ELEMENTARY TREATISE 
 
 ON THE 
 
 WAVE-THEORY OF LIGHT. 
 
 
 BY 
 
 HUMPHREY LLOYD, D.D., D.C.L.; F. R. SS. L. & E. ; V.P.R,I.A.;7?ZQ; 
 << 
 
 FELLOW OF TRINITY COLLEGE, DUBLIN. f~f.W 
 
 
 
 LONDON: 
 
 LONGMAN, BKOWN, GREEN, LONGMANS, AND ROBERTS. 
 
 1857. 
 

 DUBLIN: 
 IprintctJ at tije SInibcrsilp ^Duss, 
 
 BY M. H. GIT.L. 
 
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 comply ; and 
 he trusts that the delay in acceding to it may be ex- 
 cused 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 for- 
 mer. In preparing these additions, the author has 
 
 b 
 
VI ADVERTISEMENT. 
 
 derived much aid from the Repertoire dOptique Mo- 
 derne of the Abbe Moigno, a work which contains 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 with the ele- 
 ments 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 Sir John Herschel's Essay on 
 Light, and in Mr. Airy's Tract on the Undulatory 
 Theory of Optics. 
 
 TRINITY COLLEGE, DUBLIN, 
 March ISth, 1857. 
 
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 transpa- 
 rent 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 illumination ; photometry. (9, 10) 
 Velocity of light determined by eclipses of Jupiter's satellites by aber- 
 ration of fixed stars. (11) M. Fizeau's experiment. I (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 explained. (17) 
 Lights differ from one another in intensity and colour ; law of intensity. 
 te(18) Colour of light varies with length of wave ; limits of sensibility of 
 eye. (19) Aberration of light physically explained, ......... 1 
 
 CHAPTER II. 
 
 RKKLEXION AND REFRACTION. 
 
 (20, 21) Reflexion of light twofold ; law of reflexion. (22) Intensity of re- 
 flected light. (23, 24) Law of refraction; refractive index. (25-27) 
 ** Deviation of ray refracted by prism. (28) Refractive index determined 
 experimentally. ^(29) Newton's explanation of laws of reflexion and re- 
 fraction. (30, 31) Connexion of the phenomena ; hypothesis of the fits: 
 its insufficiency. ^(32, 33) Laws of reflexion and refraction deduced from 
 principles of wave-theory. *(34) Principle of least time. (35) Intensity 
 of reflected and refracted light. (36) Experimentum crucis between two 
 theories. (37) M. Foucault's experiment ; velocity of light in air, and in 
 water, determined experimentally. (38) No effect of Earth's motion on 
 amount of refraction Arago's experiment, 17 
 
v CONTENTS. 
 
 CHAPTER III. 
 
 DISPERSION. 
 
 PAGE. 
 
 (39, 40) Composition of solar light; seven principal colours. (41) Colour and 
 refrangibility permanent and original affections. (42) Mixture in the 
 spectrum ; modes of reducing it. Pw(43) Newton's method of determining 
 the refractive indices of several species of simple light. >-(44) Intensity of 
 light in the spectrum. (45-47) Fixed lines of spectrum ; observations of 
 Fraunhofer. (48, 49) Dispersion produced by a thin prism ; measure of 
 dispersive power^SO) Dispersive powers different in different substances ; 
 experiments of Newton and Dollond. (51, 52) Condition of achromatism ; 
 dispersive powers compared experimentally. ^53, 54) Physical explana- 
 tion of dispersion ; difficulty in wave-theory remov r ed by M. Cauchy. 
 ^(55, 56) Relation between velocity of propagation and wave-length, 
 (57, 58) Refractive index deduced in terms of wave-length in vacua. 
 (59) Limiting index, 31 
 
 CHAPTER IV. 
 
 DOUBLE REFRACTION. 
 
 v(60) Property of double refraction discovered by Bartholinus. (61, 62) Bodies 
 possessing it ; Iceland spar. (63) Ordinary and extraordinary rays. 
 (64, 65) Optic axis ; phenomena of double refraction symmetrical around it. 
 (66) Construction of Huygens for directions of refracted rays. (67) At- 
 tractive and repulsive crystals. (68) Biaxal crystals. (69) Connexion 
 between optical characters and crystalline forms, discovered by Sir D. 
 Brewster. (70) Double refraction in biaxal crystals Fresnel's theory. 
 (71) Effects of double refraction ; two images. (72, 73) Double- refracting 
 prism. Double image micrometer, 45 
 
 CHAPTER V. 
 
 INTERFERENCE OF LIGHT. 
 
 (74) Effect of coexisting vibrations. Superposition of small motions. (75) 
 * Union of two waves ; interference of light: (76) Analogous phenomena 
 in water, and in air. >(77) Interference of aerial pulses exhibited to the 
 eye. (78, 79) Grimaldi's experiment of interference ; Young's experiment. 
 X(80) Phenomena of interference of aqueous waves. (81) Explanation of 
 Young's experiment ; positions of the bands. (82, 83) Fresnel's expe- 
 ls riment of interference; positions of the fringes. (84) Interference of 
 direct and reflected lights. (85) Interference produced by obtuse prism. 
 (86) Arago's experimentum crucis displacement of the fringes produced 
 by interposed plate. (87) Rectilinear propagation of light explained on 
 * the principles of the wave- theory, 56 
 
CONTENTS. ix 
 
 CHAPTER VI. 
 
 1> I F F K A C T I O X . 
 
 Page. 
 
 (88) Diffracted fringes observed by Grimaldi. (89-91) Diffraction by a single 
 edge ; form of the fringes. (92) Diffraction byjine wire. (93, 94) Diffrac- 
 tion by narrow rectilinear aperture, &c.x(95) Newton's explanation of phe- 
 nomena. >(96) Newtonian theory refuted; effects independent of the na- 
 ture and form of the body. (97) Hypothesis of Mairan and Du Tour. 
 (98) Young's theory of diffraction incomplete. (99) Fresnel's theory. 
 (100, 101) Fringes produced by a single edge explained, and their places 
 computed. (102) Fringes within shadow of narrow opaque body explained. 
 (103) Fringes formed by narrow rectilinear aperture. (104) Diffraction 
 by small circular aperture. (105) Diffraction by opaque circular disc. 
 
 >(106) Diffracted rings produced by fibres irregularly disposed ; Young's 
 eriometer. (107, 108) Diffraction by fine gratings. (109) Fixed lines 
 in diffracted spectrum ; their intervals always proportional. (110) Wave- 
 
 ^Jengths corresponding to seven principal fixed lines. (Ill) Law of inten- 
 sity of light in diffracted spectrum. (112) Ruled and striated surfaces. 
 (113) Colours of mother of pearl. (114) Diffraction produced by diaphragms 
 in telescopes, 72 
 
 CHAPTER VII. 
 
 COLOURS OF THIN PLATES. 
 
 (115, 116) Colours of thin plates exhibited under various circumstances. 
 >(117) Newton's mode of observing them ; Newton's scale. (118) Laws of 
 the phenomena. (119, 120) Thicknesses of the plate corresponding to the 
 successive rings. v(121) Other laws of the rings experimentally proved. 
 (122) Transmitted rings. (123) Iroscope. v(l24) Newton's explanation 
 of the colours of thin plates ; Jits of easy reflexion and transmission. (125) 
 Assumption of this theory disproved. (126, 127) Speculations of Hooke 
 and Young the colours of thin plates produced by interference. (128) 
 Their laws deduced. (129, 130) Point of contact a point of discordance of 
 two waves; loss or gain of half an undulation. ^131) Transmitted rings 
 explained. (132) Imperfection of Young's theory removed by Poisson ; 
 experimentum crucis of Fresnel. (133) Phenomena of interference pro- 
 duced by two plates slightly differing in thickness, or slightly inclined. 
 (134, 135) Colours of thick plates Newton's experiment; explanation 
 (136) M. Babinet's variation of Newton's experiment. X 157 ) Colours of 
 mixed plates. (138) Review of the two theories in their application to the 
 foregoing phenomena, 100 
 
X CONTENTS. 
 
 CHAPTER VIII. 
 
 . POLARIZATION OF LIGHT. 
 
 PAG K. 
 
 (139, 140) Polarization of Light discovered by Huygens. (141) Polarization 
 by reflexion discovered by Malus. (142, 143) Characters of polarized 
 ray. (144) Angles of polarization; Brewster's law. (145) Polariza- 
 tion by plate bounded by parallel surfaces ; by pile of plates. (146) Law 
 of Malus. Common light equivalent to two polarized rays of equal in 
 tensity, whose planes of polarization are perpendicular. (147) Light par- 
 tially polarized by reflexion. (148) Light partially polarized by refrac- 
 tion ; law of Arago. (149) Light completely polarized by transmission 
 through pile of plates. (150) Polarization by transmission through plate 
 of tourmaline. (151) Phenomenon observed by M. Haidinger polar- 
 ized light distinguishable by naked eye, 122 
 
 CHAPTER IX. 
 
 TRANSVERSAL VIBRATIONS THEORY OF REFLEXION AND REFRACTION OF 
 POLARIZED LIGHT. 
 
 (152) Newton's objection to wave-theory, derived from phenomenon of polariza- 
 tion, removed. (153) Propagation of wave by transversal vibrations illus- 
 trated. (154) Evanescence of normal vibrations experimentally proved. 
 (155) Vibrations in polarized ray two hypotheses ; experimentum crucis 
 of Professor Stokes. (156) Hypothetical principles of MacCullagh and 
 Neumann; amplitudes of reflected arid refracted vibrations deduced. (157) 
 Intensity of light reflected at extreme incidences. (158) Partial polari- 
 zation by reflexion and refraction ; law of Arago. (159) Complete polariza- 
 tion of reflected light; law of Brewster. (160, 161) Change of plane of 
 polarization by reflexion and refraction, 135 
 
 CHAPTER X. 
 
 ELLIPTIC POLARIZATION. 
 
 (162) Vibration of ethereal molecules generally elliptic. (163) Elliptic vibra- 
 tion the resultant of two rectilinear vibrations. (164) Direction of axes, 
 and ratio of their lengths. (165) Elliptic polarization by total reflexion. 
 (166) Circular polarization ; Fresnel's rhomb. (167, 168) Laws of metal- 
 lic reflexion discoveries of Malus and Brewster. (169) Reflected light 
 elliptically polarized. (170) Intensity of reflected light determined expe- 
 rimentally by M. Jamin. (171, 172) Difference of phase of two compo- 
 nents. Effect of successive reflexions. (173) Action of transparent bodies 
 on light; M. Jamin's results. (174) Difference of phase of two components 
 in light reflected from transparent bodies ; angle of maximum polarization. 
 
CONTENTS. XI 
 
 PAiiE. 
 
 (175) Substances of positive and negative reflexion. (176) Elliptic vibra- 
 tion of reflected light determined experimentally. (177) Professor Haugh- 
 ton's researches; circular polarization of reflected light, 148 
 
 CHAPTER XI. 
 FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 (178, 179) Form of the wave-surface in uniaxal crystals not general. 
 (180-182) Fresnel's theory of double refraction ; surface of elasticity. 
 (183) Planes of polarization of two rays; lawofMalus. (184) Fresnel's 
 construction for wave-surface ; its equation. (185) Directions, and planes 
 of polarization, of two rays determined. (186) Optic axes; velocities of 
 two rays. (187) Velocity of both rays in biaxal crystals variable ; 
 Fresnel's experiments. (188) Sir William Hamilton's discovery of co- 
 nical refraction. (189, 190) External conical refraction established ex- 
 perimentally ; law connecting planes of polarization ; effect of linear aper- 
 ture. (191) Internal conical refraction. (192) Eeflexion and refraction 
 by crystalline media, 1 63 
 
 CHAPTER XII. 
 
 INTERFERENCE OF POLARIZED LIGHT. 
 
 (193-195) Depolarization and colour produced by crystalline plates. (196) Laws 
 of the tints discovered by M. Biot. (197) Phenomena reduced to interfer- 
 ence difficulty. (198, 199) Laws of interference of polarized light. 
 
 (200) Explanation of colours of crystalline plates; interval of retardation. 
 
 (201) Emergent light in general elliptically polarized. (202) Colours 
 produced by thick plates. (203) Rings surrounding the optic axes in 
 uniaxal crystals. (204) Rings in biaxal crystals lemniscates ; dark 
 brushes. (205, 206) Double-refracting structure developed by compression 
 or dilatation ; Fresnel's explanation of phenomenon. (207) Double refract- 
 ing structure produced by heat. (208) Effects of pressure and heat on 
 double-refracting crystals, 180 
 
 CHAPTER XIII. 
 
 KOTATORY POLARIZATION. 
 
 (209) Rotation of plane of polarization in rock-crystal. (210) Laws of rota- 
 tory polarization established by M. Biot. (211, 212) Right-handed and 
 left-handed crystals discoveries of Sir J. Herschel, Sir D. Brewster, and 
 M. Pasteur. (213) Laws of rotatory polarization in rock-crystal theoreti- 
 cally explained by Fresnel. (214) Relation between rotation and double 
 
Xll CONTENTS. 
 
 PAGE. 
 
 refraction in direction of axis. (215) Mr. Airy's discovery of elliptic pola- 
 rization in directions inclined to axis. (216) Prof. Mac Cullagh's theory 
 of refraction in quartz. (217) Rotatory polarization of liquids discove- 
 ries of MM. Biot and Seebeck. (218) Rotation of mixtures ; optical ana- 
 lysis of compounds, 199 
 
 CORRECTION. 
 
 The important experiment, the principle of which is described in Art. (37), was 
 first performed by M. Foucault. M. Fizeau was occupied with the problem at the 
 same time, although independently ; and the researches of the two experimenters 
 were communicated to the French Academy on the same day. 
 
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 power 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 luminous ; 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 9 
 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 laminae of exceeding tenuity. 
 
 (5) In the same homogeneous medium, light is propagated 
 mriffhtlineS) 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 ' 
 of diffusion, or the area of the spherical surface, is as the square 
 of its radius. Hence the intensity of the light is inversely as 
 the square of the radius, i. e. inversely as the square of the 
 distance. 
 
 Let the light be supposed to emanate from the points of 
 an uniformly luminous surface, which we shall suppose to be 
 a small portion of a sphere. Then the quantity of light 
 emitted is proportional to the quantity emitted by a single 
 point, and the number of points (or area) conjointly. Hence 
 if a denote the area of the luminous surface, and i the quan- 
 tity emitted from a single point, which is a measure of the 
 
 B 2 
 
4 PROPAGATION OF LIGHT. 
 
 absolute brightness, the intensity of the illumination, at any 
 distance d, is 
 
 ai 
 
 ~d? 
 
 (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 inclination. 
 The intensity of the illumination is, therefore, diminished in 
 the same proportion, and is expressed by the formula 
 
 ai sin 9 
 
 ~&~> 
 
 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 
 intensity. 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 -. ^ , . -, j 
 
 - and -jr z , respectively ; and these being rendered 
 
 equal in the experiment, we have 
 
 ai d z 
 
 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 sha- 
 dows 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 c 
 the Danish astronomer, Olaus Roemer. 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 conjunc- 
 tion, and therefore farthest, they happened later. He thence 
 inferred that light was propagated with a finite velocity, and 
 that the difference between the computed and observed times 
 was due to the change of distance. This difference is found 
 
PROPAGATION OF LIGHT. 
 
 i ; V-- 
 
 to be 8 13 s ; and accordingly the velocity of light is such, 
 that it traverses 192,500 miles in a second of time. 
 
 \l(/3 T > 
 
 (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"-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 ^ Jo tn P art f tne 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 
 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 
 
PROPAGATION OF LIGHT. 7 
 
 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 
 
 traversed the distance between the wheel and the mirror. If, 
 therefore, that distance be denoted by a, the velocity of light 
 will be 
 
 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, 
 
 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. 
 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 
 
8 PROPAGATION OF LIGHT. 
 
 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 1 2*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 experi- 
 ments 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 individual 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 
 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, 
 
PROPAGATION OF LIGHT. 9 
 
 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 par- 
 ticles 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 com- 
 municated 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 narrow 
 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 nf ^l^fdnn and the wave-theory. 
 /f'ln the ^r^fertEeluminous body is suppose^kTsend forth, 
 or emit, continually, material particles^pxtreme minuteness, 
 in all directions. In the.Mle^tEe 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. 
 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 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 
 
10 PROPAGATION OF LIGHT. 
 
 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 to- 
 gether, by concentrating them in the foci of lenses or mirrors, 
 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 
 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 i 
 concentrated by means of a large lens, and suffered to fall upon 
 the paper, but without any perceptible effect. 
 
PROPAGATION OF LIGHT. 11 
 
 (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 
 one and 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 be 
 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 variousj^- 
 and independent bodies of the universe, and that it acts 
 equally on all the particles of light, so as to generate in them 
 4he 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 
 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 
 
12 PROPAGATION OF LIGHT. 
 
 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 
 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 
 
PROPAGATION OF LIGHT. 13 
 
 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 
 minutely, 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 state of their vibra- 
 tion. Thus, while each particle moves only to and fro verti- 
 cally, an undulation or wave is propagated horizontally along 
 the string ; and there will be a succession of similar undula- 
 tions as long as the original disturbance continues. The 
 particles a, ', a", or the par- 
 ticles b, b' y 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 ', or between b and b' ; and the length of a wave is the 
 distance between them, estimated in the direction in which 
 the motion is propagated. It is evident from this description 
 that a wave, or undulation, comprises particles in every phase 
 of their vibration. 
 
 Now, instead of a single string, let us suppose an infinite 
 
14 PROPAGATION OF LIGHT. 
 
 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 waves 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 the place of the actual strings 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, 
 if X denote the length of the wave, T the time of vibration, 
 and v the velocity of wave-propagation, 
 
 A = v r . 
 
 (17) We have hitherto considered the propagation of 
 vibratory movement without reference to any diversity in 
 its nature. It is obvious, however, that vibrations may diifer 
 from one another in two particulars, namely, in the space of 
 vibration, and in the time. In the aerial pulses the amplitude 
 of the vibration determines the loudness of the sound; and 
 the frequency of the pulses, or the time of vibration, deter- 
 mines 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. 
 
PROPAGATION OF LIGHT. 15 
 
 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 though 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 represented. 
 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 
 liyJit is the sum of the lights produced by two luminous ori- 
 gins 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 lumi- 
 nous origin must decrease inversely as the square of the dis- 
 tance ; for, from the laws of wave propagation 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 in- 
 tensity of light is deduced from the principles of undulatory 
 propagation. 
 
 (18) The colour of the light (it has been said) depends 
 on the number of impulses which the nerves of the eye receive, 
 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 
 
16 PROPAGATION OF LIGHT. 
 
 corresponding to the ray of mean refrangibility is nearly 200 
 ten-millionths, or jouuo^ n ^ 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. 
 
 (19) The aberration of light, it has been said, 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 coexisting 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 
 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 hypo- 
 thesis, and has shown that it suffices to explain other pheno- 
 mena also, in which the Earth's motion is concerned. Profes- 
 sor 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 the Earth. 
 
CHAPTER II. 
 
 REFLEXION AND REFRACTION. 
 
 (20) WHEN light meets the surface of a new medium, a 
 portion of it is always turned back, or reflected. 
 
 The reflexion 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 therefore 
 reflect the light in various directions. 
 
 (21) 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. 
 
 (22) The intensity of the reflected light, on the other hand, 
 
18 REFLEXION AND REFRACTION. 
 
 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 1 000. 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 89i, the number 
 of rays reflected at this angle being, in both cases, 721 out of 
 1000. 
 
 (23) 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- 
 cular to the bounding surface. On the contrary, when the 
 ray passes from a denser into a rarer medium, the angle of 
 
REFLEXION AND REFRACTION. 19 
 
 incidence is less than the angle of refraction, and the devia- 
 tion is from the perpendicular. 
 
 (24) 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 Yitello in the thirteenth ; but both of these ob- 
 servers failed in discovering the connecting law. The law of 
 refraction, just stated, was discovered by Willebrord Snell, 
 about the year 1621. 
 
 If and T// 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, 
 
 sn = j. sn 
 
 H being a constant quantity. This constant is termed the 
 index of refraction ; and since > i//, 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, ^ = 1*336; for 
 crown glass, p = 1*535 ; for diamond, p = 2'487 ; and, for 
 chromate of lead, p = 3. 
 
 (25) 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 and 0' denote the angles which the inci- 
 dent and emergent rays make with the perpendiculars to the 
 faces at the points of incidence and emergence, \fj and i// the 
 
 c 2 
 
20 REFLEXION AND REFRACTION. 
 
 angles which the portion of the ray within the prism forms 
 with the same, then the deviations at incidence and emergence 
 are, respectively, - ;//, and 0' - i// ; and the total deviation 
 S = <p + 0' - (^ + !//) 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 
 vertical angle of the prism ; or, denoting this angle by a, 
 
 that 
 
 i// + i// = a ; 
 wherefore 
 
 S = (j) + 0' a. 
 
 (26) 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 
 
 <p = fjL\fj 9 $' = fjL$ ; and + ^' = /u (^ + ') = pa. 
 
 Hence 
 
 S = 0u-l)a. 
 
 (27) 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, 
 
 -f $ = a + S, ^-fi//=a. 
 
 But since, in this case, = 0', there is also i// = i// ; and con- 
 sequently 
 
 = i (a + S), ^ = ia. 
 
 Hence we have 
 
 sin ^ (a + S) = /LI sin a ; 
 
 from which a + 8, and therefore , 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- 
 
REFLEXION AND REFRACTION. 21 
 
 pearing for a moment to be stationary between the opposite 
 changes. By this principle it is easy to place a prism, experi- 
 mentally, in the position in which the refractions are equal 
 at botli sides. 
 
