror must be large enough to receive the whole of the principal pencil, or the cone of rays meeting in the principal focus of the object-mirror, but it must not be made larger than necessarj', or otherwise the brightness of the central part of the field will be impaired. The mirror will therefore be a section of the full cone of rays converging from the object-mirror to its principal focus, made by a plane at an angle of 45° to the axis ; it will therefore be in the form of an ellipse. Let the semi-vertical angle of the cone be 6, then tan 6 ■■ A ■ F' where A denotes the semi-aperture of the reflector. Suppose a section of the cone by a plane through the axis perpendicular to the plane mirror to be represented in the figure, MN being the section of the plane mirror ; then MN will be the major axis of the ellipse. Denote the two portions of MJSF, as divided in the centre A', by x and x'. Then if A'P be denoted hyd, dsind d J2ta.n 6 _ Ad >J2 X =- ■ sin (45" - 6) d sin 6 1 - tan d \/2 tan 6 F-A' Ad'Ji "sin (45° -I- 6") H-tan6' F+A' and therefore if a, h denote, the semi-axes of the ellipse, ,, AFd V2 a=-^{x + x) = j^.._^, . Let y denote the breadth of the section perpendicular to that represented in the figure, at J.' ; then by properties of the ellipse, a'' X xx' 282 Newton's telescope. [chap. xi. But y is the radius of the circular section of the cone through the point A, so that y = Ad/F; and therefore l/^r-A" a"" 2i^'"' If we give a its value, the corresponding value of b becomes Ad 0= , — ^^ . JF'-A' The aperture of the object-mirror will be small compared with its focal length, and therefore A^ may be neglected in comparison •with F'. The approximate values of a and b will therefore be Ad'^2 , Ad ^ = -F' which are in the ratio of \/2 to 1. 256. The extreme field of view in Newton's telescope is determined by joining the adjacent extremities of the eye-lens and the plane mirror. The intercepted portion of the perpen- dicular qp, raised at the principal focus of this lens, will give the whole extent of the image illuminated by any rays whatever, proceeding from the object. The field of view will be the angle subtended at the centre of the object-mirror by the corresponding image PQ. Let y be the magnitude of this image, a the semi- aperture of the eye-lens, A' the perpendicular distance of the extremity B' of the small mirror from the line A'a. Then, by similar triangles, / d-A' where d denotes the distance A'P. or A'p. Neglecting A' in comparison with d, in this result, we get _ad + A'f y- d+f ' and if @ be half the visible field, l ad + A'f F d+f ■ 255—257.] GREGORY'S TELESCOPE. 283 From the preceding investigations, A' = AdjF, very nearly. If we substitute for A' this value, and neglect / in comparison ■with d, we get A ■^ m and & = -Ja-\ — ) , as in Herschel's telescope. Gregory's Telesc 18. Two collimators of the usual construction were pointed directly towards each other and the wires of each were made by adjustment to be seen distinctly together with the images of the wires of the other ; the geometrical focal lengths being / and /', 3/ and 9/' small deviations of the positions of the wires from the geometrical foci, and D the interval between the object-glasses ; show that to a first approximation, — ^ = 4;^ , and to a nearer approximation. 3/ rv^f fV f)r 19. An object-glass of focal length F is used in combination with a four- glass erecting eye-piece in which the focal lengths of the lenses are /„ f^, f^, f^ in succession from the object-glass. The distances of the first and last pairs of lenses in the eye-piece are constant and equal to a, h, but the distance x between the middle pair is variable. If the telescope be adjusted for parallelism of the pencils of emergent rays and so that its magnifying power is greatest, the distance y between the focus of the object-glass and the lens/j will be given by the equation ^G/i-/^^<-7.)[(-;>a-i-/^>j. and X will be expressed by a fraction whose numerator and denominator involve y to the first degree only, with known quantities. CHAPTER XII. Optical Instruments and Experiments. 272. If light be admitted into a darkened room through an aperture fitted with a single convex lens or a combination of lenses, inverted images of external objects will be formed within the room at their proper distances from the lens ; and if the objects be at a considerable distance from the lens, compared with its focal length, the distances of their images will be very nearly the same and equal to that focal length. If therefore a screen be placed perpen- dicularly to the axis of the lens, at a distance from it equal to the focal length, an inverted picture of the external scene will be formed on the screen. For the purposes of drawing, it is convenient that the image should be thrown into a horizontal position. This is effected by placing between the lens and its principal focus, a plane mirror inclined at an angle of 45° to the axis of the lens. This is the principle of the portable camera obscura. A box from which external light is excluded is substituted for the darkened chamber ; for the screen may be substituted a sheet of sensitive paper, upon which the light acts chemically ; with this arrangement, an inverted picture of the external objects is printed upon the paper, and may be preserved in the form of a photograph. In the magic lantern and the solar microscope, a picture or an object is placed before a collective lens-system at a distance from it a little greater than the focal length of the system, and is then strongly illuminated by an artificial light or the light of the sun thrown into the axis of the tube by. a system of reflectors. A real inverted and magnified image is formed at a certain distance from the lens-system, and may be seen depicted on a screen in a darkened room. If the object and the screen be fixed, 302 THE CAMERA LUCIDA. [chap. XII. the adjustment may be effected by moving the lens-system back- wards or forwards in a sliding tube by means of a screw. The adjustment will always be possible, provided the distance of the screen from the object be greater than four times the focal length of the lens-system. 273. The camera lucida, invented by WoUaston, is an instru- ment of great use to the draftsman, in preparing an accurate drawing of a building or a landscape. Its essential feature is a quadrilateral prism of glass, represented in the adjoining figure. The angle j4 is a right angle, and the opposite angle is 13-5°, while the remaining angles B and D are equal ; it follows that the angles B and D are each GT^". Rays of light which are incident perpendicularly on the face AD and are reflected successively at DC and GB, will emerge perpendicular to the face AB. u B Let PRSTU be such a ray, and let PQ be a small object perpendicular to PR. Then an image qp will be formed by refraction at the plane surface AD; the rays diverging from qp will be reflected at the surface CD, and made to proceed from an equal image q^p^, symmetrically placed on the other side of CD; the rays diverging from qj)^ will be again reflected at the surface CB and another image q'p' will be formed. Finally when the rays proceeding from g''^}' are refracted again into the air, they will 272 — 273.] THE CAMERA LUCIDA. 303 proceed from an image Q'P'. Let PR the distance of the object from the first surface be denoted by x, and UP' the distance of the final image from the final surface AB by x, and let m, v, w, be the le^igths of the three portions of the path within the prism. Then 'pR = fix. Also it is easy to see that Up = /j,x + u + v + w, and therefore , ii + v + w X =x-\ . Hence the difference between the distances of the object and final image from the vertical and horizontal sides of the prism, respec- tively/, is equal to tlie length of tlie path within the prism divided by its refractive index. The prism is mounted in a brass frame and attached by its axis to the end of a brass stem, the lower extremity of which may be clamped to a table ; the length of the stem may be varied at pleasure by means of a sliding tube. The upper surface of the prism AB is furnished with an eye-stop of small aperture, which is adjusted so that the aperture is as nearly as possible bisected by the edge B; by this means only a small part of the surface AB is used, and the rest is covered. When the vertical face of the prism is turned towards the object, the observer looks downwards through the aperture and sees at the same time the image of the object through the uncovered portion of the prism, and the paper on which it is thrown through the remaining portion of the aperture. The image will be erect, since the rays from the upper part of the object proceed towards the upper part of the image. Since the dimensions of the prism are very small in comparison with the distance of the object, the distances of the object and image will be nearly equal. If the distance of the object from the prism be very different from the distance of the latter from the table, the image and the paper cannot be seen together distinctly. This may be remedied by a convex lens whose focal length is equal to the greatest distance of the prism from the table. The lens is turned up horizontally under the prism, and the paper being in the principal focus, its image is thrown to an infinite distance and therefore made to coincide with the image of a remote object formed by the prism. The same correction may be made by placing a concave lens of the same focal length vertically in front of 304 hadley's sextant. [chap. xii. the vertical face of the prism. The rays proceeding from a distant object are made to diverge from an image whose distance is equal to the focal length; this image will therefore coincide with the paper after passing through the prism. The convex lens is to be used by normally sighted persons, the concave by short-sighted persons. For near objects the adjustment of the distances is completed by varying the distance of the prism from the paper. 