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ELEMENTARY TREATISE 
 
 ON THE 
 
 WAVE-THEORY OF LIGHT. 
 
 BY 
 
 HUMPHREY LLOYD, D.D., D.C.L.; 
 
 PROVOST OF TRINITY COLLEGE, DUBLIN, 
 AND FORMERLY PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY. 
 
 irb Cbition, JUbiseb twb 
 
 il V.I ft SI TT1 
 
 CONDON: 
 ONGMANS, GREEN, AND CO 
 
 1873- 
 
DUBLIN : 
 
 POINTED AT THE UNIVERSITY PRESS, 
 BY M. H. GILL. 
 
ADVERTISEMENT 
 
 TO THE SECOND EDITION 
 
 THE former edition of this little work was given to 
 the public in the shape of Lectures, as delivered, in 
 compliance with the regulations of the Chair which 
 the author then occupied, and without any expecta- 
 tion that its use would extend much beyond the circle 
 of his immediate hearers. It has however found its 
 way elsewhere ; and the author has been urged, by 
 some of his fellow-labourers in other Universities, to 
 reprint it. With this request he could not but com- 
 ply ; and he trusts that the delay in acceding to it 
 may be excused to those who made it, by the desire 
 of the author to render the work more deserving of 
 their favourable estimation. 
 
 In the present edition some account is given of 
 the more important discoveries in Physical Optics, 
 which have been made since the publication of the 
 former. In preparing these additions, the author 
 has derived much aid from the Repertoire d'Optique 
 Moderne of the Abbe Moigno 3 a work which contains 
 
VI ADVERTISEMENT. 
 
 a full analysis, and critical discussion, of most of the 
 recent researches in Optics. He has also to acknow- 
 ledge his obligations to M. Moigno, for the favourable 
 introduction of the former edition of the present work, 
 in the pages of the " Repertoire," to the notice of Con- 
 tinental readers. 
 
 The form of Lectures has been abandoned ; but 
 the author fears that the style still retains more of the 
 traces of the lecture-room than is consistent with a 
 formal scientific treatise. His only aim has been to 
 present, to those who were conversant only with the 
 elements of Mathematics, a clear and connected view 
 of his attractive subject ; and he has been compelled, by 
 this limitation, to confine himself in many cases (as in 
 all that relates to the Dynamics of Light) to a general 
 account of methods, and of their results. Those who 
 desire a more exact acquaintance with the science will, 
 of course, study it in Herschel's Essay on Light, and 
 in Airy's Tract on the Undulatory Theory t of Optics. 
 
CONTENTS 
 
 CHAPTER I. 
 
 PROPAGATION OF LIGHT. 
 
 Page. 
 
 (1, 2) Distinction of bodies 'with respect to light ; Vision. (3) Light 
 emanates from luminous bodies in all directions. (4) Non-luminous 
 bodies transparent and opaque. (5) Light is propagated in right lines ; 
 ray of light. (6) Intensity of light emanating from a luminous point 
 varies inversely as square of distance. (7, 8) Measure of illumina- 
 tion; photometry. (9, 10) Velocity of light determined by eclipses 
 of Jupiter's satellites by aberration of fixed stars. (11) M. Fizeau's 
 experimental determination of velocity of light. ( 12) Two theories of 
 light, Des Cartes and Newton ; Hooke and Huygens. (13) Impulse 
 of light insensible. (14) Uniform velocity of light inconsistent with 
 theory of emission. (15) Rectilinear propagation of light apparently 
 opposed to wave-theory not really so. (16) "Wave-propagation ex- 
 plained. (17, 18) Mathematical laws of vibratory movement. (19) 
 Lights differ from one another in intensity and colour ; law of inten- 
 sity. (2,0) Colour of light varies with length of wave ; limits of sen- 
 sibility of the eye. (21) Aberration of light physically explained, . 1 
 
 CHAPTER II. 
 
 REFLEXION AND BEPKACTION. 
 
 (22, 23) Reflexion of light twofold ; law of reflexion. (24) Laws of inten- 
 sity of reflected light. (25, 26) Law of refraction; refractive index. 
 (27, 28) Light traversing media bounded by parallel planes. (29-31) 
 Deviation of ray refracted by prism deduced. (32) Refractive index 
 determined experimentally. (33) Newton's explanation of the laws of 
 reflexion and refraction. (34, 35) Hypothesis of the^s; its insuffi- 
 ciency. (36, 37) Laws of reflexion and refraction deduced from the 
 
 b 
 
VI CONTENTS. 
 
 Page. 
 
 principles of the wave-theory. (38) Principle of least time. (39) In- 
 tensity of reflected and refracted light. (40) Experimentum crucis 
 between the two theories. (41, 42) M. Foucault's experiment; 
 velocity of light in air, and in water, determined experimentally. 
 (43) No effect of the Earth's motion on the amount of refraction 
 Arago's experiment. (44) M. Fizeau's experiment, 20 
 
 CHAPTER III. 
 
 DISPERSION. 
 
 (45-47) Composition of solar light; seven principal colours. (48) Colour 
 and refrangibility are permanent and original affections. (49) Mix- 
 ture in the spectrum ; modes of redncing it. (50) Newton's method 
 of determining the refractive indices of the several species of simple 
 light. (51) Intensity of light in the spectrum. (52-54) Fixed lines of 
 the_spectrum ; observations of Fraunhofer. (55, 56) Dispersion pro- 
 duced by a thin prism ; measure of dispersive powers. (57) Disper- 
 sive powers are different in different substances ; experiments of 
 Newton and Dollond. (58, 59) Condition of achromatism ; disper- 
 sive powers compared experimentally. (60, 61) Physical explanation 
 of dispersion ; difficulty in the wave-theory removed by M. Cauchy. 
 (62, 63) Relation between velocity of propagation and wave-length. 
 (64, 65) Refractive index deduced in terms of the wave-length in 
 
 (66) Limiting index, ....,..;...- ... 37 
 
 CHAPTER IV. * 
 
 ABSORPTION AND EMISSION. 
 
 (67) Absorption of light ; its dependence on the thickness of the body 
 traversed. (68) Dichroism explained. (69) Phenomena of absorp- 
 tion analyzed. (70) Effects of Sun's rays threefold. Luminous, 
 calorific, and chemical effects equally absorbed. (71) Absorptive 
 properties of bodies different. (72) Absorption of light by gases. 
 (73, 74) Emission of light dependent on temperature. Light emitted 
 by solids, liquids, and gases. (75) Modes of exhibiting the phenomena 
 of emission. Lights emitted by the alkaline metals. (76, 77) Peculia- 
 rity of the light of incandescent gases ; spectral analysis. Effects of 
 temperature. (78) Reversal of the spectra, of the gases. (79) Rela- 
 tion between the emissive and absorptive powers law of Kirch off ; a 
 consequence of the laws of vibratory propagation. (80) Phenomena 
 of solar spectrum explained. Solar chemistry, ........ 53 
 
CONTENTS. Vll 
 
 CHAPTEE V. 
 
 DOUBLE REFRACTION. 
 
 Page. 
 
 (81) Double refraction discovered by Bartbolinus. (82, 83) Bodies pos- 
 sessing double refraction ; Iceland spar. (84) Ordinary and extraor- 
 dinary rays. (85, 86) Optic axis; phenomena of double refraction 
 symmetrical around it. (87) Construction ofHuygens for the direc- 
 tions of the two refracted rays. (88) Case in which both rays are 
 refracted according to the ordinary law. (89) Eefractive indices in 
 Iceland spar for the principal rays of the spectrum. (90) Case in 
 which the refracting surface, and the plane of incidence, contain the 
 optic axis. (91) Attractive and repuhice crystals. (92) Biaxal crys- 
 tals. (93) Connexion between the optical characters and the crystal- 
 line forms, discovered by Brewster. (94) Double refraction in biaxal 
 crystals. (95) Effects of double refraction ; two images. (96, 97) 
 Double-refracting prism. Double image micrometer, 63 
 
 CHAPTER VI. 
 
 INTERFERENCE OP LIGHT. 
 
 (98) Effect of co-existing vibrations. Law of superposition of small motions. 
 (99) Union of two waves ; interference of light. (100) Analogous phe- 
 nomena in water, and in air. (101, 102) Grimaldi's experiment of in- 
 terference ; Young's experiment. (103) Phenomena of interference 
 of aqueous waves. (104, 105) Explanation of Young's experiment; 
 positions of the bands calculated. (106) Fresnel's experiment of in- 
 terference ; positions of the fringes. (107) Interference of direct 
 arid obliquely-reflected lights. (108) Interference produced by an 
 obtuse prism. (109) Arago's experimentum crucis displacement of the 
 fringes produced by interposed plate. (110, 111) Laws of the com- 
 position of co-existing vibrations. (112) Rectilinear propagation of 
 light explained on the principles of the wave-theory, 78 
 
 CHAPTER VII. 
 
 DIFFRACTIO N. 
 
 (113) Diffracted fringes observed by Grimaldi. (114-116) Diffraction 
 by a single edge ; form of the fringes. (117) Diffraction by fine wire. 
 (118, 119) Diffraction by a narrow rectilinear aperture. (120) New- 
 ton's explanation of the phenomena. (121) Newtonian theory refuted ; 
 effects independent of the nature and form of the body. (122) Hypo- 
 
Page. 
 
 thesis of Mairan and Du Tour. (123) Young's theory of diffraction ; 
 incomplete. (124) Fresnel's theory. (125, 126) Fringes produced 
 by a single edge explained, and their places computed. (127) Fringes 
 within the shadow of a narrow opaqne body explained. (128) Fringes 
 formed by a narrow rectilinear'aperture. (129) Diffraction by asmall 
 circular aperture. (1 30) Diffraction by an opaque circular disc. (131) 
 Diffracted rings produced by fibres irregularly disposed; Young's 
 eriometer. (132, 133) Diffraction by fine gratings. (134) Fixed 
 lines in the diffr-acted spectrum ; their intervals always proportional. 
 
 (135) "Wave-lengths corresponding to the seven principal fixed lines. 
 
 (136) Law of intensity of light in the diffracted spectrum. (137) Euled 
 and striated surfaces. (138) Colours of mother of pearl. (139) Dif- 
 fraction produced by diaphragms in telescopes, 98 
 
 CHAPTER VIII. 
 
 COLOURS OF THIN PLATES. 
 
 (140, 141) Colours of thin plates exhibited under various circumstances. 
 (142) Newton's mode of observing them ; Newton's scale. (143) Laws 
 of the phenomena. (144-146) Laws of the rings, experimentally 
 proved. (147) Transmitted rings. (148) Iroscope. (149) Newton's 
 explanation of the colours of thin plates ; Jits of easy reflexion and trans- 
 mission. (150) Assumption of this theory disproved. (15,1) Specu- 
 lations of Hooke and Young the colours of thin plates produced by 
 interference. (152) Laws of the colours of thin plates mathematically 
 deduced. (153, 154) Point of contact a point of discordance of two 
 waves ; loss or gain of half an undulation. (155) Transmitted rings 
 explained. (156) Imperfection of Young's theory removed by Poisson ; 
 experimentum crucis of Fresnel. (157) Phenomena of interference 
 produced by two parallel plates. (158) Interference produced by two 
 plates slightly inclined. (159,160) Colours of thick plates Newton's 
 experiment ; explanation. (161) M. Babinet's variation of Newton's 
 experiment. (162) Colours of mixed plates. (163) Eeview of the two 
 theories in their application to the foregoing phenomena, 128 
 
 CHAPTER IX. 
 
 POLARIZATION OF LIGHT. 
 
 (164) Polarization of light discovered by Huygens. (165) Polarization by 
 reflexion discovered by Malus. (166, 167) Distinctive characters of a 
 
CONTENTS. IX 
 
 Page. 
 
 polarized ray. (168, 169) Angles of polarization ; Brewster' s law. 
 (170) Polarization by a plate bounded by parallel surfaces ; by a pile of 
 plates. (171) Law of Malus. Common light equivalent to two polarized 
 rays of equal intensity, whose planes of polarization are perpendicular. 
 (172) Light partially polarized by reflexion. (173) Light partially 
 polarized by refraction; Law of Arago. (174) Light completely 
 polarized by transmission through a pile of plates. (175) Polarization 
 by transmission through a plate of tourmaline. (176, 177) Modes 
 of producing a polarized ray of strong intensity by means of double 
 refraction. Nicol's prism. (178) Phenomenon observed by Haidinger 
 polarized light distinguishable by the naked eye, 151 
 
 CHAPTEE X. 
 
 TRANSVERSAL VIBRATIONS THEORY OF REFLEXION AND REFRACTION OP 
 POLARIZED LIGHT. 
 
 (179) Newton's objection to the wave-theory, derived from the phenome- 
 non of polarization, removed. (180) Propagation of wave by trans- 
 versal vibrations illustrated. (181) Evanescence of the normal 
 vibrations experimentally proved. (182) Direction of the vibrations 
 in polarized ray two hypotheses; experimentum crucis of Professor 
 Stokes. (183) Fresnel's theory of the reflexion and refraction of 
 polarized light. (184, 185) Amplitudes of the reflected and refracted 
 vibrations deduced. (186) Intensities of the reflected and refracted 
 rays. (187) Intensity of light reflected at extreme incidences. 
 
 (188) Partial polarization by reflexion and refraction; law of Arago. 
 
 (189) Complete polarization of reflected light; law of Brewster. 
 (190, 191) Change of plane of polarization by reflexion and refraction, 166 
 
 CHAPTER XI. 
 
 ELLIPTIC POLARIZATION. 
 
 (192) The vibration of the ethereal molecules is generally elliptic. 
 
 (193) An elliptic vibration the resultant of two rectilinear vibrations. 
 
 (194) Direction of axes of the ellipse, and ratio of their lengths. 
 
 (195) Elliptic polarization by total reflexion. (196) Circular polarization ; 
 Fresnel's rhomb. (137, 198) Laws of metallic reflexion discoveries 
 of Malus and Brewster. (199) Light reflected from metals elliptically 
 polarized. (200) Intensity of reflected light determined experimentally 
 by M. Jamin. (201, 202) Difference of phase of the two compo- 
 nents. Effects of successive reflexions. (203) Action of transpa- 
 
CONTENTS. 
 
 Page. 
 
 rent bodies on light; M. Jamin's results. (204) Difference of phase 
 of the two components in light reflected from transparent bodies; 
 angle of maximum polarization. (205) Substances of positive and 
 negative reflexion. (206) Elliptic vibration of reflected light de- 
 termined experimentally. (207) Professor Haughton's researches ; cir- 
 cular polarization of reflected light, 181 
 
 CHAPTER XII. 
 
 FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 (208, 209) Form of the wave-surface in uniaxal crystals not general. 
 (210-212) Fresnel's theory of double refraction; surface of elasticity. 
 (213) Planes of polarization of two rays; law of Malus. (214) Fres- 
 nel's construction for the wave-surface ; its equation. (215) Direc- 
 tions, and planes of polarization, of the two rays determined. 
 (216) Optic axes ; velocities of the two rays. Normal velocities of the 
 waves; law of Biot. (217) Case of uniaxal crystals, and of single- 
 refracting substances. (218) Velocity of both rays in biaxal crystals 
 variable ; Fresnel's experiments. (.219) Sir "William Hamilton's dis- 
 covery of Conical refraction. (220-222) External conical refraction 
 established experimentally ; law connecting the planes of polarization ; 
 effect of linear aperture. (223) Internal conical refraetion, .... 197 
 
 CHAPTER XIII. 
 
 INTERFERENCE OF POLARIZED LIGHT. 
 
 (224-226) Depolarization and colour produced by crystalline plates. 
 (227) Laws of the tints discovered by M. Biot. (228) The phenomena 
 reduced to principle of interference difficulty. (229) Laws of inter- 
 ference of polarized light ; explanation of colours of crystalline plates. 
 (230) Emergent light in general elliptically-polarized. (231-233) Cal- 
 culation of intensities and tints of the emergent pencils. Laws of the 
 colours deduced. (234) Colours produced by thick plates. (235) Rings 
 surrounding the optic axis in uniaxal crystals. (236) Rings in biaxal 
 crystals lemniscates ; dark brushes. (237-238) Double-refracting 
 structure developed by compression or dilatation ; Fresnel's explana- 
 tion. (239) Double-refracting structure produced by heat. (240) Effects 
 of pressure and heat on double-refracting crystals, 
 
CONTENTS. XI 
 
 CHAPTER XIV. 
 
 ROTATOltY POLARIZATION. 
 
 Page. 
 
 (241) Rotation^of plane of polarization in rock-crystal. (242) Laws of ro- 
 tatory polarization established by Biot. (243) Right-handed and 
 left-handed crystals discoveries of Herschel and Brewster. (244) Laws 
 of rotatory polarization in rock-crystal theoretically explained by 
 Fresnel. (245) Relation between rotation and double refraction in 
 the direction of the axis. (246) Airy's discovery of elliptic polari- 
 zation in directions inclined to the axis. (247) Mac Cullagh's theory 
 of refraction in quartz. (248) Rotatory polarization of liquids dis- 
 coveries of Biot and Seebeck. (249) Rotation of mixtures; optical 
 analysis of compounds. (250) Jellett's analyzer, and saccharometer, . 237 
 
ELEMENTS 
 
 OF 
 
 THE WAVE-THEORY OF LIGHT. 
 
 CHAPTER I. 
 
 PROPAGATION OF LIGHT. 
 
 (1) NATURAL bodies may be divided into two classes in 
 relation to Light. Some possess, in themselves, the powe'r of 
 exciting the sense of vision, and of producing the sensation of 
 light ; while others are devoid of that property. Bodies of 
 the former class are said to be toamows ; those of the latter, 
 non-luminous. The Sun and the fixed stars are all luminous 
 bodies ; terrestrial bodies are luminous, in the states of incan- 
 descence, combustion, or phosphorescence. 
 
 Non-luminous bodies acquire the power of exciting the 
 sensation of light in the presence of a luminous body. Thus, 
 a lamp or candle illuminates all the objects in a room, and 
 renders them visible ; and the light of the Sun illuminates the 
 Earth and the planets. This property of bodies is due to 
 their capacity of reflecting light, and belongs ^to them in dif- 
 ferent degrees. 
 
 (2) The foregoing distinction of bodies, obvious as it seems, 
 was not fully comprehended by the ancients. According to 
 
2 PROPAGATION OF LIGHT. 
 
 them, vision was performed by something which emanated 
 from the eye to the object ; and the sense of Sight was explained 
 by the analogy of that of Touch. In this view, then, the sen- 
 sation was represented as independent of the nature of the 
 body seen ; and all objects should be visible, whether in the 
 presence of a luminous body or not. This strange hypothe- 
 sis held its ground for many centuries. The Arabian astro- 
 nomer, Alhazen, who lived in the latter part of the eleventh 
 century, seems to have been the first to refute it, and to prove 
 that the rays which constituted vision came from the object 
 to the eye. 
 
 (3) The light of a luminous body emanates from it in all 
 directions. Thus, the light of a lamp or candle is seen in all 
 parts of a room, if nothing intervenes to intercept it ; and the 
 light of the Sun illuminates the Earth, the Planets, and their 
 satellites, in whatever position they may be placed respect- 
 ing it. 
 
 Each physical point of a luminous body is an independent 
 source of light, and is called a luminous point. 
 
 (4) Non-luminous bodies are distinguished into two classes, 
 according as they allow the light which falls upon them to pass 
 freely through their substance, or intercept it. Bodies of the 
 former kind are said to be .transparent ; those of the latter, 
 opaque. 
 
 There are no bodies in nature actually corresponding to 
 these extremes. The most transparent bodies, as air and water, 
 intercept a sensible quantity of light, when of sufficient 
 thickness ; and, on the other hand, the most opaque bodies, 
 such as the metals, allow a portion of light to pass through their 
 substance, when reduced to laminse of exceeding tenuity. 
 
 (5) In the same homogeneous medium, light is propagated 
 in right lines, whether it emanates directly from luminous 
 bodies, or is reflected from such as are non-luminous. 
 
PROPAGATION OF LIGHT. 
 
 This is proved by the fact that when an opaque body is 
 interposed in the right line connecting the eye and the lumi- 
 nous source, the light of the latter is intercepted, and it ceases 
 to be visible. The same thing is proved also by the shadows of 
 bodies, which, when received upon plane surfaces perpendi- 
 cular to the path of the light, are observed to be similar to 
 the section of the body which produces them. 
 
 This property of light was recognised by the ancients ; and 
 by means of it the few optical laws which were known to 
 them became capable of mathematical expression and reason- 
 ing. Any one of these lines, proceeding from a luminous 
 point, is called in optics a ray. 
 
 (6) In a perfectly transparent medium, the intensity of 
 the light proceeding from a luminous point varies inversely 
 as the square of the distance. 
 
 This is easily proved, if light be supposed to be a material 
 emanation of any kind. For the intensity of the light, received 
 upon any spherical surface whose centre is the luminous point, 
 is as the quantity of the light directly, and inversely as the 
 space over which it is diffused. But none of the light being lost, 
 the quantity of light received upon any spherical surface is the 
 same as that emitted, and is therefore constant ; and the space 
 
 Q_ diffusion, or the area of the spherical surface, is as the square 
 
 "*' its radius. Hence the intensity of the light is inversely as 
 9 square of the radius, i. e. inversely as the square of the 
 itance. 
 
 Let the light be supposed to emanate from the points of > 
 uniformly luminous surface, which we shall suppose to be 
 a small portion of a sphere. Then the quantity of light 
 oliutted is proportional to the quantity emitted by a single 
 jint, and the number of points (or area) conjointly. Hence 
 a denote the area of the luminous surface, and i the quan- 
 y emitted from a single point, which is a measure of the 
 
 B2 
 
4 PROPAGATION OF LIGHT. 
 
 absolute brightness, the intensity of the illumination, at any 
 distance d, is 
 
 ai 
 
 # 
 
 (7) A plane surface, whose dimensions are small in com- 
 parison with the distance, and which is perpendicular to the 
 incident light,- may, without sensible error, be considered as a 
 portion of a spherical surface concentric with the luminary. 
 The intensity of the illumination, therefore, or the quantity of 
 light received upon a given portion of such a plane, is expressed 
 by the formula of the preceding Article. 
 
 When the surface is inclined to the incident light, the 
 quantity of the light received by any given portion is dimi- 
 nished in the ratio of unity to the sine of the angle of inclina- 
 tion. The intensity of the illumination is, therefore, diminished 
 in the same proportion, and is expressed by the formula 
 
 ai sin 9 
 
 ~ir~' 
 
 9 being the inclination of the surface to the incident light. 
 
 (8) Experience proves that the eye is incapable of com- 
 paring directly two lights, so as to determine their relative 
 intensities. But, although unable to estimate degrees, the eye 
 can detect differences of intensity with much precision ; and 
 with this power it is enabled (by the help of the principles 
 just established) to compare the intensities of two lights indi- 
 rectly. 
 
 Let two portions of the same paper, or any similar reflect- 
 ing surface, be so disposed, that one of them shall be illumi- 
 nated by one of the lights to be compared, and the other by 
 the other, the light being incident upon each at the same 
 angle. Then let the distance of one of the lights be altered, 
 until there is no longer any appreciable difference in the inten- 
 
PROPAGATION OF LIGHT. O 
 
 sities of the illuminated portions. The illuminating powers of 
 the two lights will then be as the squares of their respective 
 distances ; and their absolute brightnesses as the illuminating 
 powers directly, and as their luminous surfaces inversely. For, 
 if i and i' denote the absolute brightnesses of the two lights, 
 a and a' the areas of the luminous surfaces, and d and d' their 
 distances from the paper, the intensities of illumination are 
 
 ai sin , a'i' sin , . , , . 
 
 and , respectively ; and these being rendered 
 d d 
 
 equal in the experiment, we have 
 
 ai d? 
 
 The following simple and convenient mode of practising this 
 method was suggested by Count Eumford. A small opaque 
 cylinder is interposed between the lights to be compared and 
 a screen ; in this case it is obvious that each of the lights 
 will cast a shadow, which is illuminated by the other light, 
 while the remainder of the screen is illuminated by both lights 
 conjointly. If, then, one of the lights be moved, until the 
 shadows appear of equal intensity, their illuminations are 
 equal, and, therefore, the illuminating powers of the two 
 lights are to one another as the squares of their distances 
 from the screen. 
 
 (9) Light is propagated with a finite velocity. 
 
 This important discovery was made in the year 1676, by 
 the Danish astronomer, Olaus Hoemer. Roemer observed that 
 when Jupiter was in opposition, and therefore nearest to the 
 Earth, the eclipses happened earlier than they should according 
 to the astronomical tables ; while, when Jupiter was in con- 
 junction, and therefore farthest, they happened later. He 
 thence inferred that light was propagated with a finite velo- 
 city, and that the difference between the computed and 
 
6 PROPAGATION OF LIGHT. 
 
 observed times was due to the change of distance. This 
 difference is found to be 8 m 13 s ; and accordingly the velocity 
 of light is such, that it traverses 192,500 miles in a second of 
 time. 
 
 (10) The velocity of light, combined with that of the 
 Earth in its orbit, was afterwards applied by Bradley to explain 
 the phenomenon of the aberration of the fixed stars. From the 
 theory of aberration so explained, it followed that the velocity 
 of the light of the fixed stars is to the velocity of the Earth in 
 its orbit, as radius to the sine of the maximum aberration. 
 This latter quantity the constant of aberration, as it is called 
 is now found to be 20 /Xt 36 ; and the Earth's velocity being 
 known, the velocity of the light of the fixed stars is deduced. 
 The value so obtained is 191,500 miles in a second, which dif- 
 fers from that inferred from the eclipses of Jupiter's satellites, 
 by only the 2Uo^ n P ar ^ ^ ^ ne whole. 
 
 From this it follows, that the direct light of the fixed stars, 
 and the reflected light of the satellites, travel with the same 
 velocity. 
 
 (11) The velocity of light, emanating from a terrestrial 
 source, has been recently measured by M. Fizeau, by direct 
 experiment. The first idea of this experiment was communi- 
 cated to M. Arago, by the Abbe Laborde, a few years before : 
 its principle will be understood from the following description. 
 
 Let the light of a lamp be reflected nearly perpendicularly 
 by a mirror placed at a considerable distance. Let a toothed 
 wheel, the breadth of whose teeth is equal to that of the interval 
 between them, be interposed near the luminous source ; and let 
 the mirror be so adjusted that the light passing through one 
 of these intervals is reflected to that diametrically opposite. If 
 the eye be placed behind the latter interval, the wheel being 
 at rest, it will perceive the reflected ray, which has traversed 
 a space equal to double the distance of the mirror from the 
 
PROPAGATION OF LIGHT. 7 
 
 wheel. But if, on the other hand, the wheel be made to re- 
 volve rapidly, its velocity may be such that the light trans- 
 mitted through the opening at one extremity of the diameter 
 shall not pass through the opposite aperture on its return, but 
 be intercepted by the adjacent tooth ; and it will be conti- 
 nually invisible to the eye, so long as the wheel revolves with 
 the same velocity. If the velocity of the wheel be doubled, 
 the light will be transmitted, on its return, through the suc- 
 ceeding opening, and will reappear to the eye. If the velo- 
 city be trebled, the light will be intercepted by the next tooth, 
 and there will be a second eclipse ; and so on. 
 
 It is plain that if the velocity of the wheel, correspond- 
 ing to the 1st, 2nd, 3rd, or m th eclipse, be known, the ve- 
 locity of the light may be calculated. Thus, if the wheel 
 makes n revolutions in a second, and has p teeth, the time 
 
 of passage of one tooth across the same point of space = - 
 of a second. Consequently, the first eclipse will correspond 
 
 to - of a second. But in the same time the light has twice 
 2np 
 
 traversed the distance between the wheel and the mirror. If, 
 therefore, that distance be denoted by a, the velocity of light 
 will be 
 
 V=2a*2np. 
 
 If n be the number of revolutions in a second correspond- 
 ing to the m th eclipse, the velocity of light will be given by 
 the formula, 
 
 2np 
 
 V=2a 
 
 2m -1' 
 
 The apparatus devised by M. Fizeau for this experiment 
 is ingenious and effective. It consists of two telescopes, di- 
 rected towards each other, and so adjusted that an image of 
 the object-glass of each is formed in the focus of the other. 
 
8 PROPAGATION OF LIGHT. 
 
 The light from the source is introduced laterally into the first 
 telescope, through an aperture near the eye-piece. It is then 
 received on a transparent plate, placed between the focus and 
 the eye-glass, and inclined at an angle of 45 to the axis of the 
 instrument. It is thus reflected along the axis of the first 
 telescope, having passed through one of the apertures in the 
 revolving wheel, and is received perpendicularly on the mirror 
 in the focus of the second. It then returns by the same route, 
 and is received by the eye at the eye-glass of the first tele- 
 scope. The distance of the two telescopes in M. Fizeau's ex- 
 periments was 9440 yards. The revolving disc had 720 teeth, 
 and was connected with a counting apparatus which measured 
 its velocity of rotation. The first eclipse took place when the 
 wheel made 12'6 revolutions in a second. With double the ve- 
 locity, the light was again visible; with treble the velocity, 
 there Was a second eclipse, and so on. The mean result of 
 the experiments gave 196,000 miles, nearly, for the velocity 
 of light. 
 
 (12) Let us now proceed to the physical explanation of 
 the foregoing facts. 
 
 We have seen that light travels from one point of space 
 to another in time, and with a prodigious velocity. Now, 
 there are two distinct and intelligible ways of conceiving 
 such a propagated movement. Either it is the same body 
 which is found in different times in distant parts of space ; or 
 there are a multitude of moving bodies, occupying the entire 
 interval, each of which vibrates continually within certain 
 limits, while the vibratory motion itself is communicated in 
 succession from one to another, and so advances uniformly. 
 These two modes of propagated movement may be distin- 
 guished by the names of the motion of translation and the mo- 
 tion of vibration. The former is more familiar to our thoughts, 
 and is that which we observe, when with the eye we follow 
 
PROPAGATION OF LIGHT. 
 
 the path of a projectile in the air ; or about which we reason, 
 when we determine the course of a planet in its orbit. Mo- 
 tions of the latter kind, too, are everywhere taking place 
 around us. When the surface of stagnant water is agitated 
 by any external cause, the particles of the fluid next the origin 
 of the disturbance are set vibrating up and down, and this 
 vibratory motion is communicated to the adjacent particles, 
 and from them onwards, to the boundaries of the fluid surface. 
 All the particles which are elevated at the same instant con- 
 stitute what is called a wave ; and that this wave does not 
 consist of the same particles in two successive instants may be 
 seen in the movements of any floating body, which will be 
 observed to rise and fall as it is reached and passed by the 
 wave, but not to advance, as it must necessarily do if the 
 particles of the fluid on which it rested had a progressive 
 motion. The phenomena of sound afford another well-known 
 instance of the motion of vibration. The vibratory motion 
 is communicated from the sounding body to the ear, through 
 all the intervening particles of the air, though each of the 
 aerial particles moves back and forwards through a very nar- 
 row space. 
 
 Each of these modes of propagated motion has been ap- 
 plied to explain the phenomena of light ; and hence the two 
 rival theories the theory of emission and the wave-theory. 
 In the former the luminous body is supposed to send forth, 
 or emit, continually, material particles of extreme minuteness, 
 in all directions. In the latter, the same body is supposed 
 to excite the vibrations of an elastic ether, which are commu- 
 nicated from particle to particle, to its remotest bounds. This 
 ethereal medium is supposed to pervade all space, and to be 
 of such extreme tenuity as to afford no appreciable resistance 
 to the motions of the planets. 
 
 Such are the two systems, some traces of which may 
 be found even in the recorded opinions of the ancients. 
 
10 PROPAGATION OF- LIGHT. 
 
 It is only within a period comparatively recent, however, 
 that either of them has been stated formally, or supported 
 by any show of reasoning. Descartes put forward, very 
 distinctly, the hypothesis that light consisted of 4 small par- 
 ticles emitted by the luminous body ; and he even endea- 
 voured to explain the laws of reflexion and refraction on that 
 supposition. But as Newton was the first to deduce the 
 mathematical consequences of the theory of emission, he has 
 been usually regarded as its author. The wave-theory was 
 propounded by Hooke, in the year 1664 ; and was developed 
 into several of its consequences, a few years later, by Huy- 
 gens. Let us examine each of these theories by the only test 
 to which a physical theory can be subjected, namely, the 
 accordance of its consequences with phenomena. 
 
 (13) The fundamental assumption of the theory of emis- 
 sion the hypothesis that light consists of bodies moving with 
 an immense velocity would appear to be easily submitted 
 to the test of experiment. If the weight of a molecule of 
 light amounted to but one grain, its momentum would equal 
 that of a cannon-ball, 150 pounds in weight, moving with 
 the velocity of 1000 feet in a second. The weight of a 
 single molecule may be assumed to be many millions of times 
 less than what has been here supposed ; but, on the other 
 hand, many millions of such molecules may be made to act 
 together, by concentrating them in the foci of lenses or mir- 
 rors, and the effects of their impulse might be expected in this 
 manner to be rendered evident. 
 
 This apparently easy test of the materiality of light was 
 appealed to by many experimental philosophers of the last 
 century, and with various results. The effects observed have 
 been traced, with much probability, to extraneous causes, such 
 as aerial currents produced by unequal temperature ; and it is 
 now universally conceded that no sensible effect of the impulse 
 
PROPAGATION OF LIGHT. 11 
 
 of light has been ever perceived. The experiments of Mr. Ben- 
 net seem to be decisive on this point. In these experiments a 
 slender straw was suspended horizontally by means of a single 
 fibre of the spider's thread. To one end of this delicately sus- 
 pended lever was attached a small piece of white paper, and 
 the whole was inclosed in a glass vessel, from which the air 
 was withdrawn by the air-pump. The sun's rays were then 
 concentrated by means of a large lens, and suffered to fall 
 upon the paper, but without any perceptible effect. 
 
 (14) But the actual velocity of light is not the only diffi- 
 culty which the theory of emission has to encounter at the 
 very outset. It has been further proved that this velocity is 
 the same, whether the light is directly emitted from the 
 Sun or a fixed star, or reflected from a planet or its satel- 
 lite that it is, in short, independent of the luminous source, 
 as well as of the subsequent modifications which it under- 
 goes in the celestial spaces. It is not easy to account for 
 these facts in the theory of emission. The emissive force, 
 required to produce the known velocity, is calculated to bS 
 more than a million of million times greater than the force 
 of gravity at the earth's surface ; and it can hardly be sup- 
 posed that this prodigious force is the same for all the various 
 and independent bodies of the universe, and that it acts 
 equally on all the particles of light, so as to generate in them 
 the same velocity. Yet even this assumption will not avail. 
 Laplace has shown, that if the diameter of a fixed star were 
 250 times as great as that of our sun, its density being the 
 same, its attraction would be sufficient to destroy the whole 
 momentum of the emitted molecules, and the star would be 
 invisible at great distances. With a smaller mass there will 
 be a proportionate retardation, so that the final velocities will 
 be different, whatever be the initial ones. The suggestion of 
 M. Arago seems to offer the only way of escaping the force 
 
12 PROPAGATION OF LIGHT. 
 
 of this objection. It may be supposed that the molecules of 
 light are originally projected with different velocities, but 
 that among these velocities there is but one which is adapted 
 to our organs of vision, and which produces the sensation of 
 light. 
 
 The uniform velocity of light is, on the other hand, an 
 immediate consequence of the principles of the wave-theory. 
 It follows from these principles, that the velocity with which 
 /vibratory movement is propagated in an elastic medium de- 
 pends in no degree on the exciting cause, but varies solely 
 with the elasticity of the medium and its density. If these 
 then be supposed to be uniform in the vast spaces which in- 
 tervene between the material bodies of the universe, the ve- 
 locity will be the same, whatever be the luminous origin. 
 
 (15) The rectilinear motion of light has long been urged 
 in favour of the theory of emission, and against the wave- 
 theory. If light consists in the undulations of an ethereal 
 medium (it has been said), as sound consists in the undulations 
 of the air, it should be propagated in all directions from every 
 new centre, and so bend round interposed obstacles. Thus 
 luminous objects should be visible, even when an obstacle is 
 between them and the eye, just as sounding bodies are heard, 
 though a dense body may be interposed between them and 
 the ear, and shadows could not exist. 
 
 To this objection, which was that chiefly urged by New- 
 ton himself, it might be enough to reply, that though vibra- 
 tory motion in an elastic medium is propagated in all direc- 
 tions from every new centre, yet there is no reason to conclude 
 that it is propagated with the same intensity in every direc- 
 tion, however inclined to that of the original wave. In fact, 
 analogy furnishes grounds for an opposite conclusion ; for 
 there are a multitude of facts which prove that sound is not 
 propagated with the same intensity in all directions, however 
 
PROPAGATION OF LIGHT. 13 
 
 inclined to the direction of the original motion. Now, if 
 there be any difference between the intensity of the direct and 
 lateral propagation, this difference may be ever so great ; i. e. 
 the ethereal medium may be so constituted that the intensity 
 of the laterally-propagated vibration shall be insensible. 
 
 But the solution of the difficulty rests upon more solid 
 grounds than analogy. A more minute examination of 
 the nature and laws of vibratory motion has, in fact, shown 
 this to be the case, as respects the luminiferous waves. It 
 has been proved, that whatever be the intensity of the partial 
 waves of the ether, which are propagated laterally round any 
 interposed obstacle, the total light resulting from their joint 
 action must degrade rapidly. And the luminous fringes which 
 have been observed within the shadows of bodies do, in fact, 
 represent the intensities resulting from these lateral waves, 
 when submitted to the most rigid mathematical calculation. 
 
 (16) Let us now proceed to consider, somewhat more mi- 
 nutely, the nature of a wave and its mode of propagation. 
 
 Let us conceive, then, a cord stretched in a horizontal 
 position, one end being attached to a fixed point, and the 
 other held in the hand. If the latter extremity be agitated, 
 by the motion of the hand up and down, a series of waves 
 will be propagated along the cord, each of which will advance 
 uniformly. Here it is evident that each particle of the cord 
 has merely a vibratory motion in a vertical direction. But 
 as this vibratory motion is communicated from each particle 
 to the next, along the whole length of the cord, it will 
 follow that some of the particles reach their highest posi- 
 tion when others are in the lowest; while other particles, 
 intermediate to these, are neither in their highest nor their 
 lowest position, but in some intermediate stage of their vibra- 
 tion. Thus, while each particle moves only to and fro verti- 
 cally, an undulation or wave is propagated horizontally along 
 
14 PROPAGATION OF LIGHT. 
 
 the string ; and there will be a succession of similar undula- 
 tions as long as the original disturbance continues. The 
 particles a, a', a", or the par- 
 ticles 6, 6 X , b", &c.,, are said 
 to be in similar phases of vi- 
 bration. The wave, or undulation, consists of all the par- 
 ticles between two which are in similar phases, as between 
 a and a 7 , or between b and b' ; and the length of a wave is the 
 distancev between them, estimated in the direction in which 
 the motion is propagated. It is evident from this description 
 that a wave, or undulation, comprises particle^ in^wry phase 
 of their vibration. 
 
 Now, instead of a single string, let us suppose an infinite 
 number, all diverging from the same centre ; and let us sup- 
 pose that they are each made to undulate by a disturbing 
 action at that centre, acting in a similar manner, and in the 
 same degree, on all. It is obvious, then, that an undulation 
 will be propagated along all the strings ; and that these undu- 
 lations will be equal in magnitude, and will be propagated 
 with the same velocity, provided the strings be equal in ten- 
 sion, elasticity, and other respects. In this case, then, simi- 
 lar w r aves will be propagated to points equally distant from 
 the origin of disturbance in the same time ; and all the points 
 which are in a similar phase of vibration will be situated on 
 the surface of a sphere, of which that origin is the centre. 
 
 In place of the actual strings which we have been consi- 
 dering, let us imagine rows of ethereal particles connected 
 by their mutual actions, and all that has been said will 
 apply to the propagation of light, the luminous body being 
 the source of disturbance. The length of the wave is the 
 distance, estimated in any direction from the centre, of two 
 particles which are in similar phases of vibration ; and it is 
 therefore the space through which the vibratory movement 
 is propagated in the time of a single vibration. Accordingly, 
 
PROPAGATION OF LIGHT. 15 
 
 if A denote the length of the wave, T the time of vibration, 
 and v the velocity of wave-propagation, 
 
 X = VT. 
 
 (17) In the foregoing illustration we have assumed the 
 vibratory motion of the molecules of the ether, to take place 
 in a direction perpendicular to that in which it is propagated. 
 The grounds of this assumption will be explained hereafter. 
 But, in the meantime it is important to consider, in more 
 detail, other properties of the movement. 
 
 Let us suppose, then, that each molecule performs its 
 vibrations in a right line,* passing through its position of 
 rest, and that these vibrations are all completed in the same 
 time. Let us further suppose, that the two halves of the 
 complete oscillation are perfectly similar, the deviation of the 
 molecule from its position of rest on the two sides being 
 equal at the corresponding times of the two half-oscillations. 
 These conditions will be satisfied by an equation of the 
 form 
 
 y-irin |f (*-,), 
 
 in which t is the actual time, t Q the time of the commence- 
 ment of the vibration, and T the time of the vibration itself, f 
 The displacement, ?/, vanishes when t - t Q is any multiple of 
 \ T ; and its values are equal, with opposite signs, for any 
 two values of t-t which differ by JT. The displacement 
 increases from t - t = 0, to t - t = J T, when it reaches its 
 maximum ; and it decreases from - t = J T, to t t = -J- T, 
 
 * More complex forms of vibration will be considered hereafter. 
 
 f The same conditions would be satisfied by the more complicated function 
 
 . ZTT . 4ir . 6?r 
 
 y = ! sin t + 02 sin t + a 3 sm t + &c. 
 
 and there are certain phenomena connected with the dispersion of light which 
 seem to require the more complete expression of the periodical function. 
 
16 PROPAGATION OF LIGHT. 
 
 when it vanishes. In the remaining half of the vibration its 
 values are equal at corresponding times, but with the nega- 
 tive sign, until at length it vanishes again, and the molecule 
 returns to its position of rest, when t - t = T. 
 
 (18) Now, let us suppose that, in virtue of the attraction 
 which subsists among the particles of the ether, the vibratory 
 movement is communicated from the vibrating molecule to 
 the next adjoining, from this latter to the next, and so on. 
 Each molecule will move to and fro, according to the same 
 laws as the first ; but, as the propagation of the vibratory 
 motion requires time, the successive molecules will be in dif- 
 ferent stages of their vibration. The displacements will, 
 accordingly, be expressed by the formula already given ; but 
 the time of the commencement of each will be different, and 
 proportionate to the distance traversed. Let x be the dis- 
 tance of any molecule from that corresponding to the origin 
 
 tJC 
 
 of the time, and v the velocity of- propagation ; then t = - . 
 Also, if A be the length of an undulation, or the space tra- 
 versed in the time 1 
 the formula becomes 
 
 versed in the time T, T = -. Making these substitutions, 
 
 . ; , . ^ 
 y = asm -r- (vt-x). 
 A 
 
 When x is constant, this equation gives the relation be- 
 tween the displacement and the time, or the law of the 
 vibratory motion, for any one molecule whose distance from 
 the origin is given. On the other hand, if t be constant, 
 the equation gives the relation between the ordinates and 
 abscissae of the consecutive molecules, for a given time, or the 
 equation of the wave.* 
 
 * If we take the partial differentials of this equation with respect to t and x 
 we see that 
 
 ftL-fclgift 
 
 d? dz* ' 
 
PROPAGATION OF LIGHT. 17 
 
 C19) We have hitherto considered the propagation of 
 vibratory movement without reference to any diversity of 
 its nature. It is obvious, however, that vibrations may 
 differ from one another in two particulars, namely, in the 
 space of vibration, aud in the time. In the aerial pulses the 
 amplitude of the vibration determines the loud ness of the 
 sound ; and the frequency of the pulses, or the time of vibra- 
 tion, determines its note. In like manner, the amplitude of 
 the ethereal vibrations determines the intensity of the light ; 
 and their frequency, or the period of vibration, determines 
 the colour. Thus, two lights may differ from one another in 
 intensity and colour, the former depending (according to the 
 wave-theory) on the space of vibration, and the latter on the 
 time. 
 
 But although the intensity of the light is obviously depen- 
 dent on the amplitude of the vibration, yet it does not appear, 
 a priori, by what power of the amplitude it is to be repre- 
 sented. In fact, we must define what we mean by a double, 
 triple, &c, quantity of light, before we can know how that 
 quantity is to be mathematically measured. If then we say 
 that a double light is the sum of the lights produced by two 
 luminous origins of equal intensity, placed close together, it 
 is easy to prove that the quantity of light, in general, is 
 measured by the square of the amplitude of the vibration. 
 From this it follows that the intensity of the light diverging 
 from any luminous origin must decrease inversely as the 
 square of the distance ; for, from the laws of wave propaga- 
 tion, it appears that the space of vibration diminishes in the 
 inverse simple ratio of the distance. Thus the known law 
 of the variation of the intensity of light is deduced from the 
 principles of undulatory propagation. 
 
 (20) The colour of the light (it has been said) depends 
 on the number of impulses which the nerves of the eye receive, 
 
 [UFI7BESITT 
 
18 PROPAGATION OF LIGHT. 
 
 in a given time, from the vibrating particles of the ether, 
 the sensation of violet being produced by the most frequent 
 vibrations, and that of red by the least frequent. But the 
 number of vibrations performed in a given time varies inversely 
 as the time of a single vibration ; the colour of the light, 
 therefore, varies with the time of vibration, or with the length 
 of the wave in a given medium. By experiments, which will 
 be described hereafter, it has been found that the length of a 
 wave, in air, corresponding to the extreme red of the spectrum, 
 is 266 ten-millionths of an inch, and that corresponding to the 
 extreme violet 167 ten-millionths. The length of the wave 
 corresponding to the ray of mean refrangibility is nearly 200 
 ten-millionths, or J^OTJU^ ^ an i ncn 
 
 It appears, then, that the sensibility of the eye is confined 
 within much narrower limits than that of the ear ; the ratio 
 of the times of the extreme vibrations which affect the eye 
 being only that of 1*58 to 1, which is less than the ratio of the 
 times of vibration of a fundamental note and its octave. There 
 is no reason for supposing, however, that the vibrations them- 
 selves are confined within these limits. In fact, we know that 
 there are invisible rays beyond the two extremities of the spec- 
 trum, whose periods of vibration (and lengths of wave) must 
 fall without the limits now stated to belong to the visible 
 rays. 
 
 (21) The aberration of the light of the fixed stars results 
 from the movement of the Earth in its orbit, combined with the 
 movement of light. Nothing can be simpler than its expla- 
 nation in the theory of emission. In fact, we have only to 
 combine the two co-existing motions according to the known 
 mechanical law, and the apparent direction of the star is that 
 of their resultant. The angle between this direction, and that 
 of the principal component, is called the aberration. 
 
 In order to explain this phenomenon, in accordance with 
 
PROPAGATION OF LIGHT. 19 
 
 the principles of the wave-theory, it seemed necessary to sup- 
 pose that the ether which encompasses the Earth does not par- 
 ticipate in its motion, so that the ethereal current produced 
 by their relative motion pervades the solid mass of the Earth 
 " as freely," to use the words of Young, " as the wind passes 
 through a grove of trees." Fresnel has developed this hy- 
 pothesis, and has shown that it suffices to explain other 
 phenomena also, in which the Earth's motion is concerned. 
 Professor Stokes has lately shown that the same results may 
 be deduced from a more plausible hypothesis relative to the 
 mutual dependence of the ether and of the Earth. 
 
 c2 
 
CHAPTER II. 
 
 REFLEXION AND REFRACTION. 
 
 (22) WHEN light meets the surface of a new medium, a 
 portion of it is always turned back, or reflected. 
 
 The reflection of light is twofold. Thus, when a beam of 
 solar light is admitted into a darkened chamber through an 
 aperture in the window, and is allowed to fall upon a metallic 
 mirror, a reflected beam is seen pursuing a determinate direc- 
 tion, after leaving the mirror ; and if the eye be placed in this 
 direction, it will perceive a brilliant image of the sun. This 
 beam is said to be regularly reflected, and its intensity increases 
 with the polish of the mirror. But it is observed also, that 
 in whatever part of the room the eye is placed, it can always 
 distinguish the portion of the mirror which reflects the light ; 
 some of the rays, consequently, are reflected in all directions. 
 This portion of the light is said to be irregularly reflected, and 
 its intensity decreases with the polish of the mirror. 
 
 Irregular reflexion is due, mainly, to the inequalities of the 
 reflecting surface, which is composed of an indefinite number 
 of reflecting surfaces in various positions, and which, there- 
 fore, reflect the light in various directions. 
 
 (23) The angles of incidence and reflexion or the angles 
 which the incident and reflected rays make with the perpen- 
 dicular to the reflecting surface at the point of incidence 
 are in the same plane, and are equal. This law is universally 
 true, whatever be the nature of the light itself, or that of the 
 body which reflects it. 
 
REFLEXION AND REFRACTION. 21 
 
 (24) The intensity of the reflected light, on the other hand, 
 is found to vary greatly with the medium. The following lead- 
 ing facts have been established experimentally. 
 
 I. The quantity of light regularly reflected increases with 
 the angle of incidence, the increase being very slow at mo- 
 derate incidences, and becoming very rapid at great ones. 
 Thus, water at a perpendicular incidence, according to the 
 experiments of Bouguer, reflects only 18 rays out of 1000 ; 
 at an incidence of 40 it reflects 22 rays ; at 60, 65 rays ; 
 at 80, 333 rays ; and at 89|, 721 rays. 
 
 II. The quantity of light reflected at the same incidence 
 varies both with the medium upon which the light falls, and 
 with that from which it is incident. Thus, at a perpendicu- 
 lar incidence, the number of rays reflected by water, glass, 
 and mercury, are 18, 25, and 666, respectively, the number 
 of incident rays being 1000. The dependence of the quantity 
 of the reflected light upon, the medium from which it is in- 
 cident is easily shown by immersing a plate of glass in water 
 or oil. 
 
 III. The differences in the reflective powers of different 
 substances are much more marked at small, than at great 
 incidences. Thus, water and mercury the first of which 
 reflects but the one-fiftieth part of the incident light at a 
 perpendicular incidence, while the latter reflects two-thirds 
 are equally reflective at an incidence of 89^, the number 
 of rays reflected at this angle being, in both cases, 721 out of 
 1000. 
 
 (25) When light is incident upon the surface of a trans- 
 parent medium, a portion enters the medium, pursuing there 
 an altered direction. This portion is said to be refracted. 
 
 When the ray passes from a rarer into a denser medium, 
 the angle of incidence is, in general, greater than the angle of 
 refraction, and the deviation takes place towards the perpendi- 
 
22 REFLEXION AND REFRACTION. 
 
 cular to the bounding surface. On the contrary, when the 
 ray passes from a denser into a rarer medium, the angle of 
 incidence is less than the angle of refraction, and the devia- 
 tion is from the perpendicular. 
 
 (26) The angles of incidence and refraction are in the same 
 plane ; and their sines are in an invariable ratio. 
 
 In order to verify this law experimentally, it is only neces- 
 sary to measure several angles of incidence at the surface of 
 the same medium, and the corresponding angles of refraction. 
 This was done by Ptolemy in the second century, and sub- 
 sequently by Vitello in in the thirteenth ; but both of these 
 observers failed in discovering the connecting law. The law 
 of refraction, just stated, was discovered by Willebrord Snell, 
 about the year 1621. 
 
 If u and v be employed to denote the angles which the 
 portions of the ray in the rarer and denser medium, respec- 
 tively, make with the perpendicular to the common surface, 
 the second part of the law of refraction is expressed by the 
 
 equation, 
 
 sin u = n sin 0, 
 
 JJL being a constant quantity. This constant is termed the 
 index of refraction ; and since u > v, it is always greater than 
 unity. 
 
 "When a ray of light passes into any medium from a va- 
 cuum, the index of refraction is in that case termed the abso- 
 lute index of the medium. For air, and the gases, it exceeds 
 unity by a very small fraction ; for water, fi = 1*336 ; for 
 crown glass, /m = 1*535; for diamond, ju = 2*487; and, for 
 chromate of lead, /m = 3. 
 
