LIBRARY OF THE LMVKkSITV OF CALIFORNIA. GIPT 01- MlSS ROSE WHITING. Deceived September. 1896. Accession No. &$ ]r $ 2. Class Nu AA XX> AA i - A AM A >*iv A r^W***; 2S228?2Si8^^ -^-^*r^*^*.iu..Aflc ""v^:-^^A, ^AAAAA wrw^^wws f ^^TV^ ' 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 / 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 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 - + 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. 39 PATERNOSTER Row, E.G. LONDON, April 1875. GENERAL LIST OF WORKS PUBLISHED BY MESSRS. LONGMANS, GREEN, AND Co. 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By Thomas Bull, M.D. 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