274. Hadley's Sextant is an instrument for measuring the angular distance between two distant points. It consists of a framework in the form of a sector of a circle, with a graduated arc, and two plane mirrors, whose planes are perpendicular to the plane of the sector. One of the mirrors A is moveable about an axis through the centre of the arc, and carries a pointer whose vernier slides along the graduated arc. The other mirror is fixed at F and is parallel to the mirror A when the pointer of the latter is at E, the zero of the graduated scale ; the lower part of this mirror only is silvered, so that rays of light may be transmitted directly through the upper part. The instrument is fitted with a small telescope G whose axis is directed towards the dividing line of the mirror F. To measure the angular distance between any two points P, Q, the instrument is brought into the same plane with them and the 273 — 27G.] THK HELIOSTAT. ,305 telescope G is directed towards one of them Q, which can be seen directly through the unsilvered part of the mirror F. The mirror A is then moved so that P, as seen through the telescope by a pencil reflected in succession at the mirrors A and F, appears to coincide with Q. In this arrangement, the angular distance between the points P and Q is the deviation of the axis of the pencil by the two reflexions; and this is equal to twice the inclina- tion of the mirrors. The inclination of the mirrors may be read off the graduated scale. If the arc be graduated so that every half- degree may be read as a degree, the reading will give the angular distance between the two points without any further calculation. Tlie Heliostat. 21 o. A heliostat is an instrument which will reflect the light of the sun in a fixed direction throughout the day, notwithstanding the motion of the sun. It will be supposed that the change in the Sun's declination during the day is so small that it may be neglected ; so that the sun will describe a small circle on the celestial sphere, about the pole. In all the heliostats which have been constructed, one essential feature is an axis, parallel to the axis of the earth, which is turned by clock-work with the same angular velocity as that of the sun. The simplest form of heliostat is Fahrenheit's ; in this instru- ment a plane mirror is rigidly connected with the revolving axis, in such a way that the normal to the mirror makes with the axis an angle equal to half the sun's polar distance. If the normal be ad- justed so as to have the same right ascension as the sun, they will continue to have the same right ascension throughout the day, and the mirror will continue to reflect the sun's rays in the direction of the earth's axis. By a second fixed mirror they can afterwards be reflected in any required direction. 27G. Foucault's heliostat is arranged so as to reflect the solar rays in any required horizontal direction. The point is fixed, and OA is the rotating axis. OB is a rod rigidly fixed to the revolving axis at an angle which can be adjusted so as to be equal K. 20 306 THE HELIOSTAT. [chap. XII. to the sun's polar distance; then if the right ascension of the plane OBA be properly set, OB will represent the direction of the sun's rays throughout the day. Let OC be the horizontal direction in which the sun's rays are to be reflected, and let C be the point about -which a mirror turns ; the normal to the mirror NOB is jointed to OB at a point B, such that 00 = OB. Also the rod BOD passes through a slot GD fixed to the plane of the mirror. Then since 00= OB, it follows that OB and 00 make equal angles with the normal to the mirror and are always in the same plane with it. Hence sun-light incident on the mirror parallel to OB will be reflected parallel to 00. The line 00 can be moved in its own plane to any azimuth. 277. Silbermann has constructed a heliostat which will reflect the sun's rays in any fixed direction. As before, let OA be the revolving axis, OB a rod rigidly 276—278.] LIGHTHOUSES. 307 connected with it, so that OB represents the direction of the sun's rays ; OR is a rod which can be fixed in any direction, in which it is desired to send the reflected rays. Oach is a small rhombus of jointed bars, a and h being fixed joints on the rods OB and OR. The normal to the mirror ON carries a slot, in which the angular point c of the rhombus slides. The mirror will then reflect the rays of the sun in the direction OR. Ex. If a heliostat be arranged so as to reflect the sun's rays in a fixed direction, prove that if the diurnal change in the sun's declination be neglected, the normal to the mirror, and the intersection of the mirror with the plane of reflexion, will describe cones of the second order, whose circular sections are perpendicular to the axis of the earth and to the reflected ray. Liglitlwuses. 278. The lenses used in lighthouses were first introduced by Fresnel. A lighthouse lens consists of a plano-convex lens surrounded by a series of rings forming steps outwards; it is represented in cross section by the central part of figure. The back of the compound lens is plane, and the central face is spherical in front. The convex surfaces of the rings are not spherical ; they are 20—2 308 KRESXELS ANNULAR LENS. [chap. XII. annular, generated round the axis of the lens by the revolution of circular arcs in the plane of that axis, but having their centres beyond it in a series of points which retreat further from that axis as each corresponding ring increases in diameter. Fresnel so calculated the coordinates of the respective centres of the actual arcs, so that the extreme rays are made to emerge parallel to the axis. This approximation corrects aberration almost perfectly. In large lighthouses the diameter of this lens subtends an angle of 57° at the centre of the light. 279. To find the form of the cross-section of any ring in Fresnel's lens, let us suppose that Q is the radiant point, ABGD a section of the ring, OAB being the plane side of the compound lens. Let QAPR, QSGT be the extreme rays which can pass through the ring in the plane of the figure, the emergent portions PR and CT being parallel to the axis of the lens. Let E be the centre of the circular arc GP, and let the radii EP, EG make angles , -v^ with the axis. Let a, p be the angles of incidence and refraction at A, and o- those at S, and let QO=f, BG=t, AB=b. Then 6=/(tan ;S-tan a)-|-nano-, and fi sin p = sin a, /^ sin a- = sin yS. Also the angles of incidence and emergence at P are, respectively, i>-p, ; those at (7 are A^ - o- and -yjr ; so that sin <^ = /i sin {(p - p), sin ■f = fi sin (i/^ - o-) ; 278 — 280.] fiiesnel's reflecting peisms. 309 and therefore tan,^= ^^^ tan 1^ : /i cos p — 1' /isin a- fi cos tr — 1 ' from which ^ and ■^ are determined. Draw CM parallel to J. 5 to meet AP in M. Then GJf = 6-ftanp. Also from the triangle CPM, the chord GP is easily seen to be given by the equation ^p _ C'Jlfcosp And therefore if r be the radius of curvature of the arc GP, so that OP = 2rsinH^-0). we get finally (b —t tan p) cos p ~ 2 sin ^ (i|r - ^) cos { J(^ +'^)-pY Also, the coordinates of E referred to as origin are EH =r cos ■^^ — t 0H= r sin -^ —/tan ^ — t tan o-, and therefore the curvature of the surface and the position of its centre of curvature are determined. If the section be required for a prism which is detached, t = 0. 280. Above and below the lens, there are a series of totally reflecting zones of triangular section. These zones are continued so as to leave only a small space above and below the light, where the light is not caught and sent out horizontally. We shall now find the form of the section of the reflecting zones, following Fresnel. The reflecting surface is curved ; but instead of the true curve a circular arc is necessarily adopted. Let ABG be the section of the totally reflecting zone and Q the radiant point; and let AK, GH be the extreme emerging rays, which are to be parallel to each other and to the horizon. Let the 310 FRESNEL'S reflecting prisms. [chap. XII. angle AQB be a, and the angles of incidence and refraction of the ray QA at A be 0, 6'; produce QA to D, and let the angle DAK beS. In order to avoid superfluous glass, the side AB of the prism is made to coincide with the path of the ray QB after refraction and reflexion at B, and the side AC to coincide with the path of the ray QA after refraction at A. Hence the angle BAK is equal to the angle CAQ and therefore the angle BAQ to CAK; also the angle BAG is \-n- + 6' ; and therefore adding together the several parts which make up the larger angle QAK, or ^' = 25 + S-^7r, and therefore smd = /i sin {W + S — ^ir). From this equation the angle 6 may be found. Let 4> and 4>' be the angles of incidence and refraction of tlie ray QB at B, then ^ = d — a and therefore fi sin <^' = sin {d — a). The internal incidence of the ray CH at G is equal to ff', since the emergent rays AK, GH are parallel. At B and G draw the radii BE, GE of the circular arc BG which is the reflecting surface, and join BG; then it is easily seen that the angle ABE = ^ (!■"■ + f), and AGE = \{^Tr + 6'). Also the angle BEG =BAC-{ABE+A GE), and therefore the angle BEG= ^ - ^ (5' + ') 280 — 282.] DETERMINATION OF REFRACTIVE INDICES. 311 The angles EBG and EGB are therefore each equal to therefore the angle ABG = EBG - ABE = J (tt - 0' - <^') and the angle AGB = EGB - AGE = J (tt - 361' + ^')- Now in the triangle QAB the side QA i& supposed known and the angles AQB and QAB ; therefore AB is known. Then in the triangle ABG, the two angles B and G are known and also the side A B ; therefore the side BG is known. Lastly the angle BEG has been found, and therefore the radius of curvature of the circular arc is known. 281. So far only the section of the apparatus has been described. This section may be revolved round a vertical axis passing through the light ; and this gives the form of a lighthouse apparatus for a fixed light sending light out horizontally in all directions. For a flashing light, it is usual to arrange the lenses in the form of eight panels ; each panel consists of a Fresnel's annular lens generated by the revolution of the section about a horizontal axis, together with totally reflecting zones formed by revolving the section through an angle of 45° about the vertical axis. The whole apparatus is then slowly revolved about its vertical axis. Further information on this subject may be found in a paper by Mr. James T. Chance, " On optical apparatus used in Light- houses," published in the Proc. Inst. Givil Engineers, Vol. xxvi., 1867, and also in the Article " Lighthouses," in the Encycl. Brit. Determination of Refractive Indices. 282. The general method of measuring the refractive index of a solid medium for any particular coloured ray of light, is to observe the minimum deviation of a ray of light of this colour, as it passes through a prism made out of the substance. It has been already seen that, when a ray of light passes through a prism with minimum 312 DETERMIXATIOX OF REFRACTIVE INDICES. [CHAP. XII. deviation, its path is symmetrical with respect to the prism ; so that with the usual notation Jr, 4>' = •^'. and therefore I) + i = 2^ ) If /J, be the refractive index of the medium, sin ^ = /i sin ', and therefore sin ^(JD + t) = /i, sin ^i. As soon, therefore, as we have measured i, the refracting angle of the prism, and J), the minimum deviation, we can calculate /j,. 