 (27) When a ray of light traverses a medium bounded by 
 parallel planes, and re-enters the original medium, the emer- 
 gent ray is parallel to the incident. 
 
REFLEXION AND REFRACTION. 23 
 
 Let u and u\ denote the angles of incidence and refraction 
 at the first surface. On account of the parallelism of the 
 refracting surfaces, the angle of incidence on the second will 
 be equal to i, the angle of refraction at the first. Let u z he 
 the angle of refraction at the second surface. Then 
 
 sin u sin u 2 
 
 sin -u-i sin K I 
 
 Therefore u t = n, or the incident and emergent rays are equally 
 inclined to the normals to the refracting surfaces. 
 
 ; 
 
 (28) When a ray of light traversing a medium bounded 
 by parallel planes enters a third, which is different from either 
 of the former, its course in the third medium will be the 
 same as if the light had entered it directly from the first, and 
 is, therefore, independent of the intervening medium. 
 
 This law is a consequence of the observed fact, that when 
 the light emerges from the third medium into the first, the 
 bounding surface being parallel to the former, the emergent 
 ray is parallel to the incident. Let another ray, parallel 
 to the former, be incident directly from the first into the 
 third medium, and traversing it, emerge again into the 
 first ; the emergent ray will be parallel to the incident, and 
 therefore parallel to that which has traversed the two media. 
 These two rays, consequently, will be equally refracted 
 at the second surface of the third medium, and therefore the 
 portions in the third medium will be also parallel. Hence, 
 when a ray passes from any medium, through an intervening 
 one, into a third, all being bounded by parallel surfaces, the 
 total deviation of the ray is the same as if it had passed 
 directly from the first into the third". 
 
 Let u denote, as before, the angle of incidence on the first 
 surface, u\ and u 2 the angles of refraction at the first and 
 
24 REFLEXION AND REFRACTION. 
 
 second surfaces, respectively, and let m and mi be the indices 
 of refraction at the two surfaces. Then, 
 
 sin u sin 
 
 sin Ui sin u z 
 
 and multiplying 
 
 _ sin u _ 
 
 fj. denoting the ratio of the sines from the first medium into 
 the third. 
 
 It is obvious that this result may be generalized ; and 
 that, when there is any number of successive media, bounded 
 by parallel surfaces, the index of refraction from the first into 
 the last is the continued product of the indices from the first 
 into the second, from the second into the third, and so on. 
 
 When the first medium is a vacuum, m and /u will be the 
 absolute indices of refraction of the second and third media, 
 respectively. Hence m\^ the relative index of refraction from 
 the second medium into the third, is equal to the quotient 
 arising from the division of the absolute index of the latter 
 by that of the former. 
 
 (29) When light traverses a prism, that is, a medium 
 bounded by two inclined plane surfaces, the total deviation of 
 the refracted ray is the sum of the deviations at incidence and 
 emergence. Let u and u' denote the angles which the inci- 
 dent and emergent rays make with the perpendiculars to the 
 faces at the points of incidence and emergence, v and v' the 
 angles which the portion of the ray within the prism forms 
 with the same, then the deviations at incidence and emergence 
 are, respectively, u - v, and u' - v' ; and the total deviation 
 A = u + u' - (v+ v'). Now, it is easily shown that the alge- 
 braic sum of the angles, which the portion of the ray within 
 the prism makes with the two perpendiculars, is equal to the 
 
REFLEXION AND REFRACTION. 25 
 
 vertical angle of the prism ; or, denoting this angle by A, 
 that 
 
 v + v ' = A ; 
 
 wherefore 
 
 A = u -f u' - A. 
 
 (30) When a ray of light is incident nearly perpendi- 
 cularly upon a thin prism, the total deviation is constant, and 
 bears an invariable ratio to the angle of the prism. 
 
 For in this case the angles of incidence and refraction, 
 being small, are proportional to their sines, so that 
 
 u = nv, u' = jjiv' ; and u + u' = JJL (v + #') - /u A. 
 Hence 
 
 (31) The deviation produced by a prism is easily deter- 
 mined when the angles of incidence and emergence are 
 equal. 
 
 For we have seen that, generally, 
 
 But since, in this case, u = ^^, there is also v = v' ; and con- 
 sequently 
 
 Hence we have 
 
 sin % (A + A) = ju sin ^ A ; 
 
 from which A+ A, and therefore A is determined. 
 
 It may be shown that the angle of deviation, in this case, 
 is the least possible ; and accordingly, if the prism be turned 
 slowly round its axis, the inclination of the emergent to the 
 incident ray will first decrease, and afterwards increase, ap- 
 pearing for a moment to be stationary between the opposite 
 changes. By this principle it is easy to place a prism, experi- 
 
26 REFLEXION AND REFRACTION. 
 
 mentally, in the position in which the refractions are equal 
 at both sides. 
 
 (32) We are now enabled to determine the refractive 
 index of a transparent solid experimentally. 
 
 The first step of this process is to polish two plane faces, 
 inclined to one another at a sufficient angle, and to measure 
 that angle by a goniometer. This being done, the prism is to 
 be placed, with its refracting edge vertical, before the object- 
 glass of the telescope of a theodolite, so as to refract to the 
 cross wires in its focus the rays proceeding from a distant 
 mark. The prism is then to be turned slowly round its axis, 
 and the telescope moved, until the deviation is a minimum. 
 The horizontal circle being read, and the prism removed, the 
 telescope is to be turned directly to the distant mark, and the 
 reading repeated ; the difference of the two readings is the 
 deviation. The angle of the prism and the deviation being 
 obtained, the refractive index is given by the formula, 
 
 sin|(A+A) 
 
 To determine the refractive index of a fluid, we have only 
 to inclose it in a hollow prism, whose sides are formed of 
 glass plates with parallel surfaces. For the course of the ray 
 will be the same as if it had been incident directly from the 
 air into the fluid, and had emerged similarly, without passing 
 through the glass. 
 
 (33) Let us now proceed to the physical explanation of 
 the phenomena. 
 
 To account for the phenomena of reflexion and refraction, 
 it is supposed, in the theory of emission, that the particles of 
 bodies and those of light exert a mutual action ; that, when 
 they are nearly in contact, this action is attractive ; that, at 
 
REFLEXION AND REFRACTION. 27 
 
 a distance a little greater, the attractive force is changed into 
 a repulsive one ; and that these attractive and repulsive 
 forces succeed one another for many alternations. Nothing 
 can be more reasonable than this hypothesis, granting that 
 light is material ; for the succession of attractive and repul- 
 sive forces, here assumed, is altogether similar to tha. 
 to which the known phenomena of molecular action are 
 ascribed. 
 
 On these suppositions Newton has rigorously deduced the 
 laws of reflexion and refraction. In the case of reflexion, it is 
 shown that the whole perpendicular velocity of the molecule 
 is restored to it in an opposite direction, by the operation of 
 the supposed repulsive force ; and, therefore, that the angles 
 which its path makes with the perpendicular to the surface, 
 before and after reflexion, are equal. In the case of refrac- 
 tion, it is proved that the effect of the attractive force is to 
 increase the square of the perpendicular velocity of the mole- 
 cule, by an amount which is constant for the same medium ; 
 from which it follows, that the sines of the angles which its 
 course makes with the perpendicular to the surface, before 
 and after refraction, are in the inverse ratio of the velocities 
 in the two media. This problem was the first in which the 
 effects of molecular forces were submitted to calculation ; and 
 its solution is justly regarded as forming an era in the history 
 of science. 
 
 (34) But although the theory of emission is successful 
 in explaining the laws of reflexion and refraction, considered 
 as distinct phenomena, it is by no means equally so in ac- 
 counting for their connexion and mutual dependence. When 
 a beam of light is incident on the surface of any transparent 
 medium, part is in all cases transmitted, and part reflected ; 
 the intensity of the reflexion being less, the less the differ- 
 ence of the refractive indices of the two media, and the re- 
 
28 REFLEXION AND REFRACTION. 
 
 flexion ceasing altogether when this difference vanishes. How 
 is it, then, that some of the molecules obey the influence of 
 the repulsive force, and are reflected, while others yield to 
 the attractive force, and are refracted ? To account for this, 
 Newton was obliged to have recourse to a new hypothesis. 
 The molecules of light, in their progress through space, are 
 supposed to pass continually into two alternate states, or fits, 
 which recur periodically and at equal intervals. While in one 
 of these states, called the fit of easy reflexion, they are disposed 
 to obey the repulsive or reflective forces of any body which 
 they meet ; and, on the other hand, they yield more readily 
 to the attractive or refractive forces, when in the alternate 
 state, or fit of easy transmission. Now, the molecules com- 
 posing a beam of common light are supposed to be in every 
 possible stage of these fits, when they reach the surface ; 
 some in a fit of reflexion, and others in a fit of transmission. 
 Some of them, consequently, will be reflected, and others 
 refracted, and the proportion of the former to the latter will 
 increase with the incidence. 
 
 To account for the fits themselves, Newton assumed the 
 existence of an ethereal medium, analogous to that of Huy- 
 gens, although he did not assign to it the same office. The 
 molecules of light were supposed to excite the vibrations of 
 this ether, just as a stone flung into water raises waves on its 
 surface. This vibratory motion was supposed to be propa- 
 gated with a velocity greater than that of the molecules ; so 
 as to overtake them, and impress upon them the disposition 
 in question, by conspiring with or opposing their progressive 
 motion. In one of his queries Newton has even calculated 
 the elastic force of this ether, as compared with that of air 
 in order that the velocity of propagation should exceed that 
 of light. 
 
 (35) The hypothesis of the fits has lost much of its credit, 
 
REFLEXION AND REFRACTION. " 29 
 
 since the phenomena of the colours of thin plates phenomena 
 which first suggested it, to the mind of Newton have been 
 shown to be irreconcilable with it. The explanation which 
 it yields of the facts now under consideration is alike un- 
 satisfactory. In fact, the molecules which are transmitted 
 are not all in the maximum of the fit of transmission ; but are 
 supposed to reach the surface in every possible phase of this, 
 which may be called the positive fit. But as a change of the 
 fit from positive to negative is, in general, sufficient to over- 
 come altogether the effect of the attractive force, and subject 
 the molecule to the repulsive one, it is obvious that the phase 
 of the fit must modify the effects of these forces in every inter- 
 mediate degree ; and that the molecules which do obey the 
 attractive force must have their velocities augmented in 
 different degrees, depending on their phase. Hence, as the 
 direction of the refracted ray depends on its velocity, the 
 transmitted beam should consist of rays refracted in widely 
 different angles, and should be scattered and irregular. 
 
 (36) Let us now turn to the account which the other 
 theory gives of the same phenomena, and of their laws. 
 
 The velocity of propagation, in the wave-theory of Light, 
 depends on the elasticity of the vibrating medium as com- 
 pared with its density. In the same homogeneous medium the 
 velocity will be therefore constant, and the wave propagated 
 from any centre of disturbance spherical. But when a wave 
 reaches the surface of a new medium whose elasticity is diffe- 
 rent, it will give rise to two waves, one in each medium, and 
 both differing in position from the original wave. For it is 
 obvious that, in general, the several portions of the incident 
 wave will reach the bounding surface at different moments of 
 time. Each of these portions will be the centre of two new 
 waves, one of which will be propagated in the first medium 
 with the original velocity, while the other will be propagated 
 
30 REFLEXION AND REFRACTION. 
 
 in the new medium, and with the velocity which belongs to 
 it ; so that there will be an infinite number of partial wares in 
 both media, diverging from the several points of the bounding 
 surface. But, by the principle of the coexistence of small 
 motions, the agitation of any particle of either medium is the 
 sum of the agitations sent there at the same instant from these 
 several centres of disturbance. The surfaces on which these 
 are accumulated will be the reflected and refracted waves, and 
 they are obviously those which touch all the small spherical 
 waves at any instant. 
 
 Thus, let mn be the front of 
 a plane wave, meeting the re- 
 flecting surface at m. Each 
 portion of this wave, as it 
 reaches the surface, becomes 
 the centre of a diverging sphe- 
 rical wave in the first medium, which will be propagated with 
 the velocity of the original wave. Accordingly, when the 
 portion n reaches the surface at k, the portion m will have 
 diverged into the spherical wave, whose radius, mo, is equal 
 to nk. And, in like manner, if m'n 1 be drawn parallel to 
 mn, the wave diverging from m will in the same time have 
 reached the spherical surface whose radius, m'o', is equal to 
 n'k. The surface which touches all these spheres at any 
 instant is that of the reflected wave. But, as mo and m'o' are 
 proportional to mk and m'k, it is obvious that this tangent 
 surface is plane ; and since mo = nk, and the angles at n and 
 o are right, it follows that the angles nmk and okm are equal, 
 or that the incident and reflected waves are equally in- 
 clined to the reflecting surface. 
 
 (37) The proof of the law of refraction is in all respects 
 analogous to the preceding. Let mn be the position of the 
 incident plane wave at any moment. In an interval of 
 
REFLEXION AND REFRACTION. 31 
 
 time proportional to nk, the portion n of this wave will 
 have reached the surface at k, 
 and the portions ra and m' will 
 have become the centres of di- 
 verging spherical waves in the 
 second medium, the radii 
 of these spheres, mo and m'o', 
 being to the intercepts, nk 
 and n'k, in the constant ratio 
 of the velocities of propagation in the two media. The 
 surface which touches these spheres is that of the refracted 
 wave. It is obvious, as before, that it is plane ; and, since 
 sin nmk : sin mko : ink: mo, we learn that the sines of the 
 angles which the incident and refracted waves make with 
 the refracting surface are in the constant ratio of the veloci- 
 ties of propagation. 
 
 (38) Such is the demonstration of the laws of reflexion 
 and refraction given by Huygens. The composition of the 
 grand or primary wave, by the union of the several secondary 
 or partial waves, in this demonstration, has been denominated 
 the principle of Huygens ; and it is obviously a case of the 
 more general principle of the co-existence of small motions. 
 It easily follows from this mode of composition, that the sur- 
 face of the primary wave marks the extreme limits to which 
 the vibratory movement is propagated in any given time ; so 
 that light is propagated from any one point to another in the 
 least possible time. This is the well-known law of Fermat, 
 the law of swiftest propagation ; and it will appear from 
 what has has been stated, that it holds, whatever be the 
 modifications which the course of the light may undergo by 
 reflexion or refraction. 
 
 This law may be thus enunciated: "The course pursued 
 by any reflected or refracted ray is that which would be de- 
 
32 REFLEXION AND REFRACTION. 
 
 scribed in the least possible time, by a body moving from 
 any point on the incident to any point on the reflected or 
 refracted ray." If I therefore denote the length of the path 
 described by the incident light, between any assumed point 
 and the point of incidence, I' the corresponding length of the 
 path described by the refracted light, and v and v' the velo- 
 cities of propagation in the two media, the sum of the times, 
 
 / I' 
 
 - + -, is a minimum ; or, multiplying by v, and denoting 
 
 the ratio - by ^, 
 
 /+ i 1 = minimum. 
 
 The constant factor, /j, is the refractive index of the medium. 
 In the case of reflexion, ju = l, and Z+Z' is a minimum. 
 The course pursued by a reflected ray is therefore such, that 
 the sum of the paths described between any two points and 
 the reflecting surface is the least possible. 
 
 (39) The intensity of the light, in the reflected and re- 
 fracted waves, will depend on the relative densities of the 
 ether in the two media. For we may compare the contiguous 
 strata of ether in these media to two elastic bodies of diffe- 
 rent masses, one of which moves the other by impact ; and it 
 is easy to deduce, on this principle, the intensities of the 
 reflected and refracted lights in the case of perpendicular 
 incidence. 
 
 (40) On reviewing what has been said, we cannot but be 
 struck by the remarkable fact, that theories so widely opposed 
 as the theory of emission, and that of waves, should lead 
 mathematically to the same result. According to both, we 
 have seen, the ratio of the sines of incidence and refraction is 
 equal to the ratio of the velocities of light in the two media, 
 
REFLEXION AND REFRACTION. 33 
 
 and is therefore constant. But there is this important diffe- 
 rence between them : in the wave-theory, the sines of these 
 angles are in the direct ratio of the velocities, while, accord- 
 ing to the theory of emission, they are in the inverse ratio. In 
 other words, the velocity of light in the denser medium is 
 less according to the former theory ; while, according to the 
 latter, it is greater. Here, then, the two theories are directly 
 at issue upon a point of fact, and we have only to ascertain 
 how this fact stands, in order to be able to decide between 
 them. The important experiment by which this was first 
 accomplished was made by Arago; and the result, as will be 
 shown hereafter, is conclusive in favour of the wave-theory. 
 
 (41) The conclusion deduced from the experiment here 
 referred to presupposes the laws of Interference of Light 
 laws which, in themselves, are intimately connected with the 
 principles of the wave- theory. It was desirable, therefore, to 
 deduce the same conclusion, if possible, by direct means. The 
 experiment by which this is effected has been made by 
 M. Foucault, upon a method devised by Arago ; its principle 
 will be understood from the following description. 
 
 Let a ray of light, reflected by a heliostat, be admitted 
 into a darkened chamber in a horizontal direction, and fall 
 upon a mirror which revolves about a vertical axis situated 
 in its own plane. It is manifest that, as the mirror revolves, 
 the reflected ray will move in the horizontal plane passing 
 through the point of incidence, with an angular velocity 
 double of that of the mirror itself. Now, in this plane let a 
 second mirror be placed, perpendicular to the right line 
 joining the centres of the two mirrors. Then, when the ray 
 reflected by the revolving mirror meets the fixed mirror, in 
 the course of its angular movement, it will be turned back 
 on its course, and, after a second reflexion by the revolving 
 mirror, return towards the aperture. 
 
 D 
 
34 REFLEXION AND REFRACTION. 
 
 It is plain that if the revolving mirror were for a moment 
 to rest in this position, the ray, after a second reflexion by it, 
 would return precisely by the path by which it came. But, 
 owing to the progressive movement of light, the mirror de- 
 scribes a certain small angle round its axis, in the interval 
 between the two appulses of the ray ; and the ray, after the 
 second reflexion, will deviate from its first position, by an 
 angle which is double of that described by the mirror in the 
 interval. Hence, if this angle can be observed, the velocity 
 of light is known. 
 
 For, if t be the time taken by the light to traverse the 
 interval of the two mirrors, forwards and backwards, the 
 angle described by the mirror in that time will be = w, w de- 
 noting the angle described by the mirror in the unit of time. 
 Hence, the angle described by the reflected ray in the time t, 
 or the deviation, is 2wt. Let this angle be denoted by a, and 
 there is 
 
 But the corresponding space is double the distance between 
 the two mirrors, or 2a. Consequently, the velocity of the 
 light is 
 
 M. Foucault has been enabled to observe an appreciable 
 deviation of the reflected ray, when the distance of the two 
 mirrors was 4 metres, and the revolving mirror made only 
 25 turns in a second. And as such a mirror can be made 
 to revolve 1000 times in a second, it is obvious that the 
 time taken by light to traverse even this short distance is 
 capable of being measured with precision. 
 
 (42) To apply this to the question at issue, we have only 
 
REFLEXION AND REFRACTION. 35 
 
 to interpose a column of water between the mirrors, to observe 
 the deviation, and to calculate the velocity. 
 
 Let e denote the length of the water tube, and a (as be- 
 fore) the interval of the two mirrors. Then, Y being the ve- 
 locity of the light in the compound path, determined by the 
 preceding method, it is obvious that 
 
 a _ a - e e 
 V ~ ~~~ + 7' 
 
 r and v' being the velocities in air, and in water, respectively. 
 But experiments with the revolving mirror give the values 
 of V and v ; therefore v', the velocity of light in water, is 
 known. By these means Foucault and Fizeau established 
 the fact, that the velocity of light is less in water than 
 in air, in the inverse proportion of the refractive indices. 
 The result is, therefore, decisive in favour of the wave- 
 theory. 
 
 (43) The refractive index being equal to the ratio of 
 the velocities of light in the two media, whichsoever 
 theory we adopt, it follows that any change in the velocity 
 of the incident ray must cause a variation in the amount 
 of refraction, unless the velocity of the refracted ray be al- 
 tered proportionally. Now the relative velocity of the light 
 of a star is altered by the Earth's motion ; and the amount 
 of the change is obviously the resolved part of the Earth's 
 velocity in the direction of the star. It was, therefore, a 
 matter of much interest to determine how, and in what de- 
 gree, this change affected the refraction. The experiment 
 was undertaken by Arago, at the request of Laplace. An 
 achromatic prism was attached in front of the object-glass of 
 the telescope of a repeating circle, so as to cover only a por- 
 tion of the lens. The star being then observed, directly through 
 the uncovered part of the lens, and afterwards in the direc- 
 
 D2 
 
36 REFLEXION AND REFRACTION. 
 
 tion in which its light was deviated by the prism, the dif- 
 ference of the angles read off gave the deviation. The stars 
 selected for observation were those in the ecliptic, which 
 passed the meridian nearly at 6 A. M. and 6 P. M., the velocity 
 of the Earth being added to that of the star in the former 
 case, and subtracted from it in the latter. No difference 
 whatever was observed in the deviations. 
 
 The experiment of Arago is, however, illusory. The eye 
 itself is in movement, and the aberration thus produced has 
 been shown to be equal and opposite to the change produced 
 by refraction. 
 
 (44) M. Fizeau has attempted by other means to deter- 
 mine the effect of the motion of the refracting medium upon 
 the velocity of propagation. Two long tubes of equal length, 
 filled with water, are interposed in the paths of two interfer- 
 ing pencils, and the place of the fringes observed. A rapid 
 movement is then communicated to the water in one of the 
 tubes ; and it is observed that the fringes are displaced. 
 The amount of the displacement is, in truth, very small 
 about half the breadth of a fringe in the experiments of 
 M. Fizeau. But its magnitude and direction accord suffi- 
 ciently with the results of theory, and with the hypothesis of 
 Fresnel on the relation between the velocity of the body, and 
 that of the ether connected with it. 
 
CHAPTEE III. 
 
 DISPERSION. 
 
 (45) WE have hitherto supposed light to be simple or ho- 
 mogeneous. The light of the Sun, however, and most of the 
 lights, natural or artificial, with which we are acquainted, are 
 compound, each ray consisting of an infinite number of rays 
 differing in colour and ref Tangibility . This important dis- 
 covery we owe to Newton. We shall briefly describe the 
 principal experiments by which its truth was established. 
 
 (46) When a beam of solar light is admitted into a dark- 
 ened room through a small circular aperture, and received on 
 a screen at a distance, a circular image of the Sun will be de- 
 picted there, whose diameter will correspond to that of the hole. 
 If now the light be intercepted by a prism, having its refracting 
 edge horizontal and perpendicular to the incident beam, the 
 image of the Sun will be thrown upwards by the refraction 
 of the prism, and will be no longer white and circular, but 
 coloured and oblong ; the sides which are perpendicular to the 
 edge of the prism being rectilinear and parallel, and the ex- 
 tremities semicircular. The breadth of this image, or spectrum 
 (as it is called), is equal to the diameter of the unrefracted 
 image of the Sun, but its length is much greater. 
 
 Now if the solar beam consisted of rays having all the same 
 refrangibility, the refracted image should be circular, and of 
 the same dimensions as the unrefracted image, from which it 
 should differ only in position. For the rays composing the 
 beam, being parallel at their incidence on the prism, must 
 (on this supposition) be equally refracted by it, and there- 
 
38 DISPERSION. 
 
 fore continue parallel after refraction. This not being the 
 case, we conclude that the rays composing the incident beam 
 are of different degrees of refr eligibility (the more refrangible 
 rays going to form the upper part of the spectrum, and the 
 less refrangible the lower), and that the elongation of the 
 solar image, and the variety of its colouring, arise from the 
 separation of these rays in their refraction through the 
 prism. 
 * 
 
 (47) It further appears that the rays, which differ in re- 
 frangibUity, likewise differ in colour. The spectrum is red at its 
 lowest or least refracted extremity, violet at its most refracted 
 extremity, and yellow, green, and blue, in the intermediate 
 spaces, these colours passing into one another by impercep- 
 tible gradations. Sir Isaac Newton distinguished seven 
 principal colours in the spectrum, and measured the spaces 
 occupied by each. These colours, arranged in the order of 
 their refrangibility, are red,- orangey yellow, green, blue, indigo, 
 violet ;* of which the yellow and orange are the most lumi- 
 nous, the red and green next in order, and the indigo and 
 violet weakest. 
 
 Any one of these rays may be separated from the rest by 
 transmitting it through a small aperture in a screen which in- 
 tercepts the remainder of the light. The ray thus separated 
 may be examined apart from the rest, and will be found 
 to undergo no dilatation, or change of colour, "by any subse- 
 quent refractions or reflexions. We are, therefore, warranted 
 
 * The imperfection of Newton's classification of colours has been pointed 
 out by Professor Forbes and others. The" indigo ought not to have been dis- 
 tinguished from the blue, the difference to the eye being much less, in kind, 
 than between any other two adjacent colours of the scale. We may, there- 
 fore, better distribute the colours of the spectrum into six, viz., red, orange, 
 yellow, green, blue, and violet, of which the red, yellow, and blue may be re- 
 garded as primary colours, and the orange, green, and violet as sec ondary. 
 
DISPERSION. 39 
 
 in con-eluding that solar light is compound, and consists of 
 an infinite number of simple rays, which are permanent in 
 their own nature, but differ from one another both in their 
 colour and refrangibility. 
 
 (48) The following experiment may be considered as 
 removing all doubt on this subject. Close behind the prism 
 is placed a board, perforated with a small aperture, through 
 which the refracted light is permitted to pass. This light is 
 then received on a second board, similarly perforated, at a 
 considerable distance from the first ; so that a small portion 
 of the light of the spectrum is suffered to pass through the 
 aperture in the second board, the rest being intercepted. Close 
 behind this aperture a second prism is fixed, having its refract- 
 ing edge parallel to that of the first. The first prism being then 
 turned slowly round its axis, the light of the spectrum will 
 move up and down on the second board, and the differently- 
 coloured rays will be successively transmitted through the se- 
 cond aperture, and be refracted by the prism behind it. If 
 then the places of these twice-refracted rays on the screen be 
 noted, the red will be found to be lowest, the violet highest, 
 and the intermediate colours in the same order as they are in 
 the spectrum. Here, on account of the unchanged position of 
 the two apertures, all the rays are necessarily incident upon 
 the second prism at the same angle ; and yet some of them 
 are more refracted, and others less, in the same proportion as 
 by the first prism. 
 
 We conclude, then, that the peculiar colour and refran- 
 gibility belonging to each kind of homogeneous light, are 
 permanent* and original affections, and are not generated 
 
 * Professor Stokes has recently discovered that the refrangihility of light 
 does undergo alteration in certain cases, some bodies possessing the property of 
 lowering the refrangibility of the incident light that is, of emitting rays of a 
 
40 DISPERSION. 
 
 by the changes which that light undergoes in refractions or 
 reflexions. 
 
 (49) In the experiments hitherto described, the analysis 
 of solar light, or its resolution into its simple components, is 
 far from being complete, inasmuch as there is a considerable 
 mixture of the different species of simple light in the spectrum. 
 This will be evident, if we consider that, as the several pencils 
 of homogeneous light suffer no dilatation by the prism, each will 
 depict on the screen a circular image, equal in magnitude to the 
 unrefracted image of the Sun ; and accordingly the spectrum 
 will consist of innumerable circles of homogeneous light, 
 whose centres are disposed along the same right line, and whose 
 common diameter is that of the Sun's unrefracted image. 
 Wherefore the number of such circles mixed together in the 
 spectrum, is to the corresponding number in the unrefracted 
 image of the Sun, as the interval between the centres of two 
 contingent circles, or the breadth of the spectrum, to the in- 
 terval between the centres of the extreme circles, which is the 
 length of the rectilinear sides. The mixture in the spectrum, 
 therefore, varies as the breadth of the spectrum divided by 
 its length ; and if the breadth can be diminished, the length 
 remaining the same, the mixture will be diminished in pro- 
 portion. 
 
 Newton's method of diminishing the breadth of the spec- 
 trum, or the diameter of the Sun's unrefracted image, was as 
 follows. The solar beam, admitted through a small circular 
 aperture, is received upon a lens of long focus, at the distance 
 of double its focal length from the aperture ; and at the same 
 distance beyond the lens will be formed a distinct image of 
 
 lower refrangibility, when excited by those of a higher. This property belongs 
 to the solution of sulphate of quinine, and to certain coloured glasses. Professor 
 Stokes has denominated it fluorescence. 
 
DISPERSION. 41 
 
 the hole, equal to it in magnitude. A prism being then placed 
 immediately behind the lens, this image will be dilated in 
 length, its breadth remaining unaltered, and thus a spectrum 
 will be formed whose breadth is the diameter of the hole ; 
 whereas, without this contrivance, the breadth would be equal 
 to that diameter, together with a line which (at the distance 
 of the screen. from the hole) subtends an angle equal to the 
 apparent diameter of the Sun. Thus, by diminishing the 
 diameter of the aperture, the breadth of the spectrum, and 
 therefore the mixture, may be reduced at pleasure. 
 
 If the diameter of the aperture be very small, the spec- 
 trum is reduced to a narrow line, and is unfit for examination. 
 To remedy this, Newton employed a narrow rectangular 
 aperture, whose length, parallel to the edge of the prism, may 
 be as great as we please, while its breadth is very small. In 
 this manner we obtain a spectrum as broad as we wish, and 
 whose light is as simple as before. 
 
 (50) In order to determine the laws of dispersion, it is 
 necessary to find experimentally the indices of refraction of 
 the several species of simple light, of which solar light is com- 
 posed. 
 
 Newton's method was to determine the refractive indices of 
 the extreme red and violet rays directly by means of the formula 
 of Article (32), and to deduce those of the other rays by a simple 
 proportion. When the refracting prism was of crown-glass, 
 the indices of refraction of the extreme rays were found to be 
 
 *77 *7& 
 
 and ^7-, respectively. To determine the refractive indices 
 oU oU 
 
 of the intermediate rays, it was necessary to measure the 
 spaces which they occupied in the spectrum. For this pur- 
 pose Newton delineated on paper the spectrum AHAa, and 
 distinguished it by the cross lines A0, B6, Cc, &c., drawn at 
 the confines of the several colours ; so that the space 
 
42 DISPERSION. 
 
 is that occupied by the red light, ~BGcb that by the orange, 
 CDdc the yellow, DE^ 
 
 j.i T-< ,1 * yy " g tfcSa 
 
 the green, EF/e the /"T j j ~~j r~l r\ 
 
 blue, FG-$/ the indigo, 1^1 i [ i j / 
 
 and GH% the violet. *""& ^T 1 - a - 6~s i"'' 
 He then found that, if the whole length of the rectilinear 
 side, AH, be taken as unit, the distances to the confines of 
 the several colours, AB, AC, AD, &c.,will be denoted by the 
 numbers , , , i, f , f Now the intervals AB, BC, CD, 
 &c., occupied by the several colours in the spectrum, will be 
 to one another as the corresponding variations of the index 
 of refraction. If, therefore, the whole variation of /u, or -^9 
 be divided as the line AH is in the points B, 0, D, &c., the 
 refractive indices of the rays at the confines of the several 
 colours will be as follow : 
 
 77 77| 77^ 77^- 77 77| 77| 78 
 50' 50' 60' 50' 50' 50 ' To ' 50' 
 
 771 
 
 The mean refractive index is ~, or 1 '55; and it appears 
 
 from the preceding that it belongs to the rays at the confines 
 of the green and blue. 
 
 A much more exact method of determining the refractive 
 indices of the species of simple light will be explained pre- 
 sently. 
 
 (51) The intensity of the light is very different in the 
 different parts of the spectrum. According to the experi- 
 ments of Fraunhofer, the following numbers represent the 
 intensities of the light in each of the coloured spaces, the 
 maximum intensity (which occurs at the confines of the 
 yellow and orange) being represented by 1000 ; viz., red, 
 94; orange, 640; green, 480; blue, 168; indigo, 31; 
 violet, 6. 
 
DISPERSION. 43 
 
 (52) On a minute examination of the solar spectrum, 
 when every care has been taken in making the experiment, 
 it is found that it is not, as Newton supposed, a continuous 
 band of coloured light, whose intensity is greatest about the 
 confines of the yellow and orange, and diminishes regularly 
 to the two extremities ; but that, on the contrary, there are 
 at certain points abrupt deficiencies of light, total or partial, 
 indicated by the existence of numerous dark lines or bands, 
 crossing the spectrum in the direction at right angles to its 
 sides ; while in the intermediate spaces the intensity of the 
 light does not increase or decrease continually, but varies 
 irregularly, or according to some very complex law. Solar 
 light, then, does not consist of rays of every possible refran- 
 gibility, within certain limits, for it is found that many rays 
 corresponding to certain degrees of refrangibility are wanting 
 in the spectrum. 
 
 Some of these lines are wholly black; others dark, of 
 various degrees of illumination. Again, some of them are 
 well defined and single ; others are clustered together, so as 
 to present the appearance of dark bands. They are irregu- 
 larly disposed throughout the length of the spectrum. They 
 are not, however, the result of any accidental cause ; for, when 
 solar light is used, and the refracting substance is the same, 
 it is found that they preserve the same relative position, both 
 with respect to one another and to the colours of the spec- 
 trum. On the other hand, when the refracting substance is 
 varied, their relative positions with respect to one another are 
 altered ; but their positions as referred to the colours of t]ie 
 spectrum, as also their relative breadth and intensity, remain 
 unchanged. 
 
 (53) If other kinds of light as that of the fixed stars, 
 flames, the electric spark are examined in the same way, 
 similar bands are discovered, but differing in each species of 
 
44 DISPERSION. 
 
 light in their position, &c. ; so that each species of flame, and 
 the light of each fixed star, has its own system of bands, 
 which remains unaltered under all circumstances, and 
 which, therefore, is a distinct physical characteristic of the 
 species of light to which it belongs. Thus the light of the 
 electric spark has bright bands, instead of dark ones. The 
 flames of oil, hydrogen, and alcohol, have each a brilliant line 
 between the red and the yellow. The red flames coloured by 
 nitrate of strontian exhibit a brilliant blue line, which is de- 
 tached from the rest of the spectrum ; and the salts of potash 
 give rise to a remarkable red ray, beyond the limits of the 
 ordinary red of the spectrum, and separated from it by a dark 
 interval. On the other hand, the spectrum of the flame of 
 cyanogen exhibits great regularity, as well in the distribution 
 of the dark bands, as in the intensity of the intervening 
 luminous spaces. 
 
 These bands depend on the rapidity of the combustion. 
 Thus sulphur, when burning slowly, exhibits blue and green 
 bands in the spectrum ; in rapid combustion, its light is 
 nearly homogeneous. 
 
 (54) These fixed lines, as they are called, were first 
 noticed by Wollaston, in the year 1802. They have since 
 been much more fully examined by Fraunhofer, who distin- 
 guished 590 in the solar spectrum, of which he has delineated 
 354. Of these he has selected seven principal ones, to serve 
 as standards of comparison, and has designated them by the 
 letters B, C, D, E, F, G and H. Of these, B and C are 
 single lines in the red portion of the spectrum, the former 
 near to its extremity ; D is a double line, at the confines of 
 the orange and yellow ; E is a group of fine lines in the 
 green ; F is a strongly marked black line in the blue ; Gr is a 
 group of fine lines in the indigo ; and H is a similar group in 
 the violet, clustered round one much stronger line. They 
 
DISPERSION . 45 
 
 are of the utmost importance in optical investigations. On 
 account of the sharpness of their delineation, their position 
 may be observed with an accuracy equal to that of astro- 
 nomical measurements, and the refractive indices of the rays, 
 to which they correspond, thus determined with the utmost 
 exactness. 
 
 The refractive indices of the definite rays in the glasses 
 vary with their composition. Those for distilled water are 
 the following. They may be taken, as proposed by Sir John 
 Herschel, as standards to which the refractive indices of other 
 substances may be referred : 
 
 Kay. Kefractive Index. 
 
 B . . 1-33098 
 
 C . . 1-33171 
 
 D . . 1-33358 
 
 E . . 1-33585 
 
 F . . 1-33779 
 
 G . . 1-34129 
 
 H . . 1-34418 
 
 (55) The dispersion of a ray which passes nearly perpen- 
 dicularly through a thin prism is easily expressed. 
 
 If Si and S 2 denote the deviations of the extreme red and 
 violet rays, jui and ^ the refractive indices of the prism for 
 those rays, and A its refracting angle, we have 
 
 Si = Gm -1) A, S 2 = Ou 2 -l)A; 
 whence 
 
 $2 ~ $1 = (jU 2 - jUi) A. 
 
 Accordingly the dispersion, in this case, is equal to the angle 
 of the prism multiplied by the difference of the refractive 
 indices. 
 
46 DISPERSION. 
 
 (56) The dispersive power of a substance is measured, 
 not by the absolute dispersion, which varies in general with 
 
 the angle of refraction, but by the ratio which that quantity 
 
 g _ g 
 bears to the total deviation, or by -. But, in the case of 
 
 Ci 
 
 a ray which passes nearly perpendicularly through a thin 
 prism, this ratio is constant ; for, dividing the third of the 
 equations of the preceding article by the first, 
 
 The dispersive power, therefore, is measured by the difference 
 of the refractive indices of the extreme red and violet rays, 
 divided by the refractive index of the former minus unity. 
 
 (57) Newton supposed that the dispersive powers of all 
 substances were the same. He was led to this erroneous con- 
 clusion by observing that when a prism of glass was inclosed 
 in a prism of water with a variable angle, their refracting an- 
 gles being turned in opposite directions, the emergent ray was 
 coloured when it was inclined to its original direction ; while, 
 on the other hand, it was colourless whenever, by varying the 
 angle of the water prism, the refractions of the two prisms 
 were made to compensate each other, or the ray to emerge 
 parallel to the incident ray. Hence he concluded that the 
 dispersion was always proportional to the total deviation ; 
 and that refraction could never take place without a separa- 
 tion of the refracted ray into its coloured elements. 
 
 When Newton's experiment with the two prisms was re- 
 peated a long time after, by Dollond, he found that the 
 results were exactly the opposite to those stated by New- 
 ton ; that, in fact, the emergent ray was colour -ed, when the 
 deviation was nothing, or the ray parallel to its original di- 
 rection ; and that, on the other hand, when the dispersions 
 
DISPERSION. 47 
 
 of the two prisms were made to correct each other by varying 
 the angle, so that the ray emerged colourless, their refractions 
 were no longer equal, and the ray emerged inclined to its 
 original direction. This important discovery led to the con- 
 struction of the achromatic telescope. 
 
 (58) It is easy to determine the condition of achromatism, 
 when a ray of light passes nearly perpendicularly through 
 two prisms, whose refracting angles are small. 
 
 The dispersions produced by the two prisms are (ju 2 -jui) A, 
 and (pi- pi) A x , respectively (55) ; and, therefore, when the 
 total dispersion is nothing, we must have 
 
 fa - /ii) A + fa'- p^A.'= 0, or4' = - &ZH. 
 
 A p 2 - pi 
 
 The negative sign, in the second member, indicates that 
 the angles of the two prisms must be turned in opposite 
 ways. 
 
 (59) In order to ascertain the relative dispersive powers 
 of different substances, they must be separately compared with 
 some standard substance, such, e. g., as water. For this 
 purpose a vessel must be constructed, whose opposite sides, 
 formed of parallel glass, are moveable on hinges, and may be 
 inclined to one another at any angle. It is closed on the other 
 two sides by metallic cheeks, to which the moveable sides are 
 accurately fitted. The vessel being filled with water, it is 
 evident that the transmitted ray will be refracted in the sam& 
 manner as by the inclosed water prism, the parallel plates of 
 glass producing no change in the direction of the refracted 
 ray. The substance whose dispersive power is sought being 
 formed into a thin prism, a beam of light is to be transmitted 
 nearly perpendicularly through the two prisms, with their re- 
 fracting angles turned in opposite directions ; and the angle 
 
48 DISPERSION. 
 
 of the water prism is to be varied, until the beam emerges 
 colourless. The angle of the water prism being then measured, 
 the ratio of the differences of the refractive indices, and thence 
 that of the dispersive powers, will be given by the formula 
 of the preceding Article. 
 
 (60) We now proceed to the physical explanation of the 
 foregoing phenomena. 
 
 To account for dispersion, the modern advocates of the 
 theory of emission have been forced to assume that the mo- 
 lecules of light are heterogeneous, and that the attractions 
 exerted on them by bodies vary with their nature, being 
 in this respect analogous to chemical affinities. This sup- 
 position, as Young has justly observed, is but veiling 
 our inability to assign a mechanical cause for the pheno- 
 menon. 
 
 According to the principles of the wave-theory, the 
 colour of light is determined by the frequency of the ethereal 
 vibrations, or by the length of the wave ; the longest 
 waves producing the sensation of red, and the shortest 
 that of violet. Now, observation proves that the refractive 
 index, or "the ratio of the velocities of propagation in the 
 two media, is different for the light of different colours. 
 The velocity of propagation in a refracting medium, there- 
 fore, varies with the length of the wave. Here, then, we en- 
 counter a difficulty in this theory, which was long regarded 
 as the most formidable obstacle to its reception. Analysis 
 seemed to indicate that the velocity of wave-propagation 
 depended solely on the elasticity of the medium as com- 
 pared with its density, and should therefore be the same 
 for light of all colours, as it is for sound of all notes ; and 
 that, consequently, all rays should be equally refracted. It 
 will be necessary to enter, in some detail, into the considera- 
 tion of this difficulty. 
 
DISPERSION. 49 
 
 (61) The conclusion of analysis to which we have just 
 adverted, namely, that the velocity of wave-propagation 
 is constant in the same homogeneous medium, is deduced 
 on the particular supposition, that the sphere of action of 
 the molecules of a vibrating medium is indefinitely small 
 compared with the length of a wave. If this restriction 
 be removed, we have no longer any ground for concluding 
 that the waves of different lengths will be propagated with 
 the same velocity ; and the conclusion hitherto acquiesced 
 in must be regarded but as an approximate result. It was 
 in this point of view that the question presented itself to 
 M. Cauchy . Eesuming the . problem of wave-propagation 
 with the more general equations, he has proved that there 
 exists, generally, a relation between the velocity of propagation 
 (or the refractive index in vacuo) and the length of the wave; 
 and, therefore, that the rays of different colours will be dif- 
 ferently refracted. 
 
 (62) Let us make, for abridgment, 
 
 z 2* 27T 
 
 /4 ' = 1T> s = 
 
 A T 
 
 in which X and r denote, as before, the wave-length and time 
 of vibration. M. Cauchy has proved that k and s are con- 
 nected by an equation of the form 
 
 s* = atf + ajt + a^ + &c., 
 
 in which the coefficients a^ a z , a 3) &c., vary with the medium. 
 Now the velocity of wave-propagation is 
 
 F =-=v 
 
 r k 
 consequently, 
 
 &c. 
 
50 DISPERSION. 
 
 Accordingly, the velocity of propagation is a function of the 
 wave-length, and varies with the colour. 
 
 (63) In a vacuum, and in media (such as atmospheric air) 
 which do not disperse the light, the coefficients #2, #3, &c., 
 are insensible, and we have 
 
 that is, the velocity of propagation is independent of the wave- 
 length, and the same for light of all colours. 
 
 In other media we may, as a first approximation, neglect 
 the third and following terms of the series, and we have 
 
 Hence, if Fi, F 2 denote the velocities of propagation for two 
 definite rays of the spectrum, and k i9 k^ the corresponding 
 values of k, 
 
 The truth of this formula has been verified by M. Cauchy, by 
 introducing in it the values of the refractive indices and 
 wave-lengths in certain media, as determined by Fraunhofer 
 for the seven definite rays. 
 
 (64) The general formula, above given, is unsuited to a 
 comparison with observation in its present form, inasmuch as 
 
 the variable k( = j is not independent of F. This diffi- 
 \ A / 
 
 culty is overcome by M. Cauchy by inverting the first series. 
 The result is of the form 
 
 tf = A l s*+Az s 4 +A 3 s 6 + &c. 
 M. Cauchy has shown that this series, as well as the former, 
 
DISPERSION. 51 
 
 is convergent, and that all the terms after the third may be 
 neglected. Hence, since 
 
 7 S 
 
 = = 
 
 the velocity in vacuo being unity, we have 
 
 an equation expressing the refractive index 'in terms of the 
 time of vibration, or of the wave-length in vacuo. 
 
 (65) The constants in this formula, A lt A 2 , A 3 , will be 
 determined, when we know three values of ju, with the cor- 
 responding values of s, or of the wave-length in vacuo ; and 
 the formula may be then applied to calculate the values of fj. 
 corresponding to any other values of s, which may be thus 
 compared with the results of observation. The comparison 
 has been made by Professor Powell, and by M. Cauchy him- 
 self, by means'of the observations of Fraunhofer on the refrac- 
 tive indices of water and several kinds of glass, and the 
 agreement of the calculated and observed results is within the 
 limits of the errors of observation. 
 
 But the truth of a formula, expressing the relation be- 
 tween the refractive index and the wave-length in vacuo, can 
 only be satisfactorily tested in the case of highly-dispersive 
 media ; and for such media no observations of sufficient accu- 
 racy hitherto existed. To supply this want, Professor Powell 
 undertook the laborious task of determining the refractive in- 
 dices corresponding to the seven definite rays of Fraunhofer, 
 for a great number of media, including those of a highly dis- 
 persive power, and of comparing them with the theory of 
 M. Cauchy. The result of the comparison is, on the whole, 
 satisfactory. 
 
 (66) It is an interesting consequence of the preceding for- 
 
 E2 
 
52 DISPERSION. 
 
 mula, pointed out by Professor Powell, that as s diminishes, 
 or the wave-length in vacuo increases, the value of ^ approxi- 
 mates to a fixed limit, given by the equation 
 
 which, therefore, defines the limit of the spectrum on the side 
 of the less refrangible rays. This limiting index corresponds 
 to a point not greatly below the red extremity of the visible 
 v spectrum. 
 
CHAPTEE IV. 
 
 ABSORPTION AND EMISSION. 
 
 (67) ALL bodies absorb a portion of the light which enters 
 their substance ; and this absorption is different for rays of 
 different refrangibilities, so that white light always becomes 
 coloured after passage through a medium of sufficient thick- 
 ness. There is, accordingly, no body perfectly transparent. 
 On the other hand, no known body is perfectly opaque. Gold 
 itself, which is the densest of the metals, transmits a faint 
 greenish light, when in the state of gold leaf. 
 
 The phenomena of absorption are all explained on the 
 supposition that light, which has traversed a given thickness 
 of any medium, loses by absorption the same fractional part 
 of its intensity, in passing through any given thickness of 
 the medium. According to this hypothesis, the intensity of 
 the light must diminish in geometrical progression, as the 
 thickness increases in arithmetical. For if i denote the in- 
 tensity of the light incident on the medium, and ia the 
 intensity after it has traversed the unit of thickness, the 
 intensity, after passing through two, three, &c., units, will be 
 * a 2 , i a 3 , &o. And generally, the intensity of the light, I, 
 after traversing the thickness 0, will be 
 
 The quantity a is the coefficient of transmission ; and it is 
 different for the light of different colours. Hence, when the 
 incident light is white, the elements of which it consists will 
 be absorbed in different proportions, and the emergent light 
 
 
54 ABSORPTION AND EMISSION. 
 
 will no longer contain the rays of different colours in the 
 same proportion as originally, and will, consequently, be co- 
 loured. 
 
 (68) In general the colour of the emergent light is the 
 same for all thicknesses traversed, the tint only becoming 
 deeper as the thickness is augmented. But there are media 
 in which the colour itself changes with the thickness. This 
 will be seen by the foregoing formula. For if i and i 1 be the 
 original intensities of the two colours, which we shall sup- 
 pose to predominate in the transmitted light, and a and a' 
 their coefficients of transmission, we have only to suppose 
 i > i', and a < a 7 . Under such circumstances i a 9 will be at 
 first greater than i' a /e , when the thickness is small ; but it 
 will diminish more rapidly with an increase of thickness, and 
 finally become smaller. It is easily seen that the thickness 
 at which the change occurs is given by the formula 
 
 = log- *' ~ log- *' m 
 log. a - log. a 
 
 (69) The complete analysis of the phenomena of absorp- 
 tion is effected by interposing the absorbing medium in a 
 pencil of rays of solar light which has traversed a prism, and 
 receiving the emergent rays on a screen. For the most part 
 the spectrum of the light, which has passed through absorb- 
 ing media, thus exhibits a single maximum of absorption. 
 There are, however, media whose action upon light is more 
 complex. The blue glass which is coloured by cobalt, has 
 three such maxima, dividing the spectrum into four por- 
 tions.* The solution of chlorophyll (the colouring matter 
 
 * This affords an easy method of obtaining a red light which is nearly homo- 
 geneous. "We have only to combine a plate of this glass with one of the red 
 glasses which transmit only the less refrangible rays of the spectrum, and we 
 shall obtain a red ray of considerable purity. 
 
ABSORPTION AND EMISSION. 55 
 
 of the leaves of plants) in ether, produces phenomena of the 
 same kind, the spectrum exhibiting dark bands in the red, 
 yellow, green, and violet. The absorption bands of human 
 blood in the yellow and green portions of the spectrum are 
 very marked, and are visible even when the blood contained 
 in the solution is so small as to affect but slightly the 
 colour. 
 
 The colours of natural bodies are, probably, due to ab- 
 sorption, the incident light being reflected, after having 
 penetrated the medium to some minute depth, which is of 
 course twice traversed. 
 
 (70) The rays of the Sun produce calorific and chemical 
 effects, as well as those of light and colour. The effect of 
 the solar rays on the thermometer are obvious. The chemical 
 effects consist in the power of decomposing certain chemical 
 compounds ; and it is to them that we owe all the beautiful 
 and important results of photography. These three effects, 
 the luminous, the calorific, and the chemical, all vary with 
 the refrangibility of the rays, but in different proportions ; 
 and accordingly, when the Sun's rays are dispersed by re- 
 fraction, the maxima of light, of heat, and of decomposing 
 power, occupy different parts of the spectrum. In fact, the 
 thermic and the chemical effects are found to be greatest 
 beyond the limits of the visible spectrum, the maximum of 
 the former being beyond the red rays, and that of the latter 
 beyond the violet. The power of the rays to produce these 
 different effects, therefore, varies with the refrangibility of 
 the rays, or with the rapidity of the vibrations, the slower 
 vibrations producing the effect of heat, the more rapid those 
 of light, and the most rapid the chemical effects. 
 
 It appears from the experiments of MM. Jamin and 
 Becquerel that, in a given substance, and for a ray of given 
 refrangibility, the luminous, the thermic, and the chemical 
 
56 ABSORPTION AND EMISSION. 
 
 rays are equally absorbed ; or, in other words, that the co- 
 efficient of transmission, a, is the same for all, and varies only 
 with the refrangibility of the ray, or the rapidity of the 
 vibrations of the molecules of the ether. It follows from 
 this, that these three effects cannot be separated from one 
 another by absorption, any more than by refraction, in any 
 ray of definite refrangibility. Hence also we draw the im- 
 portant conclusion, that the three properties of solar light 
 are not to be referred to separate agencies, but are only dif- 
 ferent effects of the same vibratory movement, varying with 
 its rapidity. 
 
 (71) Translucent and colourless bodies are those which 
 transmit all the luminous rays in the same proportion. 
 There is, however, in nature, no body absolutely translucent. 
 Air, which is the most so, colours the light of the Sun at first 
 yellow, and afterwards red, when the thickness of the atmo- 
 sphere traversed is greatest, viz., in the horizon. And water 
 produces the same effects in a very decided manner, the light 
 which has traversed a tube filled with water becoming first 
 yellow, then orange, and finally red, as the length of the 
 tube is increased. The absorbing effects are, however, much 
 more marked beyond the limits of the visible spectrum, 
 nearly all translucent bodies stopping either the calorific or 
 the chemical rays outside these limits. The only known 
 exception to this rule is that of rock-salt, which is accordingly 
 the only known substance perfectly transparent. White quartz 
 possesses a similar property, although in an inferior degree. 
 On the other hand, alum and ice, which are both translucent, 
 powerfully absorb the calorific rays beyond the red extre- 
 mity of the spectrum. 
 