 (28) 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 ho- 
 rizontal circle being read, and the prism removed, the tele- 
 scope 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 + S) 
 ^ sin ^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. 
 
 (29) 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 
 
22 REFLEXION AND REFRACTION. 
 
 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 repulsive forces, 
 here assumed, is altogether similar to that 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 refraction, it is proved 
 that the effect of the attractive force is to increase the square 
 of the perpendicular velocity of the molecule, 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 re- 
 garded as forming an era in the history of science. 
 
 (30) 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 difference 
 of the refractive indices of the two media, and the reflexion 
 ceasing altogether when this difference vanishes. How is it, 
 then, that some of the molecules obey the influence of the re- 
 pulsive force, and are reflected, while others yield to the 
 
REFLEXION AND REFRACTION. 23 
 
 attractive force, and are refracted ? To account for this, New- 
 ton was obliged to have recourse to a new hypothesis. The 
 molecules of light, in their progress through space, are sup- 
 posed to pass continually into two alternate states, or fits, 
 which recur periodically and at equal intervals. While in one 
 of these states, called the Jit 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 re- 
 fracted, and the proportion of the former to the latter will in- 
 crease 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, 
 
 (31) The hypothesis of the fits has lost much of its credit, 
 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- 
 
24 REFLEXION AND REFRACTION. 
 
 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 diffe- 
 rent degrees, depending on their phase. Hence, as the direc- 
 tion 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* 
 
 (32) 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 compared 
 with its density. In the same homogeneous medium the ve- 
 locity 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 
 in the new medium, and with the velocity which belongs to 
 it ; so that there will be an infinite number of partial waves 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 
 
REFLEXION AND REFRACTION. 
 
 25 
 
 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 leaves, 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 reflect- 
 ing surface at m. Each portion 
 of this wave, as it reaches the sur- 
 face, becomes the centre of a di- 
 verging spherical 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'ri be drawn parallel to mn, the wave diverging 
 from m will in the same time have reached the spherical sur- 
 face whose radius, mo', is equal to rik. The surface which 
 touches all these spheres at any instant is that of the re- 
 flected wave. But, as mo and mo are proportioned 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 inclined to the reflecting sur- 
 face. 
 
 (33) 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 mo- 
 ment. In an interval of time 
 proportional to nk, the por- 
 tion n of this wave will have 
 reached the surface at k, and the portions m and m will have 
 
26 REFLEXION AND REFRACTION. 
 
 become the centres of diverging spherical waves in the 
 second medium, the radii of these spheres, mo and m'o' 9 
 being to the intercepts, nk and ri/c, 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 nmh : sin mko : : nk : 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. 
 
 (34) 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 secon- 
 dary or partial waves, in this demonstration, has been deno- 
 minated the principle of Huygens ; and it is obviously a case 
 of the more general principle of the co-existence of small mo- 
 tions. It easily follows from this mode of composition, that 
 the surface 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 ap- 
 pear from what 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- 
 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 / therefore denote the length of the path 
 described by the incident light, between any assumed point 
 and the point of incidence, /' 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, 
 
REFLEXION AND REFRACTION. 27 
 
 / /' 
 
 - + -, is a minimum ; or, multiplying by v, and denoting the 
 v v 
 
 V . 
 
 ratio - by ju, 
 
 / + fjj! = minimum. 
 
 The constant factor, ju, is the refractive index of the medium. 
 In the case of reflexion, ju = 1, and /+ /' 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. 
 
 (35) The intensity of the light, in the reflected and re- 
 fracted waves, will depend on the relative densities of the ether 
 in Jthe two media. For we may compare the contiguous strata 
 of ether in these media to two elastic bodies of different 
 masses, one of which moves the other by impact ; and it is 
 easy to deduce, on this principle, the intensities of the re- 
 flected and refracted lights in the case of perpendicular in- 
 cidence. 
 
 (36) On reviewing what has been said, we cannot but be 
 struck by the remarkable fact, that theories so widely op- 
 posed 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 refrac- 
 tion is equal to the ratio of the velocities of light in the two 
 media, and is therefore constant. But there is this important 
 difference between them : in the wave-theory, the sines of 
 these angles are in the direct ratio of the velocities, while, 
 according to the theory of emission, they are in the inverse. 
 In other words, the velocity of light in the denser medium 
 is less according to the former theory ; while, according to the 
 latter, \i\^> 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 
 
28 REFLEXION AND REFRACTION. 
 
 them. The important experiment by which this was first ac- 
 complished was made by Arago ; and the result, as will be 
 shown hereafter, is conclusive in favour of the wave-theory. 
 
 (37) The conclusion deduced from the experiment here re- 
 ferred to presupposes the laws of Interference of Light laws 
 which, in themselves, are intimately connected with the prin- 
 ciples of the wave-theory. It was desirable, therefore, to de- 
 duce the same conclusion, if possible, by direct means. The 
 experiment by which this is effected has been recently made 
 by M. Fizeau, 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 re- 
 volving 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 to- 
 wards the aperture. 
 
 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 be- 
 tween 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. 
 
REFLEXION AND REFRACTION. 29 
 
 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 = wt 9 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, = 2w. 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 
 
 r= 
 
 M. Fizeau has been enabled to observe an appreciable devia- 
 tion 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 has been made to revolve 1000 
 times in a second, it was obvious that the time taken by light 
 to traverse even this short distance was capable of being mea- 
 sured with precision. It only remained to interpose a column 
 of water between the mirrors, to observe the deviation, and 
 to calculate the velocity. By these means M. Fizeau has 
 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. 
 
 (38) The refractive index being equal to the ratio of the 
 velocities of light in the two media (direct or inverse) it follows, 
 whichsoever theory we adopt, 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 altered 
 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 
 
30 REFLEXION AND REFRACTION. 
 
 the direction of the star. It was, therefore, a matter of much 
 interest to determine how, and in what degree, 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 re- 
 peating circle, so as to cover only a portion of the lens. The 
 star being then observed, directly through the uncovered part 
 of the lens, and afterwards in the direction in which its light 
 was deviated by the prism, the difference of the angles read 
 off gave the deviation. The stars selected for observation 
 were those in the ecliptic, which passed the meridian nearly 
 at 6A.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. 
 
 This remarkable and unexpected result can be reconciled 
 to the theory of emission, as Arago has observed, only by the 
 hypothesis already adverted to,* namely, that the molecules 
 are emitted from the luminous body with various 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. It is explained, in accordance with the prin- 
 ciples of the wave-theory, on the same hypotheses which have 
 been already made to explain the aberration of light ;f and it 
 is shown, on these suppositions, that both the laws, and the 
 amount of refraction, are independent of the Earth's motion. 
 
 * Art, (14). f Art. (19). 
 
31 ) 
 
 CHAPTER III. 
 
 DISPERSION. 
 
 (39) 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 disco- 
 very we owe to Newton. We shall briefly describe the prin- 
 cipal experiments by which it is established. 
 
 (40) 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 axis of 
 the prism being rectilinear and parallel, and the extremities 
 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 suppo- 
 sition) be equally refracted by it, and therefore continue paral- 
 lel after refraction. This not being the case, we conclude 
 
 
32 DISIERSION. 
 
 that the rays composing the incident beam are of different 
 degrees of ref Tangibility, 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. 
 
 It further appears that the rays, which differ in refran- 
 gibility, likewise differ in colour; the spectrum being 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 imperceptible 
 gradations. Sir Isaac Newton distinguished the spectrum, or 
 coloured image of the Sun, into seven principal colours, and 
 measured the spaces occupied by each. These colours, ar- 
 ranged in the order of their refrangibility, are red., orange, 
 yellow, green, blue, indigo, violet ;* of which the yellow and 
 orange are the most luminous, 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 
 in concluding that the 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. 
 
 * 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 intoszo:, viz., red, oranye, 
 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 secondary. 
 
DISPERSION. 33 
 
 (41) 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 
 axis 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. 
 
 From the foregoing we conclude, then/ that the peculiar 
 colour and refrangibility belonging to each kind of homoge- 
 neoiis light, are permanent* and original affections, and are not 
 generated by the changes which that light undergoes in refrac- 
 tions or reflexions. 
 
 (42) In the experiments hitherto described, the analysis 
 
 * Professor Stokes has recently discovered that the refrangibility of light 
 does undergo alteration in certain cases, some bodies possessing the pro- 
 perty of lowering the refrangibility of the incident light that is, of emitting 
 rays of a lower refrangibility, when excited by those of a higher. This pro- 
 perty belongs to the solution of sulphate of quinine, and to certain coloured 
 glasses. Professor Stokes has denominated it fluorescence, 
 
 D 
 
34 DISPERSION. 
 
 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 at the same distance ; so that the 
 spectrum consists 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. 
 
 There are various ways of diminishing the breadth of the 
 spectrum, or the diameter of the Sun's unrefracted image, 
 amongst which that of Newton seems as convenient in prac- 
 tice as any. 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 
 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 
 
DISPERSION. 35 
 
 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 spectrum 
 is reduced to a narrow line, and is unfit for examination. To 
 remedy this, Newton employed a narrow rectangular aper- 
 ture, whose length, parallel to the axis 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. 
 
 (43) 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 
 composed. 
 
 Newton's method was to determine the refractive indices 
 of the extreme red and violet rays directly by means of the 
 formula of (28), and to deduce those of the other rays by a 
 simple proportion. 
 
 When the refracting prism was of crown-glass, the indices 
 
 77 78 
 
 of refraction of the extreme rays were found to be , TT^ re ~ 
 
 50 50 
 
 spectively. To determine the refractive indices of the in- 
 termediate rays, it was necessary to measure the spaces which 
 they occupied in the spectrum. For this purpose Newton 
 delineated on paper the spectrum AHA, and distinguished 
 it by the cross lines Aa, h g f p <? c * , 
 B, Cc, &c., drawn at / 
 the confines of the seve- \ 
 
 ral colours ; so that the ""* S~~* 5 c B x 
 space ABba is that occupied by the red light, ~BCcb that by 
 the orange, CDdc the yellow, DRed the green, EF/6? the blue, 
 FG^/the indigo, and GRhg the violet. 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, 
 
 D 2 
 
36 DISPERSION. 
 
 J, i, |, . Now the intervals AB, BC, CD, &c., occu- 
 pied by the several colours in the spectrum, or the differences 
 of the deviations which they subtend, are to one another 
 as the corresponding variations of the index of refraction. 
 If, therefore, the whole variation of /i, or j^, be divided as the 
 line AH is in the points B, C, D, &c., the refractive indices 
 of the rays at the confines of the several colours will be as 
 follow : 
 
 77 771 771 77^. 77* 77f 77# 78 
 60' ~60"' "50"' "50"' "60"' ~50~' ~50~' 50' 
 
 The mean refractive index is p or 1-55; and it appears 
 
 oO 
 
 from the preceding that it belongs to the rays at the con- 
 fines of the green and blue. 
 
 (44) The intensity of the light is very different in the dif- 
 ferent parts of the spectrum. According to the experiments 
 of Fraunhofer s the following numbers represent the intensi- 
 ties 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. 
 
 (45) 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 con- 
 fines of the yellow and orange, and diminishes regularly to the 
 two extremities ; but that, on the contrary, there are at cer- 
 tain points abrupt deficiencies of light 9 total or partial, indicated 
 by the existence of numerous dark lines or bands, crossing 
 the spectrum in the direction of its breadth; while in the in- 
 termediate 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 
 
DISPERSION. 37 
 
 (as has been hitherto supposed) of rays of every possible re- 
 frangibility, 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 va- 
 rious 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 irregularly 
 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 spectrum. 
 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 the spectrum, 
 as also their relative breadth and intensity, remain un- 
 changed. 
 
 (46) If other kinds of light as that of infixed stars, 
 flames, the electric spark are examined in the same way, 
 similar bands are discovered, but diifering in each species of 
 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 unalterable under all circumstances, and which, there- 
 fore, 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, hydro- 
 gen, and alcohol, have each a brilliant line between the red and 
 the yellow. The red flames coloured by nitrate ofstrontian ex- 
 hibit a brilliant blue line, which is detached 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 spec- 
 trum, and separated from it by a dark interval. On the other 
 hand, the spectrum of the flame of cyanogen exhibits great re- 
 
38 DISPERSION. 
 
 gularity, 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. 
 
 (47) 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 distinguished 590 
 in the solar spectrum, of which he has delineated 354. Of 
 these he has selected seven principal ones, to serve as stan- 
 dards 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 ex- 
 tremity ; 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 ; G 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 are of the ut- 
 most importance in optical investigations. On account of the 
 accuracy of their delineation, their position may be observed 
 with an accuracy equal to that of astronomical measurements, 
 and the refractive indices of the rays, to which they correspond, 
 thus determined with the utmost exactness. 
 
 (48) 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 red and violet 
 rays, ^ and ju 2 the refractive indices of the prism for those 
 rays, and a its refracting angle, we have 
 
 1 = (/iii - 1) a, S 2 = 0*2 - 1) a ; 
 whence 
 
DISPERSION. 39 
 
 Accordingly the dispersion, in this case, is equal to the angle 
 of the prism multiplied by the difference of the refractive 
 indices. 
 
 (49) The dispersive power of a substance is measured, 
 not by the absolute dispersion, which varies in general with 
 
 the angle of reflection, but by the ratio which that quantity 
 
 g _ g 
 bears to the total deviation, or by = . But, in the case of 
 
 Oi 
 
 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 red and violet rays, divided by 
 the refractive index of the former minus unity. 
 
 (50) 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 dis- 
 persion was always proportional to the total deviation ; and 
 that refraction could never take place without a separation 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 re- 
 sults were exactly the opposite to those stated by Newton ; 
 that, in fact, the emergent ray was coloured, when the devia- 
 
40 DISPERSION. 
 
 tion was nothing, or the ray parallel to its original direction ; 
 and that, on the other hand, when the dispersions 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 was inclined to its original direction. 
 This important discovery led to the construction of the achro- 
 matic telescope. 
 
 (51) 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 (/* 2 - jUi) a, 
 and (ju 2 ' - jit/) a', respectively (48) ; and, therefore, when the 
 total dispersion is nothing, we must have 
 
 / ' >\ t r\ a 1/2 Wi 
 
 (jit 2 - //i) a + (jito - jiti ) a = 0, or = - '; ~. 
 
 CJt JU-2 """* Wl 
 
 The negative sign, in the second member, indicates that the 
 angles of the two prisms must be turned in opposite ways. 
 
 (52) 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 same 
 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 
 
DISPERSION. 41 
 
 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. 
 
 (53) 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 suppo- 
 sition, as Young has justly observed, is but veiling our in- 
 ability to assign a mechanical cause for the phenomenon. 
 
 According to the principles of the wave-theory, the colour 
 of light is determined by the frequency of the ethereal vibra- 
 tions, or by the length of the wave; the longest waves pro- 
 ducing 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 dif- 
 ferent for the light of different colours. The velocity of propa- 
 gation in a refracting medium, therefore, varies with the length 
 of the wave. Here, then, we encounter a difficulty in this 
 theory, which was long regarded as the most formidable ob- 
 stacle to its reception. Analysis seemed to indicate that the 
 velocity of wave-propagation depended solely on the elasticity 
 of the medium as compared with its density, and should 
 therefore be the same for light of all colours, as it is for sound 
 of all notes ; so that all rays should be equally refracted. It 
 will be necessary to enter, in some detail, into the considera- 
 tion of this difficulty. 
 
 (54) The conclusion of analysis to which we have just ad- 
 
42 DISPERSION. 
 
 verted, namely, that the velocity of wave-propagation is con- 
 stant 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. Resuming the prob- 
 lem of wave-propagation with the more general equations, he has 
 proved that there exists, generally, a relation between the ve- 
 locity of propagation (or the refractive index in vacuo) and the 
 length of the wave ; and, therefore, that the rays of different 
 colours will be differently refracted. 
 
 (55) Let us make, for abridgment, 
 
 2;r 2?r 
 
 k=-^-, s= 9 
 A T 
 
 in which A 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 2 = aj? -f 2 & 4 + ajk* + &c., 
 
 in which the coefficients a 19 <z 2 > #3? &c., vary with the medium. 
 Now the velocity of wave-propagation is 
 
 F-- - * 
 
 ~ T ~~k> 
 
 consequently, 
 
 Accordingly, the velocity of propagation is a function of the 
 wave-length, and varies with the colour. 
 
 (56) In a vacuum, and in media (such as atmospheric air) 
 
DISPERSION. 43 
 
 which do not disperse the light, the coefficients a 2 , a 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 
 
 F 2 = ! + a z k\ 
 
 Hence, if F 1} F 2 denote the velocities of propagation for two 
 definite rays of the spectrum, and k l3 2 , 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, as determined by Fraunhofer for the seven definite 
 rays in certain media. 
 
 (57) The general formula, above given, is unsuited to a 
 comparison with observation in its present form, inasmuch as 
 
 the variable k ( = \ is not independent of F. This diffi- 
 
 culty is overcome by M. Cauchy by inverting the first series. 
 The result is of the form 
 
 fc = A^ + A^ + A^ + &c. 
 
 M. Cauchy has shown that this series, as well as the former, 
 is convergent, and that all the terms after the third may be 
 neglected. Hence, since 
 
 the velocity in vacua being unity, we have 
 
44 DISPERSION. 
 
 an equation expressing the refractive index in terms of the 
 time of vibration, or of the wave-length in vacua. 
 
 (58) The constants in this formula, A ly A^ A 3 , will be 
 determined, when we know three values of /*, with the cor- 
 responding values of s, or of the w r ave-length in vacua ; and 
 the formula may be then applied to calculate the values of /* 
 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 ofFraunhofer 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 vacua, 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. 
 
 (59) It is an interesting consequence of the preceding for- 
 mula, pointed out by Professor Powell, that as s diminishes, 
 or the wave-length in vacua increases, the value of jm 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 
 spectrum. 
 
CHAPTER IV. 
 
 DOUBLE REFRACTION. 
 
 (60) 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, w^hile 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 Islan- 
 dici dis-diaclastici, quibus mira et insolita refractio detegitur" 
 
 A few years after the date of this publication, the sub- 
 ject was taken up by Huygens. This distinguished philo- 
 sopher 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 tf ordinary refraction (33). He was, there- 
 fore, 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 
 of phenomena, he was happily led to assign the true law of 
 extraordinary refraction. The important researches of Huy- 
 
46 DOUBLE REFRACTION. 
 
 gens on this subject are contained in the fifth chapter of his 
 " Trait^ de la Lumiere." 
 
 (61) 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. 
 
 (62) 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 rhombohedron, which is accord- 
 ingly the primitive form. The angles of the 
 
 bounding parallelograms, CAB and ABD, in 
 the rhombohedron of Iceland spar, are 101 
 55' and 78 5'. Two of the solid angles, at 
 A and O, are contained by three obtuse an- 
 gles ; while the remaining four are bounded by one obtuse 
 and two acute angles. The line AO, joining the summits of 
 the obtuse solid angles, is called the axis of the rhombohedron, 
 and is equally inclined to the three faces which meet there. 
 The angles at which the faces themselves are mutually in- 
 clined are 105 5' and 74 55'. 
 
DOUBLE REFRACTION. 47 
 
 (63) If a transparent piece of this eubstance 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. 
 
 (64) In proceeding to the consideration of this law, we 
 must observe, in the first place, that there is a certain direction 
 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 
 
48 DOUBLE REFRACTION. 
 
 related to it. There are, properly speaking, an infinite num- 
 ber 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 crystal- 
 lized 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. 
 
 (65) All the phenomena of double refraction are symme- 
 trical round this line. 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 it is found, that the ratio of the sines of inci- 
 dence and refraction of the extraordinary ray is not constant, 
 but diminishes as the inclination of the incident ray to the 
 optic axis increases ; being least of all 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*483. 
 
 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, 
 
DOUBLE REFRACTION. 49 
 
 whatever be the refracting surface. To see it, 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 seen 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 ordinary refrac- 
 tion. 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. 
 
 (66) The directions of the two refracted rays are given by 
 the following construction. 
 
 Let AC be the incident ray, 
 and CF 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 
 ordinary ray; and from the centre C, and with the radius CD, 
 let the sphere DOG be described. Let the spheroid of revo- 
 lution GE be described with the same centre, its axis of revo- 
 lution 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 
 
50 DOUBLE REFRACTION. 
 
 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, without 
 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 refraction 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. 
 
 (67) We have seen that in Iceland spar the extraordinary 
 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 nume- 
 rous, 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 
 
DOUBLE REFRACTION. 51 
 
 the generating ellipse ; or, in other words, that the spheroid 
 \sprolate instead of oblate. 
 
 (68) 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 crystals 
 possessed two optic axes. Among the most remarkable of the 
 crystals with two axes may be mentioned arragonite, mica, sul- 
 phate 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. 
 
 (69) 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 crystals. 
 
 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 crystals of 
 the third class, orbiaxal 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 
 
 E 2 
 
52 DOUBLE REFRACTION. 
 
 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, reasonably 
 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 refrac- 
 tion belong either to the rhombohedral, or to the pyramidal sys- 
 tem, systems whose fundamental forms are the rhombohedron 
 and the straight pyramid. In each of these forms there is one 
 axis of figure^ or one line round which the whole is symmetri- 
 cal : 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 refraction. 
 Finally, in the oblique pyramid ', which is the fundamental form 
 of the prismatic systems, there is no one line, or axis of figure, 
 round which the whole is symmetrical; and it is therefore 
 probable that the density of the crystal is unequal 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. 
 