283. The appai-atus used consists essentially of a horizontal graduated circle, with a horizontal telescope which can be turned round, so that its optic axis always passes through the centre of the rim. The prism is fixed with wax or cement to a levelling stand placed over the centre of the graduated circle. The light is supplied through a collimator, which consists of a fine vertical slit placed in the focus of an achromatic object-glass, so that the rays emerge from the collimator parallel to each other, the collimator being fixed so that its axis passes through the centre of the rim. The refracting angle of the prism is first measured. The prism is placed so that light from the collimator is reflected at both faces of the prism. The image of the slit as reflected at each 2S2 — 285.] DETERMINATION OF REFEACTIVE INDICES. 313 face in succession is viewed by means of the telescope, the telescope being moved round till the image falls on the cross-wires of the telescope. The angle through which the telescope must be turned from seeing the image reflected in one face, in order to see the image reflected in the other face, is read off the graduated circle. It may be shown that this angle is equal to twice the refracting angle of the prism. For let BAG be the refracting angle of the prism, and let the incident ray be in direction of PAX. Then, if AQ be the ray reflected in the face AB, AQ and AX must make equal angles with AB, so that Z BAX = i Z QAX. Similarly, if AR be the direction of the ray reflected in the face AC, Z GAX=\ Z RAX; and therefore, by addition, Z BAG = 1 Z QAR. 284. The minimum deviation for a ray of definite refrangi- bility, corresponding to any fixed line of the spectrum, is next measured. The slit is first viewed directly, the prism being turned so as not to obstruct all the light, and the telescope is moved until the line of the spectrum coincides with the cross-wires of the telescope. The prism and telescope are then moved so that an image of the slit formed by light which has passed through the prism is seen through the telescope ; and the prism is turned so as to make the image move nearer to the direction of direct light, the telescope following the image so as always to keep it in view. At length a position of the prism is obtained, such that if the prism be turned either way the image recedes from the direction of the direct light ; this position of the prism is therefore the position of minimum deviation. The telescope is moved until the line of the spectrum coincides again with the cross-wires of the telescope. The angle through which the telescope has been turned from the position of direct light is read off the graduated circle, and this angle is the minimum deviation required. 285. To measure the refractive index of a liquid, it is enclosed in a hollow piism of glass, made by cementing plates of glass 314 DETERMINATION OF THE FOCAL LENGTH OF A LENS. [CHAP. XIL together. The two sides of the plates however are never accurately parallel, and from the observed deviation it is necessary to subtract the small deviation caused by the empty prism. The refractive indices of gases in given conditions as to temperature and pressure have been measured by a similar process. They must be enclosed in a tube, the ends of which are closed by two plates of glass placed very obliquely with reference to the axis of the tube. The experiments of Biot and Arago on the refractive indices of gases showed that for gases the quantity /j.^—1 is proportional to the density of the gas, a law which had been enunciated by Newton, who deduced it from his theory of emission. 286. To find the focal length of a thin convex lens. This is usually measured by adjusting the lens and an object, until the distance between the object and the image is a minimum; this distance is then four times the focal length. For, if u, v be the distances of the object and image in front of, and behind, the lens, u V f while the distance between the object and the image is given by the equation Combining these equations, we get uv = af, and therefore (u - vf = «" — ^xf The quantity (m — vf is always positive, and therefore the least value of X is equal to 4/1 If the lens be concave, it is placed in contact with a convex lens, so that the whole combination may be collective ; the focal length of the combination may be determined as before. If /, /' be the numerical focal lengths of the two lenses, F that of the combination, F~f f which determines /when/' is known. 285—287.] THE CARDINAL POINTS OF INSTRUMENTS. 315 0)1 the expenmental determination of the focal len-gth and cardinal pointi of an optical instrument. 287. It has been seen that the position and magnitude of the image of an object formed by any symmetrical optical instrument can always be determined by geometrical construction or by simple formulae, when the positions of the two focal points and the two principal points are given. The positions of the two principal points are determined as soon as the positions of the focal points and the focal length are known, and thus it appears that the optical system is completely determined when we know the abscissai of the focal points and the focal length of the system. To determine these three quantities, three experiments will be necessary. Let g, g' be the abscissae of the two focal points, referred to any origin, ^, ^' the abscissae of the object and its image, respectively; then, if/" be the focal length of the system, (5'-f)(r-/)=r- For the sake of symmetry, suppose that all the distances con- sidered are measured from a fixed point whose abscissa is e, and let e-g = p, g' -e = q; the constants to be determined are then p, q and /. If we take three different positions of the object and determine by experiment the position of the image for each, we get the three equations ia-p){b-q)=f\ {a'-p){b'-q)=f. {a"-p){h"-q)=f\ where a', h' and a", h" refer to the second and third positions. The quantities a, h, a', h', a", h" are all determined by measuring, and therefore these three equations are sufficient to determine the values of p, q, f in terms of known values. If we eliminate /* between the first and second, and again between the first and last, Ave get two equations of the first degree in p and q, which serve to 316 THE CARDINAL POINTS OF INSTRUMENTS. [CHAP. XII. determine these quantities without ambiguity. Any one of the three equations will then determine /'. This will leave the sign of / undetermined ; but the sign of /may be found by examining the image. If the image be erect, then ^' — g' and / will have opposite signs, and if inverted, the same sign. 288. In the formulae it is supposed that for the three experi- ments the object lies on one and the same side of the lenses. If in any case the object were at the other side, the positions of the object and image may be interchanged, and then the arrangement is the same as in the preceding cases. Practically we are restricted to real images, and therefore in the case of a single lens we are limited to a collective lens, unless special methods of determining the positions of virtual foci be adopted. It is always possible, however, to combine the system with a single collective lens so as to make the whole system collective, and then, having determined the constants of the combination, to calculate those of the original system. The experiments should be chosen so as to have the three equations as different as possible, in order that the errors of obser- vation may have the smallest effect on the results. 289. For a single lens, or for an achromatic object-glass, which consists of lenses in contact, the distance between the principal points is usually small. If this distance, which we may call \, be known, two experiments are all that is necessary. For we then have the equation p + q = 2f+\ in addition to the two ia-p){b-q)=f\ ia'-p){i'-q)^f\ Eliminating p and q between these equations, we find ^* "'^^'^ ^^ _(a + 6'-X)(a'-h6-\) = 0. which is in general a quadratic to find f. But if the experiments be chosen so that a' -\-b' -a-h = 0, that is, so that the distance of the object from the image is the same in each experiment, while the lens has different positions, the equation reduces to an equa- 287 — 290.] THE CARDINAL POINTS OF INSTRUMENTS. 317 tion of the first degree. Let c be the distance between the object and the image ; then a = c — b, a =c — h', and the equation be- comes 4/(c-X) = (c-X+6'-6) {c-\-h'+h), /=i(c-X)-i-^. The values of p, q are then p=^{if+c + X-h-h'), q = \{^f-c + \ + h + h'). 290. A convenient method of arranging the experiments in this case is first to place the object at a great distance, and then the rays will converge to the second focal point, and thus q is determined immediately. Then reverse the lens, and measure the position of the image of the same object, and this will determine p. Besides these two experiments one other is required, which gives an equation, {a-p){h-q)=r, and determines f. The distance X is then given by the equation \=p + q-2j{a-p){b-q). The solution may now be regarded as complete; but if a specially exact determination is necessary, this investigation may be regarded as giving X as a preparation to the method just ex- plained. Gauss gives the following convenient method of arranging the third experiment. On a sheet of paper describe a circle of about the same radius as the lens, and make a small well-defined cross at its centre ; then let the lens be placed in contact with the paper so as to be accurately concentric with the circle, and view the cross with a small microscope with cross lines, which is capable of adjust- ment along the axis of the lens. Adjust the microscope until the image of the cross coincides with the cross in the microscope. Then remove the lens and again adjust the microscope. The distance through which the microscope has been moved is then equal to f— ^. The point of the cross may be taken as the point of reference, and then a = 0,6 = r-f 318 METHODS OF PHOTOMETRY. [CHAP. XII. Methods of Photometry. 291. It has been shown that when an element of a surface is illuminated by light proceeding from a source of intensity I, at a distance r, so that the axis of the pencil makes an angle 6 with the normal to the element of surface, then the intensity of illumination is proportional to /cos 6 It is found that the eye is of itself unable to estimate the ratio of the intensities of two sources of light, but that it is an accurate judge of the equality of illumination of two illuminated surfaces when they are placed side by side. All methods of photometry depend therefore on the equalising of two illuminations. In order to compare the intensities of two sources of light, the two halves of a piece of thin porcelain are illuminated by the two sources, respectively, in such a way that either the light falls normally on the porcelain, or the lights from the two sources make equal angles with the plane of the porcelain. The distances of the lights are then adjusted so that the two halves of the porcelain are equally illuminated. Then the intensities of the sources are in the inverse proportion of the squares of their, dis- tances from the porcelain. This is the principle of both Ritchie's and of Foucault's photometers. 