 Black bodies absorb not only the luminous rays, but also 
 the chemical rays beyond the more refrangible end of the 
 spectrum, while they transmit the thermic rays beyond the 
 
ABSORPTION AND EMISSION, 57 
 
 less refrangible end. Yellow and red glasses absorb the 
 rays of higher refrangibility, and, along with them, the 
 chemical rays. Yellow glass, which transmits light abun- 
 dantly, is therefore employed with much advantage in photo- 
 graphic workshops. On the other hand, blue glasses transmit 
 the chemical rays freely, but absorb the thermic rays beyond 
 the red. 
 
 (72) The phenomena of the absorption of light by the 
 gases are very different from those presented by solids and 
 liquids. The rays stopped by them are few and well-defined ; 
 so that when the light is analysed by a prism, the spectrum 
 exhibits a small number of fine dark lines, wholly separated 
 from one another, and variously distributed. Accordingly 
 the gases allow all the light to pass, with the exception of 
 a few rays of definite refrangibility. It will be seen from 
 the foregoing account, that the spectra of light transmitted 
 through the gases resemble the solar spectrum, only that the 
 dark lines are fewer, and the whole phenomenon less com- 
 plex. And this resemblance has led to the inference that 
 the dark lines in the solar spectrum are due to gaseous ab- 
 sorption. We have only to assume that, in place of a 
 single gas absorbing a small number of definite rays, the 
 light of the Sun traverses a mixture of many gases, each of 
 which produces a different set of lines by absorption. 
 
 (73) All solid and liquid bodies, when sufficiently heated, 
 emit heat and light, the emitted rays increasing in refrangi-, 
 bility as the temperature is augmented. Thus, at the tem- 
 perature of boiling water, the non-luminous heat-rays alone 
 are emitted. "When the temperature is raised to 525 C., 
 according to Dr. Draper, the less refrangible red rays 
 begin to appear. The more refrangible rays are developed 
 in succession, as the temperature is further increased, until, 
 
58 ABSORPTION AND EMISSION. 
 
 at a white heat, all the rays of the visible spectrum are 
 emitted. The power of emitting any definite ray depends, 
 consequently, on the temperature of the heated body. It 
 does not follow, from this, however, that the emissive powers 
 of all bodies are equal ; for bodies differ greatly from one 
 another in their capacity of receiving heat. 
 
 (74) "When the light emitted from an incandescent solid 
 or liquid body is analyzed by the prism, it is found to contain 
 all the colours of the solar spectrum, although in varying 
 proportions. Its spectrum is accordingly continuous. The 
 case is altogether different when the incandescent body is a 
 vapour, or a gas. In this case the spectrum is found to corr- 
 sist of a small number of definite rays, separated from one 
 another by broad, dark intervals ; and the band of light is 
 discontinuous. 
 
 (75) There are different modes of exhibiting these phe- 
 nomena, according to the degree of heat required for incan- 
 descence. For some substances the heat of a spirit-lamp is 
 sufficient ; while for others the heat of the oxy-hydrogen 
 light, or even that produced by the electric spark, is neces- 
 sary. The Bunsen lamp, which itself emits a very feeble 
 light, affords a very convenient mode of exhibiting the spectra 
 of the former class. When a small piece of common salt 
 (chloride of sodium) is introduced into this flame, the light 
 emitted is a bright orange-yellow ray, which is of the same 
 refrangibility as the ray D of the solar spectrum. And the 
 case is the same with all the salts of the same metal, the 
 light of the metallic vapour overpowering that of the other 
 elements of the compound. By such means Bunsen and 
 Eoscoe studied the lights of the alkaline metals, sodium, 
 potassium, lithium, calcium, strontium, and barium ; and they 
 found that the same process was applicable to the vapours of 
 
ABSORPTION AND EMISSION. 59 
 
 copper and manganese. The light of lithium vapour consists 
 of only two rays, one in the red and the other in the orange. 
 The vapour of potassium likewise exhibits two rays, one in 
 the red, near to A, and the other in the violet. 
 
 For the more refractory metals, the spark developed by 
 the Buhmkorff coil affords a convenient mode of ignition. 
 The metal to be examined may be employed as the elec- 
 trodes. It is, however, more convenient to allow the spark 
 to pass between two pieces of charcoal, one of which is 
 hollowed, and contains in the cavity a fragment of the metal 
 to be examined. 
 
 (76) In the case of the more permanent gases, the spark 
 from the coil is allowed to pass through hermetically-sealed 
 glass tubes filled with the gases to be examined, the current 
 being directed by small wires of platinum at the two extre- 
 mities of the tube. The flame of the vapour of hydrogen is 
 thus found, when analyzed by the prism, to contain three 
 definite rays only. 
 
 It thus appears that the lights of the incandescent gases 
 are not only distinguished from those of solids and liquids by 
 the discontinuity of their spectra, but, moreover, each gas or 
 vapour has its own characteristic spectrum whereby it is dis- 
 tinguished from all others. It further appears that when 
 two or more substances are present in the volatilized vapour, 
 the spectrum contains the characteristic lines of both, These 
 facts are the foundation of spectral analysis. The physicist, 
 measuring the refrangibility of the rays of such a spectrum, . 
 is enabled to detect by their means the presence of the sub- 
 stances whose vapours are so combined and volatilized ; and 
 he has even been led to the discovery of new elementary 
 substances hitherto unknown to chemistry. Thus Bunsen 
 and Hoscoe discovered the new metals rubidium and ccesium, 
 and Crooke discovered another alkaline metal, which he has 
 
60 ABSORPTION AND EMISSION. 
 
 denominated thallium, from the beautiful green ray which 
 . characterizes its spectrum. 
 
 (77) The position of the bright lines in the spectra of the 
 gases is not affected by the temperature of incandescence, but 
 their brightness is ; and it happens thus, that lines which 
 are invisible at the lower temperature, become visible when 
 the temperature is much raised ; and this process continues 
 until, at extreme temperatures, the spectrum becomes con- 
 tinuous. Thus lithium, whose spectrum consists of a single 
 red line at ordinary temperatures, exhibits an orange and a 
 blue band when the temperature is highly raised. 
 
 \ 
 
 (78) The rays of light emitted by an incandescent gas 
 are those which it absorbs at lower temperatures. This 
 property, which has been denominated the reversal of the 
 spectra of the gases, was first discovered in the spectrum of 
 sodium vapour. When a little common salt is introduced 
 into the wick of a spirit-lamp, the spectrum exhibits the 
 bright yellow double line which is characteristic of sodium ; 
 and it is found that the vapour of sodium, at the ordinary 
 temperature, stops or absorbs precisely the same ray, when it 
 is interposed in the path of the continuous light emanating 
 from an oil lamp, the light of which gives with the prism a 
 continuous spectrum. A more exact process is to introduce 
 a small piece of metallic sodium into a glass tube closed at 
 both ends, and from which the air has been expelled by in- 
 troducing hydrogen. On heating the tube by means of a 
 spirit-lamp, the metal is vaporized. By such means Bunsen 
 and Kirchoff found that the vapours of the alkaline metals, 
 lithium, potassium, strontium, and barium, when interposed 
 in the light of a continuous spectrum, absorb precisely the 
 same rays which they emit when self-luminous. 
 
 (79) The general law, of which the foregoing is a par- 
 
ABSORPTION AND EMISSION. 
 
 61 
 
 ticular case, was enunciated by Kirchoff in 1860. It may be 
 thus expressed: The relation between the power ' of emitting, 
 and the power of absorbing any given ray, is the same for all 
 bodies at the same temperature. This law applies to the 
 calorific, and to the chemical rays, as well as to those of 
 light. 
 
 The relation between the emissive and the absorptive 
 powers of any gas is determined by the difference of tempera- 
 ture of the gas in the two cases. The greater that difference, 
 the more marked is the phenomenon of reversal. 
 
 The law above stated, and which had already been 
 demonstrated, in the case of heat, to be a necessary con- 
 sequence of the conditions of equilibrium of temperature, has 
 been shown by Professor Stokes to be a necessary consequence 
 of the laws of propagation of vibratory motion. The mole- 
 cules of an incandescent gas vibrate in a definite manner, 
 producing rays of definite refrangibility. Hence the ether 
 within a gas will be set in vibration by these same rays, 
 which will thus communicate to it their vis viva, and be 
 absorbed. On the other hand, the waves whose time ^of 
 vibration is different, will not set the ether in motion, and 
 will be transmitted through it without loss. 
 
 (80) We have already adverted to the similarity of the ab- 
 sorptive spectra of the gases, to the phenomena of the solar 
 spectrum ; and we are at once led to the conclusion, that the 
 light of the Sun is that of an incandescent solid, surrounded 
 by an atmosphere of vapours, which absorb certain definite 
 rays. We have now only to compare, by the spectroscope, 
 the position of any of these rays, with those of the bright 
 lines in the spectra of the metallic vapours, and if we 
 find them to be identical, we may conclude with certainty 
 that that vapour exists in the solar atmosphere, and therefore 
 the metal itself in the body of the Sun. This comparison has 
 
62 ABSORPTION AND EMISSION. 
 
 been made by Kirchoff, and he has thereby ascertained that 
 sodium, bismuth, magnesium, chrome, nickel and iron, all exist 
 in the Sun. The metals, gold, silver, mercury, tin, lead, 
 aluminium and silicum, on the other hand, do not exist in 
 the Sun, their specific rays being wanting in the solar spec- 
 trum. 
 
( 63 
 
 CHAPTEE Y. 
 
 DOUBLE REFRACTION. 
 
 (81) It has hitherto been assumed that, when a ray of 
 light is incident upon the surface of a transparent medium, 
 the intromitted portion pursues, in all cases, a single determi- 
 nate direction. This is, however, very far from the fact. In 
 many, indeed in most cases, the refracted ray is divided 
 into two distinct pencils, each of which pursues a separate 
 course, determined by a distinct law. 
 
 This property is called double refraction. It was first dis- 
 covered by Erasmus Bartholinus, in the well-known mineral 
 called Iceland spar. After a long series of observations, he 
 found that one of the rays within the crystal followed the 
 known law of refraction, while the other was bent according 
 to a new and extraordinary law not hitherto noticed. An 
 account of these experiments was published at Copenhagen, 
 in the year 1669, under the title " Experimenta Crystalli 
 Islandici dis-diaclastici, quibus mira et insolita refractio dete- 
 gitur." 
 
 A few years after the date of this publication, the subject 
 was taken up by Huygens. This distinguished philosopher 
 had already unfolded the theory which supposes light to 
 consist in the undulations of an ethereal fluid ; and from that 
 theory had derived, in the most lucid and elegant manner, 
 the laws of ordinary refraction (37). He was, therefore, 
 naturally anxious to examine whether the new properties of 
 light, discovered by Bartholinus, could be reconciled to the 
 same theory ; and, in his desire to assimilate the two classes 
 
64 DOUBLE REFRACTION. 
 
 of phenomena, he was happily led to assign the true law of 
 extraordinary refraction. The important researches of Huy- 
 gens on this subject are contained in the fifth chapter of his 
 " Traite de la Lumiere" 
 
 (82) The property of double refraction is possessed by all 
 crystallized minerals, excepting those belonging to the tessular 
 system, i. e. those whose fundamental form is the cube. It 
 belongs likewise to all animal and vegetable substances, in 
 which there is a regular arrangement of parts ; and, in fine, 
 to all bodies whatever, whose parts are in a state of unequal 
 compression or dilatation. The separation of the two refracted 
 pencils is in some cases considerable, and the course of each 
 easily ascertained by observation ; but it is generally too mi- 
 nute to be directly observed, and its existence is only proved 
 by the appearance of certain phenomena which are known to 
 arise from the mutual action of two pencils. In Iceland spar, 
 the substance in which the property was first discovered, the 
 separation of the pencils is very striking : and, as this mineral 
 is found in considerable masses, and in a state of great purity 
 and transparency, it is well fitted for the exhibition of the 
 phenomena. 
 
 (83) Carbonate of lime, of which Iceland spar is a variety, 
 crystallizes in more than 300 different forms, all of which 
 may be reduced by cleavage to the rhombohcdron, which is 
 accordingly the primitive form. The angles 
 of the bounding parallelograms, CAB and 
 ABD, in the rhombohedron of Iceland spar, 
 are 101 55' and 78 5 7 . Two of the solid 
 angles, at A and , are contained by thre 
 obtuse angles ; while the remaining four are bounded by one 
 obtuse and two acute angles. The line AO, joining the sum- 
 mits of the obtuse solid angles, is called the axis of the rhom- 
 
DOUBLE REFRACTION. 65 
 
 boliedron, and is equally inclined to the three faces which 
 meet there. The angles at which the faces themselves are 
 mutually inclined are 105 5 X and 74 55'. 
 
 (84) If a transparent piece of this substance be laid upon 
 a sheet of white paper, on which a small black spot is marked 
 with ink, we see two images of the spot instead of one, on 
 looking through the crystal ; and if the eye be held perpen- 
 dicularly above the surface, and the crystal turned round in 
 its plane, one of these images will appear to describe a circle 
 round the other, which is immoveable, the line connecting 
 them being in the direction of the shorter diagonal of the 
 rhombic face. We may vary this experiment, by substituting 
 for the dark spot on the paper a luminous point on a dark 
 ground, as, for example, the light of the sky seen through 
 a small aperture ; but the most direct mode of performing 
 the experiment is to transmit a ray of the Sun's light 
 through the crystal, and receive the emergent pencils on a 
 screen. 
 
 If now we examine the course of the two rays within the 
 crystal, we shall find that, at a perpendicular incidence, the 
 deviation of one of them is nothing ; that, at any other inci- 
 dence, the ray is bent towards the perpendicular, the sines 
 of the angles of incidence and refraction being in the constant 
 ratio of 1 '654 to 1 ; and that these angles are always in the 
 same plane. This ray, therefore, is refracted according to 
 the known law, and is called the ordinary ray. On examin- 
 ing the other ray, however, we find that, at a perpendicular 
 incidence, the deviation, instead of vanishing, is 6 12'; that, 
 at other incidences, the refracted ray does not follow the law 
 of the sines ; and that, moreover, the angles of incidence and 
 refraction are in different planes. This ray, therefore, is re- 
 fracted according to a new and different law, and is called the 
 extraordinary ray. 
 
66 DOUBLE REFRACTION. 
 
 (85) In proceeding to the consideration of this law, we 
 must observe, in the first place, that there is a certain direc- 
 tion in every double-refracting crystal, along which if a ray 
 be transmitted, ,it is no longer divided. This line is called 
 the optic axis, and all the phenomena of double refraction are 
 related to it. There are, properly speaking, an infinite 
 number of such lines within the crystal, all parallel to one 
 another ; so that the optic axis is fixed, not in position, but 
 in direction only. It has been already mentioned that the 
 line connecting the obtuse solid angles of the rhombohedron 
 of Iceland spar is the axis of the crystal. Now if we conceive 
 a crystallized mass of this substance to be subdivided into its 
 elementary molecules, which are of this form, the axis of 
 each of these molecules will be an optic axis. The optic axis 
 of the crystallized mass, therefore, is a direction in space 
 parallel to the axes of the elementary molecules, or equally 
 inclined to the three faces containing the obtuse solid 
 angle. 
 
 (86) All the phenomena of double refraction are symme- 
 trical round the optic axis. To see this, we have only to polish 
 an artificial face on the crystal, perpendicular to the optic axis, 
 and to mark the course of the refracted rays. We shall then 
 observe that, when the ray is incident perpendicularly on this 
 face, or in the direction of the axis, it undergoes no deviation 
 by refraction, and the ordinary and extraordinary rays coin- 
 cide ; that for every other incidence the ray is divided, the 
 refracted rays being both in the plane of incidence, and the 
 deviation of the extraordinary ray being less than that of the 
 ordinary. This deviation of the extraordinary ray, and there- 
 fore the ratio of the sines, is the same for all rays equally in- 
 clined to the axis, whatever be the azimuth of the plane of 
 incidence. But when the inclination of the ray to the optic 
 axis is varied, the ratio of the sines of incidence and refraction 
 
DOUBLE REFRACTION. 67 
 
 of the extraordinary ray likewise varies, the ratio diminishing 
 as the inclination increases, and being least when the ray is per- 
 pendicular to the axis. This least value of the ratio is called 
 the extraordinary index ; in Iceland spar it is 1/4S3. 
 
 In the preceding cases, the plane of incidence contains 
 the optic axis, and the extraordinary ray continues in that 
 plane. This is generally true under the same circumstances, 
 whatever be the refracting surface. To see this, we have only 
 to look obliquely through a rhomb of Iceland spar at a point 
 on a sheet of paper : the extraordinary image will be observed 
 to revolve round the other, as the rhomb is turned, and will 
 twice arrive in the plane of incidence, namely, when that 
 plane contains the optic axis. The same coincidence of the 
 two planes occurs also when the plane of incidence is perpen- 
 dicular to the optic axis ; but in this case the ratio of the 
 sines of incidence and refraction of the extraordinary ray is 
 constant, so that this ray then satisfies both the laws of or- 
 dinary refraction. This constant ratio is the extraordinary 
 index already referred to ; it is best determined by means of 
 a prism of the crystal having its refracting edge parallel to 
 the optic axis. 
 
 (87) The directions of the two refracted rays are given by 
 the following construction. 
 
 Let AC be the incident ray, 
 and OF the section of the sur- 
 face of the crystal made by the 
 plane of incidence. Let the 
 incident ray be produced any- 
 where to B, and let BF be 
 drawn perpendicular to it, 
 meeting the surface in F. 
 Let CD : CB : : sine of refraction ; sine of incidence of the 
 
68 DOUBLE REFRACTION. 
 
 ordinary ray ; and from the centre C, and with the radius CD, 
 let the sphere DOGr be described. Let the spheroid of revo- 
 lution GrE be described with the same centre, its axis of 
 revolution being in the direction of the optic axis of the 
 crystal, and equal to the diameter of the sphere, while the 
 other axis is greater in the ratio of the ordinary to the 
 extraordinary index. Now, if through F a line be drawn 
 perpendicular to the plane of the diagram, and through that 
 line there be drawn tangent planes, FO and FE, to the 
 sphere and spheroid, the lines CO and CE, drawn from the 
 centre to the points of contact, will be the directions of the 
 ordinary and extraordinary rays. This elegant construction 
 was given by Huygens. 
 
 For this construction Newton substituted another, with- 
 out stating the theoretical grounds on which he formed it, 
 or even advancing a single experiment in its confirmation. 
 In this unsatisfactory position the problem of double refrac- 
 tion was suffered to rest for nearly a century ; and it was 
 not until the period of the revival of physical optics, in the 
 hands of Young, that any new light was thrown upon the 
 question. Young was led by the theory of waves to assume 
 the truth of the law of Huygens ; and, at his instigation, 
 Wollaston undertook the experimental examination, which 
 recalled to it the attention of the scientific world, and ended 
 in its universal admission. The French Institute soon after 
 proposed the question of double refraction as the subject of 
 their prize essay, and the successful memoir of Malus left 
 no doubt remaining as to the accuracy of the Huygenian 
 law. 
 
 (88) It is plain, from the construction of Huygens, that 
 the point of contact of the tangent plane with the sphere, 0, 
 
 /~1T> 
 
 is in the plane BCF, and that the ratio, =r, which is the 
 
DOUBLE REFRACTION. 69 
 
 ratio of the sines of incidence and refraction, is constant. 
 Hence the ray CO observes both the laws of Descartes. On 
 the other hand, the point of contact with the ellipsoid, E, is 
 not, in general, in the plane of incidence, EOF ; and further, 
 
 PT> 
 
 the ratio is not constant. The extraordinary ray, there- 
 fore, does not in general follow either of the Cartesian laws. 
 There are certain cases, however, in which the extra- 
 ordinary ray observes one, or both, of these laws. The most 
 important of these is that in which the refracting surface 
 contains the axis of the crystal, and the plane of incidence is 
 perpendicular to the axis. In this case, the axis of the 
 crystal is a right line passing 
 through C, perpendicularly to 
 the plane of incidence ; and 
 the section of the ellipsoid by 
 that plane is its equator, and 
 is therefore a circle whose 
 radius is the greater axis of 
 the generating ellipse. The 
 tangent plane passing through F touches the ellipsoid in a 
 point of that circle ; and therefore the ray CE drawn to that 
 
 CiT) 
 
 point is in the plane of incidence. Again, ^, which is the 
 
 ratio of the sines of incidence and refraction of the extra- 
 ordinary ray, is a constant quantity ; and therefore the 
 second law of Descartes is also observed. 
 
 CE being the greatest value of the radius- vector of the 
 
 P"R 
 
 ellipsoid, the ratio -^ is, in this case, the least ratio of the 
 OJii 
 
 sines of incidence and refraction for the extraordinary ray. 
 This least value is the extraordinary index. 
 
 Let n and n f denote the indices of refraction of the ordinary 
 and extraordinary rays ; then 
 
70 DOUBLE REFRACTION. 
 
 CB sin i CB sin i 
 
 CO sinr CE sin/ 
 
 Let CO, which is equal to the axis of revolution of the ellip- 
 soid, = b ; and CE, the equatorial axis, = a. Then, if CB be 
 taken= 1, there is 
 
 n = - ; ri = -. 
 b a 
 
 The numerical values of n and n' 9 for Iceland spar, for 
 the less refrangible rays, are 
 
 = 1-653; ' = 1-484. 
 
 (89) In order to verify these conclusions, we have only to 
 polish a prism of Iceland spar so that the refracting edge 
 may be parallel to the axis of the crystal. The plane of in- 
 cidence will then be perpendicular to the axis, and the two 
 refracted rays will be found to observe both the laws of 
 Descartes. The following are the indices of refraction of the 
 principal rays of the spectrum for this substance, as deter- 
 mined by Eudberg : 
 
 
 
 Ray. . n. 
 
 B . 1-65308 . ' 1-48391 
 
 C . 1-65452 . 1-48455 
 
 D . 1-65850 , 1-48635 
 
 E "'," 1-66360 .. 1-48868 
 
 F . 1-66802 . 1-49075 
 
 G . 1-67617 . 1-49453 
 
 H . 1-68330 . 1-49780 
 
 (90) When the refracting surface contains the axis of 
 the crystal, and the plane of incidence passes through that 
 axis, the section of the ellipsoid by the plane of incidence 
 will be the ellipse MEM', whose lesser axis, MM 7 , is the 
 
DOUBLE REFRACTION. 
 
 71 
 
 diameter of the circle. In this case the plane of incidence 
 cuts the ellipsoid in a diame- 
 tral plane ; the points of con- 
 tact, and E, are both in that 
 plane ; and the extraordinary 
 ray observes the first law of 
 Descartes. Again, by the known 
 property of the ellipse, the points 
 of contact, E and 0, are on the 
 same right line, POE, perpendicular to the axis ; and are 
 to one another in the constant ratio of the axes of the ellipse. 
 Hence 
 
 But 
 
 OP 
 EP 
 
 _ 
 EP ~ a 
 
 tan OOP 
 
 tanr 7 
 
 tanECP tanr' 
 
 We have therefore the remarkable relation between the 
 angles of refraction of the two rays 
 
 tan r f b n' 
 tan r a n' 
 
 This relation has been verified experimentally by Malus ; 
 and it is thus proved that the section of the surface of the 
 extraordinary wave by any diametral plane is an ellipse- 
 
 The relation between the angles of incidence and refrac- 
 tion of the extraordinary ray in this case is easily deduced. 
 For 
 
 tan r = 
 
 sin r 
 
 sin i 
 
 V 1 - sin 9 r V ri* - sin 2 i 
 
 Wherefore 
 
 n' . . 
 
 . sm i 
 
 , n f n 
 
 tan r' = tan r = . 
 
 n V n* - sin 2 i 
 
72 DOUBLE REFRACTION 
 
 (91) We have seen that in Iceland spar the extraordi- 
 nary index is less than the ordinary, and that consequently 
 the extraordinary ray is refracted from the axis. This, how- 
 ever, is not universally true of all double-refracting crystals. 
 Biot discovered that there were many crystals in which the 
 extraordinary index was greater than the ordinary, and in 
 which, therefore, the extraordinary ray is refracted towards 
 the axis. Crystals of this kind he called attractive, while 
 those of the former were denominated repulsive. Among the 
 attractive, or (as they are sometimes called) positive crystals, 
 are quartz, ice, zircon ; the repulsive or negative class is far 
 more numerous, and includes, among others, Iceland spar, 
 sapphire, ruby, emerald, beryl, and tourmaline. 
 
 The Huygenian law applies to attractive as well as to re- 
 pulsive crystals, it being observed, that in the former case the 
 axis of revolution of the ellipsoid must be the greater axis of 
 the generating ellipse ; or, in other words, that the spheroid 
 is prolate instead of oblate. 
 
 (92) It has been hitherto assumed that there is but one 
 optic axis in every crystal, or one direction only along which 
 a ray will pass without division. It was reserved for Sir 
 David Brewster to discover that the greater number of crys- 
 tals possessed two optic axes. Among the most remarkable 
 of the crystals with two axes may be mentioned arragotiite, 
 mica, sulphate of barytes, sulphate of lime, topaz, and felspar. 
 The angles range in magnitude through the entire quadrant ; 
 and they accordingly afford a new and important criterion 
 for the distinction of mineral substances. 
 
 (93) It appears from the foregoing, that crystalline bodies 
 may be divided into three classes, with respect to their action 
 upon light, namely 
 
 I. Single-refracting crystal*. 
 
DOUBLE REFRACTION. 73 
 
 II. Uniaxal crystals, or those which have one axis of 
 double refraction. 
 
 III. Biaxal crystals, or those which have two such 
 axes. 
 
 Sir David Brewster has established a connexion between 
 these diversities of optical character and the varieties of crys- 
 talline form. He has shown that all the crystals of the first 
 class, i. e. all single-refracting crystals, belong to the tessular 
 system of Mohs ; that all uniaxal crystals belong either to 
 the rhombohedral or to the pyramidal system ; and that crys- 
 tals of the third class, or biaxal crystals, belong to one or 
 other of the prismatic systems. 
 
 These important relations bear, in a very close and defi- 
 nite manner, upon the proximate cause of double refraction. 
 It has been just mentioned, that the only crystals which do 
 not possess the property of double refraction are those belong- 
 ing to the tessular system, i. e. those whose fundamental form 
 is the cube. Now in this, and its derived forms, we can assign 
 three lines at right angles to one another, round which the 
 whole figure is symmetrical ; and we may, therefore, reason- 
 ably conclude that the density and elasticity of the crystal 
 is the same in each of these directions, and consequently the 
 same throughout. Again, crystals with one axis of double 
 refraction belong either to the rhombohedral, ortotliepramidal 
 system, systems whose fundamental forms are the rhombo- 
 hedron, and the straight pyramid. In each of these forms there 
 is one axis of figure, or one line round which the whole is 
 symmetrical : and we may, therefore, assume that the density 
 of the crystal is either greater or less in this direction than in 
 others, while it is equal in all directions at right angles to it. 
 The axis of form is, in this case, the axis of double refrac- 
 tion. Finally, in the oblique pyramid, which is the funda- 
 mental form of the prismatic systems, there is no one line, or 
 axis of figure, round which the whole is symmetrical ; and it 
 
74 DOUBLE REFRACTION. 
 
 is therefore probable that the density of the crystal is un- 
 equal in all the three directions. Such crystals are found to 
 have two optic axes. 
 
 It has been stated that in uniaxal crystals the optic axis 
 is also the axis of form. In biaxal crystals, it did not at first 
 appear that the optic axes were in any manner related to the 
 lines which bound the elementary crystal. Sir David Brew- 
 ster, however, ascertained that if two lines be taken, one bi- 
 secting the acute, and the other the obtuse angle contained 
 by the optic axes, these (together with a third line at right 
 angles to both) are closely connected with the primitive 
 form. 
 
 These relations between the optical properties of crystals 
 and their external forms are so close and intimate, that any 
 change (however produced) in one of them, is found to be 
 accompanied by a corresponding change in the other. Thus, 
 if the form of a crystal be altered by mechanical compression, 
 or change of temperature, its refracting properties undergo a 
 corresponding change. 
 
 (94) It was long supposed that one of the refracted rays, 
 in every crystal, followed the ordinary law of the sines, while 
 the other was refracted according to the Huygenian law. 
 But Fresnel has proved, both from theory and by experiment, 
 that this is not the gase, and that in biaxal crystals, both rays 
 are refracted in an extraordinary manner, and according to a 
 new law. It is, in fact, a consequence of his beautiful theory 
 of double refraction, that the form of the wave, which is 
 propagated in the interior of such a crystal, is neither a 
 sphere nor spheroid, as in uniaxal crystals, but a curved 
 surface of the fourth order. This surface is composed of two 
 sheets ; and if tangent planes be drawn to these, after the 
 same manner as to the sphere and spheroid in the Huyge- 
 nian law, the points of contact determine the directions of 
 
DOUBLE REFRACTION. 75 
 
 the two refracted rays. These more general laws of double 
 refraction will be more fully considered hereafter. 
 
 (95) We may now proceed to illustrate some of the more 
 remarkable effects of double refraction. 
 
 If a rhomboid of Iceland spar, or any other double-re- 
 fracting crystal, be placed close to a small object as, for 
 example, a black spot on a sheet of paper it will be ob- 
 served that one of the images is sensibly nearer than the 
 other ; and that the difference of their apparent distances 
 changes with the thickness of the crystal, and with the ob- 
 liquity of the ray. 
 
 This effect is easily accounted for. It is a well-known 
 principle of optics, that when an object is viewed through 
 a denser medium bounded by parallel planes as, for ex- 
 ample, a cube of glass the image is nearer to the surface 
 than the object ; the difference of their distances being to the 
 thickness of the medium, ag the difference of the sines of in- 
 cidence and refraction to the sine of incidence. This interval, 
 through which the image is made to approach, increases 
 therefore with the refractive power of the medium ; thus in 
 water it is one-fourth of the thickness, in glass one-third, 
 and so for other media. Now as double-refracting crystals 
 have two refractive indices, of different magnitudes, there 
 will be two images, at different distances from the surface. 
 In Iceland spar, the ordinary index is greater than the ex- 
 traordinary, and therefore the ordinary image is nearer than 
 the other. The reverse is the case in positive crystals, such 
 as quartz, in which the extraordinary index is the greater. 
 
 (96) The refractions being equal at the two parallel sur- 
 faces of the rhomb, whether the refraction be ordinary or ex- 
 traordinary, the two rays will emerge parallel to the incident 
 ray, and therefore parallel to one another ; and the distance 
 
76 DOUBLE REFRACTION. 
 
 between them will be proportional to the thickness of the 
 crystal. But if the surfaces be inclined, so as to form a prism, 
 the deviation of the two rays will be different, and they will 
 emerge inclined to one another ; consequently the separation 
 will increase with the distance. 
 
 Such a separation is of use in many experiments. In order 
 to render the divergence of the emergent pencils greatest, the 
 prism should be cut with its edge parallel to the optic axis ; 
 so that the refraction may take place in a plane perpendicular 
 to the axis. In this case the ordinary and extraordinary 
 refractions differ by the greatest amount, and therefore the 
 difference of the deviations jof the two pencils is greatest. A 
 double-refracting prism, so cut, is usually achromatized by 
 a prism of glass, with its refracting angle turned in the oppo- 
 site way. 
 
 A better arrangement has been suggested by Wollas- 
 ton. Two prisms of the same substance, and of equal refract- 
 ing angles, are cut in such a manner, that in one the refracting 
 edge is parallel to the optic axis, and in the other perpendicu- 
 lar to it. They are then united, with their refracting angles 
 turned in opposite directions, so as to form a parallelepiped. 
 The resulting separation of the two rays, in this compound 
 prism, is the sum of the separations produced by each sepa- 
 rately. Let 0j and e t be the deviations of the ordinary and 
 extraordinary rays produced by the first prism ; o 2 and e 2 the 
 corresponding angles for the second. On account of the po- 
 sition of the axes, the ray which is refracted in the ordinary 
 manner by the first prism will be refracted extraordinarily 
 by the second ; and its angle of deviation will be QI e.y. 
 In like manner, the ray which is refracted extraordinarily by 
 the first prism will be refracted ordinarily by the second ; and 
 its deviation will be e l - o 2 . Consequently the resulting an- 
 gular separation of the two rays will be 
 
DOUBLE REFRACTION. 77 
 
 By this duplication the weak double refraction of rock crystal 
 is rendered very sensible. 
 
 (97) A prism of this kind is employed in the double 
 image micrometer, an ingenious and valuable instrument in- 
 vented by Rochon. It consists of a telescope, in which a 
 prism, such as we have described, is introduced between the 
 object-glass and its principal focus ; and thus two images are 
 
 formed in the principal focus, whose interval is greater or 
 less, according to the distance of the prism from that point. 
 When the instrument is used, the prism is moved until the 
 two images appear in contact, and its distance from the focus 
 is then read on a graduated scale. The two angles in this 
 case having the same subtense, the visual angle of the object 
 is to the deviation produced by the prism, as the distance of 
 the prism from the focus is to the focal length. Now the di- 
 vergence of the two rays is constant for a given prism, and 
 may be determined either by calculation or experiment ; 
 consequently, the visual angle is deduced from the preceding 
 proportion. By this instrument Arago has determined the 
 apparent diameters of the planets with great precision. 
 
 The same instrument has been also employed in war, to 
 determine the distance of an inapproachable object. Thus, if 
 it be required to ascertain the distance of the walls of a be- 
 sieged town, in order to know whether they are within the 
 range of shot, it is only necessary to measure by this instru- 
 ment the angle subtended by any object there whose 
 height is known approximately. The height of the object, 
 divided by the tangent of the angle, is the distance required. 
 
CHAPTEE VI. 
 
 * 
 INTERFERENCE OF LIGHT. 
 
 (98) HAVING considered the mode of propagation of a 
 luminous wave, and the modifications which it undergoes on 
 encountering the surface of a new medium, we may now pro- 
 ceed to inquire what will be the effect, when two series of 
 waves are propagated simultaneously from two near luminous 
 origins. 
 
 It is obvious that when two waves one proceeding from 
 each source arrive at any instant at the same point of space, 
 the particle of ether there will be thrown into vibration by 
 both ; and we are to consider what will be the result of this 
 compound vibration. Now, it is demonstrated by analysis, 
 that when two small vibrations are communicated at the same 
 time to a material point, each of them will subsist indepen- 
 dently of the other ; and the motion of the point will, in 
 consequence, be the resultant of the motions due to each vi- 
 bration considered separately. This principle is denominated 
 the superposition of small motions. Its nature may be made 
 clear by a simple instance. 
 
 Let a pendulous body receive an impulse in any plane 
 passing through the point of suspension : it will then, of 
 course, vibrate in that plane. Now, at the lowest point of 
 the arc of vibration, let a second impulse be given to the 
 moving body, in a direction perpendicular to the plane in 
 which it already vibrates. This impulse, if communicated to 
 the 'body at rest, would cause it to vibrate in a plane at right 
 angles to the former, and through an arc depending on the 
 magnitude of the impulse. Now it will be found, on trial, 
 
INTERFERENCE OF LIGHT. 79 
 
 that the distance of the body from the vertical, measured in 
 either of these planes, is the same at any instant as if the 
 other vibration did not exist ; so that each vibration subsists 
 independently of the other, and the result will be a compound 
 elliptical vibration. We have here supposed the coexisting 
 vibrations to take place in separate planes, in order that 
 their independence may be more distinctly recognised. When 
 the two vibrations are in the same plane? it is obvious that 
 the resulting vibration will be also in that plane ; and that 
 its amplitude will be the sum of the amplitudes of the com- 
 ponent vibrations when their directions conspire, and their 
 difference when they are opposed. 
 
 (99) Let us transfer this to the case of Light : Let us 
 suppose that two sets of waves start at the same time from two 
 near luminous origins (which, for simplicity, we shall assume 
 to be of equal intensity), and that a distant particle of ether 
 is thrown into vibration by both at the same time. Then, 
 supposing that these two vibrations are performed in the 
 same plane, it follows from what has been said, that, when 
 their directions conspire, they will be added together, and 
 the resulting space of vibration will be double of either ; and 
 that, on the contrary, they will counteract one another, and 
 the resulting vibration will be reduced to nothing, when their 
 directions are opposed. 
 
 It is evident, further, that the directions of the vibrations 
 will conspire, and therefore 
 the space of vibration be AV^N 
 doubled, when the two waves 
 arrive in the same phase ; A f "^Vj 
 and that, on the contrary, 
 their directions will be op- A v __ / 
 
 posed, and the resulting vi- 
 
 bration reduced to nothing, when they arrive in opposite 
 
* 
 
 80 INTERFERENCE OF LTGIIT. 
 
 phases. Let the waving lines AB and A'B', or AB and 
 A /X B /X , represent the two undulations, the distance of any 
 particle from its state of rest being represented by the ordi- 
 nate, or perpendicular, at the corresponding point of the 
 horizontal or mean line. Then, if the undulation A'B' be 
 superposed upon AB, the corresponding points of each being 
 in the same phase, it is evident that the distances by which 
 the particle at any point is removed from its state of rest by 
 each, mn and m'n', will be added together, and the space of 
 vibration doubled. "Whereas, if the undulations A X/ B /X and 
 AB, whose corresponding points are in opposite phases, be 
 superposed, the distances from the position of rest, mn and 
 ni"n", lie on opposite sides of the mean line, and when added 
 together destroy one another. Thus the space of vibration is 
 doubled, when the waves arrive at the same point in the 
 same phase : it is annihilated, when they arrive in opposite 
 phases. Now the intensity of the light is as the square of the 
 amplitude of vibration ; the intensity, therefore, is quadrupled 
 in the former case, and reduced to nothing in the latter. 
 
 (100) Before we proceed to examine more particularly this 
 indication of theory, we may observe that it is altogether 
 analogous to what is known to take place in other cases of 
 vibratory motion. If two waves of water arrive at the same 
 point at the same instant, in such a manner that the crest of 
 one wave coincides with that of the other, their effects will 
 be added together, and the water at that point will be raised 
 into a wave, whose height is the sum of the heights of the 
 conspiring waves. If, on the other hand, the crest of one 
 wave coincides with the sinus, or depression of the other, the 
 height of the resultant wave will be the difference of the 
 components ; and, when these are equal, the resultant wave 
 will entirely disappear. 
 
 The peculiarity of the tides in the port of Batsha fur- 
 
INTERFERENCE OF LIGHT. 81 
 
 nishes a striking instance of these effects. The tidal wave 
 reaches this port by two distinct channels, which are so un- 
 equal in length, that the time of arrival by one passage is 
 exactly six hours longer than by the other. It follows from 
 this that when the crest of the tidal wave, or the high water, 
 reaches the port by one channel, it is met by the low water 
 coming through the other ; and when these opposite effects 
 are also equal, they completely neutralize each other. At 
 particular seasons, therefore, when the morning and evening 
 tides are equal, there is no tide whatever in the port of Batsha ; 
 while at other seasons there is but one tide in the day, wiiose 
 height is the difference of the heights of the ordinary morn- 
 ing and evening tides. 
 
 Analogous phenomena take place in sound, and produce 
 the coincidences or beats in music. These effects occur when 
 the condensed part of the aerial pulse, arising from one origin 
 of sound, coincides with the rarified part of that proceeding 
 from the other. They are often heard during the playing of 
 a large organ, and give rise to the swelling and falling sounds 
 which are heard, especially among the lower notes of the 
 instrument. 
 
 (101) That two lights, then, should produce darkness, is a 
 result of the same kind as that two sounds should cause silence, 
 or that two waves should make a dead level. But we are not 
 left to analogy alone for the proof of this remarkable conse- 
 quence of the wave-theory of light. The phenomenon itself 
 has been established by the most direct and convincing ex-, 
 periments ; and we shall soon see that it is observed in a 
 multitude of cases where its existence was at first little sus- 
 pected. 
 
 This important law now known under the name of the 
 interference of light was for the first time distinctly stated 
 and established by Young, although some facts connected 
 
82 INTERFERENCE OF LIGHT. 
 
 with it were known to Grrimaldi. The latter observer had 
 even explicitly asserted that " an illuminated body may be 
 rendered darker by the addition of light" and adduced a sim- 
 ple experiment in proof of it. Grimaldi's experiment was as 
 follows. Let the Sun's light be admitted into a darkened 
 chamber through two small and equal apertures of a circular 
 form. Two diverging cones of light will be thus produced ; 
 and each of these cones will be surrounded by a penumbra in 
 which the illumination is only partial. Now let these two 
 beams be received on a screen at some distance, where the 
 penumbras of the two cones overlap. It will be then ob- 
 served, that although the greater part of this doubly illu- 
 minated space is brighter than the penumbra of one cone alone, 
 yet the boundaries of the overlapping portions are much 
 darker than the other parts of the penumbras which do not 
 overlap ; and if one of the beams be intercepted by an ob- 
 stacle, this dark part will recover the brightness of the rest. 
 Thus darkness may be produced by adding light ; and, on the 
 other hand, by withdrawing a portion of the light we may 
 augment the illumination. 
 
 (102) This interesting experiment assumed a more distinct 
 and decisive character in the hands of Young. If the two 
 apertures be reduced to a very small size, and brought close 
 together, and if the original light be homogeneous, we shall 
 observe a series of alternate bright and dark bands, formed at 
 those points where the waves proceeding from the two origins 
 conspire, or are opposed. That these alternations of light 
 and darkness are caused by the mutual action of the two 
 beams, is proved by the fact, that if one of the beams be in- 
 tercepted, the whole system of bands will disappear, and the 
 light which remains become of uniform intensity. By with- 
 drawing one of the lights, then, the dark intervals recover 
 their brightness ; so that darkness, in this case, must have 
 been produced by the action of one light on the other. 
 
 In this and other experiments of interference, the two 
 
INTERFERENCE OF LIGHT. 83 
 
 lights must emanate from a common origin, the vibrations 
 of two independent sources of light not being synchronous. 
 The simplest mode of effecting this is to condense the Sun's 
 rays into a small luminous point by means of a convex lens 
 of short focus.* 
 
 (103) We shall best understand the circumstances of this 
 phenomenon, by considering what takes place in another 
 more familiar case of interference. If two stones be flung at 
 the same instant into a pool of stagnant water, a series of 
 circular waves will be propagated from each of the two cen- 
 tres of disturbance ; and where these two sets of waves cross, 
 they will produce effects similar to those we have been de- 
 scribing in the case of light. Where the crest of one wave 
 falls upon the crest of another, they will be added together, 
 and form a higher crest, or ridge, on the surface. And, on 
 the contrary, where the crest of one wave meets the hollow, 
 or sinus, of another, they will counteract one another's effects, 
 and the water will stand at that point at its original level, as 
 if undisturbed. 
 
 It is obvious that there will 
 be several sets of consecutive 
 points of each class, or several 
 lines of double disturbance and 
 no disturbance. One line of 
 double disturbance, AA, will be 
 produced by the meeting of waves 
 equidistant from the two centres ; 
 
 * A convenient mode of performing the experiment of Young has been 
 employed by M. Billet. A convex lens, of tolerably long focal length, is cut into 
 two halves, which are reunited in such a manner that their optical centres are 
 slightly separated. Under these circumstances the rays diverging from a distant 
 point, and falling on the lens, will converge to two near points, and will be in a 
 condition to interfere after divergence. The intensity of the light is much 
 greater than that emanating from two small apertures. 
 
 G 2 
 
84 INTERFERENCE OF LIGHT. 
 
 as the first of one with the first of the other, the second of 
 one with the second of the other, &c. This line is ne- 
 cessarily straight. On either side of this there will be 
 a line, BB, B X B X , consisting of those points where the first 
 wave from one origin encounters the second from the other, 
 the second from one the third from the other, of all those 
 points, in short, whose distances from the two centres differ 
 by the length of a single wave. The next pair of lines, CO, 
 C'C', consist of those points whose distances from the two centres 
 differ by the length of two waves ; and so on. All these lines 
 are hyperbolas, and on all of them the disturbance is doubled, 
 and an elevated ridge raised on the surface. But there are 
 likewise intermediate lines, composed of those points whose 
 distances from the two centres differ by half a wave, by a icave 
 and half, by two waves and half, &c. On all these lines, the 
 crest of the wave from one origin meets the hollow of a wave 
 from the other ; and these, therefore, are the lines of no dis- 
 turbance. They are evidently hyperbolas like the former. 
 
 All that has been now said applies strictly to the phe- 
 nomena of light, in the aspect under which they are pre- 
 sented by the wave-theory. In the same medium the waves 
 of any given length are propagated with a constant velocity. 
 When therefore two series of waves of equal length diverge 
 at the same time from two centres, they will arrive at the 
 same point in the same phase, provided that the lengths of 
 the paths traversed are equal, or differ by any whole number 
 of undulations. They meet in opposite phases, on the other 
 hand, when the lengths of their paths differ by half a wave, 
 or by any odd multiple of half a wave. The central bright 
 band, then, is formed at those points where the distances tra- 
 versed are equal. The next bright band on either side is pro- 
 duced where the distances traversed differ by the length of 
 one entire wave ; the succeeding pair where the distances differ 
 by two whole waves ; and so on. In the same manner, the 
 
INTERFERENCE OF LIGHT. 
 
 85 
 
 first dark band is produced on either side of the central bright 
 one, and at points for which the distances traversed differ by 
 the length of half a wave ; the second pair of dark bands 
 where these distances differ by one wave and half; and 
 BO on. 
 
 (104) In Young's experiment, if the light diverging from 
 the two apertures and O x , 
 be received on a screen, AD, 
 it is found that the central 
 bright band is formed at the 
 point A, where the screen 
 is intersected by the line 
 PA, which bisects the line 00' and is perpendicular to it. 
 The central band, therefore, is formed where the paths traversed 
 by the two pencils are equal. There will be a dark band on 
 either side of the central bright one, and, beyond these, a pair 
 of bright bands. The difference of the distances of one of these 
 from the two apertures, BO' - BO, is a constant quantity, what- 
 ever be the position of the screen ; this difference is the length 
 of a wave. Beyond these is a second pair of bright bands, the 
 difference of whose distances from the two centres, CO X - CO, is 
 double of the preceding, or equal to two whole waves ; and in 
 like manner, the difference of the lengths of the paths, at the 
 place of each succeeding bright band, is an exact multiple of 
 the first difference, or of the length of a wave. 
 
 Similarly, for the intermediate dark bands, the difference 
 in the lengths of the paths, where the first pair is formed, is 
 half the difference, BO X - BO, or half the length of a wave. The 
 differences of the paths, at the place of each succeeding pair 
 of dark bands, are in like manner intermediate to the cor- 
 responding differences for the bright bands on either side, and 
 are odd multiples of half a wave. 
 
 The difference of the distances from the two apertures 
 
86 INTERFERENCE OF LIGHT. 
 
 being constant for the successive points of the same band, it 
 follows that these points must form an hyperbola, whose foci 
 coincide with the two apertures. It will be easily seen that 
 the curvature of these hyperbolic lines is very small, except 
 close to their vertices ; and that we may, without sensible 
 error, consider them as coincident with their asymptots. 
 
 As the distance of the fringe varies with the length of 
 the wave, it follows that when white light is employed, the 
 fringes will be coloured bands, which, after a few successions, 
 are effaced by the overlapping of the fringes of different 
 orders. "When the light, although coloured, is not perfectly 
 homogeneous, the same effect of the obliteration of the fringes 
 is produced after a greater succession. It has been found 
 possible to purify solar light so far, that 8000 fringes have 
 been counted. 
 
 (105) It is easy to determine the positions of the bands, as 
 dependent on the interval of the apertures, and on the dis- 
 tance of the screen. 
 
 The place of any bright or 
 dark band, m, is determined 
 by the condition that the dif- 
 ference of its distances from 
 the two apertures, m(5' - mO, 
 is an integer multiple of the length of half a wave. Now, 
 drawing the lines On, V, parallel to PA, and denoting the 
 distance AP by , the interval of the apertures 00' by c, and 
 the distance Am by #, the right-angled triangles Qmn, Q'mri, 
 give 
 
 Om = 
 
 Q'm = 
 
 the distance b being very great in comparison with x and c. 
 
INTERFERENCE OF LIGHT. 87 
 
 Hence 
 
 26 6 
 
 But this difference is equal to n ^, A being the length of a 
 
 wave ; we have, therefore, 
 
 nb\ 
 
 in which the even values of n correspond to the places of the 
 bright bands, and the odd values to those of the dark ones. 
 
 The preceding formula enables us to compute the length 
 of a wave of light, when the distances b, c, and x have been 
 determined by accurate measurement. It has been found in 
 this manner that the length of a wave is -0000266 of an inch 
 for the extreme red rays ; -0000167 for the extreme violet ; 
 and -0000203, or about the 30000 of an incil > for ^e mean 
 rays of the spectrum. 
 
 (106) But though the principle of Interference seemed to 
 be established by the experiments and reasonings of Young, 
 it was not freed from all question. It might be supposed 
 that the light passing by the edges of the apertures, in the 
 experiment last described, underwent modifications of some 
 kind or other which produced the observed effects. It was, 
 therefore, of importance to show that these effects were 
 wholly independent of apertures or edges ; and that any two 
 rays proceeding from the same luminous origin, and meeting 
 under a small obliquity, will interfere in the manner already 
 described, whatever be the attending circumstances. This 
 has been done by Fresnel ; and the experiment, which he de- 
 vised for the purpose, has been justly ranked among the 
 most important and instructive in the whole range of Physi- 
 cal Optics. 
 
88 INTERFERENCE OF LIGHT. 
 
 Two plane mirrors are placed so as to meet at a very 
 obtuse angle. A beam of light diverging from the focus of a 
 lens is suffered to fall upon them ; and there will be therefore 
 two reflected beams, whose directions are inclined at a very 
 small angle. Here, then, are two beams diverging from the 
 same luminous origin, separated simply by reflexion at plane 
 surfaces, without the intervention of edges, or of anything 
 accidental which can be regarded as contributing to the 
 result. These beams, however, still interfere, and produce a 
 succession of alternate bright and dark bands, in the manner 
 already explained. In order to satisfy ourselves that these 
 bands are in fact produced by the mutual action of the two 
 beams, we have only to intercept one of them, by covering 
 one of the mirrors, and the whole system instantly vanishes. 
 
 Let Qn and Qn' 
 
 represent the sec- / 
 
 tions of the two Op- ::: - --A- 
 
 mirrors, which we 
 
 shall suppose to be '" :: - ^ 
 
 perpendicular to the 
 plane of the dia- 
 gram ; and let k be the luminous origin, or the focus of the 
 lens in which the Sun's rays are concentred. Then taking 
 the points and O 7 at equal distances on opposite sides of 
 the mirrors, these points will be the foci of the two reflected 
 pencils, or the points of divergence of the two beams. Now 
 it is found, in the first place, that the bands are parallel to 
 the line of intersection of the two mirrors ; secondly, that 
 they are symmetrically placed on either side of a plane pass- 
 ing through this intersection, and through the point P, which 
 bisects the interval between the two foci and O 7 ; and 
 thirdly, that in proceeding from the mirrors, they are propa- 
 gated in hyperbolas, whose foci are and 0', and whose 
 common centre is P. 
 
INTERFERENCE OF LIGHT. 89 
 
 These results are in exact accordance with theory. 
 In fact, since On = nk, and V = n'k, the difference of the 
 paths traversed by the reflected rays, 'knm and kn'm, when 
 they meet at m, is the same as if they had reached that point 
 diverging directly from the points and O x . All, then, that 
 has been said respecting the interference of the pencils di- 
 verging from two near luminous origins, will apply to this 
 case. Since &Q = OQ = O'Q, the line QP, which bisects the 
 line 00', is also perpendicular to it, and any point of it, as 
 A, is equidistant from and 0'. The bands, therefore, are 
 symmetrically situated with respect to this line ; and the dis- 
 tance, Aw?, of the band of any order from the central band is 
 
 wAAP 
 
 equal to -QQT-. 
 
 This distance is easily expressed in terms of given quan- 
 tities. For PQ = OQ x cos OQP = A-Q x cos OQP ; and 
 00' = 20P = 2&Q x sin OQP. But since the angles QO, 
 &Q0 7 , are bisected by the lines Qw and Q, x , it is easy to see 
 that the angle OQP (or the half of the angle OQO') is equal 
 to the inclination of the mirrors. If then this inclination be 
 denoted by , and the distances &Q and QA by a and b, we 
 have 
 
 00' = 2a sin e, AP = a cos + b <= a + b ; q. p. ; 
 
 and therefore the distance of the band of the n th order from 
 the centre is expressed by the formula 
 
 (a + b) n\ 
 2a sin c 
 
 The measurement of the distances is most conveniently 
 made by means of a lens, having in its focus a finely divided 
 scale cut with the diamond upon a piece of parallel glass. 
 The fringes so measured are those formed in the focal plane 
 of the lens. 
 