DOUBLE REFRACTION. 53 
 
 (70) 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 case, 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 propa- 
 gated 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 Huygenian law, the points of 
 contact determine the directions of the two refracted rays. 
 These more general laws of double refraction will be more 
 fully considered hereafter, 
 
 (71) 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-refract- 
 ing crystal, be placed close to a small object, as, for example, 
 a black spot on a sheet of paper, it will be observed that one 
 of the images is sensibly nearer than the other ; and that the 
 difference of their apparent distances changes with the thick- 
 ness of the crystal, and with the obliquity 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 example, 
 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, as the difference of the sines of incidence 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 
 
54 DOUBLE IlEFKACTION. 
 
 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 extraordinary, and therefore 
 the ordinary image is nearer than the other. The reverse is 
 the case in positive crystals, such as quartz, in which the extra- 
 ordinary index is the greater. 
 
 (72) 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 
 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 re- 
 fractions differ by the greatest amount, and therefore the dif- 
 ference of the deviations of 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 opposite 
 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 ; 
 and the effect of this arrangement is to double the separation 
 of the images produced by either singly. By this duplication 
 
DOUBLE REFRACTION. 55 
 
 the weak double refraction of rock crystal is rendered very- 
 sensible. 
 
 (73) An achromatic prism of this kind is employed in the 
 double image micrometer, an ingenious and valuable instrument 
 invented 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 de- 
 viation produced by the prism, as the distance of the prism 
 from the focus is to the focal length. Now the divergence of 
 the two rays is constant for a given prism, and may be deter- 
 mined 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 a man, or any other object whose 
 height is known approximately. The height of the object, 
 divided by the tangent of the angle, is the distance required. 
 
CHAPTER V. 
 
 INTERFERENCE OF LIGHT. 
 
 (74) 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 independently 
 of the other; and the motion of the point will, in consequence, 
 be the resultant of the motions due to each vibration 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 mov- 
 ing 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, that the distance 
 of the body from the vertical, measured in either of these 
 
INTERFERENCE OF LIGHT. 57 
 
 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 component vibrations when their 
 directions conspire, and their difference when they are opposed. 
 
 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 re- 
 sulting space of vibration will be double of either ; and that, 
 on the contrary, they will counteract one another, and the re- 
 sulting vibration will be reduced to nothing, when their direc- 
 tions are opposed. 
 
 It is evident, further, that the directions of the vibrations 
 will conspire, and therefore the space of vibration be doubled, 
 when the two waves arrive in the same phase ; and that, on 
 the contrary, their directions will be opposed, and the result- 
 
 in f vibration reduced to no- 
 
 . 
 thing, when they arrive m A '/^" 
 
 , T . ,, 
 
 opposite phases. JLet the 
 
 waving; lines AB and A'B'. A/" 
 
 or AB and A"B", represent 
 
 the two undulations, the dis- A v 
 
 tance of any particle from its 
 
 state of rest being represented by the ordinate, or perpendicu- 
 
58 INTERFERENCE OF LIGHT. 
 
 lar, 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'ri, will be 
 added together, and the space of vibration doubled. Whereas, 
 if the undulations A"B" and AB, whose corresponding points 
 are in opposite phases, be superposed, the distances from the 
 position of rest, mn and m"n" 9 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, there- 
 fore, is quadrupled in the former case, and destroyed in the 
 latter. 
 
 We have here taken, for the sake of illustration, two of 
 the most important cases, those, namely, in which the co- 
 existing undulations are in complete accordance, or complete 
 discordance. When this is not the case, and the waves 
 meet in some intermediate stage of the vibratory movement, 
 the position of the maximum will be altered, as well as its 
 magnitude ; and the rules for the composition of the coexist- 
 ing vibrations bear a close analogy to the well-known rule 
 for the composition of forces. 
 
 (76) We learn, then, as a result of the wave-theory, that 
 two lights may either augment each other's effects ; or they 
 may partially, or even wholly, destroy one another, and thus, 
 by their union, produce complete darkness. 
 
 Before we proceed to examine more particularly this in- 
 dication of theory, we may observe that it is altogether ana- 
 logous to what is known to take place in other cases of vibra- 
 tory motion. If two waves of water arrive at the same point 
 at the same instant, in such a manner that the crest of one 
 
INTERFERENCE OF LIGHT. 59 
 
 . 
 
 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, 
 Avhose 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 re- 
 sultant wave will be the difference of the components ; and, 
 when these are equal, the resultant wave will entirely 
 disappear. 
 
 We have a magnificent example of these effects in the 
 well-known phenomena of the spring and neap tides ; the tidal 
 wave in the former case being the sum of the waves caused 
 by the action of the Sun and Moon, and in the latter, their 
 difference. 
 
 The peculiarity of the tides in the port of Batsha furnishes 
 a still more striking instance of the principle of interference. 
 The tidal wave reaches this port by two distinct channels, 
 which are so unequal in length, that the time of arrival by one 
 passage is exactly six hours longer than by the other. It fol- 
 lows 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 oppo- 
 site 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, whose height is the difference of the heights of the 
 ordinary morning 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 in- 
 strument. 
 
60 INTERFERENCE OF LIGHT. 
 
 (77) The interference of the aerial pulses may be exhibited 
 to the eye. Let a compound tube be taken, consisting of two 
 equal and similar branches terminating in a common trunk. 
 It is evident, then, that if the air be thrown into the same 
 state of vibration at the extremities of the two branches, the 
 particles going and returning simultaneously in both, a 
 double vibration will be propagated to the extremity of the 
 main trunk, and may be rendered sensible by the agitation of 
 the particles of sand on a stretched membrane. If, on the 
 other hand, the air be in opposite states of vibration at the ex- 
 tremities of the branches, these will neutralize one another in 
 the trunk, and the membrane, and the sand, will be quiescent. 
 The conditions here described are attained, by bringing the 
 ends of the branches over the parts of a vibrating plate which 
 are in similar, or in opposite states of vibration. When the 
 length of the tube is such that it is in unison with the vibrat- 
 ing plate, it will utter a distinct sound in the one case, while 
 in the other it will be silent. 
 
 The alternate augmentation and intermission of sound ob- 
 served by Young, when a tuning-fork is turned round its 
 axis at a short distance from the ear, are easily referred to the 
 same principles. 
 
 (78) 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 
 
INTERFERENCE OF LIGHT. 61 
 
 with it were known to Grimaldi. The latter writer 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 pen- 
 umbras of the two cones overlap. It will be then observed, 
 that although the greater part of this doubly illuminated space 
 is brighter than the penumbra of one cone alone, yet the boun- 
 daries 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 obstacle, 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. 
 
 (79) 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 to- 
 gether, and if the original light be homogeneous, we shall ob- 
 serve a series of alternate bright and dark bands, formed at 
 those points where the w r aves 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 intercepted, 
 the whole system of bands will disappear, and the light which 
 remains become of uniform intensity. By withdrawing 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. 
 
62 INTERFERENCE OF LIGHT. 
 
 (80) 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 centres of dis- 
 turbance ; and where these two sets of waves cross, they w^ill 
 produce effects similar to those we have been describing 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; 
 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 necessarily straight. 
 On either side of this there will 
 be a line, BB, B'B', 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', con- 
 sist 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 like- 
 wise intermediate lines, composed of those points whose dis- 
 tances from the two centres differ by half a wave, by a, wave 
 
INTERFERENCE OF LIGHT. 63 
 
 and half^ by two leaves and half, &c. On all these'lines, the 
 crest of the wave from one origin meets the sinus 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 
 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 so on. 
 
 (81) In Young's experiment, if the light diverging from 
 the two apertures O and O', 
 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 OO' 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 
 
64 INTERFERENCE OF LIGHT 
 
 side of tfie central bright one, and, beyond these, a pair of 
 bright bands. If we measure the distances of one of these 
 from the two apertures, we shall find that their difference, 
 BO' - BO, is a constant quantity, whatever 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' - 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 found to be some exact multiple of 
 the first difference, or of the length of a wave. 
 
 Performing the same measurements for the intermediate 
 dark bands, we find that the difference in the lengths of the 
 paths, where the first pair is formed, is half the difference, 
 BO' - BO, or half the length of a wave. The differences of the 
 paths, at the place of each succeeding pair of dark bands, are 
 found in like manner to be intermediate to the corresponding 
 differences for the bright bands on either side, or to be odd 
 multiples of half a wave. 
 
 The difference of the distances from the two apertures 
 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. 
 
 It is easy to determine the positions of the bands, as de- 
 pendent on the interval of the apertures, and on the distance 
 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, mO f - mO, 
 is an integer multiple of the length of half a wave. Now, 
 
*- 
 
 INTERFERENCE OF LIGHT. 65 
 
 drawing the lines On, OV, parallel to PA, and denoting the 
 distance AP by b, the interval of the apertures OO' by c, and 
 the distance Am by a?, the right-angled triangles Omn, O'mri, 
 give 
 
 Om = v/^+-c 2 ^ 4 
 
 O'm = b* + x + ic 2 = b + 
 
 
 the distance 6 being very great in comparison with x and c. 
 Hence 
 
 <* + ** - <* - 
 
 O'm - 
 
 But this difference is equal to n -, X being the length of a 
 
 26 
 -, 
 
 wave ; we have, therefore, 
 
 nb\ 
 
 x = ; 
 2c 
 
 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 j^nou ^ an mcn f r the mean 
 rays of the spectrum. 
 
 (82) 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 
 
66 INTERFERENCE OF LIGHT. 
 
 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 Physical 
 Optics. 
 
 Two plane mirrors are placed so as to meet at a very ob- 
 tuse angle. A beam of light diverging from the focus of a 
 lens is suffered to fall upon them ; and there will be there- 
 fore 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 pro- 
 duce 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 Qri 
 represent the sec- 
 tions of the two mir- 
 rors, which we shall 
 suppose to be per- 
 pendicular 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 concentered. Then taking 
 the points O and O' at equal distances on opposite sides of the 
 mirrors, these points will be the foci of the two reflected pen- 
 cils, or the points of divergence of the two beams. Now it 
 
INTERFERENCE OF LIGHT. 67 
 
 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 passing 
 through this intersection, and through the point P, which 
 bisects the interval between the two foci O and O'; and 
 thirdly, that in proceeding from the mirrors, they are propa- 
 gated in hyperbolas, whose foci are O and O', and whose com- 
 mon centre is P. 
 
 (83) These results are in exact accordance with theory. 
 In fact, since On = nk, and Q'ri = rik, the difference of the 
 paths traversed by the reflected rays, knm and krim, when they 
 meet at m, is the same as if they had reached that point di- 
 verging directly from the points O and O'. 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 OO', is also perpendicular to it, and any point of it, as 
 A, is equidistant from O and O'. The bands, therefore, are 
 symmetrically situated with respect to this line ; and the dis- 
 tance, Ar/z, of the band of any order from the central band, is 
 
 .. n\AP 
 equal to 
 
 This distance is easily expressed in terms of given quan- 
 tities. For PQ = OQ x cos OQP = Q x cos OQP ; and 
 OO' = 2OP = 2&Q x sin OQP. But since the angles &QO, 
 &QO', are bisected by the lines Qra and Q', 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 de- 
 noted by e, and the distances &Q and Q A by a and b, we have 
 OO' = 2a sin c, AP = a cos e + 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\ 
 
 2 sin c 
 
 F 2 
 
68 INTERFERENCE OF LIGHT. 
 
 (84) 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. 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. 
 
 (85) 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 be thus 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 necessary 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. 
 
 (86) 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 ; 
 
INTERFERENCE OF LIGHT. 69 
 
 and, accordingly, when neither of the pencils has experienced 
 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 dis- 
 tant 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 to- 
 wards the side of the interposed plate in the former case, and 
 from it in the latter. Here then we have a complete experi- 
 mentum cruets, 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. 
 
 The amount of the displacement of the fringes, in this 
 important experiment, depends on the thickness of the in- 
 terposed 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 va- 
 cuum, Arago has shown that the minutest changes in the 
 refractive power of the air may be observed such, for exam- 
 ple, as would arise from a variation of temperature amount- 
 ing to 5\yth f a degree centigrade. By the same method it 
 was ascertained that dry air was more refractive than air satu- 
 rated with moisture, the difference amounting, very nearly, to 
 the millionth of the refractive index. 
 
70 
 
 INTERFERENCE OF LIGHT. 
 
 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. 
 
 (87) 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 we^ 
 reply, 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 interfer- 
 ence, 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. 
 
 To see this, let abed represent a portion of a spherical wave; 
 and let O be the place 
 of the eye, and O the 
 line drawn from it to 
 the luminous centre. 
 Commencing from the 
 point , let portions 
 ab, be, cd, &c., be 
 taken, such that the 
 differences of the dis- 
 tances of their extremities from the point O 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 
 
INTERFERENCE OF LIGHT. 71 
 
 by each pair of consecutive portions to the eye, are in com- 
 plete discordance, and should wholly destroy one another if 
 their intensities were equal. It is easy to see that this is the 
 case with respect to portions, as^, gh, which are remote from 
 the point a. For the magnitudes of the waves sent by the 
 several portions to any point depend first, on the magni- 
 tudes of these portions themselves, and secondly, on the angles 
 which the line drawn from them to that point makes with 
 the front of the wave. As respects the former, it is evident 
 that ab is greater than be, be than cd, and so on ; but these 
 differences continually diminish, and the magnitudes of the 
 consecutive portions approach indefinitely to equality, as they 
 recede from the point a. The same is true of the obliquities. 
 Hence, the portions of the wave, fg, gh, remote from the 
 point a, destroy one another's effects, and the effect on the 
 eye, or on a screen at O, will be entirely due to those parts of 
 the grand wave which are in the neighbourhood of the line 
 connecting that point with the luminous origin. 
 
 Of these parts ab produces the greatest effect being 
 both the largest and the least oblique. The effect of the 
 neighbouring portions is, however, very sensible, and we 
 shall have occasion hereafter to study some important pheno- 
 mena to which they give rise. In the meantime, one remark- 
 able consequence of this explanation is obvious namely, that 
 if the alternate portions be, de, &c., whose effects are negative, 
 be stopped, the total effect will be augmented, and the re- 
 sulting light increased by intercepting a portion of the wave. 
 We shall see hereafter that this startling conclusion is con- 
 firmed by experiment. 
 
CHAPTER VI. 
 
 DIFFRACTION. 
 
 (88) IT has been shown to be a result of the wave-theory, 
 that the intensity of the light which encounters an obstacle 
 must diminish rapidly within the edge of the geometric shadow. 
 It now remains to consider the other phenomena which arise 
 under these circumstances ; and it will be found that the 
 same theory affords the most complete account, not only of 
 their general characters, but even of their most minute details. 
 
 In order to understand the theory of shadows, it is neces- 
 sary to investigate their laws in the simple case in which 
 the magnitude of the luminous body is reduced to a point. 
 The effects thus presented were first observed, and in some 
 degree explained, by Grimaldi ; and they have been since 
 studied, as a separate branch of Optical Science, under the 
 title of diffraction or inflexion. 
 
 Grimaldi found, that when a small opaque body was 
 placed in the cone of light, admitted into a dark chamber 
 through a very small aperture, its shadow was much larger 
 than its geometric projection ; so that the light suffered 
 some deviation from the rectilinear course in passing by the 
 edge. On observing these shadows more attentively, he 
 found that they were bordered with three iris-coloured 
 fringes, which decreased in breadth and intensity in the 
 order of their distances from the shadow, and which preserved 
 the same distance from the edge throughout its entire extent, 
 unless where the body terminated in a sharp angle. Similar 
 fringes were observed, under favourable circumstances, within 
 the shadows of narrow bodies. 
 
DIFFRACTION. 
 
 73 
 
 The phenomena of diffraction were subsequently examined 
 by Hooke and by Newton ; and, lastly, in the hands of Young 
 and Fresnel, they have been forced to furnish evidence in 
 favour of the wave-theory, which few who impartially ex- 
 amine it can continue to withstand. We shall first describe 
 the most important of these phenomena, and afterwards ex- 
 amine them in their bearing upon4he two theories. 
 
 (89) The most obvious of these phenomena are the modi- 
 fications which light undergoes in passing by the edge of an 
 obstacle of any kind. 
 
 Let a beam of homogeneous light, entering a dark cham- 
 ber, fall on a lens of short focal length, MN, by which it is 
 
 brought to a focus at O, and thence diverges. Let an ob- 
 stacle, PP, be placed in the diverging beam, and let the sha- 
 dow which it casts be received upon a sheet of white paper at 
 Q, or on a piece of roughened glass. We shall then observe 
 the following phenomena : 
 
 I. The line OPQ, which is the boundary of the geometric 
 shadow, is not the actual boundary of light and shade. 
 
 II. The space below this line, QS, is not absolutely dark, 
 but is enlightened by a faint light, which extends to a sensible 
 distance within the geometric shadow, and gradually fades 
 away as it recedes from the edge of this shadow at Q. 
 
 III. On the other side of the boundary of the geometric 
 shadow, at QR, the paper is not uniformly illuminated by the 
 diverging beam, but is observed to be covered with a series 
 
74 DIFFRACTION. 
 
 of alternate bright and dark bands, which are parallel to the 
 edge of the shadow. The distances of these fringes inter se, 
 and from the edge of the shadow, vary with the position of 
 the screen, and diminish indefinitely as the screen is brought 
 near the obstacle. These fringes succeed one another for 
 many alternations, becoming, however, less marked as the 
 distance from the edge of the geometric shadow increases, 
 until at length they are wholly obliterated and lost. They 
 preserve the same distances from the shadow in all parts, 
 except only where the edge of the body forms a sharp 
 angle.* 
 
 IV. The dimensions of the fringes vary with the colour 
 of the light ; being broadest in red light, narrowest in violet 
 light, and of intermediate magnitude in the light of mean re- 
 frangibility. Hence, when white or compound light is em- 
 ployed, the fringes of different colours will not be accurately 
 superposed ; and there will be formed a succession of iris-co- 
 loured fringes, the colours following the order which they have 
 in the spectrum. 
 
 (90) If we follow/the course of the fringes from their 
 origin, we shall observe that they are propagated in lines sen- 
 sibly curved, whose concave side is turned towards the sha- 
 dow. In order to obtain accurate measures of the distances 
 of the fringes from the edge of the shadow, at different dis- 
 tances from the obstacle, Fresnel viewed them directly with 
 an eye-piece, furnished with a micrometer. He thus ascer- 
 tained that the curved path of each fringe was an hyperbola, 
 whose summit coincided with the edge ol the obstacle, and 
 whose centre was the middle point of the line connecting that 
 edge with the luminous origin. * 
 
 * If this angle be salient, the fringes, instead of forming a similar angle, 
 are observed to curve round the shadow. When the angle is re-entrant, they 
 cross, and enter on the shadow at each side, without interfering with one 
 another. 
 
DIFFRACTION. 75 
 
 If we consider these hyperbolas as coincident with their 
 asymptots (which may be done without sensible error, unless 
 near the edge of the obstacle), and if we then determine the 
 angles which they make with one another, and with the edge 
 of the geometric shadow, we shall find that these angles in- 
 crease rapidly as the distance of the obstacle from the lumi- 
 nous point diminishes. When this distance is about 40 inches, 
 the fringes are very close together, the fringes of the first 
 and second order making an angle with one another of less 
 than 2' in red light. At the distance of 4 inches this angle is 
 increased to more than 5' ; and at ^ of an inch it exceeds 
 16'. Thus the fringes dilate, as the edge of the obstacle ap- 
 proaches the luminous origin. 
 
 (91) In this experiment the incident light is supposed to 
 diverge from a luminous point. If the dimensions of the 
 luminous origin had been considerable, it will be easily 
 understood that each line in it, parallel to the edge of the 
 obstacle, would give rise to a different system of fringes ; and, 
 as the dark bands of some of these systems must coincide with 
 the bright bands of others, every trace of the phenomenon 
 would be obliterated. 
 
 (92) The preceding experiments exhibit the effect of a 
 single edge. When the light diverging from the luminous 
 point is suffered to pass by two near edges, the phenomena will 
 be varied in a very interesting manner. 
 
 Let &fine wire be placed in the pencil of light diverging 
 from a luminous point, and let its shadow be received on a 
 screen, or plate of roughened glass, as before. We then ob- 
 serve, outsidethe geometric shadow, a set of parallel bands, 
 or fringes, analogous to those produced by the single edge 
 in the former experiment. These are the exterior fringes. 
 But we observe further that the whole space of the geometric 
 shadow itself is also occupied by parallel stripes, alternately 
 
76 DIFFRACTION. 
 
 bright and dark. These are the interior fringes ; and they 
 are in general closer, and more finely marked than the ex- 
 terior. When the breadth of the obstacle is considerable, 
 the interior fringes disappear, and the phenomena fall under 
 the class already examined. 
 
 The interior fringes are propagated, like the exterior, in 
 hyperbolic curves ; but their curvature is less considerable, 
 and the deviation from a right-lined course is scarcely per- 
 ceptible within the limits at which they are commonly ob- 
 served. They are also, like the exterior fringes, broader in 
 red than in violet light, and of intermediate breadths in the 
 light of intermediate refrangibility. Accordingly, in com- 
 pound or white light, the fringes of different dimensions are 
 superposed ; and the bands are no longer alternately bright 
 and black, but coloured with different tints, in the order of 
 the colours of the spectrum. 
 
 (93) It still remains to examine the effects produced by 
 two edges turned inwards, so as to form an aperture of any 
 dimensions. 
 