292. Ritchie's photometer consists of a rectangular box open at both ends. In the lid is a narrow strip of porcelain or oiled paper. The instrument is placed between the two sources to be compared, and the light is reflected up to the porcelain by two pieces of mirror (which must be cut from the same piece of glass) placed at angles of 45° to the axis of the box. The box is then moved from one source towards the other until the two halves of the porcelain are equally illuminated, and the distances of the lights measured. 291—294.] METHODS OF PHOTOMETRY. 319 293. In Foucault's photometer the lights which are to be compared act separately on two different parts of the same vertical plate of thin transparent porcelain, PQ. RS is an opaque vertical screen which separates the two illuminations from one another. If this screen be so adjusted that the vertical planes ASni, BSn which limit the regions illuminated separately by the two sources A, B, intersect just in front of the lamina PQ, the dark band inn can be made as narrow as we please. The distances of A and B are then adjusted so that the two portions of the lamina are equally illuminated. 294. In Rumford's photometer the intensities of the two shadows on a screen of a vertical rod due to the two lights are compared. The lights are arranged so that the shadows fall close together, and the shadow formed by one light is lighted by the light from the other source. The distances being so adjusted that the shadows are of equal intensity, the distances of the lights are measured, and thus the intensities of the two sources can be compared. Bunsen has invented a very simple photometer. If a spot of grease be made on a sheet of paper, then if the paper be equally illuminated on its two sides, the transparent spot cannot be seen except by close inspection. The sources of light are placed on opposite sides of the paper and their distances are so adjusted that the grease spot disappears ; then the intensities of the sources are inversely as the squares of their distances from the paper. The adjustment should first be made from the side on which one source lies, then the screen should be turned round and the adjustment made from the side on which lies the other source, the same side 320 STELLAR PHOTOMETRA'. [CHAP. XII. of the paper being observed each time. The mean of these two positions will give a fairly accurate result. 295. In all these comparisons the lights are supposed to be of the same quality, otherwise the comparison fails. A strict comparison of two compound lights of different qualities could only be arrived at after comparing the relative intensities of all the different coloured rays of the spectra given by the two lights, and tabulating the results. 296. The first steps towards stellar photometry were taken by Sir John Herschel. He received the light of the moon on a lens of short focus, so as to make a small image of the moon in the focus of the lens; this image he used as an artificial star, with reference to which the brightness of stars could be estimated. The lens could be adjusted at different distances until the brightness of the star and the image were equal. The distances of the image for different stars give a means of comparing their intensities. Dr Seidel used an instrument not very different in principle but more convenient in practice. He divided the small object- glass of a telescope into two halves, one of which could be moved in the direction of its axis. Two stars to be compared were made to appear nearly in the same direction by internal prismatic reflexion. The distance through which the half of the object-glass had to be moved in order that the images might appear of equal intensity gave sufficient data for a comparison of the brightness of the stars. 297- More recently a method of comparing the brightness of stars, depending on the fact that the absorption of light passing through a dense medium is a function of the thickness of the medium, has been used by Professor Pritchard at the Observatory at Oxford. A thin wedge of homogeneous and nearly neutral- tinted glass is interposed, so that the. star images formed in the focus of the telescope are seen through the wedge. Simple means are contrived for measuring with great exactness the several thick- nesses at which the light of these telescopic star images is extinguished. In this way the light of any star may be readily compared with that of any standard star, and a catalogue of star- magnitudes can be formed. 294—299.] VELOCITY OF LIGHT. 321 Methods of determining the Velocity of Light. 298. There are two methods of determining the velocity of light by optical experiments, the one devised by Fizeau and the other by Foucault. Fizeau's experiments were repeated in 1876 by M. Cornu, and later a modification of Fizeau's method has been used by Dr. Young and Professor Forbes in Scotland. The velocity of light has also been determined by A. A. Michelson, of the United States navy, who followed Foucault's method. 299. In Fizeau's experiments two astronomical telescopes several miles apart are arranged so that their axes are accurately parallel, the one telescope looking into the other. In one of the telescopes a mirror is then placed at the focus of the object-glass, exactly perpendicular to the axis of the instrument. The observer stands at the other telescope ; in this instrument a plate of glass, inclined at an angle of 45° to the axis of the telescope, is placed between the eye-piece and the principal focus of the object-glass. Light is admitted through the side of the instrument and reflected down the tube by the plate glass, the rays coming to a focus at the principal focus of the object-glass, so that they may emerge from the instrument in a direction parallel to its axis. These rays of light enter the object-glass of the distant telescope, are reflected back in the same direction by its mirror, and some of these rays after passing the object-glass will pass through the inclined plate of H. 21 :Y22 VELOCITY OF LIGHT. [CHAP. XIL glass and enter the eye-piece and will be received by the eye in the usual manner. A wheel with a large number of fine teeth is rotated, so that the teeth pass in front of the focus of the object- glass. When the wheel rotates comparatively slowly, but quickly enough for the intermittent light to make a continuous impression on the eye, the eye will see an image of the light ; for the time taken to travel to the distant telescope and back again is so small that light which passed through the space between two teeth at starting will have time to return through the same space before the wheel has turned appreciably. We shall suppose that the breadth of the teeth is equal to the interval between two consecu- tive teeth. If now the speed of rotation be increased, it may happen that light which passed through the space between two teeth, may on its return be stopped by the next tooth, which has moved for- wards in the interval. In this case no light will reach the eye. If the velocity of rotation be continuously increased, the image will reappear, at first faintly, then more brightly, and will again begin to disappear, and so on. Let 21 be the whole length of the path of the light as it passes from the toothed wheel back to the same point. Then if ■« be the velocity of light, the time taken for the complete journey to and fro will be 2l/v. Let m be the number of teeth in the wheel, and n the number of revolutions of the wheel per second ; then the time taken by one tooth to pass before the principal focus will be lj2mn seconds. If therefore the number of revolutions per second be such as to produce the first eclipse, 2Z_ 1_ V 2mn ' or w = 4>miil I and if n be such as to cause the _p*'' eclipse, it may easily be seen that 4fmnl '=2f^V The distance I and the number of revolutions per second are observed, and then v is determined by these formulae. The imperfection in this method is that in actual experiments a total eclipse of the reflected rays is hardly ever reached ; there is usually only a very great falling off in its intensity, and the 299—300.] VELOCITY OF LIGHT. 323 exact moment which must be taken to represent the moment of eclipse cannot be determined with very great precision. 300. Messrs. Young and Forbes used a telescope arranged with a rotating wheel, similar to the instrument described, with two distant reflecting telescopes nearly in the same line, but at different distances. The method of observation was to arrange the speed of the toothed wheel so that the brightness of the two images seen should be equal; it was found that this could be effected with considerable precision. Let E be the brightness of the image when the wheel is not in position; then when the wheel is rotating slowly the brightness will be hE. As before, let n be the number of revolutions per second, and let t be the time occupied by the double journey ; also let k be the breadth of each tooth and interval in the wheel. In the time t, the circumference of the wheel will have passed over a space 2mnM. Before the first eclipse, it is easy to see that the effect of the rotation of the wheel is the same as if the breadth of each tooth were k + 2mnkt, while the wheel revolved slowly. Thus, if / be the intensity of the light, I=lE[l-2mnt}. If N denote the number of revolutions per second at the first eclipse, l-2mtN'=0, jrr_ _}_ ^ 2mt 4>ml ' Thus in the first phase ^=i^^l-l In the second phase, that is, when n passes the value JV, I is increasing, and is represented by the formula /=i^{i-4. In the same way it may be seen that in the j)*'' phase, when p is odd, we have r'=^E. n P-w 21—2 324 and when p is even VELOCITY OF LIGHT. [chap. xn. I=hE w-p + ^}- Let E', r, t', W, r denote quantities for the second distant telescope, similar to those denoted by the same letters for the other. We shall denote the distant telescopes by A and B, and shall suppose that A is the more distant. As before, it may be proved that in the p"' phase of B or r = ^E' according as p is odd or even. By comparing the values of T and T, it may be seen that the r*^ equality may be in the r'" phase of B and the (r + 1)"" phase of A If r be even, the condition for the r"" equality will therefore be and the condition for the (r + iy equality will be If we subtract these equations, we get E{^-i2r^2)} = E'{2r-^-P-], i(« + «') {f + J'} = (r+l)E+rE'. or A very great simplification may be effected in this formula by choosing the two distances I and Z' so that I _r+l l'~ r ' where r is an even integer. In the actual experiments of Messrs. Young and Forbes, I : I' = 13 : 12. 