90 INTERFERENCE OF LIGHT. 
 
 (107) The phenomenon of interference is displayed in a 
 striking manner by the mutual action of direct and reflected 
 light ; and the experiment in this form is more manageable 
 than that of Fresnel. We have only to take a piece of plate 
 glass, or a metallic reflector, and to place it in such a posi- 
 tion that the rays diverging from the luminous origin shall be 
 reflected at an angle of nearly 90. A screen placed on the 
 other side of the mirror will receive both the direct and re- 
 flected pencils ; and as they meet under a small angle, and 
 have traversed paths differing by a small amount, they are in 
 a condition to interfere. The phenomenon is better seen, 
 however, by receiving the two pencils upon a lens, behind 
 which the eye is placed.* It will be readily seen that the sys- 
 tem of bands formed in this manner is but half of that pro- 
 duced in Fresnel's experiment. 
 
 (108) There is yet another mode of studying the funda- 
 mental phenomenon of interference, which is in some respects 
 more convenient than any of the former. It is obvious that 
 the original beam may be separated by refraction, as well as 
 reflexion ; and if the inclination of the two refracted pencils 
 be small, similar results will take place. For this purpose it 
 is only necessary to procure a prism with a very obtuse angle, 
 and to allow the beam of light to fall perpendicularly on the 
 opposite face. It is evident that this beam will be differ- 
 ently refracted, at emergence, by the two faces which con- 
 tain the obtuse angle ; and that it will thus be divided into 
 two beams, which will be slightly inclined. These beams 
 then proceed from one common origin, and meet under a 
 small obliquity, and therefore fulfil all the conditions neces- 
 
 * The details of this experiment are fully described in a Paper by the 
 Author, published in the seventeenth volume of the Transactions of the Royal 
 Irish Academy. 
 
INTERFERENCE OF LIGHT. 91 
 
 sary for their interference. It is found, accordingly, that a 
 series of alternate bright and dark bands is formed parallel 
 to the edge of the prism. 
 
 (109) It will be evident, from what has been said, that 
 the central fringe produced by the interference of two pencils 
 is the locus of those points at which they arrive in the same 
 time; and, accordingly, when neither of the pencils has ex- 
 perienced any interruption in its progress, the points of that 
 fringe will be equally distant from the two luminous origins. 
 The case is altered, however, if we interpose a thin plate of a- 
 denser medium in the path of one of the interfering rays. 
 If the light is retarded in the denser medium, it is obvious 
 that the points reached in the same time will no longer be 
 equally distant from the two centres, but will be nearer to- 
 that whose light has undergone the retardation. The reverse 
 will be the case if the light is accelerated in the interposed 
 plate ; so that the central fringe, and the whole system, will 
 be shifted towards the side of the interposed plate in the 
 former case, and from it in the latter. Here then we have a 
 complete experimentum crucis, by which to decide between the 
 theory of emission and that of waves; and its result, as 
 we have already stated, is conclusive in favour of the wave- 
 theory. 
 
 It is easy -to calculate the relation between the thickness 
 and refractive index of the plate, and the displacement of the 
 fringes. Let a and a' denote the distances of the displaced 
 central fringe from the two origins ; and let e be the thick- 
 ness of the plate interposed in the path of the pencil diverg- 
 ing from the first. Then as the times of traversing these two 
 distances are the same, we have 
 
 a-e e a' 
 v v f v ' 
 
92 INTERFERENCE OF LIGHT. 
 
 in which v and v ' are the velocities of light in air, and in the 
 plate, respectively. Hence 
 
 If the displacement be equal to the breadth of n fringes, we 
 
 qj 
 
 have a' - a = n\ ; and since = /*, the refractive index of the 
 
 plate, 
 
 e (n - 1) = n\. 
 
 The amount of the displacement of the fringes, in 
 this important experiment, depends on the thickness of the 
 interposed plate, and on its refractive index ; so that any one 
 of these quantities will be determined when the other two are 
 known. Accordingly, by observing the displacement of the 
 fringes produced by a plate of known thickness, the refractive 
 index of the plate is found. Arago and Fresnel have em- 
 ployed this method to determine the refractive powers of the 
 gases. The method is susceptible of very great precision. 
 By observing the position of the fringes formed by two rays, 
 one of which has passed through air, and the other through 
 a vacuum, Arago has shown that the minutest changes in 
 the refractive power of the air may be observed such, for 
 example, as would arise from a variation of temperature 
 amounting to ^th of a degree centigrade. By the same 
 method it was ascertained that dry air was more refractive 
 than air saturated with moisture, the difference amounting, 
 very nearly, to the millionth of the refractive index. 
 
 In connexion with these results, Arago has shown, that 
 the scintillation of the stars is a phenomenon of interference, 
 due to changes in the refractive powers of portions of the 
 atmosphere, through which different portions of light reach 
 the eye. 
 
INTERFERENCE OF LIGHT. 93 
 
 (110) We have hitherto confined our attention to the two 
 most important cases of coexisting vibrations those, namely, 
 in which they are in complete accordance, or are completely 
 opposed. There is little difficulty in calculating the result, 
 when the coexisting vibrations are in any intermediate stages 
 of the vibratory movement. 
 
 The general value of the displacement of an ethereal mole- 
 cule, produced by vibratory movement, is given in the equa- 
 tion of (18). If in this equation we make, for abridgment, 
 
 %TT 2lT 
 
 vt = 0, and -r- x - a, 
 A A 
 
 it becomes 
 
 y = a sin (0 - a). 
 
 The coefficient a, in this equation, is the amplitude of the 
 displacement, and the angle, a, is the phase. Now let two 
 parallel vibratory movements having the same period, but 
 differing in amplitude and phase, be communicated simulta- 
 neously to the molecule, and let their amplitudes be denoted 
 by i and a z , and their phases by cti and a 2 . The two dis- 
 placements are 
 
 y l = a, sin (0 - a,) ; y 2 = 2 sin (0 - a 2 ). 
 
 Now by the principle of the coexistence of small motions, 
 these will be added together ; and, in order that they may 
 compound a single vibration similar to the components, we 
 must have 
 
 V = y\ + y* = A sin (9 - p). 
 
 substituting for y\ and y 2 their values, and developing, the 
 resulting equation will be fulfilled independently of 0, pro- 
 vided that 
 
 A sin p = #1 sin a! + a^ sin a 3 ; 
 A cos p = (ii cos ai + #3 cos a 3 . 
 
94 INTERFERENCE OF LIGHT. 
 
 (Ill) These two equations contain all the laws of the 
 compound movement. Squaring and adding 
 
 A* = di* + 2aiCt z cos (ai - a 2 ) + aj. 
 
 And the amplitude of the resulting vibration is, conse- 
 quently, equal to the diagonal of a parallelogram whose sides 
 are the amplitudes of the component vibrations, and con- 
 tained angle the difference of their phases. When the am- 
 plitudes of the component vibrations are equal, or a* = a\, 
 
 A* = 2#i 2 (1 + COS (aj - a 2 ) j = 4#! 2 cos 2 -^ ( - ; 
 whence 
 
 . ~ Ct] Cf 2 
 
 A = Za\ cos JT . 
 
 Again, if we divide the former of the two equations of 
 condition by the latter, we have 
 
 a\ sin ai + # 2 sin a.? 
 
 tan p = ; 
 
 i cos cti -f 6/ 2 cos a 2 
 
 which gives the phase of the resulting vibration. This 
 equation is reducible to the form 
 
 a\ sin (p - 01) + 2 sin (p - a 2 ) = ; 
 
 from which it follows that the differences between the phase 
 of the resultant vibration, and the phases of the two compo- 
 nents, are represented by the angles made by the diagonal 
 of the parallelogram with the two sides. Accordingly, pa- 
 rallel vibrations are compounded by the same rules as two 
 forces meeting in a point. 
 
 The amplitude of the resultant vibration is a max- 
 imum, and equal to the sum of the amplitudes of the compo- 
 nent vibrations, when cos (ai - a 2 ) = + 1, or 
 
 ai 02 = 2mr. 
 
INTERFERENCE OF LIGHT. 95 
 
 It is a minimum, and equal to the difference of the amplitudes 
 
 of the components, when cos (ai - o 2 ) = - 1, or 
 
 / 
 a, - 02 = (2n + 1) TT. 
 
 Now a i - a 2 = 2ir I l ^ 2 ). Therefore the intensity of the 
 \ A / 
 
 resulting light is a maximum, when 
 Xi - x. 2 = wA, 
 
 or the difference of the lengths of the paths traversed by the 
 two waves is a multiple of the length of a wave. It is a 
 minimum, when 
 
 #1 - ^2 = (n + J) X, 
 
 or the difference an odd multiple of half a wave length. 
 When ai - a 2 = (2n + 1) -, cos (c^ - a 2 ) = 0, and 
 
 and the intensity of the resulting light is the sum of the in- 
 tensities of the components. 
 
 (112) The principle of interference furnishes the complete 
 answer to the difficulty suggested by Newton, and shows in 
 what manner the rectilinear propagation of light is reconciled 
 to the wave-theory. It had been objected, that if light con- 
 sisted in the undulations of an elastic fluid, it should diverge 
 in every direction from each new centre, and so bend round 
 interposed obstacles, and obliterate all shadow. To this the 
 reply is, that light does diverge in every direction from each 
 new centre, that it does bend round interposed obstacles ; 
 but that shadows notwithstanding exist, because the several 
 portions of this laterally-diverging light destroy one another 
 by interference, and no effect is produced, except by those parts 
 of the wave which are in the right line joining the luminous 
 origin and the eye. 
 
96 
 
 INTERFERENCE OF LIGHT. 
 
 To see this, let a bed represent a portion of a spherical wave ; 
 and let be the place 
 of the eye, and Oa the 
 line drawn from it to 
 the luminous centre. 
 Commencing from the 
 point a, let portions 
 ab, be, cd, &c., be 
 taken, such that the 
 differences of the dis- 
 tances of their extremities from the point shall be the same 
 for all, and equal to half a wave. Now we may suppose all 
 these portions of the grand wave to be so many centres of 
 disturbance ; and it is obvious that the secondary waves, sent 
 by each pair of consecutive portions to the eye, are in com- 
 plete discordance, and should wholly destroy one another if 
 their intensities were equal. It is easy to see that this is the 
 case with respect to portions, as /</, gh, which are remote from 
 the point a. For the magnitudes of the waves sent by the 
 several portions to any point depend first, on the magni- 
 tudes of these portions themselves, and secondly, on the 
 angles which the line drawn from them to that point makes 
 with the front of the wave. As respects the former, it is evi- 
 dent that ab is greater thn be, be than cd, and so on ; but 
 these differences continually diminish, and the magnitudes of 
 the consecutive portions approach indefinitely to equality, as 
 they recede from the point a. The same is true of the obli- 
 quities. Hence, the portions of the wave, fg, gh, remote from 
 the point , destroy one another's effects, and the effect 
 on the eye, or on a screen at 0, will be entirely due to those 
 parts of the grand wave which are in the neighbourhood of 
 the line connecting that point with the luminous origin. 
 
 Of these parts ab produces the greatest effect, being both 
 the largest and the least oblique. The effect of the neigh- 
 
INTERFERENCE OF LIGHT. 97 
 
 bouring portions is, however, very sensible, and we shall 
 have occasion hereafter to study some important phenomena 
 to which they give rise. In the meantime, one remarkable 
 consequence of this explanation is obvious namely, that 
 if the alternate portions be, de, &c., whose effects are negative, 
 be stopped, the total effect will be augmented, and the result- 
 ing light increased by intercepting a portion of the wave. 
 We shall see hereafter that this startling conclusion is con- 
 firmed by experiment. 
 
CHAPTEE VII. 
 
 DIFFRACTION. 
 
 (113) IT has been shown to be a result of the wave-theory, 
 that the intensity of the light which encounters an obstacle 
 must diminish rapidly within the edge of the geometric 
 shadow. It now remains to consider the other phenomena 
 which arise under these circumstances ; and it will be found 
 that the same theory affords the most complete account, not 
 only of their general characters, but even of their most minute 
 details. 
 
 In order to understand the theory of shadows, it is neces- 
 sary to investigate their laws in the simple case in which 
 the magnitude of the luminous body is reduced to a point. 
 The effects thus presented were first observed, and in some 
 degree explained, by Grimaldi ; and they have been since 
 studied, as a separate branch of Optical Science, under the 
 title of diffraction or inflexion. 
 
 Grimaldi found, that when a small opaque body was 
 placed in the cone of light, admitted into a dark chamber 
 through a very small aperture, its shadow was much larger 
 than its geometric projection; so that the light suffered 
 some deviation from the rectilinear course in passing by the 
 edge. On observing these shadows more attentively, he 
 found that they were bordered with three iris-coloured 
 fringes, which decreased in breadth and intensity in the 
 order of their distances from the shadow, and which preserved 
 the same distance from the edge throughout its entire extent, 
 unless where the body terminated in a sharp angle. Similar 
 
&*r * 
 
 ftri.iv 
 
 DIFFRACTION. 
 
 99 
 
 fringes were observed, under favourable circumstances, within 
 the shadows of narrow bodies. 
 
 The phenomena of diffraction were subsequently examined 
 by Hooke and by Newton ; and, lastly, in the hands of 
 Young and Fresnel, they have been forced to furnish evi- 
 dence in favour of the wave- theory, which few who impar- 
 tially examine it can withstand. We shall first describe the 
 most important of these phenomena, and afterwards examine 
 them in their bearing upon the two theories. 
 
 (114) The most obvious of these phenomena are the mo- 
 difications which light undergoes in passing by the edge of 
 an obstacle of any kind. 
 
 Let a beam of homogeneous light, entering a dark cham- 
 ber, fall on a lens of short focal length, MN, by which it is 
 
 brought to a focus at 0, and thence diverges. Let an ob- 
 stacle, PP', be placed in the diverging beam, and let the 
 shadow which it casts be received upon a sheet of white paper 
 at Q, or on a piece of roughened glass. "We shall then ob- 
 serve the following phenomena : 
 
 I. The line OPQ, which is the boundary of the geometric 
 shadow, is not the actual boundary of light and shade. 
 
 II. The space below this line, Q,S, is not absolutely dark, 
 but is enlightened by a faint light, which -extends to a sen- 
 sible distance within the geometric shadow, and gradually 
 fades away as it recedes from the edge of this shadow at Q,. 
 
 III. On the other side of the boundary of the geometric 
 
100 DIFFRACTION. 
 
 shadow, at QR, the paper is not uniformly illuminated by the 
 diverging beam, but is observed to be covered with a series 
 of alternate bright and dark bands, which are parallel to the 
 edge of the shadow. The distances of these fringes inter se, 
 and from the edge of the shadow, vary with the position of 
 the screen, and diminish indefinitely as the screen is brought 
 near the obstacle. These fringes succeed one another for 
 many alternations, becoming, however, less marked as the 
 distance from the edge of the geometric shadow increases, 
 until at length they are wholly obliterated and lost. They 
 preserve the same distances from the shadow in all parts, 
 except only where the edge of the body forms a sharp 
 angle.* 
 
 IY. The dimensions of the fringes vary with the colour 
 of the light ; being broadest in red light, narrowest in violet 
 light, and of intermediate magnitude in the light of mean re- 
 frangibility. Hence, when white or compound light is em- 
 ployed, the fringes of different colours will not be accurately 
 superposed ; and there will be formed a succession of iris-co- 
 loured fringes, the colours following the order which they 
 have in the spectrum. 
 
 (115) If we follow the course of the fringes from their 
 origin, we shall observe that they are propagated in lines sen- 
 sibly curved, whose concave side is turned towards the sha- 
 dow. In order to obtain accurate measures of the distances 
 of the fringes from the edge of the shadow, at different dis- 
 tances from the obstacle, Fresnel viewed them directly with 
 an eye-piece, furnished with a micrometer. He thus ascer- 
 tained that the curved path of each fringe was an hyperbola, 
 
 * If this angle be salient, the fringes, instead of forming a similar angle, 
 are observed to curve round the shadow. "When the angle is re-entrant, they 
 cross, and enter on the shadow at each side, without interfering with one 
 another. 
 
DIFFRACTION. 101 
 
 whose summit coincided with the edge of the obstacle, and 
 whose centre was the middle point of the line connecting that 
 edge with the luminous origin. 
 
 If we consider these hyperbolas as coincident with their 
 asymptots (which may be done without sensible error, unless 
 near the edge of the obstacle), and if we then determine the 
 angles which they make with one another, and with the edge 
 of the geometric shadow, we shall find that these angles in- 
 crease rapidly as the distance of the obstacle from the lumi- 
 nous point diminishes. When this distance is about 40 inches, 
 the ^fringes are very close together, the fringes of the first 
 and second order making an angle with one another of less 
 than 2' in red light. At the distance of 4 inches this angle 
 is increased to more than 5 7 ; and at T % of an inch it exceeds 
 16'. Thus the fringes dilate, as the edge of the obstacle ap- 
 proaches the luminous origin. 
 
 (116) In this experiment the incident light is supposed to 
 diverge from a luminous point. If the dimensions of the 
 luminous origin had been considerable, it will be easily 
 understood that each line in it, parallel to the edge of the 
 obstacle, would give rise to a different system of fringes ; and, 
 as the dark bands of some of these systems must coincide with 
 the bright bands of others, every trace of the phenomenon 
 would be obliterated. 
 
 (117) The preceding experiments exhibit the effect of a 
 single edge. "When the light diverging from the luminous 
 point is suffered to pass by two neqr edges, the phenomena will 
 be varied in a very interesting manner. 
 
 Let a fine wire be placed in the pencil of light diverging 
 from a luminous point, and let its shadow be received on a 
 screen, or plate of roughened glass, as before. We then ob- 
 serve, outside the geometric shadow, a set of parallel bands, 
 
102 DIFFRACTION. 
 
 or fringes, analogous to those produced by the single edge 
 in the former experiment. These are the exterior fringes. 
 But we observe further that the whole space of the geometric 
 shadow itself is also occupied by parallel stripes, alternately 
 bright and dark. These are the interior fringes ; and they 
 are in general closer, and more finely marked than the ex- 
 terior. When the breadth of the obstacle is considerable, 
 the interior fringes disappear, and the phenomena fall under 
 the class already examined. 
 
 The interior fringes are propagated, like the exterior, in 
 hyperbolic curves ; but their curvature is less considerable, 
 and the deviation from a right-lined course is scarcely per- 
 ceptible within the limits at which they are commonly ob- 
 served. They are also, like the exterior fringes, broader in 
 red than in violet light, and of intermediate breadths in the 
 light of intermediate refrangibility. Accordingly, in com- 
 pound or white light, the fringes of different dimensions are 
 superposed ; and the bands are no longer alternately bright 
 and black, but coloured with different tints, in the order of 
 the colours of the spectrum. 
 
 (118) It still remains to examine the effects produced by 
 two edges turned inwards, so as to form an aperture of any 
 dimensions. 
 
 For this purpose Fresnel employed an instrument consist- 
 ing of two metallic plates, one of which is fixed in the frame of 
 the apparatus, while the other is moveable by means of a fine 
 screw. The edges of these plates are right-lined and parallel, 
 so that they form always $> rectangular aperture ; and, by 
 means of the adjusting screw, the magnitude of this aperture 
 may be varied at pleasure. 
 
 "When a narrow rectangular aperture, thus formed, is sub- 
 stituted for the wire in the last experiment, the resulting phe- 
 nomena are very remarkable. In the first place, the luminous 
 
DIFFRACTION. 103 
 
 beam diverges considerably after passing the aperture, so that 
 the space which it occupies on the screen, or roughened glass, 
 is much wider than the geometric projection of the aperture. 
 Secondly, the entire of this space is covered with parallel bands, 
 or fringes, alternately bright and dark, distributed symmetri- 
 cally on either side of the line passing through the luminous 
 point and the centre of the aperture. 
 
 If we trace these fringes, from their origin at the aper- 
 ture to any distance, we shall find that they are propagated 
 in hyperbolas, like the former. The curvature of these 
 hyperbolic branches, and their inclination to one another, 
 depend on the breadth of the aperture, and on its distance 
 from the luminous point. Fraunhofer, who observed this 
 class of phenomena with great attention and care, found that 
 the angular distances of the successive bands of any given 
 colour from the central line formed an arithmetical progres- 
 sion, whose common difference was equal to its first term ; 
 and that, when different apertures were used, the distances 
 of one and the same band from the central line were inversely 
 as the breadths of the apertures. These fringes are broadest 
 and most widely separated in red light ; they are narrowest 
 and closest in violet light, and of intermediate magnitudes in 
 the intermediate rays of the spectrum. In white light, there- 
 fore, they present the succession of colours observed in other 
 cases. 
 
 When the aperture is formed by two straight edges 
 slightly inclined, Newton observed that the fringes were not 
 accurately parallel to the edges, but became broader as they 
 approached ; and that they finally crossed, and formed two 
 hyperbolic branches, one of whose asymptots is perpendicular 
 to the line bisecting the angle of the edges, while the others 
 are parallel to the edges themselves. 
 
 (119) Itis scarcely necessary to observe that the phenomena 
 
104 DIFFRACTION. 
 
 of diffraction may be endlessly varied, by "varying the form of 
 the diffracting edge. The preceding cases have been se- 
 lected as the most elementary. They are abundantly sufficient, 
 when pursued into numerical details, to test the truth of 
 any theory which may be applied to this class of pheno- 
 mena ; and such a theory being once established, the laws of 
 the more complex appearances are best sought for in its de- 
 ductions, We shall proceed, therefore, to consider the pre- 
 ceding phenomena in their relation to the two theories of 
 
 light. 
 
 (120) Newton conceived the rays of light to be inflected in 
 passing by the edges of bodies, by the operation of the at- 
 tractive and repulsive forces, which the molecules of bodies 
 were supposed to exert upon those of light before they ar- 
 rived in actual contact. By the operation of such forces, 
 Newton was enabled to explain the laws of reflexion and re- 
 fraction ; and it was reasonable to suppose that the same 
 forces played an important part in the phenomena now under 
 consideration. 
 
 Thus, the rays passing by the edges of a narrow opaque 
 body, such as a hair or fine wire, are supposed to be turned 
 aside by its repulsion ; and, as this force decreases rapidly as 
 the distance increases, it follows that the rays which pass at a 
 distance from the body will be less deflected than those which 
 pass close to it, as is shown 
 in the annexed diagram. 
 The caustic formed by the 
 intersection of these de- 
 flected rays will be con- 
 cave inwards ; and as none 
 of the rays pass within it, 
 it will form the boundary 
 of the visible shadow. Thus this supposition explains satisfac- 
 
DIFFRACTION. 105 
 
 torily the curvilinear termination of the visible shadow, and 
 its excess above the geometric one. 
 
 To account for the fringes which are parallel to the edge 
 of this shadow, Newton appears to have supposed the at- 
 tractive and repulsive forces to succeed one another for some 
 alternations ; and the molecules composing each ray, in their 
 passage by the body, to be bent to and fro by these forces, 
 in a serpentine course, and to be finally thrown off at one or 
 other of the points of contrary flexure. The intersection of 
 the rays thus thrown off at different points of the same ser- 
 pentine course will form a caustic or fringe ; so that each suc- 
 ceeding fringe will be produced by the rays which pass at a 
 given distance from the edge of the body. 
 
 Finally, the separation of white light into its elements is 
 explained, by supposing that the rays which differ in refrangi- 
 bility differ also in inflexibility, the body acting alike upon 
 the less refrangible rays at a greater distance, and t upon the 
 more refrangible rays at a less distance. 
 
 It is needless to comment upon the vagueness of these 
 explanations. Newton himself was dissatisfied with them, 
 and the subject fell from his hands unfinished. Still, how- 
 ever, the mere guesses of such a mind as that of Newton 
 must claim a deep interest ; and it was natural that among 
 his followers more weight should be attached to these con- 
 jectures, than he himself ever assigned to them. It seems 
 necessary, therefore, to advert to some of the circumstances 
 of the phenomena, which are not only unexplained by 
 this theory, but which seem moreover entirely at variance 
 with it, 
 
 (121) If the phenomena of inflexion be the effects of at- 
 tractive and repulsive forces emanating from the interposed 
 body, and if these forces are the same, or even analogous 
 to those to which the reflexion and refraction of light are 
 
1 06 DIFFRACTION. 
 
 ascribed in the theory of emission, it will follow that they 
 must exist in different bodies in very different degrees ; so that 
 the amount of bending of the rays, and therefore the breadth 
 of the diffracted fringes, should vary with the mass, the nature, 
 and the form of the inflecting body. Now it is clearly as- 
 certained, on the contrary, that all bodies, whatever be their 
 nature or the form of their edge, produce under the same 
 circumstances fringes identically the same ; and, in fact, the 
 partial interruption of light, caused by the interposition of 
 an obstacle of any kind, appears to be the only condition 
 essential to the phenomenon. 
 
 Gravesende seems to have first observed that the nature 
 or density of the body had no effect upon the magnitude 
 of the diffracted images ; and the fact has since been con- 
 firmed in the fullest manner by almost every inquirer in this 
 branch of experimental science. It is now admitted that the 
 inflecting forces, if such exist, must be independent of the 
 nature of the inflecting body, and altogether different from 
 those to which, in the theory of emission, the phenomena 
 of reflexion and refraction are ascribed. 
 
 To ascertain whether the form of the edge had any effect 
 upon the fringes, Eresneltook two plates of steel, the edge of 
 each of which was rounded in one half of its length, and 
 sharp in the remaining half, and placed the rounded portion 
 of each edge opposite the angular part of the other. If then 
 the position of the fringes depended on the form of the edge, 
 the effect would thus be doubled, and the fringes appear 
 broken in the midst. They were found, on the contrary, to 
 be perfectly straight throughout their entire length. 
 
 Again, the inflecting forces (though they must be supposed 
 to vary in intensity with the form and mass of the body, 
 and with the distance of the luminous molecule from the 
 edge) cannot be conceived to depend in any way upon the 
 distance previously traversed by the molecule, before it arrives 
 
DIFFRACTION. 107 
 
 in the neighbourhood of that edge ; so that the magnitude 
 and position of the fringes, in this hypothesis, cannot vary 
 with the distance of the inflecting edge from the luminous 
 point. But this conclusion is the reverse of fact. The 
 fringes dilate, and their mutual inclination is increased, as 
 the obstacle approaches the luminous origin. 
 
 The phenomena of diffraction, therefore, do not arise from 
 the operation of attractive and repulsive forces, exerted by 
 the molecules of bodies upon those of light. 
 
 (122) The same objections apply to the hypothesis of 
 Mairan and Du Tour, which ascribes these eifects to the re- 
 fraction of small atmospheres encompassing the bodies, and 
 of a different refractive power from the surrounding medium. 
 For, if such an atmosphere be retained by the attraction of 
 the body which it encompasses, and this seems to be the only 
 intelligible mode of accounting for its presence, its den- 
 sity, and its form, must vary with those of the body itself ; 
 and, consequently, its effects upon the rays of light must vary 
 also. 
 
 We are forced, then, to conclude, that the phenomena of 
 diffraction are inexplicable in the system of emission ; and we 
 proceed to examine in what manner, and with what success, 
 the principles of the wave-theory have been applied to their 
 explanation. 
 
 (123) This important step in Physical Optics was made 
 by Young, and all the complicated phenomena of diffraction 
 are now reduced to the simple principle of Interference. 
 
 The exterior fringes, formed without the shadows of bodies 
 were ascribed by Young to the interference of two portions 
 of light, one of which passed by the body, and was more or 
 less deviated, while the other was obliquely reflected from its 
 edge. The fringes formed by narrow apertures Mjere, in like 
 
108 DIFFRACTION. 
 
 mauner, supposed to arise from the interference of the two 
 pencils reflected from the opposite edges ; while the interior 
 fringes, within the shadows of narrow bodies, were accounted 
 for by the interference of the pencils which passed on either 
 side of the body, and were bent into the shadow. The ob- 
 served facts closely correspond with the calculated results of 
 this theory ; and in the case last mentioned, Young proved 
 that the phenomenon admitted no other explanation. Plac- 
 ing a small opaque screen on either side of the diffracting 
 body, so as to intercept the portion of light which passed 
 by one of its edges, the whole system of bands immediately 
 disappeared, although the light passing by the other edge 
 was unmodified. 
 
 The general laws of the fringes the dependence of their 
 magnitudes upon the length of the wave, and upon the distances 
 of the luminous origin and of the screen are fully explained 
 on these hypotheses. It is easy to infer from them that, as 
 the position of the screen is varied, the successive points of 
 the same fringe are not in a right line, but form an hyper- 
 bola ; and that, when the distance of the luminous origin is 
 lessened, the inclination of these hyperbolic branches (con- 
 sidered as coincident with their asymptots) augments, and 
 the fringes dilate. 
 
 The theory of Young, however, did not bear a closer 
 comparison with facts. If th exterior fringes arose from the 
 interference of the direct light with that obliquely reflected 
 from the edge of the obstacle, it would follow that the inten- 
 sity of the light in them should depend on the extent and 
 curvature of the edge. Fresnel found, on the contrary, that 
 the fringes were wholly independent of the form of the dif- 
 fracting edge ; the fringes formed by the back and by the 
 edge of a razor, for example, being precisely alike in every 
 respect. In the other cases of diffraction also, he perceived 
 that the r#ys grazing the edge of the body were not the 
 
DIFFRACTION. 109 
 
 only rays concerned in the production of the fringes ; but 
 that the light which passed by these edges at sensible dis- 
 tances was also deviated, and concurred in their forma- 
 tion. Fresnel was thus forced to seek a broader foundation 
 for his theory. 
 
 (124) In this theory the phenomena of diffraction are as- 
 cribed to the interference of the partial, or secondary waves, 
 which are separated from the grand wave by the interposi- 
 tion of the obstacle. In applying this principle, Fresnel sup- 
 poses the surface of the wave, when it reaches the obstacle, to 
 be subdivided into an indefinite number of equal portions. 
 Each of these portions may, by the principle of Huygens, be 
 considered as the centre of a system of partial waves ; and the 
 mathematical laws of interference enabled him to compute the 
 resultant of all these systems at any given point. This re- 
 sultant vibration, Fresnel has shown, is in general expressed 
 by means of two integrals, which are to be taken within limits 
 determined by the particular nature of the problem. Its 
 square is the measure of the intensity of the light ; and it is 
 found that its value has several maxima and minima, which 
 correspond to the intensities of the light in the bright and 
 dark bands. 
 
 The problem of diffraction was thus completely solved; 
 and its laws derived from the two principles to which the 
 laws of reflexion and refraction are themselves referred, the 
 principle of Interference, and the principle of Huygens. It only 
 remained to apply the solution to the principal cases, and to 
 compare the results with those of observation. The cases of 
 diffraction selected by Fresnel are those whose laws have 
 been already stated ; viz. the phenomena produced 1, by a 
 single straight edge ; 2, by an aperture terminated by parallel 
 straight edges ; and 3, by a narrow opaque body of the same 
 form. The agreement of observation and theory is so com- 
 
110 DIFFRACTION. 
 
 plete, that the computed places of the several bands seldom 
 differ from those observed by more than the 100th part of a 
 
 millimetre. 
 
 \ 
 
 (125) The general circumstances of these phenomena 
 may be deduced by very simple considerations from the 
 principles already laid down ; although the complete de- 
 velopment of these principles demands the aid of a compli- 
 cated analysis. 
 
 Thus, in the case of the fringes produced by a single 
 
 N. 
 
 edge, let be the luminous origin, Mo-N a diverging wave, 
 and E any point at which the illumination is sought. From 
 this point, as centre, let a circle be described, touching the 
 circle MN in #, and let the lines RM, Ita'c, &c., be drawn in 
 such a manner that the intercepts W, cc', dd', &c., are equal 
 respectively to one, two, three, &c., semi-undulations. The 
 effect produced at the point R is then, by the principle of 
 Huygens, the sum of the effects produced by each of the por- 
 tions ab, be, cd, &c., separately. But, the distances of these 
 consecutive portions from the point R differing by half a 
 wave, their effects will be opposed at that point ; so that, if 
 m denote the intensity of the light sent from the portion ab, 
 m' that from be, &c. the light sent from the indefinite wave, 
 #Mor N, being taken as unity the actual light which reaches 
 the point R will be 1, 1 + m, 1 + m - m', 1 + m - m' -f m" , &o., 
 according as the obstacle is placed at the point <?, b, c, d, &c. 
 
DIFFRACTION. Ill 
 
 And the intensity of the light when the obstacle is altogether 
 withdrawn is 
 
 1 + m - m' + m" - m'" + &o. = 2. 
 
 Now, as the terms of this series are continually decreas- 
 ing, and are affected alternately with opposite signs, it is 
 manifest that if we stop at any term, the sign of the remain- 
 der will be the same as that of its first term, and therefore 
 alternately positive and negative. Accordingly the intensities, 
 1 + m, 1 + m-m', 1 + m - m' + m x/ , &c., are alternately greater 
 and less than 2 ; and the intensity of the light sent to the 
 point E is alternately greater and less than when no obstacle 
 is interposed. 
 
 It will be easily understood, from this general explanation, 
 in what manner the magnitude of the fringes depends on the 
 length of the wave, on the distance of the luminous origin 
 from the obstacle, and on the distance of the screen. They 
 must be broadest in red light, and narrowest in violet light ; 
 and in white or compound light, the diffracted bands of dif- 
 ferent colours will occupy different positions, so as to form a 
 succession of iris-coloured bands having the violet or blue in- 
 side, and the red without. After a few successions these 
 bands wholly disappear, owing to the superposition of bands 
 of different colours. 
 
 (126) It is easy to compute the relative places of the 
 same fringe, for different positions of the luminous point, and 
 of the screen. 
 
 Let P be the edge of 
 the obstacle, PA a por- 
 tion of the wave, diverg- 
 ing from 0, which has 
 just reached that edge ; 
 and let QR be the screen, and R the place of a fringe of any 
 
112 DIFFRACTION. 
 
 given order. Then, in order that this point should belong to 
 the same fringe, for every distance of the luminous origin and 
 of the screen, it is only necessary that the interval of retarda- 
 tion, EP - EA, of the central and marginal parts of the wave 
 should be constant. For in this case the whole wave, AP, 
 may be divided into a given number of parts, such that the 
 difference of the distances of the successive points of division 
 from the point E shall be constant ; and therefore the effec- 
 tive wave consists of the same number of elementary portions 
 in the same relative state as to interference. 
 
 Now, denoting OP by a, PQ, by 6, and QE by x, we 
 have 
 
 EP = 
 since x is very small in comparison with b. Similarly, 
 
 BO = V(a + b)* + r* = a + b + 2(a + b}' 
 
 so thatEA = b + . , and EP - EA = J 
 
 2(a + b) \b a + b 
 
 But, by the condition of the question, this difference is a 
 constant quantity ; and denoting this constant by S, we have 
 
 b a + b 
 
 l2S(a + b 
 
 / - 
 \ a 
 
 When b varies, a remaining unaltered, i. e. when the po- 
 sition of the screen is varied, the value of x is the ordinate 
 of an hyperbola whose abscissa is b ; so that the successive 
 points of the same fringe belong to an hyperbola, whose sum- 
 mit is the edge of the obstacle. 
 
 (127) The interior fringes formed in the shadow of a nar- 
 row opaque body arise, it has been said, from the interference 
 

 DIFFRACTION. 113 
 
 of the two portions of the wave which pass by the edges on 
 either side. Let PP' be the section of the opaque body, PC 
 
 and PV the two portions of the diverging wave which has 
 just reached its edges, and E any point of the shadow. Then 
 if these portions be divided in the points #, 5, c, &c.,a', b\ c', &c., 
 in such a manner, that the difference of the distances of any 
 two consecutive points from the point E is equal to half 
 an undulation, the elementary wave sent from each portion 
 will be in complete discordance with those sent from the two 
 adjacent ; so that, if the several portions be equal, they 
 will neutralize one another's effects at the point E, with 
 the exception of the extreme portions, P#, PV, the halves of 
 which next the edges remain uncompensated. 
 
 Now the arcs P#, ab, be, &c., are very nearly equal, when 
 the lines drawn from their extremities to the point E are 
 sufficiently inclined to the normal, or, in other words, when 
 this point is sufficiently removed from the edge of the geo- 
 metric shadow. In this case, then, the only efficacious parts 
 of the wave are the halves of the extreme portions, P# and 
 P'o/ ; and the intensity of the light at the point E will be de- 
 termined by the difference of their distances from that point, 
 or (which comes to the same thing) by the difference of the 
 lengths of the lines connecting it with the edges of the ob- 
 stacle. The phenomena of interference are therefore the same 
 as in the case of light emanating from two near origins, already 
 considered ; and we may transfer to the present case the con- 
 
 i 
 
114 DIFFRACTION. 
 
 elusions arrived at in (106). Accordingly, if c denote the 
 breadth of the obstacle, and b its distance from the screen, 
 the distance, #, of any band from the centre of the sha- 
 dow is 
 
 nb\ 
 
 x = 
 
 2c 
 
 (128) The positions of the fringes formed by a narrow 
 rectangular aperture are determined by a similar formula. 
 
 Let PP X be the section of the aperture, PAP X the portion 
 of the wave which has just reached it, diverging from the lu- 
 minous origin at ; and let QQ' be the projection of the 
 aperture on the screen. Then, if we take the point E on this 
 screen in such a man- 
 ner, that the difference 
 of its distances from 
 the edges of the aper- 
 ture, EP'-EP, shall 
 be equal to a whole 
 number of semi-undula- 
 tions, that point will be the centre of a dark or bright band, 
 according as the assumed number is even or odd. For, in the 
 former case, the wave PAP 7 may be divided into an even 
 number of parts, such that the distances of every two conse- 
 cutive points of division from the point E differ by half an 
 undulation. The waves sent by every two consecutive por- 
 tions to the point E will therefore be in complete discordance, 
 and the total effect at that point will be null. On the other 
 hand, when the difference EP X - EP is equal to an odd num- 
 ber of semi-undulations, the number of opposing portions of the 
 wave will be odd, and as the alternate portions compensate 
 each other's effects at the point E, there will remain one por- 
 tion producing there its full effect. 
 
 The successive bands being formed at the points for which 
 
DIFFRACTION. 115 
 
 RP X - RP = n\, it is obvious that their, distances, RB, from 
 the centre of the projection of the aperture, will be given by 
 the same formula as in the case last considered, c being now 
 the breadth of the aperture, with this difference, however, 
 that the dark bands correspond to the even values of n, and 
 the bright bands to the odd values, which is the reverse of 
 what takes place in the bands formed within the shadow of an 
 opaque obstacle. We learn then, 1st, that the distances of 
 the successive fringes of any colour form an arithmetical pro- 
 gression whose common difference is equal to its first term ; 
 2ndly, that they vary directly as the distance of the screen, 
 and inversely as the breadth of the aperture ; and 3rdly, that 
 they are proportional to the length of the wave ; and there- 
 fore greatest for the extreme red rays, least for the extreme 
 violet, and of intermediate magnitude for the rays of inter- 
 mediate refrangibility. 
 
 "We have supposed the screen to be so remote that the 
 bands are entirely icitliout the projection of the aperture. 
 This will obviously be the case when UP' - QP is less than 
 half a wave. When the distance of the screen is so small that 
 QP X - QP exceeds this limit, fringes will be visible also within 
 the projection of the aperture. In this case the portions into 
 which the wave is divided are sensibly different in magni- 
 tude as well as obliquity. The reasoning above employed 
 is therefore no longer applicable ; and the points of maximum 
 and minimum brightness can only be obtained by a complete 
 calculation of the intensity of the light. 
 
 (129) The phenomena of diffraction hitherto considered 
 are of the simplest class ; but as such phenomena arise in 
 every instance in which light is in part intercepted, it is ob- 
 vious that they admit of endless modifications, varying with 
 the form of the interposed body. Some of these are too re- 
 markable to pass unnoticed. 
 
 i 2 
 
116 DIFFRACTION. 
 
 Among the most striking of these effects are those pro- 
 duced by light diverging from a luminous origin, and trans- 
 mitted through a small circular aperture; as, for example, 
 that formed by a pin in a sheet of lead. When the trans- 
 mitted light is viewed through a lens, the image of the aper- 
 ture appears as a brilliant spot, surrounded by coloured rings 
 of great vividness ; and these vary in the most beautiful 
 manner, as the distance of the aperture from the luminous 
 origin, or from the eye, is altered. When the latter distance 
 is considerable, the central spot is white, and the coloured 
 rings follow the order observed in thin plates. As the eye 
 approaches the aperture, the central white spot contracts to 
 a point, and then vanishes. The rings then close in on it in 
 order ; and the centre assumes in succession the most vivid 
 and beautiful hues, altogether similar to those of the reflected 
 rings of thin plates. 
 
 This remarkable coincidence has been shown to be an 
 exact result of theory. It has been demonstrated that the in- 
 tensity of the light of any simple colour, at the central spot, 
 and the compound tint in the case of white light, will be 
 the same as that reflected from a plate of air, whose thickness 
 bears a certain simple relation to the radius of the aperture, 
 and to its distances from the luminous origin and from the 
 eye. 
 
 The points of maximum and minimum intensity are easily 
 determined. \ 
 
 Let be the luminous I 
 
 point, and AB the line drawn 
 from it through the centre of 
 the aperture PP X ; then the 
 interval of retardation, S, of 
 
 the ray which reaches any point B on this line, coming from* 
 the edge of the aperture, is OP + PB - OB. Let 
 
DIFFRACTION. 117 
 
 AB = b, and AP =r ; then, since r is very small in compari- 
 son with a and , it is easy to see that 
 
 Now when this interval is equal to a whole number, n, of 
 semi- undulations, the aperture may be divided by concentric 
 circles, such that the rays which reach the point B, coming 
 from any two successive circumferences, shall differ by the in- 
 terval of half a wave. It follows from the preceding for- 
 mula that the squares of the radii, and therefore the superficies 
 of the successive circles thus formed, are as the numbers 
 of the natural series ; so that the annuli comprised between 
 every two succeeding circumferences are equal. But the ele- 
 mentary waves proceeding from each annulus are in complete 
 discordance with those from the two adjacent. The successive 
 annuli will therefore destroy one another's effects, and the 
 total intensity of the light at the point B will be null, or 
 equal to that of the last, according as the number of an- 
 nuli (the central circle included) is even or odd. Hence, 
 for a given aperture there will be a succession of points on 
 the axis, at which the intensity of the light is alternately 
 nothing and a maximum ; and it is obvious from the preceding 
 that the distances of these points will be the values of b given 
 by the formula 
 
 in which the points of complete darkness correspond to the^ 
 even values of n, and those of maximum brightness to the 
 odd values. 
 
 Such is the case with homogeneous light. As the points 
 of maximum and minimum intensity are different for the rays 
 of different colours, there will be no point of complete dark- 
 
 
 IT7 
 
118 DIFFRACTION. 
 
 ness in compound light, but a succession of points, at which 
 the centre of the aperture is richly coloured. 
 
 (130) The theory of Fresnel is not only in exact accord- 
 ance with facts already known, but it has also led to many 
 new and unexpected conclusions, and predicted consequences 
 which have been afterwards verified on trial. One of the 
 most remarkable of these is the phenomenon of diffraction by 
 an opaque circular disc. Poisson applied Fresnel's integrals 
 to this case ; and he was led to the startling result, that the 
 illumination of the centre of the shadow was precisely the 
 same as if the disc had been altogether removed. The prin- 
 ciples already laid down will enable the reader to satisfy 
 himself of the theoretical truth of this conclusion. Arago 
 was the first to show that it was in accordance with fact, 
 and his experiment may be repeated without much diffi- 
 culty. 
 
 (131) We have seen that when light diverging from a 
 luminous point passes by the edges of a fine hair or wire, a 
 succession of coloured bands will be formed parallel to the 
 edge of the shadow ; and the distances of these bands from 
 the shadow, and from one another, will be greater, the 
 less the diameter of the ' wire. If many such wires be ex- 
 posed to the diverging beam, and if (instead of being pa- 
 rallel) they are crossed and interlaced in every possible 
 direction, it is easy to conceive that the coloured bands 
 will be disposed in concentric circles, whose centre is the 
 luminous point. These circles resemble the halos visible 
 round the Sun and Moon in hazy weather. Their dia- 
 meters vary in the inverse ratio of the thickness of the wires 
 or fibres. 
 
 This law was applied by Young, in a very ingenious man- 
 ner, to the comparison of the diameters of fibres, or small 
 particles of any kind. 
 
DIFFRACTION. 119 
 
 A plate of metal is perforated with a small round hole, 
 about the j\yth of an inch in diameter, around which, at the 
 distance of ahout J or ^ of an inch, is a circle of smaller 
 holes. The flame of a lamp is then placed immediately 
 behind the aperture, and the luminous point viewed through 
 the substance to be examined. A ring or halo will be seen 
 surrounding the aperture; and by moving the substance 
 backwards and forwards on a graduated scale, this ring may 
 be brought to coincide with the circle of small holes pierced 
 in the plate. The distance from the aperture is then read 
 off on the scale, and varies obviously in the inverse ratio of 
 the angular diameter of the spectrum ; but the diameters 
 of the particles vary also in the same inverse ratio, so that 
 the distance on the scale at once becomes a measure of these 
 diameters. In this manner Young compared the diameters 
 of a great number of very minute substances, such as the 
 fibres of the finest wools, the globules of the blood, &c. The 
 instrument itself he called the Eriometer. 
 
 (132) In the case last mentioned we have supposed the 
 intervals of the fibres, or fine wires, to be much greater than 
 their thickness; in which case the phenomenon depends mainly 
 on the diameter of the opaque fibre. When the intervening 
 apertures are very small, the effect is influenced by their 
 magnitude, and assumes a different character. Thus, if a 
 grating be formed, by stretching a wire between two fine 
 screws of equal thread, and if this grating be held in the 
 beam diverging from a luminous point, we shall observe,, 
 on either side of the direct image, a series of spectral images 
 richly coloured with all the prismatic tints ; the spectra in- 
 creasing in breadth, and diminishing in intensity, as they 
 recede from the centre. 
 
 These phenomena are seen to most advantage by means 
 of a telescope adjusted to the luminous origin. The grating 
 
120 DIFFRACTION. 
 
 being held before the object-glass of the telescope, the spectra 
 are formed at its focus, and are there viewed, with all the ad- 
 vantages of distinctness and amplification, by means of the 
 eye-glass. Fraunhofer, who observed these phenomena with 
 much attention and care, traced no fewer than thirteen spec- 
 tra on either side of the central image ; the first pair being 
 separated from the central image, and from the second pair, 
 by intervals absolutely black. By a very accurate mode of 
 measurement he ascertained that the deviations of any one 
 colour from the central image, in the successive spectra, formed 
 an arithmetical progression ; and that the absolute amount 
 of these deviations varied inversely as the intervals^ of the 
 axes of the wires. 
 
 (133) The foregoing results flow readily from the principle 
 of interference, ^hQ first pair of spectra, on either side of the 
 central image, being produced by the interference of those rays 
 whose paths differ by one undulation ; the second pair, by those 
 whose paths differ by two undulations ; and so on. 
 
 Let the light proceeding from a very remote origin fall on 
 the grating, whose opaque parts are represented by ab, a'b', 
 a"b", &c. ; and let Q, be the place of the eye. Then, if we take 
 a portion of the grating, a' a", composed of one opaque and one 
 transparent portion, in such a manner that the difference of the 
 distances of its extremities from the point Q, Qa" - Qa', shall 
 be equal to the length of 
 a wave, it is manifest that 
 the corresponding portion 
 of the incident wave, a' a" 
 may be divided into two 
 parts very nearly equal, 
 the waves sent from which 
 to the point Q shall be 
 in complete discordance. Without the grating, therefore, the 
 
DIFFRACTION. 121 
 
 effect of that portion of the incident wave would be null at the 
 point Q, and no light from it would reach the eye. The 
 effect of the grating, however, is to intercept the whole or 
 part of one of the two interfering portions, and thus to render 
 the other visible : and this effect is greatest when the opaque 
 and transparent parts of the grating are equal. A bright 
 band will therefore be visible in the direction Q,". The same 
 thing will happen for all the similar divisions of the grating, 
 the distances of whose extremities from the point Q differ by 
 two, three, or any whole number of undulations ; and thus 
 there will be a succession of bright bands, visible at different 
 angular distances from the direct ray PQ,. 
 
 These angles are easily computed. Let a'k be the arch of 
 a circle described with the centre Q, ; then a"k = a' a" cos a'a"k 
 = a'a'sin PQa". But the interval of retardation, a^k, is equal 
 to the length of a wave ; so that, if the angle PQa" be de- 
 noted by 0, and the interval composed of an opaque and trans- 
 parent part of the grating a' a", by 6, we have 
 
 sin0 = -. 
 
 
 
 This is the angular distance of the first bright band from the 
 central one ; and it is obvious that the corresponding angle, 
 for the band of the n th order is given by the formula 
 
 The position of each ray, in these spectra, therefore de- 
 pends solely on the length of the wave and is independent of 
 the nature of the substance by which it is produced. 
 
 (134) It is a remarkable circumstance of the phenomenon 
 whose laws we have been tracing, that when the experiment 
 is performed with the requisite care, the several species of 
 
122 DIFFRACTION. 
 
 homogeneous light are so pure and unmixed in the spectra, 
 that the fixed lines may be discerned. These lines, then, are 
 wholly independent of refraction, and exist in the parts of 
 the solar beam before they are separated by the prism. The 
 phenomenon, when thus exhibited, is however distinguished 
 by a remarkable peculiarity. The distances of the fixed lines 
 in the diffracted spectrum are always proportional, whatever 
 be the diffracting substance ; while the ratios of the intervals 
 of the fixed lines (or of the breadths of the coloured spaces), 
 in the spectra formed by refraction, vary with the dispersive 
 powers of the prisms. In fact, the angle 9 being small, we 
 may make sin = 6 sin V ; so that 
 
 *.Ti 
 
 = 
 
 sin 1' 
 
 Hence, if 0i, 2 , 3 , denote the deviations corresponding to 
 any three definite points in the spectrum, and Ai, A 2 , A 3 , the 
 corresponding wave-lengths, it follows that 
 
 02 0i A 2 Ai 
 
 9 3 Oi X 3 - AI ' 
 
 or the intervals of the fixed lines of the spectrum are as the 
 differences of the corresponding wave-lengths, and are there- 
 fore in an invariable ratio. The difference in the disposition 
 of the fixed lines, in the spectra formed by diffraction and by 
 refraction, will be seen in the diagrams of Art. (136), in 
 which the points B, C, D, &c., of the horizontal line BH, 
 represent the relative positions of the principal fixed lines, in 
 the spectrum formed by a prism of flint-glass, and in the 
 diffracted spectrum, respectively. 
 
 (135) The formula of Art. (133) suggests a very simple 
 method of determining the length of the wave corresponding 
 to any given ray of the spectrum. The value of f, or the in- 
 
DIFFRACTION. 123 
 
 terval of the axes of the wires, may be ascertained with the 
 greatest ease and precision ; and we have, therefore, only to 
 measure the angular deviation, 0, of the ray of any simple 
 colour from the axis, in order to deduce the value of A. 
 Fraunhofer computed in this manner the lengths of the 
 waves corresponding to the seven principal fixed lines of the 
 spectrum ; and the resulting values are perhaps the most 
 exact optical constants we possess. 
 
 The wave-lengths, corresponding to the principal fixed 
 lines, B, C, D, E, F, Or, H, expressed in millionths of a 
 millimetre, were thus found to be 
 
 656, 589, 526, 484, 429, 393. 
 
 The wave-length corresponding to the middle point of the 
 diffracted spectrum is 553' 5 millionths of a millimetre. The 
 wave-lengths corresponding to the extreme visible points are 
 738 and 369 millionths, respectively, the former of which is 
 exactly double of the latter. 
 