 For this purpose Fresnel employed an instrument consist- 
 ing of two metallic plates, one of which is fixed in the frame of 
 the apparatus, while the other is moveable by means of a fine 
 screw. The edges of these plates are right-lined and parallel, 
 so that they form always a rectangular aperture; and, by means 
 of the adj usting 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 
 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- 
 
DIFFRACTION. 77 
 
 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 f]*)m 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. 
 
 (94) It is scarcely necessary to observe that the phenomena 
 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 
 
78 DIFFRACTION. 
 
 the more complex appearances are best sought for in its de- 
 ductions. We shall proceed, therefore, to consider the preced- 
 ing phenomena in their relation to the two theories of light. 
 
 (95) 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- 
 torily the curvilinear termination of the visible shadow, and 
 its excess above the geometric one. 
 
 To account for infringes 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 
 
DIFFRACTION. 79 
 
 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 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. 
 
 (96) If the phenomena of inflexion be the effects of attractive 
 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 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 dif- 
 fracted fringes, should vary with the mass, the nature, and the 
 form of the inflecting body. Now it is clearly ascertained, 
 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 interrup- 
 tion of light, caused by the interposition of an obstacle of an}' 
 kind, appears to be the only condition essential to the pheno- 
 menon. 
 
80 DIFFRACTION. 
 
 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 
 chemical 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, Fresnel took 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 
 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. 
 
 (97) The same objections apply to the hypothesis of 
 Mairan and Du Tour, which ascribes these effects to the re- 
 
DIFFRACTION. 81 
 
 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 density, 
 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. 
 
 (98) 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 were, in like manner, 
 supposed to arise from the interference of the two pencils re- 
 flected 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 observed facts 
 closely correspond with the calculated results of this theory ; 
 and in the case last mentioned, Young proved that the phe- 
 nomenon admitted no other explanation. Placing 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 
 
82 DIFFRACTION. 
 
 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 hyperbola ; 
 and that, when the distance of the luminous origin is les- 
 sened, the inclination of these hyperbolic branches (considered 
 as coincident with their asymptots) augments, and the fringes 
 dilate. 
 
 The theory of Young, however, did not bear a closer 
 comparison with facts. If the 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 re- 
 spect. In the other cases of diffraction also, he perceived that 
 the rays grazing the edge of the body were not the only rays 
 concerned in the production of the fringes ; but that the light 
 which passed by these edges at sensible distances was also 
 deviated, and concurred in their formation. Fresnel was thus 
 forced to seek a broader foundation for his theory. 
 
 (99) 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 
 
DIFFRACTION. 83 
 
 r 
 
 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 ofHuygens. 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 explained; 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- 
 plete, that the computed places of the several bands seldom 
 differ from those observed by more than the 100th part of a 
 millimetre. 
 
 (100) The general circumstances of these phenomena may 
 be deduced by very simple considerations from the principles 
 already laid down; although the complete development of 
 these principles demands the aid of a complicated analysis. 
 
 Thus, in the case of the fringes produced by a single 
 
 edge, let O be the luminous origin, MaN a diverging wave, 
 and R any point at which the illumination is sought. From 
 
 G 2 
 
84 DIFFRACTION. 
 
 this point, as centre, let a circle be described, touching the 
 circle MaN in #, and let the lines R#&, Rc'c, &c., be drawn in 
 such a manner that the intercepts bb', 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 
 in denote the intensity of the light sent from the portion ab, 
 m that from be, &c. the light sent from the indefinite wave, 
 aM. or aN, being taken as unity the actual light which reaches 
 the point R will be 1 , 1 + m, 1 + m - m, 1 + m - m + m", &c., 
 according as the obstacle is placed at the point , b, c, d, &c. 
 And the intensity of the light when the obstacle is altogether 
 withdrawn is 
 
 1 + m - m' + m" - m" + &c. = 2. 
 
 Now, as the terms of this series are continually decreas- 
 ing, and are affected alternately with opposite signs, it is ma- 
 nifest that if we stop at any term, the sign of the remainder 
 will be the same as that of its first term, and therefore al- 
 ternately positive and negative. Accordingly the intensities, 
 1 + m, 1 + m - m, 1 + m - m + m", &c., are alternately greater 
 and less than 2 ; and the intensity of the light sent to the 
 point R is alternately greater and less than when no obstacle 
 is interposed. 
 
 It will be easily understood, from this general explanation, 
 in what manner the magnitude of the fringes depends on the 
 length of the wave, on the distance of the luminous origin 
 from the obstacle, and on the distance of the screen. They 
 must be broadest in red light, and narrowest in violet light ; 
 and in white or compound light? the diffracted bands of dif- 
 ferent colours will occupy different positions, so as to form a 
 succession of iris-coloured bands having the violet or blue in- 
 side, and the red without. After a few successions these 
 
DIFFRACTION. 85 
 
 bands wholly disappear, owing to the superposition of bands 
 of different colours. 
 
 (101) It is easy to compute the relative places of the same 
 fringe, for different positions of the luminous point, and of the 
 screen. 
 
 Let P be the edge of 
 the obstacle, PA a por- 
 tion of the wave, diverg- 
 ing from O, which has I 
 
 just reached that edge ; 
 
 and let QR be the screen, and R the place of a fringe of any 
 given order. Then, in order that this point should belong to 
 the same fringe, for every distance of the luminous origin and 
 of the screen, it is only necessary that the interval of retarda- 
 tion, RP - RA, of the central and marginal parts of the wave 
 should be constant. For in this case the whole wave, AP, 
 may be divided into a given number of parts, such that the 
 difference of the distances of the successive points of division 
 from the point R shall be constant ; and therefore the effective 
 wave consists of the same number of elementary portions in 
 the same relative state as to interference. 
 
 Now, denoting OP by a, PQ by b, and QR by #, we 
 have 
 
 RP= frTtf=b + 
 q. /?., since x is very small in comparison with b. Similarly, 
 
 so that RA = b + ~ . and RP - RA = i x* ( *-- 
 
 , MU\A JL*IJL j-n-tx. 2 * IT 
 
 2 (a + b) \b a + b 
 
 But, by the condition of the question, this difference is a 
 constant quantity ; and denoting this constant by S, we have 
 
 b a -f b 
 
86 
 
 DIFFRACTION. 
 
 When b varies, a remaining unaltered, i. e. when the po- 
 sition of the screen is varied, the value of x is the ordinate 
 of an hyperbola whose abscissa is b ; so that the successive 
 points of the same fringe belong to an hyperbola, whose sum- 
 mit is the edge of the obstacle. 
 
 (102) The interior fringes formed in the shadow of a nar- 
 row opaque body arise, it has been said, from the interference 
 
 of the two portions of the wave which pass by the edges on 
 either side. Let PP' be the section of the opaque body, PC 
 and P'c' the two portions of the diverging wave which has 
 just reached its edges, and R any point of the shadow. Then, 
 if these portions be divided in the points , b, c, &c., a, b', c, 
 &c., in such a manner, that the difference of the distances of 
 any two consecutive points from the point R is equal to half an 
 undulation, the elementary wave sent from each portion will 
 be in complete discordance with those sent from the two ad- 
 jacent portions ; so that, if the several portions be equal, they 
 will neutralize one another's effects at the point R, with the 
 exception of the extreme portions, P, PV, the halves of which 
 next the edges remain uncompensated. 
 
 Now the arcs Pa, ab, be, &c., are very nearly equal, when 
 the lines drawn from their extremities to the point R are 
 sufficiently inclined to the normal, or, in other words, when 
 this point is sufficiently removed from the edge of the geo- 
 metric shadow. In this case, then, the only efficacious parts 
 of the wave are the halves of the extreme portions, Pa and 
 Pa; and the intensity of the light at the point R will be de- 
 
DIFFRACTION. 87 
 
 ter mined by the difference of their distances from that point, 
 or (which comes to the same thing) by the difference of the 
 lengths of the lines connecting it with the edges of the ob- 
 stacle. The phenomena of interference are therefore the same 
 as in the case of light emanating from two near origins, already 
 considered ; and we may transfer to the present case the con- 
 clusions arrived at in (81). Accordingly, if c denote the 
 breadth of the obstacle, and b its distance from the screen, the 
 distance, x 9 of any band from the centre of the shadow is 
 
 nb\ 
 
 *=-*;' 
 
 (103) The positions of the fringes formed by a narrow rec- 
 tangular aperture are determined by a similar formula. 
 
 Let PP' be the section of the aperture, PAP' the portion 
 of the wave which has just reached it, diverging from the lu- 
 minous origin at O ; and let QQ' be the projection of the 
 aperture on the screen. Then, if we take the point R on this 
 screen in such a man- 
 ner, that the difference 
 of its distances from 
 the edges of the aper- 
 ture, RP' - RP, shall 
 be equal to a whole 
 number of semi-undula- 
 tions, that point will be the centre of a dark or bright band, 
 according as the assumed number is even or odd. For, in the 
 former case, the wave PAP may be divided into an even 
 number of parts, such that the distances of every two conse- 
 cutive points of division from the point R differ by half an 
 undulation ; the waves sent by every two consecutive por- 
 tions to the point R will therefore be in complete discordance, 
 and the total effect at that point will be null. On the other 
 hand, when the difference RP' - RP is equal to an odd num- 
 ber of semi-undulations, the number of opposing portions of the 
 wave will be odd, and as the alternate portions compensate 
 
88 DIFFRACTION. 
 
 each other's effects at the point R, there will remain one por- 
 tion producing there its full effect. 
 
 The successive bands being formed at the points for which 
 HP' - HP = ri\, it is obvious that their distances, RB, from 
 the centre of the projection of the aperture, will be given by 
 the same formula as in the case last considered, c being now 
 the breadth of the aperture, with this difference, however, 
 that the dark bands correspond to the even values of n, and 
 the bright bands to the odd values, which is the reverse of 
 what takes place in the bands formed within the shadow of an 
 opaque obstacle. We learn then, 1st, that the distances of 
 the successive fringes of any colour form an arithmetical pro- 
 gression whose common difference is equal to its first term ; 
 2ndly, that they vary directly as the distance of the screen, 
 and inversely as the breadth of the aperture ; and 3rdly, that 
 they are proportional to the length of the wave ; and therefore 
 greatest for the extreme red rays, least for the extreme violet, 
 and of intermediate magnitude for the rays of intermediate 
 refrangibility. 
 
 We have supposed the screen to be so remote that the 
 bands are entirely without the projection of the aperture. 
 This will obviously be the case when QP' - QP is less than 
 half a wave. When the distance of the screen is so small that 
 QP' - QP exceeds this limit, fringes will be visible also imthin 
 the projection of the aperture. In this case the portions into 
 which the wave is divided are sensibly different in magni- 
 tude, as well as obliquity. The reasoning above employed 
 is therefore no longer applicable ; and the points of maximum 
 and minimum brightness can only be obtained by a complete 
 calculation of the intensity of the light. 
 
 (104) The phenomena of diffraction hitherto considered 
 are of the simplest class : but as such phenomena arise in 
 every instance in which light is in part intercepted, it is ob- 
 vious that they admit of endless modifications, varying with 
 
DIFFRACTION. 89 
 
 r 
 
 the form of the interposed body. Some of these are too re- 
 markable to pass unnoticed. 
 
 Among the most striking of these effects are those pro- 
 duced by light diverging from a luminous origin, and trans- 
 mitted through a small circular aperture ; as, for example, 
 that formed by a pin in a sheet of lead. When the trans- 
 mitted light is viewed through a lens, the image of the aper- 
 ture appears as a brilliant spot, surrounded by coloured rings 
 of great vividness; and these vary in the most beautiful 
 manner, as the distance of the aperture from the luminous 
 origin, or from the eye, is altered. When the latter distance 
 is considerable, the central spot is white, and the coloured 
 rings follow the order observed in thin plates. As the eye 
 approaches the aperture, the central white spot contracts to 
 a point, and then vanishes. The rings then close in on it in 
 order ; and the centre assumes in succession the most vivid 
 and beautiful hues, altogether similar to those of the reflected 
 rings of thin plates. 
 
 This remarkable coincidence has been shown to be an 
 exact result of theory. It has been demonstrated that the in- 
 tensity of the light of any simple colour, at the central spot, 
 and the compound tint in the case of white light, will be 
 the same as that reflected from a plate of air, whose thickness 
 bears a certain simple relation to the radius of the aperture, 
 and its distances from the luminous origin and from the 
 eye. 
 
 The points of maximum and minimum intensity are easily 
 determined. 
 
 Let O be the luminous 
 point, and O AB the line drawn 
 from it through the centre of ; 
 the aperture PP' ; then the 
 interval of retardation, S, of 
 
 the ray which reaches any point B on this line, coming from 
 the edge of the aperture, is OP + PB - OB. Let OA = , 
 
90 DIFFRACTION. 
 
 AB = b, and AP = r ; then, since r is very small in compari- 
 son with a and 5, it is easy to see that 
 
 Now when this interval is equal to a whole number, n, of 
 semi-undulations, the aperture may be divided by concentric 
 circles, such that the rays which reach the point B, coming 
 from any two successive circumferences, shall differ by the in- 
 terval of half a wave. It follows from the preceding for- 
 mula that the squares of the radii, and therefore the superficies 
 of the successive circles thus formed, are as the numbers 
 of the natural series ; so that the annuli comprised between 
 every two succeeding circumferences are equal. But the ele- 
 mentary waves proceeding from each annulus are in complete 
 discordance with those from the two adjacent. The successive 
 annuli will therefore destroy one another's effects, and the total 
 intensity of the light at the point B will be null, or equal to 
 that of the last, according as the number of annuli (the cen- 
 tral circle included) is even or odd. Hence, for a given aper- 
 ture, there will be a succession of points on the axis, at which 
 the intensity of the light is alternately nothing and a maximum ; 
 and it is obvious from the preceding that the distances of these 
 points will be the values of b given by the formula 
 
 - -I- T = n\ ; 
 
 in which the points of complete darkness correspond to the 
 even values of n, and those of maximum brightness to the 
 odd values. 
 
 Such is the case with homogeneous light. As the points 
 of maximum and minimum intensity are different for the rays 
 of different colours, there will be no point of complete dark- 
 ness in compound light, but a succession of points, at which 
 the centre of the aperture is richly coloured. 
 
DIFFRACTION. 91 
 
 (105) The theory of Fresnel is not only in exact accord- 
 ance with facts already known : it has also led to many 
 new and unexpected conclusions, and predicted consequences 
 which have been afterwards verified on trial. One of the 
 most remarkable of these is the phenomenon of diffraction by 
 an opaque circular disc. Poisson applied Fresnel's integrals 
 to this case ; and he was led to the startling result, that the 
 illumination of the centre of the shadow was precisely the 
 same as if the disc had been altogether removed. The prin- 
 ciples already laid down will enable the reader to satisfy him- 
 self of the theoretical truth of this conclusion. Arago was 
 the first to show that it was in accordance with fact, and his 
 experiment may be repeated without much difficulty. 
 
 (106) We have seen that when light diverging from a 
 luminous point passes by the edges of a fine hair or wire, a 
 succession of coloured bands will be formed parallel to the edge 
 of the shadow ; and the distances of these bands from the 
 shadow, and from one another, will be greater, the less the 
 diameter of the wire. If many such wires be exposed to the 
 diverging beam, and if, instead of being parallel, they are 
 crossed and interlaced in every possible direction, it is easy to 
 conceive that the coloured bands will be disposed in concen- 
 tric circles, whose centre is the luminous point. These circles 
 resemble the halos visible round the Sun and Moon in hazy 
 weather. Their diameters vary in the inverse ratio of the 
 thickness of the wires or fibres. 
 
 This law was applied by Young, in a very ingenious man- 
 ner, to the comparison of the diameters of fibres, or small par- 
 ticles of any kind. 
 
 A plate of metal is perforated with a small round hole, about 
 the 5 -\jth of an inch in diameter, around which, at the distance 
 of about J or J of an inch, is a circle of smaller holes. The 
 flame of a lamp is then placed immediately behind the aperture, 
 and the luminous point viewed through the substance to be 
 
92 DIFFRACTION. 
 
 examined. A ring or halo will be seen surrounding the aper- 
 ture ; and by moving the substance backwards and forwards 
 on a graduated ruler, this ring may be brought to coincide 
 with the circle of small holes pierced in the plate. The dis- 
 tance from the aperture is then read off on the ruler, and varies 
 obviously in the inverse ratio of the angular diameter of the 
 spectrum ; but the diameters of the particles vary also in the 
 same inverse ratio, so that the distance on the ruler at once 
 becomes a measure of these diameters. In this manner Young 
 compared the diameters of a great number of very minute 
 substances, such as the fibres of the finest wools, the glo- 
 bules of the blood, &c. The instrument itself he called the 
 Eriometer. 
 
 (107) In the case last mentioned, we have supposed the 
 intervals of the fibres, or fine wires, to be much greater than 
 their thickness ; in which case the phenomenon depends mainly 
 on the diameter of the opaque fibre. When the intervening 
 apertures are very small, the effect is influenced by their mag- 
 nitude, and assumes a different character. Thus, if a grating 
 be formed, by stretching a wire between two fine screws of 
 equal thread, and if this grating be held in the beam diverg- 
 ing from a luminous point, we shall observe, on either side of 
 the direct image, a series of spectral images richly coloured 
 with all the prismatic tints ; the spectra increasing in breadth, 
 and diminishing in intensity, as they recede from the centre. 
 
 These phenomena are seen to most advantage by means 
 of a telescope adjusted to the luminous origin. The grating 
 being held before the object-glass of the telescope, the spectra 
 are formed at its focus, and are there viewed, with all the ad- 
 vantages of distinctness and amplification, by means of the 
 eye-glass. Fraunhofer, who observed these phenomena with 
 much attention and care, traced no fewer than thirteen spec- 
 tra on either side of the central image ; the first pair being 
 separated from the central image, and from the second pair, 
 
DIFFRACTION. 93 
 
 by intervals absolutely black. By a very accurate mode of 
 measurement he ascertained that the deviations of any one 
 colour from the central image, in the successive spectra, formed 
 an arithmetical progression ; and that the absolute amount of 
 these deviations varied inversely as the intervals of the axes 
 of the wires. 
 
 (108) These results flow readily from the principle of in- 
 terference, the^rs^ pair of spectra, on either side of the cen- 
 tral image, being produced by the interference of those rays 
 whose paths differ by one undulation ; the second pair, by those 
 whose paths differ by two undulations; and so on. 
 
 Let the light proceeding from a very remote origin fall on 
 the grating, whose opaque parts are represented by ab, a'b', 
 a"b" 9 &c. ; and let Q be the place of the eye. Then, if we take 
 a portion of the grating, act", composed of one opaque and one 
 transparent portion, in such a manner that the difference of the 
 distances of its extremities from the point Q, Qa" - Qa', shall 
 be equal to the length of 
 a wave, it is manifest that 
 the corresponding portion 
 of the incident wave, a'a", 
 may be divided into two 
 parts very nearly equal, 
 the waves sent from which 
 to the point Q shall be 
 
 in complete discordance. Without the grating, therefore, the 
 effect of that portion of the incident wave would be null at the 
 point Q, and no light from it would reach the eye. The 
 effect of the grating, however, is to intercept the whole or 
 part of one of the two interfering portions, and thus to render 
 the other visible : and this effect is greatest when the opaque 
 and transparent parts of the grating are equal. A bright 
 band will therefore be visible in the direction Qa". The same 
 thing will happen for all the similar divisions of the grating, 
 the distances of whose extremities from the point Q differ by 
 
94 DIFFRACTION. 
 
 two, three, or any whole number of undulations ; and thus 
 there will be a succession of bright bands, visible at different 
 angular distances from the direct ray PQ. 
 
 These angles are easily computed. Let a'k be the arch of 
 a circle described with the centre Q ; then a"k = a' a" cos a'a'k 
 = act" sin PQa". But the interval of retardation, a"k, is equal 
 to the length of a wave ; so that, if the angle PQa" be de- 
 noted by 9, and the interval composed of an opaque and trans- 
 parent part of the grating, a f a", by t, we have 
 
 sin 9 = . 
 
 
 
 This is the angular distance of the first bright band from the 
 central one ; and it is obvious that the corresponding angle, 
 for the band of the n th order is given by the formula 
 
 . n n\ 
 
 sin O n = . 
 
 
 
 The position of each ray, in these spectra, therefore depends 
 solely on the length of the ivave> and is independent of the na- 
 ture of the substance by which it is produced. 
 
 (109) It is a remarkable circumstance of the phenomenon 
 whose laws we have been tracing, that when the experiment 
 is performed with the requisite care, the several species of 
 homogeneous light are so pure and unmixed in the spectra, 
 that the fixed lines may be discerned. These lines, then, are 
 wholly independent of refraction, and exist in the parts of 
 the solar beam before they are separated by the prism. The 
 phenomenon, when thus exhibited, is however distinguished 
 by a remarkable peculiarity. The distances of the fixed lines 
 in the diffracted spectrum are always proportional, whatever 
 be the diffracting substance ; while the ratios of the intervals 
 of the fixed lines (or of the breadths of the coloured spaces), 
 in the spectra formed by refraction, vary with the dispersive 
 powers of the prisms. In fact, the angle being small, we 
 may^make sin 9 = sin 1'; so that 
 
DIFFRACTION. 95 
 
 Hence if, 15 2 , 0^ denote the deviations corresponding to any 
 three definite points in the spectrum, and \ 19 X 2 , X 3 , the corres- 
 ponding wave-lengths, it follows that 
 
 6/3 Ui AS Xi 
 
 or the intervals of the fixed lines of the spectrum are as the 
 differences of the corresponding wave-lengths, and are there- 
 fore in an invariable ratio. The difference in the disposition 
 of the fixed lines, in the spectra formed by diffraction and by 
 refraction, will be seen in the diagrams of Art. (1 1 1), in which 
 the points B, C, D, &c., of the horizontal line BH, represent 
 the relative positions of the principal fixed lines, in the spec- 
 trum formed by a prism of flint-glass, and in the diffracted 
 spectrum, respectively. 
 