300 — 301.] VELOCITY OF LIGHT. 325 Then 1^/^= (r + l)/r, and therefore the equation of condition becomes or i\-='l+^' 2r ■ But it has been shown that N' = v/^ml' ; and therefore we get, 2ml' (n + n) r The value obtained by these experiments is 301,382,000 metres per second. The value found by Cornu, using Fizeau's method was 300,400,000 metres per second. 301. We shall next give a short account of Foucault's experi- ments to determine the velocity of light. A beam of sunlight was transmitted by means of a mirror into a dark room through a small square hole in the window-shutter, and after passing through a lens G was allowed to fall on a small plane mirror mon which was capable of rapid rotation about an axis through perpendicular to the plane of the paper. At present we shall confine ourselves to the consideration of the path of a small pencil of the incident light which diverges from a point P of the aperture. This pencil, after passing through the lens, is made to converge to the point p ; but before the rays reach p they are in- tercepted by the plane mirror mon and are reflected to the point p', when op = op'. At p' is placed a portion of a spherical mirror whose centre is o and radius op, which reflects the pencil back ;}2G VELOCITY OF LIGHT. [CHAl'. XII. again in the same direction, and if the small plane mirror be at rest the pencil will retrace its original course back to P. Between P and the lens G is placed a sheet of plate glass, inclined at an angle of 45° to the axis PC; and part of the returning pen- cil is reflected at this piece of glass and is brought to a focus at p", where it is viewed through a telescope. When the mirror inon is made to revolve slowly, the light will be returned only when the mirror mon is in a position to send light to the small mirror at p', and therefore the image p" will be intermittent ; but if the velocity of rotation be increased up to about 30 revolutions per second, the impression produced in an observer's eye is continu- ous. So long as the mirror revolves with moderate velocity, the time taken by the light to travel from o to p and back again is so short that the returning pencil reaches the mirror mon before it has appreciably changed its position ; but if the velocity of ro- tation be greatly increased, until the mirror makes several hun- dred rotations per second, the mirror will have turned through a small angle during the time occupied by the reflected light in pass- ing from to p and back again. The pencil returning from p' will be reflected by the mirror in its new position, and after re- flexion will appear to diverge from a point q, where oq = op', and after passing through the lens will be made to converge to a point Q on the line qC; the image by reflexion in the plate glass will therefore be at q' instead of p", where p" g" = PQ. Across the aperture through which the light was admitted was stretched a fine wire, whose position is represented by P, and the displacement of the image of this wire p"q' can be measured by the aid of the observing telescope. Let the value of this displacement beS. 302. Let 11 be the number of revolutions of the mirror per second ; this can be determined by means of a siren. Also let OP = a, Co = b, and let op = r. Then if v be the velocity of light, the time occupied by the light in passing from o to p' and back again will be V During this time the revolving mirror will have rotated through an angle ^irnt or ^Trni-fv. 301 — 302.] VELOCITY OF LIGHT. 327 The points p, q, p lie on the circle whose centre is o ; also the lines j^p, qp are respectively perpendicular to the two positions of the mirror, and therefore the angle pp'q is equal to the angle between the two positions of the mirror, or to 47rnr/i). It therefore follows that the arc pq subtends at the centre of the circle an angle Sirnr/v ; and therefore 2)q = . Also, by similar triangles, PQ : pq = a: (b + »•), and therefore PQ = ^^^^'^ This length PQ, being equal to p" q", has been determined by observation to be B, and therefore we get _ S-Trnr'a " ~ S (6 + r) ' an equation which expresses v in terms of quantities which can be measured. Foucault found the velocity of light by this method to be 298,000,000 metres per second. The value obtained by Michelson by a slight modification of the same method was 299,940,000 metres per second. The method employed by Foucault may be applied to the determination of the velocity of light in other transparent media, such as water. For this purpose a tube filled with the water, with its ends closed by plate glass, is placed between the revolving mirror and the small spherical mirror, so that part of the double journey is performed through water instead of air. It is found that lioht travels slower in water than in air. CHAPTER XIII. Eefraction through Media of Varying Density. Meteorological Optics. 303. When the medium varies continuously according to a given law we may regard the refractive index at any point as a given function of the coordinates of that point. Equating this function to a constant, we obtain the equation to a surface along which the refractive index is constant; the form of this surface will indicate the manner of stratification of the medium. By considering the refraction of a ray of light as it passes from one stratum of uniform refractive index to a consecutive stratum, we are led to a differential equation to the path of the ray ; the solution of this equation will determine the equation to the path. 304. "We shall first suppose the medium symmetrical about a point, that is, stratified in concentric spherical surfaces, the ray moving in a plane passing through this point. Let PQ, QR be two consecutive directions of the ray, after 303 — 305.] MEDIUM STRATIFIED IN SPHERICAL SURFACES. 329 being refracted at a spherical surface whose centre is 0. Then if (j), <})' be the angles of incidence and refraction, fji sin = fjf sin ^'. If j)< P be the lengths of the perpendiculars drawn from to the ray before and after refraction, this equation may be expressed in the form, fj-p = fj-p'- This result is true for any such refraction ; and therefore if the ray passes through a continuously changing medium, stratified in spherical surfaces whose centre is 0, the equation of the ray may be expressed by the equation where a is a constant. We now can easily find the law of the refractive index in the medium so that a given curve may be described. From the equation of the curve we can express p in terms of r, and then the law of variation of refractive power is given by the equation a fi = - . P Ex. 1. If /i varies inversely as the radius vector, show that the path of auy ray is an equiangular spiral. Ex. 2. If /x varies inversely as r""*""^, the equation to the path of a ray is r" = a''cos?i5. Ex. 3. If /x oc , the path of a ray is an epicycloid. V r^ — a^ 305. As an illustration of the way in which objects are seen in a heterogeneous medium, let us consider a medium such that b where a and h are fixed constants. This law of refractive index was suggested to Maxwell by the eye of a fish. The equation to the path of any ray is jxp = const., or „ ^ ■ = constant = x- ■ say, 330 maxwell's fish-eye medium. [chap. xin. where c is an arbitrary constant which varies as we pass from one ray to another. The equation to any ray is, therefore, 2cp = a' + r\ from which we derive the equation I'dr _ dp In words, this result shows that the radius of curvature of the curve is the same at all points along it, so that the path of the ray is a circle whose radius is c. Indeed it is easy to see that the relation between r and p corresponding to any point of a circle whose radius is c, and whose centre is at a distance k from the origin is so that a' = c' — Z;^ Thus if AOA' be a chord of the circle, through the origin, the rectangle AO . OA' = a^ This result is independent of the par- ticular ray chosen, and therefore A, A' are conjugate points. Pairs of conjugate points are therefore situated on the same line through the centre of the spherical strata, and the product of their distances from that centre is equal to o.'^ Now suppose that an eye is viewing an object through a medium of this kind. We shall suppose that the eye is placed in a small crevasse bounded by orthotomic surfaces, and that the eye is in air close to the surface of the crevasse. Let AB be a small object, A'B' its image, and let E be the position of the eye. Then when the eye is directed towards the object it will see it erect. But .S05 — 30G.] ASTRONOMICAL RKFRACTION. 331 when the eye is turned away from the object it will still see it in the position A'B', inverted. Moreover if we trace the rays by which the eye sees the latter image, it is clear that they come from the back of the object, so that it is the hack of the object which is seen inverted at A'B'. There is another peculiarity about the images thus formed. The amount of divergence in the plane of the figure will not in general be the same as that perpendicular to its plane ; the rays will therefore have a different divergence for height and breadth. 306. Astronomical refraction is the name given to the angle between the apparent direction of a star as seen through the atmosphere, and the direction in which it would appear if there were no atmosphere. Without sensible error the earth may be considered spherical, and the atmosphere stratified in spherical layers whose centre is the centre of the earth. It has already been proved that the path of a ray of light through such an atmosphere will be such that ftp = const., where j> is the perpendicular from the centre of the earth on the tangent to the path at a point where the refractive index is //.. Let X be the radius vector drawn from the centre to any point of the path ; this radius vector will be the normal to the stratum. Also let ^ be the angle between the ray and this normal, then the preceding equation may be written in the form fix sin ^ = fjLjjb sin z, where yu.„, a, z are the values which ft,, x,