 (136) But the diffracted spectrum is further distinguished 
 by the simplicity of the law which governs the intensity of the 
 light in its several parts. The intensity of the light in the or- 
 dinary spectrum (formed by a prism of flint-glass) was deter- 
 mined by Fraunhofer, for the points corresponding to the 
 principal fixed lines. These intensities are represented by the 
 ordinates of the curve in the annexed diagram. The ordinate 
 
 B c 
 at the point m, situated between the fixed lines D and E, at a 
 
 rj 
 
 distance Dm = DE, corresponds to the maximum inten- 
 sity, and divides the whole light of the spectrum into two 
 
124 DIFFRACTION. 
 
 equal parts, the areas of the two portions of the curve being 
 equal. 
 
 The law of the intensity in the diffracted spectrum was 
 deduced by Mossotti from the foregoing : it is represented by 
 the ordinates of the curve in the following diagram. We see 
 that 
 
 I. The ordinate which divides the light into two equal 
 portions corresponds to the middle point of the spectrum. 
 
 II. This ordinate is a maximum ; and the curve is sym- 
 metrical with respect to it as an axis. 
 
 Accordingly, the intensity of the light of the diffracted spec- 
 trum is a maximum at the middle of its length, and decreases 
 thence symmetrically on either side. It is evanescent, when 
 the wave-length increases, or decreases, by about one-third 
 of the value corresponding to the maximum intensity. Hence 
 while in the spectra formed by refraction the ratios of 
 the spaces occupied by the several colours, and the intensi- 
 ties of the light at the several points, vary with the refracting 
 substance, they are, on the other hand, invariable in the dif- 
 fracted spectrum. The latter spectrum, accordingly, must 
 be regarded as the normal one, to which all others are to be 
 referred. 
 
 (137) Gratings producing these effects may be formed in 
 several ways as, for example, by tracing a number of paral- 
 lel lines on glass with a fine diamond point. Fraunhofer suc- 
 ceeded by such means in forming ruled surfaces in which the 
 striae were actually invisible under the most powerful micro- 
 scopes, the interval of the grooves being only the ^oiou of an 
 inch. 
 
 Analogous phenomena may be produced by reflexion. If 
 
DIFFRACTION. 125 
 
 a great number of parallel lines be engraved at very small and 
 equal intervals upon a polished surface, the light reflected 
 from the intervals of the grooves will interfere in a manner 
 precisely analogous to that admitted through the apertures 
 of the gratings ; and will, by their interference, produce the 
 most brilliant spectra. In some of the grooved metallic sur- 
 faces constructed by Mr. Barton, there are 10,000 lines to the 
 inch. With surfaces so minutely divided, the spectra pro- 
 duced are as perfect as those formed by the finest prisms ; and 
 the colours which they display are little inferior to those of 
 the diamond. 
 
 Similar appearances may be observed on metallic surfaces 
 which have been polished with a coarse powder, the powder 
 leaving minute striae which produce the effects we have been 
 describing. They may also be very simply produced by pass- 
 ing the finger over the surface of a piece of glass moistened 
 by the breath. 
 
 (138) The beautiful colours of mother of pearl are natural 
 instances of the same phenomena. This substance is com- 
 posed of a vast number of very thin layers, which are gra- 
 dually and successively deposited within the shell of the 
 oyster, each layer taking the form of the preceding. "When 
 it is wrought, therefore, the natural joints are cut through 
 in a great number of sinuous lines ; and the resulting sur- 
 face, however highly polished, is covered by an immense 
 number of undulating ridges, formed by the edges of the 
 layers. These strise may be observed by the aid of a 
 powerful microscope, although they are sometimes so close 
 that 5000 of them occupy an inch. That they are the 
 causes of the brilliant colours displayed by this substance 
 has been placed beyond doubt by an experiment of Sir 
 David Brewster. This experiment consisted simply in 
 taking the impression of the surface of the pearl on wax, 
 
126 DIFFRACTION. 
 
 the impressed surface displaying all the colours of the ori- 
 ginal body. In fact, the colours of striated surfaces indicate 
 their structure, perhaps more unerringly than any other 
 means : Sir David Brewster has made a very ingenious 
 use of their laws, in investigating the curious and compli- 
 cated structure of the crystalline lens in fishes and other 
 animals. 
 
 (139) There remains another class of phenomena pro- 
 duced by diffraction, which it is important to notice. 
 
 "We have already seen the effects produced, when light 
 diverging from a luminous point is transmitted through a 
 narrow aperture, and received on a screen. But if we 
 vary the experiment, by placing a lens of considerable focal 
 length as the object-glass of a telescope immediately be- 
 hind the aperture, and receive the image on a screen at 
 the conjugate focus, the appearances displayed are altered 
 in a remarkable manner, and differ more widely from those 
 produced in the former case, as the aperture is greater. 
 In fact, the phenomena of diffraction are thus produced with 
 apertures of considerable dimensions, and were observed by 
 Sir William Herschel with the undiminished object-specula 
 of his great telescopes : they are rendered more distinct, 
 however, when the aperture of the telescope is limited by a 
 diaphragm of moderate size. When a star is viewed through 
 a telescope of high power, having its object-lens thus limited, 
 its image is encompassed with a system of diffracted rings 
 slightly coloured, succeeding one another at equal intervals, 
 the diameters of the rings being inversely as the diameters of 
 the apertures. The phenomena vary in a very curious 
 manner when the form of the aperture is changed. Thus, 
 when a triangular diaphragm is substituted for the circular 
 one, the disc of the star appears surrounded by a black ring, 
 from which diverge six rays at equal intervals. 
 
DIFFRACTION. 127 
 
 These phenomena have been examined in detail by Sir 
 John Herschel and by M. Arago. Their mathematical ex- 
 planation has been given by Mr. Airy, in his valuable tract 
 on the Undulatory Theory ; and the deductions of theory are 
 found to be in complete accordance with the observed facts. 
 
( 128 ) 
 
 CHAPTEE VIII. 
 
 COLOURS OF THIN PLATES. 
 
 (140) THE colours of thin plates were first noticed by 
 Boyle and Hooke. They are displayed in every instance in 
 which transparent bodies are reduced to films of great tenuity. 
 Boyle succeeded in blowing glass so thin as to exhibit the 
 phenomena : they are more readily developed in mica, and 
 some other transparent minerals which possess a lamellar 
 structure ; but the most familiar instance of their exhibition 
 is in the froth of liquids, the fluid envelopes of the bubbles 
 which compose it being in general of extreme thinness. 
 
 These colours vary with the thickness of the film, and dis- 
 appear altogether when it passes certain limits. "When the 
 film exceeds a certain thickness, all the colours are equally 
 reflected, and the reflected light is therefore white. On the 
 other hand, when the thickness falls below a certain limit, no 
 light whatever reaches the eye, and the surface of the film 
 appears absolutely black. 
 
 (141) The foregoing facts may be observed in the common 
 soap-bubble, when properly defended from the disturbing in- 
 fluence of currents of air. If the mouth of a wine-glass be 
 dipped in water, which has been rendered somewhat viscid 
 by the mixture of soap, the aqueous film which remains in 
 contact with it after emersion will display the whole suc- 
 cession of these phenomena. "When held in a vertical plane, 
 it will at first appear uniformly white over its entire surface ; 
 but, as it grows thinner by the descent of the fluid particles, 
 
COLOURS OF THIN PLATES, 129 
 
 colours begin to be exhibited at the top, where it is thinnest. 
 These colours arrange themselves in horizontal bands, and 
 become more and more brilliant as the thickness diminishes ; 
 until finally, when that thickness is reduced to a certain 
 limit, the upper part of the film becomes completely black. 
 When the bubble lias arrived at this stage of tenuity, cohesion 
 is no longer able to resist the other forces which are acting 
 on its particles, and it bursts. 
 
 Similar phenomena may be observed when a drop of oil is 
 let fall on water. As the oil spreads rapidly over the surface, 
 it is soon reduced to a very thin film, which displays the 
 spectral colours. 
 
 Every one has noticed the fact that steel and other metals, 
 when polished, acquire various shades of colour by exposure 
 to the air. These colours are produced by a thin coating of 
 metallic oxide, which is gradually formed on the surface. The 
 formation of this oxide is greatly accelerated by an augmenta- 
 tion of temperature ; and the colour thus formed is so inva- 
 riably connected with the thickness of the film, and this latter 
 with the degree of heat, that artists are in the habit of mea- 
 suring the temperature by the colour developed. Thus steel, 
 in the process of tempering, is said to have received a yellow 
 heat, a blue heat, &o. 
 
 The same appearances are displayed in a still more strik- 
 ing manner by air itself, or even by a vacuum. If two 
 plates of glass be pressed together by the fingers, we shall 
 observe, round the point of nearest approach, a succession of 
 coloured bands of great brilliancy, which dilate as the 
 pressure is increased, and the inclosed plate reduced in thick- 
 ness. 
 
 (142) In order to observe these phenomena, in such a 
 manner as to be enabled to trace their laws, we must follow 
 Newton. Newton's experiment consisted simply in laying a 
 
 K 
 
130 COLOURS OF THIN PLATES. 
 
 convex lens of glass upon a plane surface of the same mate- 
 rial. The thickness of the inclosed plate of air increases as 
 the square of the distance from the point of contact, and is, 
 therefore, the same at all equal distances from that point ; 
 and, as the reflected colour depends on the thickness, the 
 bands of the same colour will be arranged in concentric 
 circles, of which that point is the centre. The same suc- 
 cession of colours is produced when any other transparent 
 fluid is inclosed between the glasses. The colours, how- 
 ever, are more vivid, the more the refractive power of the 
 thin plate differs from that of the substances within which it 
 is inclosed. 
 
 When we look attentively at these rings, the light being 
 reflected always at the same angle, we observe that the cen- 
 tral one is not a mere annulus, but a complete circle of nearly 
 uniform colour. If then we diminish the thickness of the 
 plate of air, by pressing the two glasses more closely together, 
 this central circle is observed to dilate, and a new circle of a 
 different colour to spring up in its centre. This will dilate 
 in turn, driving the former before it, and another circle ap- 
 pear within it ; until at length a black spot shows itself in 
 the centre of the system, after which no further diminution 
 of thickness will alter the succession. "When the black spot 
 makes its appearance, we have obtained a plate of air so 
 thin as no longer to reflect any colours, and the phenomenon 
 is complete. Newton traced seven coloured rings round 
 this spot, the colours of which are said to be of the first, se- 
 cond, third, &c., order, according to the order of the ring to 
 which they belong. Thus, the red of the third order is the 
 red in the third ring from the central black, &c. The whole 
 succession of colours is called Newton' 's scale. 
 
 (143) The principal laws of these phenomena are included 
 in the following propositions : 
 
COLOURS OF THIN PLATES. 131 
 
 I. In homogeneous light, the rings are alternately bright 
 and black ; the thicknesses corresponding to the bright rings 
 of succeeding orders being as the odd numbers of the natural 
 series, and those corresponding to the black rings as the in- 
 termediate even numbers. 
 
 II. The thickness corresponding to the ring of any given 
 order varies with the colour of the light, being greatest in 
 red light, least in violet, and of intermediate magnitude in 
 light of intermediate refrangibility. In white or compound 
 light, therefore, each ring will be composed of rings of 
 different colours, succeeding one another in the order of their 
 refrangibility. 
 
 III. Tha thickness corresponding to any given ring varies 
 with the obliquity of the incident light, being very nearly 
 proportional to the secant of the angle of incidence. 
 
 IV. The thickness varies with the substance of the re- 
 flecting plate, and in the inverse ratio of its refractive 
 index. 
 
 (144) In order to establish the first of these laws, it is 
 necessary to employ homogeneous light. This may be ob- 
 tained by means of the prism : or we may adopt the method 
 suggested by Mr. Talbot, and illuminate the glasses with a 
 spirit lamp having a salted wick. The light of such a lamp 
 being a yellow of almost perfect homogeneity, the rings will 
 be alternately black and yelloiv ; and their number is so great 
 as to baffle any attempt to determine it. 
 
 The law of the thicknesses corresponding to the succes- 
 sive rings is easily established. Let be the point of con- 
 tact of the plane and spherical surfaces, and aa', W, cc', &o. 
 the diameters of the successive 
 rings formed round that point c/& a ale 
 
 as a centre. It is evident that /^P' * /3 
 
 the thicknesses of the plate of 
 
 K2 
 
132 COLOURS or THIN PLATLS. 
 
 air at the points where these rings are formed, #a, If3, cy, 
 &c., are as the squares of the distances 0#, 0&, Oc, &c., or as 
 the squares of the diameters of the rings. To determine the 
 law of the thicknesses, therefore, we have only to measure 
 these diameters. This was done by Newton with great ac- 
 curacy, and it was found that the squares of the diameters 
 were in arithmetical progression; consequently, the thick- 
 nesses corresponding to the successive rings formed a similar 
 progression. 
 
 (145) But Newton did not stop here : he ascertained 
 further the absolute thickness of the plate of air at which each 
 ring was formed. It is manifest that if the thickness of the 
 plate be determined for any one ring, that corresponding to 
 the others will be given by the law just stated. Newton, 
 accordingly proceeded to ascertain this thickness for the dark 
 ring of the fifth order. This was done by measuring its 
 diameter accurately, and determining the radius of the spheri- 
 cal surface from the focal length of the lens and its refractive 
 index. The thickness is thence immediately deduced ; for it 
 is equal to the square of the radius of the ring divided by 
 the diameter of the spherical surface. The value thus de- 
 duced being suitably corrected, it was found that the thick- 
 ness of the plate of air was the yy^o o ^ an i ncn > a ^ the 
 dark ring of the fifth order ; and this thickness being decuple 
 of that corresponding to the first bright ring, it followed 
 that the thickness of the plate of air, at the place of the 
 first bright ring, was the iWooo f an i ncn - Thus the 
 bright rings of the successive orders are formed at the thick- 
 nesses 
 
 * 3 5 _7 &c 
 1.78000' 178000' 178000, 178000 
 
 -and the intermediate dark rings at the thicknesses 
 
 2 4 6 i 
 
 _ __ __ ___ 
 
 178000 1 USuuO 1 178000' 178000 
 
 &c . 
 
COLOURS OF THIN PLATES. 133 
 
 These determinations belong to the most luminous rays 
 of the spectrum, or those at the confines of the orange and 
 yellow. 
 
 (146) The variation of the diameters of the rings (or of 
 the thicknesses of the plate of air at which they are exhibited) 
 with the colour of the light, may be observed by illuminating 
 the glasses with different portions of the spectrum in succes- 
 sion, or, yet more simply, by looking at the rings through 
 coloured glasses ; and it is found that the magnitude of the 
 rings is greater, the less the refrangibility of the light. This 
 being understood, it is easy to comprehend the cause of the 
 succession of colours in each ring, when white or compound 
 light is used. For the rings, in this case, are the aggregate 
 of the rings of different colours ; and these being of different 
 magnitudes, the compound ring will be variously coloured, 
 the more refrangible rays occupying the interior, and the 
 less refrangible the exterior parts of the Ting. It is easy 
 to see also, that all phenomena of colour must disappear 
 after a few successions, the rings of different colours, belong- 
 ing to different orders, being at length superposed. 
 
 The variation of the rings (and therefore of the thick- 
 nesses) with the obliquity of the incident light may be observed 
 by depressing the eye. The rings are then seen to dilate 
 rapidly with the obliquity of the reflected pencil; the 
 thicknesses of the plate of air at which they are exhibited 
 being nearly as the secants of the angles of incidence or re- 
 flexion. 
 
 The fourth and last law, which expresses the depend- 
 ence of the thickness at which any ring is formed upon the 
 refractive power of the plate, is easily verified by introducing 
 a drop of water between the glasses. The rings are then ob- 
 served to contract ; and if we compare their diameters in air 
 
134 COLOURS OF THIN PLATES. 
 
 and in water, it will be found that the corresponding thick- 
 nesses are as four to three, or in the inverse ratio of the refrac- 
 tive indices. 
 
 (147) We have hitherto spoken only, of the reflected rings. 
 There is another system of rings formed by transmission, but 
 much fainter than the former. The transmitted rings are 
 found to observe the same laws as the reflected rings, with 
 this remarkable exception, that the colour transmitted at any 
 particular thickness of the plate is always complementary to 
 that reflected at the same thickness ; so that, in homogeneous 
 light, the bright transmitted ring is always at the same dis- 
 tance from the centre as the corresponding dark one of the 
 reflected system. 
 
 (148) The phenomena of thin plates are exhibited, under 
 a modified form, in the following experiment : 
 
 A little fine soap is spread upon a plate of black glass, 
 and is distributed uniformly by rubbing the surface lightly 
 with a piece of soft leather. If then we blow on the surface, 
 thus prepared, through a short tube, taking care to direct 
 the tube always to the same part of the plate, the vapour 
 of the breath will be deposited in a thin film, whose thick- 
 ness diminishes regularly from the point to which the tube 
 is directed. This film will accordingly display a series of 
 coloured rings analogous to those formed by the plate of 
 air between two object-glasses, with this difference, how- 
 ever, that the order of the rings is reversed, the outermost 
 ring corresponding to the centre of Newton's scale. This 
 little apparatus, contrived by Mr. Head, is denominated by 
 him an iroscope. 
 
 (149) For the physical explanation of these phenomena. 
 
COLOURS OF THIN PLATES. 135 
 
 it has been already stated, Newton framed the hypothesis 
 of the fits of easy reflexion and transmission already referred to. 
 Its application is obvious. The molecule of light is in a, fit of 
 easy transmission in its passage through the first surface ; this 
 is succeeded by a fit of easy reflexion, and so on alternately, 
 the spaces traversed during the continuance of the fits being 
 all equal. On arriving at the second surface, therefore, the 
 molecule will be in a fit of easy transmission, or easy reflex- 
 ion, according as the interval of the surfaces is an even or 
 an odd multiple of the length of the fit. Thus the alter- 
 nate succession of bright and dark rings, and the arithmetical 
 progression of the thicknesses at which they are exhibited, 
 are explained. 
 
 To explain the second law, it is necessary to suppose that 
 the length of the fits varies ivith the colour of the light, being 
 greatest in red light, least in violet, and of intermediate mag- 
 nitude in light of intermediate refrangibility. Newton de- 
 termined the absolute lengths of these fits for the rays of 
 each simple colour, and found that they bore a remarkable 
 numerical relation to the lengths of the chords sounding the 
 octave. 
 
 To account for the two remaining laws, Newton was con- 
 strained to make new suppositions, and to attribute pro- 
 perties to the fits which are inconsistent with every physical 
 account which has been given of them. Thus, to explain 
 the dilatation of the rings with the obliquity, he assumed 
 that the length of the fits augmented with the incidence, and 
 nearly in the ratio of the square of the secant of the angle of 
 incidence. This assumption is inconsistent with the physical 
 theory. If the fits are produced by the vibrations of 
 the ether which are propagated faster than the lumi- 
 nous molecules, and which alternately conspire with and 
 oppose their progressive motion, their lengths should 
 
COLOURS OF THIN PLATES. 
 
 continue the same in the same medium, whatever be the inci- 
 dence. 
 
 The fourth law appears to be also irreconcileable with 
 theory. The thicknesses of the plates of different media, at 
 which the same tint is exhibited, being in the inverse ratio of 
 the refractive indices, it was necessary to suppose that the 
 lengths of the fits varied in the same proportion ; and since, 
 in the Newtonian theory, the refractive indices are directly 
 as the velocities of propagation, it would follow that, as the 
 velocities augmented, the spaces traversed by the ray in the 
 interval of its periodical states must diminish, and in the same 
 proportion. 
 
 (150) Newton seems to have regarded this hypothesis as 
 the mere expression of a physical fact ; and in this light it 
 was long considered. It cannot be denied that, as the thick- 
 ness of the plate increases, the light appears by reflexion 
 and transmission alternately ; and it is of no moment, it may 
 be said, by what name these alternate states are called. 
 But if we look more narrowly into the theory, we shall find 
 that it assumes the alternate appearance of the light, in the 
 reflected and transmitted pencils, to be the effect of an alter- 
 nate reflexion and transmission at a single surface, that surface 
 being the second surface of the plate. Now it can be shown 
 that this supposition is inconsistent with fact ; that light is 
 reflected from both surfaces of the plate ; and that the con- 
 currence of these two reflected pencils is an essential condi- 
 tion of the phenomenon. 
 
 To show this, let us employ (instead of common light) 
 light which is polarized in a plane perpendicular to the plane 
 of incidence ; and let it fall upon a plate of air inclosed be- 
 tween two transparent surfaces of different refractive powers. 
 Under these circumstances it is found that the intensity of 
 
COLOURS OF THIN PLATES. 137 
 
 the light in the rings varies with the incidence; and that 
 the whole system disappears in two cases, namely, when the 
 incidence corresponds to the polarizing angle of either of the 
 media. 
 
 To understand the conclusion to which this leads, we 
 must assume a property of light which will be hereafter 
 established namely, that when light, thus polarized, is in- 
 cident upon a transparent surface at what is called the 
 polarizing angle, it is wholly transmitted, and no portion of 
 it whatever reflected. We see then, from the experiment, 
 that the rings disappear when the light reflected from either 
 of the two surfaces of the plate vanishes ; and we are there- 
 fore warranted in concluding, that the light reflected from 
 both surfaces is essential to their production. 
 
 (151) The preceding experiment, and the conclusion 
 drawn from it, lead us to the very threshold of the true 
 theory. In fact, the light incident on the first surface of the 
 plate is in part reflected, and in part also transmitted. The 
 transmitted portion undergoes a similar subdivision at the 
 second surface ; and part of the light reflected at that surface 
 will emerge through the first, and reach the eye along with 
 that reflected there. Thus the reflected light consists of two 
 portions, one reflected at the upper, and the other at the 
 lower surface of the plate ; and these two portions will 
 interfere, and reinforce or weaken each other's effects, ac- 
 cording as they reach the eye in the same or in opposite 
 phases. 
 
 This mode of explaining the phenomena of thin plates 
 was pointed out by Hooke, in a remarkable passage in 
 his Micrographia, some years before the subject was taken 
 up by Newton. In this passage he very clearly describes 
 the manner in which the rings of successive orders depend 
 
138 COLOURS OF THIN PLATES. 
 
 on the interval of retardation of the second " pulse," or 
 wave, with respect to the first, and therefore on the thick- 
 ness of the plate. But he does not seem to have had any 
 distinct idea of the principle of interference itself ; and his 
 conception of the mode in which the colours resulted from 
 this "duplicated pulse" is entirely erroneous. Euler was 
 the next who attempted to connect the phenomena of thin 
 plates with the wave-theory of light ; but the attempt, like 
 all the physical speculations of this great mathematician, 
 was signally unsuccessful, and the subject remained in this 
 unsettled state, until the principle of interference was dis- 
 covered by Young. When this principle was combined 
 with the suggestion of Hooke, the whole mystery vanished. 
 The application was made by Young himself, and all the 
 principal laws of the phenomena were readily and simply 
 explained. 
 
 (152) Let man be the course of a ray reflected at the first 
 surface of a plate ; mopo'n' that of the 
 ray reflected at the second surface, 
 and twice transmitted through the 
 first. From the point o' let fall the 
 perpendicular o'r' upon the reflected 
 ray on ; it will be also perpendicular 
 to 0V, and will therefore be parallel 
 to the front of the two reflected waves. 
 Now let us conceive a wave reflected at the first surface, in 
 the position o'r', to meet at the same place an anterior wave 
 reflected at the second surface, and let us calculate the origi- 
 nal interval between them. From the time that they reached 
 the first surface at 0, one has travelled over the space or', 
 and the other over the space op + po'. But, if we let fall 
 the perpendicular or upon pb', it is evident from the law of 
 refraction that the spaces or' and o'r are traversed in the 
 
COLOURS OF THIN PLATES. 139 
 
 same time in the two media ; and, consequently, that the 
 interval of retardation is the time of describing op + pr. 
 Now pr = op cos 2opq, and therefore op+pr = op (1 + cos 2opq) 
 = 2op cos 2 opq. But op cos opq =pq ; and, consequently, the 
 interval is 2pq cos opq. Or, if we denote that interval 
 by 8 ; the thickness of the plate pq, by t ; and the angle 
 opq by 0, 
 
 $ = 2t cos 9. 
 
 The two waves are in complete accordance or discordance 
 when the interval of retardation is an exact multiple of the 
 length of half a wave : i. e. when 
 
 n being any number of the natural series. Equating these 
 values of S, therefore, we have, for the values of the thick- 
 ness of the plate which will produce a complete accordance 
 or discordance of the two waves, 
 
 t = n\ sec 0. 
 
 We learn then, 1st, that the successive thicknesses of the 
 plate, for which the intensity of the reflected light is greatest 
 or least, are as the numbers of the natural series ; 2ndly, that 
 for different species of simple light these thicknesses are 
 proportional to the lengths of the waves ; 3rdly, that for dif- 
 ferent obliquities they vary as the secant of the angle of in- 
 cidence on the exterior medium ; and, 4thly, that for plates 
 of different substances they are proportional to X, and therer 
 fore in the direct ratio of the velocities of propagation, or in 
 the inverse ratio of the refractive indices of the substances of 
 which the plate is composed. 
 
 (153) There is one part of the preceding explanation 
 which demands a little further consideration. The two waves 
 
140 COLOURS OF THIN PLATES. 
 
 being in complete accordance when the interval of retarda- 
 tion is an even multiple of the length of halt a wave, and in 
 complete discordance when that interval is an odd multiple of 
 the same quantity, it would seem, from the foregoing account, 
 that the bright rings should be formed at all those points for 
 which n is an even number in the formula above given (or 
 the thickness an even multiple of jX sec 0,) and the dark 
 rings at those points for which it is odd. If this were true, 
 the point of contact should be a point of accordance, and the 
 rings should commence from a bright centre, instead of a dark 
 one. 
 
 This apparent discrepancy is explained by the fact, that 
 the two reflexions take place under opposite circumstances, 
 one of the rays being reflected at the surface of a denser, and 
 the other at that of a rarer medium. 
 
 The effect of this difference will be best understood by a 
 simple illustration. When one elastic ball strikes another at 
 rest, it communicates motion to it in all cases ; but its own 
 condition after the shock will depend on the relative masses 
 of the two balls. If the balls be equal, the first will remain 
 at rest after the shock. If they be unequal, it will move ; 
 and its motion will be in the direction of its former motion, 
 when its mass exceeds that of the second ball, it will be in 
 the opposite direction when it is less. This will help us to 
 understand what passes when a wave reaches the surface 
 separating two media. The particles of ether next the 
 bounding surface communicate motion to the adjacent par- 
 ticles of the second medium, and thus give rise to the re- 
 fracted wave. But the former particles will not remain at 
 rest afterwards, unless the density and elasticity of the ether 
 be the same in the two media. When this is not the case, 
 the particles of the first medium will move, after communi- 
 cating motion to those of the second, and, in moving, give 
 rise to the reflected wave. Thus refraction is always accom- 
 
TJ1U71. 
 
 COLOURS OF THIN PLATES. 141 
 
 panied by reflexion ; and this reflexion is greater, the greater 
 the difference of the densities of the ether in the two media. 
 It appears also, from what has been said, that the direction 
 of the motions of the particles of the first medium, after they 
 communicate motion to those of the second, will be different 
 according as the ether is denser or rarer in the first medium. 
 In the former case the vibration of the particles is in the 
 same direction that it was before ; in the latter it is in the 
 opposite direction. Thus there will be a reflected wave in both 
 cases ; but in one case this reflected wave is caused by a 
 vibration in the same direction as that of the incident wave ; 
 in the other, by a vibration in an opposite direction. 
 
 The result of this difference is obviously the same as if 
 one of the systems of waves were to gain or lose half an 
 undulation on the other ; so that when the waves reflected 
 from the two surfaces of the plate should be in complete ac- 
 cordance, as far as depended on the difference of the lengths 
 of their paths, they will actually be in complete discordance, 
 and vice versa. Thus the dark rings will be formed where 
 the thickness of the plate is any even multiple of X sec 0, 
 and the bright ones where that thickness is an odd multiple 
 of the same quantity ; and the facts and the theory are re- 
 conciled. This explanation of the phenomenon was given by 
 Young. 
 
 (154) The principle which we have been illustrating has 
 been experimentally established by M. Babinet, by an inde- 
 pendent method. A pencil of rays diverging from a narrow 
 aperture is separated into two, slightly inclined to one 
 another, by means of the obtuse prism (108). These are 
 allowed to fall on a thick plate of parallel glass, whose second 
 surface is quicksilvered in one-half of its extent ; and in such 
 a manner that both may be reflected by the transparent portion 
 of that surface, or both by the opaque portion, or one by the 
 
142 COLOURS OF THIN PLATES. 
 
 former and the other by the latter. These two portions will 
 interfere, and produce fringes after reflexion ; and it is found 
 that in the two former cases the central band is white, the 
 two waves being in complete accordance : in the third case 
 i. e. when one of the pencils is reflected from the rarer, and 
 the other from the denser medium the central band is a 
 black one ; the two waves are, therefore, in complete discord- 
 ance, and their phases differ by half an undulation. 
 
 It follows from the preceding, that in the system of rings 
 formed between two object-glasses, the central spot will be 
 white, if the thin plate is of a density intermediate to those of 
 the two glasses ; for it is evident that the reflexion takes 
 place under the same conditions at the two surfaces i. e. in 
 both cases at the surface of a rarer, or in both at that of a 
 denser medium. Young verified this anticipation by inclos- 
 ing oil of sassafras between two object-glasses, one of which 
 was of flint-glass, and the other of crown-glass. 
 
 (155) We have spoken of another set of rings visible by 
 transmission. These are produced by the interference of the 
 rays directly transmitted through the plate with those which 
 penetrate it after two interior reflexions. It follows from the 
 preceding considerations that they should be complementary 
 to those seen by reflexion ; and this is observed to be the 
 case. The extreme paleness of the transmitted rings arises 
 from the great difference in the intensities of the interfering 
 pencils. 
 
 (156) The theory of thin plates, as it came from the 
 hands of Young, laboured under an imperfection, which was 
 however soon removed. It is obvious that the intensities of 
 the two portions of light, reflected from the upper and under 
 surfaces of the plate, can never be the same, the light inci- 
 dent on the second surface being already weakened by partial 
 
COLOURS OF THTN PLATES. 143 
 
 reflexion at the first. These two portions, therefore, can 
 never wholly destroy one another by interference, and the 
 intensity of the light in the dark rings can never entirely 
 vanish, as it appears to do when homogeneous light is em- 
 ployed. 
 
 Poisson was the first to point out, and to remedy, this 
 defect in the theory. It is evident, in fact, that there must 
 be an infinite number of partial reflexions within the plate, 
 at each of which a portion is transmitted ; and that it is the 
 sum of all these portions, and not the two first terms of the 
 series only, which is to be considered in the calculation of the 
 effect. When the problem is taken up in this more general 
 form, it is found that, where the effective thickness of the 
 plate is an exact multiple of the length of half a wave, the 
 intensities of the reflected and transmitted lights will be the 
 same as if it were removed altogether, and the bounding 
 media placed in absolute contact. Hence, when these media 
 are of the same refractive power, the reflected light must 
 vanish altogether, and the transmitted light be equal to the 
 incident. 
 
 Here then we have reached a point, with respect to which 
 the two theories are completely opposed. According to both, 
 a certain portion of light is reflected from the first surface of 
 the plate. This portion, in the Newtonian theory, is left in 
 all cases to produce its full effect, and there should there- 
 fore be a considerable quantity of light in the dark rings ; 
 while, in the wave-theory, it is, at certain intervals, wholly 
 destroyed by the interference of the other portions, and the 
 dark rings should be absolutely black in homogeneous light. 
 
 The latter of these conclusions seems to accord with 
 phenomena, while the former is obviously at variance with 
 them. This is clearly shown by an experiment of Fresnel. A 
 prism was laid upon a lens having its lower surface blackened, 
 a portion of the base of the prism being suffered to extend 
 
144 COLOURS OF THIN PLATES. 
 
 beyond the lens. The light reflected from this portion, 
 according to the Newtonian theory, should not surpass that 
 of the dark rings in intensity. The roughest trial is suffi- 
 cient to show that the intensities of the light in the two 
 cases are widely different, and thus to prove that the dark 
 rings cannot arise, as they are supposed to do in the theory 
 of the fits, from the suppression of the second reflexion. 
 
 (157) When a pencil of light falls upon two plates in suc- 
 cession, some of the many portions into which it is divided 
 by partial reflexion at the bounding surfaces, are frequently 
 in a condition to interfere, and to give rise to the phenomena 
 of colour. 
 
 Thus, when light is transmitted through two parallel 
 plates, slightly differing in thickness, the colour is the same 
 as that produced by transmission through a single plate, 
 whose thickness is the difference of their thicknesses, and is 
 found to be independent of the interval of the plates. This 
 phenomenon was observed by Nicholson ; and it has been 
 shown by Young to arise from the interference of two pencils, 
 one of which is twice reflected within the first plate, and the 
 other twice reflected in the second, It is obvious, in fact, 
 that if t be the thickness of the first plate, and t' that of the 
 second, the first pencil will have traversed the thickness 
 3^ + t' in glass, and the second the thickness 3^ -f t, the 
 spaces traversed in air being the same ; so that the interval 
 of retardation is the time of describing the space 2(t- t') in 
 glass. Sir David Brewster observed a similar case of inter- 
 ference, produced by two plates of equal thickness slightly 
 inclined; the thickness traversed in the two plates being 
 altered by their inclination. 
 
 (158) In the foregoing cases, the interfering pencils are 
 mixed up with, and overpowered by, the light directly trans- 
 
COLOURS OF THIN PLATES. 145 
 
 mitted, and some contrivance is necessary to make the colours 
 visible. The phenomena are^ much more obvious in the light re- 
 flected from both plates, which, on account of their inclination, 
 is thus separated from the direct light. It is obvious, in fact", 
 that the direct image of a luminous object, seen through two 
 glasses slightly inclined, will be accompanied by several 
 lateral images, formed by 2, 4, 6, &c. reflexions. These 
 images Sir David Brewster observed to be richly coloured ; 
 the bands of colour being parallel to the line of junction of 
 the two glasses, and their breadth being greater, the less the 
 inclination of the plates. The colours in the first lateral 
 image are produced by the interference of the two pencils 
 ABCDEFGH, ABCdefgh, into 
 
 which the ray is divided at the / 
 
 first surface of the second plate ; 
 
 one of these portions being re- v/\ 
 
 fleeted externally by the second 
 plate, and internally by the first, 
 while the other is reflected in- 
 ternally by the second, and exter- 
 nally by the first. The routes of 
 these portions are obviously equal 
 
 when the plates are parallel, and differ in length only by 
 reason of their inclination. 
 
 (159) The two preceding cases of interference may be 
 produced with plates of any thickness. What are termed the 
 colours of thick plates, however, are phenomena of another 
 kind, and arise in circumstances wholly different. These 
 phenomena were first observed by Newton. 
 
 In Newton's experiment a beam of light is admitted 
 through a small aperture, and received on a concavo-convex 
 mirror with parallel surfaces, the hinder of which is silvered. 
 A screen of white paper being then held at the centre of the 
 
 L 
 
146 
 
 COLOURS OF THIN PLATES. 
 
 mirror, having a hole in the middle to let the beam pass and 
 repass, a set of broad coloured rings will be depicted on it, 
 similar to the transmitted rings of thin plates. The diameters 
 of these rings vary inversely as the square roots of the thick- 
 nesses of the mirrors. 
 
 When the mirror is inclined a little, so as to throw the 
 reflected image a little to one side of the aperture, the rings 
 are formed as before ; but their centre is in the middle of the 
 line joining the aperture and its image. At this centre is a 
 spot, which changes its appearance in a remarkable manner 
 as the image recedes from the aperture, being alternately 
 dark and bright in homogeneous light, and in white light 
 assuming every gradation of tint in rapid succession. 
 
 (160) These phenomena have been shown to arise from 
 the interference of the two portions of light, which are irre- 
 gularly scattered in the passing and repassing of the ray 
 through the refracting surface. 
 
 Thus, let be the aperture through which the beam is ad- 
 mitted, and allowed to 
 fall perpendicularly on 
 a reflecting plate at A 
 and B. A portion of 
 the incident light will 
 be irregularly scat- 
 tered, in the passage of 
 the ray OAB through 
 
 'the first surface of the plate ; and this portion will diverge 
 from the point A in all directions. Let AC be one of the 
 rays which compose this scattered portion : this is reflected 
 at the second surface of the plate in the direction CD, and 
 emerges in the direction DM. But the direct ray, AB, 
 which is reflected back in BA, will again be partially scat- 
 tered in repassing through the first surface. Let AM be one 
 
COLOURS OF THIN PLATES. 147 
 
 of the rajs of this second pencil, meeting the ray DM of the 
 first at the point M ; and let us calculate their interval of re- 
 tardation. The latter has traversed the space AB + BA in 
 glass, and AM in air ; while the former has described the 
 space AC + CD in glass, and DM in air. The interval of 
 retardation is therefore the time of describing AM - DM in 
 air, plus the time of describing 2 (AB - AC) in glass ; and it 
 is easy to prove that the corresponding space in air is equal 
 to 
 
 a (2b + fia) 9 
 
 in which a denotes the distance, OA, of the screen from 
 the plate ; b the thickness, AB ; and y the distance, OM, of 
 any point on the screen from the aperture. Equating this 
 
 to n -, it follows that the successive bright and dark rings 
 will be formed where 
 
 When a is very great in comparison with 6, as is usually the 
 case, we have simply 
 
 tf 
 
 2t> 
 
 (161) The phenomena of the colours of thick plates have 
 been reproduced by M. Babinet under a more instructive 
 form. 
 
 The rays proceeding from a luminous point are refracted 
 by a lens, and are then received upon a transparent plate 
 with parallel surfaces, interposed between the lens and its 
 focus. If now this plate be slightly tarnished, or covered 
 with powder, a series of concentric rings will be formed 
 around the focal image. The innermost of these is white ; 
 and this is followed by a series of coloured rings in the order 
 
 L2 
 
148 COLOURS OF THIN PLATES. 
 
 of Newton's scale. Their diameters increase with, the distance 
 of the plate from the focus, and diminish as the thickness of 
 the plate increases. They are formed, as before, by the in- 
 terference of the two pencils, which are scattered in passing 
 through the two surfaces of the plate. 
 
 The phenomenon is reduced to its simplest conditions, 
 by receiving the light diverging from a narrow rectilinear 
 aperture upon two polished wires, stretched parallel to the 
 slit, and nearly in the same plane with it. If then the eye, 
 fortified by a lens, be placed so that the sum of the dis- 
 tances of the aperture and of the eye from each wire is the 
 same, a series of coloured fringes will be seen, formed by 
 the interference of the pencils irregularly reflected by the 
 two wires. 
 
 (162) When the interval between two glasses is filled 
 with different substances, such as water and air, or water 
 and oil, in a finely subdivided state, the portions of light 
 which have traversed them are in a condition to interfere, 
 the interval of retardation depending on the difference of the 
 velocities of light in the two media. Accordingly, coloured 
 rings will be seen, when a luminous object is viewed through 
 the glasses, the rings being similar to those usually seen by 
 transmission, but much larger. But when a dark object is 
 behind the glasses, and the incident light somewhat oblique, 
 the rings immediately change their character, and resemble 
 those of the ordinary reflected system ; one of the portions, 
 in this case, being reflected, and therefore suffering a loss of 
 half an undulation. 
 
 These phenomena were observed and explained by Young, 
 and were denominated by him the colours of mixed plates. 
 Young observed some similar effects in an unconfined medium. 
 Thus, when the dust of the lycoperdon is mixed with water, 
 the mixture exhibits a green tint by direct light, and a purple 
 
COLOURS OF THIN PLATES. 149 
 
 tint when the light is indirect ; and the colours rise in the 
 series, when the difference of the refractive densities is lessened 
 by adding salt to the water. The interval of retardation, 
 in this case, depends on the . magnitude of the transparent 
 particle. 
 
 (163) In concluding this review of the two theories, in 
 their application to the laws of unpolarized light, it should 
 be observed that any well-imagined hypothesis may be 
 accommodated to phenomena, and seem to explain them, if 
 only we increase the number of its postulates, so as to 
 embrace each new class of phenomena as it arises. In a 
 certain sense such an explanation may be said to be true, so 
 long as it is thus made to represent all known facts ; but it 
 is no longer a theory, of whose very essence it is to ascend in 
 simplicity, at the same time that it rises in generality : 
 it is " a mob" of hypothetical laws, without connexion, order, 
 or dependence. 
 
 These remarks apply to the explanation of the pheno- 
 mena of thin plates in the theory of emission. These 
 phenomena, Newton saw, could not be accounted for on 
 the bare hypothesis of molecules shot from the luminous 
 body, and subjected to the attractive and repulsive forces of 
 the bodies which they met in their progress. He was com- 
 pelled to add a new property to light, to endow the mo- 
 lecules with dispositions which seemed wholly alien to their 
 other properties, and which could only be connected with, 
 them by assuming a material mechanism much more com- 
 plicated than was at first proposed. But this was not all. 
 Each of the laws of thin plates was found to require a new 
 property in the fits to which they were referred ; and none 
 of these properties were in any way related to the rest, 
 or to the mechanism on which they were supposed to de- 
 pend. 
 
150 COLOURS OF THIN PLATES. 
 
 These imperfections of the emission-theory are still more 
 glaring when we pass from one class of phenomena to 
 another. The phenomena of diffraction, for example, are 
 referred to principles altogether different from those which 
 seemed to be required in explaining the colours of thin 
 plates ; and the two classes of phenomena, in this way of 
 accounting for them, bore no relation of any kind to each 
 other. 
 
 All this is otherwise in the wave-theory. Here the 
 several classes of phenomena are deduced from a common 
 principle, and are, therefore, mutually related. The prin- 
 ciple of interference is a necessary consequence of the nature 
 of a vibration ; and this one principle, as we have seen, ex- 
 plains in the most complete manner the laws of the other 
 phenomena. 
 
 But it is not merely in their reference to a common 
 origin that these phenomena are thus related : they are 
 even connected numerically. The simple laws of inter- 
 ference, the laws of diffraction, and those of thin plates, 
 are all dependent upon a single constant for each kind of 
 light,^-the length of a wave in each medium; and this 
 constant being inferred from any one experiment, in any 
 one class of phenomena, we can compute numerically the de- 
 tails of all the rest, and compare them with the results of 
 measurement. The agreement is found to be complete. 
 
CHAPTER IX. 
 
 POLARIZATION OF LIGHT. 
 
 (164) IN the various phenomena which take place when 
 a ray of light encounters the surface of a new medium, it has 
 been supposed that the direction and intensity of the several 
 portions into which it is subdivided will continue the same, 
 on whatever side of the ray the surface is presented, provided 
 that the angle and the plane of incidence continue unchanged. 
 In other words, it was taken for granted that a ray of light 
 had no relation to space, with the exception of that dependent 
 on its direction ; that, around that direction, its properties 
 were on all sides alike ; and that, if the ray be made to revolve 
 round that line as an axis, the resulting phenomena would 
 be unaltered. 
 
 Huygens was the first to prove that this was not always 
 the case. In the course of his researches on the law of double 
 refraction, he found that when a ray of solar light is received 
 upon a rhomb of Iceland crystal, in any but one direction, it 
 is always subdivided into two of equal intensity. But, on 
 transmitting these rays through a second rhomb, he observed 
 that the two portions into which each of them was subdivided, 
 were no longer equally intense ; that their relative brightness 
 depended on the position of the second rhomb with regard to 
 the first ; and that there were two positions in which one of 
 the rays vanished altogether. 
 
 On analyzing the phenomena, it is found that they de- 
 pend on the relative positions of the planes, passing through ^S 
 the axes of the crystals, and perpendicular to the refracting 
 
152 POLARIZATION OF LIGHT. 
 
 surfaces. Such planes are denominated principal sections. 
 "When these sections are parallel, the ray which has under- 
 gone ordinary refraction by the first crystal will be also re- 
 fracted ordinarily by the second ; and the ray which has been 
 extraordinarily refracted by the first will be also extraordi- 
 narily refracted by the second. On the contrary, when the 
 principal sections of the two crystals are perpendicular, the 
 ray which has suffered ordinary refraction by the first crystal, 
 will undergo extraordinary refraction by the second ; and 
 the extraordinary ray of the first will be refracted according 
 to the ordinary law in the second. In the intermediate 
 positions of the two principal sections, each of the rays re- 
 fracted by the first crystal will be divided into two by the 
 second; and these two pencils are in general different in 
 intensity, their intensities being measured by the squares of 
 the cosines of the distances from the position of greatest in- 
 tensity. 
 
 From this " wonderful phenomenon/' as Huygens justly 
 called it, it appears that each of the rays refracted by 
 the first rhomb has acquired properties which distinguish it 
 altogether from solar light. It has, in fact, acquired sides ; 
 and it is evident that the phenomena of refraction depend, 
 in some unknown manner, on the relation of these sides to 
 certain planes within the crystal. 
 
 (165) The phenomenon discovered by Huygens remained, 
 for more than 100 years, a unique fact in science ; and the 
 kindred phenomena the properties which light acquires in 
 a greater or less degree in almost every modification which 
 it undergoes remained unnoticed until the beginning of the 
 present century. 
 
 In the year 1808, while Malus was engaged in the 
 experimental researches by which he established the Huy- 
 genian law of double refraction, he happened to turn a 
 
POLARIZATION OF LIGHT. 153 
 
 double-refracting prism towards the windows of the Luxem- 
 bourg palace, whioh then reflected the rays of the setting 
 sun. On turning round the prism, he was astonished to find 
 that the ordinary image of the window nearly disappeared 
 in two opposite positions of the prism ; while in two other 
 positions, at right angles to the former, the extraordinary 
 image nearly vanished. Struck with the analogy of this 
 phenomenon to that which is observed when light is trans- 
 mitted through two rhomboids of Iceland spar, Malus at 
 first ascribed it to some property which the light had ac- 
 quired in its passage through the atmosphere : but he soon 
 abandoned this idea, and found that this new property was 
 impressed upon the light by reflexion at the surface of the 
 glass. 
 
 Pursuing the subject, he was led to the important disco- 
 very, that when a ray of light is reflected from the surface of 
 glass, or water, or any other transparent medium, at certain 
 angles, the reflected ray acquires all the characters which 
 belong to the light which has undergone double refraction. 
 "When received upon a rhomb of Iceland spar, or a double- 
 refracting prism, one of the two pencils into which it is di- 
 vided vanishes in two positions of the rhomb, namely, when 
 the principal section of the crystal is parallel or perpendicular 
 to the plane of reflexion ; while, in intermediate positions 
 these pencils vary in intensity- through every possible gra- 
 dation. 
 
 A ray, then, may acquire sides or poles may, in short, 
 be polarized by reflexion at the surface of a transparent 
 medium, as well as by double refraction. The plane of po- 
 larization is the plane of reflexion at which the effect is pro- 
 duced ; and it is experimentally known by its relation to the 
 principal section of a double-refracting crystal, the ray 
 undergoing ordinary refraction only, when the principal sec- 
 tion is parallel to the plane of polarization. 
 
154 POLARIZATION OF LIGHT. 
 
 (166) But a polarized ray possesses other characters. 
 "When a ray of light, proceeding directly from a self-luminous 
 body, is received upon a reflecting surface at a given angle, 
 the intensity of the reflected beam will be unaltered, whether 
 the surface be above or below, on the right or on the left of the 
 incident beam. The case however is different, when, instead 
 of the direct light of a self-luminous body, we submit to the 
 same trial light which has been already polarized. It is then 
 no longer indifferent on which side of the ray the new sur- 
 face is presented. The inclination of the reflected or trans- 
 mitted ray will, indeed, remain unaltered, on whatever side 
 the surface be presented, but its intensity will be very dif- 
 ferent ; and a ray which is reflected most intensely when the 
 new surface is presented at one side, under a certain angle, 
 will be wholly transmitted when it is offered to another, all 
 other circumstances being identical. 
 
 It is evident then, that the ray which has suffered reflexion 
 at the first surface, in this experiment, has in consequence 
 acquired properties wholly distinct from the original light. 
 The latter is equally reflected in every azimuth of the plane 
 of reflexion ; while, on the other hand, the intensity of the 
 twice-reflected ray diminishes, as the angle between the re- 
 flecting planes increases ; and it vanishes altogether, and the 
 ray is wholly transmitted, when the plane of reflexion at the 
 second surface is perpendicular to that at the first. These 
 sides, or poles, once acquired, are retained by the ray in all 
 its future course, provided it undergoes no further modifica- 
 tion by reflexion or refraction ; for, whether the plates be an 
 inch or a mile asunder, the phenomena are the same, 
 
 (167) A polarized ray, then, is distinguished by the fol- 
 lowing characters : 
 
 I. It is not divided into two pencils by a double-refracting 
 crystal, in two positions of the principal section with respect 
 
POLARIZATION OF LIGHT. 155 
 
 to the ray ; being refracted ordinarily, when the plane of 
 polarization coincides with the principal section, and extra- 
 ordinarily when it is perpendicular to it. In other cases it 
 gives rise to two pencils, which vary in intensity according 
 to the position of the principal section. 
 
 II. It suffers no reflexion at the polished surface of a trans- 
 parent medium, when this surface is presented to it at a cer- 
 tain angle, and in a plane of incidence perpendicular to the 
 plane of polarization ; while it is partially reflected, when the 
 reflecting surface is presented in other planes of incidence, or 
 under different angles. 
 
 The apparatus best fitted for the exhibition of these phe- 
 nomena is that devised by M. Biot. It consists of a tube, 
 furnished at its extremities with two graduated rings, which 
 are capable of revolving in a plane perpendicular to its axis. 
 Each of these rings carries a plate of glass set in a frame, 
 and held by two projecting arms. These plates are capable 
 of revolving round a transverse axis, so that their incli- 
 nation to the axis of the tube may be varied at pleasure ; 
 and a small graduated circle, attached to one of the arms, 
 measures the inclination. The whole apparatus is connected 
 with a vertical pillar, by a moveable joint, so that the tube 
 may be inclined to the horizon at any angle. In this form of 
 the apparatus, it is arranged to exhibit the properties of polar- 
 ized light dependent on reflexion ; in order to show the other 
 properties, one of the glass plates may be removed, and a double- 
 refracting prism, or a plate of tourmaline, substituted in its place. 
 
 (168) The angles of incidence at which light is polarized 
 are called the angles of polarization. They are in general dif- 
 ferent for different substances : thus the angle of polarization 
 of glass is 57, and that of water ^ 53. 
 
 The connexion between the polarizing angles and the other 
 properties of substances with regard to light was discovered 
 
156 POLARIZATION OF LIGHT. 
 
 by Sir David Brewster. In the year 1811 he commenced 
 an extensive series of experiments, with the view of deter- 
 mining the angles of polarization of different media, and of 
 connecting them by a law. The result of these investiga- 
 tions was the simple and remarkable principle, that " the 
 index of refraction of the substance is the tangent of the angle of 
 polarization" Hence, when the refractive index is known, the 
 angle of polarization is inferred, and vice versa ; and we learn 
 from the law that this angle ranges in different substances from 
 45 upwards, increasing always with the refractive power. 
 
 The refractive index being different, for the differently 
 coloured rays which compose solar light, it follows that all 
 the rays of the spectrum are not polarized at the same angle ; 
 so that if a beam of solar light be reflected successively by 
 two glass plates, whose planes of reflexion are at right angles, 
 the reflected beam will never be wholly extinguished, but 
 will be coloured red or blue, according as the angle of in- 
 cidence is the polarizing angle of the more, or of the less re- 
 frangible rays. When the angle of incidence is the angle of 
 polarization corresponding to the most luminous portion of the 
 spectrum, the reflected light is of a purplish tint, formed by 
 the union of the remaining rays in different proportions. 
 These effects are, of course, most observable in highly dis- 
 persive substances. 
 
 (169) The law of Brewster may be presented in another 
 form. We may say that the angle of polarization is such, that 
 the reflected and refracted rays form a right angle. In fact, if 
 this angle be denoted by TT, and the corresponding angle of 
 refraction by p, we have 
 
 sin TT sin TT 
 
 tan TT = iu, or = : 
 
 cos TT sin p 
 
 therefore, cos TT = sin /o, and 
 
POLARIZATION OF LIGHT. 157 
 
 Now the angle of reflexion is equal to the angle of incidence, 
 TT; consequently the angles of reflexion and refraction are 
 complementary, and the reflected and refracted rays are per- 
 pendicular. 
 