 (110) The formula of Art. (108) suggests a very simple 
 method of determining the length of the wave corresponding 
 to any given ray of the spectrum. The value of 6,.or the in- 
 terval of the axes of the wires, may be ascertained with the 
 greatest ease and precision ; and we have, therefore, bujjy to 
 measure the angular deviation, 9, of the ray of any simple 
 colour from the axis, in order to deduce the value of X. Fraun- 
 hofer computed in this manner the lengths of the waves cor- 
 responding to the seven principal fixed lines of the spectrum ; 
 and the resulting values are perhaps the most exact optical 
 constants we possess. 
 
 The wave-lengths, corresponding to the principal fixed 
 lines, B, C, D, E, F, G, H, expressed in millionths of a milli- 
 metre, were thus found to be* 
 
 688, 656, 589, 526, 484, 429, 393. 
 
 * M. Mossotti has pointed out a curious relation between these numbers 
 and the lengths of the chords which produce the notes of the diatonic scale. 
 
96 
 
 DIFFRACTION. 
 
 The wave-length corresponding to the middle point of the 
 diffracted spectrum is 553*5 millionths of a millimetre. 
 The wave-lengths corresponding to the extreme visible points 
 are 738 and 369 millionths, respectively, the former of which 
 is exactly double of the latter. 
 
 (Ill) But the diffracted spectrum is further distinguished 
 by the simplicity of the law which governs the intensity of the 
 light in its several parts. The intensity of the light in the or- 
 dinary spectrum (formed by a prism of flint-glass) was deter- 
 mined by Fraunhofer, for the points corresponding to the 
 principal fixed lines. These intensities are represented by the 
 ordinates of the curve in the annexed diagram. The ordinate 
 
 at the point m (situated between the fixed lines D and E, at a 
 distance Dm = DE) corresponds to the maximum inten- 
 sity, and divides the whole light of the spectrum into two 
 equal parts, the areas of the two portions of the curve being 
 equal. 
 
 The law of the intensity in the diffracted spectrum was 
 deduced by Mossotti from the foregoing : it is represented by 
 the ordinates of the curve in the following diagram. We see 
 that . 
 
 I. The ordinate which divides the light into two equal 
 portions corresponds to the middle point of the spectrum. 
 
 II. This ordinate is a maximum ; and the curve is sym- 
 metrical with respect to it as an axis. 
 
DIFFRACTION. 97 
 
 Accordingly, the intensity of the light of the latter spec- 
 trum is a maximum at the middle of its length, and decreases 
 thence symmetrically on either side. It is evanescent, when 
 the wave-length increases, or decreases, by about one-third of 
 the value corresponding to the maximum intensity. 
 
 Hence while, in the spectra formed by refraction, the ratios 
 of the spaces occupied by the several colours, and the intensities 
 of the light at the several points, vary with the refracting sub- 
 stan<:e 9 they are, on the other hand, invariable in the diffracted 
 spectrum. The latter spectrum, accordingly, must be re- 
 garded as the normal one, to which all others are to be re- 
 ferred. 
 
 (112) Gratings producing these effects may be formed in 
 several ways as, for example, by tracing a number of paral- 
 lel lines on glass with a fine diamond point. Fraunhofer suc- 
 ceeded by such means in forming ruled surfaces in which the 
 striae were actually invisible under the most powerful micro- 
 scopes, the interval of the grooves being only the ^Q Jo o f an 
 inch. 
 
 Analogous phenomena may be produced by reflexion. If 
 a great number of parallel lines be engraved at very small and 
 equal intervals upon a polished surface, the light reflected 
 from the intervals of the grooves will interfere in a manner 
 precisely analogous to that admitted through the apertures 
 of the gratings ; and will, by their interference, produce the 
 most brilliant spectra. In some of the grooved metallic sur- 
 faces constructed by Mr. Barton, there are 10,000 lines to the 
 inch. With surfaces so minutely divided, the spectra pro- 
 duced are as perfect as those formed by the finest prisms ; and 
 the colours which they display are little inferior to those of 
 the diamond. 
 
 Similar appearances may be observed on metallic surfaces 
 which have been polished with a coarse powder, the powder 
 leaving minute striae which produce the effects we have been 
 
 H 
 
98 DIFFRACTION. 
 
 describing. They may also be very simply produced by pass- 
 ing the finger over the surface of a piece of glass moistened 
 by the breath. The striaa thus formed in the coating of vapour 
 display very brilliant colours, which vary with the position of 
 the eye. 
 
 (113) The beautiful colours of 'mother of pearl are natural 
 instances of the same phenomena. This substance is com- 
 posed of a vast number of very thin layers, which are gra- 
 dually and successively deposited within the shell of the 
 oyster, each layer taking the form of the preceding. When 
 it is wrought, therefore, the natural joints are cut through 
 in a great number of sinuous lines ; and the resulting sur- 
 face, however highly polished, is covered by an immense 
 number of undulating ridges, formed by the edges of the 
 layers. These striae may be observed by the aid of a 
 powerful microscope, although they are sometimes so close 
 that 5000 of them occupy an inch. That they are the 
 causes of the brilliant colours displayed by this substance 
 has been placed beyond doubt by an experiment of Sir 
 David Brewster. This experiment consisted simply in 
 taking the impression of the surface of the pearl on wax, or 
 any other substance fitted to receive it : it was found that 
 the impressed surface displayed all the colours of the ori- 
 ginal body. In fact, the colours of striated surfaces indicate 
 their structure, perhaps more unerringly than any other 
 means : Sir David Brewster has made a very ingenious, 
 use of their laws, in investigating the curious and compli- 
 cated structure of the crystalline lens in fishes and other 
 animals. 
 
 (114) There remains another class of phenomena pro- 
 duced by diffraction, which it is important to notice. 
 
 We have already seen the effects produced, when light 
 diverging from a luminous point is transmitted through a 
 
DIFFRACTION. 99 
 
 narrow aperture, and received on a screen. But if we 
 vary the experiment, by placing a lens of considerable focal 
 length (as the object-glass of a telescope) immediately be- 
 hind the aperture, and receive the image on a screen at 
 the conjugate focus, the appearances displayed are altered 
 in a remarkable manner, and differ more widely from those 
 produced in the former case, as the aperture is greater. 
 In fact, the phenomena of diffraction are thus produced 
 with apertures of considerable dimensions, and were observed 
 by Sir William Herschel with the undiminished object-spe- 
 cula of his great telescopes : they are rendered more dis- 
 tinct, however, when the aperture of the telescope is limited 
 by a diaphragm of moderate size. When a star is viewed 
 through a telescope of high power, having its object-lens 
 thus limited, its image is encompassed with a system of dif- 
 fracted rings slightly coloured, succeeding one another at 
 equal intervals ; the diameters of the rings varying in- 
 versely as those of the apertures. The phenomena vary in 
 a very curious manner, when the form of the aperture is 
 changed. Thus, when a triangular diaphragm is substituted 
 for the circular one, the disc of the star appears surrounded 
 by a black ring, from which diverge six rays at equal in- 
 tervals. 
 
 These phenomena have been examined in detail by Sir 
 John Herschel and by M. Arago. Their mathematical ex- 
 planation has been given by Mr. Airy, in his valuable tract 
 on the TJndulatory Theory ; and the deductions of theory are 
 found to be in complete accordance with the observed facts. 
 
 H 2 
 
CHAPTER VII. 
 
 COLOURS OF THIN PLATES. 
 
 (115) THE colours of thin plates were first noticed by 
 Boyle and Hooke. They are displayed in every instance in 
 which transparent bodies are reduced to films of great tenuity. 
 Boyle succeeded in blowing glass so thin as to exhibit the 
 phenomena : they are more readily developed in mica, and 
 some other transparent minerals, which possess a lamellar 
 structure ; but the most familiar instance of their exhibition 
 is in the froth of liquids, the fluid envelopes of the bubbles 
 which compose it being in general of extreme thinness. 
 
 These colours vary with the thickness of the film, and dis- 
 appear altogether when it passes certain limits. When the film 
 exceeds a certain thickness, all the colours are equally reflected, 
 and the reflected light is therefore white. On the other hand, 
 when the thickness falls below a certain limit, no light what- 
 ever reaches the eye, and the surface of the film appears abso- 
 lutely black. 
 
 (116) The foregoing facts may be observed in the common 
 soap-bubble, when properly defended from the disturbing in- 
 fluence of currents of air. If the mouth of a wine-glass be 
 dipped in water, which has been rendered somewhat viscid 
 by the mixture of soap, the aqueous film which remains in 
 contact with it after emersion will display the whole suc- 
 cession of these phenomena. When held in a vertical plane, 
 it will at first appear uniformly white over its entire surface ; 
 but, as it grows thinner by the descent of the fluid particles, 
 colours begin to be exhibited at the top, where it is thinnest. 
 These colours arrange themselves in horizontal bands, and 
 
COLOURS OF THIN PLATES. 101 
 
 become more and more brilliant as the thickness diminishes ; 
 until finally, when the thickness is reduced to a certain 
 limit, the upper part of the film becomes completely black. 
 When the bubble has arrived at this stage of tenuity, cohesion 
 is no longer able to resist the other forces which are acting on 
 its particles, and it bursts. 
 
 Similar phenomena may be observed when a drop of oil is 
 let fall on water. As the oil spreads rapidly over the surface, 
 it is soon reduced to a very thin film, which displays the 
 spectral colours. 
 
 Every one has noticed the fact that steel and other metals, 
 when polished, acquire various shades of colour by exposure 
 to the air. These colours are produced by a thin coating of 
 metallic oxide, which is gradually formed on the surface. The 
 formation of this oxide is greatly accelerated by an augmenta- 
 tion of temperature ; and the colour thus formed is so inva- 
 riably connected with the thickness of the film, and this latter 
 with the degree of heat, that artists are in the habit of mea- 
 suring the temperature by the colour developed. Thus steel, 
 in the process of tempering, is said to have received a yellow 
 heat, a blue heat, &c. 
 
 The same appearances are displayed in a still more strik- 
 ing manner by air itself, or even by a vacuum. If two 
 plates of glass be pressed together by the fingers, we shall 
 observe, round the point of nearest approach, a succession of 
 coloured bands of great brilliancy, which dilate as the pressure 
 is increased, and the inclosed plate reduced in thickness. 
 
 (117) In order to observe these phenomena, in such a 
 manner as to be enabled to trace their laws, we must follow 
 Newton. Newton's experiment consisted simply in laying a 
 convex lens of glass upon a plane surface of the same mate- 
 rial. The thickness of the inclosed plate of air increases as the 
 square of the distance from the point of contact, and is, there- 
 fore, the same at all equal distances from that point ; and, as 
 
102 COLOURS OF THIN PLATES. 
 
 the reflected colour depends on the thickness, the bands of the 
 same colour will be arranged in concentric circles, of which 
 that point is the centre. The same succession of colours is 
 produced when any other transparent fluid is inclosed between 
 the glasses. The colours, however, are more vivid, the more 
 the refractive power of the thin plate differs from that of the 
 substances within which it is inclosed. 
 
 When we look attentively at these rings, the light being 
 reflected always at the same angle, we observe that the cen- 
 tral one is not a mere annulus, but a complete circle of nearly 
 uniform colour. If then we diminish the thickness of the 
 plate of air, by pressing the two glasses more closely together, 
 this central circle is observed to dilate, and a new circle of a 
 different colour to spring up in its centre. This will dilate in 
 turn, driving the former before it, and another circle appear 
 within it ; until at length a black spot shows itself in the 
 centre of the system, after which no further diminution of 
 thickness will alter the succession. When the black spot 
 makes its appearance, we have obtained a plate of air so thin 
 as no longer to reflect any colours, and the phenomenon is 
 complete. Newton traced seven coloured rings round this 
 spot, the colours of which are said to be of the first^ se- 
 cond^ third, &c., order , according to the order of the ring to 
 which they belong. Thus, the red of the third order is the 
 red in the third ring from the central black, &c. The whole 
 succession of colours is called Newton's scale. 
 
 (1 18) The principal laws of these phenomena are included 
 in the following propositions : 
 
 I. In homogeneous light, the rings are alternately bright 
 and black ; the thicknesses corresponding to the bright rings 
 of succeeding orders being as the odd numbers of the natural 
 series, and those corresponding to the black rings as the inter- 
 mediate even numbers. 
 
 II. The thickness 'corresponding to the ring of any given 
 
COLOURS OF THIN PLATES. 103 
 
 order varies with the colour of the light, being greatest in red 
 light, least in violet, and of intermediate magnitude in light of 
 intermediate refrangibility. In white or compound light, there- 
 fore, each ring will be composed of rings of different colours, 
 succeeding one another in the order of their refrangibility. 
 
 III. The thickness corresponding to any given ring varies 
 with the obliquity of the incident light, being very nearly pro- 
 portional to the secant of the angle of incidence. 
 
 IV. The thickness varies with the substance of the reflect- 
 ing plate, and in the inverse ratio of its refractive index. 
 
 (119) In order to establish the first of these laws, it is 
 necessary to employ homogeneous light. This may be ob- 
 tained by means of the prism : or we may adopt the method 
 suggested by Mr. Talbot, and illuminate the glasses with a 
 spirit lamp having a salted wick. The light of such a lamp 
 being a yellow of almost perfect homogeneity, the rings will be 
 alternately black and yellow ; and their number is so great as 
 to baffle any attempt to determine it. 
 
 The law of the thicknesses corresponding to the succes- 
 sive rings is easily established. 
 Let O be the point of contact ?Tjt- 
 
 of the plane and spherical sur- /y$' 
 faces, and aa, bb', cc, &c. the 
 
 diameters of the successive rings formed round that point as a 
 centre. It is evident that the thicknesses of the plate of air 
 at the points where these rings are formed, a, 6/3, cy, &c., 
 are as the squares of the distances Oa, Ob, Oc, &c., or as 
 the squares of the diameters of the rings : to determine the 
 law of the thicknesses, therefore, we have only to measure 
 these diameters. This was done by Newton with great ac- 
 curacy, and it was found that the squares of the diameters 
 were in arithmetical progression ; consequently, the thick- 
 nesses corresponding to the successive rings formed a similar 
 progression. 
 
104 COLOURS OF THIN PLATES. 
 
 (120) But Newton did not stop here : he ascertained 
 further the absolute thickness of the plate of air at which each 
 ring was formed. It is manifest that if the thickness of the 
 plate be determined for any one ring, that corresponding to 
 the others will be given by the law just stated. Newton, 
 accordingly, proceeded to ascertain this thickness for the dark 
 ring of the fifth order. This was done by measuring its 
 diameter accurately, and determining the radius of the spheri- 
 cal surface from the focal length of the lens and its refractive 
 index. The thickness is thence immediately deduced ; for it 
 is equal to the square of the radius of the ring divided by 
 the diameter of the spherical surface. The value thus de- 
 duced being suitably corrected, it was found that the thick- 
 ness of the plate of air was the y-^oo f an inch, at 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 yy^Voo ^ an mcn - Thus the bright rings of 
 the successive orders are formed at the thicknesses 
 
 &c. 
 
 378000 178000 178000 178000 
 
 and the intermediate dark rings at the thicknesses 
 
 _2__ _JL_, ^ 6 _ __?_ 
 178000* 178000 178000* 178000' 
 
 These determinations belong to the most luminous rays 
 of the spectrum, or those at the confines of the orange and 
 yellow. 
 
 (121) 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- 
 s i OI1) O r 5 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 
 
COLOURS OF THIN PLATES. 105 
 
 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 re- 
 frangible the exterior parts of the ring. It is easy to see also, 
 that all phenomena of colour must disappear after a few succes- 
 sions, the rings of different colours, belonging 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 thick- 
 nesses of the plate of air at which they are exhibited being 
 nearly as the secants of the angles of incidence or reflexion. 
 
 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 
 and in water, it will be found that the corresponding thick- 
 nesses of the plate are as four to three, or in the inverse ratio 
 of the refractive indices. 
 
 (1 22) We have hitherto spoken only of the reflected rings. 
 There is another system of rings formed by transmissioji, 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. 
 
106 COLOURS OF THIN PLATES. 
 
 ( 1 23) 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 thickness 
 diminishes regularly from the point to which the tube is di- 
 rected. 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, however, that the 
 order of the rings is reversed, the outermost ring corresponding 
 to the centre of Newton's scale. This little apparatus, con- 
 trived by Mr. Read, is denominated by him an iroscope. 
 
 (124) It is now time that we should enter on the physical 
 account of these phenomena. 
 
 For their explanation, it has been already stated, Newton 
 framed the hypothesis of the Jits of easy reflexion and trans- 
 mission already referred to : its application to the phenomena 
 of thin plates is obvious. The molecule of light is in &jit of 
 easy transmission in its passage through \hsjirst 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 with the colour of the liyht^ being 
 greatest in red light, least in violet, and of intermediate mag- 
 
COLOURS OF THIN PLATES. 107 
 
 nitude in light of intermediate refrangibility. Newton deter- 
 mined the absolute lengths of these fits for the rays of each 
 simple colour, and found that they bore a remarkable numeri- 
 cal 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 properties 
 to the fits, which are inconsistent with every physical account 
 which has been given of them. Thus, to explain the dilata- 
 tion of the rings with the obliquity, he assumed that the 
 length of the fits augmented with the incidence, and nearlyi n 
 the ratio of the square of the secant of the angle of incidence. 
 This assumption is at entire variance with the physical theory. 
 If the fits are produced by the vibrations of the ether which 
 are propagated faster than the luminous molecules, and which 
 alternately conspire with and oppose their progressive motion, 
 their lengths should continue the same in the same medium, 
 whatever be the incidence. 
 
 The fourth law appears to be also irreconcileable with the 
 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 ve- 
 locities augmented, the spaces traversed by the ray in the 
 interval of its periodical states must diminish, and in the same 
 proportion. 
 
 ( 1 25) 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 thickness 
 of the plate increases, the light appears by reflexion and trans- 
 mission 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 
 
108 COLOURS OF THIN PLATES. 
 
 the alternate appearance of the light, in the reflected and 
 transmitted pencils, to be the effect of an alternate 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 untrue ; that light is reflected from both surfaces 
 of the plate ; and that the concurrence of these two reflected 
 pencils is an essential condition 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 between 
 two transparent surfaces of different refractive powers. Under 
 these circumstances it is found that the intensity of 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 incident 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 therefore warranted in concluding, 
 that the light reflected from both surfaces of the plate is essen- 
 tial to their production. 
 
 (1 26) 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 por- 
 tions, one reflected at the upper, and the other at the lower sur- 
 
COLOURS OF THIN PLATES. 109 
 
 face of the plate ; and these two portions will interfere, and 
 reinforce or weaken each other's effects, according as they 
 reach the eye in the same or in opposite phases. 
 
 (127) 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 on the interval 
 of retardation of the second " pulse," or wave, with respect to 
 the first, and therefore on the thickness 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 re- 
 mained in this unsettled state, until the principle of Interfer- 
 ence was discovered 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 sim- 
 ply explained. 
 
 ( 1 28) Let mon be the course of a ray reflected at the first 
 surface of a plate ; mopo'ri that of the 
 
 ray reflected at the second surface, 
 and twice transmitted through the 
 first. From the point o' let fall the 
 perpendicular or 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 
 
110 COLOURS OF THIN PLATES. 
 
 the position 0V, 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 o, one has travelled over the space or', and 
 the other over the space op +po'. But, if we let fall the per- 
 pendicular or upon po' 9 it is evident from the law of refraction 
 that the spaces or and o'r are traversed in the same time in 
 the two media ; and, consequently, that the interval of retar- 
 dation 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 6. 
 
 The two waves are incomplete accordance or discordance, when 
 the interval of retardation is an exact multiple of the length 
 of half a wave : i. e. when 
 
 X 
 
 g = W 2' 
 
 n being any number of the natural series. Equating these 
 values of 8, therefore, we have, for the values of the thickness 
 of the plate which will produce a complete accordance or dis- 
 cordance 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 there- 
 fore in the direct ratio of the velocity of propagation, or in the 
 
COLOURS OF THIN PLATES. 111 
 
 inverse ratio of the refractive index of the substance of which 
 the plate is composed. 
 
 (129) There is one part of the preceding explanation 
 which demands a little further consideration. The two waves 
 being in complete accordance when the interval of retarda- 
 tion is an ev en multiple of the length of half 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 J A 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 se- 
 parating two media. The particles of ether next the bound- 
 ing surface communicate motion to the adjacent particles of 
 the second medium, and thus give rise to the refracted wave. 
 But the former particles will not remain at rest afterwards, 
 unless the density and elasticity of the ether be the same in 
 
112 COLOURS OF THIN PLATES. 
 
 the two media. When this is not the case, the particles of 
 the first medium will move, after communicating motion to 
 those of the second, and, in moving, give rise to the reflected 
 wave. Thus refraction is always accompanied 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 vi- 
 bration 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 un- 
 dulation on the other ; so that when the two 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 incomplete discordance, 
 and vice versa. Thus the dark rings will be formed where 
 the thickness of the plate is any even multiple of JX 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. 
 