' = dr. The law of refraction is /J, sin <}> —(fj, + d/ju) sin (i^ — dr), or fji sin = (fj, + dfi) (sin = 0, dr tan nd/j, tan (f) fj, so that dr= — . n To determine the limits of integration of this equation, it may be supposed that at a great distance from the earth's surface the air becomes so rarefied that its action on the path of light may be neglected. If 6 be the value of ^ for this straight path, we get sin5 = (— ) sing. Also by integration Jen n^ ' and therefore finally, substituting for Q, we get the value of the astronomical refraction, 1 r . _, /sin z\ This is Simpson's formula; of refraction. 306 — 310.] BRADLEY'S FORMULA. 333 308. Bradley expressed this formula in another form. Simp- son's formula may be written siu (z — nr) 1^0 and therefore sin z — sin {z — nr) _//.„" — 1 sin z + sin (z — nr) /i„" + 1 ' whence tan -^ = ^ — tan ( z - ^ or approximately, r = - -„ — - tan (^ — ^ 1 . Bradley wrote this formula in the form r = ^ tan(z —fr) and found that for a mean state of the air, corresponding to the barometer 29'6 inches, and thermometer 50° Fahrenheit, we can express the observed refractions very closely, by taking g=57"m6,f=3. 309. It has been proved by experiments of Biot and Arago, that if fi be the refractive index and p the density of the atmosphere, /i" — 1 = 4\_^ d^ dfji. dy ds ' dy ds dx ds ' But the equation of refraction may be written where p is the radius of curvature of the path of the ray ; and therefore the intrinsic equation of the path of the ray is /jL _dfji dx dfj, dy p dy ds dx ds ' Since — # and -=- are the direction cosines of the normal, ds ds measured in the direction of p, this equation may be written p, dp, p dn' where dn is an element of the normal to the curved ray ; or finally Id.. . - = -r- (log a)- p dn^ ° '^' 311. If the medium be stratified in horizontal layers, the refractive index is a function of y only, and the angle i^ is zero. The previous investigation then gives 9/i cosO = fx, dd sin 6, or d(jj. cos 6) = 0. By integration, we get p. cos 6 = c, :33G THEORY OF MIRAGE. [CHAP. XIII. where c is a constant ; an equation which might have been deduced directly from the law of refraction. The differential equation of the path is therefore dx dy or — = + ' As soon as the form of fi is given in terms of y, this equation can be integrated, and the equation of the path determined. The form of the curve is symmetrical about an axis parallel to that of y. To find the position of the vertex, we have only to make the tangent horizontal, or dx ' we then find 2 2 fl = c . If we make the ray pass through a point (0, b), as for instance through the eye, we find a locus of vertices. Writing ^ {y) for /Lt", the equation to a ray passing through the point (0, b) is dy '-< But the vertex of this curve is found by combining its equation with the equation <^(3/)=c^ and therefore if (f, rj) be the vertex of any ray passing into the eye, dy ^ = Jiv)j] >"Ji.y)-4>{'n)' To find where an object close to the horizon would be seen, the eye being on the same level as the object, we must trace the curve of vertices of all the rays passing into the eye, and find the points where it is met by a vertical half-way between the eye and the object; each of the points of intersection is a vertex of the path of a ray by which the object can be seen. When the curve of vertices at one of these points leans forwards towards the eye, two 311 — 312.] THEOllY OF MIltAGE. 337 contiguous rays cross each other and an inverted image is seen ; but if the curve of vertices leans away from the eye, the contiguous rays do not cross each other and the image is seen erect. 312. Usually the density of the air decreases with the height above the ground ; but often in countries where there are large tracts of hot sand, the air is heated and rare close to the ground, and for a small distance the density increases as we rise from the ground, but afterwards diminishes. At the height where the density is a maximum, we shall have fi stationary, so that j- = ; and from the equation to the ray we infer that -^^ is zero, so that the path of the ray has a point of inflexion. If S be an object and the observer's eye, both situated above the layer of maximum density, a ray passing from S to by the upper air will be concave to the horizon. If we consider rays proceeding from /S at a less inclination to the horizon, some of them will remain concave, but those more inclined to the horizon may have a point of inflexion, and in this case, if the ray be not stopped by the ground, it may reach the eye by another path. Thus the observer will see the object directly and erect by the upper path, in the direction OS", and an inverted image in the direction OS' due to the lower path. The appearance will be the same as if an upper erect object at S" were reflected in a mirror or lake. At sea this phenomenon is often seen turned upside down. The density of the air decreases rapidly from the surface of the water upwards. An image of a distant ship or shore is H. 22 338 THEORY OP ANY CONTINUOOS MEDIUM. [CHAP. XIII. thus often seen erect through the nearly uniform lower strata of the air, while just above them is seen an inverted image, formed by rays which travel along paths passing through the upper strata. These phenomena are known as mirage, and the explanation was first given by Monge. 313. Let the refractive index be defined according to any con- tinuous law. The path of any ray will be such as to make J/uds a minimum. Let V=Jfjds, taken between any two points A, B. Then, if the path be varied slightly, dV=jdiids+JiJid{ds). Also, since ds = J{dxf + (dyf + (dz)", S(ds)=f^d{da>)+f^didy) + ^d{dz), and therefore the second integral may be written But 3 (dx) = d (9a;) ; hence, integrating by parts, this becomes Collecting the terms, the value of 9 F becomes + (S-s(''s)H*'- By the principles of the Calculus of Variatious. dV must vanish 312—314.] THE RAINBOW. 339 for all indefinitely small variations of the path ; so that all along the ray we get the relations d / dx\ _ dfi ' dsydsj dx dsyds) dy d / dz\ d/i s\ daij dz The terminal condition is at each end. If we suppose that the terminal points A, B are con- strained to move on surfaces for which V is constant, this equation expresses the fact that at each end, the ray is perpendicular to the surfaces V = constant. Any two of the above general equations, when integrated, will give the path of the ray. The direction cosines of the radius of curvature of the path are d^x d^y d^z normal. ds^- If dn be an element of this principal dfi ((Px dfi d^y dfi d'z dfi dn^" \ds^ dx ds^ dy ds^ dz (/d^xV (d^yV (d'z\\ dfi \dx <^x dyd'y dzd^z '^f'TsXdsds^'^Jsd^^dsds', p" Hence we get the equation which is the differential equation of the path. The Rainbow. 314. The first satisfactory explanation of the rainbow was given by Antonius de Dominis, archbishop of Spalatro, in a work 22—2 340 THE RAINBOW. [CHAP. XIII. De Radiis Visus et Lucis, published in 1611. He shows that the inner bow is formed by two refractions and one intermediate re- flexion of the sun's light in drops of rain ; and the outer bow by two refractions and two intermediate reflexions. This explanation was adopted by Descartes and was confirmed by experiments made with glass globes filled with water, and arranged so as to exhibit the colours of the two bows. It remained for Newton to complete the theory by explaining the colours. The complete theory in- volves considerations which belong to Physical Optics and was effected by Sir G. Airy ; we must confine ourselves to the approxi- mate theory. 315. When the parallel rays of the sun strike a drop of water, part of the light will be scattered at the outer surface of the drop and serve to render the drop visible, and part will enter the drop by refraction ; of those rays which enter the drop part will be re- fracted out of the drop at the incidence on the second surface of the drop, and part will be reflected back into the drop, and so on, for any number of incidences. Let us consider the rays which are incident in a plane of symmetry and which pass out of the drop by refraction after one internal reflexion ; it is clear that they will not all emerge in the same direction, for the deviation will depend on the angle of incidence. Moreover, if the angle of incidence in- crease uniformly the deviation will vary sometimes rapidly, some- times more slowly ; and the more slowly the deviation changes the less will be the divergence of the emergent rays. If therefore the emergent rays be received on a screen, the band will not be uni- formly bright, but will be brightest in those parts where the diver- gence is least, that is, where the deviation change most slowly. Now the changes of the deviation are slowest near a maximum or a minimum, and therefore at the spot where the deviation is a minimum the band will be much brighter than anywhere else. Within the direction of minimum deviation there will be no light transmitted. If instead of a single drop, a shower of drops be illuminated by the rays of the sun, those drops whose positions are such that the rays emerge in the direction of the eye with minimum deviation will appear more brilliant than the others, and will be marked out against the cloud as specially bright. This phenomenon is the 314—316.] THE RAINBOW. 341 same in all planes which pass through the line joining the sun and the observer's eye, and therefore the assemblage of bright drops will form an arc of a circle whose centre is on this line, and whose angular radius as seen by the eye only depends on the refractive index of the light. The refractive index is not the same for all the rays of a solar beam, being greatest for the violet and least for the red rays, and therefore the position of the bright arc will not be the same for all the coloured rays of the solar beam. There will therefore be a series of coloured bands exhibit- ing the colours of the solar spectrum. This is the principle of the explanation of the rainbow. 316. Let SP be a ray of light incident on the drop of water at P, PQ the ray refracted into the drop ; part of the light will pass out by refraction at Q along the line QQ', while another part will be reflected at Q along the line QR, where part will pass out by refraction and part be reflected, and so on. Let ' the angle of refraction, so that sin (f>=fi sin — '. When the ray is incident at Q, the angle of incidence is — (p' in the same direction as before is produced. But for the part reflected at Q, the deviation is tt — 2(^', and where the ray meets the surface again at R the angle of incidence is again ^'. If therefore the ray under- 342 THE RAINBOW. [CHAP. XIII. goes n internal reflexions and then passes out by refraction, the whole deviation will be D = 2(<^-f)+w(7r-2^')- The most efficacious rays, as we have seen, are those which make the deviation a maximum or minimum. To find the angle of incidence for these rays, we must equate to zero the first differential coefficient of B with regard to ; we therefore get 0.1-(. + l)f. From the equation, sin (/> = /^ sin ^', we find cos ^ = /tt cos (/)' -^ , and therefore, eliminating the differential coefficient, /i cos ^' = (w + 1) cos 4>- Besides this we have the equation of refraction, H sin <\>' = sin ; squaring and adding both members of this equation, we get /a' = (n + 1)" cos" ^ + sin' ^, or cos ^ Since ^ lies between and ^tt, there is no ambiguity in this value of ^. 