 (170) The law of Brewster applies to the case of light 
 reflected from the surface of a rarer, as well as that of a 
 denser medium ; and it follows from it, that the two angles of 
 polarization, at the bounding surface of the same two media, 
 are complementary. For the index of refraction, from the 
 denser into the rarer medium, is the reciprocal of the index 
 when the light proceeds in the contrary direction ; conse- 
 quently, the tangents of the angles of polarization are re- 
 ciprocals, and the angles themselves complementary. 
 
 It follows from this, that when a beam of light falls upon 
 a medium bounded by parallel planes, and at its polarizing 
 angle of the first surface, the portion which enters the 
 medium will meet the second surface also at its polarizing 
 angle, and be completely polarized by reflexion there. For 
 the ray being incident upon the first surface at the polarizing 
 angle, the angle of refraction will be the complement of the 
 angle of incidence, and will be therefore equal to the angle of 
 polarization at the second surface. But the surfaces being 
 parallel, the angle of refraction at the first surface is equal to 
 the angle of incidence at the second ; the ray will therefore 
 fall upon the second surface at its polarizing angle. 
 
 From the same principles it follows, that if several plates 
 of glass, or of any transparent substance, be arranged parallel 
 to one another, and a ray of light be incident upon the first 
 surface at the polarizing angle, the several portions which 
 reach the succeeding surfaces will meet them also at their 
 polarizing angles, and the portions reflected at each will be 
 completely polarized. Such a piles of plates is highly useful 
 
 jp^$S Off THU^^^^k 
 
 IESIT7I 
 
158 POLARIZATION OF LIGHT. 
 
 as a polarizer ; for the reflected beam is necessarily far more 
 intense than that produced by a single surface. 
 
 (171) It has been shown, that when a beam of light 
 polarized by reflexion, is suffered to fail upon a second re- 
 flecting surface at the polarizing angle, the intensity of the 
 twice-reflected beam will vary with the inclination of the 
 planes of reflexion, being greatest when these planes are 
 coincident, and vanishing when they are perpendicular. In 
 all cases, the intensity varies as the square of the cosine of the 
 angle formed by the two planes of reflexion. This law was at first 
 conjecturally assumed by Malus ; its truth has since been 
 verified by the experiments of Arago. 
 
 It follows, as a consequence of this law, that a ray of com- 
 mon light may be conceived to be composed of two polarized 
 rays of equal intensity, whose planes of polarization are per- 
 pendicular.* For if light, thus composed, is incident on a 
 reflecting surface, and if a, and 90 - a, denote the angles 
 which the plane of reflexion makes with the planes of polari- 
 zation of the two pencils, the intensity of the reflected light in 
 one of these rays will be I cos 2 a, and that in the other I sin 2 a, 
 denoting the intensity of each of the incident pencils ; and 
 the sum of these, or the total intensity of the reflected light, 
 is 
 
 I (cos 2 a + sin 2 a) = I. * 
 
 * This is not be undertood as describing the actual physical character 
 of ordinary, or unpolarized light. This may be more correctly represented 
 as polarized light, whose plane of polarization is incessantly changing ; so that, 
 in a given time, there are as many polarized rays in any one plane as in any 
 other at right angles to it, This agreement has been verified experimentally 
 by Professor Dove, by impressing mechanically a rapid motion of rotation upon 
 the plane of polarization of the light ; the phenomena presented by the resulting 
 light agreeing in all respects with ordinary or unpolarized light. 
 
POLARIZATION OF LIGHT. 159 
 
 The intensity is therefore constant, and independent of the 
 position of the plane of reflexion with respect to the ray ; and 
 this, we have seen, is the distinctive character of common, or 
 unpolarized light. 
 
 (172) We now proceed to consider the effects which take 
 place, when the light is incident upon the reflecting surface 
 at an angle different from the polarizing angle. 
 
 Malus observed, that when the angle of incidence was 
 either greater or less than the polarizing angle, the proper- 
 ties already described were only in part developed in the re- 
 flected light ; that neither of the two pencils into which it 
 was divided by a rhomb of Iceland spar ever wholly vanished ; 
 but that they varied in intensity between certain limits, these 
 limits being closer the more remote the incidence from the 
 polarizing angle. From this he naturally concluded that, in 
 these circumstances, a portion only of the reflected light had 
 received the modification to which he had given the name of 
 polarization, that portion increasing as the incidence ap- 
 proached the polarizing angle ; and that the remaining portion 
 was unmodified, or in the state of common light. Partially 
 polarized light, then, according to Malus, is composed of two 
 portions, one of which is wholly polarized, while the other 
 is in the state of ordinary or unpolarized light. . In this 
 supposition Malus has been followed by most subsequent 
 philosophers. 
 
 If this partially polarized light be reflected at a second 
 surface in the same plane, and at the same angle, the reflected 
 pencil is found to contain a greater portion of polarized light ; 
 and by increasing the number of successive reflexions, the light 
 may become, as to sense at least, wholly polarized. This fact 
 was first observed by Sir David Brewster ; and it was found 
 that light may be polarized at any incidence, by a sufficient 
 number of reflexions, the number of reflexions necessary to 
 
160 POLARIZATION OF LIGHT. 
 
 produce this result increasing as the incidence is more re- 
 moved from the polarizing angle. 
 
 (173) It remains to describe the modification which light 
 undergoes in refraction. 
 
 When common light is suffered to fall upon a plate of 
 glass, a portion of it in all cases enters the plate, and is re- 
 fracted ; and this refracted portion is found to be partially 
 polarized. The quantity of polarized light in the refracted 
 pencil increases with the incidence, being nothing at a per- 
 pendicular incidence, and greatest when the incidence is most 
 oblique. The plane of polarization is not (as in the case of 
 reflected light) coincident with the plane of incidence, but 
 perpendicular to it. 
 
 The connexion between the light thus polarized, and that 
 polarized by reflexion, is very intimate, the two portions being 
 always of equal intensity. This remarkable law was dis- 
 covered by Arago, and may be thus enunciated " When 
 an unpolarized ray is partly reflected at, and partly transmitted 
 through a transparent surface, the reflected and transmitted por- 
 tions contain equal quantities of polarized light ; and their planes 
 of polarization are at right angles to each other. 9 ' 
 
 (174) ,If the transmitted light be received upon a second 
 plate parallel to the first, the portion of common light which 
 it contains undergoes a new subdivision ; and so continually, 
 whatever be the number of plates. Hence, when that num- 
 ber is sufficiently great, the transmitted light will be, as to 
 sense, completely polarized, the plane of polarization being 
 perpendicular to the plane of incidence. These facts were 
 discovered by Malus. 
 
 It is a remarkable consequence of these principles, that 
 when a ray is incident upon a pile of parallel plates at the 
 polarizing angle, after passing a certain number the intensity 
 
POLARIZATION OF LIGHT. 161 
 
 of the transmitted light will undergo no further diminution. 
 For the light becoming wholly polarized in the plane per- 
 pendicular to the plane of incidence, no portion of it will be 
 reflected by any of the succeeding plates ; it is therefore trans- 
 smitted without diminution through them, whatever be their 
 number. The case is different, when the light is incident 
 on the pile at any other than the polarizing angle ; and it 
 follows therefore that the intensity of the light transmitted 
 through a thick pile is greatest, when it is incident at the 
 polarizing angle. 
 
 (175) There are certain crystals which, like the pile of 
 transparent plates, possess the property of polarizing the 
 transmitted light. This property depends upon a peculiarity 
 in the absorbing powers of double-refracting crystals, 
 namely, that the absorption of a polarized ray varies with the 
 position of its plane of polarization with respect to the crystal. 
 Thus, tourmaline absorbs a polarized ray more rapidly when 
 the plane of polarization is parallel to the axis, than when it 
 is perpendicular to it. Now a ray of common light, which 
 falls upon a plate of this crystal cut parallel to the axis, may 
 be resolved into two, one polarized in a plane passing 
 through the axis, and the other in a plane perpendicular 
 to it ; and the former of these being absorbed more rapidly 
 than the latter, the transmitted light will be partially polar- 
 ized in the plane perpendicular to the axis of the crystal. When 
 the plate is sufficiently thick, the latter portion alone will be 
 sensible, and the ray emerges wholly polarized in the per- 
 pendicular plane. 
 
 The tourmaline, accordingly, is of much use in experi- 
 ments on polarized light, not only in affording a ready test 
 of polarization, but also in producing a polarized beam. It 
 has the disadvantages, however, that the polarization of the 
 
162 POLARIZATION OF LIGHT. 
 
 emergent light is never perfect, and that its intensity is much 
 weakened by absorption both the rays being absorbed in 
 their passage through the crystal, although with unequal 
 energies. 
 
 The polarization produced by double refraction is the 
 most complete of any ; while the intensity of the polarized 
 pencils is greater than in any other case, being very nearly 
 one-half of the intensity of the original light. The intensity 
 of the light reflected from a plate of glass, at the polarizing 
 angle, is not more than the one-twelfth part of that of the 
 incident light. 
 
 (176) From the foregoing it would appear that the most 
 effective mode of producing a polarized ray of strong inten- 
 sity is, to transmit a ray of common light through a rhomb 
 of Iceland spar, and to shut off one of the two refracted 
 pencils by means of an opaque plate on' the second surface of 
 the rhomb. Here however there is a difficulty. The two 
 rays will overlap, unless the aperture by which the light is 
 admitted to the first surface of the rhomb is very small, or 
 the thickness of the rhomb is very considerable. The former 
 condition reduces the quantity of the light ; the latter is 
 difficult of attainment, on account of the rarity of large 
 specimens of the crystal of sufficient purity. 
 
 These difficulties may be evaded, by receiving the light 
 upon a lens of short focus, placed immediately behind the 
 rhomb. The rays which have passed through the lens and 
 the rhomb will converge to two foci, one of which may be 
 covered by a diaphragm, and the rays diverging from the 
 other suffered to proceed alone. These rays may be reduced 
 to parallelism, when required, by receiving them upon a 
 second lens, placed at its own focal distance from the point 
 of divergence. The aperture of this lens will determine the 
 breadth of the beam. It will be convenient to employ a 
 
POLARIZATION OF LIGHT. 
 
 163 
 
 plano-convex as the first lens, and to cement it with Canada 
 balsam to the rhomb. 
 
 (177) A more usual method is to stop one of the pencils 
 by total reflexion. A long rhomb of Iceland spar is taken, 
 and cut obliquely, in such manner that the refracting edges 
 of the two resulting prisms shall be parallel to the axis of 
 the crystal. The cut faces are then polished, and reunited 
 in their original positions by Canada balsam. Now the index 
 of refraction of Canada balsam is less than that of the 
 ordinary ray of Iceland spar, while it is greater than that of 
 the extraordinary ray. Hence, when the incidence is suffi- 
 ciently oblique, the ordinary ray will be totally reflected, 
 while the extraordinary ray is transmitted. The transmitted 
 light is therefore polarized. 
 
 The angle of incidence at which total reflexion takes 
 place is easily calculated. The refractive index of the ordinary 
 ray in Iceland spar = 1*654 ; and that of Canada balsam 
 = 1*549. Hence the ratio of the sines of incidence and re- 
 fraction from the crystal into the balsam, for that ray, 
 
 1*549 
 
 0'937. Accordingly, when the refracted ray just 
 
 grazes the surface, the sine of incidence = 0-937 ; and the 
 corresponding angle = 69 30'. The ordinary ray therefore will 
 be wholly reflected, when it is incident on the separating sur- 
 face at this or any greater angle, or when that surface is 
 inclined to the axis of the prism at any angle less than 20 30 X . 
 This ingenious contrivance is denominated, after the in- 
 ventor, Nicol's prism. 
 
 The foregoing arrangement has been much improved 
 by M. Foucault, by dispensing with the Canada balsam 
 altogether, and shortening the rhomb. The sines of the 
 angles of total reflexion in passing from the crystal into 
 
 M2 
 
164 POLARIZATION OF LIGHT. 
 
 air, are, for the ordinary ray, sin r- t K - 0'605, and' for 
 
 I'DOo 
 
 the extraordinary ray, sin r = TTOT = 0*674 ; and the cor- 
 
 responding angles are 37 14' and 42 23'. Hence, if the 
 length of the compound prism be so adjusted that the ray 
 shall fall upon the bounding surface at an incidence interme- 
 diate to these two angles, the ordinary ray will be wholly 
 reflected, while the extraordinary ray will be transmitted. 
 It is obvious that the length of the rhomb which will give 
 this incidence is much shorter than in Nicol's prism. 
 
 (178) M. Haidinger has observed a remarkable pheno- 
 menon of polarized light, by which it may be recognised by 
 the naked eye, and its plane of polarization ascertained. This 
 phenomenon consists in the appearance of two brushes, of a 
 pale orange-yellow colour, the axis of which coincides always 
 with the trace of the plane of polarization ; these are accom- 
 panied, on either side, by two patches of light, of a comple- 
 mentary or violet tint. In order to see them, the plane of 
 polarization of the light must be turned quickly from one 
 position to another, so as to shift the position of the brushes. 
 Thus they may be observed by looking for a few moments 
 at one of the images of a circular aperture, formed by a 
 rhomb of Iceland spar, and then at the other, and so alter- 
 nately. They gradually disappear when the eye continues 
 directed to them in the same position ; but they may be 
 made to reappear by shifting that position, or the plane of 
 polarization on which it depends. 
 
 The most probable explanation of this phenomenon seems 
 to be that given by M. Jamin, in which it is ascribed to the 
 refracting coats of the eye. When polarized light falls upon 
 a pile of parallel plates, the proportion of the refracted to the 
 incident light varies with the plane of polarization, being a 
 
POLARIZATION OF LIGHT. 165 
 
 minimum when that plane coincides with the plane of inci- 
 dence, and a maximum when it is perpendicular to it. These 
 variations are nothing at a perpendicular incidence : they are 
 greatest when the angle of incidence is equal to the angle 
 of polarization. Accordingly, when the polarized light is 
 incident obliquely on the plates, the refracted light should 
 exhibit two dark brushes, enlarging from the centre to the 
 circumference, in the plane of polarization ; and two bright 
 brushes in the perpendicular plane. 
 
 In the preceding explanation, the incident light is sup- 
 posed to be homogeneous. When white light is used, the inten- 
 sities of its several components, in the refracted pencil, will 
 vary with the refractive indices, and consequently the brushes 
 will be coloured. M. Jamin has shown that the effect of a 
 single refracting surface will be to produce two yellow brushes, 
 whose axis is in the plane of polarization. 
 
( 166 ) 
 
 CHAPTER X. 
 
 TRANSVERSAL VIBRATIONS THEORY OF REFLEXION AND RE- 
 FRACTION OF POLARIZED LIGHf. 
 
 (179) HAVING in the preceding chapter stated the prin- 
 cipal facts of polarization, we may proceed to consider their 
 connexion with the physical theory. 
 
 It is strange that the department of optics, in which the 
 wave-theory now stands unrivalled, should be the very one 
 which Newton selected as affording the most decisive evidence 
 against it : "Are not," says he, " all hypotheses erroneous, 
 in which light is supposed to consist in pressure, or motion, 
 propagated through a fluid medium ? .... for pressures 
 or motions, propagated from a shining body through an uni- 
 form medium, must be on all sides alike ; whereas it appears 
 that the rays of light have different properties in their diffe- 
 rent sides/' In this objection Newton seems to have had his 
 thoughts fixed upon that species of undulatory propagation 
 whose laws he himself had so sagaciously unfolded. When 
 sound is propagated through air, the vibrations of the particles 
 of the air are performed in the same direction in which the 
 wave advances ; and if the vibrations of the ether which con- 
 stitute light had been of the same kind, the objection would 
 be insuperable. For, if the particles of the ether vibrated 
 in the direction of the ray itself, the ray could not bear a 
 different relation to the different parts of the. surrounding 
 space. 
 
 But the case is altered, if the vibrations of the ethereal 
 particles be performed in a transverse direction. Let us sup- 
 
TRANSVERSAL VIBRATIONS. 167 
 
 pose the direction of the vibrations to be perpendicular to 
 that of the ray : then it is obvious that if that direction be 
 vertical, for example, while the ray advances horizontally, 
 the ray will bear a relation to the parts of space above and 
 below , different from that which it bears to those parts which 
 are on the right hand and on the left. Such is, in fact, the 
 mode of vibration which is now assumed to belong to the 
 ether, in the wave-theory, the ethereal molecules being sup- 
 posed to vibrate in the plane of the wave ; and we shall find 
 that, with the help of this assumption, all the complicated 
 phenomena of polarization and double refraction are explained 
 in the fullest and most complete manner. 
 
 The principle of transversal vibrations, as it is called, seems 
 to have first occurred to Hooke, and was announced, in 1672, 
 in his Micrographia. Young and Fresnel arrived at the same 
 principle independently ; and the latter has reared upon its 
 basis the noblest fabric which has ever adorned the domain 
 of physical- science, Newton's system of the universe alone 
 excepted. 
 
 (180) In order to conceive the manner in which an un- 
 dulation may be propagated by transversal vibrations, let us 
 imagine a cord stretched in a horizontal position, one end 
 being attached to a fixed point, and the other held in the 
 hand. If the latter extremity be made to vibrate, by moving 
 the hand up and down, each particle of the cord will, in suc- 
 cession, be thrown into a similar state of vibration ; and a 
 series of waves will be propagated along it with a uniform 
 velocity. The vibrations of each succeeding particle of tHe 
 cord, being similar to that of the first, will all be performed 
 in the same plane, and the whole will represent the state of 
 the ethereal particles in a polarized ray. 
 
 Now if, after a certain number of vibrations in the verti- 
 cal plane, the extremity of the cord be made to vibrate in 
 
168 TRANSVERSAL VIBRATIONS : 
 
 another plane, and then in another, and so on, in rapid 
 succession, each particle of the cord will, after a time pro- 
 portional to its distance from the extremity, assume in suc- 
 cession all these varied vibrations; and the whole cord, 
 instead of taking the form of a plane curve, as in the last 
 case, will be thrown into a species of helical curve, depending 
 on the nature of the original disturbance. Such is the con- 
 dition of the ethereal particles in a ray of common, or un- 
 polarized light. 
 
 "When, therefore, we admit a connexion to subsist among 
 the particles of the ether, such as that which holds among 
 the particles of the cord, there -is no difficulty in conceiving 
 how a vibration may be propagated in a direction perpendi- 
 cular to that in which it is executed. It is true, the particles 
 of the ether are not chained together by cohesive forces, like 
 those of the cord ; but the attractive forces which subsist 
 among them are of the same kind, and may be shown to pro- 
 duce a similar effect. In fact, let us conceive the ether to 
 be composed of separate molecules, which act on one another 
 according to some law varying with the distance. When any 
 row or line of such molecules is similarly displaced, through a 
 space which is small compared with the separating intervals, 
 the molecules of the succeeding row will be moved in the 
 same direction by the forces developed with the change of 
 distance ; so that the vibrations of the particles composing 
 the first row will be communicated to those of the second, and 
 thus the vibratory motion will be propagated in a direction 
 perpendicular to that in which it takes place. The rapidity 
 of the propagation will depend on the magnitude of the force 
 developed by the displacement. 
 
 To account for the fact, that there are no sensible vibra- 
 tions in a direction normal to the wave, we have only to sup- 
 pose the repulsive force between the molecules to be very 
 great, or the resistance to compression very considerable. 
 
REP'LEXION AND REFRACTION OF POLARIZED LIGHT. 169 
 
 For, in this case, the force which resists the approach of two 
 strata of the fluid is much greater than that which opposes 
 their sliding on one another. 
 
 (181) But the existence of transversal vibrations and of 
 transversal vibrations only is a necessary consequence of the 
 laws of interference of polarized light, if the theory of waves 
 be admitted at all. It has been experimentally proved, by 
 Fresnel and Arago, that two rays oppositely polarized com- 
 pound a single ray whose intensity is constant, whatever be 
 the phases of vibration in which they meet. But theory shows, 
 that the intensity of the light resulting from the union of two 
 rays oppositely polarized will be constant, and independent 
 of the phase, only ivhen the vibrations normal to the wave are 
 evanescent. 
 
 This conclusion is easily extended to the case of common, 
 or unpolarized light. In unpolarized light, therefore, as in 
 polarized, the vibrations are only in the plane of the wave ; 
 but in the latter, these vibrations are all parallel to a fixed 
 line, while in the former they take place in every possible 
 direction in the plane of the wave. The phenomenon of po- 
 larization consists simply in the resolution of these vibrations 
 into two sets, in two rectangular directions, and the subse- 
 quent separation of the two systems of waves thus produced. 
 When the resolved vibrations are not separated, but one of 
 them is diminished in any ratio, the light is said to be partially 
 polarized. 
 
 (182) We have stated that the vibrations of the molecules 
 of the ether, in a polarized ray, are all parallel to a fixed 
 direction in the plane of the wave. This fixed direction may 
 be either parallel or perpendicular to the plane of polarization; 
 and there is nothing in the phenomena, hitherto dis- 
 covered, to determine the choice between these two positions. 
 
170 TRANSVERSAL VIBRATIONS : 
 
 Hence, contrary suppositions have been made respecting it. 
 In the theories of Fresnel and of Cauchy, the vibrations are 
 assumed to be perpendicular to the plane of polarization, in 
 those of MacCullagh and Neumann, to be parallel to it. 
 
 Professor Stokes has however arrived at a result, in the 
 dynamical theory of diffraction, which seems to afford the 
 means of deciding between these hypotheses. When a polar- 
 ized ray is diffracted, the plane of vibration of the diffracted 
 ray should differ from that of the incident, the positions of 
 the two planes being connected by a very simple relation. 
 This relation may be deduced in the following elementary 
 manner. 
 
 "When a polarized ray is incident perpendicularly upon a 
 fine grating, the direction of its vibrations is (by the principle 
 of transversal vibrations) in the plane of the grating, when 
 the wave reaches it. Let a denote the angle formed by that 
 direction with the lines of the grating : then, if the ampli- 
 tude of the incident vibration be taken equal to unity, it may 
 be resolved into two, namely, cos a, parallel to the lines of 
 the grating, which will be unaltered by diffraction ; and sin a, 
 perpendicular to them. The second component is to be re- 
 solved again, in the direction of the diffracted ray, and per- 
 pendicular to it, respectively ; and of these the latter portion 
 alone is propagated as light. Its value is sin a cos 9, 
 6 being the angle which the diffracted ray makes with the 
 incident. Hence the two components of the diffracted ray 
 are 
 
 cos a, and sin a cos 9 ; 
 
 and their ratio is equal to the tangent of the angle which 
 the direction of the vibration in the diffracted ray makes 
 with the lines of the grating. Denoting this angle by a', 
 we have therefore, 
 
 tan a' = tan a cos 9. 
 
REFLEXION AND REFRACTION OF POLARIZED LIGHT 171 
 
 Accordingly, the angle which the direction of the vibration 
 makes with the lines of the grating is less in the diffracted 
 than in the incident ray. 
 
 It remains, therefore, only to measure the angles which the 
 planes of polarization of the incident and diffracted rays make 
 respectively ;with the lines of the grating. If the latter is less than 
 the former, the vibrations are parallel to the plane of polari- 
 zation ; if it be greater, they are perpendicular to it. The 
 experiment has been made by Professor Stokes himself ; and 
 he has drawn the conclusion that the latter is the fact, and, 
 consequently, that the hypothesis of Fresnel is the true one. 
 
 The foregoing conclusion seems to be fully confirmed by 
 the phenomena of double refraction. In a uniaxal crystal, the 
 ordinary ray is polarized in the plane passing through the ray 
 and the axis of the crystal, and the extraordinary ray in the 
 perpendicular plane. But the velocity of the ordinary ray 
 is constant, and therefore also the elasticity of the medium 
 in the direction of the vibrations ; while, in the extraordi- 
 nary ray, both are variable with the direction of the ray. 
 Hence the direction of the vibrations, in the ordinary ray, 
 must be perpendicular to the plane passing through the ray 
 and the axis of the crystal ; while, in the extraordinary ray, 
 the direction of the vibrations is in the plane passing through 
 the ray and the axis. In both, therefore, the direction of 
 the vibrations is perpendicular to the plane of polariza- 
 tion. 
 
 (183) We now proceed to consider the application of the 
 principle of transversal vibrations to the problem of reflexion 
 and refraction. 
 
 The direction of the light reflected and refracted at the 
 surface of a uniform medium is a simple consequence of the 
 theory of waves ; and we have already explained Huygens* 
 demonstration of the laws which govern this direction a 
 
1 72 TRANSVERSAL VIBRATIONS I 
 
 demonstration which holds good, whatever be the magnitude 
 and direction of the propagated vibration, or, in other words, 
 whatever be the intensity and plane of polarization of the 
 light. The problem which we have now to consider is that 
 which proposes to determine the latter quantities, or to 
 deduce the intensities and planes of polarization of the re- 
 flected and refracted pencils, those of the incident pencil being 
 given. 
 
 This important problem was solved by Fresnel, upon the 
 following principles : 
 
 I. The movements of the molecules of the ether, in a di- 
 rection parallel to the separating surface of the two media, 
 are unaltered ; and, consequently, the resolved parts of 
 the vibrations parallel to that surface are equal in the two 
 media. 
 
 II. The known law of the vis viva holds in the reflexion 
 and refraction of light ; from which it follows, that the masses 
 of ether put in motion, multiplied by the squares of the 
 amplitudes of vibration, are the same before and after re- 
 flexion. 
 
 III. The densities of the ether in the two media are as 
 the squares of their refractive indices. 
 
 (184) Let v and w denote the amplitudes of the reflected 
 and refracted vibrations, that of the incident vibration being 
 taken as unity. When the light is polarized in the plane of 
 incidence, these three vibrations are all parallel to the bound- 
 ing surface, and to one another. Hence, the first principle 
 gives, in this case, the simple condition 
 
 1 + v = w. 
 
 On the other hand, the law of the vis vim furnishes the 
 relation 
 
 m 
 
REFLEXION AND REFRACTION OF POLARIZED LIGHT. 173 
 
 m and m' denoting the corresponding masses of the ether in 
 motion in the two media. Eliminating between these equa- 
 tions we have 
 
 m + m' 1 m + m' 
 
 expressions which are remarkable as being identical with those 
 of the velocities of two elastic balls after impact. 
 
 In order to determine m and m' 9 let BA, AC represent 
 the velocities and directions of the incident and refracted 
 rays ; AA X the separating surface of the two media ; and 
 BB X , CC', lines parallel to that surface. Then the corres- 
 ponding volumes of the ether in motion in the two media are 
 as the parallelograms A X B, A X C ; 
 that is, as AB sin A X AB to 
 AC sin A X AC, or in the ratio of 
 
 sin i cos i : sin r cos r, 
 
 A \ 
 i and r being the angles of inci- V 
 
 dence and refraction. But by the V 
 
 third of the foregoing principles, c 
 
 the densities of the ether in the two media are in the ratio 
 
 of 1 to ju 2 , or as 
 
 sinV : sin 2 ^. 
 
 Combining these ratios, the ratio of the masses is 
 
 m / _ tan i 
 m tan r 
 
 Substituting this value of in the second of the two equa- 
 tions of condition, and dividing by the square of the first, w is 
 eliminated, and we find 
 
 1 tan i 
 
 1 + v~ tan r 
 Wherefore 
 
 sin (i -r) - 2 cos i sin r 
 
 V= : -r-. (> t?.l40 : 7: r-' 
 
 sin ( i + r) sin (i + r) 
 
174 TRANSVERSAL VIBRATIONS ! 
 
 (185) In the case of light polarized perpendicularly to the 
 plane of incidence, the vibrations are all in the plane of inci- 
 dence ; and, since they are also perpendicular to the rays 
 themselves, the condition given by the first principle is 
 
 (1 + 0') cos i = w' cos r ; 
 
 the other conditions remaining the same as in the former case. 
 Eliminating, as before, we have 
 
 1 - v _ sin i cos i 
 1 + v sin r cos r 
 "Whence 
 
 ^^ /= tanfc'-r) ^ = sin 2 i 
 
 tan (i + r)' sin (i+ r) cos (i - r)' 
 
 (186) The intensity of the light is measured by the vis viva, 
 or by the mass of the ether put in motion, multiplied by the 
 square of the amplitude of the vibration. Hence, for light 
 polarized in the plane of incidence, the intensities of the inci- 
 dent, reflected, and refracted rays will be m, mv z , and w'p 2 , 
 respectively ; or, if we take the intensity of the incident 
 light as unity, 
 
 1, v*, and 1 - v 2 ; 
 
 since, by the law of the vis viva, m (1 - v 2 ) = m'w n . Simi- 
 larly, for light polarized in the perpendicular plane, the in- 
 tensities in the three pencils are 
 
 1, v /2 , and 1-V 2 . 
 
 (187) Confining our attention for the present to the re- 
 flected vibration, it will be easily seen that, for light polar- 
 ized in the plane of incidence, the value of the amplitude v, 
 and consequently the intensity of the light, increases with 
 
REFLEXION AND INFRACTION OF POLARIZED LIGHT. 175 
 
 the incidence, being least when the light is incident perpen- 
 dicularly, and greatest when it is most oblique. On the 
 other hand, the value of 0', for light polarized perpendicu- 
 larly to the plane of incidence, diminishes from the perpen- 
 dicular incidence to the polarizing angle, at which it va- 
 nishes, and afterwards increases with the incidence up to 
 t = 90. 
 
 "When i = 0, the values of v and v' are each equal to 
 
 - =-, fj, being the refractive index ; and the intensity of the 
 light reflected perpendicularly is 
 
 This remarkable expression was first given by Young. 
 
 On the other hand, when i = 90, or when the ray grazes 
 the surface, the intensity of the reflected light is equal to 
 that of the incident ; or the whole of the light is reflected, 
 whatever be the reflecting substance. 
 
 (188J We have seen that a ray of common light is equiva- 
 lent to two polarized rays of equal intensity, whose planes of 
 polarization are at right angles. Now, let such a ray, whose 
 intensity = 1, be incident upon the surface of a transparent 
 
 medium ; and let it be resolved into two, each equal to -, po- 
 
 Z 
 
 larized respectively in the plane of incidence, and in the per- 
 pendicular plane. Each of these polarized rays will give 
 rise to a reflected and refracted ray; so that the actual 
 reflected and refracted rays will consist of two portions, 
 one polarized in the plane of incidence, and the other in 
 the perpendicular plane. If these portions were of equal 
 intensity, as they are in the incident light, the reflected and 
 
176 TRANSVERSAL VIBRATIONS : 
 
 refracted rays would be unpolarized : but this, in general, is 
 not the case. 
 
 In the case of the reflected beam, the inten sities of the 
 two portions are 
 
 /*; 
 
 the whole intensity of the reflected light is their sum, or 
 
 J= 1 (; + t/ 8 ). 
 
 Now, the first thing to be observed is, that the quantities 
 v 2 and v fz are unequal ; or, that the two portions of which the 
 reflected light consists, and which are polarized in opposite 
 planes, are different in intensity. Hence the reflected light 
 will not be of the nature of common, or unpolarized light ; 
 but will have -an excess of light polarized in the plane of inci- 
 dence, the former quantity being always greater than the 
 latter. This is otherwise expressed by saying, that the light 
 is partially polarized in the plane of incidence. The quantity 
 of polarized light is measured by the difference of the two 
 portions, or by 
 
 Again, the intensities of the two refracted portions are 
 
 As the latter of these quantities is greater than the former, 
 the refracted beam always contains an excess of light po- 
 larized perpendicularly to the plane of incidence. Their dif- 
 ference, \ (tf - v' 2 ), is the same as in the former case ; and 
 accordingly, the reflected and refracted pencils contain equal 
 quantities of oppositely polarized light. Thus, the experi- 
 mental law of Arago is a necessary consequence of theory. 
 
REFLEXION AND REFRACTION OF POLARIZED LIGHT. 177 
 
 (189) The reflected light will be completely polarized, 
 when one of the portions of which it consists vanishes ; for, 
 in this case, the whole of the reflected light will be polarized 
 in a single plane. It is easy to see that the first portion, 
 which is polarized in the plane of incidence, can never vanish. 
 The second part vanishes, when 
 
 i + r = 90, 
 
 the denominator of the fraction becoming infinite ; the re- 
 flected light then contains only the other portion, and is 
 therefore completely polarized in the plane of incidence. From 
 the foregoing relation it follows that 
 
 sin i i i 
 
 cos ^ = sm r = , and tan i =ju. 
 
 /* 
 
 i. e. the tangent of the angle of polarization is equal to the re- 
 fractive index. Thus the beautiful law, which Brewster had 
 inferred from observation, is deduced as an easy consequence 
 of Fresnel's theory. 
 
 When i + r is greater than 90, i. e. when the angle 
 of incidence exceeds the polarizing angle, the expression for 
 the amplitude of the reflected vibration w, changes sign, the 
 light being polarized perpendicularly to the plane of inci- 
 dence. This change of sign is equivalent to an alteration 
 of the phase of the reflected vibration by 180, as the in- 
 cidence passes the polarizing angle ; and the circumstance 
 explains the remarkable fact noticed by Arago, namely, that 
 when Newton's rings are formed between a lens of glass v 
 and a metallic reflector (the incident light being polarized 
 perpendicularly to the plane of reflexion), the rings change 
 their character as the incidence passes the polarizing angle 
 of the glass, the black centre being transformed into a white 
 one, and the whole system of colours becoming complementary 
 to what it was before. Sir Gr. B. Airy was led to anticipate this 
 
 N 
 
178 TRANSVERSAL VIBRATIONS : 
 
 result from a consideration of the formula, and to show 
 that a similar change must take place in the rings formed 
 between two transparent -substances of different refractive 
 powers, as the incidence passes the polarizing angle of either 
 substance. 
 
 (190) When a polarized ray undergoes reflexion, the 
 reflected light is still polarized, but its plane of polarization 
 is changed, the amount of the change depending on the in- 
 cidence. When the angle of incidence is nothing, or the 
 ray perpendicular to the reflecting surface, the new plane of 
 polarization is inclined to the plane of incidence by the same 
 angle as the old, but on the opposite side. As the angle of in- 
 cidence increases, the plane of polarization of the reflected ray 
 approaches the plane of incidence, and finally coincides with 
 it, when the incidence reaches the polarizing angle. As the 
 angle of incidence still further increases, the plane of polariza- 
 tion of the reflected ray crosses the plane of incidence, and 
 therefore lies on the same side of it with the original plane ; 
 and the two planes of polarization finally coincide, when the 
 angle of incidence is 90. 
 
 The azimuth of the plane of polarization of the reflected 
 ray may be deduced from the theory we have been consider- 
 ing. For, let the vibration of the incident ray, a, be re- 
 solved into two, one in the plane of incidence, and the other 
 in the perpendicular plane. If a denote the angle which it 
 makes with the plane of incidence, these resolved portions 
 are a cos a, and a sin a. After reflexion they become,* re- 
 spectively, 
 
 sin (i-r) - tan (i-r) t 
 
 - a cos a _ : _ -, a sin a - - ; 
 sin (i + r) tan (i+r) 
 
 * In order to explain the facts above mentioned, the values of v and v' 
 (184, 5) must be affected with opposite signs; and there are other phenomena 
 
REFLEXION AND REFRACTION OF POLARIZED LIGHT. 179 
 
 and they compound a single vibration, inclined to the plane 
 of incidence at an angle whose tangent is the ratio of the 
 component vibrations. If, then, this angle be denoted by 
 a', we have 
 
 tan a' = - tan a 
 
 cos (i + r) 
 cos (i - r) 
 
 The truth of this formula has been verified by the obser- 
 vations of Fresnel himself, and more fully since by those of 
 Arago and Brewster. 
 
 When a = 0, a! = ; and when a = 90, a' = 90. Accord- 
 ingly, when the plane of polarization of the incident ray 
 coincides with, or is perpendicular to, the plane of incidence, 
 it is unchanged by reflexion. When i + r = 90, a' = 0, and 
 the plane of polarization of the reflected ray coincides with 
 the plane of incidence, whatever be the azimuth of the inci- 
 dent ray. 
 
 (191) The plane of polarization of a polarized ray is 
 changed by refraction, as well as reflexion, but in an opposite 
 direction, the plane being removed farther from the plane 
 of incidence, instead of approaching it. This movement of 
 the plane of polarization increases with the incidence ; being 
 nothing when the ray falls perpendicularly upon the re- 
 fracting surface, and greatest when the incidence is most 
 oblique. The law of the change is expressed by the simple 
 formula, 
 
 cot a' = cot a cos (i-r) ; 
 
 which indicate that the former quantity is negative, and the latter positive (see 
 Professor Powell's paper " On the Demonstration of Fresnel's Formulas," Phil, 
 Mag. Aug. 1856). This is equivalent to saying, that one of the waves gains, or 
 loses, half an undulation in the act of reflexion. We shall see hereafter that the 
 complete theory of reflexion includes a progressive change of phase ; and that 
 the conclusions of Arts. (,184, 5) are only approximate. 
 
 N2 
 
180 TRANSVERSAL VIBRATIONS. 
 
 in which a and a' denote (as before) the angles which the 
 planes of polarization form with the plane of incidence, be- 
 fore and after refraction. This law was discovered experi- 
 mentally by Sir David Brewster : it is a necessary conse- 
 quence of the theory already given, and is deduced by a 
 process exactly similar to that of the preceding article. 
 
 When the refracted ray meets a second surface parallel 
 to the former, and emerges into air, the azimuth of the plane 
 of polarization of the emergent ray will be given by the 
 formula 
 
 cot a" = cot a f cos (r - i) ; 
 
 wherefore, since cos (r - i) = cos (i-r), we have, for the 
 change of the plane of polarization, produced by transmission 
 through a plate bounded by parallel surfaces, 
 
 cot a" = cot a cos 2 (i - r). 
 
CHAPTEE XI. 
 
 ELLIPTIC POLARIZATION. 
 
 (192) WHEN an ethereal molecule is displaced from its 
 position of equilibrium, the forces of the neighbouring mole- 
 cules are no longer balanced, and their resultant tends to drive 
 the particle back to its position of rest.* The displacement 
 being supposed to be very small, in comparison with the in- 
 tervals between the molecules, the force thus excited will be 
 proportional to the displacement ; and from this it follows, 
 according to known mechanical principles, that the trajectory 
 described by the molecule will be an ellipse, whose centre 
 coincides with the position of equilibrium. Hence the vibra- 
 tion of the ethereal molecules is, in general, elliptic, and the 
 nature of the light depends on the direction and relative mag- 
 nitude of the axes. By the principle of transversal vibrations, 
 these elliptic vibrations are all in the plane of the wave ; their 
 axes, however, may either preserve constantly the same direc- 
 tion in that plane, or they may be continually shifting. In 
 the former case, the light is said to be polarized ; in the lat- 
 ter, it is unpolarized, or common light. 
 
 The relative magnitude of the axes of the ellipse deter- 
 mines the nature of the polarization. When the axes are 
 equal, the ellipse becomes a circle, and the light is said to be 
 circularly polarized. On the other hand, when the lesser axis 
 vanishes, the ellipse becomes a right line, and the light is 
 
 * This is not strictly true, except in homogeneous or uncrystallized 
 media. 
 
182 ELLIPTIC POLARIZATION. 
 
 plane-polarized, the vibrations being in this case confined 
 to a single plane passing through the direction of the ray. 
 In intermediate eases, the polarization is called elliptical; and 
 its character may vary indefinitely between the two extremes 
 of plane polarization and circular polarization. 
 
 (193) An elliptic vibration may be regarded as the re- 
 sultant of two rectilinear vibrations, at right angles to one 
 another, which differ in phase. 
 
 For, let x and y denote the distances of the molecule of 
 the ether from its position of rest, in the two rectangular 
 directions ; a and b the amplitudes of the component vibra- 
 tions ; and t the time. Then 
 
 x = a sin (vt -a), y = b sin (vt - j3) ; 
 whence 
 
 a - j3 = arc f sin = |) - arc (sin = X \ 
 
 Taking the cosines of both sides, and clearing the result of 
 radicals, we obtain 
 
 This is the equation of an ellipse referred to its centre. 
 
 When the .component rays are in the same phase, or 
 a = j3, the equation is reduced to 
 
 which is the equation of a right line passing through the 
 origin. In this case, therefore, the vibration becomes recti- 
 linear, and the light is plane-polarized. 
 
 When the component vibrations are equal in amplitude, 
 and differ 90 in phase y 
 
 a = b and a-3 = 90 
 
ELLIPTIC POLARIZATION. 183 
 
 and the preceding equation becomes 
 
 y* + x* = a*. 
 
 The path described by the molecule is then a circle. The 
 same thing is true, when a - /3 is any odd multiple of 90. 
 
 (194) The nature of the elliptic polarization is completely 
 denned, when we know the direction of the axes of the ellipse, 
 and the ratio of their lengths. 
 
 These may be determined experimentally. In fact, when 
 the elliptically-polarized ray is transmitted through a double- 
 refracting prism, whose principal section is parallel to one 
 of the axes of the ellipse, it is resolved into two plane-po- 
 larized rays, one of which has the greatest possible intensity, 
 and the other the least. Accordingly, the direction of the 
 principal section, for which the two pencils are most unequal, 
 is the direction of one of the axes ; and the square roots of 
 the intensities are in the ratio of their lengths. 
 
 The direction of the axes of the ellipse may be more con- 
 veniently determined by turning the prism until the two 
 pencils are of equal intensity : the principal section is then 
 inclined at an angle of 45 to each of the axes. 
 
 (195) When a plane-polarized ray undergoes reflexion, 
 the reflected light is, generally, elliptically polarized. For a 
 plane-polarized ray may be resolved into two, polarized re- 
 spectively in the plane of incidence, and in the perpendicular 
 plane ; and we shall presently see that the effect of reflexion 
 is, in general, to alter the phases of these two portions, and 
 by a different amount. Hence the reflected light is com- 
 pounded of two plane-polarized rays, whose vibrations are at 
 right angles, and whose phases are no longer coincident ; it 
 is therefore elliptically polarized (193). 
 
 The first case in which this effect was observed was that 
 of total reflexion. 
 
384 ELLIPTIC POLARIZATION. 
 
 
 
 When the angle of incidence exceeds the angle of total 
 reflexion (the light passing from the denser into the rarer 
 medium), the expressions for the amplitudes of the reflected vi- 
 brations, given in (184, 5), "become imaginary. But it is obvious 
 that, in this case, the intensity of the reflected light is simply 
 equal to that of the incident, there being no refracted pencil. 
 How, then, are the imaginary expressions to be interpreted ? 
 They signify, according to Fresnel, that the periods of vibra- 
 tion of the incident and reflected waves, which had been 
 assumed to coincide at the reflecting surface, no longer 
 coincide there when the reflexion is total ; or, in other words, 
 that the ray undergoes a change of phase at the moment of 
 reflexion. The amount of this change has been deduced 
 by Fresnel, by an ingenious train of reasoning, based upon 
 the interpretation of imaginary formulae. It varies with the 
 incidence ; and is different for light polarized in the plane of 
 incidence, and in the perpendicular plane. 
 
 In the case of light polarized in any azimuth, we have only 
 to conceive the incident vibration resolved into two, one in 
 the plane of incidence, and the other in the perpendicular 
 plane. The phases of these vibrations being differently 
 altered by reflexion, the reflected vibration will be the re- 
 sultant of two vibrations at right angles to one another, and 
 differing in phase, the amount of the difference depending 
 upon the angle of incidence : this vibration, consequently, 
 will be elliptic, and the reflected light elliptically-polarized. 
 When the azimuth of the plane of polarization of the incident 
 ray is 45, the amplitudes of the resolved vibrations will be 
 equal ; and if, moreover, their difference of phase is a quarter 
 of an undulation, the ellipse will become a circle, and the 
 light will be circularly-polarized. 
 
 (196) Eeducing his formulae to numbers, in the case of 
 St. Grobain's glass, Fresnel found that the difference of phase 
 
ELLIPTIC POLARIZATION. 185 
 
 of the two portions of the reflected light amounted to one- 
 eighth* of an undulation, when the angle of incidence was 
 54 37 X . Polishing, therefore, a parallelepiped of this glass, 
 whose faces of incidence and emergence were inclined to the 
 other faces at these angles, 
 it followed that a ray 
 RR'R^R"', incident per- 
 pendicularly on one of 
 these sides, and once re- 
 flected at each of the others, at R' and R", would emerge 
 perpendicularly at the remaining side, the difference of phase 
 in the two portions of the twice-reflected ray amounting to a 
 quarter of an undulation. If, then, the incident ray be pola- 
 rized in a plane inclined at an angle of 45 to the plane of 
 reflexion, the emergent light will be circularly-polarized. 
 This was found to be the case on trial, and the theory 
 thereby verified. The parallelepiped described is well known 
 under the name of Fresnel's rhomb ; and is of essential 
 service in all experiments relating to circular and elliptic 
 polarization. 
 
 If the circularly-polarized ray be made to undergo two 
 more total reflexions, in the same plane and at the same 
 angle, by transmitting it through a second rhomb placed 
 parallel to the first, it will emerge plane-polarized ; and its 
 plane of polarization will be inclined 45 on the other side of 
 the plane of reflexion. In fact, the two additional reflexions 
 increase the difference of phase of the two portions, into which 
 the light was originally resolved, from 90 to 180 ; and we 
 know that two equal vibrations, whose phases differ by 180, 
 compound a single right-lined vibration, whose direction 
 
 * In order to produce a difference of phase of ajjuarter of an undulation by 
 a single reflexion, the refractive index should bey 
 
 ^ . , 
 
186 ELLIPTIC POLARIZATION. 
 
 bisects the supplement of the angle formed by their direc- 
 tions. 
 
 This property enables us to distinguish a circularly-po- 
 larized ray from a ray of common light. On the other hand, 
 it is at once distinguished from plane-polarized light, by the 
 circumstance that it is divided into two rays of equal inten- 
 sity by a double-refracting crystal, whatever be the position 
 of the plane of the principal section. 
 
 (197) The effects produced upon light by reflexion at the 
 surfaces of metals were first observed by Malus. 
 
 Malus found that metals differed from transparent bodies, 
 in their action upon light, in this, that common light was 
 never completely polarized by reflexion at their surfaces. The 
 phenomenon of polarization was, however, partially produced; 
 and the effect increased to a maximum at a certain angle of 
 incidence. 
 
 When the incident light was polarized in a plane inclined 
 at an angle of 45 to the plane of incidence, Malus observed 
 that the reflected light was completely depolarized, the pencil 
 being divided into two by a double-refracting prism in every 
 position of its principal section. 
 
 (198) The subject of metallic reflexion was next studied 
 by Sir David Brewster, and the laws of the phenomena 
 investigated in much detail. These laws may be reduced to 
 the following : 
 
 I. When a ray of common light is incident upon a me- 
 tallic reflector, the reflected light is partially polarized, the 
 amount of polarized light in the reflected pencil increasing 
 up to a certain incidence, which is thence called the angle of 
 maximum polarization. 
 
 II. When the light is reflected several times in succes- 
 
ELLIPTIC POLARIZATION. 187 
 
 sion, in the same plane and at the same angle, the proportion 
 of polarized light in the reflected pencil is increased ; and by 
 a sufficient number of reflexions the light becomes, as to 
 sense, wholly polarized in the plane of incidence. 
 
 III. When a ray polarized in the plane of incidence, or 
 in the perpendicular plane, falls upon a metallic reflector, it 
 is polarized in the same plane after reflexion. 
 
 IY. A ray polarized in any other plane is, in general, 
 partly depolarized by reflexion, the effect produced being 
 greatest at the angle of maximum polarization. * * "\ 
 
 Y. When light, so depolaVized^uSd^rgoes a second re- 
 flexion in the same plane, and at the same angle, its polariza- 
 tion is restored. The new plane of polarization lies on the 
 opposite side of the plane of incidence from the original 
 plane, and its azimuth is changed. 
 
 (199) From the last of the foregoing laws it is evident, 
 that the light produced by the reflexion of a polarized ray by 
 a metallic surface is not common light. Neither is it plane- 
 polarized light, since it does not vanish in any position of 
 the analyzing rhomb. It is eUipticatty-polarized ; and all the 
 phenomena are explicable on the hypothesis, that the two 
 oppositely polarized rays, into which the incident ray is re- 
 solved, differ in phase after reflexion, the difference amount- 
 ing to 90 at the angle of maximum polarization. For it 
 is plain that the effect of a second reflexion, in the same plane 
 and under the same angle, will be to double the difference 
 of phase, which thus becomes 180 ; and the resulting light 
 will be plane-polarized, the plane of polarization lying at the 
 opposite side of the plane of incidence. 
 
 It is easy to see, from the foregoing, that the problem of 
 metallic reflexion is reduced to the determination of the inten- 
 sity, and the phase, of the reflected vibrations, in the case of 
 light polarized in the two principal planes. For any polarized 
 
188 ELLIPTIC POLARIZATION'. 
 
 ray may be resolved into two, polarized respectively in the 
 plane of incidence and in the perpendicular plane ; and these 
 planes, by the third of the preceding laws, are unaltered by 
 reflexion. The two components, however, undergo changes 
 both of intensity and phase ; and when these are known, the 
 character of the reflected pencil is completely determined. 
 
 This problem has been solved experimentally, by M. Ja- 
 min, in the most complete manner. 
 
 (200) The intensity of the light reflected by a metal at 
 different incidences is determined by M. Jamin by comparison 
 with the intensity of light reflected from glass under the same 
 angle, which latter is known by Fresnel's formulas (184, 5 ). A 
 plate of metal, and one of glass, are placed in the same plane, 
 and in contact, and the light is allowed to fall partly upon 
 each. When the incident light is polarized in the plane of 
 incidence, the light reflected from the metal, as well as from 
 the glass, continues polarized in that plane. If, therefore, 
 the two reflected pencils be received on a double-refracting 
 prism, whose principal section coincides with the plane of 
 incidence, each of then\ will furnish a single refracted pencil. 
 But if the principal section of the prism be inclined to the 
 plane of incidence at any angle, a, each of the reflected 
 pencils will furnish two refracted pencils, whose intensities 
 will vary with the azimuth of the principal section according 
 to the known law of Malus. 
 
 Let I be the intensity of the light reflected from the metal, 
 and I 7 that of the light reflected from the glass. The inten- 
 sities of the ordinary and extraordinary pencils, into which 
 the former is subdivided by the prism, are respectively 
 
 I cos 2 a, I sin 2 a ; 
 
 and those of the corresponding pencils, derived from the latter, 
 are 
 
 I' COS 2 a, I' sin 2 a. 
 
ELLIPTIC POLARIZATIOX. 189 
 
 Hence, if the prism be turned, until the ordinary image of 
 the light reflected from the metal is equal, in intensity, to 
 the extraordinary image of the light reflected from the glass, 
 I cos 2 a = I x sin 2 a, and 
 
 The azimuth of the principal section, a, is measured by means 
 of a graduated circle attached to the prism ; and the value 
 of I 7 for each incidence is given by Fresnel's formulas. 
 
 A second measure is obtained, by turning the prism until 
 the extraordinary image of the light reflected from the metal 
 is equal to the ordinary image of the light reflected from the 
 glass ; and similar processes are followed in the case of light 
 polarized in the perpendicular plane. 
 
 The results of these observations prove, that when light 
 polarized in the plane of incidence is reflected by a metal, 
 the intensity of the reflected light increases continually, as 
 the incidence increases from to 90, the total variation, 
 however, being very small. In the case of light polarized 
 perpendicularly to the plane of incidence, on the other hand, 
 the intensity of the reflected light diminishes from a perpen- 
 dicular incidence, up to the angle of maximum polarization, 
 and afterwards increases. The values found by experiment 
 accord satisfactorily with the results of M. Cauchy's dynamical 
 theory. The intensities of the reflected light, in the two 
 oases, are equal at the extreme incidences : at all other in- 
 cidences the intensity of the reflected light is less in the case 
 of light polarized perpendicularly to the plane of incidence ; 
 and the inequality is greatest at the angle of maximum 
 polarization. 
 