 (130) 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 (85). These are al- 
 lowed to fall on a thick plate of parallel glass, whose second 
 surface is quicksilvered in one-half of its extent ; and in such 
 
COLOURS OF THIN^PLATES. 113 
 
 a manner as to be both reflected by the transparent portion of 
 that surface, or both by the opaque portion, or one by the for- 
 mer 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 dis- 
 cordance, 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. This anticipation of theory was verified by Young, 
 by inclosing oil of sassafras between two object-glasses, one 
 of which was of flint-glass, and the other of crown-glass. 
 
 (131) 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. 
 
 (132) The theory of thin plates, as it came from the hands 
 of Young, laboured under an imperfection, which was, how- 
 ever, soon removed. It is obvious that the intensities of the 
 two portions of light, reflected from the upper and under sur- 
 faces of the plate, can never be the same, the light incident 
 
 i 
 
114 COLOURS OF THIN PLATES. 
 
 on the second surface being already weakened by partial 
 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 calcula- 
 tion of the effect. When the problem is taken up in this 
 more general form, it is found that, where the effective thick- 
 ness 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 bound- 
 ing media placed in absolute contact. Plence, 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 be- 
 
COLOURS OF THIN PLATES. 115 
 
 yond the lens. The light reflected from this portion, accord- 
 ing to the Newtonian theory, should not surpass that of the 
 dark rings in intensity. The roughest trial is sufficient 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. 
 
 (133) 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 
 3t + t' in glass, and the second the thickness 3t' -t- 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 in- 
 clined ; the thickness traversed in the two plates being altered 
 by their inclination. 
 
 In the foregoing cases, the interfering pencils are mixed 
 up with, and overpowered by, the light directly trans- 
 mitted, and some contrivance is necesssary to make the 
 colours visible. The phenomena are much more obvious in 
 
 i 2 
 
116 COLOURS OF THIN PLATES. 
 
 the light reflected 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 accom- 
 panied by several lateral images, formed by 2, 4, 6, &c. re- 
 flexions. 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- 
 flected 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. 
 
 (134) 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 
 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 diame- 
 
COLOURS OF THIN PLATES. 
 
 117 
 
 ters of these rings vary inversely as the square roots of the 
 thicknesses 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. 
 
 (135) 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 O be the aperture through which the beam 
 is admitted, and let it 
 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 O AB 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 scattered in 
 repassing through the first surface. Let AM be one of the 
 rays of this second pencil, meeting the ray DM of the first at 
 the point M; and let us calculate their interval of retardation. 
 The latter has traversed the space AB + BA in glass, and 
 AM in air ; while the former has described the space 
 
118 COLOURS OF THIN PLATES. 
 
 AC + CD in glass, and DM in air. The interval of retar- 
 dation 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 
 
 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 b, as is usually the 
 case, we have simply 
 
 (136) 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 of Newton's 
 scale. Their diameters increase with the distance of the plate 
 from the focus, and diminish as the thickness of the plate in- 
 creases. They are formed, as before, by the interference of the 
 two pencils, which are scattered in passing through the two 
 surfaces of the plate. 
 
COLOURS OF THIN PLATES. 119 
 
 The phenomenon is reduced to its simplest conditions, by 
 receiving the light diverging from a narrow rectilinear aper- 
 ture upon two polished ivires, 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 distances 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. 
 
 (137) 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 inter- 
 val 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 im- 
 mediately 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 undu- 
 lation. 
 
 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 
 tint when the light is indirect ; and the colours rise in the 
 series, when the difference of the refractive densities is les- 
 sened by adding salt to the water. The interval of retarda- 
 tion, in this case, depends on the magnitude of the transparent 
 particle. 
 
 (138) In concluding this review of the two theories, in 
 
120 COLOURS OF THIN PLATES. 
 
 their application to the laws of unpolarized light, it should be 
 observed, that any well-imagined hypothesis may be accommo- 
 dated to phenomena, and seem to explain them, if only we in- 
 crease the number of its assumed principles, 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, 
 whose very essence it is to ascend in simplicity, at the same 
 time that it rises in generality : it is " a mob" of hypothe- 
 tical laws, without connexion, order, or dependence. 
 
 These remarks apply to the explanation of the phenomena 
 of thin plates adopted in the theory of emission. These phe- 
 nomena, Newton saw, could not be accounted for on the bare 
 hypothesis of molecules shot from the luminous body, and sub- 
 jected to the attractive and repulsive forces of the bodies 
 which they met in their progress. He was compelled to add 
 a new property to light, to endow the molecules with dis- 
 positions which seemed wholly alien to their other proper- 
 ties, and which could only be connected with them by as- 
 suming a material mechanism much more complicated 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; audnone of these properties were 
 in any way related to the rest, or to the mechanism on which 
 they were supposed to depend. 
 
 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 prin- 
 ciple, and are, therefore, mutually related. The principle of 
 
COLOURS OF THIN PLATES. 121 
 
 interference is a necessary consequence of the nature of a vi- 
 bration ; and this one principle, as we have seen, explains 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 bound 
 by the chain of number. The simple laws of 'interference, the 
 laws ofdiffraction, 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 details of all the rest, and compare 
 them with the results of measurement. The agreement is 
 found to be complete. 
 
( 122 ) 
 
 CHAPTER VIII. 
 
 POLARIZATION OF LIGHT. 
 
 (139) 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 observe 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 was sur- 
 prised to observe 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 such 
 positions in which one of the rays vanished altogether. 
 
 On analyzing the phenomenon, it is found that these two 
 positions are those in which the principal sections of the two 
 crystals are parallel or perpendicular. When these sections are 
 parallel, the ray which has undergone ordinary refraction by 
 the first crystal will be also refracted ordinarily by the second ; 
 
POLARIZATION OF LIGHT. 123 
 
 and the ray which has been extraordinarily refracted by the 
 first will be also extraordinarily refracted by the second. On 
 ' the contrary, when the principal sections of the two crystals 
 are perpendicular, the ray which has suffered ordinary refrac- 
 tion 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 refracted 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 square of the 
 cosine of the distance from the position of greatest intensity. 
 
 (140) 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 cer- 
 tain planes within the crystal. Such was the conclusion of 
 Newton. " This argues," says he, "a virtue or disposition 
 in those sides of the rays, which answers to, and sympathizes 
 with, that virtue or disposition of the crystal, as the poles of 
 two magnets answer to one another." 
 
 Although the phenomenon discovered by Huygens was 
 one of such importance, in the mind of Newton, as to force him 
 to admit the existence of properties in the rays of light which, 
 until then, had never been imagined, yet the result 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. 
 
 (141) In the year 1808, while Malus was engaged in the 
 
124 POLARIZATION OF LIGHT. 
 
 experimental researches by which he established the Huyge- 
 nian law of double refraction, he happened to turn a double- 
 refracting prism towards the windows of the Luxembourg 
 palace, which 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 oppo- 
 site 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 transmitted through two 
 rhomboids of Iceland spar, Mai us at first ascribed it to some 
 property which the light had acquired in its passage through 
 the atmosphere : but he soon abandoned this idea, and found 
 that this new property was impressed upon the light by re- 
 flexion 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 divided 
 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 under- 
 going ordinary refraction only, when the principal section is 
 parallel to the plane of polarization. 
 
POLARIZATION OF LIGHT. 125 
 
 (142) 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, if, 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 what 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. 
 
 (143) A polarized ray, then, is distinguished by the follow- 
 ing characters : 
 
 I. It is not divided into two pencils by a double-refracting 
 crystal, in two positions of the principal section with respect 
 to the ray ; being refracted ordinarily, when the plane of 
 
126 POLARIZATION OF LIGHT. 
 
 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. 
 
 (144) 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 
 by Sir David Brewster, In the year 1811 he commenced 
 
POLARIZATION OF LIGHT. 127 
 
 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. 
 
 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 ?r, and the corresponding angle of refrac- 
 tion by p t we have 
 
 Sin 7T SHlTT 
 
 tan TT = it, or = -. ; 
 
 cos TT sin p 
 
 therefore, cos ?r = sin p, and TT + p = 90. 
 
 Now the angle of reflexion is equal to the angle of incidence, 
 TT ; consequently the angles of reflexion and refraction are 
 
128 POLARIZATION OF LIGHT. 
 
 complementary, and the reflected and refracted rays are per- 
 pendicular. 
 
 (145) The law ofBrewster 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 recipro- 
 cals, 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 the polariz- 
 ing 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 pile of plates is highly useful 
 as a polarizer ; for the reflected beam is necessarily far more 
 intense than that produced by a single surface. 
 
 (146) It has been shown, that when a beam of light is 
 
POLARIZATION OF LIGHT. 129 
 
 polarized by reflexion, is suffered to fall upon a second reflect- 
 ing 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 in- 
 tensity varies as the square of the cosine of the angle formed 
 by the two planes of reflexion. This law was at first conjec- 
 turally 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 , and 90 - , 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 , and in the other I sin 2 , I 
 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 ) = I. 
 
 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. 
 
 * This is not to be understood as describing the actual physical cha- 
 racter of ordinary, or unpolarized light. This may be more correctly re- 
 presented 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 phe- 
 nomena presented by the resulting light agreeing in all respects with ordi- 
 nary or unpolarized light. When analyzed by a double refracting prism, 
 the two images were of equal intensities in all azimuths, so as to have 
 similar properties in all planes passing through the ray. 
 
 K 
 
130 POLARIZATION OF LIGHT. 
 
 (147) 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 supposi- 
 tion 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 
 produce this result increasing as the incidence is more re- 
 moved from the polarizing angle. 
 
 (148) 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- 
 
POLARIZATION OF LIGHT. 131 
 
 fracted ; and this refracted portion is found to be partially 
 polarized. The quantity of polarized light in the refracted 
 light increases with the incidence, being nothing at a perpen- 
 dicular 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 trans- 
 mitted through, a transparent surface, the reflected and trans- 
 mitted portions contain equal quantities of polarized light ; and 
 their planes of polarization are at right angles to each other" 
 
 (149) 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 per- 
 pendicular to the plane of incidence. These facts were dis- 
 covered by Malus. The laws of the phenomena have been 
 since investigated, in much detail, by Sir David Brewster ; 
 and he has drawn the conclusion, that when a ray of light is 
 transmitted through any number of plates, in the same plane 
 of incidence, the polarization will be complete, when the sum 
 of the tangents of the angles of incidence is equal to a certain 
 constant. Hence, when the plates are parallel, and the inci- 
 dence therefore the same on all, the tangent of the angle of 
 complete polarization is inversely as their number. 
 
 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 
 
 K2 
 
132 POLARIZATION OF LIGHT. 
 
 of the transmitted light will undergo no further diminution. 
 For, when the transmitted light becomes wholly polarized, 
 no portion of it whatever will be reflected by any of the suc- 
 ceeding plates, its plane of polarization being perpendicular 
 to the plane of incidence ; it is therefore transmitted without 
 diminution through them, whatever be their number. The 
 case is different, however, 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. 
 
 (150) There are certain crystals which, like the pile of 
 transparent plates, possess the property of polarizing the trans- 
 mitted 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, tour- 
 maline absorbs a polarized ray more rapidly when the plane 
 of polarization is parallel to the axis, than when it is perpendi- 
 cular to it. But a ray of common light falling upon this crys- 
 tal may be divided into two, one polarized in a plane passing 
 through the axis, and the other in a plane perpendicular to it ; 
 and as the former of these is absorbed more rapidly than the 
 latter, the transmitted light will be partially polarized 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 perpendicular 
 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 emer- 
 gent light is never perfect, and that its intensity is much 
 weakened by absorption both the rays being absorbed in 
 
POLARIZATION OF LIGHT. 133 
 
 their passage through the crystal, though with unequal ener- 
 gies. 
 
 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 ^th part of that of the incident light. 
 
 (151) 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 po- 
 larization 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 ti few moments at one of 
 the images of a circular aperture, formed by a rhomb of Ice- 
 land spar, and then at the other, and so alternately. 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 
 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 
 
134 POLARIZATION OF LIGHT. 
 
 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 supposed 
 to be homogeneous. When white light is used, the intensities 
 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 eifect of a single 
 refracting surface will be to produce two yellow brushes, whose 
 axis is in the plane of polarization. 
 
( 135 ) 
 
 CHAPTER IX. 
 
 TRANSVERSAL VIBRATIONS THEORY OF REFLEXION AND RE- 
 FRACTION OF POLARIZED LIGHT. 
 
 (152) 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- 
 pose the direction of the vibrations to be perpendicular to 
 that of the ray : then it is obvious that if that direction be 
 
136 TRANSVERSAL VIBRATIONS : 
 
 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 
 supposed to vibrated 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. 
 
 (153) 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 succession, be 
 thrown into a similar state of vibration; and a series of waves 
 will be propagated along it with a uniform velocity. The vi- 
 brations 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 
 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 sue- 
 

 REFLEXION AND REFRACTION OF POLARIZED LIGHT. 137 
 
 cession all these varied vibrations ; and the whole cord, in- 
 stead of taking the form of a curve lying all in one plane (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 condition of the ethereal particles in a ray of common, 
 or impolarized 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. 
 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. 
 
138 TRANSVERSAL VIBRATIONS : 
 
 (154) 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 when 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. 
 
 (155) We have stated that the vibrations of the molecules 
 of the ether, in a polarized ray, are all parallel to a fixed di- 
 rection in the plane of the wave : this fixed direction may 
 be either parallel or perpendicular to the plane of polariza- 
 tion ; and there was nothing in the phenomena, hitherto dis- 
 covered, to determine the choice between these two positions. 
 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 Mac Cullagh and Neumann, to ^parallel to it ; and 
 
REFLEXION AND REFRACTION OF POLARIZED LIGHT. 139 
 
 this difference in one of the postulates of the different theories 
 has necessarily led to others, especially as respects the relative 
 densities of the ether in different media. 
 
 Professor Stokes has recently 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 0, 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 ; 
 
 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 0. 
 
 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. 
 
140 TRANSVERSAL VIBRATIONS I 
 
 It would appear, therefore, that we had only to measure the 
 angles which the planes of polarization of the incident and dif- 
 fracted 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 polarization ; if it be greater, they are per- 
 pendicular to it. The experiment has been made by Professor 
 Stokes himself; and he has drawn the conclusion that the 
 latter is the fact, and, therefore, that the original hypothesis 
 of Fresnel is the true one. An opposite result has been since 
 obtained by M. Holtzmann, on repeating the same experiment 
 under somewhat different circumstances ; and the question 
 must therefore be regarded as still undetermined.* 
 
 (156) 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 
 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 reflected and 
 refracted pencils, those of the incident pencil being given. 
 
 This important problem was first solved by Fresnel. In 
 the attempt to generalize his theory, and to apply it to reflex- 
 ion and refraction at the surfaces of crystallized media, Pro- 
 fessor MacCullagh and Mr. Neumann were led to modify the 
 
 * In retaining, therefore, the demonstration of the laws of reflexion and 
 refraction of polarized light, which was adopted in the former edition of this 
 work, the author does not wish to be considered as giving a preference to the 
 principles upon which it depends. 
 
REFLEXION AND REFRACTION OF POLARIZED LIGHT. 141 
 
 principles of Fresnel. The principles, so modified, are the fol- 
 lowing : 
 
 I. The vibrations of polarized light are parallel to the 
 plane of polarization. 
 
 II. The density of the ether is the same in all bodies as 
 in vacua. 
 
 III. The vis viva is preserved ; 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 
 reflexion. 
 
 IV. The resultant of the vibrations is the same in the two 
 media ; and, therefore, in singly-refracting media, the refracted 
 vibration is the resultant of the incident and reflected vibra- 
 tions. 
 
 When the light is polarized in the plane of incidence, the 
 fourth principle, alone, is sufficient to determine the magni- 
 tudes of the reflected and refracted vibrations. For, the di- 
 rections of the vibrations, being in the plane of incidence and 
 perpendicular to the rays, are necessarily inclined to one 
 another at the same angles as the rays themselves ; these 
 angles therefore are 
 
 20, 0-0', + 0', 
 
 and 0' denoting the angles of incidence and refraction. 
 Hence, if v and v denote the amplitudes of the reflected and 
 refracted vibrations, that of the incident vibration being taken 
 as unity, we have 
 
 sin (0 - 0') ,_ sin 20 
 
 " sin (0 + 0')' : sin (0 + 0') ' 
 
 When the light is polarized perpendicularly to the plane of 
 incidence, the vibrations in the incident, reflected, and refracted 
 pencils are all parallel. The law of equivalent vibrations 
 therefore gives, in this case, the following simple relation 
 
 among them, 
 
 1 + w = w 
 
142 
 
 TRANSVERSAL VIBRATIONS! 
 
 and another relation is required, in order to deduce the values 
 of w and w. This second relation is furnished by the principle 
 of the vis viva, and is 
 
 m ( 1 - w 2 ) = m'w' 2 , 
 
 m and m' denoting the masses of the ether in motion in the 
 two media. Eliminating between these equations, we find 
 
 2m 
 
 w 
 
 m - m 
 m + m 
 
 w' 
 
 m + m 
 
 expressions which are remarkable as being identical with those 
 for the velocities of two elastic balls after impact. 
 
 Let B A, AC represent the ve- 
 locities and directions of the inci- 
 dent and refracted rays ; A A' the 
 separating surface of the two me- 
 dia ; and BB', CC', lines parallel 
 to that surface. Then the masses 
 of ether in motion in the two me- 
 dia are to one another as the pa- 
 rallelograms A'B, A'C ; that is, 
 
 m : m : : AB sin A'AB 
 : : sin 9 cos : sin 6' cos 9' 
 
 AC sin A' AC 
 : sin 29 : sin 29'. 
 
 Subtituting this ratio, therefore, in the expressions for w and 
 w' 9 given above, 
 
 tan (0-0') sin 20 
 
 w 
 
 w = 
 
 tan (0 + 0'/ sin (0 + 0') cos (0 - 0')' 
 
 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 raV 2 , 
 respectively ; or, if we take the intensity of the incident light 
 as unity, 1 , v 2 , and 1 - # 2 ; since, by the law of the vis viva, 
 m ( 1 - v 2 ) = m'v*. Similarly, for light polarized in the perpendi- 
 
REFLEXION AND REFRACTION OF POLARIZED LIGHT. 143 
 
 cular plane, the intensities in the three pencils are 1, w 2 , and 
 1 - v:\ 
 
 (157) Confining our attention for the present to the re- 
 flected vibration, it will be seen that its amplitude, and con- 
 sequently the intensity of the light, increases with the inci- 
 dence, whether the light be polarized in the plane of incidence, 
 or in the perpendicular plane, being least when the light is 
 incident perpendicularly, and greatest when it is most oblique. 
 
 In the former case, i. e. when 9=0, the values of v and w 
 
 are each equal to - - , p being the refractive index ; and the 
 intensity of the light reflected perpendicularly is 
 
 -1 
 
 + 1 
 
 This remarkable expression was first given by Young. 
 
 On the other hand, when - 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, what- 
 ever be the reflecting substance. 
 
 (158) 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 J, po- 
 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 refracted rays would be un- 
 polurized : but this, in general, is not the case. 
 
144 TRANSVERSAL VIBRATIONS : 
 
 In the case of the reflected beam, the intensities of the 
 
 two portions are 
 
 iv 2 , iw 2 ; 
 
 and the whole intensity of the reflected light is their sum. 
 Now the first thing to be observed is, that these two quan- 
 tities are unequal; or, that the two portions of which the re- 
 flected 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 expression 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 por- 
 tions, or by 
 
 0* - to 8 )- 
 
 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 polarized 
 perpendicularly to the plane of incidence. Their difference, 
 J (u 2 - ^ 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 experimental law of 
 Arago is a necessary consequence of theory. 
 
 (159) 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 + & = 90, the denominator of the 
 fraction becoming infinite ; the reflected light then contains 
 only the other portion, and is therefore completely polarized in 
 
REFLEXION AND REFRACTION OF POLARIZED LIGHT. 145 
 
 the plane of incidence. Since, in this case, + 0' =90, we 
 
 have 
 
 . sin 
 
 cos 8 = sin 6 = , and tan 6 = u ; 
 
 M 
 
 i. e. the tangent of the angle of polarization is equal to the re- 
 fractive index. Thus the beautiful law, Avhich Brewster had 
 inferred from observation, is deduced as an easy consequence 
 of Fresnei's theory. 
 
 When 6 + 0' is greater than 90, i. e. when the angle 
 of incidence exceeds the polarizing angle, the expression for 
 the amplitude of the reflected, w, vibration changes sign, the 
 light being polarized perpendicularly to the plane of incidence. 
 This change of sign is equivalent to an alteration of the phase 
 of the reflected vibration by 180, as the incidence passes 
 the polarizing angle ; and the circumstance explains the re- 
 markable fact noticed by Arago, namely, that when New- 
 ton's rings are formed between a lens of glass 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 sys- 
 tem of colours becoming complementary to what it was before. 
 Mr. Airy was led to anticipate this result, from a consider- 
 ation of the formula ; and to show that a similar change must 
 take place in the rings formed between two transparent sub- 
 stances of different refractive powers, as the incidence passes 
 the polarizing angle of either substance. 
 
 (160) When a polarized ray undergoes reflexion, the re- 
 flected light is still polarized, but its plane of polarization is 
 changed, the amount of the change depending on the inci- 
 dence. 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- 
 
146 TRANSVERSAL VIBRATIONS: 
 
 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 polarization 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, , be resolved 
 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,* respectively, 
 
 sin (0-6') tan (9-9') 
 
 - a cos a -r 7^ 5-, a sin a - -~ ^; 
 sin (9 + 9) tan (9 + 9) 
 
 and they compound a single vibration, inclined to the plane 
 of incidence at an angle whose tangent is the ratio of the com- 
 ponent vibrations. If, then, this angle be denoted by a', we 
 
 have 
 
 cos (9 + 9') 
 
 tan a = - tan a c 7^. 
 
 cos (9-9) 
 
 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. 
 