317. The value of /i for water is about |, and in order that the value of ^ may be real, the numerator must be less than the denominator in the expression for cos ^ ; and therefore (n + 1)" must be > /i', or {n + 1) must be greater than ^. Thus n must be equal to 1 at least, and the light emerging from the drop at Q does not possess either minimum or maximum deviation, and therefore forms no rainbow. There is no superior limit to the value of 71, and, theoretically, bows may be formed after any number of internal reflexions. 318. So far we have not enquired whether the value of <^ as 316 — 319.] THE RAINBOW. 343 found above gives a maximum or minimum value of D or neither ; we must examine the sign of the second differential coefficient. By successive differentiation, ^ = 2-2(. + l)^, It was shown that d^' cos ^ d fjL cos ^' ' from this we derive ,j , — ti cos + fi sin ' sin — sin (^' cos --^- dd) Substituting for -~ its value, this becomes Ctip /:* cos' ^' sin — cos" <^ sin ' yu. cos <^ Now, since ^' is always less than \ir, the denominator of this fraction is positive, and the fraction takes the sign of ^ (1 — sin" ^') sin — (1 — sin" ^') sin ^', . , sin always positive, D will be a minimum for any number of internal reflexions. The deviation will of course be in different directions according as the incident ray falls on the upper or lower half of the drop. 319. We must next consider the order of the coloured rays by examining the changes in the direction of the most efficacious 344 THE PRIMARY RAINBOW. [CHAP. XIII. rays for different refractive indices. If A denote this minimum deviation, A is determined by the equations A = 7i'7r + 2(/)-2(n + l)^', jj, cos (f) ={n + 1) cos . From the first of these equations we find dfj, [dfi a/^j Also, differentiating the equation sin ^ = //. sin <^', . dd) •idii' , ■ ,1 cos

' — (n + 1) cos ^) t— _ 2 sin ^' cos^ ' that is, -J— = - tan S. dfJL fJL This shows that -j— is positive ; therefore A increases with fi, and the minimum deviation is greatest for the violet rays and least for red rays. . 320. It has been shown that in order to produce a rainbow, at least one reflexion inside the drop is necessary. At each subsequent reflexion part of the light will be lost, and the corresponding rain- bows will be fainter. The rainbow produced by one internal reflexion is called the primary rainbow. The angle of incidence corresponding to the most efficacious rays is given by the formula, cos ^ = V —3—' and the deviation by the equation i) = 2(^-f)-|-7r-2f. The refractive indices of water for red and violet rays, respect- ively, are ^f and ^. If these values be substituted for ft in the preceding formula, we find by the aid of trigonometrical tables 319 — 320.] THE PRIMARY RAINBOW. 345 the values of the deviations corresponding to these rays to be, D^ = 137° 58' 20", Z)^=1^43'20". Let be the eye of the spectator, and SOS' a line drawn in the direction of the sun's rays; then, if we make the angle S'OR equal to the supplement of Dr, that is, equal to 42° 1' 40", RO will be the direction in which the most efficacious red rays will enter the eye. Similarily, if an angle S'OV be constructed equal to the supplement of Dy, that is, equal to 40° 16' 40", VO will be the direction in which the most efficacious violet rays will enter the eye, and the intermediate coloured rays will enter in directions inter- mediate between RO and VO. And, further, if the lines OR, OF revolve round the line OS' as an axis, it is clear that all the drops on the conical surface -V generated by the revolution of RO will transmit red rays copiously to the eye, and similarly for the other colours. To the eye there- fore will appear a series of coloured arches with the violet rays innermost. The effect of the rays which strike the eye with greater deviation, will be to light up the cloud within the bow with faint light, while no light will reach the eye from drops lying outside the bow. The separation of the colours is not perfect, but they overlap each other, so that some of the colours can scarcely be recognised. The reason of this, just as in Newton's experiment with the prism, is that the sun has an angular diameter of 33', and as each point of the sun sends out rays we get a series of rainbows due to the dif- ferent elements of the sun's surface all superimposed and confused together. There is yet another set of rays which pass through the drop with minimum deviation, those which strike the drop on its lower side at the same angle of incidence as before. These are directed 346 THE SECONDARY RAINBOW. [CHAP. XIII. after refraction away from the earth, and are not seen by an observer on the earth ; though they give bows which have some- times been observed during balloon ascents, or on the summits of high mountains which lie above the clouds. When the sun ia suflBciently near the horizon a complete circle may sometimes be seen in this manner. 321. When the rays undergo two internal reflexions they form a rainbow called the secondary rainbow. If we make m = 2, and substitute the same values of fi as before, we find 7)^ = 230° 58' 50", D^=234° 9' 20". These deviations being greater than 180°, it is easy to see that the rays which reach the eye of an observer stationed on the earth are incident on the lower half of the drop. Let 80S' be a line drawn through the observer's eye, in the direction of the sun's rays, and let angles 8 OR, S'OV be con- structed, respectively equal to Dn — 180°, By — 180°, that is, to 50° 58' 50", and 54" 9' 20". Then RO, VO will be the directions of the most efficacious red and violet rays, respectively, and the phenomenon of the secondary rainbow may be deduced by revolving the lines OR, OV about the line 08' as before. The order of the colours is inverted in this bow, the violet being outside and the red inside. The rays which reach the eye with greater deviation serve to light up the cloud outside the bow. The secondary bow will be less bright than the primary bow, for two reasons ; first, the light has undergone two internal reflexions and has thereby been weakened, and secondly, there is a greater angular dispersion of the rays in this rainbow than in the primary bow. 322. These two rainbows are the only ones which are usually 320—323.] HALOS. 347 perceived, although the higher bows exist in theory. The third and fourth bows could never be seen except under special circum- stances. For if we make w = 3, we find for red rays D = 318° 24' = 360° — 41° 36'. The direction of the rays will therefore pass be- hind the cloud, and to an observer stationed there it would be lost in the much brighter direct light from the sun. If ?z = 4, D = 360° + 44° 13'. The case of four internal reflexions therefore differs little from the last; the efficacious rays will be incident on the upper half of the drop and will fall behind the cloud as before. For the fifth arc, D = 360° -1- 126°, and the bow will have an an- gular radius of 54° and may be seen outside the secondary bow, especially in waterfalls where the drops are near the eye. The higher bows have never been seen except in laboratories under carefal experimental conditions. Halos and similar phenomena. 323. Besides the rainbow, which owes its existence to re- fractions and reflexions of the solar light by drops of water present in the air, there are other phenomena of a similar nature which are due to the presence in the air of ice-crystals, which re- flect and refract the solar rays. These phenomena we now proceed to enumerate and explain. The most frequent of them are called halos ; these are coloured circles which appear round the sun, and sometimes also about the moon. The halo which is seen the most frequently has an angular radius of 22 degrees. The colours range from red inside to violet outside. This phenomenon, very common in northern regions, is not rare even in our climate; several of them are noted weekly at meteorological observatories. Another circle, whose angular radius is 46 degrees, surrounds the former and presents the colours in the same order ; this is called the halo of 46 degrees. After these, the next phenomenon in order of frequency is a circle of white light passing through the sun parallel to the horizon. This is called the parhelic circle. On the parhelic circle are seen several white or coloured images 348 HALOS. [chap. XIII. of the sun ; at the points where the circle meets the inner halo are two coloured images of the sun which are red on the inside. These images are clearly defined when the sun is on the horizon ; when the sun has a greater altitude, they are seen a little outside the points of intersection. They are called parhelia. More rarely are seen two similar images, situated also on the parhelic circle, at the points of intersection of that circle with the outer halo. Rarer still are seen points on the parhelic circle which mani- fest a sudden increase of brightness. These points are not fixed ; they are found between 90 and 140 degrees from the sun. They are called paranthelia. The anthelion is a white image which appears on the parhelic circle just opposite to the sun. Outside the parhelic circle are sometimes found curves less simple than the halos and the parhelic circle. From the parhelia belonging to the inner halo there proceed two oblique arcs, called the arcs of Lowits. At other times, at the upper and lower parts of each halo, are seen tangential arcs which, for the inner halo, are occasionally pro- longed and form a sort of elliptic halo ; the halo of 46 degrees also has tangential arcs, but these arcs are never prolonged. Finally, at the sides of the halo of 46° supra-lateral and infra-lateral tangential arcs are sometimes seen. 324. These phenomena cannot be explained by the action of small drops of water. Most of them are coloured and are therefore due to refractions. Also they appear in our climate more frequently in winter than in summer, and in northern countries they shine with an intensity and frequency unknown in our country. Marriotte explained some of these appearances by the existence in the atmosphere of ice-crystals, and the others have been attributed to the same cause. Some of the assumptions are arbitrary, but Galle and Bravais have established the theory so as to leave little doubt of its validity. The crystals of ice have been carefully observed and it is found that one crystalline form occurs more frequently than all others ; this is the form of a hexagonal prism, which presents itself under two aspects, either much elongated like a needle, or very flat like a thin plate. 