 (201) It remained to determine the difference of phase of 
 the two component pencils corresponding to any incidence. 
 For this purpose two mirrors of the same metal were 
 
190 ELLIPTIC POLARIZATION. 
 
 placed parallel to one another, with their reflecting surfaces 
 opposed ; and their distance was adjusted by means of a 
 screw. A ray of light, incident upon one of the mirrors, 
 will, after reflexion, fall upon the other in the same plane, 
 and under the same angle. It will then return to the first, 
 its plane and angle of incidence being unaltered ; and will 
 thus undergo a series of similar reflexions between the 
 mirrors, the number of which depends on their distance, and 
 on the angle of incidence. 
 
 Now the incident ray, polarized in any plane, may be 
 resolved into two, polarized respectively in the plane of inci- 
 dence, and in the perpendicular plane. The planes of 
 polarization of these two components are unchanged by re- 
 flexion : but their phases are altered, and that unequally ; 
 and the reflected light, composed of them, is therefore ellipti- 
 cally-polarized. 
 
 "When there are several reflexions in the same plane, and 
 under the same angle, the two components undergo the same 
 modification of phase at each successive reflexion, and the 
 difference of phase produced is equal to that produced by a 
 single reflexion, multiplied by the number of reflexions. But 
 the resulting light will be plane-polarized, when the difference 
 of phase becomes a multiple of 1 80 : we have, therefore, only 
 to increase the number of reflexions at the same incidence* 
 until the light is plane-polarized, and the difference of phase 
 produced by a single reflexion will be known. For, if e 
 denote the difference of phase sought, ne will be that pro- 
 duced by n reflexions. And, when the resulting light is 
 plane-polarized, ne = WITT, m being any integer number ; con- 
 sequently 
 
 mir 
 
 = . 
 
 n 
 
 * In practice it is more convenient to producu this effect by increasing the 
 incidence, the number of reflexions remaining unchanged. 
 
ELLIPTIC POLARIZATION. 191 
 
 It follows from these researches, that the phase of the ray 
 polarized perpendicularly to the plane of incidence is always 
 retarded, relatively to the other. The difference of phase 
 increases regularly with the incidence, being equal to TT at a 
 perpendicular incidence, and to 2n- at an incidence of 90. 
 At the angle of maximum polarization, E = f TT. This angle 
 is, of course, different for different metals : it is, however, 
 not far in any from 75. 
 
 (202) It follows from the preceding, that there are n - 1 
 incidences between and 90, for which the ray is restored 
 to the condition of plane polarization by successive reflexions. 
 For the ray becomes plane-polarized, as often as the differ- 
 ence of phase of the two components is a multiple of TT. 
 But, with a single reflexion, the difference of phase increases 
 by TT between and 90 of incidence. Consequently with 
 n reflexions, the difference increases by mr ; and between 
 these limits of incidence there are n - 1 multiples of TT, and 
 therefore n - 1 angles of incidence at which the polarization 
 is restored. 
 
 The plane of polarization of the restored ray will be on 
 the same side of the plane of incidence, as the plane of polari- 
 zation of the incident ray, or on the opposite, according as the 
 difference of phase is an even or odd multiple of TT. 
 
 (203) It would appear from the foregoing, that metals 
 differ from transparent bodies, in their action upon light, in 
 two particulars namely, 1st, that they do not polarize com- 
 mon light completely at any incidence ; and 2ndly, that they 
 alter plane-polarized light by reflexion into light elliptically 
 polarized. It will be seen, presently, that these differences 
 are only differences in degree. 
 
 It was long since observed by M. Biot, that diamond and 
 sulphur did not polarize the light completely at any angle ; and 
 
192 ELLIPTIC POLARIZATION. 
 
 the property was extended, by Sir John Herschel, to all trans- 
 parent bodies possessing an adamantine lustre. Sir GK B. Airy 
 has proved, that plane-polarized light becomes elliptically 
 polarized by reflexion from diamond. And, finally, Mr. 
 Dale and Professor Powell have shown that these two pro- 
 perties, supposed to be peculiar to metals, belonged to all 
 transparent bodies having a high refractive power. 
 
 In this state of the question, the problem of reflexion by 
 transparent bodies was taken up by M. Jamin, and received 
 at his hands its complete experimental solution. The con- 
 clusions deduced by M. Jamin from his observations may be 
 summed up as follows : 
 
 I. All transparent bodies polarize the light incompletely 
 by reflexion, the polarization of the reflected light becoming 
 a maximum at a certain angle of incidence. 
 
 II. They transform plane-polarized light into light el lip- 
 ticatty polarized. 
 
 III. The difference of phase which they impress upon 
 light, polarized in the two principal planes, undergoes the 
 same variations as in metallic reflexion, within certain limits 
 of incidence. 
 
 (204) It is necessary to enter a little more minutely into 
 the consideration of this third law, which (it is obvious from 
 the preceding) virtually includes the two others. 
 
 According to Fresnel's theory, when a ray polarized in 
 any plane falls upon a transparent body, the reflected light 
 continues polarized. But its plane of polarization is changed ; 
 and lies at the opposite side of the plane of incidence, when 
 the incidence is less than the polarizing angle, and at the 
 same side when it is greater (190). It follows from this, 
 that the two components of the reflected ray, polarized 
 respectively in the plane of incidence, and in the perpendicu- 
 lar plane, agree in phase at all incidences above the angle of 
 
ELLIPTIC POLAKIZATION. 193 
 
 polarization, while they differ 180 in phase at all incidences 
 below it. According to this theory,- therefore, the difference 
 of phase changes abruptly, from TT to 2?r, at that critical inci- 
 dence. On the other hand, in reflexion from metals, the 
 difference of phase of the two components increases continuously 
 from TT to 2?r, as the incidence increases from to 90. 
 
 Now M. Jamin has shown that the latter is generally 
 true for all bodies, whether opaque or transparent ; and that 
 the distinction of these bodies, as to their effects upon re- 
 flected light, consists only in this, that in transparent bodies 
 the variation of phase is insensible, except in the neighbour- 
 hood of the angle of maximum polarization. 
 
 In transparent substances, accordingly, the difference of 
 phase is nearly constant, at low and at high incidences; and 
 passes from TT to 2-rr (not abruptly, as we are required to sup- 
 pose in Fresnel's theory, but) between two incidences, one 
 lower and the other higher than the angle of maximum pola- 
 rization. The elliptic polarization of the reflected light will 
 be sensible only within the same limits of incidence ; and 
 beyond them the light is (as to sense) plane-polarized. In 
 substances of low refractive power, these limiting incidences 
 differ from one another, and from the angle of maximum po- 
 larization, by a small amount ; and for these, therefore, the 
 change of phase (although not instantaneous) is very rapid, 
 and Fresnel's laws are approximately true. 
 
 o 
 
 When the difference of phase = T TT, the ellipticity of the 
 
 reflected ray is greatest. The angle of incidence at which v 
 this occurs is the angle of maximum polarization in the case 
 of common light, and is called the principal incidence. It is 
 theoretically different from the angle given by Brewster's 
 law ; but the difference is in all cases small. 
 
 (205) M. Jamin has shown, further, that transparent 
 
 o 
 
194 ELLIPTIC POLARIZATION. 
 
 bodies may be distinguished into two classes, with respect to 
 their action upon reflected light. In some of them, as in 
 opal, the phase of the component in the plane of incidence is 
 accelerated, relatively to the other component ; in others, as' 
 hyalite, it is retarded. The bodies of these classes are deno- 
 minated, by M. Jamin, substances of positive and of negative 
 reflexion, respectively. Intermediate to these two classes we 
 should expect to find a third, characterized by the property 
 that the phase is unaltered by reflexion, and for which, there- 
 fore, Fresnel's laws are accurately true. This class is very 
 small, the only bodies observed to belong to it being menilite 
 and alum. 
 
 These distinctions appear to be connected with the refrac- 
 tive power. Thus all bodies, whose refractive index is greater 
 than 1*46, accelerate the phase of vibration in the plane of in- 
 cidence ; those whose refractive index is less than 1'46, retard 
 it ; while those bodies, for which JJL = 1'46, reflect according 
 to Fresnel's laws. 
 
 (206) The elliptical vibration of the reflected light will 
 be completely known, when we know the difference of phase 
 of the two principal components, and the ratio of their inten- 
 sities. The difference of phase is determined experimentally 
 by M. Jamin, by the process which restores the light to the 
 condition of plane polarization ; while the azimuth of the 
 plane of polarization of the restored ray gives the ratio of 
 the intensities of the two components. The results obtained 
 have been compared with the formulas given by M. Cauchy 
 for the case of diamond ; and the agreement has been found 
 to be satisfactory. These formulas involve two constants, 
 the refractive index, and the coefficient of ellipticity ; and these 
 are determined, when we know the principal incidence, and 
 the ratio of the amplitudes of the two vibrations at that 
 incidence. 
 
ELLIPTIC POLARIZATION. 195 
 
 The fundamental difference between this theory, and that 
 of Fresnel, consists (we have seen) in including a change of 
 phase of the reflected vibration, varying with the incidence. 
 This change of phase is due, according to M. Cauchy and 
 Mr. Green, to the normal vibration, which though evanes- 
 cent at a short distance from the surface modifies the 
 phase. 
 
 (207) Professor Haughton has followed up the researches 
 so ably commenced by M. Jamin, and has obtained some new 
 and interesting results. The more important of these are 
 comprised in the following laws : 
 
 I. If plane-polarized light be incident on a transparent 
 reflecting body, and the incidence be gradually increased 
 from to 90, the ratio of the axes of the reflected ellipti- 
 cally-polarized light diminishes from infinity, at 0, to a 
 minimum at the principal incidence ; and increases again to 
 infinity at 90. 
 
 II. The minimum ratio of the axes varies with the plane 
 of polarization of the incident light, and diminishes as the 
 azimuth of that plane increases, until the latter reaches a cer- 
 tain value ; after which the ratio again increases. 
 
 III. When the azimuth of the plane of polarization of 
 the incident light reaches this jalue, the ratio of the axes 
 becomes equal to unity, and the reflected light is circularly 
 polarized. 
 
 This last conclusion is one which might have been antici- 
 pated. M. Jamin had shown that the difference of phase of 
 the two principal components of the reflected light was equal 
 to 270, at the principal incidence ; so that the light reflected 
 at this incidence must be circularly-polarized, when the am- 
 plitudes of the two components are equal. This equalization 
 of the reflected components can always be effected by varying 
 the azimuth of the plane of polarization of the incident ray. 
 
 o2 
 
196 ELLIPTIC POLARIZATION. 
 
 Now a denoting this azimuth, the amplitudes of the two compo- 
 nents are cos a and sin a, that of the original vibration being 
 unity ; so that if v and v', denote (as before) the ratios of the 
 amplitudes of the reflected and incident vibrations in the two 
 principal planes, the amplitudes of the two components in 
 the reflected ray will be v cos a, and v' sin a. These will be 
 equal, and therefore the reflected light circularly-polarized, 
 when 
 
 v cos (i - r) 
 tan a = -, 
 
 v ooa(*+r) 
 
 If the principal incidence were the same as the angle 
 given by Brewster's law, cos (i + r) = 0, and a = 90. But 
 this not being the case, cos (i + r) is not actually evanescent ; 
 and the azimuth, a, at which the light is circularly-polarized, 
 is a few degrees less than 90. 
 
CHAPTEE XII. 
 
 (208) IT has been stated (81, 87), that soon after the dis- 
 covery of double refraction in Iceland crystal, Huygens 
 succeeded in embracing its laws in the theory of waves, by a 
 bold and happy assumption. He had already shown that 
 the form of the wave which gives rise to the ordinary re- 
 fracted ray, in glass and other uncrystallized substances, was 
 the sphere ; or, in other words, that the velocity of undula- 
 tory propagation was the same in all directions. One of the 
 rays in Iceland crystal, too, was found to obey the same law ; 
 and judging that the law which governed the other, though 
 not so simple, was yet next in simplicity, he assumed the 
 form of its wave to be the spheroid; that is, he supposed the 
 velocity of propagation to be different in different directions, 
 in accordance with the following construction : " Let an 
 ellipsoid of revolution be described round the optic axis, 
 having its centre at the point of incidence; and let the 
 greater axis of the generating ellipse be to the less in the 
 ratio of the greatest to the least index of refraction : then the 
 velocity of any ray will be represented by the radius vector 
 of the ellipsoid which coincides with it in direction." 
 
 (209) The law of Huygens was found to hold in many 
 crystals besides that to which it was originally applied ; and 
 in all of' these there was one optic axis, or one line along 
 which a ray of light passed without division. But when the 
 researches of Brewster made known a class of crystalline 
 
198 FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 bodies, having two optic axes, or two lines of no separation, 
 Huy gens' law was found not to be general ; and it was ascer- 
 tained that one of the rays, at least, in biaxal crystals, fol- 
 lowed some new and unknown law. 
 
 In this state of the question the problem of double re- 
 fraction was taken up by Fresnel ; and by the aid of a na- 
 tural and simple hypothesis, combined with the principle of 
 transversal vibrations, he has been conducted to its complete 
 solution, a solution which not only embraces all the known 
 phenomena, but has even outstripped observation, and pre- 
 dicted consequences which were afterwards verified by ex- 
 periment. 
 
 (210) Fresnel sets out from the supposition, that the 
 elastic force of the vibrating medium, in every crystal, is 
 different in different directions. It is easy to see, generally, 
 that the phenomenon of double refraction is a necessary con- 
 sequence of this hypothesis, and of the principle of transversal 
 vibrations. 
 
 Let us take, for example, the simple case of a ray of 
 light proceeding from an infinitely distant point, and falling 
 perpendicularly on the surface of a uniaxal crystal, cut 
 parallel to the axis. The incident wave being plane, and 
 parallel to the surface of the crystal, the vibrations are also 
 parallel to the same surface ; and we may conceive them to 
 be composed of vibrations parallel and perpendicular to the 
 axis of the crystal. Now, the elasticity brought into play by 
 these two sets of vibrations being different, they will be 
 propagated with different velocities ; and there will be two 
 waves within the crystal, in which the vibrations are pa- 
 rallel to two fixed directions at right angles to one another, 
 or two rays oppositely polarized. If the second face of the 
 crystal be parallel to the first, the two rays will emerge 
 perpendicularly ; and the only effect produced will be, that 
 
FRESNELS THEORY or DOUBLE REFRACTION. 
 
 one will be retarded more than the other, in its progress 
 through the crystal. But if the second face be oblique to- 
 the direction of the rays, they will be both refracted at 
 emergence, and differently ; and they will therefore diverge 
 from one another. 
 
 (211) To return to the general theory. Let us suppose a 
 disturbance to be produced in a medium such as we have 
 been considering, and any particle of the medium to be dis- 
 placed from its position of rest. The resultant of all the 
 elastic forces which resist the displacement will not, in ge- 
 neral, act in the direction of the displacement (as would be 
 the case in a medium uniformly elastic), and therefore will 
 not drive the displaced particle directly back to its position 
 of equilibrium. Fresnel has shown, however, that there are 
 three directions at right angles to each other, in every elastic 
 medium, in each of which the elastic forces do act in the di- 
 rection of the displacement, whatever be the nature or laws 
 of the molecular action ; and he assumes that these three di- 
 rections are parallel throughout the crystal. In fact, the 
 first principles of crystallization oblige us to admit, that the 
 arrangement of the molecules of the crystalline body is similar 
 in all parallel lines ; and the same property must belong to 
 the ether itself, if (as we have reason to presume) its 'elasti- 
 city be dependent on that of the crystal. These three direc- 
 tions Fresnel denominates axes of elasticity ; and he concludes 
 that they are also axes of symmetry, with respect to the 
 crystalline form. 
 
 If on each of these axes, and on every line diverging 
 from the same origin, portions be taken, which are as the 
 square roots of .the elastic forces in their directions, the 
 locus of the extremities of these portions will be a surface, 
 which Fresnel denominates the surface of elasticity. Its equa- 
 tion is 
 
 r* = a 2 cos 2 a + 6 7 cos 8 /3 + c 2 cos* 7 : 
 
200 FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 a 3 , 6 2 , c 2 , being the elasticities in the directions of the three 
 axes ; r the radius- vector of the surface ; and a, ]3, 7, the 
 angles which it makes with the axes. 
 
 This surface determines the velocity of propagation of 
 the wave, when the direction of its vibration is given. For, 
 the ethereal molecule vibrating in the direction of any radius 
 vector, r, of this surface, the elastic force which governs its 
 vibration will be proportional to r z ; and, since the velocity 
 of wave-propagation is as the square root of the elastic force, 
 it must in this case be represented by r. Hence, if we con- 
 ceive the vibration in the incident wave to be resolved into 
 two within the crystal, performed in two determinate direc- 
 tions, these will be propagated with different velocities ; and, 
 as a difference of velocity gives rise to a difference of refrac- 
 tion, it follows that the incident ray will be divided into two 
 within the crystal, which will in general pursue different 
 paths. Thus, the bifurcation of a ray, on entering a crystal, 
 presents no difficulty, provided we can explain in what man- 
 ner the vibration comes to be resolved. 
 
 (212) To understand in what manner this takes place, 
 let us conceive a plane wave advancing within the crystal. By 
 the principle of transversal vibrations, the movements of the 
 ethereal molecules are all parallel to the wave. But the mo- 
 tion of each molecule, when thus removed from its position of 
 equilibrium, is resisted by the elastic force of the medium ; 
 and that force is, in general, oblique to the direction of the 
 displacement. If the plane containing the direction of the force 
 and that of the displacement were normal to the plane of the 
 wave, the force would be resolvable into two, one perpen- 
 dicular to the plane of the wave, which (by the principle of 
 transversal vibrations) can produce no effect ; and the other 
 in the direction of the displacement itself, which will thus 
 be communicated from particle to particle without change. 
 
201 
 
 But this, in general, is not the case. Fresnel has shown, 
 however, that the displacement may be resolved in two direc- 
 tions in the plane of the wave, at right angles to one another, 
 such that the elastic force called into action by each compo- 
 nent will be in the plane passing through the component, 
 and normal to the wave ; and thus each component will give 
 rise to a wave, in which the direction of the vibrations is pre- 
 served, and which therefore will be propagated with a con- 
 stant velocity. 
 
 The two directions, above alluded to, are those of the great- 
 est and least diameters of the section of the surface of elasticity 
 made by the plane of the wave ; so that if the original dis- 
 placement be resolved into two, parallel to these directions, 
 each component will give rise to a plane wave, in which the 
 vibrations preserve constantly the same direction. The velo- 
 city of propagation being represented by the radius-vector of 
 the surface of elasticity in the direction of the displacement, 
 the velocities of the two parts of the wave will be proportional 
 to the greatest and least diameters of the section of the surface 
 of elasticity, to which the vibrations are parallel. Thus it 
 appears that an incident plane wave, in which the vibrations 
 are in any direction, will be resolved into two within the 
 crystal ; and these will be propagated with different veloci- 
 ties, and consequently assume different directions. 
 
 (213) The vibrations in these waves being parallel to two 
 fixed lines, namely, the greatest and least diameters of the 
 section of the surface of elasticity, it follows that the two re r 
 fracted rays are polarized, and that their planes of polarization 
 are at right angles, being the planes passing through the di- 
 rection of the ray and these two lines. Hence it follows, that 
 the plane of polarization of one of the rays bisects the dihedral 
 angle made by the two planes, which pass through the nor- 
 mal to the ware and the normals to the two circular sections of 
 

 202 
 
 the surface of elasticity ; and that the plane of polarization of 
 the other is perpendicular. This coincides, very nearly, with 
 the rule previously given by M. Biot, namely, that the plane 
 of polarization of one of the pencils bisects the dihedral angle 
 formed by planes drawn through the ray and the two optic axes ; 
 while that of the other is perpendicular, or bisects the supple- 
 mental dihedral angle. 
 
 Thus the two fundamental facts of crystalline refraction 
 namely, the bifurcation of the ray, and the opposite po- 
 larization of the two pencils are completely accounted 
 for. 
 
 Further, the amplitudes of the resolved vibrations are 
 represented by the cosines of the angles winch the direction 
 of the original vibration contains with the two fixed rec- 
 tangular directions ; and, as the squares of these amplitudes 
 measure the intensities of the two pencils, the law of Malus 
 respecting these intensities is a necessary consequence. 
 
 (214) The velocity of propagation of a plane wave in any 
 direction being known, the form of the wave, diverging from 
 any point within the crystal, may be found. For, if we con- 
 ceive an indefinite number of plane waves, which, at the com- 
 mencement of the time, all pass through the point which is 
 considered as the origin of the disturbance, the wave-surface 
 will be that touched by all these planes at any instant. Fresnel 
 has given the following elegant construction for its determi- 
 nation : "Let an ellipsoid be conceived, whose semiaxes are 
 a, b y c (the same as those of the surface of elasticity), and let 
 it be cut by any diametral plane. At the centre of this sec- 
 tion let a perpendicular be raised ; and on this line let two 
 portions be taken, whose lengths (measured from the centre) 
 are equal to the greatest and least radii of the section. The 
 extremities of these perpendiculars will be the loci of the 
 double wave." 
 
FRESNEL'S THEORY OF DOUBLE REFRACTION. 203 
 
 The equation of the wave surface is of the fourth order ; 
 it has been thrown into the following symmetric form by Sir 
 William Hamilton, 
 
 
 (215) The form of the wave-surface being known, the 
 directions of the two refracted rays . are determined by tan- 
 gent planes drawn to the two sheets of the surface, ac- 
 cording to the construction of Huygens. Conceive three 
 surfaces, having their common centre at the point of incidence, 
 and representing, respectively, the simultaneous positions of 
 three waves diverging from that point, the first in air, 
 which is a sphere ; and the other two within the crystal, 
 which are the two sheets of the surface we have been con- 
 sidering. Let the incident ray be produced to meet the 
 sphere, and at the point of intersection let a tangent plane 
 be drawn. Through the line of intersection of this plane 
 with the refracting surface, let two planes be drawn touch- 
 ing the two sheets of the refracted wave ; the lines connect- 
 ing the centre with the points of contact are the directions 
 of the two refracted rays.* 
 
 * If, in place of the ellipsoid mentioned above, we take that whose semi- 
 
 axes are,., the three principal refractive indices of the medium, the 
 a b c 
 
 surface derived from it by the same construction will represent the normal 
 slowness of the waves, and is connected with the wave-surface by a remark- 
 able relation of reciprocity. The properties of this surface lead to the fol- 
 lowing elegant construction for the directions of the refracted rays, a con- 
 struction which is, in many cases, more convenient than that given above : 
 " With the point of incidence, as a common centre, construct tbe surfaces 
 of wave-slowness belonging to air and to the crystal, respectively. Let the 
 incident ray be produced to meet the sphere, which represents the normal 
 slowness of the wave in air ; and from the point of intersection let a per- 
 pendicular be drawn to the refracting surface. This will cut the surface of 
 
204 FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 It may be shown that the direction of the vibratory move- 
 ment, at any point of the surface of the wave, coincides with 
 the projection of the radius vector upon the plane which 
 touches the surface at that point. Hence, if perpendiculars 
 be let fall from the centre, on the tangent planes to the two 
 sheets of the wave-surface, the lines connecting their feet with 
 the points of contact are the directions of the vibrations in 
 the two rays ; and therefore determine their planes of po- 
 larization. The perpendiculars themselves measure the 
 velocities of propagation of the waves, while the radii- vectores 
 represent those of the rays. 
 
 (216) From the construction of the wave-surface, above 
 given, it follows that there are two directions, namely, the 
 normals to the two circular sections of the ellipsoid, in which 
 the two sheets of the wave-surface have a common radius 
 vector, and therefore the two rays a common velocity. If to 
 and a/ denote the angles which any line drawn from the centre 
 of the wave makes with these lines, v and v 1 the radii-vectores 
 in its direction terminating in the two sheets of the wave- 
 surface, the equation above given may be reduced to the fol- 
 lowing remarkable polar forms : 
 
 tr* = (V 2 + ar 2 ) + J (c- 2 - <r 2 ) cos (w -i- a/), 
 t,'-* = 1 (<r + a' 2 ) + i (c- 2 - a' 2 ) cos (w - a/). 
 
 Since the radius- vector of the wave-surface measures the 
 velocity of the ray in its direction, the velocities of the two 
 
 slowness of the refracted waves, in general, in two points. The lines con- 
 necting these points with the centre will represent the direction and normal 
 slowness of the waves; while the perpendiculars from the centre on the 
 tangent planes at the same points will represent the direction and slowness 
 of the rays:' This construction was given by Sir William Hamilton and Pro- 
 fessor Mac Cullagh. 
 
205 
 
 rays are given by the preceding formulas. If we subtract 
 the latter from the former, we find (after a simple' trigono- 
 metrical reduction), 
 
 r 2 - v'-* = (a-* - c~ 2 ) sin w sin a/. 
 
 Hence the difference of the squares of the reciprocal veloci- 
 ties, in the two rays, is proportional to the product of the 
 sines of the angles made by their common direction with 
 the lines in which the two rays have a common velocity. 
 
 The normal velocities of the waves are given by analo- 
 gous expressions 
 
 w = 1 ( a * + c 2 ) + (a 3 - c 2 ) cos (o> + a/). 
 u' 2 = + <* + l a* - o 2 cos w - iK 
 
 (D and a/ in this case denoting the angles which the common 
 normals to the two sheets of the wave-surface make with the 
 lines of single normal velocity, or with the true optic axes. 
 
 Subtracting the latter of these expressions from the for- 
 mer, we find 
 
 u - u /z = (a? -c 2 ) sin w sin a/. 
 
 Thus the remarkable law of Brewster and Biot is a necessary 
 consequence of FresnePs theory. 
 
 (217) The two sets of lines referred to in the preceding 
 article the lines of single ray -velocity, and single wave-velocity 
 are situated in the plane of the axes of greatest and least 
 elasticity, the lines of each pair making equal angles with 
 the axis of greatest elasticity on either side. The tangents 
 of these angles are respectively, 
 
Hence, when b* = c*, or b* = a 2 , these angles become 0, or 90 ; 
 and the two optic axes unite> coinciding in the former case, 
 with the axis of greatest elasticity, and in the latter with that 
 of the least. y u~/^^ / ^^ 1 -' 
 
 In -eaeh of these cases, then, o>= a/, and the equations of 
 the wave-surface become 
 
 v~* = cr 2 sin 2 (t) + c~ 2 cos 2 cu, v' = c ; 
 
 the former of which is the equation of the ellipsoid of revolu- 
 tion, and the latter that of the sphere. Accordingly, the 
 wave-surface resolves itself into the sphere and spheroid of the 
 Huygenian law ; and the form of the wave in uniaxal crystals, 
 assumed by Huygens, is deduced as a simple corollary from 
 the general theory of Fresnel. 
 
 Finally, when the three elasticities are all equal, it will 
 appear at once from the preceding equations that the spheroid 
 becomes a sphere. The velocity is accordingly the same in 
 all directions, and the law of refraction is reduced to the 
 known law of Snell. 
 
 (218) It has been stated (94) that, as soon as a class of 
 double-refracting substances was discovered, possessing two 
 optic axes, the construction of Huygens was found not to be 
 general. It was still supposed, however, that the velocity of 
 one of the rays in every crystal was constant ; or, in other 
 words, that one of the rays was refracted according to the 
 ordinary law of the sines. According to Fresnel's theory, 
 however, the velocity of neither of the rays in biaxal crystals 
 was constant, and the refraction of both was performed ac- 
 cording to a new law. It was, therefore, a matter of much 
 interest to decide this question by accurate experiment. 
 This experimental problem was solved by Fresnel himself, 
 and the result was decisive in favour of his theory. 
 
FRESNEL'S THEORY OF DOUBLE REFRACTION. 207 
 
 It has been already shown (104^ that when light, diverg- 
 ing from a luminous origin, passes through two near aper- 
 tures in a screen, the two pencils into which it is thus divided 
 will interfere, and produce fringes, the central fringe being 
 the locus of those points at which the two rays have tra- 
 versed equal paths. Now, if two plates of glass, cut from the 
 same plate, and of exactly the same thickness, be placed per- 
 pendicularly, one in the path of each ray, the two rays will be 
 equally retarded, and the central fringe will remain undis- 
 placed. But if, instead of glass plates, we employ plates cut 
 in different directions from the same biaxal crystal, the plates 
 being of exactly the same thickness, the fringes produced by 
 the interference of the two rays in question will remain still 
 undisplaced, if the velocity of these rays is the same in the 
 two plates ; while, on the other hand, if the velocities be dif- 
 ferent, the fringes will be shifted from their original position. 
 On trial, the result was found to be as Fresnel had antici- 
 pated: the fringes were displaced; and the amount of that 
 displacement agreed with the calculated difference of velocity, 
 which had been previously deduced from theory. 
 
 (219) There are two remarkable cases of Fresnel's theory, 
 which have since furnished a very striking confirmation of 
 its truth. 
 
 If we make y = 0, in the equation of the- wave-surface, so 
 as to obtain its intersection with the plane of xz, the result- 
 ing equation is reducible to the form 
 
 # 2 -t s 2 - t> z 3 x 7 - + c*z*- a 2 c 8 = 0. 
 
 This equation is manifestly resolvable into the two follow- 
 ing : 
 
 x 2 + 2 = b*, a 2 x" + c 2 s 2 = a* c z ; 
 so that the surface intersects the plane of xz in a circle and 
 
208 
 
 ellipse. As these two curves have a common centre, and 
 as the radius of the circle, 5, is of intermediate magnitude 
 to the semiaxes of the ellipse, it follows that they must 
 intersect in four points, as is represented in the annexed 
 diagram. 
 
 Now, when two rays pass within the 
 crystal in any common direction, as 
 OAB, their velocities are represented 
 by the radii vectores of the two parts of 
 the wave, OA and OB ; and their direc- 
 tions, at emergence, are determined by 
 the positions of the tangent planes at the 
 points A andB. But in the case of the 
 ray OP, whose direction is that of the 
 line joining the centre with one of the 
 four cusps, or intersections just mentioned, the two radii 
 vectores unite, and the two rays have the same velocity. 
 There are still, however, two tangents to the plane section 
 at the point P; so that it might be supposed that the 
 rays proceeding with this common velocity within the crystal 
 would still be divided at emergence into two, and two only, 
 whose directions are determined by the tangent planes. This 
 seems to have been Fresnel's view of the case. Sir William 
 Hamilton has shown, however, that there is a cusp at each of 
 the four points just mentioned, not only in this particular 
 section, but in every section of the wave-surface passing 
 through the line OP ; or, more properly, that there is a con- 
 oidal cusp on that surface at the four points of intersection of 
 the circle and ellipse, and consequently an infinite number of 
 tangent planes, which form a tangent cone of the second 
 degree. Hence, a single ray, such as OP, proceeding 
 within the crystal in one of these directions, should be di- 
 vided into an infinite number of rays at emergence, whose 
 
209 
 
 directions and planes of polarization are determined by the 
 tangent planes. 
 
 Again, it is evident that the circle and ellipse have four 
 common tangents, such as MN ; and the planes passing through 
 these tangents, and perpendicular to the plane of the section, 
 are perpendicular to the optic axes of the crystal. Fresnel 
 seems to have thought that these planes touched the wave 
 surface in the two points just mentioned, and in these only ; 
 and, consequently, that a single ray, incident upon a biaxal 
 crystal in such a manner that one of the refracted rays should 
 coincide with an optic axis, OM, will be divided into two 
 within the crystal, OM and ON, determined by the points 
 of contact. But Sir William Hamilton has shown that the 
 four planes of which we have spoken touch the wave-surface, 
 not in two points only, but in an infinite number of points, 
 constituting each a small circle of contact ; and, consequently, 
 that a single ray of common light, incident externally in the 
 above-mentioned direction, should be divided into an infinite 
 number of refracted rays within the crystal. 
 
 Here, then, are two singular and unexpected conse- 
 quences of Fresnel's theory, not only unsupported by any 
 facts hitherto observed, but even opposed to all the analogies 
 derived from experience ; here are two remote conclusions of 
 that theory, deduced by the aid of a refined analysis, and in 
 themselves so strange, that we are inclined at first to reject 
 the principles of which they are the necessary consequences. 
 They accordingly furnish a test of the truth of that theory of 
 the most trying nature that can be imagined. 
 
 (220) Naturally anxious to submit the wave-theory to 
 this test, and to establish or disprove its new results, Sir 
 William Hamilton requested the author of this volume to 
 examine the .subject experimentally. The result of this ex- 
 
210 FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 animation has been to prove the existence of both species of 
 conical refraction. 
 
 The first case of conical refraction is that called by Sir 
 William Hamilton external conical refraction, and was ex- 
 pected to take place, as we have seen, when a single ray 
 passes within the crystal in the direction of either of the lines 
 of single ray-Telocity . These lines coincide nearly, but not 
 exactly, with the optic axes of the crystal ; and, in the case 
 of arragonite (the crystal submitted to experiment), contain 
 an angle of nearly 20. The plate of arragonite employed had 
 its faces perpendicular to the line bisecting the optic axes ; con- 
 sequently, the lines above mentioned were inclined to the per- 
 pendicular at an angle of about 10 on 
 either side. Let these lines be repre- 
 sented by OM and ON, equally in- 
 clined to the perpendicular OP. A 
 ray of common light traversing the 
 crystal in the direction OM or MO, 
 should emerge in a cone of rays, as 
 represented in the figure ; the angle 
 of this cone depending on the relative 
 
 magnitude of the three elasticities of the crystal, a z , b z , c*. In 
 the case of arragonite this angle is considerable, and amounts 
 to 3 very nearly. 
 
 A thin metallic plate, perforated with a very minute 
 aperture, was placed on each face of the crystal; and these 
 plates were so adjusted, that the line connecting the two aper- 
 tures should coincide with the line MO, or any parallel line 
 within the crystal. The flame of a lamp was then brought 
 near one of the apertures, and in such a position that the 
 central part of the beam converging from its several points 
 to the aperture should have an incidence of 15 or 16. When 
 the adjustment was completed, a brilliant annulus of light 
 
FRESNEL'S THEORY OF DOUBLE REFRACTION. 211 
 
 appeared, on looking through the aperture 
 in the second surface. When the aperture 
 in the second plate was ever so slightly 
 shifted, so that the line connecting the two 
 apertures no longer coincided with the line 
 MO, the phenomenon rapidly changed, and 
 the annulus resolved iself into two separate 
 pencils. 
 
 The incident converging cone was also formed by a lens 
 of short focus, placed at the distance of its own focal length 
 from the surface ; and in this case, the lamp was removed to a 
 distance, and the plate on the first surface dispensed with. 
 The same experiments were repeated with the sun's light; 
 and the emergent rays were even thrown on a screen, and 
 thus the section of the cone observed at various distances from 
 its summit. 
 
 In the first experiments there was a considerable discre- 
 pancy between the results of observation and theory, both as 
 to the magnitude of the cone, and some other circumstances 
 of its appearance. These discrepancies were found to arise 
 from the sensible magnitude of the little aperture on the second 
 surface of the crystal, which suffered rays to pass which were 
 inclined to the line OM at small angles. Accordingly, the 
 magnitude of the observed cone required a correction before 
 it could be compared with the results of theory : when this 
 correction was applied, the agreement of the observed and 
 theoretical angles was found to be nearly complete. 
 
 (221) The rays which compose the emergent cone are all po- 
 larized in different planes ; and these planes were found to be 
 connected by the following law, namely, " the angle between 
 the planes of polarization of any two rays of the cone is half the 
 angle between the planes containing the rays themselves and the 
 axis" This remarkable law was, in the first instance, dis- 
 
212 FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 covered by observation. On reference to Fresnel's theory, it 
 was found to yield the same result. 
 
 (222) A remarkable variation of the phenomenon took 
 place, on substituting a narrow linear aperture for the small 
 circular one, in the plate next the lamp, in the first-mentioned 
 mode of performing the experiment, the line being so ad- 
 justed, that the plane passing through it and the aperture on 
 the second surface should coincide with the plane of the optic 
 axes. In this case, according to the hitherto received views, 
 all the rays transmitted through the second aperture should be 
 refracted doubly in the plane of the optic axes, so that no part 
 of the line should appear enlarged in breadth, on looking 
 through this aperture ; while, according to Sir "William Ha- 
 milton, the ray which proceeds in the direction OM should 
 be refracted in every plane. The latter was found to be the 
 case : in the neighbourhood of each of the optic axes, the lu- 
 minous line was bent, on either side of the plane of the axes, 
 into an oval curve. This curve, it is easy to show, is the con- 
 choid of Nicomcdes, whose asymptot is the line on the first 
 surface. 
 
 (223) The other case of conical refraction called internal 
 conical refraction by Sir William Hamilton was expected to 
 take place when a single ray has been incident externally upon 
 a biaxal crystal, in such a manner that one of the refracted 
 rays may coincide with an optic axis. The incident ray in 
 this case should be divided into a cone of rays within the 
 crystal, the angle of which, in the case 
 
 of arragonite, is equal to l a 55'. The 
 rays composing this cone will be re- 
 fracted at the second surface of the crys- 
 tal, in directions parallel to the ray inci- 
 dent on the first, so as to form a small 
 cylinder of rays in air, whose base is the 
 
FRESNEL'S THEORY OF DOUBLE REFRACTION. 213 
 
 section of the cone made by the surface of emergence. This 
 is represented in the annexed diagram, in which NO is the 
 incident ray, aOb the cone of refracted rays within the 
 crystal, and aa'Vb the emergent cylinder. 
 
 The minuteness of this phenomenon, and the perfect ac- 
 curacy required in the incidence, rendered it much more dif- 
 ficult to observe than the former. A thin pencil of light, 
 proceeding from a distant lamp, was suffered to fall upon the 
 crystal, and the position of the latter was altered with extreme 
 slowness, so as to change the incidence very gradually. "When 
 the required position was attained, the two rays suddenly 
 spread out into a continuous circle, whose diameter was ap- 
 parently equal to their former interval. The same experiment 
 was repeated with the sun's light, and the emergent cylinder 
 received on a small screen of silver paper, at various distances ; 
 no sensible enlargement of the section was observable on in- 
 creasing the distance. The angle of this minute cone within 
 the crystal was found to agree, within very narrow limits, 
 with that deduced from theory, the observed angle being 
 1 50', and the theoretical angle 1 55'. 
 
 The rays composing the internal cone are all polar- 
 ized in different planes ; and the law connecting these 
 planes is the same as in the case of external conical refrac- 
 tion. 
 
( 214 ) 
 
 CHAPTEE XIII. 
 
 INTERFERENCE OF POLARIZED LIGHT. 
 
 (224). HAVING considered the theory and laws of double 
 refraction, we are prepared to study the phenomena of inter- 
 ference which take place when polarized light is transmitted 
 through crystalline substances. The effects displayed in such 
 cases are probably the most splendid in optics ; and when it 
 is considered that, through them, an insight is afforded into 
 the very laboratory of Nature itself, and that we are thus en- 
 abled almost to view the interior structure and molecular 
 arrangement of bodies, the subject will hardly be thought in- 
 ferior in importance to any other in the science. 
 
 The first discoveries in this attractive region were made 
 by Arago, who presented a memoir to the Institute, in the 
 year 1811, on the colours of crystalline plates when exposed 
 to polarized light. The subject has since been prosecuted 
 with unremitting ardour by the first physical philosophers of 
 Europe, and among the foremost by Biot, Brewster, and 
 Fresnel. 
 
 (225) It has been shown (166), that when a beam of 
 light, polarized by reflexion, is received upon a second re- 
 flecting plate at the polarizing angle, the twice-reflected light 
 will vanish, when the plane of the second reflexion is per- 
 pendicular to that of the first. The first reflector, in any 
 apparatus of this kind, is called the polarizing plate, and the 
 second (for reasons which will presently appear), the analyz- 
 ing plate. Now, if between the two reflectors we interpose a 
 
INTERFERENCE OF POLARIZED LIGHT. 215 
 
 plate of any double-refracting substance, the capability of re- 
 flexion at the analyzing plate is suddenly restored, and a por- 
 tion of the light is reflected, whose quantity depends on the 
 position of the interposed crystal. The light is thus said 
 (though improperly) to be depolarized by the crystal ; and it 
 was by this property that the double-refracting structure was 
 detected by Malus in a vast variety of substances, in which 
 the separation of the two rays was too small to be directly 
 perceived. 
 
 In order to analyze this phenomenon, let the crystalline 
 plate be so placed as to receive the polarized beam perpendi- 
 cularly, and let it be turned round in its own plane. We 
 shall then observe that there are two positions of the plate in 
 which the reflected ray vanishes, just as if no crystal had 
 been interposed. These two positions are at right angles to 
 one another ; and they are those in which the principal sec- 
 tion of the crystal coincides ivith the plane of the first reflexion, 
 or is perpendicular to it. When the plate is turned round 
 from either of these positions, the light gradually increases ; 
 and it becomes a maximum, when the principal section is 
 inclined at an angle of 45 to the plane of the first reflexion. 
 
 (226) In these experiments the reflected light is white. 
 But if the interposed crystalline plate be very thin, the most 
 gorgeous colours appear, which vary with every change of 
 inclination of the plate to the polarized beam. 
 
 Mica and sulphate of lime are very fit for the exhibition of 
 these beautiful phenomena, 'being readily divided by cleavage 
 into laminae of extreme thinness. If a thin plate of either of 
 these substances be placed so as to receive the polarized beam 
 perpendicularly, and be then turned round in its own plane, 
 the tint does not change, but varies only in intensity ; the 
 colour vanishing altogether when the principal section of the 
 crystal coincides with the plane of primitive polarization, or 
 
 09 
 
 SttSIVBRSITVl 
 
216 INTERFERENCE OF POLARIZED LIGHT. 
 
 is perpendicular to it, and, reaching a maximum, when it 
 is inclined to the plane of primitive polarization at an angle 
 of 45. 
 
 If, on the other hand, the crystal be fixed, and the ana- 
 lyzing plate turned, so as to vary the inclination of the plane 
 of the second reflexion to that of the first, the colour will be 
 observed to pass, through every grade of tint, into the com- 
 plementary colour ; it being always found that the light re- 
 flected in any one position of the analyzing plate is comple- 
 mentary, both in colour and intensity, to that which it reflects 
 in a position 90 from the former. This curious relation will 
 appear more evidently, if we substitute a double-refracting 
 prism for the second reflector ; for the two pencils refracted 
 by the prism have their planes of polarization one coincident 
 with the principal section of the prism, and the other at right 
 angles to it, and are therefore in the same condition as the 
 light reflected by the analyzing plate, with its plane of re- 
 flexion successively in these two positions. In this manner 
 the complementary lights are seen together, and may be easily 
 compared. But the accuracy of the relation is completely 
 established by making the two pencils partially overlap ; for, 
 whatever be their separate tints, it will be found that the part 
 in which they are superposed is absolutely white. 
 
 (227) When laminae of different thicknesses are interposed 
 between the polarizing and analyzing plates, so as to receive 
 the polarized beam perpendicularly, the tints are found to 
 vary with the thickness. The colours produced by plates of 
 the same crystal, of different thicknesses, follow the same law 
 as the colours reflected from thin plates of air , the tints rising 
 in the scale as the thickness is diminished ; until finally, when 
 this thickness is reduced below a certain limit, the colours 
 disappear altogether, and the central space appears black, as 
 when the crystal is removed. The thickness producing cor- 
 
INTERFERENCE OF POLARIZED LIGHT. 
 
 responding tints is, however, much greater in crystalline plates 
 exposed to polarized light, than in thin plates of air, or any 
 other uniform medium. The black of the first order appears 
 in a plate of sulphate of lime, when the thickness is the 
 2 oV oth of an inch. Between 2^0 o^ n an( ^- jV^ n ^ an i ncn > we 
 have the whole succession of colours of Newton's scale ; and 
 when the thickness exceeds the latter limit, the transmitted 
 light is always white. The tint produced by a plate of mica, 
 in polarized light, is the same as that reflected from a plate 
 of air of only the ^J n th part of the thickness. 
 
 Pursuing the examination of the same subject for oblique 
 incidences, M. Biot found that, in uniaxal crystals, the tint 
 developed or rather the number of periods and parts of a 
 period belonging to a ray of given refrangibility was deter- 
 mined by the length of the path traversed by the light within 
 the crystal, and by the square of the sine of the angle which 
 its direction made with the optic axis, jointly. In biaxal 
 crystals we must substitute, for the square of the sine, the 
 product of the sines of the angles which the ray makes with 
 the two axes. 
 
 (228) Let us now turn to the physical theory, and see in 
 what manner it explains the appearances. 
 
 We have seen that the wave incident upon a crystal 
 is resolved into two sets of waves within it, which traverse 
 it in different directions, and with different velocities. One 
 of these waves, therefore, will lag behind the other, and they 
 will be in different phases of vibration at emergence. When 
 the plate is thin, this retardation of one wave upon the other 
 will amount only to a few undulations and parts of an undu- 
 lation ; and it would therefore appear that we have here all 
 the conditions necessary for their interference, and the con- 
 sequent production of colour. 
 
 But here we are met by a difficulty. So far as this expla- 
 
218 INTERFERENCE OF POLARIZED LIGHT. 
 
 nation goes, the phenomena of interference and of colour 
 should be produced bj the crystalline plate alone, and in 
 common light, without either polarizing plate or analyzing 
 plate. Such, however, is not the fact, and the real difficulty 
 in this case is, not so much to explain how the phenomena 
 are produced, as to show why they are not always produced. 
 
 In seeking for a solution of this difficulty, we perceive 
 that the two rays, whose interference is supposed to produce 
 the observed results, are not precisely in the condition of 
 those whose interference we have hitherto examined. They 
 are polarized, and polarized in opposite planes. We are led 
 then to inquire, whether there is anything peculiar to the in- 
 terference of polarized rays which may influence these results ; 
 and the answer to this inquiry will be found to complete the 
 solution of the problem. 
 
 (229) The subject of the interference of polarized light 
 was examined, with reference to this question, by Fresnel and 
 Arago, and its laws experimentally developed. It was found 
 that two rays of light, polarized in the same plane, interfere 
 and produce fringes, under the same circumstances as two 
 rays of common light ; that when the planes of polarization 
 of the two rays are inclined to each other, the interference is 
 diminished, and the fringes decrease in intensity ; and that, 
 finally, when the angle between these planes is a right angle, 
 no fringes whatever are produced, and the rays no longer in- 
 terfere at all. These facts may be established by taking a 
 plate of tourmaline which has been carefully worked to a 
 uniform thickness, cutting it in two, and placing one-half in 
 the path of each of the interfering rays. It will be then 
 found that the intensity of the fringes depends on the rela- 
 tive position of the axes of the two tourmalines. When these 
 axes are parallel, the fringes are best defined ; they decrease 
 in intensity, when the axes of the tourmalines are inclined to 
 
INTERFERENCE OF POLARIZED LIGHT. 219 
 
 one another ; and finally, they vanish altogether when the 
 axes form a right angle. 
 
 In this law we find the account of the fact which hitherto 
 perplexed us, namely, that no phenomena of interference or 
 colour are produced, under ordinary circumstances, by the 
 two rays which emerge from a crystalline plate, and which 
 are polarized in opposite planes ; and we learn that, to pro- 
 duce these phenomena, the planes of polarization of the two 
 rays must be brought to coincidence by the analyzer. 
 
 The non-interference of rays, polarized in planes at right 
 angles to one another, is a necessary result of the mechanical 
 theory of transversal vibrations. In fact, it is a mathematical 
 consequence of that theory, that the intensity of the resultant 
 light in that case is constant, and equal to the sum, of the in- 
 tensities of the two component lights, whatever be the phases 
 of vibration in which they meet. 
 
 Fresnel and Arago discovered, further, that two op- 
 positely-polarized rays will not interfere, even when their 
 planes of polarization are made to coincide, unless they belong 
 to a pencil, the whole of which was originally polarized in one 
 plane ; a ad that, in the interference of rays which had under- 
 gone double refraction, half an undulation must be supposed 
 to be lost or gained, in passing from the ordinary to the extra- 
 ordinary system, just as in the transition from the reflected 
 to the transmitted system, in the colours formed by thin 
 plates. 
 
 The principle of the allowance of half an undulation is a 
 simple consequence of the theory of transversal vibrations. 
 In fact, the vibration of the wave incident on the crystal is 
 resolved into two within it, at right angles to one another, one 
 in the plane of the principal section, and the other in the per- 
 pendicular plane. Each of these must be again resolved, in 
 two fixed directions which are also perpendicular ; and it will 
 easily appear from the process of resolution, that of the four 
 
220 INTERFERENCE OF POLARIZED LIGHT. 
 
 components into which the original vibration is thus resolved, 
 the pair in one of the final directions must conspire, while in 
 the other they are opposed. Accordingly, if the vibrations of 
 the one pair are coincident, those of the other differ by half an 
 undulation. Hence, when the plane of reflexion of the ana- 
 lyzing plate coincides successively with these two positions, 
 the colours (which result from the interference of the portions 
 in the plane of reflexion) will be complementary. 
 
 The former of the two laws explains the office of the 
 polarizing plate in the phenomena. To account mechani- 
 cally for the non-interference of the two pencils, when the 
 light incident upon the crystal is unpolarized, we may regard 
 a ray of common light as composed of two rays of equal inten- 
 sity, oppositely polarized,* and whose vibrations are therefore 
 perpendicular. Each of these vibrations, when resolved into 
 two within the crystal, and these two again resolved in the 
 plane of reflexion of the analyzing plate, will exhibit the phe- 
 nomena of interference. But the interval of retardation will 
 differ by half a wave in the two cases ; the tints produced will 
 therefore be complementary, and the light resulting from their 
 union will be of a uniform whiteness.! 
 
 * More properly, a ray of common light must be regarded as composed of an 
 indefinite number of rays polarized in all azimuths ; so that if any two planes 
 be assumed at right angles, there will be an equal quantity of light actually po- 
 larized in each. Ordinary light, in fact, consists of a series of systems of 
 waves, in each of which the vibrations are different ; the different systems suc- 
 ceeding one another so rapidly, that, in a moderate time, as many vibrations 
 take place in any one plane, as in another at right angles to it. But the phe- 
 nomena of interference, exhibited by common light, compel us also to admit (as Sir 
 G. B. Airy has observed) that the vibrations do not change continually ; and that 
 in each system of waves there are, probably, several hundred vibrations which 
 are all similar, although the vibrations constituting one system bear no rela- 
 tion to those of another, and the different systems succeed one another with such 
 rapidity as to obliterate all trace of polarization. 
 
 f "We have here supposed the resulting light to be simply the sum of the 
 lights derived from each of the portions into which the original light was sup- 
 
INTERFERENCE OF POLARIZED LIGHT. 221 
 
 These laws of interference being kept in mind, the rea- 
 son of all the phenomena is apparent. The ray is origi- 
 nally polarized in a single plane, by means of the polarizing 
 plate. It is then divided into two within the crystal, which 
 are polarized in opposite planes ; and these are finally reduced 
 to the same plane by means of the analyzing plate. The two 
 pencils will therefore interfere ; and the resulting tint will 
 depend on the difference of retardation of the two waves, 
 produced by the difference of the velocities within the 
 crystal. 
 
 (230) It is plain that the light issuing from the crystal 
 is, in general, ellipticatty-polarized, inasmuch as it is the re- 
 sultant of two waves, in which the vibrations are at right 
 angles, and differ in phase. Hence, when homogeneous light 
 is used, and the emergent beam is analyzed with a double- 
 refracting prism, the two pencils into which it is divided vary 
 in intensity as the prism is turned, neither, in general, ever 
 vanishing. 
 
 When the thickness of the crystal is such, that the differ- 
 ence of phase of the two rays is an exact number of semi-un- 
 dulations, they will compound a plane-polarized ray at emer- 
 gence, the plane of polarization coinciding with the plane of 
 primitive polarization, or making an equal angle with the prin- 
 cipal section of the crystal on the other side, according as the 
 difference of phase is an even or odd multiple of half a wave. 
 Accordingly, one of the pencils into which the light is divided 
 
 posed to be resolved, without reference to their phase. The justice of this as- 
 sumption depends upon the fact adverted to in the preceding note, namely, 
 that the two oppositely polarized portions, into which we have supposed com- 
 mon light to be resolved, differ in phase, that difference continually varying. 
 The same thing is true, therefore, of the final components ; so that these must 
 be regarded as lights proceeding from different sources, and compound a light 
 equal in intensity to the sum of the components. 
 