 * In order to explain the facts above mentioned, the values of v and w 
 (156) must be affected with opposite signs, at all incidences below the po- 
 larizing angle ; and there are other phenomena which indicate that the for- 
 mer quantity is negative, and the latter positive (see Professor Powell's pa- 
 per " 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 con- 
 clusions of Art. (156) are only approximate. 
 
REFLEXION AND REFRACTION OF POLARIZED LIGHT. 147 
 
 When a = 0, a = ; and when a = 90, a' = 90. Accord- 
 ingly, when the plane of polarization of the incident ray coin- 
 cides with, or is perpendicular to, the plane of incidence, it is 
 unchanged by reflexion. When + 0' = 90, a = 0, and the 
 plane of polarization of the reflected ray coincides with the 
 plane of incidence, whatever be the azimuth of the incident 
 ray. 
 
 (161) 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 no- 
 thing when the ray falls perpendicularly upon the refracting 
 surface, and greatest when the incidence is most oblique. 
 The law of the change is expressed by the simple formula, 
 
 cotan a = cotan a cos (6 - 0') ; 
 
 in which a and a' denote (as before) the angles which the 
 planes of polarization form with the plane of incidence, before 
 and after refraction. This law was discovered experimentally 
 by Sir David Brewster : it is a necessary consequence of the 
 theory already given, and is deduced by a process exactly 
 similar to that of the preceding article. 
 
 L 2 
 
( 148 ) 
 
 CHAPTER X. 
 
 ELLIPTIC POLARIZATION. 
 
 (162) 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 latter, 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 
 plane-polarized the vibrations being in this case confined 
 to a single plane passing through the direction of the ray. 
 
 * This is not strictly true, except in homogeneous or uncrystallized 
 media. 
 
ELLIPTIC POLARIZATION. 149 
 
 In intermediate cases, the polarization is called elliptical ; and 
 its character may vary indefinitely between the two extremes 
 of plane polarization and circular polarization. 
 
 (163) 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 di- 
 rections; a and b the amplitudes of the component vibra- 
 tions ; and t the time. Then 
 
 x = a sin (vt - a ), y = b sin (vt - /3) ; 
 whence 
 
 f . y\ f . x 
 
 a - p = arc I sin = ^ 1 - arc ( sin = 
 
 Taking the cosines of both sides, and clearing the result of 
 radicals, we obtain 
 
 This is the equation of an elliqse referred to its centre. 
 
 When the component vibrations are equal in amplitude, 
 and differ 90 in phase, 
 
 a = b, and a-/3 = 90; 
 and the preceding equation becomes 
 
 y 2 + x* = a 2 . 
 The path described by the molecule is then a circle. 
 
 (164) The nature of the elliptic polarization is completely 
 defined, 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 
 
150 ELLIPTIC POLARIZATION. 
 
 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. 
 
 (165) 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 compounded 
 of two plane-polarized rays, whose vibrations are at right 
 angles, and whose phases are no longer coincident; it is there- 
 fore elliptically polarized (163). 
 
 The first case in which this effect was observed was that of 
 total reflexion. 
 
 When the angle of incidence exceeds the angle of total re- 
 flexion (the light passing from the denser into the rarer me- 
 dium), the expressions for the intensity of the reflected light, 
 given in (156), 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 vibration 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 
 
ELLIPTIC POLARIZATION. 151 
 
 amount of this change has been deduced by Fresnel, by a 
 most ingenious train of reasoning, based upon the interpreta- 
 tion of imaginary formula?. 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 resultant 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 ellip- 
 tic, 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 undu- 
 lation, the ellipse will become a circle, and the light will be 
 circularly polarized. 
 
 (166) Reducing his formula? to numbers, in the case of St. 
 Gobain's glass, Fresnel found that the difference of phase of 
 the two portions of the reflected light amounted to one-eighth* 
 of an undulation, when the angle of incidence was 54 37'. 
 Polishing, therefore, a parallelopiped of this glass, whose faces 
 of incidence and emergence were inclined to the other faces 
 at these angles, it followed 
 
 that a ray RRR'R", inci- &" , B, 
 
 dent perpendicularly on 
 
 one of these sides, and 
 
 once reflected 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 
 
 * In order to produce a difference of phase of a quarter of an undulation 
 by a single reflexion, the refractive index should be = 4-142. 
 
152 ELLIPTIC POLARIZATION. 
 
 the twice-reflected ray amounting to a quarter of an undula- 
 tion. If, then, the incident ray be polarized 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 parallel opiped 
 described is well known under the name of Fresnel's rhomb ; 
 and is of essential service in all experiments relating to circu- 
 lar 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 
 bisects the supplement of the angle formed by their direc- 
 tions. 
 
 This property enables us to distinguish a circularly polar- 
 ized ray from a ray of common light. On the other hand, it is 
 at once distinguished from plane-polarized light, by the cir- 
 cumstance that it is divided into two rays of equal intensity 
 by a double-refracting crystal, whatever be the position of the 
 plane of the principal section. 
 
 (167) 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. 
 
ELLIPTIC POLARIZATION. 153 
 
 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. 
 
 (168) The subject of metallic reflexion was next studied 
 by Sir David Brewster, and the laws of the phenomena inves- 
 tigated in much detail. These laws may be reduced to the fol- 
 lowing : 
 
 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 'maxi- 
 mum polarization. 
 
 II. When the light is reflected several times in succes- 
 sion, in the same plane and at the same angle, the proportion 
 of polarized light in the reflected pencil is increa^d ; 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. 
 
 IV. 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. 
 
 V. When light, so depolarized, undergoes a second reflex- 
 ion in the same plane, and at the same angle, its polarization 
 is restored. The new plane of polarization lies on the oppo- 
 site side of the plane of incidence from the original plane, and 
 its azimuth is changed. 
 
 ( 1 69) From the last of the foregoing laws it is evident, 
 that the light produced by the reflexion of a polarized ray is 
 not common light. Neither is it plane-polarized light, since 
 it does not vanish in any position of the analyzing rhomb. 
 
154 ELLIPTIC POLARIZATION. 
 
 It is elliptically polarized ; and all the phenomena are expli- 
 cable on the hypothesis, that the two oppositely polarized rays, 
 into which the incident ray is resolved, differ in phase after re- 
 flexion, the difference amounting to 90 at the angle of maxi- 
 mum 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 inci- 
 dence. 
 
 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 
 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. 
 
 (170) 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 formulae (156). 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 tw T o reflected pencils be received on a double-refracting 
 prism, whose principal section coincides with the plane of 
 incidence, each of them will furnish a single refracted pencil. 
 
ELLIPTIC POLARIZATION. 155 
 
 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' 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 3 a, I sin 2 a ; 
 
 and those of the corresponding pencils, derived from the latter, 
 
 are 
 
 I cos 2 a, I' sin 2 a. 
 
 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' sin 2 a, and 
 
 I = I' tan 2 a. 
 
 The azimuth of the principal section, a, is measured by means 
 of a graduated circle attached to the prism ; and the value 
 of I' for each incidence is given by Fresnel's formulae. 
 
 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 
 t\\Q 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, how- 
 ever, being very small. In the case of light polarized per- 
 pendicularly to the plane of incidence, on the other hand, the 
 intensity of the reflected light diminishes from a perpendicular 
 incidence, up to the angle of maximum polarization, and after- 
 
156 ELLIPTIC POLARIZATION. 
 
 wards 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 
 cases, 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 polari- 
 zation. 
 
 (171) 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 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 po- 
 larization of these two components are unchanged by reflexion : 
 but their phases are altered, and that unequally ; and the re- 
 flected light, composed of them, is therefore elliptically polar- 
 ized. 
 
 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 dif- 
 ference of phase produced is equal to that produced by a sin- 
 gle reflexion, multiplied by the number of reflexions. But 
 the resulting light will be plane-polarized, when the differ- 
 ence of phase becomes a multiple of TT : we have, therefore, 
 only to increase the number of reflexions at the same inci- 
 
ELLIPTIC POLARIZATION. 157 
 
 dence* until the light is plane-polarized, and the difference of 
 phase produced by a single reflexion will be known. For, 
 if denote the difference of phase sought, wt will be that pro- 
 duced by n reflexions. And, when the resulting light is 
 plane-polarized, nt = mir, m being any integer number ; conse- 
 quently 
 
 rrnr 
 n 
 
 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 2;r at an incidence of 90. 
 At the angle of maximum polarization, c = |TT. This angle 
 is, of course, different for different metals : it is, however, 
 not far in any from 75. 
 
 (172) 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 difference of 
 phase of the two components is a multiple of?r. 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 WTT ; and between these limits of inci- 
 dence 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 IT. 
 
 (173) It would appear from the foregoing, that metals 
 
 * In practice it is more convenient to produce this effect by increasing the 
 incidence, the number of reflexions remaining unchanged. 
 
158 ELLIPTIC POLARIZATION. 
 
 differ from transparent bodies, in their action upon light, in 
 two particulars namely, 1st, that they do not polarize common 
 light completely at any incidence ; and 2nd, that they alter 
 plane-polarized light by reflexion into light elliptically polar- 
 ized. 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 the property was extended, by Sir John Herschel, to all 
 transparent bodies possessing an adamantine lustre. Mr. 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 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 ellipti- 
 cally polarized. 
 
 III. The difference of phase which they impress upon light, 
 polarized in the two principal planes, undergoes the same va- 
 riations as in metallic reflexion, within certain limits of inci- 
 dence. 
 
 (174) 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 con- 
 
ELLIPTIC POLARIZATION. 159 
 
 tinues 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 (160). It follows from this, that the 
 two components of the reflected ray, polarized respectively in 
 the plane of incidence and in the perpendicular plane, agree in 
 phase at all incidences above the angle of polarization, while 
 they differ 180 in phase at all incidences below it. Accord, 
 ing to this theory, therefore, the difference of phase changes 
 abruptly, from ?r to 2?r, at that critical incidence. On the 
 other hand, in reflexion from metals, the difference of phase 
 of the two components increases continuously fromir to 2?r, as 
 the incidence increases from to 90. 
 
 Now M. Jainin 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 reflected 
 light, consists only in this, that in transparent bodies the va- 
 riation of phase is insensible, except in the neighbourhood 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?r, (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 polari- 
 zation. 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 polarization, 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. 
 
 Q 
 
 When the difference of phase = g 71 "' the ellipticity of the 
 reflected ray is greatest. The angle of incidence at which 
 
160 ELLIPTIC POLARIZATION. 
 
 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. 
 
 (175) M. Jamin has shown, further, that transparent 
 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 /m = 1-46, reflect according to 
 Fresnel's laws. 
 
 (176) 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 inten- 
 sities of the two components. The results obtained have been 
 compared with the formulae given by M. Cauchy for the case 
 of diamond; and the agreement has been found to be satis- 
 
ELLIPTIC POLARIZATION. 161 
 
 factory. These formulae involve two constants, the refrac- 
 tive index, and the coefficient ofellipticity ; and these are de- 
 termined, when we know the principal incidence, and the ratio 
 of the amplitudes of the two vibrations at that incidence. 
 
 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. 
 
 (177) 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 com- 
 prised 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 elliptically 
 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 value, 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 
 
 M 
 
162 ELLIPTIC POLARIZATION. 
 
 the azimuth of the plane of polarization of the incident ray. 
 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 w 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 w sin a. These will be equal, 
 and therefore the reflected light circularly polarized, when 
 
 v cos (0 - 0') 
 
 tan a = = -r?\ 7v\' 
 
 w cos (6 + 6 ) 
 
 If the principal incidence were the same as the angle given 
 by Brewster's law, cos (0 + 0') = 0, and a = 90. But this not 
 being the case, cos (0 + 0') is not actually evanescent ; and the 
 azimuth, a, at which the light is circularly polarized, is a few 
 degrees less than 90. 
 
( 163 ) 
 
 CHAPTER XI. 
 FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 (178) IT has been stated (60, 66), that soon after the dis- 
 covery of double refraction in Iceland crystal, Huygens suc- 
 ceeded 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 refracted 
 ray, in glass and other uncrystallized substances, was the 
 sphere ; or, in other words, that the velocity of undulatory 
 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 accord- 
 ance 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." We have already seen that 
 the construction for the direction of the rays, derived from this 
 assumption, was verified by experience ; and we have here 
 another instance, to which the history of science affords many 
 parallels, of the value of analogical principles in directing sci- 
 entific research. 
 
 (179) The law of Huygens was found to hold in many 
 crystals besides that to which it was originally applied ; and in 
 
 M 2 
 
164 FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 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 bodies, having 
 two optic axes, or two lines of no separation, Huygens's law 
 was found not to be general ; and it was ascertained that one 
 of the rays, at least, in biaxal crystals, followed 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 expe- 
 riment. 
 
 (180) Fresnel sets out from the supposition, that the 
 elastic force of the vibrating medium, in every crystal, is 
 different in different directions. This is, in fact, the most 
 general supposition that can be made ; and whether we sup- 
 pose that the vibrating medium is the ether within the crys- 
 tal, or that the molecules of the body itself partake of the 
 vibratory movement, there will be obviously such a connexion 
 and mutual dependence of the parts of the solid and those of 
 the medium in question, that we cannot hesitate to admit for 
 the one, what has been already established on the clearest 
 evidence for the other. 
 
 It is easy to see, generally, that the phenomenon of double 
 refraction is a necessary consequence 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 
 
\, 
 
 FRESNEL'S THEORY OF DOUBLE REFRACTION. 165 
 
 the same surface ; and we may conceive them to be composed 
 of vibrations parallel and perpendicular to the axis of the crys- 
 tal. Now, the elasticity brought into play by these two sets 
 of vibrations being different, they will be propagated with diffe- 
 rent velocities ; and there will be two waves within the crystal, 
 in which the vibrations are parallel 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 pro- 
 duced will be, that one will be retarded more than the otlier, 
 in its progress through the crystal. But if the second face be 
 oblique to the direction of the rays, they will be both re- 
 fracted at emergence, and differently ; and they will therefore 
 diverge from one another. 
 
 (181) 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 displaced 
 from its position of rest. The resultant of all the elastic forces 
 which resist the displacement will not, in general, act in the 
 direction of the displacement (as would be the case in a me- 
 dium uniformly elastic), and therefore will not drive the dis- 
 placed 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 direction of the dis- 
 placement, whatever be the nature or laws of the molecular 
 action. He assumes that these three directions are parallel 
 throughout the crystal. In fact, the first principles of crys- 
 tallization oblige us to admit, that the arrangement of the 
 molecules of the crystalline body is similar in all parallel lines 
 throughout the crystal ; and the same property must belong 
 to the ether within it, if (as we have reason to presume) its 
 elasticity be dependent on that of the crystal itself. These 
 three directions Fresnel denominates axes of elasticity ; and 
 
166 FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 he concludes that they are also axes of symmetry, with re- 
 spect 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 equation is 
 
 r 3 = a 2 cos 2 a 4 b 2 cos 2 j3 + c 2 cos 2 7 : 
 
 2 , # 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 an- 
 gles which it makes with the axes. 
 
 This surface determines the velocity of propagation of the 
 wave, when the direction of its vibrations is given. For, the 
 ethereal molecule vibrating in the direction of any radius vec- 
 tor, r, of this surface, the elastic force w r hich governs its vibra- 
 tion will be proportional to r 2 ; and, since the velocity of wave- 
 propagation is as the square root of the elastic force, it must, 
 in this case, be represented by the radius vector of the surface 
 of elasticity in the direction of the vibrations. Hence, if we 
 conceive the vibration in the incident wave to be resolved into 
 two within the crystal, performed in tw r o 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 manner the 
 vibration comes to be resolved. 
 
 (182) 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 
 
FRESNEL'S THEORY OF DOUBLE REFRACTION. 167 
 
 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 
 icave, 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. 
 But this, in general, is not the case. Fresnel has shown, how- 
 ever, 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 velocities, 
 and consequently assume different directions. 
 
 (183) The vibrations in these waves being parallel to two 
 
168 FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 fixed lines, namely, the greatest and least diameters of the 
 section of the surface of elasticity, it follows that the two re- 
 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 wave and the normals to the two circular sections of 
 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 tlirough the ray and the two optic axes; 
 while that of the other is perpendicular, or bisects the supplemental 
 dihedral angle. 
 
 Thus the two fundamental facts of crystalline refraction 
 namely, the bifurcation of the ray, and the opposite polariza- 
 tion of the two pencils are completely accounted for. 
 
 Further, the amplitudes of the resolved vibrations are re- 
 presented by the cosines of the angles which the direction of 
 the original vibration contains with the two fixed rectangular 
 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. 
 
 (184) The velocity of propagation of a plane wave in any 
 direction being known, \\ieform 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, c (the same as those of the surface of elasticity), and let 
 
FRESNKL'S THEORY OF DOUBLE REFRACTION. 169 
 
 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." 
 
 The equation of the wave surface is of the fourth order ; 
 it has been thrown into the following symmetric form by Sir 
 William Hamilton, 
 
 - c 2 
 
 ~~ 
 
 (185) The form of the wave surface being known, the di- 
 rections of the two refracted rays are determined by tangent 
 planes drawn to the two sheets of the surface, according to 
 the construction of Huygens. Conceive three surfaces, hav- 
 ing their common centre at the point of incidence, and repre- 
 senting, 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 considering. Let the incident ray 
 be produced to meet the sphere, and at the point of intersec- 
 tion let a tangent plane be drawn. Through the line of inter- 
 section of this plane with the refracting surface, let two planes 
 be drawn touching the two sheets of the refracted wave ; the 
 lines connecting 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 6 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 the surfaces 
 of wave-slowness belonging to air and to the crystal, respectively. Let the 
 
170 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 polarization. 
 The perpendiculars themselves measure the velocities of pro- 
 pagation of the waves, while the radii vectores represent those 
 of the rays. 
 
 (186) 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 w 
 and W denote the angles which any line drawn from the centre 
 of the wave makes with these lines, v and v 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 : 
 
 v~ 2 = a~ z sin 2 -J (w + a/) + c" 2 cos 2 J (w + &/), 
 t>'- 2 = #-2 S i n 2 l ( w - w ') -f c~ 2 cos 2 i (w - a/). 
 
 Since the radius vector of the wave surface measures 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 
 slowness of the refracted waves, in general, in two points. The lines con- 
 necting these points with the centre will represent the direction and normal 
 slowness of the waves ; while the perpendiculars from the centre on the 
 tangent planes at the same points will represent the direction and slowness 
 of the rays." This construction was given by Sir William Hamilton and 
 Professor Mac Cullagh. 
 
FRESNEL'S THEORY OF DOUBLE REFRACTION. 171 
 
 velocity of the ray in its direction, the velocities of the two 
 rays are given by the preceding formulae. If we subtract the 
 latter from the former, we find (after a simple trigonometrical 
 reduction), 
 
 Hence the difference of the squares of the reciprocal velocities, 
 in the two rays, is proportional to the product of the sines of 
 the angles made by their common direction with the lines in 
 which the two rays have a common velocity. In all known 
 crytals, these lines deviate very little from the optic axes, or 
 the lines in which the two parts of the wave have a common 
 velocity ; and thus the remarkable law, to the discovery of 
 which M. Biot was led by analogy, and which has been also 
 shown to flow from the constructions for the velocity given by 
 Sir David Brewster, is a necessary consequence of Fresnel's 
 theory. 
 
 The two sets of lines above alluded to 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 respec- 
 tively, 
 
 Hence, when b* = c 2 , or fr = 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. 
 
 In each of these cases, then, o> = a/, and the preceding 
 equations become 
 
 tr 2 = a~ 2 sin 2 w 4- c' z cos 2 w, i/ = c; 
 the former of which is the equation of the ellipsoid of revolu- 
 
172 FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 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, 
 which was assumed by Huygens, is deduced as a simple corol- 
 lary 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. 
 
 (187) It has been stated (70) that, as soon as a class of dou- 
 ble-refracting substances was discovered, possessing two optic 
 axes, the construction of Huygens was found not to be gene- 
 ral. It was still thought, 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 con- 
 stant, and the refraction of both was performed according 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 de- 
 cisive in favour of his theory. 
 
 It has been already shown (81) 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 
 
FRESNEL'S THEORY OF DOUBLE REFRACTION. 173 
 
 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 ordinary* rays will remain still un- 
 displaced, if the velocity of these rays is the same in the two 
 plates ; while, on the other hand, if the velocities be differ- 
 ent, the fringes will be shifted from their original position. 
 On trial, the result was found to be as Fresnel had anticipated : 
 the fringes were displaced ; and the amount of that displace- 
 ment agreed with the calculated difference of velocity, which 
 had been previously deduced from theory. 
 
 In a second experiment, two prisms were cut in different 
 directions from the same crystal of topaz, cemented together, 
 and ground to the same angle ; and the compound prism thus 
 formed was achromatized by a prism of glass. On looking 
 through the combination at a line of light, Fresnel found that 
 the ordinary image of the line was broken at the junction of the 
 two prisms, thus showing that the ray was unequally re- 
 fracted in different directions. 
 
 (188) 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 resulting 
 equation is reducible to the form 
 
 (a? + 3* _. tf) [ 2 x* + c 2 z 2 - a 2 c 2 ] = 0. 
 
 This equation is manifestly resolvable into the two follow- 
 ing: 
 
 & + z* = ft 2 , a- x 1 + c*-z* = a 2 c 2 ; 
 
 so that the surface intersects the plane of xz in a circle 
 
 * The ray whose velocity varies the least, in biaxal crystals, is some- 
 times, though improperly, called for distinction the ordinary ray. 
 