323—326.] HALOS. 349 From these forms of the ice-crystals it follows that there will be three different refracting angles to consider. Two adjacent faces will be inclined at 120°, two faces not adjacent at 60°, and finally, the sides of the prism will form an angle of 90° with the base. 325. The halo of 22° was explained by Marriotte. If we suppose that the air contains prisms of ice distributed in all directions in space, there will always occur prisms whose edges will be perpendicular to the plane drawn through the sun and the observer's eye. Now the minimum deviation for a ray of light traversing a prism of ice whose refracting angle is 60° is exactly equal to 22°. It appears then that in all the directions which make an angle of 22° with the line joining the eye and the sun, there will be seen a maximum of light. Also, the angle of minimum deviation is smaller for red rays than for violet rays, and therefore it is clear that the halo will be coloured and will appear red inside and violet outside. Cavendish explained the halo of 46°; he attributed it to the refraction of light across faces inclined to each other at 90 degrees ; the minimum deviation for such a refraction is found by calculation to be 46°. This phenomenon is explained just as before, and the order of the colours is the same. But as the refracting angle is larger than for the inner halo, the refracted rays will be more scattered ; it follows therefore that the halo of 46° is less luminous, for the light is spread over a ring of double the radius and double the breadth. 326. The two halos are the only phenomena which can be explained by supposing the ice-crystals to be distributed in all directions. But it will readily be imagined that certain directions will predominate; for the needle-shaped crystals, under the influence of their weight, will tend to assume a vertical position, while the flat crystals will direct themselves so that their bases are vertical. The reflexion of light on the prisms of ice placed in all directions with their reflecting faces vertical causes the parhelic circle. If these vertical planes are very numerous they will produce on the eye an impression of a complete circle. The 350 THE PARHELIA. [CHAP. XIII. reflexion at the vertical faces of the flat prisms will give rise to the same appearance. This explanation is due to Young. 327. The parhelia were explained by 'Marriotte. They are due to the presence in the air of vertical needle-shaped crystals. Suppose that there are a large number of vertical prisms whose refracting angles are 60°. If the sun be on the horizon the solar rays fall on the principal section of the prisms; the minimum deviation for such rays is equal to 22°, so that the parhelia are not only on the parhelic circle but on the halo of 22°. When the sun is above the horizon the solar rays are not in a principal plane ; but when they emerge from the prisms they will all make the same angle with the refracting edges, that is with the vertical, as when they enter ; so that the rays will appear to enter the eye from points on the parhelic circle. There will be a minimum of deviation in azimuth which will be greater than 22°, and which will depend on the altitude of the sun. As the minimum deviation will not be the same for different colours, it follows that the colours will form a spectrum, the red being nearest to the sun; further away from the sun the rays are superimposed so as to form a tail of white light which extends along the parhelic circle for a space of 10 to 20 degrees. The parhelia are more brilliant than the halos, because the vertical prisms are more numerous than those distributed in all directions. The oblique arcs observed by Lowitz, have been explained by Galle and Bravais, as due to small oscillations of the vertical prisms about their vertical mean position ; but the consequences of the theory have only been imperfectly verified by observations of the phenomena. The parhelia of 46° are very rare, and their position is not very accurately known. M. Bravais regards them ds produced at 44° by the parhelia of 22° which act like the sun. 328. To explain the paranthelia, which are points on the parhelic circle which manifest a greater intensity of light than the rest of the circle, we must enquire into forms of prisms which will produce a constant deviation on rays of light. It is a well-known theorem that this may be brought about by two reflexions at plane surfaces; for if a ray of light be reflected at each of two 326—328.] THE PARANTHELIA. 351 plane surfaces the deviation produced is equal to twice the angle between the reflecting surfaces. Now prisms of ice whose axes are vertical and such that two faces are in contact will present externally two reflecting faces inclined at 120° to each other. Rays reflected at these surfaces will be turned through an angle of 240° ; this will give rise to two white images of the sun on the parhelic circle, each at an angular distance of 120° from the sun. The same effect may be produced by reflexion at the interior faces of a prism. For when a ray enters across the face a, is reflected at the face h, and again at the face c, and finally emerges across the face d, if we call the three angles between these faces, (ah), (6c), ipd), it is easy to see that provided that (ah) + (cd) = (he) the angles of incidence and emergence at the faces a and d are equal with opposite sign, and therefore the deviation is equal to twice the angle (6c), which is constant. The arrangement in question may be produced in several ways, by prisms of triangular or stellate section. The first figure represents two prisms of triangular section in contact, and the other a stellate crystal with six points. In each case the deviation produced is 240°, and therefore the arrangement 352 THE PARANTHELIA. [CHAP. XIII. will produce a white image of the sun on the parhelic circle at a distance of 120° from the sun. Next, let us consider the path of a ray of light incident on a crystal whose right section is an equilateral triangle ; we suppose the light to pass through in a principal plane, and that the ray is incident on the base, and is reflected by the two sides of the triangle in succession and finally emerges through the base. It is easy to see that the total deviation by this arrangement is equal to <^ + 1^, where — , — j— > respectively. 5. A ray is propagated through a medium of variable density in a plane (.j^) which divides the medium symmetrically ; prove that the projection of the radius of curvature at any point of the path of the ray on the normal to the surface of equal density through the point is ix AW<4} CHAT". XIII.] EXAMPLES. 355 6. Show that in a refracting medium of index /i, the path of a ray ia two dimensions, when referred to polar coordinates, can be put in the form di ds' d6 -'>di'- dy. ' dr dnf.de ds^^ ds' d^ ~rd6' , . dr d6 where | = /.^,, = ,.r^^. Of all the rays that can pass through a given point the one of minimum curvature there cuts orthogonally the refracting stratum at the point. 7. A ray passes through a medium of variable density whose refractive index /i varies as y~"; show that the intrinsic equation to the path of the ray is , »-i as , — , ^=-=A;cos » rf>. d(j) 8. If the refractive index of a medium at any point be proportional to its distance from a fixed plane, prove that the path of the ray will be the curve 2x e - a -~ — = -e"+-e " , a a c a and c being constants. 9. A point of light is placed at the origin of coordinates in a medium where the refractive index is given by ;i=e~*J'. Show 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 whose focal lines lies on the axis of x, and the other at a distance v from the eye, where Jfi j,2 ekx cos^ iy = 2 (cosh hx — cos Icy), while the axis of the pencil makes an angle ^ with the axis, where cot \^ sin /fcy = cos ky-e~ **. 10. A medium is bounded by the planes of x and y, the refractive index at any point being given by the equation log n =xylc^ ; two rays are incident on it parallel to the axes, respectively, and at equal distances c from the origin ; show that if they intersect, it will be at an angle whose circular measure is !!"_£! 2 a^' 11. The path of a ray through a medium of variable density is an arc of a circle in the plane of xy ; prove that the refractive index at any point {x, y) must be / ( r ) , where /is an arbitrary function, and (a, 6) the centre of the circle. 12. If the path of a ray of light be y = aee, show that the index of re- fraction at any point is determined from the equation /x=(y'+c^)V(^+f)- 350 EXAMPLES. [chap. XIII. 13. If a small pencil of light pass directly through a plate of thickness b, the index of refraction being f(x/c), x being measured from the plane of in- cidence, and c varying slightly with the colour of the light, show that the chromatic aberration on emergence is /6_1 [ \ dx \, \cf{blc) },cf{xlc)r' f (0/c) being supposed equal to unity. 14. A ray of light is incident perpendicularly on one of the faces of a prism whose density varies in such a manner that the coefficient of refraction at any point is yje^, where ^ is constant, and 6 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 in- cidence on the second face, show that <^ is determined by the equation, cos <^ — sin <^=e<^-2« 15. The density of a prism at any point 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 point of incidence and emergence being a, and its nearest approach to the edge being c, show that the deviation is given by the equation, Dc sin ^ («4)' = sin;3eZ(a-c<:oaP), where 2/3 is the vertical angle of the prism. 16. A pencil diverging from a point and originally a quadric cone passes through a heterogeneous medium ; show (1) that the section of the pencil made by a plane perpendicular to the axis of the pencil is an ellipse whose axes do not necessarily lie in the principal planes of the pencil ; (2) that there will be no cirde of least confusion of the pencil, but that the ratio of the axes of any section will be least when the section is made at a distance h from the face, where ^~P2 _ P2 o p,, p2 being the distances of the foci from the face, and /3 the ratio of the axes of the face which lie in the principal planes. Show also that the greatest section in area between the focal lines is in every case at a distance ^{pi+p^ from the front. 17. If 6, h' be the breadths of the pth and qth rainbows, respectively, and 8 the sun's apparent diameter, show that '■-'=[{i^y-']<'-''>--- 18. 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