222 INTERFERENCE OF POLARIZED LIGHT. 
 
 by the analyzing prism will vanish in two positions of its 
 principal section ; and it is manifest that the successive thick- 
 nesses of the crystalline plate at which this takes place form 
 a series in arithmetical progression. 
 
 On the other hand, when the difference of phase is a 
 quarter of a wave length, or an odd multiple of that quantity, 
 and when, at the same time, the principal section of the 
 crystal is inclined at an angle of 45 to the plane of primitive 
 polarization the emergent light will be circularly polarized. 
 This is one of the simplest means of obtaining a circularly- 
 polarized beam ; but it has the disadvantage, that the required 
 interval of phase is only exact for waves of one particular 
 length, and that therefore the circular polarization is perfect 
 only for one colour. 
 
 (231) We may now calculate the intensities and the 
 tints of the two emergent pencils, when a parallel pencil of 
 polarized light is incident upon a thin crystalline plate, and 
 the emergent pencils transmitted through an analyzing rhomb 
 of Iceland spar, or any other double-refracting crystal. 
 
 Let the first surface of the crystalline plate be imagined to 
 lie in the plane of the paper, and from the' point of incidence, 
 0, let the line OP be the section with that plane, of the 
 plane of primitive polarization of the ray, and OA its inter- 
 section with the principal plane of the crystal ; and let the 
 angle which it forms with the plane of primitive polariza- 
 tion, POA = a. Finally, let OB be the intersection with the 
 same plane of the principal plane of the analyzing rhomb ; 
 and let the angle which it makes with the plane of primitive 
 polarization, POB = /3. On entering the crystalline plate, 
 the polarized ray will be divided into the ordinary ray, 0, 
 whose amplitude of vibration is represented by cos a, the 
 amplitude of the original vibration being taken as unit ; and 
 the extraordinary ray, E, whose amplitude is proportional to 
 
INTERFERENCE OF POLARIZED LIGHT. 223 
 
 sin a. Each of these pencils will be again subdivided into 
 two by the analyzing prism, which we shall designate by 
 the symbols 4 0', + E', E + O x , E + E x ; and it is readily 
 seen that their respective amplitudes of vibration are 
 
 for the ray + O x , cos a cos (a - /3) ; 
 
 + E', cos a sin (a - /3) ; 
 
 E+0', sin asin(a-jS); 
 
 E + E x , - sin a COS (a - )3 j. 
 
 The first and third of these compound the ordinary emergent 
 ray, and the second and fourth the extraordinary. 
 
 The amplitudes of the vibrations in these two resultant 
 pencils are given by the formula 
 
 A 2 = 2 + a' 2 + 2aa' cos 2ir 
 
 \ A J 
 
 in which a and a' are the amplitudes of the component vi- 
 brations above given, and o - e the difference of the lengths 
 of the paths traversed by them. Hence the square of the 
 amplitude of the resulting ordinary ray, or the intensity of 
 the light, will be 
 
 cos 2 a cos 2 (a - ]3) + sin 2 a sin 2 (a - P) 
 + 2 cos a sin a cos (a - /3) sin (a - j3) cos 2?r ( -y- j. 
 
 Adding and subtracting 2 cos a sin a cos (a - |3) sin (a-/3), 
 
 this is reduced to 
 
 cos 2 ]3 - sin 2a sin 2 (a - /3) sin 2 TT i r \ 
 
 Compounding in like manner the second and fourth pen- 
 
224 INTERFERENCE OF POLARIZED LIGHT. 
 
 oils, + E' and E + E', we have, for the intensity of the ex- 
 traordinary pencil, 
 
 sin 2 ]3 + sin 2a sin 2 (a - j3) sin 2 TT f -r- Y 
 
 (232) The preceding formulas give the intensities of the 
 two resulting pencils, in terms of the angles a and /3, and of 
 the difference of the paths traversed by the two rays within 
 the crystal, for any given wave-length i. e. for any species 
 of homogeneous light. When white light is used, we must 
 take the sum of the expressions corresponding to the dif- 
 ferent values of A. Now, the angles a and /3 are independent 
 of A, and therefore the same for all the components of the 
 pencil ; and the same may be said of the difference of the 
 paths, o - e, which is independent of A, except in crystals of 
 extraordinary dispersive power. Hence, when the incident 
 light is compound, the intensities of the two rays will be 
 
 cos 2 ]3 - sin 2a sin 2 (a - /3) S sin 2 TT (&* ] ; 
 
 \ A J 
 
 sin 2 /3 + sin 2a sin 2 (a - /3) S sin 2 * \- 
 
 (233) The foregoing formulas contain the whole theory 
 of the colours of crystalline plates in polarized light. 
 
 1. The sum of the two intensities is equal to 1, or to the 
 intensity of the incident light. Accordingly, when the 
 incident light is white, the colours of the two pencils are 
 complementary. The truth of this is seen by partially 
 superposing the two images, the portions superposed being 
 always white. 
 
 2. The two pencils become white, whatever be the in- 
 terval of retardation within the crystal, when 
 
 sin 2a sin 2 (a - 13) = 0. 
 
INTERFERENCE OF POLARIZED LIGHT. 225 
 
 This equation is fulfilled, when sin 2a = 0, i. e., when 
 a=0, or a = 90; 
 
 or the principal section of the crystal either parallel or per- 
 pendicular to the plane of primitive polarization. 
 
 The same condition is fulfilled, independently of a, when 
 sin 2 (a-/3) = 0, i. e., when 
 
 a -|3 = 0, ora-j3 = 90; 
 
 or the principal section of the analyzing prism parallel, or 
 perpendicular to, the principal section of the crystal. 
 
 These consequences of the formulas are also evident from 
 the consideration, that in each of the cases referred to, one 
 of the resultant pencils vanishes, and there is, consequently, 
 no interference. 
 
 The coloration is greatest when the coefficient of the quan- 
 tity sin 2 TT ( r j, is a maximum, or sin 2a sin 2 (a - /3) = 1, 
 
 i. e., when 
 
 a = 45 ; 
 
 /3 = 0, or /3 = 90. 
 
 To produce the most brilliant effects of colour, therefore, 
 the principal section of the crystal should form half a right 
 angle with the plane of primitive polarization, while the 
 principal section of the analyzing prism should be either 
 parallel, or perpendicular to, that plane. The intensities of 
 the ordinary and extraordinary pencils in this case are 
 
 (o -e 
 
 COS 2 7T -r 
 
 V A 
 
 When the angles a and /3 are unchanged, the intensity 
 of the light of the two pencils varies only with the quantity 
 
226 INTERFERENCE OF POLARIZED LIGHT. 
 
 ,0 O 
 
 r , in which o - e is the interval of retardation of the two 
 A 
 
 pencils. This interval is proportional to the thickness of the 
 crystal. Hence, when that thickness is increased, so that the 
 interval o - e is augmented by a whole number of undula- 
 tions, the intensity resumes its original value. 
 
 (234) It has been stated (226) that the phenomena of co- 
 lour are only produced when the crystalline plate is thin. In 
 thick plates, where the difference of phase of the two pencils 
 contains a great many wave-lengths, the tints of different 
 orders come to be superposed (as in the case of Newton's 
 rings where the thickness of the plate of air is considerable), 
 and the resulting light is white. The phenomena of colour may 
 however be produced in thick plates, by superposing two of 
 them in such a manner, that the ray which has the greater ve- 
 locity in the first shall have the lesser velocity in the second. 
 "We have only to place the plates with their principal sections 
 perpendicular or parallel, according as the crystals to which 
 they belong are of the same, or of opposite denominations. 
 Thus, if the crystals be uniaxal, and both positive, or both 
 negative, they are to be placed with their principal sections 
 perpendicular ; and, on the other hand, these sections should 
 be parallel, when one of the crystals is positive, and the other 
 negative. The reason of this is evident. 
 
 (235) Let us now consider the effects produced when a 
 converging or diverging pencil of rays traverses a uniaxal 
 crystal, in various directions inclined to the axis at small 
 angles: and let us suppose, for simplicity, that the crys- 
 talline plate is cut in a direction perpendicular to the 
 axis. 
 
 Let ABCD be the plate, and E the place of the ,eye. 
 
INTERFERENCE OF POLARIZED LIGHT. 
 
 227 
 
 The visible portion of the emergent beam will form a cone, 
 AEB, whose summit coin- 
 cides with the place of the 
 eye, and axis, EG, with the 
 axis of the crystal. The ray 
 which traverses the crystal 
 in the direction of the axis, 
 POE, will undergo no 
 change whatever; and con- 
 sequently will be reflected, or not, from the analyzing plate, 
 according as the plane of reflexion there coincides with, or is 
 perpendicular to, the plane of the first reflexion. But the 
 other rays composing the cone will be modified in their pas- 
 sage through the crystal ; and the changes which they 
 undergo 'will depend on their inclination to the optic axis, 
 and on the position of the principal section with respect to 
 the plane of primitive polarization. 
 
 Let the circle represent the section of the emergent cone 
 of rays made by the second surface of the crystal ; and let 
 MM 7 and NN X be two lines drawn 
 through its centre at right angles, 
 being the intersections of the plane 
 of primitive polarization, and of 
 the perpendicular plane, respec- 
 tively, with the surface. Now 
 the rays which emerge at any 
 point of these lines will not be di- 
 vided into two within the crystal, 
 nor will their planes of polarization 
 
 be altered, because the principal section of the crystal, for 
 these rays, in the one case coincides with the plane of pri- 
 mitive polarization, and in the other is perp endicular to it. 
 These rays therefore will be reflected, or not, from the ana- 
 lyzing plate, according as the plane of reflexion there coin- 
 
 Q2 
 
228 INTERFERENCE OF POLARIZED LIGHT. 
 
 cides with, or is perpendicular to, the plane of the first 
 reflexion. In the latter case, therefore, a black cross will be 
 displayed on the field, and in the former a white one, as is 
 represented in the annexed diagrams. 
 
 But the case is different with the rays which emerge at 
 any other point, such as L. The principal section of the 
 crystal for this ray, OL, neither coincides with, nor is per- 
 pendicular to, the plane of primitive polarization ; and con- 
 sequently the incident polarized ray will be divided into two 
 within the crystal, whose planes of polarization are parallel 
 and perpendicular to OL, respectively. The vibrations in 
 these two rays are reduced to the same plane by means of 
 the analyzing plate : they will therefore interfere, and the 
 extent of that interference will depend on their difference of 
 phase. 
 
 Now the difference of phase of the two vibrations depends 
 on the interval of retardation. When this interval is an odd 
 multiple of half an undulation, the two vibrations are in 
 complete discordance ; and, on the other hand, they are in 
 complete accordance when it is an even multiple of the same 
 quantity. We have seen (216) that, in general, the diffe- 
 rence of the squares of the velocities of propagation of the 
 two waves within a crystal is proportional to the product of 
 
INTERFERENCE OF POLARIZED LIGHT. 
 
 the sines of the angles made by the normal to their front with 
 the two optic axes. But, in the case of 'a parallel plate, it 
 may be shown, the difference of the squares of the velocities 
 is proportional to the thickness of the plate multiplied by 
 the interval of retardation. Hence in the case of uniaxal 
 crystals, for a given plate, the interval of retardation is pro- 
 portional to the square of the sine of the angle which the ray 
 makes with the optic axis within the crystal. It may be 
 easily shown that the sine of this angle is very nearly pro- 
 portional to the sine of the angle LEO (see first fig. p. 227), 
 which the emergent ray makes with the axis ; and this latter 
 to LO, the distance of the point of emergence from the cen- 
 tre. The retardation therefore varies as the square of the 
 distance LO ; and consequently the successive dark and 
 bright lines will be arranged in circles (as represented in the 
 preceding diagrams), the squares of whose radii are in arith- 
 metical progression. 
 
 We have been speaking hitherto of homogeneotis light. 
 When white or compound light is used, the rings of different 
 colours will be partially superposed, and the result will be a 
 series of iris-coloured rings separated by dark intervals. All 
 the phenomena, in fact, with the exception of the cross, are 
 similar to those of Newton's rings ; and we now see that 
 they are both cases of the fertile principle of interference. 
 These rings are exhibited even in thick crystals, because the 
 difference of the velocities of the two pencils is very small 
 for rays slightly inclined to the optic axis. 
 
 (236) Let us now consider briefly the case of biaml crys- 
 tals. 
 
 Let a plate of such a crystal be cut perpendicularly to the 
 line bisecting the optic axes, and let it be interposed, as be- 
 fore, between the polarizing and analyzing plates. In this 
 case the bright and dark bands will no longer be disposed in 
 
230 INTERFERENCE OF POLARIZED LIGHT. 
 
 circles, as in the former, but will form curves which are sym- 
 metric with respect to the lines drawn from the eye in the 
 direction of the two axes ; the points of the same band 
 being those for which the interval of retardation of the two 
 waves is the same. Now this interval is proportional to the 
 product of the sines of the angles which the normal to the 
 front of the waves makes with the optic axes (235 ) ; and 
 these sines are, very nearly, as the distances of the points of 
 emergence (measured on the face of the crystal) from the 
 projections of the optic axes. Hence the product of these 
 distances will be constant for all the points of the same 
 curve. The curve formed by each band is therefore the lem- 
 niscata of James Bernouilli, the fundamental property of 
 which is, that the product of the radii-vectores, drawn from 
 any point to two fixed poles, is a constant quantity. 
 
 The exactness of this law has been verified, in the most 
 complete manner, by the measurements of Sir John Herschel. 
 
 The constant varies from one curve to another, being pro- 
 portional to the interval of retardation, and increasing there- 
 
INTERFERENCE OF POLARIZED LIGHT. 231 
 
 fore as the numbers of the natural series for the successive 
 dark bands. For different plates of the same substance, the 
 constant is inversely as the thickness. 
 
 The diagrams above given represent the systems of rings 
 in a biaxal crystal whose axes form a small angle with one 
 another, in two positions of the crystalline plate, the planes 
 of polarization of the polarizing and analyzing plates being 
 at right angles. 
 
 The form of the dark brushes, which cross the entire sys- 
 tem of rings, is determined by the law which governs the 
 planes of polarization of the emergent rays. There is no dif- 
 ficulty in showing, on the principles of Fresnel's theory, that 
 two such dark curves, in general, pass through each pole ; 
 and that they are rectangular hyperbolas, whose common cen- 
 tre is the middle point of the line which connects the pro- 
 jections of the two axes. 
 
 (237) The phenomena of depolarization and of colour, 
 impressed by double-refracting substances upon the trans- 
 mitted light, are, we have seen, the necessary results of the 
 interference of the two pencils into which the light is divided 
 within them. These properties, therefore, become distinctive 
 characters of the double-refracting structure ; and thus enable 
 us to discover the existence, and to trace the laws of that 
 structure, even in substances in which the separation of the 
 two pencils is too minute to be directly observed. By such 
 means it has been discovered that a double-refracting struc- 
 ture may be communicated to bodies which do not possess it 
 naturally, by mechanical compression or dilatation. Thus Sir 
 David Brewster observed, that when pressure was applied to 
 the opposite faces of a parallelepiped of glass, it developed 
 a tint in polarized light, like a plate of double-refracting 
 crystal ; and the tint descended in the scale as the pressure 
 
232 INTERFERENCE OF POLARIZED LIGHT. 
 
 was augmented. Single-refracting crystals, such as muriate 
 of soda, and fluor spar, acquired the property of double 
 refraction by the same means. 
 
 The opposite effects of compression and dilatation may be 
 very well seen, and studied, in a thick plate of glass bent by 
 an external force. The entire mass of the plate is thus thrown 
 into an altered state of density, the parts towards the convex 
 side of the plate being dilated, and those towards the concave 
 side compressed; while, about the middle of the thickness, 
 there is a surface in which the particles are in their natural 
 state. Accordingly, when this body is interposed between 
 the polarizing and analyzing plates, so as to form an angle 
 of 45 with the plane of primitive polarization, two sets of 
 coloured bands are seen, separated by a neutral line ; and 
 these vanish altogether when the compressing force is with- 
 drawn. The parts towards the convex or dilated side of the 
 neutral line are found to have acquired a positive double- 
 refracting structure, and those on the concave, or compressed 
 side, a negative one. 
 
 In these cases of induced double refraction, the pheno- 
 mena are related to the form of the entire mass ; and the 
 axes of double refraction are single lines within the sub- 
 stance, fixed in position, as well as direction. In this respect 
 the phenomena are essentially different from those produced 
 by regular crystals, in which the laws of the double refraction 
 depend solely on the direction^ and are the same in all parts 
 of the substance. 
 
 (238) The phenomena described in the preceding article 
 are in perfect accordance with the wave-theory. Owing to 
 the connexion of the vibrating medium with the solid in 
 which it is contained, its elasticity is rendered unequal in 
 different directions by the effects of compression, the maxi- 
 
INTERFERENCE OF POLARIZED LIGHT. 233 
 
 mum and minimum of elasticity corresponding to the di- 
 rections of greatest and least pressure. Accordingly the 
 vibrations of the ray, on entering the substance, are resolved 
 into two in these directions, and these are propagated with 
 unequal velocities. 
 
 These results of theory were experimentally confirmed by 
 Fresnel ; and he found that the velocity with which a ray 
 traversed the glass was greater or less, according as its plane 
 of polarization coincided with, or was perpendicular to, the line 
 in which the pressure was exerted. The double refraction of 
 the ray is a necessary consequence of this d inference of velo- 
 cities : but this was also established by Fresnel by direct 
 experiment. A series of glass prisms were placed together, 
 with their refracting angles alternately in opposite directions, 
 and the ends of the alternate prisms were powerfully pressed 
 by screws. A ray transmitted through the combination was 
 found to be divided into two oppositely polarized. The 
 compressed prisms, in this arrangement, acquired a double- 
 refracting structure, the axis of pressure being also the axis of 
 double refraction ; and their refracting angles being all turned 
 in the same direction, the divergence of the two rays was in- 
 creased in proportion to their number, and thus rendered 
 sensible. The intermediate prisms served to correct the de- 
 viation, and to render the combination achromatic. 
 
 (239) The effects of unequal density and elasticity may 
 be much more regularly produced by the application of heat. 
 These effects may be studied by applying a bar of hot iron to 
 the edge of a rectangular plate of glass, and placing it in the 
 polarizing apparatus, so that the heated edge may form an 
 angle of 45 with the plane of primitive polarization. At the 
 end of some time, the whole surface of the plate will be 
 observed to be covered with coloured bands, the parts near 
 the opposite edges having acquired a positive double-re- 
 
234 INTERFERENCE OF POLARIZED LIGHT. 
 
 fracting structure, and those near the centre a negative one. 
 The effects are reversed when a plate of glass, uniformly 
 heated, is rapidly cooled at one of its edges ; and all the ap- 
 pearances vanish when the glass acquires the same tempera- 
 ture throughout. 
 
 If we transmit heat from the surface to the axis of a 
 glass cylinder, by immersing it in heated oil, it will display a 
 system of rings similar to those of a negative crystal with one 
 axis, the axis of the cylinder being also the axis of double 
 refraction. "When the heat reaches the axis the double 
 refraction begins to weaken ; and the colours disappear alto- 
 gether when the glass is uniformly heated. Again, if the 
 cylinder, when in this state, be made to cool rapidly by sur- 
 rounding it with a good conductor of heat, it will transiently 
 assume the opposite character of a positive double-refracting 
 crystal ; and when it is restored to a uniform temperature 
 throughout, all traces of double refraction again disappear. 
 If we employ an elliptic cylinder, instead of a circular one, 
 in the experiment just described, it will exhibit the coloured 
 curves formed by a biaxal crystal : and the phenomena may 
 be endlessly varied by varying the form of the glass to which 
 the heat is applied. 
 
 If now, by any means, the glass be arrested in one of 
 these transient states, it will acquire a permanent double- 
 refracting structure. This has been accomplished by raising 
 it to a red heat, and then cooling it rapidly at the edges. 
 For as the outer parts, which are thus more condensed, as- 
 sume a fixed form in cooling, the interior parts must accom- 
 modate themselves to that form, and therefore retain a state 
 of unequal density. The law of densit}^, and therefore the 
 double-refracting structure, will depend on the external 
 form ; and it is accordingly found that the coloured bands 
 and patches, which such bodies display in polarized light, 
 assume a regular arrangement varying with the shape of the 
 mass. 
 
INTERFERENCE OF POLARIZED LIGHT. 235 
 
 (240) As the double-refracting structure is communicated 
 to bodies which do not possess it naturally, by mechanical 
 compression or unequal temperature so, by the same means, 
 that structure may be altered in the bodies in which it already 
 resides. Thus Sir David Brewster and M. Biot found that 
 the double refraction of regular crystals may be altered, and 
 the tints they display made to rise or descend in the scale, by 
 simple pressure. 
 
 But the changes induced by heat are more remarkable. 
 Professor Mitscherlich discovered the important fact, that 
 heat dilates crystals differently in different directions, and so 
 alters their form ; and their double-refracting properties are 
 found to undergo corresponding changes. Thus, Iceland 
 spar is dilated by heat in the direction of its axis ; while it 
 actually contracts, by a small amount, in directions perpen- 
 dicular to it. The angles of the primitive form thus vary, 
 the rhomboid becoming less obtuse, and approaching the 
 form of the cube, in crystals of which form there is no 
 double refraction (93). Professor Mitscherlich accordingly 
 conjectured that the double-refracting energy of the crystal 
 must, in these circumstances, be diminished ; and the conjec- 
 ture was verified by experiment. In fact, the extraordinary 
 index in Iceland spar is found to increase considerably with 
 the temperature, while the ordinary index undergoes little or 
 no change. 
 
 We have seen (217) that the inclination of the optic axes, 
 in biaxal crystals, is a simple function of the three principal 
 elasticities of the vibrating medium, and that the plane of 
 the axes is that of the greatest and least elasticities. IF, 
 then, these elasticities be altered by heat in different propor- 
 tions, the inclination of the axes will likewise vary ; and it 
 may even happen that the plane of the axes will shift to a 
 position at right angles to that which it formerly occupied. 
 All these variations have been actually observed. Professor 
 
236 INTERFERENCE OF POLARIZED LIGHT. 
 
 Mitscherlich found that, in sulphate of lime, the angle be- 
 tween the axes (which is about 60 at the ordinary tempe- 
 rature) diminishes on the application of heat; that, as the 
 temperature increases, these axes approach until they unite ; 
 and that, on a still farther augmentation of heat, they again 
 separate, and open out in a perpendicular plane. Heat is found 
 to dilate this crystal more in one direction than in another 
 perpendicular to it. 
 
< 237 ') 
 
 CHAPTEE XIY. 
 
 ROTATORY POLARIZATION. 
 
 (241) IN the phenomena hitherto considered the changes 
 in the plane of polarization, which a polarized ray undergoes 
 in reflexion or refraction, are determinate in amount, and are 
 wholly independent of the distances traversed by the ray in 
 either medium. There are certain cases, however, in which 
 the change of the plane of polarization increases with the 
 thickness of the medium traversed ; and this plane is made 
 to revolve, sometimes from left to right (like the hands on 
 the dial-plate of a clock), and sometimes in the opposite 
 direction. This remarkable phenomenon was first observed 
 by Arago. 
 
 When a polarized ray, of any simple colour, traverses a 
 plate of Iceland spar, beryl, or any other uniaxal crystal, in 
 the direction of its axis, it suffers no change of any kind. 
 But when the ray traverses in the same manner a plate 
 of rock-crystal, its plane of polarization is found to be altered 
 at emergence ; and the change increases with the thickness of 
 the plate. In some crystals of this substance, the plane 
 of polarization is turned from left to right, while in others it 
 is turned in an opposite direction ; and the crystals themselves 
 are called right-handed or left-handed, according as they pro- 
 duce one or other of these effects. 
 
 (242) The phenomena of rotatory polarization in rock- 
 crystal were analyzed with great diligence and success by 
 M. Biot, and were reduced by him to the following general 
 laws. 
 
238 ROTATORY POLARIZATION. 
 
 I. In different plates of the same crystal, the rotation of 
 the plane of polarization is always proportional to the thick- 
 ness of the plate. The same thing holds, very nearly, in 
 plates of different crystals. 
 
 II. When two plates are superposed, the effect produced 
 is, very nearly, the same as that which would be produced by 
 a single plate, whose thickness is the sum or difference of the 
 thicknesses of the two plates, according as they are of the 
 same or of opposite denominations. 
 
 III. The rotation of the plane of polarization is different 
 for the different rays of the spectrum, and increases with 
 their refrangibility . For a given plate, the angle of rotation 
 is inversely as the square of the length of the wave. Thus the 
 angle of rotation, produced by a plate of rock-crystal whose 
 thickness is a millimetre, is 17| for the extreme red of the 
 spectrum, 30 for the rays of mean refrangibility, and 44 
 for the extreme violet. 
 
 Since the rays of different colours emerge polarized in 
 different planes, it follows that if a beam of white light be let 
 fall upon the crystal, and be received after emergence upon 
 an analyzing plate, this will reflect a portion of the light in 
 every position of the plane of reflexion ; and this light will 
 be beautifully coloured, the colour varying with the thick- 
 ness of the crystal, and the position of the analyzing plate. 
 For the analyzing plate will reflect the rays of different 
 colours in different proportions, depending on the positions 
 of their planes of polarization with respect to the plane of 
 reflexion ; and the resulting colour will be a compound tint, 
 which can be easily estimated. 
 
 (243) The curious distinction which was found to sub- 
 sist between different specimens of rock-crystal, has been con- 
 nected by Sir John Herschel with a difference of crystalline 
 form. The ordinary form of the crystal of quartz is the 
 six-sided prism terminated by the six-sided pyramid. The 
 
ROTATORY POLARIZATION. 239 
 
 solid angles, formed at the junction of the pyramid and prism, 
 are sometimes replaced by small secondary planes, which are 
 oblique with reference to the original planes of the crystal ; 
 and the form of the crystal is then called plagiedral. In the 
 same crystal these planes lean all in the same direction ; and 
 it is found that, when that direction is to the right (the apex 
 of the pyramid being uppermost), the crystal is right-handed ; 
 and that, on the contrary, it is left-handed, when the planes 
 lean in the opposite way. These relations between the ro- 
 tation of the plane of polarization, and the crystalline form, 
 have been found by M. Pasteur to hold in other crystals. 
 
 Sir David Brewster subsequently discovered that the ame- 
 thyst, or violet quartz, is made up of alternate layers of right- 
 handed and left-handed quartz. This remarkable structure 
 may be traced in the fracture of the mineral ; for the edges 
 of the layers crop out, and give to the fracture the undulating 
 appearance which is peculiar to this mineral. But the 
 structure in question is displayed in the most beautiful man- 
 ner, when we expose a plate of this substance to polarized 
 light, 
 
 The colours exhibited in polarized light likewise reveal 
 the existence of crystals of quartz penetrating others in va- 
 rious directions, when no striae, or other external appearances, 
 indicate their presence. 
 
 (244) The phenomena of rotatory polarization in rock- 
 crystal have been accounted for by the interference of two 
 circularly polarized pencils, which are propagated along the 
 axis with unequal velocities, one revolving from left to right, 
 and the other in the opposite direction. 
 
 For a plane polarized ray is equivalent to two circularly- 
 polarized rays of half the intensity, in which the vibrations 
 are in opposite directions. When a plane-polarized ray, 
 therefore, is incident perpendicularly upon a plate of rock- 
 
240 ROTATORY POLARIZATION. 
 
 crystal, cut perpendicularly to the axis, it may be resolved 
 into two such circularly-polarized rays ; and as these are sup- 
 posed to be transmitted with different velocities, one of them 
 will be in advance of the other when they assume a common 
 velocity at emergence. They then compound a single ray, 
 polarized in a single plane ; and this plane, it can be shown, 
 is removed from the plane of primitive polarization by an 
 angle proportional to the interval of retardation, and there- 
 fore to the thickness of the crystal. 
 
 Thus the laws of rotatory polarization are completely ex- 
 plained ; and it only remains to prove the truth of the 
 assumption, that two circularly-polarized pencils, whose 
 vibrations are in opposite directions, are actually transmitted 
 along the axis of quartz with different velocities. This sup- 
 position is easily put to the test of experiment, since such a 
 difference of velocity must produce a difference of refraction 
 when the surface of emergence is oblique to the direction of 
 the ray. According to this hypothesis, therefore, a polarized 
 ray transmitted through a prism of rock-crystal, in the di- 
 rection of the optic axis, should undergo double refraction at 
 emergence ; and the two pencils into which it is divided 
 should be circularly-polarized. This has been completely 
 verified by Fresnel, by means of an achromatic combination 
 of right-handed and left-handed prisms, arranged so as to 
 double the separation; and he has shown that the two 
 pencils are neither common nor plane-polarized light, but 
 possess all the physical characters of light circularly-po- 
 larized. 
 
 (245) The relation between the rotation and double re- 
 fraction of rock-crystal, in the direction of its axis, has been 
 very simply deduced by M. Babinet. 
 
 Let v and v' denote the velocities of the ordinary and 
 extraordinary waves in the direction of the axis of the 
 
ROTATORY POLARIZATION 241 
 
 crystal ; ^ and // the corresponding refractive indices ; 
 then 
 
 But, if 9 be the thickness of the crystal, and S the interval of 
 retardation of the two waves after traversing it, the second 
 member of the preceding equation is obviously equal to 
 
 $ 
 
 g ^ or to 1 + - r 8 being very small in comparison to 0. 
 
 We have therefore 
 
 Now the angle of rotation is proportional to the interval of 
 retardation of the two circularly-polarized pencils ; and when 
 that interval is equal to the length of a wave in vacuo, the 
 angle of rotation is 180. Hence the interval of retardation 
 of the emergent rays corresponding to any angle of rotation, 
 
 /o, will be X r-- , X denoting the length of the wave ; and the 
 
 corresponding interval within the crystal is equal to the same 
 quantity, multiplied by the velocity of propagation, or di- 
 vided by the refractive index. Hence, if p be the rotation 
 corresponding to the thickness of the crystal, 0, we have 
 
 and substituting this value in the preceding formula, 
 
 ,_ X P 
 
 ^ P 180 
 
 it 
 
242 ROTATORY POLARIZATION. 
 
 This difference is extremely small. When is one milli- 
 metre, the angle of rotation, p, corresponding to the rays of 
 mean refrangibility, = 30. But for these rays, A = *0005 of 
 a millimetre ; and therefore ju'-ju = -00008, 
 
 (246) The phenomena hitherto described take place only 
 in the direction of the axis of the crystal. Sir Gr. B. Airy disco- 
 vered that when a plane-polarized ray is transmitted through 
 rock-crystal in any direction inclined to the axis, it is divided 
 into two pencils which are elliptical-polarized ; the elliptical 
 vibrations in the two rays being in opposite directions, and 
 the greater axes of the ellipses coinciding respectively with 
 the principal plane, and with the perpendicular plane. The 
 ratio of the axes in these ellipses varies with the inclination 
 of the ray to the optic axis, being a ratio of equality when 
 the direction of the ray coincides with the axis, and in- 
 creasing indefinitely with its inclination to that line. With 
 respect to the course of the refracted rays, Sir Gr. B. Airy found 
 that it was still determined by the Huygenian law; but 
 that the sphere and spheroid, which determine the velocities 
 and directions of the two rays, do not touch, as in all other 
 known uniaxal crystals the latter surface being contained 
 entirely within the former. This is a necessary consequence 
 of the fact, that the interval of retardation of the two pencils 
 does not vanish, with the inclination of the ray to the optic 
 axis. 
 
 Sir Gr. B. Airy has given an elaborate calculation, founded 
 on these hypotheses, of the forms of the rings, &c., displayed 
 by rock-crystal in plane-polarized and circularly-polarized 
 light ; and he has found a striking agreement between the 
 results of calculation and experiment. Among the most re- 
 markable of the phenomena whose laws are thus developed, 
 is that produced by the superposition of two plates of rock- 
 
ROTATORY POLARIZATION. 243 
 
 crystal, of the same thickness, one of them being right- 
 handed, and the other left-handed. 
 
 In order to complete the experimental investigation of this 
 subject, it remained to determine the velocities of the two 
 elliptically-polarized rays, and the ratio of the axes of the 
 ellipses, as dependent on the inclination of the rays to the 
 axis of the crystal. This has been effected by M. Jamin, by 
 measuring the amplitudes, and the differences of phase of the 
 two component pencils, when the incident light is polarized 
 in the plane of a principal section. From these data the 
 quantities sought are deduced by calculation. 
 
 (247) All these complicated facts have been linked toge- 
 ther, and their laws deduced, by Professor Mac Cullagh. In 
 this remarkable investigation the author sets out by assuming 
 the form of the differential equations of vibratory motion in 
 rock-crystal ; and from this assumed form he has deduced the 
 elliptical polarization of the two pencils; the law of the 
 ellipticity as depending on the inclination of the ray to the 
 axis ; the interval of retardation in the direction of the 
 axis ; and the peculiar form of the wave-surface. 
 
 The ratio of the axes of the two ellipses is found to be 
 equal to unity in the direction of the axis of the crystal. In 
 all other directions it is given by a quadratic equation 
 whose constant term is equal to unity ; so that this ratio 
 has two values, one of which is the reciprocal of the other. 
 Hence the ratio of the axes is the same in both ellipses ; and 
 the greater axis of one coincides with the smaller axis of the 
 other. 
 
 When the ray traverses the axis of the crystal, the rota- 
 tion of the plane of polarization is given by the formula 
 
 CO 
 
244 ROTATORY POLARIZATION. 
 
 which comprises all the experimental laws of M. Biot (242). 
 The sign of the constant factor, (7, determines the direction of 
 the rotation. 
 
 It is a striking peculiarity of this theory, that it contains 
 (in addition to the two refractive indices) but one constant, 
 and that this constant having been determined, from the 
 known angles of rotation when the ray traverses the axis of 
 the crystal, the ratio of the axes of the ellipses may be calcu- 
 lated when the ray is inclined by any angle to the axis. 
 The author has applied this calculation to the observations of 
 Sir Gr. B. Airy, and has found the calculated and observed 
 results to agree. 
 
 (248) MM. Biot and Seebeck discovered that some of the 
 liquids, and even of the vapours, possess the same property as 
 quartz in the direction of its axis, and impress a rotation on 
 the plane of polarization of the intromitted ray, which is 
 proportional to the thickness of the substance traversed. The 
 fact is easily observed by transmitting a polarized ray through 
 a long tube filled with the liquid, and closed at each end by 
 parallel plates of glass, and analyzing the emergent ray by 
 a double-refracting prism. Among the liquids possessing 
 this property are oil of turpentine, oil of lemon, solution of 
 sugar in water, solution of camphor in alcohol, &c. The first- 
 mentioned of these liquids is right-handed, and the others 
 left-handed. They all possess the property in a much feebler 
 degree than quartz ; so that the ray must traverse a much 
 greater thickness of the substance, in order to have its plane 
 of polarization altered by the same amount. Thus a plate of 
 rock-crystal, whose thickness is one millimetre, rotates the 
 plane of polarization of the red ray through an arc of about 
 18 ; a plate of oil of turpentine, of the same thickness, 
 turns the plane of polarization only through a quarter of a 
 degree. 
 
ROTATORY POLARIZATION. 245 
 
 The rotatory liquids do not lose their peculiar power 
 (except in degree) by dilution with other liquids not pos- 
 sessing the property ; and they retain it even in the state 
 of vapour. From these and other facts, M. Biot concluded 
 that this property, in liquids, is inherent in their ultimate 
 particles. In this respect the rotatory liquids are essentially 
 distinguished from rock-crystal, which is found to lose the 
 property when it loses its crystalline arrangement. Thus 
 Sir John Herschel observed, that quartz held in solution by 
 potash (liquor of flints) did not possess the rotatory power ; 
 and the same thing has been remarked by Sir David Brewster 
 with respect to fused quartz. 
 
 (249) When two or more liquids possessing this property 
 are mixed together, the rotation produced by the mixture is 
 always the sum, or the difference, of the rotations produced 
 by the ingredients (in thicknesses proportional to the volumes 
 in which they enter the mixture), according as the liquids are 
 of the same, or of contrary denominations. The same law 
 holds good in many cases in which the liquids are chemically 
 united. 
 
 M. Biot has made an important application of this prin- 
 ciple to the analysis of compounds containing a substance 
 possessing the rotatory power combined with others which 
 are neutral, the quantity of which in the compound may 
 (by the principle just stated) be determined, by observing 
 the optical effects of the mixture. This application has 
 been found of much industrial value, in the case of the sac- 
 chorine solutions; and a very ingenious apparatus, called 
 the saccharometer, has been devised by M. Soleil for the 
 purpose. This instrument is founded upon the principle that 
 the rotatory solutions follow the same laws as rock-crystal, 
 in their action upon the light of different colours ; so that it 
 is possible to compensate the effect of the solution by a plate 
 
246 ROTATORY POLARIZATION. 
 
 of rock-crystal of a suitable thickness, and of the opposite 
 action. 
 
 (250) The quantity to be measured, in all such experi- 
 ments, being the angle through which the plane of polarization 
 has been caused to rotate, it is necessary to determine with 
 precision the position of that plane before, and after, the 
 interposition of the rotating liquid. Now this determination 
 cannot be made with the required accuracy by means of Ni- 
 col's prism, or any of the other analyzers heretofore in use ; 
 for, in all of these, the position in which the intensity of the 
 light is evanescent, or a minimum, can only be known by 
 the comparison of two lights which are not simultaneously 
 observed. To remedy this defect, Mr. Jellett has devised an 
 analyzer, in which the two lights to be compared are seen 
 at the same time, and in juxtaposition. A rhomboid of 
 Iceland spar, about two inches in length, is cut by two planes 
 perpendicular to the longitudinal edges, so as to form a right 
 prism. This prism is then divided by a plane parallel to 
 the edges, and making a small angle with the longer dia- 
 gonal of the base. One of the two halves is then reversed, 
 and the two parts are cemented together with the surfaces 
 of section in contact. Finally the two ends are ground and 
 polished. 
 
 Now let a cylindrical pencil of plane-polarized light tra- 
 verse such a prism in a direction parallel to the edges, so as 
 to be equally divided by the plane of section, each half of the 
 cylinder traversing one-half of the compound prism. Each of 
 these semi-cylindrical pencils will be doubly-refracted, and 
 the two extraordinary pencils will be made to deviate nearly 
 in opposite directions. These two pencils are to be stopped 
 by diaphragms suitably placed. On the other hand, the two 
 ordinary pencils are undeviated, and will therefore compose 
 a cylindrical pencil which is equally divided by the plane of 
 
ROTATORY POLARIZATION. 247 
 
 section. The planes of polarization of these two portions are 
 however different, and form an angle nearly equal to double 
 the angle which that plane contains with the longer diagonal 
 of the base of the prism. Hence the intensities of the two 
 halves will be unequal, unless when the plane of the section 
 coincides with the plane of primitive polarization. The ex- 
 periment then consists in equalizing the intensities by the 
 rotation of the prism ; and as the eye is capable of detecting 
 with much accuracy differences of intensities of two lights 
 observed simultaneously, the position of the plane of primi- 
 tive polarization will be determined with great exactness. 
 With a well-constructed prism of this kind, the error of ob- 
 servation need not exceed 1', that of the ordinary method 
 being about 30 '. 
 
 Mr. Jellett has contrived a saccharometer in which a 
 prism, such as has been above described, is used to determine 
 the amount of rotation of the plane of polarization produced 
 by a saccharine solution, and therefore the strength of the 
 solution itself. He estimates the probable error of the result 
 of a single observation, with such an instrument, to be only 
 0-02 of a grain in a cubic inch of the solution. 
 
 
 THE END. 
 
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INDEX 
 
 Acton's Modern Cookery 40 
 
 Aird's Blackstone Economised 39 
 
 Alpine Club Map of Switzerland 34 
 
 Alpine Guide (The) 34 
 
 Amoss Jurisprudence 10 
 
 Primer of the Constitution 10 
 
 Anderson's Strength of Materials 20 
 
 Armstrong's Organic Chemistry 20 
 
 Arnold's (Dr.) Christian Life 30 
 
 Lectures on Modern History 2 
 
 Miscellaneous Works 13 
 
 School Sermons 30 
 
 Sermons 30 
 
 (T. ) Manual of English Literature 13 
 
 A mould's Life of Lord Denman 7 
 
 Atherstone Priory 36 
 
 Autumn Holidays of a Country Parson ... 14 
 
 Ay re's Treasury of Bible Knowledge 39 
 
 Bacons Essays, by Whately n 
 
 by Hun ter n 
 
 Life and Letters, by Spedding ... 10 
 
 Works 10 
 
 Bain's Mental and Moral Science 12 
 
 on the Senses and Intellect 1 1 
 
 Baker's Two Works on Ceylon 33 
 
 Ball's Guide to the Central Alps 34 
 
 Guide to the Western Alps 34 
 
 Guide to the Eastern Alps 34 
 
 Bancroft's Native Races of the Pacific 23 
 
 Box ley's Spain 33 
 
 Beckers Charicles and Gallus 35 
 
 Black's Treatise on Brewing 40 
 
 Blackley s German- English Dictionary 15 
 
 Blaine's Rural Sports 37 
 
 Bloxams Metals 20 
 
 Book of Psalms, by E. F 32 
 
 Boultbee on 39 Articles 29 
 
 Bournes Catechism of the Steam Engine . 28 
 
 Handbook of Steam Engine 28 
 
 Treatise on the Steam Engine ... 28 
 
 Improvements in the same 28 
 
 Bawdier s Family Shakspeare 36 
 
 Bramlcy-Moore's Six Sisters of the Valley . 36 
 Brande's Dictionary of Science, Literature, 
 
 and Art 23 
 
 Bray's Manual of Anthropology 23 
 
 Philosophy of Necessity 12 
 
 Brinkley's Astronomy 18 
 
 Browne' s Exposition of the 39 Articles 29 
 
 Brunei s Life of Brunei 7 
 
 Buckle's History of Civilisation 3 
 
 ' Posthumous Remains .. ,12 
 
 Bucktons Lessons on Man and Animals... 25 
 
 Bull's Hints to Mothers 40 
 
 Maternal Management of Children . 40 
 
 Burgomaster's Family (The) 36 
 
 Burke' s Rise of Great Families 8 
 
 Vicissitudes of Families 8 
 
 Busk's Folk-lore of Rome 35 
 
 Valleys of Tirol 33 
 
 Cabinet Lawyer 40 
 
 Campbell's Norway 35 
 
 Gates' s Biographical Dictionary 8 
 
 and Woodward's Encyclopaedia ... 5 
 
 Changed Aspects of Unchanged Truths ... 14 
 
 Chesney s Indian Polity , 3 
 
 Modern Military Biography 3 
 
 Waterloo Campaign 3 
 
 Chubb on Protection 40 
 
 dough's Lives from Plutarch 4 
 
 Colenso on Moabite Stone &c 32 
 
 's Pentateuch and Book of Joshua. 33 
 
 Speaker's Bible Commentary ... 33 
 
 Collins' s Mineralogy of Cornwall 28 
 
 Perspective 27 
 
 Commonplace Philosopher in Town and 
 
 Country, by A. K. H. B 14 
 
 Comte's Positive Polity 8 
 
 Congreve's Essays 9 
 
 Politics of Aristotle 10 
 
 Conington's Translation of Virgil's yEneid 37 
 
 . Miscellaneous Writings 13 
 
 Contanseaiis Two French Dictionaries ... 15 
 Conybcare and Howson's Life and Epistles 
 
 of St. Paul 31 
 
 Cottons Memoir and Correspondence 7 
 
 Counsel and Comfort from a City Pulpit... 14 
 
 Cox's (G. W.) Aryan Mythology 4 
 
 Crusades 6 
 
 History of Greece 4 
 
 Tale of the Great Persian 
 
 War 4 
 
 Tales of Ancient Greece ... 36 
 
 and Jones's Teutonic Tales 36 
 
 Crawley's Thucydides 4 
 
 Creasy on British Constitution 3 
 
 Cresy's Encyclopaedia of Civil Engineering 27 
 
 Critical Essays of a Country Parson 14 
 
 Crookes's Chemical Analysis 24 
 
 Dyeing and Calico-printing 29 
 
 Cullcy's Handbook of Telegraphy 27 
 
 Cusack's Student's History of Ireland 3 
 
 D'Aubiene's Reformation in the Time of 
 
 Calvin <> 
 
42 
 
 NEW WORKS PUBLISHED BY LONGMANS & CO. 
 
 Davidsons Introduction to New Testament 32 
 
 Dead Shot (The), by Marksman 38 
 
 De Caisne and Le Maout's Botany 24 
 
 De Morgans Paradoxes 13 
 
 De Tocqucvillcs Democracy in America... 9 
 
 Disraeli's Lord George Bentinck 7 
 
 Novels and Tales 36 
 
 Dobson on the Ox 38 
 
 Doves Law of Storms 18 
 
 Doyle's Fairyland 25 
 
 Drews Reasons of Faith.... 30 
 
 Eastlake's Gothic Revival 26 
 
 Hints on Household Taste 27 
 
 Edivardss Rambles among the Dolomites 33 
 
 Elements of Botany 23 
 
 Ellicott's Commentary on Ephesians 30 
 
 Galatians 30 
 
 Pastoral Epist. 30 
 
 Philippians.&c. 30 
 
 Thessalonians . qo 
 
 Lectures on Life of Christ 29 
 
 Epochs of History 6 
 
 Evans's Ancient Stone Implements 23 
 
 Ewalds History of Israel 31 
 
 Fairbairn's Application of Cast and 
 
 Wrought Iron to Building... 29 
 
 Information for Engineers 28 
 
 Treatise on Mills and Millwork 28 
 
 Farrars Chapters on Language 13 
 
 Families of Speech 13 
 
 Fitz-wygram on Horses and Stables 38 
 
 Forsyth's Essays 9 
 
 Fowler's Collieries and Colliers 39 
 
 Francis's Fishing Book 38 
 
 Freeman's Historical Geography of Europe 5 
 
 Freshficld's Italian Alps 33 
 
 From January to December 13 
 
 Froudes English in Ireland 2 
 
 History of England 2 
 
 Short Studies 12 
 
 Gairdners Houses of Lancaster and York 6 
 
 Gamgce on Horse-Shoeing 38 
 
 Ganot's Elementary Physics 19 
 
 Natural Philosophy 19 
 
 Gardiner's Buckingham and Charles 3 
 
 Thirty Years' War 6 
 
 Gilbert and Churchill's Dolomites 34 
 
 Girdlestone's Bible Synonyms 30 
 
 Goodeve' s Mechanics 20 
 
 Mechanism 20 
 
 Grant's Ethics of Aristotle 10 
 
 Graver Thoughts of a Country Parson 14 
 
 Greville's Journal I 
 
 Griffin s Algebra and Trigonometry 20 
 
 Griffith's Sermons for the Times 29 
 
 Grove on Correlation of Physical Forces ... 19 
 
 Gwilt's Encyclopaedia of Architecture 27 
 
 HalliwelFs Illustrations of Shakespeare ... 8 
 
 Hare on Election of Representatives 13 
 
 Harrison s Order and Progress 9 
 
 Hartley on the Air 19 
 
 Hartwig's Aerial World 22 
 
 Polar World 22 
 
 Sea and its Living Wonders ... 22 
 
 Subterranean World 22 
 
 Tropical World 22 
 
 Haughton's Animal Mechanics 20 
 
 Hayward's Biographical and Critical Essays 7 
 
 Heath on Energy 21 
 
 Heer's Switzerland 22 
 
 Helmhotz on Tone 23 
 
 's Scientific Lectures 19 
 
 Helmsley's Trees, Shrubs, and Herbaceous 
 
 Plants 24 
 
 Hcrschel's Outlines of Astronomy 18 
 
 Holland' s Fragmentary Papers 21 
 
 Recollections 7 
 
 Howitt's Rural Life of England 34 
 
 Visits to Remarkable Places 35 
 
 Humboldfs Life 7 
 
 s3iys 12 
 
 Treatise on Human Nature 12 
 
 Ihnes History of Rome 5 
 
 Ingelow's Poems 37 
 
 Jameson s Legends of Saints and Martyrs . 26 
 
 Legends of the Madonna 26 
 
 Legends of the Monastic Orders 26 
 
 Legends of the Saviour 26 
 
 Jclf on Confession 31 
 
 'Jenkins Electricity and Magnetism 20 
 
 Jerram's Ly cidas of Milton 36 
 
 Jerrold's Life of Napoleon i 
 
 Johnston s Geographical Dictionary 17 
 
 KaliscKs Commentary on the Bible 31 
 
 Keith's Evidence of Prophecy 31 
 
 Kenyan's (Lord) Life 7 
 
 Kerls Metallurgy, by Crookes and Rb'hrig. 28 
 
 Kingdon on Communion 31 
 
 Kirby and Spence's Entomology 21 
 
 Knatchbull-Hugessen's Whispers from 
 
 Fairy-Land 35 
 
 Landscapes, Churches, &c. by A. K. H. B. 14 
 
 Lang's Ballads and Lyrics 36 
 
 Lathams English Dictionary 15 
 
 Laughton's Nautical Surveying 18 
 
 Lawlor's Centulle 35 
 
 Lawrence on Rocks 22 
 
 Lecky's History of European Morals 5 
 
 Rationalism 5 
 
 Leaders of Public Opinion 7 
 
 Leisure Hours in Town, by A. K. H. B.... 14 
 
 Lessons of Middle Age, by A. K. H. B.... 14 
 
 Lewes' s Biographical History of Philosophy 5 
 
 Liddell and Scott's Greek-English Lexicons 16 
 
 Lindhy and Moore's Treasury of Botany... 23 
 
NEW WORKS PUBLISHED BY LONGMANS & CO. 
 
 43 
 
 Lloyd's Magnetism 21 
 
 Wave-Theory of Light 21 
 
 Longmans Chess Openings 
 
 . Edward the Third 
 
 Lectures on History of England 
 Old and New St. Paul's 
 
 London's Encyclopaedia of Agriculture ... 
 
 Gardening 
 
 Plants 
 
 40 
 
 2 
 2 
 
 27 
 
 2 9 
 2 9 
 22 
 
 Lowndess Engineer's Handbook 28 
 
 Lubbock's Origin of Civilisation 22 
 
 Lyra Germanica 26, 32 
 
 Macaulays (Lord) Essays 2 
 
 History of England - . 2 
 
 Lays of Ancient Rome 26 37 
 
 Miscellaneous Writings 12 
 
 Speeches 12 
 
 Works 2 
 
 McCnllocKs Dictionary of Commerce 16 
 
 Maclcod's Principles of Economical Philo- 
 sophy 10 
 
 Theory and Practice of Banking 39 
 
 Malleson s Genoese Studies 3 
 
 Markhams History of Persia 4 
 
 Marshall's Physiology 25 
 
 Todas 22 
 
 Marshmans History of India 3 
 
 Life of Havelock 8 
 
 Martincau's Christian Life 32 
 
 Hymns 32 
 
 Maunders Biographical Treasury 39 
 
 Geographical Treasury 39 
 
 Historical Treasury 39" 
 
 Scientific and Literary Treasury 39 
 
 Treasury of Knowledge 39 
 
 Treasury of Natural History ... 39 
 
 Maxu<dFs Theory of Heat 20 
 
 May's History of Democracy 2 
 
 History of England 2 
 
 Melville's Digby Grand 36 
 
 General Bounce 36 
 
 Gladiators 36 
 
 Good for Nothing 36 
 
 Holmby House 36 
 
 Interpreter 36 
 
 Kate Coventry 36 
 
 Queen's Maries 36 
 
 Mendelssohn's Letters 
 
 Menzies Forest Trees and Woodland 
 
 Scenery 24 
 
 Merivale's Fall of the Roman Republic ... 4 
 
 Romans under the Empire 4 
 
 Mcrrifields Arithmetic and Mensuration... 20 
 
 Magnetism 18 
 
 Miles on Horse's Foot and Horse Shoeing 38 
 
 on Horse's Teeth and Stables 38 
 
 Mill (J.) on the Mind 10 
 
 (J. S.) on Liberty 9 
 
 Subjection of Women 9 
 
 on Representative Government 9 
 
 Utilitarianism 9 
 
 's Autobiography 6 
 
 Dissertations and Discussions 10 
 
 Essays on Religion &c 30 
 
 Hamilton's Philosophy 10 
 
 - System of Logic 9 
 
 Mill's Political Economy 9 
 
 Unsettled Questions 9 
 
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