174 FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 and ellipse. As these two curves have a common centre, 
 and as the radius of the circle, b, is of intermediate magni- 
 tude 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, O A and OB ; and their direc- 
 tions, at emergence, are determined by 
 the positions of the tangent planes at the 
 points A and B. 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 conoidal 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 divided into an infinite number of 
 rays at emergence, whose directions and planes of polarization 
 are determined by the tangent planes. 
 
 Again, it is evident that the circle and ellipse have four 
 
FRESNEL'S THKOUY OF DOUBLE REFRACTION. 175 
 
 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, consti- 
 tuting 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. 
 
 (189) Here, then, are two singular and unexpected con- 
 sequences 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. 
 
 Being naturally anxious to submit the wave- theory to this 
 test, and to establish or disprove its new results, Sir William 
 Hamilton requested the author to examine the subject expe- 
 rimentally. The result of this examination 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 
 
176 FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 of single ray-velocity. 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 1 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 2 , b\ c 2 . 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 cen- 
 tral 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 appeared, on look- 
 ing through the aperture in the second surface. 
 When the aperture in the second plate was 
 ever so slightly shifted, so that the line con- 
 necting the two apertures no longer coin- 
 cided with the line MO, the phenomenon ra- 
 pidly changed, and the annulus resolved itself into two sepa- 
 rate pencils. 
 
FRESNEL'S THEORY OF DOUBLE REFRACTION. 177 
 
 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 complete. 
 
 The rays which compose the emergent cone are all polar- 
 ized in different planes. It was discovered by observation that 
 these planes are 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 law was found to be in 
 accordance with theory. 
 
 (190) 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 
 
178 FRESNEL'S THEORY OF DOUBLE REFRACTION. 
 
 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 ofNicomedes, whose asymptot is the line on the first 
 surface. 
 
 (191) 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 1 55'. The 
 rays composing this cone will be re- 
 fracted at the second surface of the crys- 
 cal, 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 
 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'b'b the 
 emergent cylinder. 
 
 The minuteness of this phenomenon, and the perfect ac- 
 curacy required m 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 
 
FRESNEL'S THEORY OF DOUBLE REFRACTION. 179 
 
 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 
 was received on a small screen of silver paper, at various dis- 
 tances from the crystal ; and no sensible enlargement of the 
 section was observable on increasing 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 polarized in 
 different planes ; and the law connecting these planes is the 
 same as in the case of external conical refraction. 
 
 (192) We have seen that the problem to find the direction 
 and magnitude of the reflected and refracted vibrations, when 
 those of the incident vibration are given, was solved by Fres- 
 nel in the case of ordinary media. In the year 1831, M. 
 Seebeck generalized, to a certain extent, the solution of Fres- 
 nel, and extended it to the case of reflexion by uniaxal crystals 
 in the principal plane. The general solution of the problem 
 of reflexion and refraction by crystalline media was obtained, 
 a few years later, by Professor Mac Cullagh and M. Neumann 
 upon other principles (156) ; and the memoir of the former is 
 distinguished for the beauty and elegance of its geometrical 
 laws. This solution, like that of Fresnel for ordinary media, 
 does not include the change of phase, which is now proved to 
 take place in reflexion at the bounding surfaces of all media 
 (174). Its results, accordingly, are only approximately true, 
 the degree of approximation being probably the same as in the 
 case of Fresnel's laws themselves. 
 
 N 2 
 
( 180 ) 
 
 CHAPTEK XII. 
 
 INTERFERENCE OF POLARIZED LIGHT. 
 
 (193) 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. 
 
 (194) It has been already shown (142), that when abeam 
 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 
 plate of any double-refracting substance, the capability of re- 
 flexion at the analyzing plate is suddenly restored, and a por- 
 
INTERFERENCE OF POLARIZED LIGHT. 181 
 
 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 placed so 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 light totally disappears, and the reflected ray va- 
 nishes, 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 section of the crystal coincides 
 with the plane of the first reflexion, or is perpendicular to it. 
 When the plate is turned round from either of these posi- 
 tions, the light gradually increases ; and it becomes a maxi- 
 mum, when the principal section is inclined at an angle of 45 
 to the plane of the first reflexion. 
 
 (195) 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 lamina 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 
 is perpendicular to it, and, reaching a maximum, when ifc 
 is inclined to the plane of primitive polarization at an angle 
 of 45. 
 
182 INTERFERENCE OF POLARIZED LIGHT. 
 
 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. 
 
 ( 1 96) 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 vf 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- 
 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 
 
INTERFERENCE OF POLARIZED LIGHT. 183 
 
 of an inch. Between ^oo^ 1 an d yuth f 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 ^fa 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 crys- 
 tals 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. 
 
 (197) 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 
 undulation ; and it would therefore appear that we have here 
 all the conditions necessary for their interference, and the con- 
 sequent production of colour. Such was the sagacious con- 
 jecture of Young. 
 
 But here"we are met by a difficulty. So far as this expla- 
 nation goes, the phenomena of interference and of colour should 
 be produced by 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, 
 
184 INTERFERENCE OF POLARIZED LIGHT. 
 
 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. 
 
 (198) 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 relative posi- 
 tion of the axes of the two tourmalines. When these axes are 
 parallel, and consequently the two rays polarized in the same 
 plane, the fringes are best defined ; they decrease in intensity, 
 when the axes of the tourmalines are inclined to 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 
 
INTERFERENCE OF POLARIZED LIGHT. 185 
 
 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 produce 
 these phenomena in perfection, 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. 
 
 But though the intensity of the light does not vary with 
 the phase of the component vibrations, the character of the 
 resulting vibration will. It appears from theory, that two 
 rectilinear and rectangular vibrations compound a single vi- 
 bration, which will be also rectilinear when the phases of the 
 component vibrations differ by an exact number of semi-un- 
 dulations ; that, in all other cases, the resulting vibration will 
 be elliptic ; and that the ellipse will become a circle, when the 
 component vibrations have equal amplitudes, and the differ- 
 ence of their phases is an odd multiple of a quarter of a wave. 
 These results of theory have been completely confirmed by 
 experiment. 
 
 (199) Fresnel and Arago discovered, further, that two 
 oppositely 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; and that, in the interference of rays which had un- 
 dergone 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 
 
186 INTERFERENCE OF POLARIZED LIGHT. 
 
 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 
 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 
 
 * 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 polarized 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 succeeding 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 phenomena of interference, exhibited by common light, compel us also 
 to admit (as Mr. Airy has observed) that the vibrations do not change con- 
 tinually ; and that in each system of waves there are, probably, several hun- 
 dred vibrations which are all similar, although the vibrations constituting 
 one system bear no relation to those of another, and the different systems 
 succeed one another with such rapidity as to obliterate all trace of polari- 
 zation. 
 
INTERFERENCE OF POLARIZED LIGHT. 187 
 
 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.* 
 
 (200) These laws of interference being kept in mind, the 
 reason of all the phenomena is apparent. The ray is origi- 
 nally polarized in ar 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 retardation of one of the rays behind the other, 
 produced by the difference of the velocities with which they 
 traverse the crystal. 
 
 It has been shown, that the difference between the re- 
 ciprocals of the squares of the velocities, with which the two 
 rays traverse the crystal, is proportional to the product of the 
 sines of the angles which their direction makes with the optic 
 axes ; or, that if v and v denote the velocities of the two rays, 
 to and W the angles which their direction makes with the two 
 
 axes, 
 
 v~ 2 - v'~ 2 = c sin to sin a/. 
 
 But if t and t' denote the times occupied by the two rays in 
 traversing the crystal, and the thickness actually traversed, 
 
 * 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 
 supposed to be resolved, without reference to their phase. The justice of 
 this assumption depends upon the fact adverted to in the preceding note, 
 namely, that the two oppositely polarized portions, into which we have sup- 
 posed common light to be resolved, differ in phase, that difference continu- 
 ally 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. 
 
188 INTERFERENCE OF POLARIZED LIGHT. 
 
 or the thickness of the plate multiplied by the secant of the 
 angle of refraction, 
 
 t z - t' 2 t + t' t -t' 
 
 Now the first factor of this product is very nearly constant ; 
 we have, therefore, 
 
 t - t' = const x 9 sin w sin a/ ; 
 
 or, the interval of retardation is proportional to the product of 
 the sines of the angles which the direction of the rays makes 
 with the two axes, and to the thickness of the crystal tra- 
 versed, jointly. When the two axes coalesce, or the crystal 
 becomes uniaxal, the retardation is proportional to the square 
 of the sine of the angle which the direction makes with the 
 axis. But the tint developed is measured by the interval of 
 retardation ; accordingly the laws of the tints, discovered ex- 
 perimentally by M. Biot, flow immediately from the theory. 
 
 (201) It is plain that the light issuing from the crystal 
 is, in general, elliptically 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-re- 
 fracting 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 
 by the analyzing prism will vanish in two positions of its 
 principal section ; and it is manifest that the successive thick- 
 
INTERFERENCE OF POLARIZED LIGHT. 189 
 
 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-po- 
 larized 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. 
 
 (202) It has been stated (195) 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 re- 
 sulting light is white. The phenomena of colour may still, 
 however, be produced in thick plates, by superposing two of 
 them in such a manner, that the ray which has the greater 
 velocity in the first shall have the less in the second. We 
 have only to place the plates with their principal sections per- 
 pendicular 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 perpendi- 
 cular ; 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. 
 
 (203) 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 
 
190 
 
 INTERFERENCE OF POLARIZED LIGHT. 
 
 angles ; and let us suppose, for simplicity, that the crystal- 
 line plate is cut in a direction perpendicular to the axis. 
 
 Let ABCD be the plate, and E the place of the eye. 
 The visible portion of the emergent beam will form a cone, 
 AEB, whose summit coin- 
 cides with the place of the 
 eye, and axis, EO, 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' and NN' 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 primi- 
 tive polarization, and in the other is perpendicular to it. 
 
 M 
 
INTERFERENCE OF POLARIZED LIGHT. 191 
 
 These rays therefore will be reflected, or not, from the ana- 
 lyzing plate, according as the plane of reflexion there coincides 
 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 crys- 
 tal for this ray, OL, neither coincides with, nor is perpendi- 
 cular to, the plane of primitive polarization ; and consequently 
 the incident polarized ray will be divided into two within the 
 crystal, whose planes of polarization are parallel and perpendi- 
 cular 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 interfer- 
 ence will depend on their difference of phase. 
 
 Now the difference of phase of the two rays depends on the 
 interval of retardation. When this interval is an odd multiple 
 of half an undulation, the two rays are in complete discord- 
 ance ; and, on the other hand, they are in complete accord- 
 ance when it is an even multiple of the same quantity. 
 We have seen (200) that, for a given plate, the interval of 
 retardation is proportional 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 
 
192 INTERFERENCE OF POLARIZED LIGHT. 
 
 is very nearly proportional to the sine of the angle LEO 
 (see first fig. p. 190), which the emergent ray makes with 
 the axis ; and this latter to LO, the distance of the point of 
 emergence from the centre. The retardation therefore varies 
 as the square of the distance LO ; and consequently the suc- 
 cessive dark and bright lines will be arranged in circles, (as 
 represented in the preceding diagrams) the squares of whose 
 radii are in arithmetical progression. 
 
 We have been speaking hitherto of homogeneous 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. 
 
 (204) Let us now consider briefly the case ofbiaxal 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 
 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 
 rays, t t', is the same. Now this interval is proportional 
 to the product of the sines of the angles which the direc- 
 tion of the rays makes with the optic axes (200) ; and these 
 sines are, very nearly, as the distances of the points of emer- 
 gence (measured on the face of the crystal) from the projec- 
 tions of the optic axes. Hence the product of these distances 
 
INTERFERENCE OF POLARIZED LIGHT. 
 
 193 
 
 will be constant for all the points of the same curve. The 
 curve formed by each band is therefore the lemniscata 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- 
 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 annexed diagrams 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 polari- 
 zation 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 ; 
 
 o 
 
194 INTERFERENCE OF POLARIZED LIGHT. 
 
 and that they are rectangular hyperbolas, whose common centre 
 is the middle point of the line which connects the projections 
 of the two axes. 
 
 (205) 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 structure 
 may be communicated to bodies which do not possess it na- 
 turally, 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 crys- 
 tal ; and the tint descended in the scale as the pressure was 
 augmented. Single-refracting crystals, such as muriate of 
 soda, and fluor spar, acquired the property of double refrac- 
 tion 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 co- 
 loured bands are seen, separated by a neutral line ; and these 
 vanish altogether when the compressing force is withdrawn. 
 
INTERFERENCE OF POLARIZED LIGHT. 195 
 
 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 ne- 
 gative 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 substance, fixed 
 in position, as well as direction. In this respect the pheno- 
 mena 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. 
 
 (206) 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- 
 mum and minimum of elasticity corresponding to the direc- 
 tions of greatest and least pressure. Accordingly the vibra- 
 tions of the ray, on entering the substance, are resolved into 
 two in these directions, and these are propagated with unequal 
 velocities. The incident wave will therefore be separated into 
 two within the medium, one of which will be in advance of 
 the other, and these will be in different phases of vibration 
 at emergence. The resolved parts of the vibrations, in the 
 plane of reflexion of the analyzing plate, will accordingly in- 
 terfere, and the tint developed will be determined by the in- 
 terval of retardation. 
 
 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 difference of velo- 
 
 o 2 
 
196 INTERFERENCE OF POLARIZED LIGHT. 
 
 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 com- 
 pressed prisms, in this arrangement, acquired a double-refract- 
 ing 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 increased 
 in proportion to their number, and thus rendered sensible. 
 The intermediate prisms served to correct the deviation, and 
 to render the combination achromatic. 
 
 (207) 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 ob- 
 served to be covered with coloured bands, the parts near the 
 opposite edges having acquired a positive double-refracting 
 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 appear- 
 ances vanish when the glass acquires the same temperature 
 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 re- 
 fraction begins to weaken ; and the colours disappear altoge- 
 ther when the glass is uniformly heated. Again, if the cylin- 
 
INTERFERENCE OF POLARIZED LIGHT. 197 
 
 der, when in this state, be made to cool rapidly by surrounding 
 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, assume a fixed 
 form in cooling, the interior parts must accommodate them selves 
 to that form, and therefore retain a state of unequal density. 
 The law of density, and therefore the double-refracting struc- 
 ture, 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 vary- 
 ing with the shape of the mass. 
 
 (208) 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 
 
198 INTERFERENCE OF POLARIZED LIGHT. 
 
 is dilated by heat in the direction of its axis ; while it actually 
 contracts, by a small amount, in directions perpendicular 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 (69). 
 Professor Mitscherlich accordingly conjectured, that the dou- 
 ble-refracting energy of the crystal must, in these circum- 
 stances, be diminished; and the conjecture 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 (186) that the inclination of the optic axes, in 
 biaxal crystals, is a simple function of the three principal elas- 
 ticities of the vibrating medium, and that the plane of the axes is 
 that of the greatest and least elasticities. If, then, these elasti- 
 cities be altered by heat in different proportions, 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 Mitscherlich found that, 
 in sulphate of lime, the angle between the axes (which is about 
 60 at the ordinary temperature) diminishes on the applica- 
 tion of heat ; that, as the temperature increases, these axes 
 approach until they unite ; and that, on a still further augmen- 
 tation of heat, they again separate, and open out in a perpen- 
 dicular plane. Heat is found to dilate this crystal more in 
 one direction than in another perpendicular to it. 
 
( 199 ) 
 
 CHAPTER XIII. 
 
 ROTATORY POLARIZATION. 
 
 (209) 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 the 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. 
 
 (210) 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. 
 
 I. In different plates of the same crystal, the rotation of 
 the plane of polarization is always proportional to the thickness 
 
200 ROTATORY POLARIZATION. 
 
 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 thickness 
 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. 
 
 (211) 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 
 solid angles, formed at the junction of the pyramid and prism, 
 are sometimes replaced by small secondary planes, which are 
 
ROTATORY POLARIZATION. 201 
 
 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. 
 
 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 struc- 
 ture in question is displayed in the most beautiful manner, 
 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 various 
 directions, when no striae, or other external appearances, in- 
 dicate their presence. 
 
 (212) The connexion between the rotation of the plane of 
 polarization and the crystalline form, discovered by Sir John 
 Herschel in quartz, has since been observed in other sub- 
 stances. M. Pasteur has recently found that tar tar ic acid, and 
 the tartrates, which are all plagiedral in the same direction, 
 likewise deviate the plane of polarization to the same side. 
 On the other hand, para-tartaric acid, and the para-tartrates, 
 which have the same general form, are for the most part not 
 plagiedral ; while, in those salts of this class which are so, the 
 facettes of the crystals are inclined sometimes to the right, 
 and sometimes to the left, and this difference is found to exist 
 even in crystals belonging to the same specimen. M. Pasteur 
 has found, accordingly, that the salts of the former class have 
 no effect upon the plane of polarization ; while those of the 
 latter deviate the plane of polarization in the same direction 
 as the facettes of the crystal. 
 
202 ROTATORY POLARIZATION. 
 
 This remarkable distinction among the para-tartrates has 
 been traced by the same observer to their chemical composi- 
 tion. He has discovered that para-tar taric acid is composed of 
 two distinct acids, which have the same general crystalline 
 form ; but which differ in this, that in one of them the facettes 
 of the crystals are inclined to the right, and in the other to the 
 left. These acids (one of which is the ordinary tartaric acid) 
 accordingly deviate the plane of polarization the former to the 
 right, and the latter to the left, and by the same amount ; and 
 the difference in the optical properties of different specimens of 
 the compound acid, and its salts, arises from the predominance 
 of one or other of the component elements. 
 
 (213) 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-crystal, cut 
 perpendicularly to the axis, it may be resolved into two such 
 circularly polarized rays ; and as these are supposed to be trans- 
 mitted with different velocities, one of them will be in advance 
 of the other when they assume a common velocity at emer- 
 gence. They then compound a single ray, polarized in a sin- 
 gle 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 therefore 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 as- 
 sumption, that two circularly polarized pencils, whose vibra- 
 tions are in opposite directions, are actually transmitted along 
 
HOTATORY POLARIZATION. 203 
 
 the axis of quartz with different velocities. This supposition 
 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 trans- 
 mitted through a prism of rock-crystal, in the direction 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 polarized. 
 
 (214) 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 extra- 
 ordinary waves in the direction of the axis of the crystal ; p. 
 and p the corresponding refractive indices ; then 
 
 But, if 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 
 
 n $ 
 
 T -, or to 1 + -, being very small in comparison to 0. 
 u o u 
 
 We have therefore 
 
 Now the angle of rotation is proportional to the interval of re- 
 tardation of the two circularly polarized pencils ; and when that 
 interval is equal to the length of a wave in vacua, the angle 
 
204 ROTATORY POLARIZATION. 
 
 of rotation is 180. Hence the interval of retardation of the 
 emergent rays, corresponding to any angle of rotation, p, will 
 
 be X y^pj X denoting the length of the wave ; and the cor- 
 responding interval within the crystal is equal to this, multi- 
 plied by the velocity of propagation, or divided by the refrac- 
 tive index. Hence, if p be the rotation corresponding to the 
 thickness of the crystal, 0, we have 
 
 X D 
 
 8-- 
 
 IUL 180 ' 
 and substituting in the preceding formula, 
 
 '_ =^ p 
 ** ** 6 180' 
 
 This difference is extremely small. When 6=1 milli- 
 metre, the angle of rotation, p, corresponding to the rays of 
 mean refrangibility, = 30. But for these rays, X = '0005 of a 
 millimetre; and therefore p - ju = '00008. 
 
 (215) The phenomena hitherto described take place only 
 in the direction of the axis of the crystal. Mr. 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 elliptically 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 increasing 
 indefinitely with its inclination to that line. With respect to 
 the course of the refracted rays, Mr. 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 
 
ROTATORY POLARIZATION. 205 
 
 two rays, do not touch, as in all other known uniaxal crystals, 
 the latter surface being contained entirely wit/tin the former. 
 This is a necessary consequence of the fact, that the interval 
 of retardation of the two pencils does not vanish, with the in- 
 clination of the ray to the optic axis. 
 
 Mr. 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 the most 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- 
 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 quanti- 
 ties sought are deduced by calculation. 
 
 (216) 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 ellip- 
 ticity 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 
 
206 ROTATORY POLARIZATION. 
 
 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 
 
 _C0 
 p ~ A 2 ' 
 
 which comprises all the experimental laws of M. Biot (210). 
 The sign of the constant factor, C, 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 
 Mr. Airy, and has found the calculated and observed results 
 to agree. 
 
 (217) 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 pro- 
 portional 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 
 
ROTATORY POLARIZATION. 207 
 
 quartz ; so that the ray must traverse a much greater thickness 
 of the substance, in order to have its plane of polarization al- 
 tered 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 tur- 
 pentine, of the same thickness, turns the plane of polariza- 
 tion only through a quarter of a degree. 
 
 The rotatory liquids do not lose their peculiar power (ex- 
 cept in degree) by dilution with other liquids not possessing 
 the property; and they retain it, even in the state of vapour. 
 From these and other facts, M. Biot concludes that this pro- 
 perty, 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 Her- 
 schel 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 re- 
 spect to fused quartz. 
 
 (218) 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- 
 
208 ROTATORY POLARIZATION. 
 
 charine solutions ; and a very ingenious apparatus, called the 
 saccharimeter, has been devised by M. Soleil for the purpose. 
 This instrument is founded on 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 of rock-crys- 
 tal of a suitable thickness, and of the opposite action. 
 
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